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H A N D B O O K OF

THE GEOMETRY OF BANACH S PACES

This Page Intentionally Left Blank

H A N D B OOK OF THE GEOMETRY OF B ANAC H S PACE S

Volume 1 Edited by

W.B. J O H N S O N Texas A&M University, College Station, Texas, USA

J. L I N D E N S T R A U S S The Hebrew University of Jerusalem, Jerusalem, Israel

2001

ELSEVIER Amsterdam

9L o n d o n

9N e w

York

9O x f o r d

9P a r i s

9Shannon

9T o k y o

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 EO. Box 211, 1000 AE Amsterdam, The Netherlands 9 2001 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Global Rights Department, PO Box 800, Oxford OX5 1DX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-maih [email protected]. You may also contact Global Rights directly through Elsevier's home page (http://www.elsevier.nl), by selecting 'Obtaining Permissions'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) 978 7508400, fax: (+1) 978 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works: Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage: Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier Science Global Rights Department, at the mail, fax and e-mail addresses noted above. Notice: No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. ISBN: 0-444-82842-7 First edition 2001

Library of Congress Cataloging-in-Publication Data Handbook of the geometry of Banach spaces / edited by W.B. Johnson, J. Lindenstrauss.-1 st ed. p. cm. Includes indexes. ISBN 0-444-82842-7 (v.1 : alk. paper) 1. Banach spaces. I. Johnson, W. B. (William B.), 1944- II. Lindenstrauss, Joram, 1936QA322.2.H36 2001 515'.732-dc21 2001023427

British Library Cataloguing in Publication Data Handbook of the geometry of Banach spaces Vol. 1 edited by W.B. Johnson, J. Lindenstrauss 1. Banach spaces I. Johnson, William B., 1944- II.Lindenstrauss, J. 515.7'32 ISBN0444828427 G The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

Preface 1. Introduction The aim of this Handbook is to present an overview of the main research directions and results in Banach space theory obtained during the last half century. The scope of the theory, having widened considerably over the years, now has deep and close ties with many areas of mathematics, including harmonic analysis, complex analysis, partial differential equations, classical convexity, probability theory, combinatorics, logic, approximation theory, geometric measure theory, operator theory, and others. In choosing a topic for an article in the Handbook we considered both the interest the topic would have for non-specialists as well as the importance of the topic for the core of Banach space theory, which is the study of the geometry of infinite dimensional Banach spaces and n-dimensional normed spaces with n finite but large (local theory). Many of the leading experts on the various aspects of Banach space theory have written an exposition of the main results, problems, and methods in areas of their expertise. The enthusiastic response we received from the community was gratifying, and we are deeply appreciative of the considerable time and effort our contributors devoted to the preparation of their articles. Our expectation is that this Handbook will be very useful as a source of information and inspiration to graduate students and young research workers who are entering the subject. The material included will be of special interest to researchers in Banach space theory who may not be aware of many of the beautiful and far reaching facets of the theory. We ourselves were surprised by the new light thrown by the Handbook on directions with which we were already basically familiar. We hope that the Handbook is also valuable for mathematicians in related fields who are interested in learning the new directions, problems, and methods in Banach space theory for the purpose of transferring ideas between Banach space theory and other areas. Our introductory article, "Basic concepts in the geometry of Banach spaces", is intended to make the Handbook accessible to a wide audience of researchers and students. In this chapter those concepts and results which appear in most aspects of the theory and which go beyond material covered in most textbooks on functional and real analysis are presented and explained. Some of the results are given with an outline of proof; virtually all are proved in the books on Banach space theory referenced in the article. In principle, the basic concepts article contains all the background needed for reading any other chapter in the Handbook. Each article past the basic concepts one is devoted to one specific direction of Banach space theory or its applications. Each article contains a motivated introduction as well as

vi

Preface

an exposition of the main results, methods, and open problems in its specific direction. Most have an extensive bibliography. Many articles, even the basic concepts one, contain new proofs of known results as well as expositions of proofs which are hard to locate in the literature or are only outlined in the original research papers. The format of the chapters is as varied as the personal scientific styles and tastes of the contributors. In our view this makes the Handbook more lively and attractive. There are many strong and intricate interconnections between the different subjects treated in the various articles, so some articles contain many cross references to other articles. But, as mentioned above, no article depends on anything past the material treated in the basic concepts article. Because of this lack of interdependence, we chose to order the chapters after the basic concepts article alphabetically according to the first author of the chapter. We included in volume 1 the chapters which came first in this order among those articles which were ready by the deadline we set for this volume. The second volume of this Handbook will contain the chapters which come later in the alphabetical order as well as those chapters which were finished only after the deadline for volume 1. This volume ends with a subject index and an author index for the material of the present volume. Volume 2 will contain a subject and author index for the entire Handbook.

William B. Johnson and Joram Lindenstrauss

List of Contributors Abramovich, Yu.A., Indiana University- Purdue University, Indianapolis, IN (Ch. 2) Aliprantis, C.D., Purdue University, West Lafayette, IN (Ch. 2) Alspach, D., Oklahoma State University, Stillwater, OK (Ch. 3) Ball, K., University College London, London (Ch. 4) Bourgain, J., Institute for Advanced Study, Princeton, NJ (Ch. 5) Burkholder, D., University of lllinois, Urbana, IL (Ch. 6) Casazza, E, University of Missouri, Columbia, MO (Ch. 7) Davidson, K.R., University of Waterloo, Waterloo (Ch. 8) Delbaen, E, E.T.H., Zurich (Ch. 9) Deville, R., University of Bordeaux, Bordeaux (Ch. 10) Diestel, J., Kent State University, Kent, OH (Ch. 11) Dilworth, S.J., University of South Carolina, Columbia, SC (Ch. 12) Enflo, E, Kent State University, Kent, OH (Ch. 13) Figiel, T., Institute of Mathematics, Sopot (Ch. 14) Fonf, V.E, Ben Gurion University, Beer-Sheva (Ch. 15) Gamelin, T., The University of California, Los Angeles, CA (Ch. 16) Ghoussoub, N., University of British Columbia, Vancouver (Ch. 10) Giannopoulos, A.A., University of Crete, Heraklion (Ch. 17) Godefroy, G., Universite Paris VI, Paris (Ch. 18) Jarchow, H., University of Zurich, Zurich (Ch. 11) Johnson, W.B., Texas A&M University, College Station, TX (Chs. 1, 19) Kisliakov, S., POMI, Saint Petersburg (Chs. 16, 20) Koldobsky, A., University of Missouri, Columbia, MO (Ch. 21) K6nig, H., University ofKiel, Kiel (Chs. 21, 22) Lindenstrauss, J., The Hebrew University of Jerusalem, Jerusalem (Chs. 1, 15) Lomonosov, V. Kent State University, Kent, OH (Ch. 13) Milman, V.D., Tel-Aviv University, Tel-Aviv (Ch. 17) Odell, E.W., The University of Texas, Austin, TX (Ch. 3) Phelps, R.R., University of Washington, Seattle, WA (Ch. 15) Pietsch, A., Jena University, Jena (Ch. 11) Schachermayer, W., Vienna University of Technology, Vienna (Ch. 9) Schechtman, G., The Weizmann Institute of Science, Rehovot (Ch. 19) Szarek, S., Case Western Reserve University, Cleveland, OH and Universit~ Paris VI, Paris (Ch. 8) Wojtaszczyk, E, Warsaw University, Warsaw (Ch. 14)

vii

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Contents P refa c e List of Contributors

1. Basic concepts in the geometry of Banach spaces W.B. Johnson and J. Lindenstrauss 2. Positive operators Y.A. Abramovitch and C.D. Aliprantis 3. Lp spaces D. Alspach and E. Odell 4. Convex geometry and functional analysis K. Ball 5. A p-sets in analysis: Results, problems and related aspects J. Bourgain 6. Martingales and singular integrals in Banach spaces D.L. Burkholder 7. Approximation properties P. G. Casazza 8. Local operator theory, random matrices and Banach spaces K.R. Davidson and S.J. Szarek 9. Applications to mathematical finance F. Delbaen and W. Schachermayer 10. Perturbed minimization principles and applications R. Deville and N. Ghoussoub 11. Operator ideals J. Diestel, H. Jarchow and A. Pietsch 12. Special Banach lattices and their applications S.J. Dilworth 13. Some aspects of the invariant subspace problem P. Enflo and V. Lomonosov 14. Special bases in function spaces T. Figiel and P. Wojtaszczyk 15. Infinite dimensional convexity V.P. Fonf J. Lindenstrauss and R.R. Phelps 16. Uniform algebras as Banach spaces T. W. Gamelin and S. V. Kislyakov

V

vii

85 123 161 195 233 271 317 367 393 437 497 533 561 599 671

x

Contents

17. Euclidean structure in finite dimensional normed spaces A.A. Giannopoulos and V.D. Milman 18. Renormings of Banach spaces G. Godefroy 19. Finite dimensional subspaces of L p W.B. Johnson and G. Schechtman 20. Banach spaces and classical harmonic analysis S. V. Kislyakov 21. Aspects of the isometric theory of Banach spaces A. Koldobsky and H. Kdinig 22. Eigenvalues of operators and applications H. KOnig

707

Author Index

975

Subject Index

993

781 837 871 899 941

CHAPTER

1

Basic Concepts in the Geometry of Banach Spaces William B. Johnson* Department of Mathematics, Texas A &M University, College Station, TX 77843, USA E-mail: johnson @math.tamu.edu

Joram Lindenstrauss t Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Notations and special Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Classical spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Banach lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Geometry of the norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Analysis in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Finite dimensional Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Local structure of infinite dimensional spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Some special classes of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 6 13 20 30 35 43 53 61 74 81 83

*Supported in part by NSF DMS-9623260 and DMS-9900185, Texas Advanced Research Program 010366-163, and US-Israel Binational Science Foundation. +Supported in part by US-Israel Binational Science Foundation and by NSF DMS-9623260 as a participant in the Workshop in Linear Analysis & Probability at Texas A&M University. H A N D B O O K OF THE GEOMETRY OF BANACH SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved

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Basic concepts in the geometry of Banach spaces 1. Introduction In this introductory chapter we present results and concepts which are often used in Banach space theory and will be used in articles in this Handbook without further reference. The material we treat, while familiar to experts in Banach space theory, has not made its way into introductory courses in functional analysis. The main purpose of this article is to make the subsequent articles accessible to anyone whose background includes basic graduate courses in analysis and functional analysis. Each section of this article is devoted to one aspect of Banach space theory. Although this article can in no way be considered as an introductory course in Banach space theory, we do include indications of proof of some basic results in the hope that this will help the reader understand and get a feeling for the various concepts which are discussed. We reference only some of the (mostly introductory) books which treat the basic material we describe. Original sources are referenced in these books. We also mention some results which are not yet in elementary books on Banach spaces but which help to clarify the general picture. These generally were either discovered recently or are more difficult than the rest of the material. In these cases we refer to specific articles in this Handbook which treat the topic. In general, we do not attach names to theorems (except when experts generally refer to the theorem with a name attached, such as the Hahn-Banach theorem, the Krein-Milman theorem, Rosenthal's s theorem . . . . ) or give any historical background. Instead we refer to the books in the references as well as the articles in this Handbook.

2. Notations and special Banach spaces Banach spaces will have either real or complex scalars. When the scalar field matters (for example, in results involving spectral theory or in theorems of an isometric nature or when analyticity plays a r61e), the scalar field is mentioned explicitly, but in the notation for special spaces the scalars are not specified. Operators between Banach spaces are bounded and linear. An invertible operator T is called an isomorphism. Two norms on a vector space are called equivalent if the identity operator on X (with the two given norms) is an isomorphism. If IIT l] = 1 = I1T-1 ]], T is called an isometric isomorphism or simply an isometry and the domain and range of T are said to be isometric. We write X ~ Y to denote that the spaces are isomorphic. To denote isometry we use the equal sign. An isomorphism from a Banach space onto itself is called an automorphism. A Banach space Y is said to be a quotient of the Banach space X if Y is isometric to X / Z for some closed subspace Z of X. By the open mapping theorem, Y is isomorphic to a quotient of X if there is an operator from X onto Y. If X ~ Y, d(X, Y) denotes the Banach-Mazur distance between the spaces, defined to be the infimum of IITII lIT -1 II as T ranges over all isomorphisms from X onto Y. So d(X, Y) = 1 if X and Y are isometric; the converse is true for finite dimensional spaces but not for infinite dimensional spaces. Note that the "triangle inequality" for the Banach-Mazur distance is submultiplicative rather than subadditive; that is, d(X, Y) <~

d (X, Z)d (Z, r).

W.B. Johnson and J. Lindenstrauss A projection is an idempotent operator. A subspace Z of X is said to be complemented if there is a projection from X onto Z. This is the case if and only if Z is closed and there is a closed subspace W of X so that W A Z = {0} and X = W + Z; we then write X = W 9 Z and say that X is the direct sum of W and Z. In this case W is isomorphic to X / Z by the open mapping theorem. We regard X as a subspace of X** under the canonical embedding. Note that there is always a projection of norm one from X*** onto X* (restrict the functionals on X** to X). An operator T : X ~ Y between Banach spaces is compact; respectively, weakly compact; if the image T Bx of the unit ball Bx of X has compact; respectively, weakly compact; closure in Y. If either X or Y is reflexive, then every operator from X to Y is weakly compact. The identity operator on X is compact if and only if X is locally compact if and only if X is finite dimensional. A s e q u e n c e {Xn}n~ in a Banach space X is called weakly Cauchy provided {X*(Xn)}n~=l converges for every x* in X*. Identifying a Banach space with a subspace of its bidual, and taking into account that the uniform boundedness principle implies that a weakly Cauchy sequence is bounded, we see that {Xn }nO~ 1 is weakly Cauchy in X if and only if it converges weak* in X**. The space X is said to be weakly sequentially complete provided every weakly Cauchy sequence {Xn}neC__1 in X converges weakly, which is the same as saying that the weak* limit of {Xn}n~176 1 in X** is in X itself. Here is a list of special classical Banach spaces and other objects. The elementary theory of these can be found in beginning texts in real and functional analysis. A more detailed list of symbols, including some notation undefined in the text because we regard it as "standard", is contained in Section 12.

N R C T C(K;X)

C(K) Lp(lZ) L~(#)

Lp(T) ~p(F) ep n ,ep C

co(F)

-= = =

The natural numbers. The real numbers. The complex numbers. The unit circle in the complex plane. Continuous functions f on the (usually) compact Hausdorff space K taking values in the (usually) Banach space X, normed by Ilfll = suPtcg IIf(t)ll. C (K; X) when X is the scalar field. -- The #-measurable functions f for which Ilfllp -- ( f Ifl p d/z) 1/p < ~x~. Here 0 < p < cxz. The #-measurable essentially bounded functions, with norm IIf I1~ -infuz=0 sup Ifl~l. Lp(O, 1) -- Lp(#) w h e n / z is Lebesgue measure on the unit interval. -- Lp (#) when # is normalized Lebesgue measure on the unit circle. - Lp(lZ) when # is counting measure on the set F . - ~ p ( F ) when F = N. --gp(/-') when F = {1, 2 . . . . . n}. = The subspace of eoc of scalar sequences which have a limit. - The closure in goc ( F ) of the scalar sequences which have finite support. -

-

-

-

-

Basic concepts in the geometry of Banach spaces

= co(l-') when F = 1~. -- The closed unit ball of the Banach space X. = The closed ball of radius r with center x in the Banach space X; denoted also B(x, r) when X is understood.

c0

Bx Bx(x,r)

When 0 < p < 1, the space Lp(#) is not a Banach space except in the trivial cases that it is zero or one dimensional. The metric on L p(#), 0 < p < 1, is given by p(x, y) =

IIx- yll p. If {X~}n~__l is a sequence of Banach spaces and 1 <<.p <<.cxz, (y~ Xn)p is the Banach space of all sequences {Xn}n~=l with Xn E Xn and II{xn}~ II "-II{llXnll}~l lip < cx~. ( ~ Xn)o is the closed subspace of (Y~ Xn)~ for which Ilxnll -+ 0. When all Xn are the same space X we sometimes write ep(X) instead of (Y~ X)p. While this standard notation conflicts with the (also standard) notation g~p(F) introduced above, no confusion should arise. Sometimes also uncountable direct sums will be considered (with the obvious mean1 1 ing). The elementary duality relations c o* = el and ~p* - - ~q , 1 ~< p < cx~, 7 -+- q = 1, yield X n ) q for 1 ~< p < cx~. When the product that (y~ X~) 0* --- (Y~ X*)l and ( ~ X~)p -~X r Y (called the direct sum of X and Y) is given the norm II(x, y ) l l - - II(llx II, IJyll)llp, we write the resulting space X r Y. All these norms on X G Y are equivalent. When we are interested in X r Y only up to isomorphism we do not specify a particular norm. Probability theory has had a profound impact on Banach space theory. Consequently, we shall use in places probabilistic terminology. For example, we call the function which is one on a set A and zero on the complement of A the indicator function and denote it by 1a. We recall now some other standard concepts and results in probability theory which can be found in most introductory textbooks in the subject; in particular, in [10]. A random variable is a real valued measurable function on a probability space. A random variable g is said to be Gaussian or to have Gaussian distribution or normal distribution if its distribution function IP[g <~t] is equal to

1 /

e -(x-a)2/(2~r2) dx.

Here ~ denotes the underlying probability measure. The constant a is the mean or expectation or expected value of g, defined by Eg - f g d/1~. The quantity cr 2 is the variance of g, defined to be E(g - a) 2. If a -- 0 and cr -- 1, g is called a standard Gaussian variable. A finite collection {f~ }N of random variables on the same probability space is called n=l independent provided that N

N

]P n [ f n E An] = I-I ~[fn E An] n---1

n=l

for all Borel sets An. So E ( f - g) = E f E g if f and g are independent. An arbitrary collection of random variables is independent provided each finite subcollection of the collection is independent. Given a sequence {fn }n~_-i of random variables with fn defined on the probability space (S2n, ~n), one can construct a sequence {gn }n~_-i of independent random variables so that g~ has for each n the same distribution as fn. Indeed, let (~2,/1~) be the infinite

W.B. Johnson and J. Lindenstrauss product of the probability spaces (I-2n, I~'n) and for co = {con}nC~_1 in 1-2 set gn(co) = fn(con). In particular, it is possible to define a sequence of independent random variables having standard Gaussian distribution on the infinite product (0, 1)N of (0, 1) (with the product of Lebesgue measure). By the isomorphism theorem for separable measure algebras [ 18, p. 399], (0, 1)r~ can be replaced by (0, 1) itself, but it is often more convenient to work on the product space. The characteristic function of the random variable g is the function ~0:R --+ C defined by ~0(t) = Ee itg. Useful algebraic identities include ~0(-t) = ~0(t); qgag+b(t) -- e ibt~O(at); and, especially, qgf+g = qgf~Og, valid when f and g are independent. The characteristic function of a standard Gaussian random variable is e -t2/2. A key fact is that two random variables (possibly defined on different probability spaces) have the same distribution if and only if they have the same characteristic function. This is proved by an inversion formula [ 10, 2.3.a], another simple consequence of which is that the characteristic function of a random variable g is real valued if and only if g is symmetric; that is, g and - g have the same distribution. A (by no means immediate) consequence of the inversion formula [10, 2.7] is that for 0 < r < 2 the function e -Itlr is the characteristic function of a (necessarily symmetric) random variable, called a symmetric r-stable random variable. The tail distribution ~[Igl > t] of a symmetric r-stable random variable g is like t -r as t ~ c~, so that Ilgllp < ~ for p < r, but Ilgllr - cx~. Another probabilistic notion that plays an important role in Banach space theory is that of martingale. First we recall the notion of conditional expectation. Let ~ be a probability measure on a a-algebra/3. If ,4 is a sub cr-algebra of 13 and f is a ~-integrable function, then by the Radon-Nikod3~m theorem there is a A-measurable function g so that for each A in A, fA f dip -- fA g d~. The function g is called the conditional expectation of f given A and is sometimes denoted by E ( f I A) 9 Suppose now that { f n } F~/ = 0 is a sequence of random variables on the same probability space and/3n is the smallest cr-algebra for which fo, f l , . . . , fn are measurable. The sequence {fn }n~_0 is called a martingale provided that for each n, fn -- E(fn+l I/3n). The sequence {dn}n~=0 of differences; do = fo, dn - fn fn-1; is then called a martingale difference sequence. Notice that a sequence {dn}n~=0 of independent random variables is a martingale difference sequence if and only if dn has mean zero for all n ~> 1. A simple but important example of a martingale difference sequence which is not a sequence of independent random variables is the Haar system, discussed in Section 4. One basic theorem about martingales, the martingale convergence theorem (see [10, 4.2.10]), states that every L1 bounded martingale converges a.e., which means that if {fn }n=0 is a martingale of P-measurable functions and SUPnElfnl < c~, then {fn }n~__0converges a.e. Moreover, if the martingale is uniformly integrable (see Section 4, then it also converges in L1 (~) (see [10, 4.5.3]).

3. Bases Excepting [2] and [12], which are oriented to Banach space theory, few introductory texts in functional analysis treat Schauder bases. Nevertheless, bases are a very useful tool for investigating properties of Banach spaces.

Basic concepts in the geometry of Banach spaces We prove few statements in this section since most of the results are proved in [2] and [12]. The book [14] often contains only sketches of proofs. Chapter 2 of [21] contains enough details to be pleasant reading for the mature student or experienced mathematician who is not an expert in Banach space theory. All of these books contain exercises. Many in [21 ] are challenging even after peeking at the hints. Exercises in [ 14], as in this introductory article, are scattered throughout the text and are sometimes flagged by such expressions as "clearly", "hence", and the like. A Schauder basis or simply a basis for a Banach space X is a sequence {Xn}n~=l of vectors in X such that every vector in X has a unique representation of the form ~ OlnXn with each C~n a scalar and where the sum is converges in the norm topology. The mapping x w-> Otn then defines for each n a linear functional x n on X. One checks that the expression !xV9= SUPn IIE~=I xk (x)x~ll defines a stronger complete norm on X, so that !.! and II, II are equivalent by the open mapping theorem. One deduces from this that the biorthogonal functionals for a basis are necessarily continuous. Moreover, the biorthogonal functionals are a basic sequence in X*; that is, they form a basis for their closed linear span. When it is useful to specify the biorthogonal functionals, we sometimes refer to the "basis" {Xn, x n }n=l" A sequence {Xn}n~=l is called normalized if each vector Xn has norm one and {Xn}n~__l is called seminormalized if it is bounded and bounded away from zero. For most purposes it is sufficient to consider normalized or at least seminormalized basic sequences 9 The partial sum projections of a basis {Xn,'A n*l~ln__l' defined for n = 1, 2 , . . . by Pnx -Y~.inl x * ( x ) x i , are necessarily uniformly bounded and converge strongly to the identity; that is, Ilx - PnX II --+ 0 for each x in X. The supremum of the norms of these partial sum projections is called the basis constant. This quantitative notion is of interest also if we just consider finite basic sequences or bases for finite dimensional spaces. A sequence {x,},~__l of nonzero vectors is basic with basis constant at most C if and only if for all n < m (and all scalars O/i) the inequality IIY~in~__l OliXi ]l ~ C IIZ i % l Oli Xi II holds. A block basis {yj }j= 1 +1 of the basis {x~}~__l is a sequence of nonzero vectors of the form yj m ~-'~nJ z--~k=nj+l ~kxk for some sequence n l < n2 < .- .. The basis constant of a block basis of {Xn }n=~1 is no larger than the basis constant of {xn }~ n=l" A basis is called monotone provided that its basis constant is one. One can change to an equivalent norm, Ill'Ill, in which the basis is monotone" just set ]llxlll- sup~ IlPnxll. In fact, if one defines instead Illxlll = SUPn<m IIPmx - Pnxll one gets an equivalent norm which is even bimonotone; that is, the basis is monotone and also all the complementary projections I - Pn have norm one. Often one constructs a basic sequence {Xn}n~=l by induction and checks that it is basic by n

~-~n+ 1

verifying for each n the inequality IIY~./=I ~ixilJ <<.Cnllz_..i=l ~ixill with y ~ n = l ( C n - 1) < oo cx~; the basis constant of {x,}~_ 1 is then at most I-I,=l C~. Here is one classical construction, which yields, for example, that every infinite dimensional Banach space contains a basic sequence. Given any sequence Cn as above and having selected x l . . . . . Xn, take a finite set S - S, of norm one linear functionals on X which almost determine the norm on the linear span Xn of X l . . . . . Xn in the sense that for each x e Xn, Ilx II ~< Cn maxx.~S Ix*(x)l. Now Xn+~ can be any nonzero vector in the finite codimensional subspace S• of X. This construction allows considerable " p l a y " - given ~ > 0, as long as

W.B. Johnson and J. Lindenstrauss

you select Xn+l SO that maxx,cS [[Xn+l [[-1 [X*(Xn+l)[ is sufficiently small, the resulting sequence {Xn}n~__l will have basis constant less than e + I-In~__l Cn. This yields, for example, that every weakly null, non-norm null sequence has a basic subsequence. This last result can be improved substantially. Two basic sequences {Xn }n~__l and {Yn }n~=l are equivalent provided that the map Txn = Yn extends to an isomorphism from the closed span of {Xn}nC~=l onto the closed span of {Yn}n~ K-equivalent if [[TllllT-1[[ ~< K (so, strangely, {Xn}n~__l is 1-equivalent to {2Xn}n~__l, but usually basic sequences are normalized). The principle o f small perturbations says that if {Xn}n~__l is a basic sequence in X and IlXn - Yn II--+ 0 sufficiently quickly, then {Yn}n~__l is a basic sequence which is equivalent to {Xn}n~=l. To prove this, let {Xn*}n~__lbe Hahn-Banach extensions of the functionals biorthogonal to {Xn}n~__l to functionals in X* and define an operator S on X by Sx -- Y ~ x * ( x ) ( y n - X n ) , SO [[S[[ ~ E [[x*[[[[yn --Xn[[. If I]sll < 1, elementary considerations yield that I + S is an automorphism of X which maps Xn to Yn. This also yields that if { n}n=l is a basis of X, then so is {Yn}n~_l. From this one sees that if {Xn}n~__l is a basis for X with biorthogonal functionals ttx n*J~n=l' if {Yn}n~=l is a seminormalized sequence in X, and if for each k, x k (Yn) -+ 0 as n --+ cx~, then {y~}n~__l has a subsequence which is equivalent to some block basis {Zn}n~__l of {Xn}n~__l. From this it follows that every infinite dimensional subspace Y o f a Banach space X with a basis {Xn}n~__l contains, f o r every K > 1, a normalized basic sequence {Zn}n~__l which is K-equivalent to a normalized block basis {Yn }n~--_1 o f {Xn }~__1. Moreover, the small perturbation argument indicated above shows that the generated isomorphism which maps {Yn}n~=l into Y extends to an automorphism on X. To see how these basic facts about bases might be used, suppose that T is a noncompact operator from a subspace X0 of a Banach space X with basis {Xn, x~ }n~__l into a Banach space Y with basis {Zn}n= 1. Then there is a sequence {Yn}n~__l in the unit ball of X0 such that for some E > 0 and all n ~ m, IITyn - Tym I[ > ~. By passing to a subsequence of differences of {Yn}n~=l it can be assumed that for each k, x k (Yn) --+ 0 as n --+ ~ . In view of what was discussed in the previous paragraph, by passing to another subsequence it can be assumed that {Yn}n~=l is an arbitrarily small perturbation of a block basis of {Xn}n~__l. Repeating the same argument for {Tyn}n~=l in Y and normalizing at the end, we conclude that if T is a noncompact operator from a subspace Xo o f a Banach space X with basis {Xn}n~__l into a Banach space Y with basis {Zn}n~=l, then there are automorphisms U on X and V on Y and a normalized block basis {Yn}n~__lof {Xn}n~ with Yn E U-1Xo and such that {VTUyn}n~__l is a seminormalized block basis o f {Zn}n~__l. This last mentioned result gives considerable information in cases where all block bases of a basis can be characterized. The simplest examples of bases are provided by the unit vector basis {en}n~__l for s 1 <~ p < cxz, and co. It is evident that in these spaces every normalized block basis of {en}n~=l is 1-equivalent to {en}nC~__1 and every seminormalized block basis of {en}n~__l is equivalent to {en}n~__l. Thus we can conclude that every operator from a subspace o f s into s 1 <~ p < r < cx~, is compact. This shows in particular that no infinite dimensional subspace o f s isomorphically embeds into s p when 1 <~ p, r < c~ and p ~ r (an isomorphism between infinite dimensional spaces cannot be compact since no infinite dimensional space is locally compact).

Basic concepts in the geometry of Banach spaces We should mention that, in contrast to the case of gp and ~r, it generally is quite difficult to determine whether two spaces are isomorphic even when both are presented concretely as similar but different function spaces. Another consequence of the principle of small perturbations is that g l has the Schur property, which means that every weakly convergent sequence in ~ j is norm convergent. Indeed, otherwise there would be a weakly null sequence {Xn }n~=l in g l which is bounded away from zero. The sequence {Xn}n~=l is necessarily bounded, so the sequence {Xn}n~_l would have a subsequence {Xnk }~=l which is equivalent to a block basis of the unit vector basis of g 1 and thus equivalent to the unit vector basis of g 1. This is a contradiction because the unit vector basis of ~l is not weakly null. An easy formal consequence of the fact that g l has the Schur property is that every weakly Cauchy sequence in f j is norm convergent. In particular, el is weakly sequentially complete. The most natural basis for Lp(O, 1), 1 ~ p < oo, is the Haar system {hn}~_0, where h0 = 1 [0,1), and for n - 2 j 4- k with j = 0, 1 . . . . and k = 0, 1 . . . . . 2 j - 1,

hn = l[k2-J,(2k+l)2-J-1 ) -- l[(2k+l)2-j-~,(k+l)2-j ). It is easy to check that if 1 ~< p ~< oo, then for each n and all scalars Oti the inequality n X--~n+l ~ihillp holds, which means that the Haar system is a monotone II~,i=o~ihillp <~ Ilz~i=0 basic sequence in L p (0, 1). Since the linear span of the Haar system is dense in L p (0, 1) when p < oo, the Haar basis is a monotone basis for L p (0, 1) when 1 <~ p < oo. The sequence {f,}~-0, where f0 = 1[0,1] and for n >~ 1, f , is the indefinite integral of the Haar function h , _ j , forms a monotone basis for C[0, 1]. When normalized in the supremum norm, this is called the Faber-Schauder basis. The trigonometric system {ein~ ~176 forms a basis for L p ( ~ ) when it is ordered 0, 1, - 1 , 2, - 2 . . . . and 1 < p < oo. This follows from the boundedness of the Riesz projection in this range. The Riesz projection is defined formally by 7~(f) = ]Y~'n~_0/(n)e in~ w h e r e / ( n ) = (2•) -1 f2~r f ( t ) e _ i n t dt is the nth Fourier coefficient of the function f . The unboundedness of the Riesz projection on L l(T) and C(~) yields that the trigonometric system is not a basis for these spaces. A series ~ x, in a Banach space is said to converge unconditionally provided every rearrangement of the series converges. This is equivalent to ~ xk, converges for each subsequence of {x,}~~176and also to ~ i x , converges for each choice of signs +. Except in finite dimensional spaces, unconditional convergence is weaker than absolute convergence; that is, convergence of ~ IIx~ II (see Section 8). A basis {x, }~-I is said to be an unconditional basis provided that ~ UnXn converges unconditionally whenever it converges. This is equivalent to saying that every permutation of {x,}~~176is also a basis. If {xn}~~176is an unconditional basis for the Banach space X and 0 - IO,}n~=l is a sequence of + l ' s , define So "X --+ X by S o ( ~ c ~ , X n ) - Y]~OnotnX,. The supremum over all such 0 of ]1So 11is finite, and is called the unconditional constant of the basis. One can define an equivalent norm 111.Ill on X for which the unconditional basis has unconditional constant one by setting [[Ixll] = sup 0 []Sox[]. Such a norm is said to be unconditionally monotone. When the norm is unconditionally monotone, all permutations of the basis are monotone (sometimes this weaker condition is referred to as "unconditionally monotone").

10

W.B. Johnson and J. Lindenstrauss

A sequence which is an unconditional basis for its closed linear span is said to be unconditionally basic. Since a block basis of an unconditional basis is clearly unconditional (with unconditional constant no larger than that of the basis itself), the perturbation principle described above yields that every infinite dimensional subspace of a space with unconditional basis contains an unconditionally basic sequence. The unit vector bases for ep, 1 ~< p < cx~, and co are the simplest examples of unconditional bases. For 1 < p < c~, the Haar system forms an unconditional basis for L p (0, 1). The "modem" proof of this proceeds via martingale theory. The exact unconditional constant of the Haar system in Lp(O, 1) is computed in [23]. In Section 8 we point out that the trigonometric system is an unconditional basis in Lp(7~) only for p -- 2. A basis {Xn }n~__1 for X is shrinking provided the linear span of the biorthogonal functionals {Xn}n= * ~ 1 is (norm) dense in X*, which is to say that {Xn*}n~__lis a basis for X*. Notice that a bounded shrinking basis necessarily converges weakly to zero. The basis {Xn }n~__l is boundedly complete provided that whenever the sequence {y~in__101iXi }nC~_1 is bounded, then it is convergent. If a basis is shrinking, then its biorthogonal functionals are a boundedly complete basis for X*. Conversely, if the biorthogonal functionals form a boundedly complete basis for their closed linear span Y, then the basis is shrinking (and hence Y = X*). Similarly, a basis is boundedly complete if and only if its biorthogonal functionals are a shrinking basis for their closed linear span. If {Xn }n~__l is a boundedly complete basis for X and Y is the normed closed linear span of the biorthogonal functionals {xn* }n=l' ~ the shrinkingness of {Xn*}n~__limplies that the natural evaluation mapping from X into Y* is a surjective isomorphism (which is easily seen to be an isometry if the basis is monotone). Consequently, a space with a boundedly complete basis is isomorphic to a separable conjugate space. From these facts it follows, in particular, that a Banach space X with a basis is reflexive if and only if some (or every) basis for X is both shrinking and boundedly complete. If X has an unconditional basis, then some (or every) unconditional basis is boundedly complete if and only if X has no subspace isomorphic to co, while some (or every) unconditional basis is shrinking if and only if no (or no complemented) subspace of X is isomorphic to el. Consequently, a space with unconditional basis is reflexive if and only if no subspace is isomorphic to either co or to ~1 if and only if its second dual is separable. In contrast, there is a nonreflexive space which has both a shrinking basis and a boundedly complete basis which is of codimension one in its bidual, [ 14, 1.d.2]. While most of the separable spaces encountered in classical analysis have bases, many do not have an unconditional basis; in particular, the spaces L1 (0, 1) and C[0, 1]. Indeed, co does not embed isomorphically into L1 (0, 1) (for example, because L1 (0, 1) is weakly sequentially complete and co is not; see Section 4), so any unconditional basis for L1 (0, 1) would have to be boundedly complete and L 1(0, 1) would be isomorphic to a separable conjugate space. But L 1(0, 1) does not even embed isomorphically into a separable conjugate space (since if L 1(0, 1) C X* and for t E (0, 1) x~ is a weak* cluster point of the sequence n - l l(t,t+l/n), then it can be checked that there is an uncountable subset S of (0, 1) and 6 > 0 so that for all t :~ s in S, IIx? - x* II > 6). A more sophisticated (but actually technically easier) argument shows even that L1 (0, 1) does not embed isomorphically into a space with unconditional basis; see [14, 1.d.1]. Since, as mentioned in Section 4,

Basic concepts in the geometry of Banach spaces

11

every separable space embeds isometrically into C[0, 1], C[0, 1] also does not embed isomorphically into a space with unconditional basis. It is a deep result that there exist infinite dimensional spaces which do not contain an unconditionally basic sequence (see [37]). The example uses Banach spaces whose norm is not defined explicitly by a formula but by an implicit procedure. This method of defining a Banach space (or a norm) is very useful in many contexts. A basis {Xn }n~=1 is called symmetric provided every permutation of {Xn }n~__1 is equivalent to {Xn}n~176. In particular, every permutation of {Xn}n~__l is a basis, so a symmetric basis is unconditional. A basis {Xn}n~__l is called subsymmetric provided it is unconditional and equivalent to each subsequence of itself. A symmetric basis is subsymmetric [14, 3.a.3]. It is evident that if a basis is unconditional, symmetric, or subsymmetric, then the same is true for the biorthogonal functionals (in their closed linear span). The unit vector bases for ~p, 1 ~< p < ec, and co are symmetric, while the L pnormalization of the Haar system is not a symmetric basis for L p (0, 1) if p 7~ 2. In fact, L p(O, 1) has no subsymmetric basis if p 7~ 2. We have already mentioned that L 1(0, 1) does not have even an unconditional basis. That L p(O, 1), 2 < p < ec, has no subsymmetric basis follows from the dichotomy principle for sequences in the space, which is discussed in Section 8. This principle says that if {xn}~~ 1 is a seminormalized basic sequence in Lp(O, 1), 2 < p < ec, then {X~}nOC=lhas a subsequence which is equivalent to the unit vector basis for either s or g2. Since, as noted in Section 4, Lp(O, 1), p ~ 2, is not isomorphic to either gp or ~2, Lp(O, 1), 2 < p < ec, has no subsymmetric basis. That L p(0, 1), 1 < p < 2, has no subsymmetric basis then follows by duality. It is interesting that symmetric bases are nevertheless prevalent. For example, any space which has an unconditional basis is isomorphic to a complemented subspace of a space which has a symmetric basis (this will be discussed further in Section 11). In fact, there is even a space Y which has a symmetric basis and also an unconditional basis {Zn}n~ which is universal for unconditional bases in the sense that every unconditional basis for any Banach space is equivalent to a subsequence of {Zn },c__1 ! The space Y and the universal unconditional basis is easy to construct, given the fact, to be proved in Section 4, that every separable Banach space is isometric to a subspace of C[0, 1]. Indeed, one takes a dense sequence {X~}neC=l of nonzero vectors in C[0, 1] and lets Y be the completion of the finitely supported scalar sequences under the norm II{an } ~ l l l " - sup+ IIZn -+-anXn IIC[0,11. The unit vector basis in Y is a universal unconditional basis. See [14, p. 129] or [41] for further discussion and for the proof that the space Y has a symmetric basis. While unconditional bases, symmetric bases, and subsymmetric bases are all useful strengthenings of the notion of basis, there are equally useful weakenings. A Schauder decomposition for a Banach space X is a sequence {X~ }n~__l of nonzero closed subspaces of X such that every vector x in the space has a unique representation of the form x -- ~ Xn with Xn ~ Xn. If each Xn is finite dimensional, the decomposition is called a finite dimensional decomposition, or simply an FDD. A basis can be regarded as an FDD {Xn } ~ 1 such that every space Xn is one dimensional. As in the case of bases, a Schauder decomposition {Xn}~~1761 for a Banach space X determines a sequence {Pn }~~176 1 (called the partial sum projections of the decomposition) of commuting projections with increasing ranges which converge strongly to the identity operator on X; namely, for x = y~ Xn with x~ E Xn and n - 1, 2 . . . . . Pnx - ~-~inl xi. Conversely, if {Pn}n~__l is a sequence of commuting projec-

12

W.B. Johnson and J. Lindenstrauss

tions with increasing ranges which converge strongly to the identity operator on X, then {(Pn - Pn-1)X}n~=l (P0 "-- 0) is a Schauder decomposition for X for which {Pn }n~_-i is the sequence of partial sum projections. The biorthogonal functionals for a basis are replaced in the case of a Schauder decomposition by the sequence { (P* - Pn*l)X* }n~=l of (even weak*) closed subspaces of X*, which form a Schauder decomposition for their closed span. The supremum of the norms of the partial sum projections determined by a Schauder decomposition is finite and is called the decomposition constant of the decomposition, and the Schauder decomposition is called monotone if its decomposition constant is one. The definitions of unconditional, shrinking, and boundedly complete bases generalize immediately to Schauder decompositions. The structure theory for bases goes over with little difficulty to FDD's. More importantly, there are useful concepts involving FDD's which are not merely generalizations of notions about bases. For example, a blocking of a Schauder decomposition {Xn}n~=l is a sequence of the form {Xnk "-[-''" + Xnk+l-1}L1 with 1 = n 1 < n2 < . . . . Any blocking of a basis (that is, of a Schauder decomposition into one dimensional spaces) produces an FDD which is not a basis. While bases and block bases are fine in situations where passage to subsequences or subspaces involves no loss, blockings of FDD's sometime provide the proper framework for investigations where it is important to maintain global control. For example, it was mentioned above that every normalized basic sequence in ep, 1 < p < cx~, has a subsequence equivalent to the unit vector basis of ep, but one can prove that every FDD for a subspace of gp, 1 < p < ~ , has a blocking which is an ep decomposition [14, 2.d.1]. (A Schauder decomposition {Xn}n~__l is said to be an g.p decomposition provided that for Xn ~ Xn, the series y~ Xn converges if and only if y~ [[Xn[Ip < oo.) A separable Banach space is said to have the bounded approximation property or simply BAP provided there is a sequence {Tn }n~__l of finite rank operators on X which converges strongly to the identity. If the operators can be chosen to commute, the space has the commuting bounded approximation property (CBAP). In either case the supremum of [[Tn [[ is finite; if at most )~, we say that the space has the )~-BAP or )~-CBAP, or when )~ = 1 the metric approximation property (MAP) or commuting metric approximation property (CMAP). Finally, A Banach space X has the approximation property or simply AP provided that for every compact subset K of X and E > 0, there is a finite rank operator T on X such that [Ix - Tx l[ < ~ for every x ~ K. If a space X has the AP, then every compact operator S into X is the norm limit of a sequence of finite rank operators. Indeed, take for n = 1, 2 . . . . a finite rank operator Tn on X so that for each x in the image under of the unit ball of the domain of S the inequality IIx - Tnxll <<.1In holds. Then IIS - ZnSII <<.1/n. The converse is also true [ 14, 1.e.4]. The implications basis =, FDD =, CBAP =, BAP =, AP are all easy or obvious. None are reversible except possibly CBAP =, BAP; this is discussed in [24], as is the existence of spaces which fail the AP and connections among these various approximation conditions. There are some surprises, such as the AP implies the MAP in reflexive spaces; the MAP implies CMAP; and a space which has the BAP is isomorphic to a complemented subspace of a space which has a basis. Incidentally, while spaces failing the AP or other approximation properties might be thought pathological, they arise naturally in various contexts (for example, the nonseparable space B(s of operators on e2 fails the AP, as do some separable C* algebras). Moreover, the construction of spaces failing approximation

Basic concepts in the geometry of Banach spaces

13

conditions leads to nice theorems which do not involve any notion of approximation; in particular, isomorphic characterizations of Hilbert space (see [35]). The existence of spaces failing the AP as well as other considerations lead to the study of other basis-like structures in Banach spaces. Probably the most useful of these is that of Markuschevich basis. A Markuschevich basis for a Banach space X is a biorthogonal sysfor which {x• }y~r is fundamental (that is, the span of the x• is dense tem {x• x•* }• * in X) and {X•215 is total; that is, the x•*'s separate the points of X. The trigonometric system in L I(~') or C(~') is a natural example of a Markuschevich basis which is not a (Schauder) basis in any order (but this last statement is not easy to verify). Markuschevich bases do not always exist in the nonseparable setting, but are a useful tool for investigating nonseparable spaces in situations where they exist (see [44]). It is easy to see that any separable space X admits a Markuschevich basis. One starts with linearly independent sequences {Yn}nCX~1 in X and {'Yn*'CX~ln--Iin X* with {Yn}n~1_ fundamental and {yn*}n~_l_ total and applies a Gram-Schmidt type procedure to biorthogonalize the sequences (alternating between working in X and X*) to produce a Markuschevich basis {Xn,X*}n~__l so that _ and span{xn*}n~__l = s-an'* ~ 1 (see [14, 1.f.3]). A deeper span{Xn}n~=l -- span{yn}n~_l p [.Ynln= fact [14, 1.f.4] is that a separable space contains a Markuschevich basis { X n , X n*} n~: 1 for which SUPn IIxn II IIxn*ll ~< 20, and it is known that twenty can be replaced by any number larger than one. We shall see in Section 8 that a finite dimensional space has a basis N {Xn, x n, }n=l for which IIxn II IIx* II - 1 for each n, but it is an open problem whether every separable space has a Markuschevich basis with this property. One separable theorem in which Markuschevich bases are used but do not appear in the statement is that every separable space X has a subspace Y such that both Y and X~ Y have an FDD [ 14, 1.g.2]. Incidentally, it is open whether the conclusion can be improved to "both Y and X~ Y have a basis".

4. Classical spaces In this section we present some basic facts concerning the structure of the classical spaces C(K) and Lp(#), their classification, and the relations among the spaces. Most books on real analysis, such as [ 18], contain some elementary results about the structure of these spaces. Beyond the most basic material, [21 ] is accessible to students and has the additional advantage that some of the relations between these spaces and other spaces encountered in analysis are touched on. The more narrowly focused book [7] is also directed at students, contains several structural results omitted from [21], and the author's off-the-wall style makes the book fun to read. A normalized sequence {Xn}n~=l of disjointly supported functions in Lp(#), 1 <<,p < cx~, is 1-equivalent (see Section 3) to the unit vector basis for gp and the closed span of {Xn}~__l is the range of a norm one projection. The first statement is clear and the projection is defined by x w-> Y~n(f XlXnlP-lsignxn d#)xn. In particular, every block basis of the unit vector basis for gp, 1 ~< p < cx~, spans a space isometric to g p onto which there is a norm one projection. This last statement is also true if g p is replaced by co. Hence by the principle of small perturbations discussed in Section 3, if Y is an infinite dimensional subspace of X, X = g p for 1 ~ 0 there is a subspace Z of Y with d (Z, X) < 1 § ~ and so that there is a projection of norm less than 1 + ~ from X onto Z.

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W.B. Johnson and J. Lindenstrauss

Although general subspaces of X, X = s for 1 ~ p 7~ 2 < cc or X = co can be bad in that they can fail the approximation property and there are many of them, in a sense that is made precise in [24], the isomorphism theory of complemented subspaces of X is simple: An infinite dimensional complemented subspace Y of X, X = g.p for 1 <<.p < e~ or co, is isomorphic to X. To see this, write X ~ Y @ V for some V. As mentioned in the previous paragraph, also Y ~ X @ W for some W. Hence (we write the formula for s for co the proof is identical but the notation changes) ~ (s 9 s 0 . . . ) p ~ (Y @ Y @ . . . ) p 9 (V @ V G . . ")p Y G (Y G Y q ) ' " ) p @ (V G V q ) . . . ) p ~ Y ~ s .~ W O s

s

q) gp "~ Y.

This method of proof, which is useful in many contexts, is called the decomposition method. It is also true that every infinite dimensional complemented subspace of s is isomorphic to ~ but the proof requires an additional argument [ 14, 2.a.7]. Let 1 ~< p < r < ec. In Section 3 we saw that every operator from a subspace of s into g.p is compact. There are noncompact operators from s into s such as the formal identity operator Ip,r. When there exist "nonsmall" operators from one space to another (here "nonsmall" means "noncompact", but elsewhere it can have another meaning), it is helpful to have one "nonsmall" operator which is typical of the genre. In fact, Ip,r is typical of the noncompact operators from s into s in a sense that can be made precise: Say that an operator T : X ~ Y factors through an operator S: W --+ Z provided there exist operators U :X --+ W and V : Z --+ Y so that the diagram W

X

s

>Z

1

~Y

commutes; that is, T -- V SU. Now we can make precise the statement about Ip,r" If T is a noncompact operator from a subspace of g.p into g-r, 1 <. p < r < e~, then Ip,r factors through T. Indeed, this is an immediate consequence of the comments in Section 3 on noncompact operators from a subspace of a space with a basis and the remark in this section that a block basic sequence of the unit vector basis of g p spans a complemented copy of g p in ep. It is easily seen that the only separable L ~ (#) spaces are those for which the measure /z is purely atomic and has finitely many, say n, atoms, in which case L ~ ( # ) is isometric to ~ since it is obvious that the indicator functions of n distinct atoms are, in Lot(#), 1-equivalent to the unit vector basis for ~nOO" The isometric classification of the separable Lp(#)-spaces, 1 ~< p < oo, is also simple. First, the measure # can be assumed to be a-finite. Secondly, any L p ( # ) , 1 <<.p <~ c~, with # ~r-finite, is isometric to a space of the same form with # replaced by a probability measure. Indeed, take g positive a.e. [#] with integral one and define a measure v by dv = g d#. Then the mapping f v-+ f . g-1/p defines an isometry from Lp(lZ) onto L p ( v ) . (For p -- oo the isometry is just the formal identity.)

Basic concepts in the geometry of Banach spaces

15

Next, if Lp(l z) is separable with # a purely nonatomic probability measure, then the isomorphism theorem for separable measure algebras [18, p. 399] yields that Lp(tX) is isometric to L p(O, 1). On the other hand, if the measure # is purely atomic with atoms {A• then the function which maps for each y the unit vector ey to lZ(Ay) -1/p times the indicator function of A• extends to an isometry from g.p(F) onto Lp(lZ). N o w if x and y are in Lp(#), 1 ~ p ~ 2 < oc, then x and y are disjointly supported if and only if Ilx + yll p = IlXllpp + Ilyll p [18, p. 416]. This implies that an isometry from one Lp space onto another preserves disjointness and therefore also atoms. All these remarks combine to show that g.np,~p, Lp(O, 1), s @p Lp(O, 1), ~pn @p Lp(O, 1), n -- 1, 2 . . . . , is a complete listing, up to isometry, of the separable Lp(#) spaces when 1 ~< p 7~ 2 < co, and these are all mutually nonisometric. Of course, in the Hilbertian case p - 2, s n -- 1, 2 . . . . ; ~2 is the appropriate listing. The decomposition method then yields that s s L p(O, 1) is a complete listing, up to isomorphism, of the separable L p spaces. Later in this section it is noted that s is not isomorphic to L p (0, 1) when 1 ~< p 7~ 2 < ec, so this is a listing of nonisomorphic spaces for these values of p. In order to study the structure of subspaces of L p (0, 1) as well as to investigate many other questions about L p for one fixed value of p, it is convenient to consider the scale of L p spaces as p varies. One reason for this is that (as we shall see) there are spaces of functions X on (0, 1) on which the norm II" II p, is equivalent to II" IIP2 with Pl < P2. Note that when this occurs, since I1" IIp ~< I1" IIr when p ~< r, all the norms I1" II p are equivalent on X for pl ~< p ~< p2. In fact, the extrapolation principle says that then all the norms I1" IIp are equivalent on X for p ~< p2. Indeed, suppose C is a constant so that IIx IIp2 ~< C IIx IIpl for x E X, 0 < p < p l , and 0 < )~ < 1 is defined by the formula pl = )~p + (1 - )Qp2. Then, by H61der's inequality, for x E X we have IIx IIp~ ~ IIx II~pIIx II(1-~)p= ~ C 1-~ IIx IIp~l-&IIx II~p and hence C 1-1/z IIx Ilpl ~ IIx lip ~ IIx IIp~. The preceding yields particularly good information in case 2 < p and X is a closed subspace of L p(O, 1) which is closed in Lr (0, 1) for some r ~: p. By the open mapping theorem, the I1" IIp and I1" IIr norms are equivalent on X, and hence so is the I1" 112 norm. This means that X must be isomorphic to a Hilbert space. Moreover, the orthogonal projection P from L2 (0, 1) onto X induces a bounded linear projection/3 from L p (0, 1) onto X. The extrapolation principle then says that I1" lip*, 1/p + I/p* = 1, and I1" 112 are equivalent on X. The orthogonal projection onto X extends to the bounded projection 15, from Lp, (0, 1) onto X. Suppose that, on the other hand, X is a closed subspace of L p (0, 1), 1 ~< p < ~ , and for some (or, equivalently, every) 0 < r < p the I1" IIp and I1" Ilr norms are not equivalent on X. H61der's inequality then yields that for any M < cx~ the infimum of Ilxl[ixl<~M]llp as x ranges over the the unit sphere of X is zero. This means that X contains unit vectors which are an arbitrarily small perturbation of vectors whose supports have arbitrarily small measure, which in turn yields that X contains a sequence of unit vectors which is an arbitrarily small perturbation of a sequence of disjointly supported vectors in L p (0, 1) and hence, by

W.B. Johnson and J. Lindenstrauss

16

the principle of small perturbations, that X contains a subspace which is isomorphic to s and complemented in L p (0, 1). These observations yield a dichotomy principle for subspaces of L p(O, 1), 2 1, X contains a subspace C-isomorphic to s and C-complemented in L p (0, 1). This is stated for Lp(O, 1), but is valid for general L p ( # ) by the isometric classification of Lp spaces discussed earlier. Notice also that the comments in Section 3 about s now yield that if s embeds isomorphically into L p(#), 0 < r < cx~ and 2 < p < oo, then either r = p or 1~ - - - 2 .

The natural examples of infinite dimensional function spaces which are closed in L p for more than one value of p come from probability theory. Suppose that {gn}n~=l is a sequence of independent standard Gaussian random variables, discussed in Section 2. From the form of the distribution of g it is evident that ]lg]lp is finite for all finite p and {gn}n~=l is an orthonormal sequence in L2 (0, 1). A characteristic function argument yields that if n Y ~n= l [otk]2 -- 1, then Y~k= 1 ctk gk is again a standard Gaussian and hence has the same norm in Lp(O, 1) as g. This means that for all 0 < p < oo, the mapping en ~ ][g][plgn, n = 1, 2 . . . . . extends to an isometry from s onto a subspace X of L p (0, 1) where X does not even depend on p, and the orthogonal projection onto X defines a bounded projection from L p (0, 1) onto X if 1 < p < oe (in Section 10 we explain why no isomorph of s in L 1(0, 1) or L ~ (0, 1) can be complemented). In particular, the dichotomy principle for subspaces of L p, 2 < p < oo, is not a monochotomy principle. This also implies, in view of the structure theory of s discussed earlier, that s and L p (0, 1) are not isomorphic for 0 < p :/: 2 < o~. A second natural example from probability theory of a subspace of L p, 0 < p < o0, which is isomorphic (but not isometric for p # 2) to s comes from considering a sequence {en}n~=l of independent random variables each taking on each of the values 1 and - 1 with probability 1/2. Such a sequence {en}n~__l is called a Rademacher sequence. Classically the Rademacherfunctions {rn }n~__l are the concrete realization of such a sequence on [0, 1] m

V ~2n - 1

defined by rn " z_,~=o hz,+k, where {hn}n~__l is the Haar system. Khintchine's inequality says that a Rademacher sequence is equivalent, in the Lp norm, 0 < p < o~, to the unit vector basis of s for future reference we write the inequality explicitly with the best constants labeled Ap and Bp:

Since a Rademacher sequence is orthonormal,

llz .ll it follows that Ap =- 1 for p ~> 2 and Bp = 1 for p ~< 2, and the exact values of Ap and Bp are known. The most elementary proof of Khintchine's inequality proceeds by checking the right inequality in (1) for p an even integer. This clearly gives the result for all 2 ~< p < oo and one then obtains the result for 0 < p < 2 from the extrapolation principle. The

Basic concepts in the geometry of Banach spaces

17

less computational modern proof of Khintchine's inequality gives a vector valued version of Khintchine's inequality called the Kahane-Khintchine inequality. This will be done in Section 8. The existence of an r-stable variable g (see Section 2) for 0 < r < 2 shows that the subspace structure of L p (0, 1), 0 < p < 2, is much more complicated than that of L p (0, 1), 2 < p < cx~. Indeed, just as in the Gaussian case, there exists for 0 < r < 2 a sequence {gn}n~l of symmetric r-stable random variables defined on (0, 1). If 0 < p < r, then these random variables are in Lp(O, 1). Now if ~ k -n- 1 I ~ l r 1, then ~ = l otkgk is again symmetric r-stable and hence has the same norm in L p(O, 1) as g. This means that for all 0 < p < r, the mapping en ~ Ilgllp-1 gn, n -- 1, 2 . . . . . extends to an isometry from er into L p (0, 1). In Section 9 we explain how to derive from this the fact that for 1 ~< p < r < 2, Lr (0, 1) embeds isometrically into L p(O, 1). It turns out that this covers all the cases in which er embeds isomorphically into Lp(iZ ) with p and r finite. The remaining cases are discussed in Section 8. In the reflexive range 1 < p < cx~, information about subspaces of Lp,(#), 1/p + 1/p* - 1, given above gives information on quotients of Lp(#). It turns out that the case p = 1 is quite different: Every separable Banach space X is isometric to a quotient of el. Indeed, if {Xn}n~=l is dense in the unit ball of X, the linear extension of the map en ~ Xn (where {en }~--1 is the unit vector basis for el) maps the unit ball of el onto a dense subset of the unit ball of X and hence extends to a quotient mapping from el onto X. Similarly, if the Banach space X has density character x, then X is isometric to a quotient of el ( F ) when F has cardinality x. Another useful and interesting property of el (which also holds for e l ( F ) for any set F ) is the lifting property: If T is an operator from a Banach space X onto el then there is a lifting S of T; that is, an operator S :el --+ X for which T S = Ie~. Indeed, by the open mapping theorem there are Xn in X with Txn = en and )~ :=- supllxn II < cx~. The mapping en ~ Xn then extends to an operator S : e 1 ---+ X with [[Sll - ;~ satisfying T S = le~. As noted in Section 3, e 1 has the Schur property; that is, weakly convergent sequences in el are norm convergent. An immediate consequence is that every weakly compact subset of el is norm compact. The weakly compact subsets of L 1(0, 1) are more complicated since L 1(0, 1) contains infinite dimensional reflexive subspaces. There is however a nice characterization of subsets of L1 (/z) which have weakly compact closure when /z is a finite measure. First, if X is any Banach space and W is a subset of X such that for each E > 0 there exists a weakly compact set S so that W C S + EBx, then W has weakly compact closure (use the fact that a bounded subset of X is weakly compact if its weak* closure in X** is a subset of X). Next, given W C L1 (#) with # a finite measure, set for k E N, a k - a k ( W ) "--sup{llxllxl~klll" x ~ W}. Clearly {ak}~= 1 decreases to some a = a(W) ~ O. The set W is called uniformly integrable if a = 0. If a = 0 then W has weakly compact closure because W is a subset of kBL~(~) + akBLl(~) and BL~(u) is weakly compact in L1 (/z). If a(W) > 0, the set W does not have weakly compact closure and in fact even contains a sequence equivalent to the unit vector basis of el. To see this, define two further numerical parameters for a subset W of L 1(#); namely, b(W) := sup l i m n ~ { Ilxn 1An II1 }, where the supremum is over all sequences {Xn}n~__l in W and {An}~=l of measurable sets with #An --+ 0; and c(W), defined the same way as b(W) except that the supremum is

18

W.B. Johnson and J. Lindenstrauss

over all sequences of disjoint measurable sets. It is an elementary exercise in measure theory to verify that a(W) -- b(W) = c(W). Now if c(W) > 0, take a sequence {Xn}n~=l in W and a sequence {An}n~=l of disjoint measurable sets so that 0 < [IXn1an II1 ~ c(W). Clearly c({xn 1Xn}) = 0, so {Xn 1X, } has weakly compact closure. The sequence {Xn 1An }nCC_--I is equivalent to the unit vector basis of e 1 since the An's are disjoint and IlXn1An II is bounded away from zero. This implies that a subsequence of {Xn}nCC=l is also equivalent to the unit vector basis of el. (Remark: If {Yn}n~ is bounded in any L1 (IX) space and there is a sequence {Bn}n~__l of disjoint measurable sets and b > 0 so that for all n, Y-~k#n llynlsk 111 ~< b/2 < b <<, I[ynls, Ill, then {Yn}n~ is equivalent to the unit vector basis of el. Moreover, the closed span Y of {Yn}n~=l is complemented in L1 (IX). Indeed, set B "-- Un=l oc Bn. The sequence {ynls}n~__l is a small perturbation of the disjoint sequence {ynlBn}neC=l and hence there is a projection P from LI(IX) onto the closed span Z of {Yn 18 }n~ 1 9The mapping Yn 18 ~-+ Yn extends to an operator T from Z into the closed span of {Yn}neC__1 since {ynls}n~=l is equivalent to the unit vector basis of el. Then f w-~ T P ( f 18) defines a projection onto Y.) To apply this remark one uses a combinatorial argument to verify that there are n l < n2 < ... so that yk := Xnk, Bk "-- Ank satisfy the above conditions for any 0 < b < c(W). It follows from the observations made in the beginning of this section that any separable subspace of any L 1 (IX) space embeds isometrically into L 1(0, 1). Consequently, a bounded subset of L l (Ix) either has weakly compact closure or contains a sequence equivalent to the unit vector basis of el. From this dichotomy principle and the fact that a weakly compact subset of a Banach space is weakly sequentially compact it follows that L l (IX) is weakly sequentially complete. It also follows that a nonreflexive subspace of L I(Ix) contains a subspace which is isomorphic to e 1 and is complemented in L 1(Ix). There is a dichotomy principle, called Rosenthal's el theorem, for general Banach spaces which generalizes the one above for L l" A bounded sequence in a Banach space either has a weakly Cauchy subsequence or a subsequence which is equivalent to the unit vector basis ofel. See [30] or [14, 2.5.e] or [7, Chapter XI]. A space Z is called )~-injective provided that for all Banach spaces X, every operator T from every subspace of X into Z has an extension to an operator T from X into Z for which IIT [I ~< )~llT [I- The spaces e ~ ( F ) are 1-injective. This is proved by composing an operator T into Z with each of the evaluation functionals at points of F and applying the usual Hahn-Banach theorem. More generally, L~(Ix) is 1-injective if Ix is a-finite. That is because L~(Ix) is a complete lattice (see Section 5) in this case and the usual proof of the scalar Hahn-Banach theorem carries over. It is obvious that ec,c embeds isometrically into L ~ ( 0 , 1), and L ~ ( 0 , 1) embeds isometrically into e ~ because its predual, L 1(0, 1), is a quotient of el, the predual of e ~ . Since both ecc and Loc (0, 1) are 1-injective, each space is isomorphic to a complemented subspace of the other. A simplified version of the decomposition method then shows that e ~ and Loc (0, 1) are isomorphic. Surprisingly, there is no known operator which exhibits explicitly the isomorphism between these two concrete spaces. The space co has a weaker injectivity property: it is separably 2-injective, meaning that for all Banach spaces X and every operator T from a subspace Y of X with X~ Y separable, there is an extension of T to an operator 7~ from X into co for which II7~II ~< 211T II. Indeed, since eoc is 1-injective, T extends to an operator 7~ from X into eoc having the same norm

Basic concepts in the geometry of Banach spaces

19

as T and the range of T is separable because X~ Y is separable. Thus it is enough to check that if Z is a separable space which contains co, then there is a projection of norm at most two from Z onto co. To see this, first let zn* be an extension of the nth unit vector in ~ 1 = Co* to a norm one element of Z*. Let d(., .) be a translation invariant metric on Z* which induces the weak* topology on the unit ball of Z*; for example, if {Xn}n~ is dense in the unit sphere of Z the metric can be defined by d(x*, y*) - y~n~__l 2-n I(x * - y*)(Xn)l. Since every weak* limit point of {Zn*}n~ belongs to the unit ball B of the annihilator of co in Z*, :r it follows that d (Zn*, B) --+ 0. Thus we can choose w n* in B so that d (z*, w n) --+ O, which means that Zn* - - Wn* --+ 0 weak*. The formula P z "-- {(z* - w,)(z)* }n=l ~ defines a projection of norm at most two from Z onto co. That the space co is not separably )~-injective for any )~ < 2 can be seen by considering P 1 for any projection P from c onto co. More interesting is that the separability assumption is needed: There is no projection from s onto co. Indeed, otherwise we would have s ~ co (9 g~cc/co. But s is not isomorphic to a subspace of s This can be seen by proving that c0(R) embeds into got/co but not into s c0(R) admits no countable separating family of linear functionals, so it does not embed into s To see that c0(R) embeds into s take a family {Ar}rER of infinite subsets of N so that the intersection of any two is finite (replace N by the rationals and for each r E IR consider a sequence of rationals which converges to r). Let Xr be the image in the quotient space s of the indicator function of Ar. It is easy to check that { X r } r E I R is 1-equivalent to the unit vector basis of c0(R). The space co is the only (up to isomorphism) separable space which is separably injective [43]. General C (K) spaces do have another useful into extension property" A compact operator T from a subspace Y o f a Banach space X into C ( K ) has an extension T to a compact operator from X into C ( K ) . Indeed, since C ( K ) spaces have the BAR T can be approximated in the operator norm by operators of finite rank and hence we can write T = ~--~=0 oc Tn with each Tn of finite rank and IIZn II < 2 -n for n ~> 1. Therefore it is enough to observe that if S ' X --+ C ( K ) has finite rank, then S extends to a finite rank operator S" X --+ C ( K ) with IISII ~< (1 + e)llSII (where e > 0 is arbitrary). But by using partitions of unity and the principle of small perturbations one checks that there is a subspace E of C (K) so that S X C E and d (E, g n ) < 1 + ~, where n -- dim E < ec. Then E is (1 + e)injective and hence the desired extension of S exists. The same argument works when C ( K ) is replaced by any/2oc space (defined in Section 9). Every Loc(#) is isometric to C ( K ) for some compact Hausdorff space K. This follows from Gelfand theory, since Loc(#) is a commutative B* algebra with unit (see [19, Chapter 11]). Alternatively, it follows from lattice characterizations of C ( K ) spaces (see Section 5). The C (K) spaces play a special r61e in Banach space theory because they are a universal class: Every Banach space X is isometric to a subspace o f some C ( K ) space. This can be seen by embedding X into goc ( F ) for F appropriately large and applying the comment in the previous paragraph. Alternatively, X isometrically embeds via evaluation into C ( K ) with K the unit ball of X* with the weak* topology. This approach is preferable because the unit ball of X* is weak* metrizable when X is separable. In fact, every separable Banach space X embeds isometrically into C[0, 1]. First, C ( A ) embeds isometrically into C[0, 1], where A is the usual "middle thirds" Cantor set in [0, 1], by extending a continuous

20

W.B. Johnson and J. Lindenstrauss

function on A affinely on the component intervals of the complement of A in [0, 1]. Next, using that A is homeomorphic to {0, 1}r~, it can be shown that every compact metric space K is the continuous image of A. Build a tree structure K ( i l , ie . . . . . in): {il, ie . . . . . in} {0, 1}n; n = 1, 2 . . . . } of nonempty closed subsets of K so that each K ( i l , i2 . . . . . in) is the union of K (il, ie . . . . . in, 0) and K (il, i2 . . . . . in, 1), K -- K0 U K1, and the maximum diameter of K ( i l , i2 . . . . , in) as {il, ie . . . . . in} varies over {0, 1}n tends to zero as n --+ cx~. This is not hard to do using compactness of K. The map defined by {ij}7-1 ~ K ( i l ) N

K ( i l , i2) N K ( i l , ie, i3) N . . . is then a continuous surjection from A = {0, 1}N onto K. Having gotten a continuous mapping g from A onto K, one embeds C ( K ) into C ( A ) by the formula T x = x o g, x E C (K). The isometric classification of C ( K ) spaces is easy and can be found in beginning texts (such as [12, Theorem 231]): The space C ( K ) is isometric to C ( H ) if and only if K is homeomorphic to H. This follows from the identification of the extreme points of the unit ball of C ( K ) * as the evaluation functionals at points of K (as well as multiples of these functionals by scalars of magnitude one). The much harder problem of the isomorphic classification of C (K) spaces has been accomplished in the separable case (that is, for compact metric spaces): If K is an uncountable compact metric space then C ( K ) is isomorphic to C (0, 1). If K is a countable compact metric space then K is homeomorphic to the space [ 1, or] of all ordinals up to the ordinal ot for some countable ordinal c~ in the order topology. C (1, or) is isomorphic to C (1, t ) when ot < fl if and only if fl < c( ~ See [40] for further discussion of the separable case. It seems a hopeless task to get an isomorphic classification of general C ( K ) spaces, but some information is contained in [44]. We have already mentioned inexplicitly that the dual of an L l(#) space is isometric to a C (K) space. Similarly, the dual of a C (K) is isometric to L 1 (//~) for some measure/x. The usual representation (see [18]) of the dual of C ( K ) is the space M ( K ) of finite signed measures on the sigma algebra of Baire subsets of K (that is, the sigma algebra generated by the closed G~ subsets of K). It is a routine exercise using the Radon-Nikod3)m theorem to verify that if {/x• }• is a maximal family of mutually singular Baire probability measures on K, then M ( K ) is isometric to (Y-~yer L1 (/zy))l. Once the duals of C ( K ) and of LI(/Z) are classified, it is natural to ask what are the preduals of C ( K ) and L1 (/z)? It turns out that every isometric predual of a space C ( K ) is isometric to L1 (/z) for some measure # and that all preduals of C (K) are mutually isometric. On the other hand, the isomorphic preduals of, for example, g ~ , form a rich class of spaces which are still not well understood. Preduals of L1 (/z) spaces are even less well understood. Here we mention only that the space ~1 has uncountably many mutually nonisomorphic preduals among the C ( K ) spaces ( C ( K ) * is isometric to el if K is a countable compact space).

5. Banach lattices Most of the Banach spaces that appear naturally in analysis carry structure in addition to their structure as Banach spaces. It tums out that this additional structure can affect properties which are defined purely in terms of Banach space concepts (such as linear operators and duality). An extra structure that classical spaces such as Lp and C (K) spaces

Basic concepts in the geometry of Banach spaces

21

possess is that of a Banach lattice, which is a Banach space over the reals that is equipped with a partial order ~< for which x v y and x A y exist for all vectors x, y, and such that the positive cone is closed under addition and multiplication by nonnegative real numbers and the order is connected to the norm by the condition that Ixl ~< lYl =r Ilxll ~< IlYlI, where the absolute value is defined by Ix l = x v ( - x ) . A linear mapping from a Banach lattice to a Banach lattice is positive if it carries positive vectors to positive vectors. This is equivalent to saying that the mapping is order preserving. It is easy to see that a positive linear mapping is continuous, so we call them positive operators. If a positive operator preserves the lattice operations, it is called a lattice homomorphism. It is clear that a one-to-one surjective positive operator T whose inverse is positive is a lattice isomorphism; that is, both T and T-1 are lattice homomorphisms. The dual of a Banach lattice X is again a Banach lattice under the standard ordering on X* in which the positive linear functionals form the positive cone. With this definition it is easy to see that the canonical mapping from X into X** is positive [ 15, 1.a.2]. While there are interesting functional analytical topics (such as positive operators and lattice homomorphisms) which are special to Banach lattices, we are mostly interested here in the Banach space properties of Banach lattices and concentrate on those lattice properties which affect the linear topological or geometry of the underlying Banach space. The basic reference for this aspect of Banach lattices is [ 15]. For a general introduction to Banach lattices which covers the basic theory as well as some fairly recent material see [1 ]. The simplest examples of Banach lattices are the spaces with a monotonely unconditional basis under the pointwise order on the coefficients, which as we have seen are better behaved, or at least more regularly behaved, than general Banach spaces. It turns out that much of the structure theory for these spaces carries over to Banach lattices. It was already mentioned that the Lp and C(K) spaces are Banach lattices. Other examples are the Orlicz spaces and the Lorentz spaces. An Orliczfunction is an even convex function on R which is zero at zero and tends to infinity at infinity. If # is a measure and M is an Orlicz function, the space LM(/Z) is the collection of all/z-measurable functions f for which there exists C > 0 so that f M ( f / C ) d / z < ~ . Then IIf IIM is defined to be the infimum of those C > 0 for which f M ( f / C ) d/z < 1. This is a norm on LM(/z) which makes LM(/z) into a Banach lattice. I f / z is counting measure on N, LM(/z) is called an Orlicz sequence space and is denoted by gM. If 1 ~< p < ~ and W is a positive nonincreasing continuous function on (0, ~ ) so that W(t) --+ ~ as t ~ 0 + , W(t) --+ 0 as t --+ c~,

f l W(t) dt -- 1, and f o W(t) dt -- cx~, the Lorentz space LW, p(/z) is the space of all/z measurable functions f for which IlflIw, p "-(f~xz f , ( t ) P W ( t ) d t ) l / p ' where f * is the decreasing rearrangement of Ifl. The space Lw, p(/z) is a Banach lattice under the norm IIf IIW,p. Lorentz sequence spaces are defined in an analogous fashion. The Orlicz spaces L M (/z) and the Lorentz spaces L w, p (/z), like the spaces Lp (/z), are symmetric lattices which are ideals in the lattice of all/z-measurable functions. Here X is symmetric means that if x E X and y is a/z-measurable function for which x* = y*, then y is in X and IlYllx = Ilxllx. A subspace Y of a lattice X is an ideal provided x 6 X, y E Y, and Ix l ~< lYl =~ x E Y. Symmetric lattice ideals play an important role in interpolation theory; see Section 11 for a short introduction. (Some sources call a symmetric lattice a rearrangement invariant space; other references use "rearrangement invariant space" to

22

W.B. J o h n s o n a n d J. L i n d e n s t r a u s s

mean "symmetric lattice ideal", sometimes with extra conditions. That is why we avoid the term "rearrangement invariant" in this article.) It is no accident that in the examples of Banach lattices given above the lattice ordering is just the natural pointwise a.e. ordering on a space of (equivalence classes of) real valued functions on a measure space (for C(K) the measure is counting measure on K and for spaces with unconditional basis the measure is counting measure on N). We call such a Banach lattice a Banach lattice of functions or a Banach lattice of #-measurable functions if it is important to specify the measure. There are representation theorems which say that there is essentially no loss of generality in considering only Banach lattices of functions. The most classical of these representation theorems, called the Kakutani representation theorem, gives abstract Banach lattice characterizations of L1 (/z) and C (K) spaces. A Banach lattice X is an abstract Lp space, 1 ~< p < cx~, provided IIx + yll p = Ilxll p + Ilyll p whenever x and y are disjoint; that is, Ixl A lyl - 0. The Lp version of the Kakutani representation theorem says that an abstract Lp space is lattice isometric to Lp(#) for some measure lz (see [ 15, 1.b.2] or [ 1, Theorem 12.26]). Moreover, the measure/x can be chosen to be a finite measure if the abstract L p space X has a weak order unit; that is, a vector u ) 0 so that u A Ix l - 0 only when x is the zero vector. A Banach lattice X is an abstract M space provided IIx + Yll - Ilxll v IlYll whenever x and y are disjoint. The C(K) representation theorem says that an abstract M space is lattice isometric to a sublattice of C(K) for some compact Hausdorff space K (see [15, 1.b.6]). Moreover, if the abstract M space has a strong order unit (that is, a vector u ~> 0 such that the unit ball of X is the order interval [-u, u] := {x: - u ~< x ~< u}), then M is lattice isometric to C(K) itself via an isomorphism which maps u to Ilu II1K. Suppose that the Banach lattice X admits a strictly positive functional; that is, a positive linear functional u* such that u*(Ixl) > 0 for every nonzero vector x in X. Define an (inequivalent) norm I1" Ilu* on X by Ilxllu* := u*(Ixl). This is obviously an (incomplete) abstract L1 norm on X from which it follows that the completion Xu, of (X, IIx Ilu*) is an abstract L 1 space. Since the formal inclusion operator from X to Xu, is an injective lattice homomorphism, the Kakutani representation theorem yields that X can be thought of as a Banach lattice of functions. If X is separable, so is Xu,, and it follows from comments made in Section 4 that Xu, is isometric and order equivalent to a sublattice of L l (0, 1), so that X can be represented as a Banach lattice of Lebesgue measurable functions on the unit interval. Not every Banach lattice admits a strictly positive functional (co(F) with F uncountable is a counterexample), but every separable Banach lattice X does. Indeed, take {Xn }n~=l dense in the unit sphere of X and for each n pick a norm one functional x n* which achieves its norm at Xn. It is easy to check that u* " - ~ 2 -n Ix~*l is a strictly positive functional. From this comment and the representation above it follows that any lattice inequality involving finitely many vectors which is true for lattices of functions must be true in a general Banach lattice. This eliminates the tedium of verifying "obvious" inequalities (for example, V(

~ i =" 1

~:iXi 9 ~i - -

-+'1} -~- ff-~i=l ]Xi[) directly from the axioms for a Banach lattice.

The representation of Banach lattices as lattices of functions suggests that abstract L1 spaces are particularly important lattices. The abstract M spaces also arise naturally in the general theory. If u is a positive vector in a Banach lattice X and Xu is the linear span of the order interval [ - u , u] with [ - u , u] taken as the unit ball, then Xu is easily seen to be

Basic concepts in the geometry of Banach spaces

23

an abstract M space with strong order unit u (Xu is complete because [ - u , u] is closed in the Banach space X). A Banach lattice is order complete or Dedekind complete if every nonempty subset which is bounded above has a least upper bound. A dual Banach lattice is order complete; in fact, any norm bounded upwarded directed net in a dual Banach lattice converges weak* to the least upper bound of the net. Suppose that X is order complete and u ~> 0 with llu I] = 1. We look a bit more closely at the C ( K ) space which is lattice isometric to the abstract M space Xu. Given x ~> 0, define the support o f x by S(x) := supn(nx)/x u. The supremum exists because X is order complete. The support S ( x ) is a component of u. (A vector 0 ~< y ~< u is called a component of u [or simply a component if u is understood] provided y is disjoint from u - y.) If T : X , --+ C ( K ) is a lattice isometry with T u = 1K, then T x is an indicator function if and only if x is a component. Of course, if A C K, 1a is in C (K) if and only if A is clopen (i.e., both open and closed). So the components form a Boolean algebra (a fact which is also easy to verify directly) which is complete because X is order complete. Thus the clopen subsets of K are also a complete Boolean algebra. An important fact is that the clopen subsets of K form a base for the topology of K. One way to see this is to observe that if 0 ~< x with x in Xu and t > 0 satisfies y := (x - tu) v 0 ~ 0, then 0 ~< t S ( y ) <. x. The interpretation of this fact in C ( K ) is that for all 0 ~< f in C ( K ) with f ~ 0, the set [ f > 0] contains a nonempty clopen subset. Since K is compact this implies that the clopen subsets of K form a base for the topology. Using this and the completeness of the Boolean algebra of clopen sets it is a simple exercise to prove that C (K) is itself an order complete Banach lattice. As was pointed out in Section 4, this implies that C (K) is 1-injective. From the discussion in the previous paragraph we can deduce: I f E is afinite dimensional subspace o f an order complete lattice X and ~ > O, then there is a finite dimensional sublattice F o f X and an automorphism T o f X so that E C T F and ]lI - T ]l < e. Given the subspace E, take positive vectors x l . . . . , xn in X whose span contains E and normalized so that u := max/xi has norm one. Then F C Xu and Xu is isometric to a C ( K ) space for which the clopen subsets of K form a base for the topology, which means that the span of indicator functions of clopen sets is dense in C ( K ) . Thus fixing a basis yl . . . . . yk for E and 3 > 0, we get a subspace F of Xu spanned by disjoint vectors and a vector u in F with ]lull = 1 so that for each 1 ~< i ~< k, there is a vector Xi SO that ]Xi -- Zi] ~ (~U (and hence Ilxi - zi [I ~< ~). Now apply the principle of small perturbations. A Banach lattice is order continuous if every downward directed net whose greatest lower bound is zero converges in norm (or weakly; it is the same) to zero. This is equivalent to saying that every order bounded increasing sequence converges in norm (necessarily to the least upper bound of the sequence) (see [15, 1.a.8] or [1, Theorem 12.9] or the beginning of the argument below). It is also easy to check that an order continuous Banach lattice is order complete [15, 1.a.8]. A Banach lattice is not order continuous if and only if it contains a sequence o f disjoint positive vectors which is equivalent to the unit vector basis f o r co and is bounded above. The "if" direction is clear. If X is not order continuous, one gets an upward directed net of positive vectors which is bounded above by, say, x, with ]ix ]l = 1. The net cannot converge in norm, so one gets 0 ~< x l ~< x2 ~< ... ~< x so that

24 a := infn

W.B. Johnson and J. Lindenstrauss Ilxn+~ -

Yn >~ O,

Xn

II > 0. Let Yn := Xn+l -- Xn ~ O. So we have for every n: IlYn II ~ a,

~

Yk ~< x.

(2)

k--1 (Although we do not need it here because we want to "disjointify" the yn'S, it is worth noticing that (2) implies that Yn --+ 0 weakly and hence {Yn}n~__l has a basic subsequence, and that any basic subsequence of {yn}n~__l is equivalent to the unit vector basis for co.) Take Yn* in the unit sphere of X* so that Y * ( Y n ) = [[Yn[[. By replacing y~* with [Yn*[ if necessary, we may assume that y~* ~> 0. For each n the nonnegative sum Y~m Y,~ (Ym) is at most I[x [I. It is an elementary exercise in combinatorial reasoning to deduce from this that for any E > 0 there is a subsequence {zk}~__l "-- {Ynk }~--1 of {yn}n~__l so that for each k > j , * Thus for each n, Z k*(Zj) < 2-J6 2, where z k* "-- Ynk"

(Zn+l--6-1~Zk)k=l V0

Zn+ 1 Zn+ l -- 6

1

Zk

>~a - e.

k=l

This means that the zn's have big disjoint pieces. More precisely, let e = a / 4 and set

(Z+l

and

Wn := (Vn -- EX) V O.

Then the Wn'S are pairwise disjoint positive vectors smaller than x with norms bounded away from zero. It is easy to see that {Wn}n~__l must be equivalent to the unit vector basis of co. This completes the proof, but note that if X is order complete the mapping en ~ Wn from co into X extends to a mapping from the positive cone of s to X by defining {~n6n }n~__l sup n an Wn; this mapping extends to an order isomorphism from s into X. Other useful characterizations of order continuous Banach lattices are given by the following (see [15, 1.b.16] or [1, Theorem 12.9]): X is order continuous if and only if every order interval is weakly compact if and only if X is an ideal in X**. Since c is not order continuous but is isomorphic to the order continuous Banach lattice co, there is not a linear topological characterization of order continuity. While it is only a sufficient condition for order continuity of X that co not embed isomorphically into X, this condition is only a bit too strong as arguments similar to those given above show that if co embeds into X then it embeds as a sublattice (see [1, Theorem 14.12]). The functional representation theorems for order continuous Banach lattices are stronger and more useful than the representation already mentioned for Banach lattices which have a strictly positive functional. Here we just indicate what is going on and refer to [15] for details. However, the motivated reader is encouraged to work out the details for himself or herself (partly because the discussion in [15] wanders unnecessarily). First, it is not hard to show that an order continuous Banach lattice which has a weak order unit u also admits a strictly positive functional u* (see [15, 1.b.15] or [1, 12.14]) and thus can be thought of as

Basic concepts in the geometry of Banach spaces

25

a space of integrable functions on a finite measure space (S-2, #) with u* (x) = f x d # for x in X. One can assume that u* (u) = 1 = Ilu II and Ilu* II ~< 2. Since X is order continuous, one has for each x ~> 0 in X that (nu)/x x converges to x as n --+ cx~, so that Xu is dense in X. By replacing the underlying a-algebra with the smallest cr-algebra for which all the functions in X are measurable, it can be assumed that X is dense in L1 (#). Then necessarily u > 0 a.e. By replacing the measure # with u d # and functions f in X by f / u , one can assume u -= 1. Now it is possible to recover the/z-measurable sets and the measure #. From the density of X in L 1 (/Z) it follows that the indicator function of a measurable set is an element x in Xu which is a component of u. From this it is essentially obvious that the L ~ ( # ) norm on Xu agrees with its abstract M space norm. If we represent Xu as a C ( K ) space, then the sets of positive/z measure are mapped onto the nonempty clopen subsets of K in an obvious way so that we can represent X and L l ( # ) as function spaces on K and transfer the measure # to K. We then get that C ( K ) C X C L1 (#) with both inclusions having dense range. Moreover, C (K) = L ~ (#) and every/z-measurable set is equal/z-a.e. to a clopen subset of K. Finally, X is an ideal in L1 (/z) (use again order continuity and the fact that X contains all indicator functions). One consequence of this representation is: Let X be an order continuous Banach lattice which has a weak order unit. A closed subspace Y of X either embeds into L 1(/z) for some measure/z or contains a normalized basic sequence which is equivalent to (even equal to a small perturbation of) a disjoint sequence. For a proof see [ 15, 1.c.8] or modify the proof of the dichotomy principle discussed in Section 4. By applying L1 theory discussed in Section 4 one gets that a subspace of an order continuous Banach lattice is reflexive if and only if it does not contain a subspace isomorphic to either s or co and that a nonreflexive Banach lattice has a sublattice which is order isomorphic to g l or co. The representation can also be used to prove that a Banach lattice which does not contain a copy of co must be weakly sequentially complete [15, 1.c.4]. Although we have used here the representation theorem for abstract L 1 spaces, it should be mentioned that since an abstract L1 space is necessarily order continuous (every normalized disjoint sequence is obviously 1-equivalent to the unit vector basis of ~ 1), some of the ideas (particularly building components in X) can be used to prove Kakutani's representation of an abstract L 1 space which has a weak order unit as a space L 1 (/z) for some probability measure/z. Expressions such as ( E n L 1 ]Xn]P) 1/p, 1 <, p <<,C~, have meaning in a general Banach U ]Xn] p) 1/ Pto be lattice. (When p - - c ~ we follow the convention of interpreting ( )--~n=l

N Vn=l IXnl.)Letting 1/p + 1 / p * - - 1, one can define

Ix.I p

n=l

-

a~x~"

n=l

Otn p*- 1 ,

n=l

but one must justify that this supremum of an infinite set of vectors must exist in the (possibly lattice incomplete) Banach lattice. By taking the supremum over finite sets of scalars satisfying Y~n--10tn N p * = 1 one obtains an upward directed net which is bounded above by ~nN1 IXn I; it is enough to check that this net converges in X. This is not hard to do N p)l/p directly, but note that if X i s a C ( K ) space the net obviously converges to (En--1 Ix~l

26

W.B. J o h n s o n a n d J. L i n d e n s t r a u s s

By working in the abstract M space Xu with u " - - ~--~nL 1 Ixnl, the C ( K ) space case gives the general case. We can then interpret scalar inequalities involving such expressions in general lattices (rather than just in lattices of functions). For example, Khintchine's inequality, discussed in Section 4, reads 1/2

Ap

~

IXn I2

~<

E

Bp

6nXn

n=l

[Xn[ 2

(3)

n=l

n=l

Auxiliary lattices such as X (~p) can now be defined and the "obvious" dualities checked. X (g2) is especially noteworthy, since it can be regarded as the complexification of X by regarding ~2 as C and defining multiplication by complex scalars in the obvious way (see [15, p. 43] for details). Starting from the scalar identity Iotl~ 1-~ = infc>oOCl/~ + (1 -O)C~/(~ one gets in a similar way an interpretation for the expression Ixl~ ~-~ and that

IIIxl~176

~ Ilxll~

1-~

(4)

Also, if 1 ~< p < q ~< cx~, 1/r " - - O / p + (1 - O ) / q positive scalars, then for any collection x l . . . . . xu that

of

with 0 < 0 < 1 and Ctl . . . . . OtN are vectors in a Banach lattice we have

(1-O)/q

otnlxnl r

<~

--

~

~

p

otnlxnl q

n--1

(5)

--

Inequality (5) is obvious for lattices of functions, as is the following" If 1 ~< p <~ cx~, XN are vectors in a Banach lattice X and x 1 . . . . . XN are in X*, then

X 1.....

1/p*

x~ (x~) ~ }2*

n=l

Ix~l p*

Ix*l r n=l

(6)

n=l

A more sophisticated version [ 15, 1.d. 1] of the argument given above shows that for any continuous homogeneous ( f ()~x) = )~f (x) if )~ >~ 0) function f on ~n and x l . . . . . Xn, the expression f ( x l , . . . , xn) can be d e f n e d in such a way that any lattice inequality that is true in ]~n is true in X; that is, if g is another continuous homogeneous function on ]1~n and f (a) ~< g (a) for all a = (ot l . . . . . an) in It~n , then f (x I . . . . . Xn) <~ g (x 1. . . . . Xn) for all xl . . . . . Xn in X. For 1 ~< p ~< cx~, a linear mapping T from a Banach space into a Banach lattice is called p-convex if there exists a constant M so that for all finite sets of vectors in the domain space the following inequality holds: 1/p [rxnl p n=l

<~ M

Ilxnll p n=l

(7)

Basic concepts in the geometry of Banach spaces

27

The smallest such M is denoted by M(P)(T). Clearly M(1)(T) -- IITII. Similarly, if for a linear mapping T from a Banach lattice into a Banach space the inequality

(i

1/p IlTx. IIp

~M

(8)

Ixnl p

n=l

n--I

always holds for some constant M, then T is called p-concave and the smallest such M is denoted by M(p)(T). Clearly M(~)(T) = [[TI[. Evidently if T is p-convex [p-concave] then it is bounded and M(P)(T) ~ [[TI[ [M(p)(T) ~ [[TI[]. It is not hard to check the identities M(P)(T *) = M(p,)(T) and M(P)(T) -- M(p,)(T*), where 1/p + 1/p* = 1. As functions of p, M(P)(T) is nondecreasing and M(p)(T) is nonincreasing (see [15, 1.d.5] or write down a proof for Lp(lZ ) and see that the H61der type inequalities (4), (5) allow a translation to the lattice setting). A Banach lattice X is called p-convex (p-concave) if the identity operator Ix on X is pconvex (p-concave) and we then define M(P)(X) := M(P)(Ix) and M(p)(X) :-- M(p)(Ix). These constants are called the p-convexity and p-concavity constants of X. So X is pconvex (p-concave) if and only if X* is p*-concave (p*-convex). A p-convex and r-concave Banach lattice can be renormed with an equivalent lattice norm so that the p-convexity and r-concavity constants are both one [ 15, 1.d.8]. In particular, a lattice which is both p-convex and p-concave is lattice isomorphic to an abstract Lp space. If T : X --+ Y is a positive operator between Banach lattices it is easy to check (see [ 15, 1.d.9]) the inequality

(i

1/p ITx, Ip

<~ Ilrll

(9)

Ixnl p

n=l

n=l

From this it follows that M(P)(T) <~ I[TIIM(P)(X) and M(p)(T) <~ IITIIM(p)(Y). There are characterizations [ 15, 1.d. 10] of general operators which are p-convex or p-concave which involve the notion of p-summing operator, to be discussed in Section 10. When p = 2, the inequality (9) is valid for general operators (except that the right hand side must be multiplied by a constant; see Section 10 or [15, 1.s The square function ( ~ n N= l Ixnl 2) 1/2 of vectors Xl . . . . . XN in a p-concave, 1 ~< p < oc, lattice X is very useful for studying unconditional basic sequences and other things as well. The reason is that by taking norms in the lattice Khintchine inequality (3) we get

Ix, I2

A1

E i

n=l

EnXn

n--1

<. E i n=l

e,x,

i

n--1

Pl 1/p

EnXn

W.B. Johnson and J. Lindenstrauss

28

By p-concavity, this last quantity is dominated by

p) lip M(p)(X)

1/2

M(p) (X) Bp

Ixnl 2

This gives the equivalence of the norm of the square function of X l . . . . , Xn with a certain Rademacher average:

A1

(5)

p) 1/p

Ixnl 2

n=l

<~ M(p)(X) Bp

Ixnl 2

(10)

n=l

Notice that the monotonicity of M(p)(X) then gives that if X is q-concave for some finite q, then for all finite p there is a constant Cp so that the inequality (EII Y'~nN__lenXn IIp) lip < N Cp(E IlY-~n=l enXn I[) holds; this is a special case of the Kahane-Khintchine inequality that will be discussed in Section 8. Inequality (10) implies that if {Xn}n~_l is an unconditional basic sequence in a Banach N

lattice which is p-concave, then the norm II~n= 10lnXnII of a linear combination is equivaX I~nXn 12) 1/2 II; more precisely, lent to the norm of the square function I](Y~n=l

AI K -1

IOtnXnl2 n=l

<~ M(p)(X)BpK

N

[OlnXn12

(11)

where K is the unconditional constant of {Xn}~_l. Since the left side of (11) is valid in any lattice, a duality argument yields that if {Xn}nCC=l is an unconditional basis for N

a complemented subspace of any Banach lattice, then []~n--101nXn]] is equivalent to [ ] ( ~ n -N- 1 [OlnXn]2)1/2 ][. We write the equivalence in a slightly different way in which it is easier to keep track of the constants. Suppose that T : Y ~ X, S : X --+ Y are operators with ST = Iy (so that T S is a projection from X onto SY), Y has a monotonely unconditional basis {Yn }~--1' and X is a Banach lattice. Then: N AIlITI1-1 n--1

)1/2 [OtnTyn 12 n--1

]~nTynl 2

~< A]-111SII n--1

(12)

29

Basic concepts in the geometry of Banach spaces

Let {y*}neC__lbe the functionals in Y* biorthogonal to {Yn}nCC=l. By composing S with (con-

tractive) projections onto the span of initial segments of {Yn }oc n=l' it can be assumed that Y is finite dimensional, in which case {Yn*}is a monotonely unconditional basis for Y*. To prove the right inequality in (12), given y~nN=lOtnYn, choose a norm one functional Z n L 1/3nyn, in y , so that II~--~=l N ot~y~ I I - Y~'~=l/~ N y ~, ( ~-~=l N ot, y~). Using (6) and the left inequality in (10), we then get

•

N OtnYn

-- Z 13notnS*y* (Tyn) n--1

n=l

<~

]I(N

I~nS*y*

n=l

21

]otn Tyn l2 n=l

N

Ioenryn le

<~ A I ' E Zen/3nS*Y* n=l

IotnTynl 2

~< AI~IIS*II ~ / ~ Y S n=l

n=l

1

which is the right side of (12). Continuing in the situation considered in the previous paragraph, we analyze further the case when X is a C(K) space and IlYnll - 1 for all n. Since {Yn}n~=l is unconditionally monotone, we have maxn Ioenl ~< II ~U_-i otny~ II. From (12), H61der's inequality, and properties of the supremum norm we get

•

N

~n Yn ~< A7111SII (maxlol.ry.l 1/2)

)1/2

Y~l~.ry.l n=l

n=l

IIN

1/2

A ll llall (max loenl llZYn ll) ~/2 rrmx ~__-+-otnTyn n=l

1/2

<~ a-jlllallllTII

Otnyn n--1

maxlotnl 1/2, gl

so that

• n=l

OtnYn

(A-~l llZ ll llS II) 2 max loenl. fl

Thus {Yn}n~=l is K-equivalent (K "-- (A]-111TII IISII)2) to the unit vector basis of co. Therefore, a seminormalized unconditional basis for a complemented subspace of a C (K) space is equivalent to the unit vector basis for co. By duality one obtains that a seminormalized

W.B. Johnson and J. Lindenstrauss

30

unconditional basis for a complemented subspace of an L1 (/Z) space is equivalent to the unit vector basis for g~l. Since C(0, 1) and L 1(0, 1) contain subspaces isomorphic to ~2 while co and ~1 do not, this gives another proof that neither C (0, 1) nor L 1(0, 1) has an unconditional basis. By using a p-convexification procedure, it is possible to build a scale of Banach lattices starting with a lattice X in a manner analogous to how the scale of L p(/z) spaces is constructed from L1 (/Z). For simplicity, we assume that X is a lattice of/z-measurable functions (for the general case see [15, p. 53]). For 1 < p < oc, the p-convexification of X is the space X (p) of all/z-measurable functions x for which IxlPsign(x) is in X. This is easily seen to be a Banach lattice under the norm

.

-

It

1/p

ILx 9

The space X (p) is p-convex with M (p) (X (p)) - - 1. More generally, if X is r-convex and sconcave, then X (p) is pr-convex and ps-concave with moduli M (pr) (X (p)) ~ M (r) (X) 1/p a n d M(ps)(X (p)) <~M(s)(X) 1/p. If the lattice X is itself p-convex, one can also define the p-concavification of X to be the space X(p) of all/z-measurable functions x for which

[xll/Psign(x) is in X. The expression 11[xll/PllP is a norm if M(p)(X) = 1, which, as we mentioned, can be achieved with an equivalent renorming of X.

6. Geometry of the norm One interesting and fruitful line of research, dating from the early days of Banach space theory, has been to relate analytic properties of a Banach space to various geometrical conditions on the norm of the Banach space. The simplest example of such a condition is that of strict convexity: A Banach space X is strictly convex provided that the equality [Ix + yll = Ilxll + Ilyl[ with y nonzero implies that x is a scalar multiple of y. This just means that the unit sphere of X does not contain any line segment or that every point of the unit sphere is an extreme point of the unit ball. Related to strict convexity is smoothness: A smooth point of the unit ball of X is a point x in the unit sphere for which there is only one norm one linear functional which achieves its norm at x. The space X is called smooth provided every point in its unit sphere is a smooth point of the unit ball. The spaces Lp(/Z), 1 < p < oe, are strictly convex and smooth, while the spaces Ll (#) and C ( K ) are neither strictly convex nor smooth except in the trivial case when they are one dimensional. It is easy to check that if X* is strictly convex (respectively, smooth), then X is smooth (respectively, strictly convex). The converse is true for reflexive spaces but not in general. In order to understand the analytic meaning that a point x in the unit sphere of X is a smooth point, consider any other vector y in X and a linear functional x* on X with [Ix*l[ = x*(x) -- 1. Then for any t > O, 1 + tx*(y) = x*(x + ty) <. [Ix + tyll. That is,

IIx + ty II - IIx II

Basic concepts in the geometry of Banach spaces

31

By using the triangle inequality one checks that the function Ilx+tyll-Ilxllt is an increasing function of t on (0, ec) and thus

x*(y) <. lim

IIx + ty II - IIx II

t-+O+

t

Similarly, x* (y) ~> lim t-+O+

IIx - ty II - IIx II -t

Thus if these two limits coincide, the value of x* (y) is uniquely determined. On the other hand, if these two limits differ, it follows from the H a h n - B a n a c h theorem that for any X satisfying lim t-+O+

IIx - ty II - IIx II IIx + ty II - IIx II ~ )~ <~ lim -t t-+O+ t

there is a linear functional x* for which IIx*ll = x*(x) = 1 and x*(y) = ~.. Consequently, x is a smooth point if and only if IIx + tyll + IIx - tyll - 2 = o(t) for every y in X. Quantitative versions of strict convexity and smoothness are of particular importance in Banach space theory. A Banach space X is said to be uniformly convex provided for every > 0 there exists 6x(e) = 3(E) > 0 so that

sup{ II(x + y)/21l Ilxll-

I l y l l - 1; IIx - y l l - ~} - 1 - 6(e).

The function 6x(e) is called the modulus of convexity of X. It is geometrically obvious (and even true) that in the definition of 3(e), the equalities can be replaced by the inequalities Ilxll ~< 1, Ilyll ~< 1, IIx - yll/> ~ without changing the value of 6(e). One of the first theorems to relate the geometry of the norm to linear topological properties is the following. A uniformly convex space is reflexive. In order to see this, we may assume as well that X is separable. If X is not reflexive, then for any )~ > 0 there is x** in the unit sphere of X** whose distance to X exceeds 1 - )~. Let A be a countable subset of the unit sphere of X* which determines the norm of Y -- s p a n X U {x**}; that is, for each y** in Y, Ily**ll - sup{y**(x*)" x* 6 A}. Let {Xn}neC__lbe a sequence in the unit sphere of X so that x*(xn) --+ x**(x*) for every x* in A. Then limn,m-+,c Ilxn + xmll -- 2, while for every n, liminfm-+ec Ilxn -- Xm II ~> 1 - )~. This argument shows that for nonrettexive X, ~x(e) = 0 for every 0 < e < 1 and in particular that X is not uniformly convex. A Banach space X is said to be uniformly smooth if the function

px(r)

-

-

p ( r ) - - s u p / I I x + rYll + Ilx - ryll

/

2

- 1. IIx II - Ily II - 1 }

satisfies p ( r ) = o ( r ) as r --+ 0. Again one checks easily that a uniformly smooth Banach space is reflexive.

32

W.B. Johnson and J. Lindenstrauss

There is a complete duality between uniform convexity and uniform smoothness. The space X is uniformly convex if and only if X* is uniformly smooth. This follows from a formula which connects the moduli 8x and Px*" px, (r) -- sup{rE/2 - 8x(~)" 0 <~ E <~ 2}. To prove this identity, let x and y be in the unit sphere of X with Ilx - y II -- E and pick x*, y* in the unit sphere of X* so that x* (x + y) - IIx + y II and y* (x - y) -- IIx - y II. Then

2px,(r)

)

IIx* + ry*ll + IIx* - r y * l l -

2

~> (x* + r y * ) ( x ) + (x* - r y * ) ( y ) - 2 = x*(x+y)+ry*(x-y)-2-11x+yll+re-2.

This shows that px, (r) ~> sup{rE/2 - 8x(~): 0 ~< ~ ~< 2}. Conversely, given r > 0 and x*, y* in the unit sphere of X*, take x and y in the unit sphere of X so that (x* + r y * ) ( x ) = IIx* + rY*ll and (x* - ry*)(x) = IIx* - ry*ll (if one wishes to avoid reflexivity at this stage one can choose x and y where the norms of the functionals are almost attained). Then

IIx* + ry*ll + IIx* - ry*ll ~< x*(x + y) + r y * ( x - y) IIx + yll + rllx - yll - 28x(~) + ~r, where E - IIx - y II. This proves the reverse inequality. The moduli of convexity and smoothness of a Hilbert space H can be easily computed from the parallelogram identity: E2 1

4

E2 =

8

)

2 pH(r)

--

v/1 + "r2 -

1= r

T +

Hilbert spaces are the "most uniformly convex" and "most uniformly smooth" spaces in the sense that if X is any space whose dimension is at least two, then 8/-/(E)/> 8x(E) and p/-/(r) ~< p x ( r ) . For infinite dimensional X this is an immediate consequence of Dvoretzky's theorem, to be discussed in Section 8, but a direct 2-dimensional geometrical argument gives the general case. The moduli of convexity and smoothness of the L p spaces can be computed exactly. It is somewhat easier to describe their asymptotic behavior near zero, and this is what really matters for Banach space theory. The result is E2

(p-

p2P '

1)--~- + o(e2),

if 1 < p ~< 2, i f 2 ~ < p < oo,

33

Basic concepts in the geometry o f B a n a c h spaces

z'P jr_ O(rP), p

-

IOL p ( r: ) --

i f l < p~<2,

_

7:2

i f 2 ~ e q / c and p x ( r ) ~< C r p for some constant C = C ( p , q ) > 0 (see [15, If.l]). Since it is easier to do analysis on a Banach space which has a norm with good geometric properties than on a general space, it is important to know which Banach spaces can be equivalently renormed so as to become strictly convex or smooth or uniformly convex or uniformly smooth, and it is useful to have in these last cases moduli which are as good as possible. The simplest, but nevertheless useful, renorming technique is as follows. Suppose that T is an injective operator from a Banach space X into a strictly convex Banach space Y. Then it is easy to check that IIIxlll := Ilxll + IITxll is an equivalent strictly convex norm on X. Moreover, if T is an isomorphism and Y is uniformly convex, then II1"III is an equivalent uniformly convex norm on X. Now if X is separable, then there is an injective operator T from X into ~2, so Ilxlll = IIx II + IITx II is an equivalent strictly convex norm on X. Also, there is an operator S from ~2 into X with dense range, so S* is an injective operator from X* into ~ --~2 and IIx*l12- IIx*ll + IIS*x*ll defines an equivalent strictly convex norm on X*. Since the adjoint operator S* is weak* to weak continuous, I1" 112 is dual to a (necessarily smooth) norm I1" 112 on X. If X is smooth and T is an injective operator from X into ~2 then (llx II2 + IIZ x II2) ~/2 is an equivalent norm which is both strictly convex and smooth. Hence every separable Banach space has an equivalent norm which is both strictly convex and smooth. For certain nonseparable spaces; in particular, ~ ( F ) with F uncountable (see [6, Chapter II.7]), there may be no equivalent strictly convex or smooth norm. If we are interested in obtaining a uniformly convex or smooth equivalent norm we have to restrict attention to reflexive spaces. However, not every reflexive space can be so renormed. For example, if ~ , = 1 g l )2 had an equivalent uniformly convex norm II II and I1" II, denotes the restriction of I1" Ilto the nth coordinate space ~ , then the expression IIIxIII := lim IIx II, (where "lim" is interpreted to be a limit over some free ultrafilter on N or a Banach limit or the limit along an appropriate subsequence) defines an equivalent norm on the finitely supported vectors in el which extends uniquely to an equivalent uniformly convex norm on ~1, but this is impossible. There is a characterization of those spaces on which there is an equivalent uniformly convex norm. These spaces, called superreflexive spaces, are discussed in Section 9. The superreflexive spaces are also the class of spaces on which there is an equivalent uniformly smooth norm. (These deep facts are discussed in [29].) If a space X has an equivalent uniformly convex norm II1"III, then the equivalent uniformly convex norms are dense in the metric space of equivalent norms (considered as bounded functions on the unit sphere of X). Take, for example, I1" II +E II1"111.The uniformly convex equivalent norms also form a G~ set since those whose modulus of convexity at 1/n is positive forms an open set. Thus the equivalent uniformly convex norms on X is a dense G~ in the space of equivalent norms on X. Since, as we mentioned, X* also admits an equivalent uniformly convex norm, it OO

/7

.

34

W.B. Johnson and J. Lindenstrauss

follows by duality that the equivalent uniformly smooth norms on X is also a dense G~ in the space of equivalent norms on X, hence so is the family of equivalent norms which are simultaneously uniformly convex and uniformly smooth. In Section 8 it is pointed out that an infinite dimensional L p (#) space cannot be equivalently renormed so as to have a better modulus of convexity or smoothness than that of the natural norm. There has long been a desire to describe reflexivity geometrically (that is, to show that a space is reflexive if and only if there is an equivalent norm on the space that satisfies some geometrical condition), and the notion of uniform convexity pushed in that direction. This problem was recently solved (at least for separable spaces) and is discussed in [29]. Early in a first course in functional analysis a student learns that a Banach space is reflexive if and only if its closed unit ball is weakly compact. There is a beautiful and useful characterization of weak compactness, called James' theorem, which does not explicitly involve the weak topology: A nonempty closed convex subset C of a Banach space X is weakly compact if and only if every x* in X* attains its maximum on C. The only if part is of course trivial. For the hard direction see [11, Theorem 79] or [26] for an accessible proof when X is separable. This theorem says that on a nonreflexive space X there exist linear functionals which do not attain their norm on the unit ball of X. Nevertheless, the functionals which attain their norm on the unit ball is a rich set: The Bishop-Phelps theorem says: Let C be a nonempty closed bounded subset of a Banach space X. Then the functionals which attain their maximum on C is (norm) dense in X*. This theorem, which is the starting point of the theory of optimization on Banach spaces, has many extensions and applications (see [25]). We outline the proof of the Bishop-Phelps theorem. First note that if f is a continuous bounded function on a complete metric space (U, d), then there is, for every E > 0, a point u0 in U so that f ( u ) <<,f ( u o ) + ~d(u, uo) for every u in U. Indeed, define a partial order on U by u < < v if f ( u ) <, f ( v ) - ~d(u, v). The boundedness of f and completeness of U imply via Zornication that there is a maximal element u0 in this partial order. This u0 evidently has the desired property. Suppose now that C is a nonempty closed bounded convex subset of X, e > 0, and x* is in X*. By the remark above there is a point x0 in C so that x*(xo) >~x*(x) - ellx - x0l[ for all x in C. Consider two convex sets KI and K2 in X • IR:

K1 := {(x, t): x ~ C; x*(x) >1 t}, K2 := {(x, t): x E X; t ~ x*(xo) + ~ l l x - x011}. The set K2 has a nonempty interior which is disjoint from K1, so the separation theorem gives a nonzero point (u*, c~) in X* 9 • and a fl so that u* (x) + a t ~> fl for (x, t) in Kl and u*(x) + at <. fl for (x, t) in K2. Clearly ot must be negative, so by normalizing we may assume that ot = - 1 . Since (xo, x*(xo)) is in Kl ('1K2, it follows that fl = u*(xo) - x*(xo). By considering K1 we see that x* - u* attains its maximum on C at xo and by considering K2 we deduce that 11u* 11~< E. The proof of the Bishop-Phelps theorem uses heavily the order in the real field. In fact, the Bishop-Phelps theorem does not hold in the complex case. There is a nonempty

Basic concepts in the geometry of Banach spaces

35

bounded closed convex set C in a complex Banach space X so that for every x* in X*, Ix*l does not attain its maximum on C. This is discussed in [26]. We conclude this section by mentioning the geometric meaning of the Radon-Nikod3)m property or RNP, an analytical concept that will be discussed in Section 7. A slice of a closed bounded convex set C is a set of the form S ( C , x * , ~ ) = {x E C: x*(x) >1 SUpyec x*(y) - c~} with x* in X* and ot > 0. C is called dentable provided that for each E > 0 there is a slice of C which has diameter smaller than ~. In [8, Th. V 3.9] and [3, Th. 5.8] it is shown that X has the RNP if and only if every nonempty closed bounded convex subset of X is dentable. Here we show how dentability can be used to derive other geometric conditions, in particular, a version of the Krein-Milman theorem valid for noncompact subsets of a space which has the RNE If X has the RNP, then every closed bounded convex subset C of X is the closed convex hull of its extreme points. Recall that a face of a convex set C is a nonempty (necessarily convex) subset F such that if )~x + (1 - ;~)y E F with x, y in C and 0 < )~ < 1 then x and y are in F. Extreme points are faces consisting of a single point. From the definition of slice it is obvious that if I(x* - Y*)(Y) I ~< 3 for all y in C then S(C, y*, ~ - 23) C S(C, x*, ~) as long as ot > 23. We first show that C has an extreme point. Take any slice S(C, x*, c~) of C whose diameter is less than one. By the observation above and the Bishop-Phelps theorem, there is a y* arbitrarily near x* which attains a maximum on C and so that the points P1 in C at which y* attains its maximum is contained in the slice S(C, x*, ~) and thus has diameter less than one. Evidently P1 is a closed face of C and of course is dentable since X has the RNE By induction we get a sequence {Pn}~~ so that Pn+l is a face of Pn and diamPn ~< 1/n. Since a face of a face is a face, all the Pn's are faces of C and their intersection is an extreme point of C which is in the slice S(C, x*, ~). To finish the proof, let K be the closed convex hull of the extreme points of C. If K were properly contained in C, then the separation theorem and the Bishop-Phelps theorem would yield a functional x* which attains a maximum on C and so that the set of points P in C at which x* attains its maximum is disjoint from K. By what we have proved P has an extreme point which is afortiori an extreme point of C, which contradicts the definition of K. It is open whether this extreme point property for every nonempty closed bounded convex subset of a space actually characterizes spaces with the RNE The statement about extreme points is however valid in a stronger form which trivially implies that the dentability condition is equivalent to the RNE A point x in C is called a exposed point of C if there is x* in X* which attains its maximum on C exactly at the single point x. If also the diameter of the slice S(C, x*, c~) tends to zero as ot --+ 0 + , then x is called a strongly exposed point of C. For example, every point on the unit sphere of a uniformly convex space is a strongly exposed point of the unit ball. The stronger statement to which we alluded above is: Every

nonempty closed bounded convex subset of a space with the RNP is the closed convex hull of its strongly exposed points.

7. Analysis in Banach spaces In this section we describe the basic facts concerning integration and differentiation in Banach spaces as well as the connection of these topics to convexity. Our "point of view"

36

W.B. Johnson and J. Lindenstrauss

is to take as known the scalar theory but not assume that the reader has any familiarity with the vector valued theory. Much of what we treat in this section is contained either in [8] or in [ 11,12]. Almost everything is in [3]. Let (s #) be a complete a-finite measure space and X a Banach space. A function from s --~ X of the form ~in=l xi 1ai with each Ai a measurable subset of s is called a simple function. A function f : ~2 --+ X is called strongly measurable or just measurable if it is the limit almost everywhere of a sequence of simple functions. A function f is called scalarly measurable provided x * f is measurable for each x* in X*. A function f is measurable if and only if it is scalarly measurable and the range of f is essentially separable, which means that for some set A of measure zero, f[X2 ~ A] is separable. The only if assertion is clear. To verify the if part, we may assume that X is separable and # select a sequence ~x*/cc in the unit ball of X* so that Ilxll - SUPn Xn (X) for every x in t n "n=l X. Thus for each fixed x in X, the function IIf - x II is a measurable real valued function on Y2. Since X is separable, given e > 0 there is a sequence {xn}~ec__1 in X and a sequence {An}n~l of disjoint measurable subsets of s so that I l f - ~nXnlAnll < ~. That is, f can be approximated uniformly by functions taking only countably many values and having all level sets measurable and hence f is the limit a.e. of a sequence of simple functions. We next define the notion of the Bochner integral of a vector valued function. This is the "strongest" of the various vector valued integrals which have been considered and is the one most useful for the topics treated in this Handbook. If f is a simple function supported on a set of finite measure, define f f d/x = ~x~X Iz[f = x]x. This is of course a finite sum. It is easy to see directly, and also follows from the scalar case by composing with linear functionals, that the integral is linear on this class of simple functions. If f is the a.e. limit of a sequence {fn }n~_-i of simple functions supported on sets of finite measure and limn,m f IlL - fm IId # = 0, then f f d # is defined to be limn f f~ d~. It is easy to check that f f d # does not depend on the particular sequence {fn }n~_-i and that f f d # exists if and only if f is measurable and f Ilfll d # < oe. That the integral has the expected properties either follows from the scalar case by composing with linear functionals or is easy to check directly; see [8, pp. 44-52]. In particular, IIf f d # II ~
(u,r)

Iif(v) - f(u)II dv = 0

(13)

and thus also f(u) - limr~om(B(O, r)) -1 fB(u,r) f(v) dv (here integration is with respect to Lebesgue measure m on Rn). Indeed, we may assume that X is separable and {Xn}n~ is dense in X. Then by the scalar theorem, lim m(B(O, r)) -1 s r--+O

(u,r)

IIf ( v ) - xnll dv = IIf(u) - Xnll

Basic concepts in the geometry of Banach spaces

37

for a.e. u and every n. For a point u for which this holds for every n we have

limsupm(B(O,

r)) -1 fB

r--+O

lif(v) - f(u)ll dv (u,r) fll

<~limsupm(B(O,r))-l f r--+O

I ] / ( v ) - Xni[ + [iXn --/(u)[[ dv

J B(u,r)

= 21if (u) - x. ll. Since {Xn}n~

is dense in X we get (13), as desired. In particular, if f " 1R --+ X is integrable

and F(t)-- fo f then F ' ( s ) - f (s) a.e. The Banach space L p (#, X), 1 ~< p < ec, is defined to be the space of all measurable X valued functions for which Ilflip := ( f ]if liP d#) 1/p < oo (with the usual modification when p = oc). The simple functions which are supported on sets of finite measure are dense in L p (#, X), 1 ~< p < cxz. W h e n / z is counting measure, L p (#, X) = gp (X) =

(X~ x) p. We mention two important notions of differentiability for a function f from a Banach space X into a Banach space Y. f is said to be Gfiteaux differentiable or G-differentiable at a point x0 in X if there is an operator T from X into Y so that for every u in X limt~o(f(xo + tu) - f(xo))/t = Tu. If this limit exists uniformly with respect to u in the unit sphere of X the function f is said to be Frdchet differentiable or F-differentiable at x0. That is, f is F-differentiable at x0 provided f (x0 + u) -- f(xo) + Tu + o(llul]). The operator T is unique if it exists and is called the G or F derivative of f at x0 and is denoted by Df(xo). If f is F-differentiable at x0 it is continuous there and standard two dimensional examples show that G-differentiability does not imply continuity. However, if f is Lipschitz and X is finite dimensional, then G-differentiability is easily seen to imply F-differentiability. This is false for infinite dimensional spaces and this is one of the main causes for the difficulty of the subject of derivatives in infinite dimensions. For example, the function f (u) = sin u is an everywhere G-differentiable Lipschitz map from X = L2(0, 1) into itself with Df(u)(v) = cosu 9v, but f is nowhere F-differentiable (for example, at u0 = 0 , sin l(0,t) - l(0,t) = (sinl - 1)l(0,t) is not o(ll l(0,t)il2) as t --+ 0). In general, familiar finite dimensional theorems (chain rule, inverse function theorem, implicit function theorem, etc.) go over to F-derivatives with the same proofs, while for G-derivatives one must proceed with care. On the other hand, the existence theorems for G-derivatives are quite satisfactory while F-derivatives often do not exist and the question when they exist leads to difficult problems. See [39] for a discussion of this topic. A direct application of the scalar mean value theorem yields that if the G-derivative Df(x) exists in a neighborhood of x0 and is continuous at x0, then it is actually an Fderivative. Thus the notion of C 1 function is the same in the context of G or F derivatives. For C l functions we can speak of the second derivative in either the G or F sense. The second derivative is then an operator from X into the space of operators from X to Y or, alternatively, a bounded bilinear map from X r X to Y. Similarly, one can speak of C" or C ec functions. One can define analytic functions when X and Y are complex spaces as those C ec functions f for which the Taylor expansion converges in some neighborhood and represents f there. A useful fact that is proved in elementary textbooks (such as [ 19,

38

W.B. Johnson and J. Lindenstrauss

Chapter 3]) is that a function f from the complex plane into Y is analytic if y* f is analytic for each y* in Y*. One important Banach space theory property that comes from differentiation theory is the R a d o n - N i k o d j m property or RNP. One way to define this notion is the following: The space X has the R N P if every Lipschitz function from N into X is differentiable a.e. Here we do not need to specify G~teaux or Fr6chet differentiability since the notions obviously coincide for all functions from R. The RNP is usually defined by the requirement that the Radon-Nikod~m theorem holds for X valued measures of finite total variation (see, for example, [8]). We shall discuss that aspect and other equivalences of the RNP later in this section. Here just note that a Banach space valued Lipschitz function f on [0, 1] has separable range, so its a.e. derivative f l , if it exists, is also separably valued and thus is a measurable function since x * f is measurable for each x* in X*. First let us get a feeling for which spaces have the RNP. It is clear from the definition that a subspace of a space with the RNP has the RNP and that a space all of whose separable subspaces have the RNP has the RNP as well. The mapping t w-~ l(0,t) from (0, 1) into L 1(0, 1) shows that L 1(0, 1) fails the RNP. Also co fails the RNP. This is seen by considering the function f ' R --+ co @~ co defined by f ( t ) - {fo sinnsds}n~--1 9 {fo cosns ds}n~__l. That f maps into co follows from the Riemann-Lebesgue lemma but the only possible candidate for a derivative is {sinns}n~__l 9 {cosns}n~__l which is not in co for any s. Separable conjugate spaces have the RNP and thus all reflexive spaces have the RNP (incidentally, this yields another proof of the fact mentioned in Section 3 that L 1(0, 1) does not embed into a separable conjugate space). To see this, let f be a Lipschitz function into a separable conjugate space Z* and let {Zn}ne~=lbe dense in Z. For all n, the scalar Lipschitz function f ( t ) ( Z n ) is differentiable for a.e.t. At a point to where all of these functions are differentiable, f ( t o ) ( z ) is differentiable for every z in Z (observe that h - l ( f ( t o -+- h) f (to))(z) - k -1 ( f (to + k) - f (to))(z) --+ 0 as h, k --+ 0 because the difference quotient is uniformly bounded since f is Lipschitz and tends to zero on the dense set {Zn }n~=l). From this we conclude that the limit g(t) := l i m h _ _ , o h - l ( f ( t + h) - f ( t ) ) exists a.e., but the limit is only in the weak* sense. This is all that can be said using just the separability of Z. However, since Z* is separable, we deduce that g is measurable (g has separable range and Jig(t) - z* II is clearly measurable for every z* in Z*; this is all that was used in the proof of measurability in the beginning of this section). Also g is bounded by the Lipschitz constant of f , so the Bochner integral G(s) " - f o g ( t ) d t is well defined for all s. Evaluating both sides at an arbitrary z in Z we see from the scalar theory that G ( s ) ( z ) = ( f ( s ) - f ( 0 ) ) ( z ) so that f ( s ) -- G(s) + f ( O ) and hence by what we proved on the differentiation of the integral we conclude that f ' ( s ) = G'(s) = g(s) a.e. also in the sense that Ilh -1 ( f ( s + h) f ( s ) ) - g(s)ll ~ 0 as h -+ 0. Thus every subspace of a separable conjugate space has the RNP and a space with the RNP has no subspace isomorphic to co or L 1(0, 1). These facts give useful criteria for the RNP and in certain cases, such as spaces with an unconditional basis, allow a complete determination whether a space has the RNR However, there do exist separable spaces with the RNP which do not embed into a separable conjugate space and there are spaces failing the RNP which do not contain isomorphic copies of either co or L 1(0, 1) (see [3, Section 3.4]).

Basic concepts in the geometry of Banach spaces

39

Before stating the usual definition of the RNP we need to introduce the notion of what is termed in [8] a "countably additive vector measure on a a-algebra". If X is a Banach space, an X valued measure r is a countably additive function from a a-algebra into X. For this to make sense, the series y~ r ( A , ) must converge unconditionally for each sequence {An}n~__l of disjoint sets in the o--algebra. One defines the total variation It] of an X valued measure by ]r ](A) = sup ~ n lit (A,)11, where the supremum is over all partitions of A into finitely many (or countably many; it is the same) disjoint sets in the cr-algebra (s The measure r is said to be offinite variation provided ]r 1(s < cx~. If r is of finite variation, then It] is a finite scalar measure. For example, if s = 1~,/3 is the collection of all subsets of I~t, and {x,},~__l is a sequence in the Banach space X, then the assignment r{n} " - Xn extends to an X valued measure on/3 if and only if ~ n x. converges unconditionally. The measure is then of finite variation if and only if ~.n ]]Xn 11 < c~. An example of an L p (0, 1), 1 ~< p < cxz, valued measure is gotten by taking the Lebesgue measurable subsets of [0, 1] and defining r ( A ) = 1a. This measure has finite variation only if p = 1. The theory of X valued measures and more general vector measures is exposed in [8]. In addition to their importance for the geometry of Banach spaces, vector measures are centrally important for spectral theory; the ones encountered there typically do not have finite variation. An X valued measure r is absolutely continuous with respect to thefinite scalar measure # provided r (A) = 0 for every set of # measure zero. Because # is finite, this is equivalent to saying that for every ~ > 0 there is ~ > 0 so that Ilr(A)]l < ~ whenever # ( A ) < 6 (see [8, p. 10]). If r is an X valued measure of finite variation it is clear that r is absolutely continuous with respect to its total variation It] and even satisfies (14) below (with # :--

Irl). An X valued measure r is differentiable with respect to a scalar measure lz provided that there is an X valued measurable function g so that r ( A ) = fa g d # for every measurable set A. We say that the Radon-Nikodym theorem holds in X provided that if r is an X valued measure of finite variation and r is absolutely continuous with respect to a finite scalar measure #, then r is differentiable with respect to #. If X satisfies this condition only for all separable finite scalar measures, we say that the separable Radon-Nikodym theorem holds in X (a measure # is called separable provided L1 (#) is separable). The usual definition is that a Banach space X has the RNP provided the Radon-Nikodym theorem holds in X and this is equivalent to saying that the separable Radon-Nikod3~m theorem holds in X (see [8, Chapter III]). Later we prove this equivalence for separable X, but first we show a general space X has the RNP if and only if the separable Radon-NikodSm theorem holds in X. Suppose that the separable Radon-Nikod3)m theorem holds in X and let f ' [ 0 , 1] ~ X be a Lipschitz function. One defines a linear mapping T from the step functions on [0, 1] into X by setting Tl[a,b] := f ( b ) - f ( a ) for a subinterval of [0, 1] and extending linearly. Since f is a Lipschitz function, the mapping T is continuous when the step functions are given the L 1(0, 1) norm, and hence T uniquely extends to an operator (also denoted by T) from L1 (0, 1) into X. The assignment r ( A ) :-- T1A obviously defines an X valued measure of finite variation which is absolutely continuous with respect to Lebesgue measure m, so we get an X valued measurable function g on [0, 1] for which r ( A ) = f a g dm for every Lebesgue measurable subset of [0, 1]. In particular, f (t) -- f0 g d m + f (0) and thus

W.B. Johnson and J. Lindenstrauss

40

by what we proved in the beginning of this section f ' ( t ) exists a.e. on [0, 1] (and is equal to g(t)). Suppose that X has the RNE To see that the separable Radon-Nikod3~m theorem holds in X, suppose first that the scalar "control measure" Ix is Lebesgue measure on [0, 1] and that the X valued measure r satisfies

[[r(A)I[ <~ Ix(A)

(14)

for all Lebesgue measurable subsets of [0, 1]. Define f ( t ) = r[0, t]. Evidently f is Lipschitz and so by assumption is differentiable a.e. on [0, 1] and f~ is measurable. That r (A) = fA f~ d# for all Lebesgue measurable sets now follows from the scalar theory by composing with linear functionals. By the measure isomorphism theorem already used in Section 4, we get that if the finite control measure # is separable and the X valued measure r satisfies (14), then r is differentiable with respect to #. Now suppose that v is a finite separable measure and r is an X valued measure which is absolutely continuous with respect to v. Then # := Irl is a finite scalar measure which is absolutely continuous with respect to v, so by the scalar Radon-Nikod3)m theorem there is a v-measurable function f >~ 0 so that #(A) = fA f dv for every v-measurable set A. Of course, IX is then also a separable measure and, as we have already remarked, r satisfies (14), so from what we already have proved there is an X valued Ix-measurable function g so that r(A) = f A g dix for every Ix-measurable set A. Then f . g is v-measurable and r(A) = fA f " g dv for every v-measurable set A. Observe that the simple argument reducing the study of an X valued measure which is absolutely continuous with respect to a finite control measure to the case where the X valued measure satisfies (14) yields another characterization of the RNP; more precisely, that the (separable) Radon-Nikodym theorem holds in X if and only iffor each operator T

from an L 1(Ix) space with tx finite (and separable) into X there is an X valued measurable function g so that T f = f f g dix for all f in L I(IX). From this it is easy to see that if X is separable and the separable Radon-Nikod3)m theorem holds in X then the RadonNikod2~m theorem holds in X. Indeed, let IX be a finite measure on a cr-algebra B and let T'LI(Ix) --+ X be an operator. Since X is separable there is a sequence {X*}n~ in X* which separates the points of X. Let A be a countably generated sub ~r-algebra of 13 so , oc separates points of that all the Loc(Ix) functions T* x n* are A-measurable. Since {Xn}n=l X, T f = T E ( f I A) for each f in L1 (Ix). The restriction of Ix to A is a separable measure since A is countably generated. Thus we get an X valued A-measurable function g so that T f = f f . g dIx for each A-measurable function f in L l (IX). But then for a general f in L1 (Ix) we have T f = T E ( f I A) = f E ( f I A)g dIx = f f . g dIx since g is A-measurable. One of many places where the RNP arises naturally is in the study of vector valued L p spaces. There is a natural isometric identification of Lp,(Ix, X*), 1/p + 1/p* = 1, with a subspace of Lp(Ix, X)*, and for 1 ~ p < (x~, Lp(Ix, X)* = Lp,(Ix, X*) for allfinite (or cr-finite) measures Ix if and only if X* has the RNP (see [8, Chapter IV]). There are other important analytic characterizations of spaces with the RNP in terms of martingales. In particular, the RNP spaces are exactly those Banach spaces in which the martingale convergence theorem is valid in the sense that X has the RNP if and only if every L1 bounded X valued martingale converges a.e. (see [8, Chapter V]).

Basic concepts in the geometry of Banach spaces

41

It turns out that in many places where one might assume reflexivity in order to use weak compactness of the unit ball it suffices to assume that the space has the RNP. We now discuss the differentiability of (real valued) convex continuous functions on a Banach space X. Part of the importance of this topic derives from the fact that the norm is a convex continuous function and differentiability of the norm is intrinsically related to its smoothness. The most elementary reference for the differentiability of convex functions and related topics is probably [11, Chapter 5]. An easy consequence of the definition is that a locally bounded convex function is continuous and even locally Lipschitz. By using the separation theorem in X | IR it follows that whenever f is convex and continuous in a neighborhood of a point x0 the set (called the subdifferential o f f ) Of(XO) :"- {X* E X*: x * ( x -- xo) ~ f ( x ) -- f ( x o ) for all x 6 X} is nonempty. From the theory of convex functions on R we know that for each u the right and left derivatives of the function t ~ f (xo + tu) exist at t = 0. These one-sided derivatives agree for every u (that is, all directional derivatives exist at x0) if and only if Of(xo) is a single point which is then necessarily the G-derivative of f at x0. Consequently f is G-differentiable at x0 if and only if for every u, f ( x o + tu) + f ( x o - tu) - 2 f ( x 0 ) -- o(t) as t --+ 0. By considering f ( x ) = Ilxll we recover the fact mentioned in Section 6 that the norm is G-differentiable at x0 in the unit sphere of X if and only if x0 is a smooth point of the unit ball of X. It also follows that f is F-differentiable at x0 if and only if f ( x o + u) + f (xo - u) - 2 f ( x 0 ) = o(llull) as Ilull -+ 0. I f a convex function f is F-differentiable in a neighborhood ofxo then D f (x) is continuous there; that is, F-differentiability of a convex function on an open set implies that it is C 1 there. Indeed, suppose that Xn --+ xo and set tO* " - D f ( x o ) ; u* "-- D f ( x n ) . Given E > 0 there is 6 > 0 so that f ( x o + y) - f ( x o ) w*(y) <. ~IlYll for IlYll ~< ~. Pick y/7 with IlY, II = ~ and (u* - tO*)(Yn) >/(~/2)llu~* - w*ll for all n. Then since u,* - D f ( x n ) , we have by the convexity of f that

6 + tO* (Yn) + f (xo) >~ f (xo + y/7) >~ u* (Yn - Xn + xo) + f (Xn) or

(~/2)llun* --

tO*

II ~< (Un* --

tO*

)(Yn) <. ~ -- f (Xn) + f (xo) + Ilu* I1" IlXn -- X011.

Since Ilu n II is bounded by the local Lipschitz constant of f , we conclude that lim sup IIw* - u~* II ~ 2~. /7

Specializing again to f ( x ) = Ilxll, we see that the norm is G-differentiable at every nonzero x if and only if X is smooth (the norm is clearly never differentiable at zero); that is, for each x in the unit sphere of X there is a unique unit functional jx of norm one which achieves its norm at x. The norm is then F-differentiable at every nonzero x if and only if the duality mapping j is a norm-to-norm continuous mapping. It follows from this and the Bishop-Phelps theorem that if I1" II is F-differentiable away from zero then the density character o f X* is equal to that o f X. In particular, X* is separable if X is.

42

W.B. Johnson and J. Lindenstrauss

The space X is uniformly smooth if the norm is uniformly F-differentiable on the unit sphere; that is, if the limit limt~0 t -1 (llx + tull - Ilxll) exists uniformly in both x and u on the unit sphere. The classical G~teaux differentiability theorem for convex functions says: A continuous convex function f on a separable Banach space X is G-differentiable on a dense G~ is dense in X then the set of G-differentiability of f is the set set. Indeed, if {Xn }n--1 ~ ('~n,m Gn,m, where Gn,m is the set of points x in X for which there exists 6 > 0 so that f ( x + 6Un) + f ( x - gUn) - 2 f ( x ) ~< 6 / m . It is readily verified that each Gn,m is open and dense. For Fr6chet differentiability, the situation is much different even for norms. The norm of el is G-differentiable at any point all of whose coordinates are nonzero but is nowhere F differentiable. This is typical in the sense that if X is separable and X* is nonseparable then X admits an equivalent norm that is nowhere F-differentiable (see [11, Theorem 106]). On the other hand: I f X* is separable then every convex function f on X is F-differentiable on a dense G~. That the set of points of F-differentiability is a G~ is easy to check (and for this no separability assumption is needed); it is equal to ("In Gn where Gn is the set of points x in X for which there exists 6 > 0 so that suPllull<~ 1 f ( x + 6u) + f ( x - 6u) 2 f ( x ) ~< g/n; the Gn's are obviously open. That each Gn is dense is not as obvious; see [ 11, Theorem 114]. We now consider the differentiability of Lipschitz (or just locally Lipschitz) functions. The theorem of Lebesgue on the a.e. differentiability of Lipschitz functions on the line has a classical extension (called Rademacher's theorem) which says that every Lipschitz mapping between finite dimensional spaces is differentiable a.e. The proof goes over to show that the finite dimensional assumption on the range of the Lipschitz map can be replaced by the assumption that the range has the R N E The main problem weakening the assumption on the domain space in Rademacher's theorem is that one must come up with an appropriate notion of a.e. or negligible set in an infinite dimensional space. It turns out that there are several nonequivalent ways to do this and we shall discuss one of them. A Borel subset A of a separable Banach space X is said to be Haar null provided there exists a probability measure # on the Borel subsets of X so that # ( x + A) - - 0 for every x in X. The Haar null sets form a o--ring and they coincide with the usual Borel sets of Lebesgue measure zero if X is finite dimensional. In an infinite dimensional space X, every compact set A is Haar null since there is a direction so that every line in this direction cuts A in a set of linear measure zero and thus # can be any probability measure supported on a line in this direction which is equivalent to linear Lebesgue measure. Once one has the notion of Haar null set, the classical proof of Rademacher's theorem can be modified to prove: Every locally Lipschitz mapping from a separable Banach space into a Banach space with the R N P is G-differentiable off a Haar null set. For a fuller discussion and a discussion of other notions of negligible set, see [39] or [3]. This theorem does not hold for F-differentiability. Even ~2 has an equivalent norm whose set of points of F-differentiability is a Haar null set (see [3, Example 6.46]). For locally Lipschitz functions one deep and useful result is: Every real valued locally Lipschitz function on a space whose dual is separable is F-differentiable at a dense set o f points (again, see [39] or [3]). It is not known if this theorem remains true for Lipschitz functions taking values in the plane.

Basic concepts in the geometry of Banach spaces 8. F i n i t e d i m e n s i o n a l

43

Banach spaces

Since all finite dimensional spaces of the same dimension over the same scalar field are mutually isomorphic, for results on finite dimensional spaces to be meaningful they must be of a quantitative nature. The notion of Banach-Mazur distance is of central importance in this context. Evaluating or even estimating the distance between spaces is often hard since it generally is quite difficult to find an operator T for which IIT II IIT-1II is minimal or close to minimal. We illustrate the computation of the Banach-Mazur distance by evaluating (or in some cases just giving the order of magnitude) the quantity d (s s While relatively easy, even in this simple situation it is by no means trivial to calculate or even closely estimate the distance when p and r are on different sides of two. The topic of Banach-Mazur distances between finite dimensional space is treated in many more involved situations in [20]. Denote the formal identity mapping from ~ to ~n by Ip,r. It is trivial to check that

IIIp,rll = 1 when p ~ r. Consequently, if 1 ~< p < r ~< ec, then d(gp, ~ ) ~< n 1/p-1/r. The simplest and most important case occurs when p (or r) is two. Either by induction from the case n -- 2, where the following equality is just the parallelogram law, or by using the orthonormality of a Rademacher sequence and taking the vectors xi to be in, e.g., L2(0, 1) and exchanging the order of integration over [0, 1] with the expectation, one sees that vectors x l . . . . . x/7 in a Hilbert space satisfy the identity

~6iXi i=1

2 Ilxi ll2.

-

(15)

i--1

Suppose that 2 < r ~< cx~ and T ' s --+ ~ is an isomorphism normalized to satisfy IIT-1II = 1 (so that IlTx[]2 /~ []Xl[r for all x). Denoting as usual the unit vectors basis as {ei}, we see from (15) that n <<, y~in___l IlTeill 2 - E l l ~ i n = l e i T e i l l 2 max• lIT (Y~'~i=ln +ei) II2 ~< IlTll2n 2/r, which is to say that [[TII ~> n 1/2-1/r. From the equality d(g.~, ~ ) - n 1/2-1/r and the (multiplicative) triangle inequality for the Banach-Mazur distance one gets that d(s ~ ) -- n 1/p-1/r if 2 ~< p ~< r ~< cx~. By duality (or by repeating essentially the same argument) one gets the same is true if 1 ~< p ~< r <~ 2. It turns out that when p < 2 < r the identity operator Ip,r does not give even a good approximation to the Banach-Mazur distance. For the upper estimate it is enough to work with complex scalars. This is because complex s /7 as a real space is isometric to the real n ~2)p which for all p has distance at most ~ to g 2n. space (Y-~l p Using de(', ") for the complex Banach-Mazur distance and dR (., .) for the real distance, we thus have for even dimensions dR(gzn /7 and for odd d i me n s i o n s di~(~ 2n+l,_rs p , gr2n ) ~< 2de ( ~ , gr) ~ 1+

2dc( , eT). The reason that the complex case is simpler is that there exists for every n a complex unitary matrix Vn of order n all of whose entries have absolute value 1/v/n (take for Vn the matrix whose j, k entry is n -1/2 exp(2Jrikj/n)). Denote the norm of an operator IIT II from g np to gr/7 by IIT IIp,r. Using the obvious identity I I T I ] - max/IITei[I for operators T with domain ~ , we have that IlVn Ill,~ = n -1/2.

W.B. Johnson and J. Lindenstrauss

44

By considering Vn* as the composition I2,1V*Icc,2 we see that IlVn*ll~,l ~< n. Since Vn* = Vn 1, it follows that d(g.~, gn) <
/7

d(g.p, g.p,) ~ Ilgnllp,p, llVn*[lp,,p ~ nl/2-1/Plllll2,pllg*ll2,2[llllp,,2 = nl/2-1/pn 1/p-1/2 91 9n 1~p-I~2 = n 1~p-l~2. In the general case 1 ~< p < 2 < r ~< oc, by replacing the pair {p, r} by {r*, p*} if necessary, we can assume that p* ~< r. Then the triangle inequality for the B a n a c h - M a z u r distance gives us d(g.np, gn) <
d(g.np, e n) ~ max{nl/p-1/2, n 1/2-1/r }

(16)

in the case of complex scalars and something slightly worse in the case of real scalars (e.g., the same except that the right side of (16) is multiplied by three). To show that the estimate (16) is accurate (up to a constant) one uses again an argument of averaging against a Rademacher sequence. The argument works in both the real and complex case. The inequality needed, called the cotype 2 inequality for L p, will be discussed later in this section. It states that for 1 ~< p ~< 2, any collection X l . . . . . Xn of vectors in an Lp(l z) space satisfies the inequality

E

6iXi i----1

2),j2 (n)j2 Ap

p

~

][xi l[2

(17)

i=1

where the expectation is with respect to the Rademacher sequence {en }n~__l. Applying (17) to the image of the unit vector basis of g~, we get for r > 2 that d(g~n, E) >1 Apn 1/2-1/r for every subspace E of an Lp(#) space. Since Ap >/A1, this shows that the upper estimate of (16) is precise up to an absolute constant. Since an n dimensional Banach space can be identified with IRn (or C n) with some symmetric (or balanced) closed bounded convex body as the unit ball, it is natural that geometric arguments often occur in the study of finite dimensional spaces. The notion of volume is of particular importance in this regard. Suppose that X and Y are both n dimensional Banach spaces, which we regard as R/7 (or C n) with some norm, and c~(.) is a norm on the space B(X, Y) of operators from X to Y. The natural algebraic dual of B(X, Y) can be identified with B(Y, X) under the trace duality (S, T) := trace ST and we denote by or* (.) that norm on B(Y, X) which makes (B(Y, X), or*) the dual of (B(X, Y), c~) under trace duality. By compactness of the unit ball of a finite dimensional space, there exists among all operators T : X ~ Y with co(T) ~ 1 one which maximizes the volume of TBx. Lewis' lemma states: If vol(TBx) achieves a maximum subject to the constraint

45

Basic concepts in the geometry of Banach spaces

ot(T) <~ 1 at T -- To, then To is invertible a n d o t * ( T o 1) -- n. This means n -1 To 1 is a norm one functional on ( B ( X , Y), ot) which achieves its norm at To. Before giving the proof of Lewis' lemma, we mention one consequence of the idea of maximizing volume, Auerbach's lemma, which does not use Lewis' lemma: I f Y is , /7 an n dimensional Banach space then there is a basis { x k , x k }k=l so that f o r each k, Ilxkll - 1 - Ilxffll. We take X - e~ and let ot(.) be the operator norm: Having gotten T0"e~ ---> Y to maximize the volume of the image of the g~ ball as above, it is obvious that n works, where xk 9-- Tek , x k* (x) "= det(T0) 1 det(Tk), and T k ' g ln ---->Y the basis {Xk, x k9}k=l is defined by letting Tkei be xi when i --/=k and Tkek = x. We turn to the proof of Lewis' lemma. Since the volume of T C for any measurable set is a constant multiple of [det(T) Ivol(C), for any operator S : X ---> Y we have

det(To+S) ot(r0 + S)

~< Idet(T0) I.

(18)

Certainly To must be invertible, so by dividing (18) by Idet(T0)l we can rewrite (18) as Idet(Ix + T o ' S ) l <~ o~(To + S)".

(19)

Since ot(T0) ~< 1 we get from (19) that for all e > 0 and all operators S : X --> Y: Idet(Ix + e T o ' S ) l ~< (1 +Eot(S)) n.

(20)

Obviously det(Ix + eT) = 1 + e(trace T) + o(e) as e --+ 0 (that is, the trace functional is the derivative of the determinant at the identity operator), so from (20) we conclude that trace T0-1S <~ not (S) for all operators S" X --+ Y which in view of the trace duality between ot(.) and ot*(.) means that ot*(T0--1) ~< n. The first application of Lewis' lemma comes from applying it when ot(.) is the operator norm. (This case, known before the lemma, see, e.g., [20, p. 137], in fact motivated the discovery of the lemma.) The norm .Af(.) on B(Y, X ) which is dual to the operator norm on B ( X , Y) for finite dimensional X and Y is called the nuclear norm. To identify the nuclear norm on B(Y, X), observe first that if IlY*II = 1 -- Ilxll with y* in Y* and x in X, then N'(y* | x) = 1, where y* N x is the rank one operator from Y to X defined by (y* N x ) ( z ) = y* (z)x. Next note that for any operator T :X --~ Y, I l r l l - - s u p { y * ( T x ) " IlY*II- 1 - - I l x l l } - - s u p { t r a c e ( y * @ x ) T " JV'(y* @ x ) -

1}.

Applying the bipolar theorem we conclude that the unit ball of (B(Y, X ) , N ' ( . ) ) is the convex hull of the closed set W := {y* | x: Ily*[[ = 1 = [[xl[}. From this description of the unit ball of (B(Y, X ) , N ' ( . ) ) one obtains that for any S in B(Y, X ) , N ' ( S ) = N , 9 min Y~'~k=l [[Y~ [l" Ilxk II, where the minimum is over all representations S -- Y-~kN1Yk @ xk. Caratheodory's theorem says that the minimum occurs already with N at most one plus the dimension of B(Y, X ) as a vector space over the reals; that is, N ~< n 2 + 1 in the case of real scalars and N ~< 2n 2 + 1 when the scalars are complex. (By being slightly more careful before applying Caratheodory's theorem, one obtains the best estimates N ~< n 2 and

46

W.B. Johnson and J. Lindenstrauss

N ~< 2n2; see [20, 8.6].) Having done these preliminaries, we can state a nice geometric reformulation of the case or(.) = I1" II in Lewis' lemma. Again identify both spaces X and Y with ~n or C n. When one convex body contains another, a point x is called a contact point of the bodies if it is in the intersection of their boundaries (so if the bodies are unit balls associated with two norms, a contact point is a point in the intersection of the two unit spheres). Assume that B x C By and vol(Bx) ~> v o l ( T B x ) f o r every operator T on ~n [C n] f o r which T B x C Br. Then there exist contactpoints xl . . . . . XN o f B x and B r and contact points x 1 . . . . , x N o f B x , and Br', and ck ~ 0 so that I = ~ = l ckx~ | xk. Also, N <<.n 2 in the real case and N <~ 2n 2 in the complex case. The most important applications of Lewis' lemma occur when X -- ~ . In this case observe that the operator To is unique up to a unitary operator on ~ . Indeed, suppose that T1 is another operator from ~ for which or(T1) -- 1 and ot*(Tl l) = n. By preceding Tl with a unitary we can assume that T o 1T1 is a positive operator D with eigenvalues 0/1 ~ "'" ~ Oen > 0. Then Y ~ = l ot~ -- traceD ~< a * ( T o l ) a ( T 1 ) <<.n while Z ~ = I 1/otk -trace D -1 <~ ot*(Tll)c~(T0) ~ n, SO that c~k -- 1 for all k. Specializing to the case of the operator norm, we obtain a classical result called John's theorem: Let X be R n or C n under some norm II" II. Then B x contains a unique ellipsoid o f maximal volume. This ellipsoid is the Euclidean ball B "-- Be~ if and only if there exist contactpoints {uk}~v=l between B and B x and ck > 0 so that f o r all x in X,

N

x-~c~(u~,x)uk.

(21)

k=l Moreover, N <~ n(n + 1)/2 in the real case and N <<.n 2 in the complex case. To see that John's theorem follows from the geometric version of Lewis' lemma, write the identity on X as Ix = Y'~=l ckxk | xk with the xk contact points between B and B x , the x k contact points between B and B x , , and the ck positive real numbers which sum to n. Here B x , is identified with those y in X for which I(Y, x)l ~< 1 for all x in B x and the duality pairing is given by the usual inner product. By taking traces we see that (x~, xk) = 1 for all k and hence, since the Euclidean norm of x k* and xk are both one, that x~ is the functional (., xk). This gives John's theorem except for the improved estimate on N, which is easy but we omit (see [20, 8.6]). An important consequence of John's theorem for Banach space theory is: I f X is n dimensional then d ( X , ~ ) <<.x/ft. Indeed, by (21) since (uk, .) is a norm one functional on X, we have for all x in X that [Ixl[ 2 - (x,x) - Z~=~ Ckl(Uk,X)[ 2 ~ Z L 1 C k -- n. This says that B x C ~/-ffB. For general spaces of dimension n we thus have d ( X , Y) <<.d ( X , g~)d(g~, Y) <<.n. It turns out that this estimate is precise up to a constant independent of n. There exist for

Basic concepts in the geometry of Banach spaces

47

each n spaces Xn and Yn of dimension n so that i n f d ( X n , Yn)/n > 0. If one puts 2n pairs of symmetric points on the unit sphere of ~ , where the points are chosen independently and are distributed uniformly on that sphere, and then takes the symmetric convex hull of the union of these points with the unit vector basis, one obtains a unit ball for a (random) space. If one takes two of these random spaces, then with big probability (that is, with probability tending to one as n --+ cx~) the Banach-Mazur distance between the two spaces exceeds gn for some constant g > 0 independent of n. Although this construction is easy to describe, the computations are delicate and the reader is referred to [20, 38.1 ] for details. The probabilistic approach has many other applications. For example, from the estimate d ( X , s <<,x/-ff it follows immediately that an n dimensional space has a basis with basis constant at most ~/-n. Spaces of the type Xn mentioned above exist (even with big probability) so that any basis in Xn has basis constant at least ~ x/ft. A discussion of these examples and their applications to infinite dimensional theory is given in [36]. The maximal volume ellipsoid appears also in another geometrical result which is a weak version of the classical Dvoretzky-Rogers lemma (see [28]). Again let X be R n or C n under some norm for which the Euclidean ball is the ellipsoid of maximum volume contained in Bx. Then there exist [n/2] orthonormal vectors Xk so that Ilxkllx ~ 1~2for n all k. Rather than deduce this from John's theorem, we prove it directly. Construct {X k}k= l inductively so that IIx~+~ IIs is maximized subject to the constraints that Ilxl12 = 1 and x is orthogonal to Xl . . . . . Xk. Fix any k < n, set 13 = Ilxk IIs, and let Xk be the span of x l . . . . . xk. Define an ellipsoid D by D'--{x+y

9x E X k , 9 y E X k• , 211xll2 + 2/3211yll2 ~ 1}.

One checks easily that D C Bx. By comparing the volume of D to that of B one gets that 2-n/2fl -(n-k+l) ~ 1 which gives the estimate Ilxkllx/> 2 -n/2(n-k+l). A geometric interpretation of the Dvoretzky-Rogers lemma is that whenever X has dimension 2n there is a subspace Y of dimension n so that in an appropriate coordinate system the unit ball of Y contains the Euclidean ball and is contained in twice the unit cube BeL. Since s is 1-injective, this implies that the identity I2,~'s -+ s can be written as a product Tl T2 with [[T2"s ~ X II - 1 and IIT I ' X --+ ~ II ~< 2. A result which goes much further is Dvoretzky's theorem, which says that every infinite dimensional Banach spaces contains for every n subspaces whose Banach-Mazur distance to s is arbitrarily close to one. The finite dimensional quantitative version of Dvoretzky's theorem from which the infinite dimensional statement above follows immediately says: For every k and e > 0 there exists no - no(e, k) so that if dim(X) ~ no then X contains a subspace Y with d(Y, ~ ) < 1 + e. There are known good estimates on n0(e, k) (and even better estimates for special classes of spaces). For general spaces n0(e, k) ~< exp(otk/e 2) for some constant a. Dvoretzky's theorem is treated in detail in several books, including [9, 19.1], [17, 4.3], [16, 1.5.8]. An exposition of Dvoretzky's theorem and related results is given in [28]. There are many proofs of Dvoretzky's theorem and in most of them the Dvoretzky-Rogers lemma is the first step. From Dvoretzky's theorem and the technique for constructing basic sequences discussed in Section 3 one gets in any infinite dimensional Banach space a basic sequence {Xn}n~=l .2k+l

so that for each k, {xj }j__Zk§ 1 is 1/k-equivalent to an orthonormal basis in a Hilbert space.

W.B. Johnson and J. Lindenstrauss

48

An easy consequence of this is that for every square summable sequence {C~n}n~__l, of scalars, every infinite dimensional Banach space contains an unconditionally convergent series ~ n Yn such that for each n, IlYn II = I~nl. One can also easily deduce this from the simple Dvoretzky-Rogers lemma; in fact, historically it was this application that motivated the discovery of the Dvoretzky-Rogers lemma. A result related to Dvoretzky's theorem is Krivine's theorem:

If {Xn }n~__l is a basic sequence in a Banach space, then there exists 1 <<.p ~ ~ so that f o r every k and every ~ > 0 there is a block basis of {Xn}n~=l of length k which is (1 + ~)equivalent to the unit vector basis of gkp. This statement can be "localized" to obtain a statement about all sufficiently long finite basic sequences having a specified basis constant (or even biorthogonality constant). We do not do so here for two reasons: (1) The finite statement follows formally from the infinite statement via a localization technique that will be mentioned in Section 9, and (2) Unlike the situation in Dvoretzky's theorem, the quantitative estimates in the finite statement are so poor that there are no known consequences of them beyond what can be deduced formally from the infinite version of Krivine's theorem stated above. For a proof of Krivine's theorem see [16, Chapter 12] or [3, Chapter 12]. In the setting of Banach lattices, Krivine's theorem yields that for every infinite dimensional Banach lattice there exists 1 ~ 0 there is a lattice isomorphism T from g np into X with II T II II T-1 II < 1 + ~. From Krivine's theorem and the fact discussed in Section 4 that ~2 embeds isometrically into L p (0, 1) it is easy to deduce Dvoretzky's theorem. This is not the recommended route to Dvoretzky's theorem as it is difficult to navigate through Krivine's theorem. More importantly, the tight quantitative estimates obtainable from direct proofs of Dvoretzky's theorem have many applications. If we apply Krivine's theorem to a basic sequence which is equivalent to the unit vector basis for ~r we see immediately that the only p that is obtainable is p = r. Thus by applying Krivine's theorem to a disjoint sequence in an infinite dimensional L p (#) space we infer that any equivalent renorming of L p ( # ) contains for every k and ~ > 0 a subspace whose B a n a c h - M a z u r distance to g kp is less than 1 + E This implies that an infinite dimensional L p ( # ) space cannot be given an equivalent norm which has a better modulus of convexity or smoothness than its natural norm. Two notions that are very important for both the finite dimensional and infinite dimensional theories are that of type and cotype. A Banach space X is said to have type p provided there is a constant C so that for every sequence x l . . . . . xn in X,

E

• i=1

1/p 6iXi

<~ C

Ilxi IIp

(22)

i:1

(with the usual modification for p = cx~). The expectation is with respect to the Rademacher sequence {en}n~__l. Similarly, X is said to have cotype q provided there is

Basic conceptsin the geometryof Banachspaces a constant C so that for every sequence Xl . . . . .

Xn in

49

X,

2) 1/2 i=1

[[xillq

(23)

The best constants in (22) and (23) are denoted by Tp (X) and Cq(X). By the parallelogram identity characterization of Hilbert space, a space X is isometric to a Hilbert space if and only if Tz(X) = 1 = C2(X). Even a one dimensional space does not have type p for any p > 2 or cotype q for any q < 2. For every space X, 7"1(X) = 1 and Coo(X) -- 1 by convexity of the norm. As functions of p and q, Tp(X) is nondecreasing and Cp(X) is nonincreasing. The inequalities Tp(X) <~d(X, Y)Tp(Y) and Cq(X) <, d(X, Y)Cq(Y) are evident, so for an n dimensional X, in the range 1 <~ p ~< 2 ~< q <~ oc the quantities Tp (X) and Tq(X) cannot exceed ~ by the distance estimate of X to g~. A Banach lattice X which is p-convex and q-concave has type p0 := P/x 2 and cotype q0 :-- q v 2. The type assertion follows from (10) and monotonicity of M (t) and M(s):

2) 1/2 ~n 8nxn

E

) 1/2

m(qo)(X) Bqo ~j ]XnI2 n

(

)1,o

<~ M(q)(X)BqoM(P)(X) Z IIx~IIP~ 17

For the lattice cotype assumption we need Khintchine's inequality (1) as well as monotonicity of the convexity and concavity parameters:

(

~ IIx~1[q~

l/qo

<~ M(qo)(X)

Ix~lq~

r/

n

(

<~ M(q)(X) ~lxnl 2 /7

Po) 1/po ~n 8nxn <~ M(q)(X)Ap---o1M(P~ (X) (IE ~n 8nxn

Po) 1/po 2) 1/2.

<~ M(q)(X)Ap-:M(P)(X)(E ~n 6nxn Specializing to

Lp(#)

spaces, we see that for 1 ~< p ~< 2,

Tp(Lp) =

1 and

C2(Lp) <~

A p 1(~< A ]- 1), while for 2 ~< p < cx), T2 (L p) ~< Bp and Cp (L p) = 1. Notice that this implies that for 1 <~ r < p ~ 2, gr is not isomorphic to a subspace of any Lp(#) space. By considering a disjoint sequence in the space, we see that an infinite dimensional L p(tZ) space with 1 ~ 1. Similarly, an infinite dimensional Lq (#) space for 2 ~< q ~< oo does not have cotype smaller than q. If {Xn }n=l ~ is a seminormalized unconditionally basic sequence in Lq(lZ), 2 < q < oo, with # a probability measure and inf IlXn 112 > 0, then since L q ( # ) has type 2 and L2(#) has cotype 2 and I1" 112 ~ I1" IIq we infer that {Xn}n~=l is, in Lq(#), equivalent to the unit vector basis of g2. Thus such a sequence cannot have dense linear span in L q ( # ) . In particular, the trigonometric system is not unconditional in Lq (0, 1) for 2 < q ~< c~ and, by duality, also not in L p(O, 1), 1 <<,p < 2. If {Xn}n~=l is a seminormalized sequence in Lq(O, 1), 2 < q < cx~, and inf IlXn ll2 > 0, then we saw in Section 4 that {Xn}n~=l has a subsequence which is equivalent to the unit vector basis for gq. Since Lp(O, 1), 1 < p < ~ , has an unconditional basis (the Haar system), it follows from the discussion in Section 3 that every seminormalized weakly null sequence in L p (0, 1) has an unconditionally basis subsequence. Combining these comments with those in the previous paragraph, and using the fact that every separable subspace of L p (#) embeds isometrically into L p (0, 1), we conclude that every seminormalized weakly null sequence in Lq (0, 1), 2 < q < ~ , has a subsequence which is equivalent to the unit vector basis f o r either g-2 or g.q. The second moment in the definitions of type and cotype can be replaced with any other moments (but of course then the constants change in (22) and (23)). This is a consequence of the Kahane-Khintchine inequality, which says that for every 0 < p < c~ there are constants A p and B p so that for every sequence X l . . . . . Xn in a Banach space,

A,(E 2) j2 ( 8iXi

i--1

• 2)

1/2

~< E ~ 8 i X i

i=1

,t j, (

<~Bp E

6i Xi

(24)

i=1

Here the expectation is with respect to the Rademacher sequence {8n}n~176The inequality (24) follows via extrapolation from the following hypercontractivity inequality: for every 2 0 SO that for every sequence x, x l . . . . . Xn in a Banach space, n

2) 1/2.

X + y ~ 8iXi i=1

i=1 To prove (25), first note that there is r

> 0 so that for all 0 < t ~< 1,

11 + Crptl p + I1 - Crptl p ~ lip 2

J

(25)

~< (1 + t2) 1/2.

(26)

This follows, for example, from L'H6pital's rule. The best constant is O'p -- (p - 1) -1/2 (see [15, I.e.14]). Secondly, deduce a vector valued version of (26): (EIIx + el~pyll p) 1/p ~ (EIIx + ~lYll 2) 1/2.

(27)

51

Basic concepts in the geometry of Banach spaces One can assume that I[x + YI[ + [Ix - YI[ - 2 with t 1 ~Crp Ilx + opey II ~<

IIx + eyl[ +

"-

]]x+yll-llx-y[] ~ 2

O. Then for e -- +1,

1 - Crp 2 I[x - e y l [ - 1 + eCrpt.

Therefore

(EII + Finally, need to that the to ek+l

+

,/1 + t 2 - (EII + lyl12)

iterate (27) to obtain (25). Formally, for k -- 1 . . . . . n define Sk "-- ~ki= 1 6 i X i ; we show that (EIIx + o'pSkllP) 1/p <~ (EI[X -Jr-Skll2) 1/2. For k -- 1 this is (27). Assume desired inequality is true for k and let Ek+l be conditional expectation with respect and E k conditional expectation with respect to el . . . . . ek. Then: (Ells + crpgk+l liP) 2/p

= (EkEk+l II(x + ~pgk) + ~k+l~pXk+l liP) 2/p <~ (Ek(Ek+lll(x +crpSk) + ek+lXk+lllZ)P/2) 2/p ~< Ek+l (]]{kll(X -}- 8k+lXk+l) -}- crpgkllP) zip

by (27)

by Minkowski

<~ ]Ek+l E k II(x -Jr-ek+lXk+l) -Jr-Skll 2 -- ]]{llx Jr- Sk+l II2. The theory of type and cotype is quite extensive. In the rest of this section we mention without proof a sample of results in this theory. Expositions of this theory can be found in several books, in particular [15-17,20]; and see the article [38]. We already mentioned that a space isomorphic to a Hilbert space is of type 2 and cotype 2. The converse is true and this is one of the basic isomorphic characterizations of Hilbert space. There is a connection between the theory of cotype and Dvoretzky's theorem. If X is an n dimensional space then X has a subspace Y of dimension k ~> nZ/P/N(p, Cp(X), ~) with d (Y, gk) ~< 1 + ~. In particular, if X is a subspace of L 1(~) then the k above is proportional to n, with a constant depending just on E. See [17, 4.15] or [16, 5.3]. A connection between type and cotype theory and Krivine's theorem is provided by the Maurey-Pisier theorem:

If X is an infinite dimensional Banach space and Po : - sup{p: X has type p} and qo := inf{q: X has cotype q}, then for any ~ > 0 and any k there are subspaces Yk and Zk of X k ) <, 1 + ~, and d(Zk, gpo k ) <, 1 + ~. so that d(Yk, gpo A proof of this is given in [16, II. 13.2]. The special cases P0 = 1 and q0 = oc of the Maurey-Pisier theorem are especially important (and much easier to prove; they do not require Krivine's theorem). If X does not have type p for any p > 1, then X contains almost isometric copies of g~ (the converse is true and obvious). Spaces which do not contain almost isometric copies of g~ for all n

52

W.B. Johnson and J. Lindenstrauss

arose first in the study of probability in Banach spaces and are sometimes called B-convex (see [34]). Similarly, a space does not have finite cotype if and only if it contains for every n arbitrarily close copies of s Since ek1 is a subspace of ~ (isometrically in the case of real scalars and up to constant in the complex case), we deduce that if X is B-convex then it has cotype q for some q < cx~. Since a uniformly convex space cannot have subspaces of arbitrary dimension which are arbitrarily close to s it follows that a uniformly convex space is B-convex; that is, has nontrivial type and cotype. The relation of B-convexity to reflexivity is not so simple but has been clarified. A Bconvex space need not be reflexive-there is even a nonreflexive space of type 2. Though nonreflexive Banach spaces need not contain almost isometric copies of s they do contain configurations of the following type for every n and ~ > 0: norm one vectors x i . . . . , Xn so that for every k < n, Ilxl + " - + Xk - (Xk+l + ' ' ' - ~ - xn)ll >~ n - E. In particular, taking n : 2 we see that: Every nonreflexive space contains real subspaces arbitrarily close to real s That is, a real space whose unit ball does not have a two dimensional section arbitrarily close to a square must be reflexive. See [2, 4.111] for proofs of these results. We next discuss the duality theory of type and cotype. It is simple that if X has type p then X* has cotype p* and C p , ( X * ) <~ T p ( X ) . Indeed, given X*l, " " " ' Xn* in X* and E > 0, take X l . . . . . Xn in X so that

,

1/p

p*

IIx~ II

Xi (Xi) ~ (1 -- ~) i=1

i=1

i=1

Using (4), we get

~ X ? ( Y i ) -- E ( ~ S i X ; ) ( ~ S i X i ) i=1

i--1

i=1 2) 1/2

E

eiX?

E

i=1

<~ Tp(X)

6ixi i=1

~ Ilxil[p

E i=lgiX* 2)1/2

i=1 In the opposite direction this does not work: L1 has cotype 2 but L ~ does not have type 2. Since L2(X*) is a subspace of L2(X)*, we have

6ix i

E

i=1

8iX*

= sup g

f : Ilfllt2(x)<<, 1 .

i=1

The problem is the passage from a general f to a function of the

form

~-'~in__l

s i x i . This

would be possible if the Rademacher projection P :-- P | I x from Lz(X) into itself

Basic concepts in the geometry of Banach spaces

53

would be bounded, where P is the orthogonal projection onto the closed linear span of the Rademacher sequence {en}~_l (a space X for which P is bounded is said to be K-convex). /5 is defined explicitly for f/ in L2(#) and xi in X by ['(~i~=l fixi) -- Y~"i=l (P~)xi" It is a deep fact (see [16, II.14], [17, 2.4], and [38]) that: X is K-convex if and only if X is B-convex. Thus the presence of copies of ~ is exactly the factor which hinders a clean duality between type and cotype. If X is B-convex and of cotype p* then X* is of type p. For finite dimensional X it is important to estimate the norm of the Rademacher projection (called the K-convexity constant of X and usually denoted by K ( X ) ) . A useful estimate (which is valid also for isomorphs of infinite dimensional Hilbert space) is K ( X ) <~ K(1 + logd(X, H)) for some constant K, where H is a Hilbert space with the same dimension as X. When dim(X) = n > 1, the distance estimate from John's theorem gives that K (X) ~< K l log n. These estimates are sharp up to the values of the constants K and Kj. For a discussion see [20, pp. 86-92], [17], [16, II.14], or [38].

9. Local structure of infinite dimensional spaces In this section we describe results and techniques whose purpose is to relate the structure of an infinite dimensional Banach space with the structure of its finite dimensional subspaces. We will be particularly interested in properties of a space which depend only on its family of finite dimensional subspaces and not on the way these finite dimensional spaces are "glued together" to form the infinite dimensional space. A Banach space X is said to be L-representable in a Banach space Y if for every finite dimensional subspace E of X and every ~ > 0 there is a finite dimensional subspace F of Y with d(E, F) <<,)~ + ~. If this holds with )~ = 1 we say that X isfinitely representable in Y. In this terminology we can rephrase some results discussed in the previous section as follows: The space ~2 is finitely representable in every infinite dimensional Banach space Y (Dvoretzky's theorem). A Banach space Y has cotype q for some q < ~ if and only if co is not finitely representable in Y. A Banach space Y has type p for some p > 1 if and only if ~1 is not finitely representable in Y. It is a trivial fact that every Banach space is finitely representable in co. For every Banach space X its second dual X** is finitely representable in X. In fact the following stronger statement, called the principle of local reflexivity, is true:

Let E be a finite dimensional subspace of X**, F a finite dimensional subspace of X* and > O. Then there is an operator T from E into T E C X so that IIT II l[T-1 ]l ~< 1 + ~, TIE•X is the identity and x** (x*) = x* (Tx**) for every x* ~ F and x** ~ E. Before giving a proof of the principle itself let us make some simple observations. If S : Y --+ Z is an operator with closed range then so is S + R if R is a finite rank operator. Also if S is as above and there is a y** c Y** such that S**y** ~ Z (i.e., the canonical image of Z in Z**) then there is for every 6 > 0 a y ~ Y such that IIYl[ ~< IIy** [I(1 + 6) and S** y** = Sy. (This reduces immediately to the case where S is a quotient map and in this case it is obvious.) Another simple observation is the following: for every E there is a 3(~)

54

W.B. J o h n s o n a n d J. L i n d e n s t r a u s s

so that if (1 + 3 ) - l ~< IISy II ~ 1 + 6 for every y in a 6-net in the unit sphere of Y, and some operator S, then (1 + ~ ) - l [ly II ~< IlSy II ~< (1 + ~)IlY II for all y E Y. We pass to the proof of the principle of local reflexivity. Let e > 0 and let 6 = 6(e) be as above. Let E and F be finite dimensional subspaces of X**, respectively, X*. We pick {ijj}j__ 1 , m in the unit ball of X* so that the set contains an algebraic basis of F and so that IIx**ll ~< (1 + 6 ) m a x j Ix**(v~)l for every x** E E. Let {wi**}/n__lbe a 6-net in the unit sphere of E so that {w**}~ 1 is a basis of E N X and is a basis of E for some k ~< r < n " Write w i** = Z h = l )r~ i h W h** for r < i ~ n. For r < i ~< n put # i , h - - Xi,h if h ~ r, ~ i i = - - 1, and #i,h = 0 for r < h # i. Consider the operator {//)i** } ri = 1

n-r+k

/7

d~

s. (2 e . . . 9 x)~

--, ( x ~ i . . 9 x ) ~

defined by

S(xl .....

(

Xn)--

x1 . . . . .

Xk,

#r+l,hXh ..... h-1

#n,hXh h-1

)

9

By the choice of # i , h for h > r it follows that S is onto. Hence the operator n-r+h

/7

~. (x e . . . 9 x)~

defined by S ( x l . . . . . Xn)

--

--, ( x e . . . 9 :~ 9 R ' m ) ~

(S(xl . . . . .

Xn),

vj(xh)),

1 ~ j <. m , **

1 ~ h <. n , has closed

/7

range. Note that by the definition of # i , h and the { w i }i=1' n-r+k d~

s * * ( w T * ,W 2** . . . . .

Wn** ) - - (W ** 1 ,11) 2** . . . . .

W k** . .o. . . o

.) 9 ( x ~ . . .

9 ~() oc.

Hence by one of the observations above applied to the operator "S~ there exist { W i } i/7= 1 ** ** in X with sup/IIw/ll ~< 1 + 3 so that S ( w l , w e . . . . . Wn) - S**(w~( * w e . . . . . w n ) and w * * ( v ~ ) - v ~ ( w i ) for all i and j . Define now T ' E --> X by T w * * - wi for 1 ~< i ~< r. t h a t T w i** - - w i - - w i** for 1 ~< i ~< k. If r < i ~< n then Y'~h=l # i , h W h -- wi - - the ith component of S ( w l , w e . . . . . Wn) - - the ith component of S**(w~( *, w e . . . . . w r ) = Z r h = l # i , h W h** -- W i** -- 0. Hence Tw** - - w i also for r < i ~ n. That T is an 1 + ~ isometry follows from the choice of 6 and the fact that for all i,

Note

1+3

~

11//9i 11 = IITw;*II

~ suplvff(w/) I -suplw~*(v~) I ~ J

(1 +3)-'.

J

We describe next a useful construction in Banach space theory (having roots in mathematical logic) which is related to the notion of finite representability.

Basic concepts in the geometry of Banach spaces

55

Recall that a family b / o f subsets of a set I is called a filter if it is closed under finite intersections, does not contain the empty set, and whenever A C B with A E b / t h e n B b/. A maximal (with respect to inclusion) filter is called an ultrafilter. By Zorn's lemma every filter is contained in an ultrafilter. An ultrafilter is called free (or nontrivial) if the intersection of all sets in b / i s empty. An indexed family {Xi}i6I in a topological space is said to converge to x with respect to a filter bt (in symbols, x -- l i n ~ xi ) provided for every open set G containing x the set {i: xi ~ G} belongs t o / g . A Hausdorff space is compact if and only if every indexed family {xi }i~I converges (to a unique point) for every free ultrafilter b / o n I. Assume now that I is a set and b / i s a free ultrafilter on I; assume also that for all i, Xi is a Banach space. We define a seminorm Ill'Ill on (Y-~i Xi)ec by IIIxlll = lirr~ Ilxi II where x -- {xi }i~/with xi ~ Xi for all i. The limit exists since a closed bounded interval on the line is compact. The quotient of (~-~i Xi)ec with respect to the closed subspace of all x with ]llx Ill --- 0 with its obvious norm is a Banach space, called the ultraproduct of the Xi (with respect to b/), and is denoted by (1--IiXi)cr If all the Xi are the same space X we call the space thus obtained an ultrapower of X, denoted also by Xu. Ultraproducts of Banach spaces are treated in detail in [9, Chapter 8]. Given two families {Xi}icI and {Yi}i~I of spaces and operators 7 ) : X i ~ Yi with sup/ ]]Ti ]] < cx:), there is a natural operator T:(1-Ii Xi)lg -'+ (Hi Yi)bt called the ultraproduct ofthe operators 7). It maps an element in (1-Ii xi)cr represented by x = {xi}i6I in (~_~X i ) ~ into the element in (]-Ii Yi)Cr represented by y = {~xi}is/. The ultraproduct of one dimensional spaces is one dimensional and more generally if dim Xi = n < ec for all i then (I-Ii xi)cr is also n-dimensional. On the other hand if I = N and limu (dim Xi) = ec then (I-Ii x i ) u is already nonseparable. The ultraproduct of Banach lattices is again a Banach lattice if we take as the positive cone in (Hi xi)cr the set of all elements which have representatives x = {xi}iEl in (Y]~i Xi)oc with xi ~ 0 for all i. If all the Xi are abstract Lp spaces for some fixed p, 1 ~< p < ec, then (Hi x i ) u is again an abstract Lp space and hence is isometric to Lp(#) for some measure # by the L p version of the Kakutani representation theorem. Similarly, if all the Xi are C (Ki) spaces for some compact Hausdorff Ki then so is (I-Ii xi)cr However, for other families of Banach spaces Xi (even, e.g., if all are Orlicz spaces with the same Orlicz function q)) the determination of the nature of (1-[i Xi)u is not an easy task. As a first application of ultraproducts we shall prove now a fact mentioned already in Section 4: If 1 <~ p <~r <. 2 then Lr (0, 1) is isometric to a subspace of L p ( 0 , 1). It was shown in Section 4 that for every integer n there is an isometry T,'g n --+ L p(O, 1). The ultraproduct T of the 7) is an isometry from (I-In grn)cr into Lp(O, 1)t4 where L/is any free ultrafilter on the positive integers. The space (1-In g~)br contains Lr(0, 1) as a subspace. Indeed, by using a sequence of partitions of [0,1] into n intervals (with each partition refining the previous one and so that the maximum length of the intervals tend to 0) we can represent Lr (0, 1) as LJn Fn where/7, C F,+I and F, isometric to ~r" for every n. For every g ~ Fn let ~, = (0, 0 . . . . . 0, g, g . . . . . g . . . ) E (y]~ F , ) ~ (n zeros). The closure of the images of all these ~, in (I-In F , ) u is isometric to Lr (0, 1). Thus Lr (0, 1) is isometric to a subspace of the abstract Lp(#) space Lp(O, l)b/. The space Lp(lZ) is nonseparable but, as in any Lp(#) space, any separable subspace of it is isometric to a subspace of Lp(O, 1). Every Banach space X is isometric in a natural way to a subspace of Xt4; map x into the element represented by the element {xi }iex with xi = x for all i. If X is a conjugate space

56

W.B. Johnson and J. Lindenstrauss

(and in particular a reflexive space) there is a contractive projection from X u onto the canonical image of X in it. Map {Xi}iEI t o w*-lingr xi. As for duality, the space (I-Ii x * ) u can be in a natural way identified with a subspace (usually proper) of the dual of (1--Ii xi)ct. We now discuss the relation between ultraproducts and )~-representability. It follows directly from the definitions that any ultrapower Xct of a Banach space X is finitely representable in X. There is also a converse, in a sense, to this statement. A Banach space X is ,k-representable in Y if and only if there is a subspace Z of some ultraproduct of Y so that d(X, Z) <, )~. The only if part is clear from the remark above. The if part is an easy generalization of the remark made above (for Lr (0, 1)) that if X is contained in the closure of an increasing union of subspaces {Xn }~~ 1 then X is isometric to a subspace of (I-In x n ) u for any free ultrafilter on the integers. Indeed, let I be the set of pairs (E, 6) with E a finite dimensional subspace of X and 6 > 0. Introduce a partial order on I by ( E l , 61) < (E2, 62) if E1 C E2 and 61 > 62, and let///be an ultrafilter on I containing for all i E I the set {j E I: i < j}. By assumption there is for every (E, 6) 6 1 an operator TE,~ from E into Y so that IIx II ~< IITE,~x II ~< ()~ + 6)Ilxll for all x E E. For every x 6 X let~ = {xE,~} 6 (Y~'~iY)~ be defined by xE,~ = TE,~x i f x 6 E and xE,~ = 0 i f x ~ E. The image of all these s in YU is easily seen to be a subspace Z which satisfies d (X, Z) ~< )~. From the principle of local reflexivity it follows that for every Banach space X the space X** is isometric to a norm one complemented subspace of a suitable ultraproduct of X. Without the complementation assertion this is a special case of the result above. To get the complementation assertion we have to modify the proof above. Now let I be the set of triples (E, F, 6), where E is a finite-dimensional subspace of X**, F a finitedimensional subspace of X*, and 6 > 0. Introduce a partial order on I by (El, F1,61) < (E2, F2, 62) if El C E2, F1 C F2, and 61 > 62, and let/g be an ultrafilter on I which refines the partial order filter. For every (E, F, 6) E I let TE,F,e :E ~ X be the operator given by the principle of local reflexivity. Using these operators we define as above an isometry T from X** into XU. Define a map S from X u into X** by S({xi }) = w*-limu xi. From the properties of {TE,F,~} one deduces easily that ST is the identity on X**, so that T S is a projection of norm 1 from XU onto T X**. A property (P) of Banach spaces is called a super property provided that if X satisfies (P) and Y is finitely representable in X, then Y satisfies (P). In particular, a super property passes from a space X to all closed subspaces of its ultraproducts. So if (P) is a hereditary property (i.e., passes to closed subspaces), a Banach space X has super (P) if and only if every ultrapower of X has (P). For example, X is superreflexive if every ultrapower of X is reflexive. An explicit local property which characterizes superreflexivity is the following: A Banach space X is superreflexive if and only if for every 6 > 0 there is an integer N(6) so that any 6-separated dyadic tree in the unit ball of X has height ~< N (6). By an 6-separated dyadic tree of height N we mean a set of points {Xi,n: 1 <<.i <~ 2 n, 1 <<.n <<.N} so that Xi,n -- (XZi-l,n+l -~- X2i,n+l)/2 and [ [ X Z i - l , n + l - - X2i,n+l[[ ~ 6 for every 1 ~< i ~< 2 n and every n < N. That this condition is necessary for superreflexivity is easy to see. A reflexive space cannot contain a bounded 6-separated dyadic tree {Xi,n } of height 0o since the closed convex hull of such a tree is not dentable. The other direction of this implication follows from what we stated in Section 6 concerning B-convexity. If X is nonreflexive it contains, 4 for example, vectors {Xj}~__1 of norm 1 so that for every i, [IZ~-I X j -- ~-'~j=i+l xjl[ >~ 3. The setXl+X2+X3+X4 4 ' Xl+X2 2 ' x3+x4 2 , xl, x2, x3, x4 is a 1-separated dyadic tree of height three

Basic concepts in the geometry of Banach spaces

57

in the unit ball of X. In a similar manner we get in every nonreflexive space a 1-separated dyadic tree of an arbitrary finite height in the unit ball. It is not hard to show that the existence of arbitrarily tall e-separated trees in the unit ball (for some e > 0 or for e = 1) is a selfdual property and thus so is superreflexivity. Much deeper is the fact that a space is superreflexive if and only if it has an equivalent uniformly convex norm. Since uniform convexity is defined by an inequality involving four vectors the if part is obvious. The hard part is the only if part; that is, the construction of a uniformly convex equivalent norm in a space which does not have e-separated dyadic trees of large height in its unit ball. The notion of dyadic trees reminds one of martingales and in fact the most elegant way to prove the only if part is to use vector-valued martingales. The proof shows in particular that if X is uniformly convex then there is an equivalent norm whose modulus of convexity ~(~) satisfies ~(e) ~> C~ q for some C > 0 and q < oe. The proof of this fact and related material can be found in [2, 4.IV] and [6, IV.4] (see also [29]). We introduce next a class of Banach spaces defined in terms of their finite dimensional subspaces which are closely related to L p ( # ) spaces. Let 1 ~ 1. A Banach space X is called an 12p,z space if for every finite dimensional subspace E of X there is /7 a further subspace F D E with d (F, gp) ~< )~ where n -- dim F. A space is called an s space if it is an s space for some )~ < oo. It is evident that every L p ( # ) space, 1 ~ 0 (take as F a small perturbation of a subspace spanned by a finite number of suitable disjoint indicator functions). Similarly, every C ( K ) space is an E ~ , l + ~ space for every e > 0 (use partitions of unity). By the principle of local reflexivity, if X** is an s space then X is an s space for every ~ > 0. From ultraproduct arguments used already above, it follows that: Any s space is isomorphic to a subspace of Lp(lZ) for some measure # (1 ~< p ~< ~ ) . In particular, for 1 < p < oe every s space is reflexive and any/22 space is isomorphic to a Hilbert space. It is possible to make a stronger statement. Assume first that 1 gp(E), S E ' g ~ (E) ---> E so that IITEII ~< 1, IISEII ~< ~ and SETE -- Ie (n(E) - - d i m E ) . L e t L / b e an ultrafilter on I which refines the order filter and put Y -- (FI gp(E))z4- Define T from X to Y by mapping x to the class represented by {TEx}Ecl (in view of the choice o f / 2 it does not matter that TEx is defined only for E which contain x). Define S: Y -+ X by S{yE} = w-lim~ SEyE (recall that X is reflexive). Then ST = Ix and Y is an L p ( # ) space. Consequently: Every E,p space X, 1 < p < oo, is isomorphic to a complemented subspace ofan L p ( # ) space. If X is separable we deduce that X is isomorphic to a complemented subspace of L p (0, 1). Similar considerations (starting with the fact proved in Section 4 that g2 is isometric to a complemented subspace of Lp(O, 1)) yield that every Hilbert space is isometric to a complemented subspace of some L p (#) space when 1 < p < oe. In the cases p -- 1 and p -- cx~ we can reason similarly, but now define S ' Y --+ X** by S{yE} -- w*-limu SEyE and get that ST -- Jx, the natural inclusion of X into X**. By considering T**" X** ----> Y** and S**" Y** ----> X (iv) and recalling that there is a norm one projection from X (iv) onto X** (see Section 2), it follows that if X is an 121 space then X** is isomorphic to a complemented subspace of an L l ( # ) space. Since L1 (#)* is

58

W.B. Johnson and J. Lindenstrauss

injective and since there is a norm one projection from X*** onto X* we deduce that if X is an E1 space then X* is an injective Banach space. Similarly, if X is an s space then the constructed ultraproduct Y is a C ( K ) space and X** is isomorphic to a complemented subspace of the injective space Y**. That is, if X is an 12~ space then X** is an injective Banach space. The converse of the previous statements essentially hold. First we show: Let 1 < p < ~ . A Banach space X is isomorphic to a complemented subspace of an L p ( # ) space if and only if X is an s space or X is isomorphic to a Hilbert space. To see the "only if" direction, assume that there is a projection Q from Y = L p ( # ) onto a subspace X which is not isomorphic to a Hilbert space. By the dichotomy principle for L p spaces, 2 0. There is a finite dimensional subspace F of Y (a small perturbation of the span of disjoint indicator functions) containing E so that d(F, ep) <~ (1 + ~) (m - - d i m F ) and so that there is a projection P from Y onto F with IIP II ~< 1 + ~. The space R Q F is a finite dimensional subspace of Z. Hence since every element in e p has essentially a finite support it follows that there is a subspace B of Z so that d(B, ep) <<.K and max{llxll, Ilyll} <~ K Ilx + y II for every x 6 Q F and y E B (K is a constant depending only on d (Z, ep); in fact, K ~< 1 + ~ is possible). Let r : F ~ B be an isomorphism from F onto B with l]r [I ~< 1 and IIr - 1 ]l ~< (1 + E) K. Define T : F --+ X by T = QIF § r ( I -- P QIF). Then the restriction of T to E is the identity (since this is true of the restrictions of P and Q to E). A simple computation shows that for every x 6 F, (1 + E ) - l ( 1 + K ) - l K - 1 l l x l l <<,IlTxll <~ (1 + (2+~)ll Qll)llxll. Hence G - T F is a subspace containing E with d(G, ep) <<,(1 + ~ ) ( 1 + K ) K ( 1 + (2 + E)II QII). Consider now the case p = 1. Every infinite dimensional complemented subspace of an L1 (/z) contains a subspace isomorphic to el. This is a consequence of the fact that such a space is nonreflexive (to be proved in the next section) and the structure of nonweakly compact sets in L l(#) discussed in Section 4. Hence if X* is injective then X**, being a complemented subspace of L l(#) for some #, contains a copy of e 1. The proof given above for 1 < p carries over verbatim and yields that X** and hence X is an E1 space. It is also true (see [21, II.D]) that an infinite dimensional complemented subspace of a C (K) space (and hence every infinite dimensional injective space) contains a copy of co. The same proof then yields that every space whose second dual is injective is a n / 2 ~ space. Instead of using this moderately difficult structure theorem for C ( K ) one can argue that if X** is injective then X*** is complemented in some L1 (#) space, hence X*** contains a complemented subspace isomorphic to el, whence X (iv) contains a subspace isomorphic to co. One consequence of the principle of local reflexivity (see [24]) is: If X* has the BAP then X has the BAP. Since every L p ( # ) space, 1 <~ p ~< cx~, has the BAP and the BAP passes to complemented subspaces, we conclude: Every s space has the BAP. A stronger statement is mentioned at the end of this section. From the results above it follows immediately that any complemented subspace of an 12p space 1 ~< p <~ cx~ is again an s space unless it is isomorphic to a Hilbert space (for p = 1 or p = ~ it cannot be a Hilbert space). We also get that: the dual ofan Ep space is an s space. This fact is not evident from the definition since Ep spaces are defined by

Basic concepts in the geometry of Banach spaces

59

the structure of their finite dimensional subspaces and by passing to the duals we get direct information only on the finite dimensional quotient spaces. The s spaces give a nice local description of the complemented subspaces of Lp(O, 1) for 1 < p < cx~. From the global point of view these complemented subspaces are very hard to describe. Some global structure theorems are known (e.g., every separable s space 1 ~< p ~< ~ has a Schauder basis, and every separable infinite dimensional s space 1 < p < cx~ which does not contain a copy of e2 is isomorphic to e p) but many natural questions on them are still open. For example, it is unknown whether every separable s space, 1 < p < cx~ (p :/: 2), has an unconditional basis. There are uncountably many distinct isomorphism types among the separable s spaces, 1 ~< p ~< ~ (p r 2). The simplest examples for 1 < p 7~ 2 < oc (besides L p ( # ) spaces) are ep | e2 and (e2 (~ e2 (~'" ")p. A discussion of all of these questions is contained in [22]. Among the s e p a r a b l e / 2 ~ spaces there are the C ( K ) spaces with K countable compact metric which, as was pointed out in Section 4, form uncountably many distinct isomorphism types. There are however spaces X such that X* is isometric to e l but X is not isomorphic to a C ( K ) space. Such a space X is an s space for every e > 0. In fact, a space X is s l+~ for every E > 0 if and only if X* is isometric to L I ( # ) for some measure #. It is known that such an X has a subspace isometric to co. In contradistinction to this, there are s spaces with 2. > 1 which have the RNP (and thus in particular do not contain a subspace isomorphic to c0). For a discussion of s spaces see [40]. There are a continuum of isomorphism classes of separable s spaces. The simplest example (besides el and L 1(0, 1)) is the kernel of a quotient map T :el --+ L1. The isomorphism type of this kernel turns out not to depend on the choice of T. This kernel is an s space which is not isomorphic to a complemented subspace of an L l ( # ) space. For a discussion of s spaces, see [22]. There is also a "local view" of Banach lattices. A Banach space X is said to have Gordon-Lewis local unconditional structure (GL-l.u.st.) if there is a constant 2. so that for every finite dimensional subspace E of X there is a space Y with an unconditional basis and operators: T : E --+ Y, S : Y ~ X so that S T = IE and []TI[ ]]Slluc(Y) <<,)~ where uc(Y) denotes the unconditional constant of a basis in Y. Note that by a simple perturbation argument we can assume that Y is always finite dimensional (we may have to replace )~ by any )~ > )~). If always Y can be chosen to be a subspace of X which contains E, X is said to have local unconditional structure (1.u.st.). Then of course T and S can be taken to be the inclusion maps. It is evident that 1.u.st. implies GL-l.u.st., but whether the converse is true is open. Later in this section we mention a partial converse. Any Banach lattice X has 1.u.st. where the X in the definition can be taken as any constant > 1. Indeed, if X is an order complete lattice, then we saw in Section 5 that for any finite dimensional subspace E and every ~ > 0 there are vectors {xl . . . . . Xm } in X whose span contains E and which is an e-perturbation of a sequence of disjointly supported vectors. Since a disjointly supported sequence in a lattice is 1-unconditional and the bidual of a Banach lattice is an order complete lattice, the general case follows from the principle of small perturbations and the principle of local reflexivity. It is clear from the definition that s spaces, 1 ~< p ~< ec, have 1.u.st. The close relationship between 1.u.st. and lattice structure is expressed in the following result: A Banach space X has GL-l.u.st. if and only if X** is isomorphic to a complemented

60

W.B. Johnson and J. Lindenstrauss

subspace ofa Banach lattice. Indeed, since by the definition a complemented subspace of a space with GL-l.u.st. has GL-l.u.st., it follows that if X** is a complemented subspace of a lattice it has GL-l.u.st. By the principle of local reflexivity we deduce that also X has GL-l.u.st. To prove the other direction we use ultraproducts. Assume X has GL-l.u.st. with some constant )~. For every finite dimensional subspace E of X there are a space YE with 1unconditional basis and operators Tt; : E --+ Yt; and St; :Y t; --+ X with l[Tt; 11~< 1, liSt; 11<~ )~ and St; Tt; = It;. Let I be the directed set of all finite dimensional subspaces of X and let L / b e an ultrafilter on I which refines the order filter. Let T :X ~ Y = (I-I Yt;)u be defined by Tx = {Tt;x} (as before, because of our assumption on b/it does not matter that Tt;x is defined just for the E containing x). Let S : Y --+ X** be defined by S{yt;} = w* lim St;yE. The space Y is a Banach lattice and ST = Ix. Hence T** is an isomorphism from X** into the lattice Y** and P S** is a projection from Y** onto T** (X**) where P is a projection from X (iv) onto X** (see Section 2). An immediate consequence of the claim just proved is that X has GL-l.u.st. if and only if X* has GL-l.u.st. Indeed, if X has GL-l.u.st. then X** is complemented in a Banach lattice. Hence the same is true for X*** and thus X* has GL-l.u.st. Conversely, if X* has GL-l.u.st. then X***, hence also X (iv), is complemented in a Banach lattice. Since X** is complemented in X (iv), it follows that also X has GL-l.u.st. The argument given earlier that if a complemented subspace X of an L p (#) space is not isomorphic to a Hilbert space then it is an s space depended on the fact that an L p ( # ) space is finitely representable in e p and that e p is, for some )~, )~-finitely representable in every finite codimensional subspace of X. The same argument shows that if Y is complemented in a Banach lattice X and there is )~ < cx~ so that X is )~-finitely representable in every finite codimensional subspace of Y, then Y has 1.u.st. From this and the characterization of GL-l.u.st. mentioned above we deduce: X has GL-l.u.st. if and only if X E3 co has l.u.st. Among the spaces which fail to have GL-l.u.st. important examples are the spaces of compact and bounded operators on ~2. In fact, the only unitary ideal of operators on ~2 (a notion discussed in Section 10) which has GL-l.u.st. is the space of Hilbert-Schmidt operators, which is itself a Hilbert space. Spaces of analytic functions, such as the disc algebra, typically do not have GL-l.u.st. For a proof of most of these statements see [9, Chapter 17], [20], or [27]. We conclude this section by mentioning briefly a quantitative notion of the approximation property and its relation to ultrapowers. A Banach space X is said to have the uniform approximation property (UAP) if there is a constant )~ and a uniformity function ~0(.) on the natural numbers so that for every finite dimensional subspace E of X there is T E B (X, X) so that IIT II ~< z, TIE = It;, and dim T X ~< ~0(dim E). A Banach space X has the UAP ifand only if every ultrapower of X has the BAP. It follows from this that every s space, 1 ~< p <~ cx~, has the UAP. It also follows, in view of results about ultrapowers presented earlier, that if X** fails the BAP then X fails the UAP. From this it can be deduced (see [24]) that there are Banach spaces which have the BAP but not the UAP (in fact, K (~2, ~2) is such a space). While it is usually easy to check whether a concrete separable space has the BAP, it is often quite difficult to decide whether a special space has the UAP. For example, it is open

Basic concepts in the geometry of Banach spaces

61

whether the disk algebra has the UAE As a matter of fact, the only general class of spaces other than L p spaces which are known to have the UAP are the reflexive Orlicz spaces 9

10. Some special classes of operators In this section we shall discuss some special classes of linear operators between Banach spaces and their relation to the geometry of Banach spaces. We shall also discuss some other topics in operator theory which are relevant to the study of the structure of Banach spaces. Most of the classes of operators we consider have the ideal property, meaning that the operators from a fixed X into a fixed Y having this property form a linear space and that whenever T : X --+ Y belongs to this class then for every bounded U : Z --+ X and V : Y --+ W the operator V T U belongs to this class. The three most elementary operator ideals are the bounded operators, the compact operators, and the weakly compact operators. Since B(X, Y), K ( X , Y), and WK(X, Y) are all Banach spaces under the operator norm, the operator norm is the natural norm to use for operators in these classes. A Banach space X has the Dunford-Pettis (DP) property provided that every weakly compact operator with domain X maps weakly compact sets into norm compact sets. It is clear that if T E WK(X, Y), S E WK(Y, Z), and Y has the DP property, then ST is a compact operator. In particular, if X has the DP property and P is a weakly compact projection on X, then p = p2 is compact. This means that the only complemented reflexive subspaces of X are the finite dimensional ones. A subset of a Banach space is relatively weakly compact if and only if it is relatively weakly sequentially compact. Therefore, a space X has the DP property if and only if every weakly compact operator with domain X maps sequences which converge weakly to zero into sequences which converge in norm to zero. The following is an elegant characterization of spaces having the DP property. X has the DPproperty if and only if whenever {Xn}n~__l in X tends weakly to 0 and {x*}n~=l in X* tends weakly to 0 the sequence of scalars {x* (Xn) }n~__l tends to O. Indeed, assume that X has ~<

11)

DP and x n > 0 9The operator T" X ~ co defined by Tx -- (x*1 (x), x 2* (x) . . . . ) is a weakly compact operator (it is easier to check that T* is weakly compact). Hence, if Xn --+ 0 weakly, Ix*(xn)[ ~< IITxnl[ ~ 0. Conversely, assume the condition on the sequences is satisfied, T : X --+ Y is weakly compact, and x~ w> 0. If IITxn IIr 0 we may assume, by passing to a subsequence, that IITxn II >~ 6 for some 6 > 0 and all n. Let y* E X* be such that y*(Txn) -- IlTxn[I and IlYn*l[ - 1 for all n. Since T* is weakly compact as well we may to

assume (passing again to a subsequence if needed) that T* y*

0-

lim(T*y* n

x*)(Xn) --limy*(Txn) --lim /7

> x* for some x*. Then IITxn II

n

a contradiction. It follows from the criterion we just proved that X has the DP property if X* has this property. (The converse is false: ( Y~n=l ~ ) 1 has the DP property but ( Y~m= er 1 e2) n oo contains a complemented copy of ~2 and hence fails the DP property.) It is also clear that a

W.B. Johnson and J. Lindenstrauss

62

complemented subspace of a space with the DP property has the same property. We shall verify shortly that C(K) spaces have the DP property. Consequently we get that: All E ~ and s spaces have the DP property. To show that C(K) has the DP property let {xn},~__l be a sequence in the unit ball of C ( K ) so that x,(t) --+ 0 for every t 6 K. Let {lZn},e~=l be a sequence in the unit ball of C(K)* which tends weakly to 0. All the # , can be considered as elements in L1 (#) where -- ZnCX~=l I#, 1/2 n. By the analysis done in Section 4 of sets in Ll (#) which have weakly compact closure it follows that for every E > 0 there is a & > 0 so that whenever E is a Borel set in K with # ( E ) < 6 then I#nl(E) < ~ for all n. By Egoroff's theorem there is a set E with # ( E ) < & so that IXn(t)[ <<.~ for all t outside E and n / > n(~). Hence for such n,

f K Xn d#n

xnd#n

that is, f K Xn dlZn ~ O. We also want to single out the strictly singular operators. An operator T : X -+ Y is strictly singular if it is not an isomorphism on any infinite dimensional subspace of X. Of course, any compact operator is strictly singular but the converse is false. The formal identity map from g r into s where p < r is a simple example of a strictly singular operator which is not compact. Note that, by the discussion in Section 3, an operator from e p, 1 <~ p < cx~, into itself (or from co into itself) is strictly singular if and only if it is compact. A more sophisticated and interesting example of a strictly singular operator is the formal identity operator Ip: C(K) --~ L p ( # ) , 1 ~< p < e~, where # is a finite measure on the compact set K. The operator Ip is not compact if, for example, K = [0, 1] and/z is Lebesgue measure (in fact, lp is compact only if # is purely atomic). Observe that if X is a subspace of C(K), the statement that the restriction of Ip to X is an isomorphism means that the supremum norm and the L p ( # ) norm are equivalent on X. If this occurs for some p < e~, then the extrapolation argument used in Section 4 shows that the supremum norm and the Lr (#) norm are equivalent on X for all 1 ~< r < cxz. Thus it is enough to check, e.g., that 11 or I2 is strictly singular. That I2 is strictly singular is an exercise in textbooks (e.g., [ 18, Chapter 10 # 41, # 55]), but we prove it later in this section when we discuss p-summing operators. Unlike the situation with compact or weakly compact operators, the dual of a strictly singular operator need not be strictly singular. If T is a quotient map from el onto a separable space X not containing a subspace isomorphic to el (e.g., X = e2) then T is strictly singular but T* is an isometric embedding of X* into s Although strictly singular operators need not be compact they have compact restrictions to infinite dimensional subspaces. Indeed, by the same method used in Section 3 to prove the existence of a basic sequence in any Banach space one can prove the following. If T : X --+ Y is not an isomorphism on any subspace of X of finite codimension then there is for every E > 0 a normalized basic s e q u e n c e {Xn}ne~ 1 in X (with basis constant two, say) so that IlTxn[] <<.4-nE for every n >~ 1. The restriction of T to the span of {Xn}n~=l is a compact operator of norm ~< e.

Basic concepts in the geometry of Banach spaces

63

It follows from this observation that the sum of two strictly singular operators is strictly singular and thus the class of strictly singular operators has the ideal property. It is also easy to check that SS(X, Y) is complete in the operator norm. Two Banach spaces X and Y are called totally incomparable if there is no infinite dimensional space Z which is isomorphic to subspaces of both X and Y. If every operator from X to Y is strictly singular (and in particular if X and Y are totally incomparable)and

Z is a complemented subspace of X • Y then there is an automorphism T of X 9 Y so that T Z is of the form T Z = Xo 9 Yo where Xo (respectively Yo) is a complemented subspace of X (respectively Y). This result is not as simple as the preceding results in this section. Its proof can be found in [ 11, 2.c. 13]. Another class of operators is that of Fredholm operators. An operator T : X --+ Y is called a Fredholm operator if or(T) := dimker T < c~ and T X is closed w i t h / ~ ( T ) := d i m Y / T X < cx~. The number i(T) = or(T) - / 3 ( T ) is called the index of T and has a significant role in many applications. As a matter of fact i (T) can be defined also for nonFredholm operators provided T X is closed and at least one of the numbers or(T) or/3(T) is finite (in this case the index can be of course either +cxz or -cx~). The class of Fredholm operators is not closed under addition or composition. However if T1 and T2 are Fredholm operators so is TiT2 (provided it is properly defined) and i(T1T2) = i(T1) + i(T2). Also i(T*) = - i (T). These facts are simple exercises in linear algebra. Somewhat harder to prove is that Fr(X, Y) is preserved by strictly singular perturbations. If T, S : X --+ Y with T Fredholm and S strictly singular then T + S is also Fredholm with i(T + S) -- i(T) (see [11, 2.c.9]). An operator T ' X --+ Y is called absolutely summing if whenever EiC~=l xi converges unconditionally in X the series Ei~l Txi converges absolutely. From the closed graph theorem one deduces easily that T is absolutely summing if and only if there is a constant C so that for any choice of {xi }in=l in X

E I I Txi ]l ~ C sup i---1

Ix*(xi)]" I[X* II ~ 1 .

i--1

More generally, for every 0 < p < cx~ we can define the class of p-summing operators as those operators for which there is a constant C so that for all choices of {xi }i=1 n in X

IlTxill p i:1

Ix*(xi)l p

<~c sup

" IIx*ll ~ 1 .

(28)

i:1

Thus 1-summing is the same as absolutely summing. The smallest constant C for which (28) holds is denoted by 7cp(T). If T is not p-summing we put :rrp(T) -- ~ . The set of all T : X ~ Y with 7rp(T) < cx~ is denoted by 17p(X, Y). It is trivial to check that for 1 <<,p < cx~, 17p(X, Y) is a Banach space with 7rp(T) as the norm. It is also clear from the definition that whenever ST is defined and one of the operators S, T is p-summing so is their product and 7rp(ST) <~ IlSllTrp(T) and Top(ST) <~ :rrp(S)llTl[. In particular the class of p summing operators has the ideal property. Sometimes one uses also the notation zr~(T). It naturally means just the usual operator norm IITII of T.

W.B. Johnson and J. Lindenstrauss

64

Without further mention we shall henceforth assume that p ~> 1 when discussing psumming operators. However, there are important applications to Banach space theory of p-summing operators for 0 < p < 1 (see, e.g., the article [32]), and much of the theory of p-summing operators, 1 ~< p < oo, goes over to the range 0 < p < 1. In particular, there is a version of the Pietsch factorization theorem (discussed below when 1 ~ 1 is needed for this). Thus top(T) can also be defined by

Zcp(T)

-

-

[[TVenl] p

sup

" V "g.p, --~ X, []VI] ~< 1 .

(29)

Equation (29) implies that 7rp(T) is the supremum of rcp(TV) as V varies over the norm one operators from s into X. In contrast to the classes of weakly compact or strictly singular operators, the class of psumming operators is determined by the behavior of T on the finite dimensional subspaces of X. Because of the quantitative nature of its definition the notion of p-summing norm plays also an important role in the study of finite dimensional spaces. If K is a compact Hausdorff space and/z a regular probability measure on K then the identity operator Ip from C ( K ) into L p ( # ) is easily seen to be a p-summing operator with p-summing norm one. It turns out that this simple example is the prototype of general p-summing operators. This is the content of the Pietsch factorization theorem. Let X be a subspace of C (K). An operator T : X --+ Y is p-summing, 1 <<,p < cx~, if and only if there is a regular probability measure lz on K and a constant C so that f o r all x in X,

IITxll ~ c

Ix(t)[ p d#(t)

(30)

Moreover, the smallest C for which (30) holds is 7rp(T). Note also that the "only if" part can always be applied with K -- B x , under the weak* topology, or with K the weak* closure of the extreme points of B x , , again in the weak* topology. In diagrammatic form, the Pietsch factorization can be stated as follows. Let X be a subspace of C ( K ) . If T is p-summing then there is a regular probability measure # on K and an operator S from Xp := I p X into Y so that T = Slplx; moreover, []SI] = 7rp(T).

C(K)

Ip

u X

> Lp(#) u

lplx

>Xp

Y

Basic concepts in the geometry of Banach spaces

65

(For p -- 2, the operator S can be extended to an operator from L p (#) into Y, but for p :/: 2 this is not in general the case.) To prove the Pietsch factorization theorem, assume that 7rp(T) = 1. Consider the following subsets of C(K) G = { f e C(K): sup{f(t)" t 6 K} < 1}, F = conv{f E C(K): f(-)-

Ix(.)l z',

IITxll- 1}.

The sets F and G are convex with G open and, since 7rp(T) = 1, F M G = 0. By the separation theorem there is a regular signed measure # on K and a positive )~ so that ffd#<)~ for f E G and f f d # ~ > ) ~ for f E F . Since w h e n e v e r g ~ < f and f E G also g E G it follows that the measure/z has to be positive so after normalization we may assume that it is a probability measure. Also since G contains the open unit ball of C(K) we must have )~ >~ 1. Hence for every x ~ X, IITx[Ip <~fK [x(t)lP d#(t). From this factorization theorem we can easily deduce several interesting facts. First, whenever 1 ~< p ~< r < ~x~, 7rr (T) ~< Sop(T) and thus FIp(X, Y) C FIr(X, Y). (This is trivially true also for r =- oc.) Next, since Lp(#) is reflexive for 1 < p < ec, every operator in FIp(X, Y) is weakly compact. By the preceding remark this is true also for p = 1. A third consequence is that the identity operator on an infinite dimensional space X is never psumming for any p < oo. Indeed, by the Pietsch factorization theorem, this is equivalent to the assertion that Ip:C(K) ~ Lp(l z) is strictly singular for any probability measure # on any compact Hausdorff space K. As we mentioned earlier, if Ip were not strictly singular for some p < ec, then there would exist an infinite dimensional subspace X of C(K) so that for all 1 ~< p < ec, the Lp(#) norm is equivalent to the supremum norm. So X would be isomorphic to an infinite dimensional Hilbert space and the identity operator on X (hence also the identity operator on g2) would be 1-summing. But obviously there exists in s an unconditionally converging series which is not absolutely summable. Recall that we already proved in Section 8 that the identity operator on an infinite dimensional space is not absolutely summing (in fact, a stronger statement was proved). The two proofs we have given for this result are very different but have in common that Hilbert spaces enter naturally into them. There are many other places where the Pietsch factorization theorem is useful, in particular, for proving the composition inequality for p-summing operators, which says if T E FIp(X, Y) and S E FIq(Y, Z), then ST E FIr(X, Z) and 7rr (ST) ~< 7rp(T)Jrq(S), where 1/r := 1/x ( l / p + 1/q). See [9, 2.22]. The Pietsch factorization theorem also makes it easy to check that 7r:p(T**) <~7rp(T) for any operator T (see [9, 2.19]). This fact can also be proved using the principle of local reflexivity. Next we examine ideals of operators on g2 and how they are related to p-summing operators. If T is a compact operator on a Banach space then we can define a (finite or infinite) sequence {)~n(T)}n~ consisting of the nonzero eigenvalues of T, repeated according to multiplicity, and ordered so that ILl (T)I >/])~2(T)] > / . . . . If the sequence is infinite, then necessarily )~n(T) --+ O. For the Banach space ~2, the spectral theorem for compact op-

66

W.B. Johnson and J. Lindenstrauss

erators says that if T ~ K ( H , H), then there is an orthonormal basis {Xn}nC~1 SO that the self-adjoint operator T*T is represented as

T * T x -- Z )~n(T*T)(x,xn)Xn nEP

(31)

with )~n(T*T) > 0 for n 6 P and T * T x n - 0 for n r P. Now if X is a Banach space with a 1-symmetric basis S - {en}n~__l, let S(62) be those compact operators on 62 for which ~ ~/)~n(T*T) en converges in X, and, for T 6 S(62), set

Then (S(62), o's) is a Banach space (verifying the triangle inequality is a bit tricky). It is less difficult to check that S(62) satisfies the unitary ideal property

crs(UTV) <<.IIUII~s(T)IIVII for all U, V ~ B(H, H) and T 6 S(62). When S is the unit vector basis for 6p, 1 ~< p < c~, the resulting ideal, called the Schatten-von Neumann p-class, is denoted by (Sp, crp). The ideal ($2, ~r2), which is a Hilbert space, is called the Hilbert-Schmidt class. The Schatten-von Neumann classes are discussed in many books, including [9] and [21 ]. Given the representation (31) for T*T with T ~ K ( H , H), it is evident that {Txn}n~=l is orthogonal and IlTxn[[ 2 - )~n(T*T), and hence c r p ( T ) - (~--~n%l ]]TxnIIP) 1/p for 1 ~< p < oo. In the Hilbert-Schmidt case, ff2(T) -- (Y~n=l c~ IlTynll 2)1/2 for any orthonormal basis {Yn}n~=l (just write IlTym[[ 2 -- (T*Tym, Ym) and expand Ym with respect to {Xn}n~__l). From this and the ideal property of $2 it follows easily that/72(62, 62) - $2 with equality of the respective norms. Indeed, if T 6 K(62) and {Xn}n~=l is an orthonormal basis, then by (29) ff2(T) 2 -- ~--~n%l [[Txnl[ 2 <~ 7rz(T) 2" On the other hand, if V'62 ---> 62 and IIVI[ ~< 1, then (Zn%l IlTWxnl[2) 1/2 <<-ff2(TV) ~< cr2(T), so (29) gives 7r2(T) ~< o'2(T). For an operator T'62 --+ X, it is not true that 7r2(T) 2 -- y~ [[Txn[] 2 for every orthonormal basis {Xn}n~=l, but it is not very hard to show that 7rz(T) 2 is the supremum over all orthonormal bases of ~ IlTxn [12. More importantly, there is a finite version [20, 18.4] of this which is useful for studying finite dimensional spaces. This lemma says" If T is an

operator from 6~ into some Banach space, then there is an orthonormal basis {xi }in=l so that 7rz(T) 2 ~< 2 zin_l I[Txi[[ 2. We next discuss p-summing operators on 62 for other values of p: For every 1 <<,p < oo, the p-summing operators on 62 coincide with the Hilbert-Schmidt operators, and of course Zrp(.) is equivalent to cr2(.). First a preliminary remark. Given T ~ K ( H , H) with T*T represented as in (31), define [T[x -- ~ x/)~n(T*T)(x,xn)Xn. There is an obvious partial isometry (i.e., an orthogonal projection followed by an isometry) U so that T -U IT[. This is the polar decomposition of the compact operator T. Suppose now that T is Hilbert-Schmidt. We want to see that T is 1-summing (and hence p-summing for every 1 ~ 0 so that Ten -- )~nen for every n and 1 -- (re(T) -~nC~=l)~n.2 Let {~n}~_|_ be a Rademacher sequence. Using Khintchine's inequality (1) we get

I(xi,x)[" Ilxll ~< 1 , ~> sup

sup i--1

-+-

xi, Z - l - ) ~ j e j i--1

j--1

>>.~ E ~_~)~j~j(xi, e~) >1el ~_~ i=1

j--1

i=1

I,~j(xi, ej)l

2

j=l

II Zxi II.

-- A1

i=1

This shows that 7rl (T) ~< A11 crz(T). To show the converse, it is enough to check that if T 6 / / p ( ~ 2 , ~2) with 2 <~ p < cx~ then T is 2-summing. More generally" If T E lip (X, Y) with 2 <<.p < c~ and Y has cotype 2 then T E 172 (X, Y). Since T is p-summing, the Pietsch factorization theorem kindly supplies a probability measure # on B x, so that for all x in X,

]x*(x)] p d/z(x*)

IlTxl] ~< top(T) X*

Using Khintchine's inequality (1) in the last step, we get 2)1/2 IITxi l]2 i=1

~< C2(Y) E ~ ~ i T x i i--1

p ) 1/p ~< C2(Y) ( E

1/p

[x*(xi)l 2

<~ C2(Y)rcp(T)Bpsup

" IIx*ll ~< 1 ,

i=1

so that 7r2(T) ~< BpC2(Y)zrp(T). We prove next an important inequality, called Grothendieck's inequality, which is an essential tool in studying p-summing operators as well as other topics. The inequality states" There is a universal constant KG so that whenever A - (aij)inj=l is a scalar tl

matrix so that I~in.j=l aijtisj[ <~ 1 for all {ti}inl and {sj}j=| of absolute value <<. 1 17 then I~'_,in,j=| aij(xi, Yj)I <<.KG for all choices of vectors {xi}in=| and {yj}j=| in the unit ball of a Hilbert space H. A more conceptual (but obviously equivalent) way of stating Grothendieck's inequality is the following. Suppose that the matrix A - (aij)inj= 1 has

W.B. Johnson and J. Lindenstrauss

68

norm at most one when considered as an operator from s to s Let H be a Hilbert space. Then IIA @ In II ~< KG as an operator from s (H) to s (H), where

(A | IH)(yl ..... Yn) "=

aljyj ..... j=l

anjyj j=l

for (Yl . . . . . Yn) in s To prove Grothendieck's inequality fix n and a matrix A - (aij)in,j=l with [IAll " IIAs ---> s 1 as above, and put !A! := [IA | I H ' s --->s We have to find an estimate on !A! independent of n and A. Note first that for all ui and vj in a Hilbert space

aij (ui, vj) ~< !A! max Ilui II max Ilvj II. i

i,j=l

j

Since all Hilbert spaces are created equal, we can use for H any (infinite dimensional or even 2n-dimensional) Hilbert space. For the purpose of proving Grothendieck's inequality, some Hilbert spaces are more equal than others! We use for H the subspace of L2(0, 1) mentioned in Section 4 consisting just of functions having a Gaussian distribution with mean 0. To prove Grothendieck's inequality it is enough to consider norm one vectors {Xi}in=l a n d {YJ}j=I in H. Given 0 < 6 < 1/2 there is an M -- M(6) so that for any norm one function f in H , Ilf - f M 112 --" • where

fM (t) - ] f (t)

if If(t)l ~< M, if If(t)[ > M.

[ M sign f(t)

Note that by our assumption on the matrix A, for any choice of functions f/ and g j in L2 (0, 1) which are uniformly bounded by M,

Zaij(fi,gj) i,j

f01

Z aij f i ( t ) g j ( t ) t,j 9

dt ~< M 2.

.

Hence

Z aij (xi, yj )

Zai,j(XiM, y M} + Z a i j ( x i ,

i,j

i,j

Yj -- Y M)

i,j

+ Z aij(xi- x M, yM) i,j

<~ M 2 + 2!A!3. Since the supremum of the left hand side (over all choices of Xi and y j) is by definition !A l, M2

we deduce that !A! ~ ~ .

This completes the shortest proof we know for Grothendieck's

Basic concepts in the geometry of Banach spaces

69

inequality. With the optimal choice of 6 this proof yields that K c < 8.69 (or KG < 8.55 if one exercises more care in the computation), but it is known that KG < 1.79. Incidentally, although the best value for K c is unknown, it is known that for complex scalars it is smaller than for real ones. The first application of Grothendieck's inequality is: Every bounded linear operator T from s to ~2 is absolutely summing and rrl(T) ~ KGIITII. Indeed, l e t {ej}7_ 1 be the unit vector basis for s and let ui -- Y'~jm=1 aijej be n vectors in s for some m so that Ein=_ 1]x , (ui)l ~< 1 for every unit vector x* in s m 1 of scalars of For any choice {Sj}j= be defined by x * ( e j ) - s j .

absolute value <~ 1 letx* e s

~ j

E aijtisj i

~_~ltil y ~ aijx* (ej) i J

Then for all {ti}in=l with ]ti] <~ 1

i

Hence, if yi e g2 with (Tui, Y i ) - IlTuill and ]Iyi I I - 1 for all i, we get

~_~ llTuill - E ( T u i , i i

Yi) = E ~__aij(rej, Yi) <~ KGIITII. j i

By the local nature of the definition of the 1-summing norm it follows immediately that if X is an s space and Y an s space then for every r :X --+ Y, 7rl ( r ) ~ )~#KGIITII. As a second application of Grothendieck's inequality, we prove a result alluded to in Section 5: If T is an operator from a Banach lattice X into a Banach lattice Y, then for all {Xi };=~ n c X

ITxil 2 i--1

~ KGIITll

I1( )"2 ~

Ixil 2

(33)

i--1

The first step in the proof is to use Grothendieck's inequality to prove (33) when X -- s and Y = s for some m. Having done that, the case where X is a C(K) space and Y is an L1 (/z) spaces follows immediately by approximation. Finally, we use the lattice theory discussed in Section 5 to reduce the general case to the case when X -- C(K) and Y = L1 (#). Before proving (33), observe that since L l(#) spaces are 2-concave we can deduce the following from (33): If 7' is an operator from a C(K) space into an L1 (#) space, then T is 2-summing and Jrz(T) <~ KclITII. Since Lp(#), 1 <~ p ~< 2, embeds isometrically into L~ (v) for some v, the local nature of the 2-summing norm gives: Every operator T from an s space into a subspace of a ~.p,r space, 1 <~ p <~ 2, is 2-summing with 7r2(T) ~< )~rKGIITII. We turn to the proofof (33) when T is an operator from s to s Let {ei}im=l be the unit vector basis for s and let Xk -- Y~im_~bkiei be in s with I1(~-~ Ixk12)l/211~ ~< 1. Let n {ui }inn be orthonormal in some Hilbert space H, set for 1 <. i <, m, Yi -- ~k--1 bkiUk, and

70

W.B. Johnson and J. Lindenstrauss

define in g~c ( H ) the vector 2 " - (yl . . . . . y m ) . T h e n l l x l l e ~ ( . ) Similarly,

__

n II( Y~k=l Ixkl2) 1/2 I1~ ~< 1.

-II T o

ITxkl 2 k=l

and this last quantity is at most KGIITII by (the conceptual form of) Grothendieck's inequality. This gives (33) when X - ~ , Y - g~n and hence also when X = C ( K ) , Y = L1 (#). Suppose now that X and Y are general Banach lattices and T :X --+ Y has norm one. Let x l . . . . . Xn be in X with II( ~ i =n 1 Ixil 2) 1/2 I I - 1 and set u -- (y~in__l Ixil2) 1/2 Since we are interested only in estimating II(Y~i~I ITxi 12) 1/2ll, we can assume by replacing X by the (separable) sublattice generated by x l . . . . . Xn that X is separable. The space T X is then separable, so we can similarly assume that Y is separable. Given any ~ > 0, as seen in Section 5 there is a strictly positive functional y* E Y* with 1/2

y*

ITxil 2 i=1

i:,

and IlY*II ~ 1 -q- ~. As mentioned in Section 5, the space Xu is lattice isometric to a C ( K ) space and Yy, is lattice isometric to an L l(#) space. The natural lattice homomorphisms J1 : Xu --+ X, J2 : Y --+ Yy* satisfy IIJ1 II = Ilu II = 1 and IIJ2 II = IlY*II ~< 1 + ~. We can then apply (33) to the operator J2 T J1 to obtain that 1/2

Irxil 2 i=1

Yv*

Since

ITxil 2 i=1

~< (1 +~)KGIITII.

<~ KGIIJ2TJmll

),j2 Yy:r

X u

1/2

ITxil 2

=y* i=1

this completes the proof. We give now one of the many applications of the preceding results to the geometry of Banach spaces. Every space X which is isomorphic to a quotient space o f an E.~ space and is as well a subspace o f an E.1 space must be isomorphic to a Hilbert space. Indeed, if we compose the map from an 12~ space into X with the embedding of X into an 121 space we get an operator from an 12~ space to an E1 space which is thus 2-summing. Like any 2-summing operator this operator factors through a Hilbert space. It follows that X is a quotient space of a Hilbert space and thus isomorphic to a Hilbert space. A quantitative inspection of the argument above shows that if X is a quotient of a C ( K ) space and a subspace of an L1 space then d ( X , ~2) ~ K6 (provided X is separable and

Basic concepts in the geometry of Banach spaces

71

infinite dimensional). Note that ~2 is isometric to a subspace of L1 (0, 1) (see Section 4) and thus ~ -- ~2 is also a quotient space of C(0, 1) because ~2 is reflexive and C(0, 1), considered as a subspace of L ~ (0, 1) -- L 1(0, 1)*, determines the norm of L1 (0, 1). (If Y is a reflexive subspace of a Banach space X and Z is a norm closed subspace of X* which determines the norm of X, then the restriction operator z* ~ z~y is a quotient mapping from Z onto Y*.) We will next give a glimpse into the connections of p-summing operators to the theory of finite dimensional spaces discussed in Section 8. If X is the space g~ then 7f2 ( I x ) = since this is the Hilbert-Schmidt norm of the identity in g~. It is quite surprising that: For any X with dim X = n, re2 (Ix) = ~/-ff. To see this take any norm one operator V from ~2 into X. By projecting onto the orthogonal complement of the kernel of V, we can factor V as V1P where V I " ~ n -+ X, P ' s --+ ~n, II P II - 1, IIV1 II - II V II, and m ~< n. Then

yr2(Ix g) ~ IIIxll IIV~ll~2(IeT)llPII ~ v/-n, Taking the supremum over all such V gives Yr2( I x ) ~ ~/-n. To prove the other inequality, we get from the Pietsch factorization theorem a probability measure # on B x . and T : L 2 ( # ) --+ X so that TI21x = Ix and IITII = 7rz(Ix), where as usual I 2 : C ( B x . ) --+ L2(#) is the formal identity and X is canonically embedded into C ( B x . ) . The space X2 := I2X is an n-dimensional Hilbert space and IzT is the identity on X2. Hence --- 7F2(Ix2) ~

7r2(I2)llTII

= 7r2(Ix).

As a consequence of the preceding we deduce: The projection constant ofan n-dimensional space X is at most v/if; that is, whenever Y contains X there is a projection P from Y onto X with I[P [[ ~< v/-~. Indeed, since fez(Ix) = V/if, the Pietsch factorization theorem yields that the identity operator on X can be represented as X J > Loo(/z) 12> L2(#) v > X, with J an isometry, 12 the formal identity, and II vii- v/if, Since L ~ ( # ) is 1-injective the operator J can be extended to a norm one operator T from Y into L ~ ( # ) . The operator P :-- V12 T is a projection from Y onto X with norm at most v/ft. Notice also that the factorization used above gives another proof of the result proved in Section 8 that d ( X , ~ ) ~ x/~. The estimate of x/~ for the projection constant of an n-dimensional space is essentially sharp. This is discussed in [33]. The p-integral operators form a class of operators which are closely related to the psumming operators. An operator T :X --~ Y is said to be p-integral, 1 ~ p ~ cx) (in symbols T E Z r ( X , Y)), provided that the composition J r T of T with the canonical embedding Jy :Y --~ Y** factors through the formal identity I~,p : L ~ ( # ) ~ L n ( # ) for some probability measure #: I~, p

L~(#)

AI X

T

> Lp(#)

JY >Y

~B > Y**

(34)

72

W.B. Johnson and J. Lindenstrauss

The p-integral norm ip(T) is then defined to be the infimum over all such factorizations of [[A [[ [[B [[. By taking ultraproducts one sees that this infimum is really a minimum. The space (Zp(X, Y), ip) is easily seen to be a Banach space and ip satisfies the ideal property i p ( S T U ) ~ []S[[ip(T)[[U[[. If T is in Zp(X, Y) and X is a subspace of C ( K ) , with K a compact Hausdorff space, then there is a probability measure v on K and an operator S : L p ( v ) ---+ Y** with [[S[[ ---ip(T) which makes the following diagram commute:

Ip

C(K)

> Lp(v) (35)

X

T

>Y

Jr

> Y**

Indeed, if (34) holds, A can be extended to an operator A" C(K) --+ L ~ ( # ) because L ~ ( # ) is 1-injective. By the Pietsch factorization theorem there is a probability measure v on K so that for each x E C (K),

[[BI~,pAx[] ~ 7rp(Bl~,pA)

[x[ p dv

and the desired conclusion follows from

7rp(BIc~,pA) ~ IIBllTrp(l~,p)llS, II =

IIBlllIAII.

It is evident that zrp(T) ~ ip(T) and from the Pietsch factorization theorem it follows that Zcp(T) = ip(T) if the domain of T is a C ( K ) space. As was mentioned implicitly in the discussion of 2-summing operators, 7r2(T) -- i2(T) for any operator. One reason for defining p-integral via a factorization of Jy T rather than T is that this forces i p(T) -ip(T**) (use the fact that a dual space is norm one complemented in its bidual). The 1-injectivity of C(K)** gives that a p-summing operator T into a C ( K ) space is p-integral with ip(T) = 7rp(T). The equality il (T*) = il (T) follows from the observation that the adjoint I~ of 11 "C(K) --+ Ll(v) is l ~ , l ' L ~ ( v ) --+ Ll(v) followed by the identification of L1 (v) with the norm one complemented subspace of C(K)* consisting of the finite signed measures which are absolutely continuous with respect to v. For other values of p, the adjoint of a p-integral operator need not be strictly singular (see [9, 5.12]) and hence need not be q-summing for any q < c~. For each 1 ~< p ~< cx~, p -r 2, there exist p-summing operators which are not p-integral (see [9, 5.13]). The case p -- 1 is particularly easy to deduce from the theory we have presented. We saw that every operator T :~1 --+ ~2 is 1-summing. If T :~1 --+/~2 is 1-integral, A

I~, 1

B

then T has a factorization ~1 > L ~ (#) > L1 (/z) > ~2. But then B and also I~, 1 are 1-summing, hence B I~, l, whence also B I~, 1A = T, are compact (use the fact that L 1(/z) has the DP property). For p = c~ and Y reflexive the infinity integral operators from X to Y are exactly those which factor through some L ~ ( # ) space (with the integral norm equal to the best factorization). In particular for X reflexive the infinity integral norm of the identity of X is

Basic concepts in the geometry of Banach spaces

73

finite if and only if X is finite dimensional (and is equal to the projection constant of X in that case). Thus also for p -- cx~ it is evident that summing and integral norms can be very different. The main reason for introducing p-integral operators is that they are needed for the duality theory of lip(X, Y). For simplicity, we restrict to the case where X and Y are finite dimensional. Following the notation used in Section 8, for finite dimensional X, Y and ot a norm on B(X, Y) we represent the dual of (B(X, Y), or) as (B(Y, X),ot*), where the pairing is given by (S, T) = trace TS ( = t r a c e S T ) . Then for all 1 ~< p ~< cx~, Hp(X, Y)* = Zp,(Y, X) when X and Y are finite dimensional. This just means that for each S E B ( Y , X ) , ip,(S) = sup{traceTS: T E B(X,Y),rcp(T) <<.1}. In view o f t h e discussion in Section 8 that (B(X, Y), II 9II)* = A/'(Y, X) for finite dimensional spaces, we have (since zr~ = II 9 II) that the identity i l (S) = A/'(S) for operators between finite dimensional spaces is equivalent to the assertion that zr~9 - i l . To prove that i l (S) - A/'(S), first note (it is more or less immediate from the definition of A/'(S)) that A/'(S) is the infimum of IIA II 11,6 II IIn II over all factorizations of S of the form

>e 1 AI Y

S

(36)

~B ~X

with A a diagonal operator. From this it follows easily that il (S) ~< A/'(S). For the reverse inequality, given a factorization

L ~ (#)

> L1 (#)

A'I

S

(37)

~B,

Y

>X

one has for E > 0 an operator A~ :Y --~ L ~ ( # ) with [[A1 - Aell < ~ so that A~Y is contained in the simple functions. This gives a factorization of B11~,l A~ of the form (36) with IIBIIIIAIIIIAII ~< IIB~ II III~,~ IIIIA~ II. Setting N := dim Y, we get A[(S - BII~,IAE) <<. Nil S - B11~, 1A~ II ~< NIIA 1 - Ae II < N~. Letting e ~ 0, we conclude that A/'(S) ~< il (S). Having seen that i l (S) = H ( S ) for operators between finite dimensional spaces, one :r can easily deduce the identity rrp -- l p, for 1 < p < c~ from the composition inequality il(ST) <<.Zcp(T)ip,(S). In fact, a more general inequality is true for operators between general spaces: If T ~ Zp(X, Y) and S ~ 17q(Y, Z) then ST ~ Zr(X, Z) and 1 1/x (71 + q)" This inequality follows from the composition inequality for p - s u m m i n g operators mentioned in Section 10. Indeed, consider the following commutative diagram:

ir(ST) ~ ip(T)rcq(S), where •

Ip

C(K)

AI X

r

-

> Lp(#)

JY >Y

-

+B > Y**

S**

> Z**

W.B. Johnson and J. Lindenstrauss

74

The operator S** B Ip satisfies

ir(S**BIp) -- 7cr(S**BIp) <~zrp(Blp)zcq(S**) ~ IlBllzrp(Ip)zrq(S) = IIBIl~rq(S) and

i r ( J z S T ) -- ir(S**JyT) -- ir(S**BIpA) <~ IIBIIzrq(S)llAII. The desired composition inequality follows by taking the infimum over p-integral factorizations of T. A finite dimensional Banach space X is said to have enough symmetries provided that the only operators on X which commute with each isometry on X are the scalar multiples of the identity operator. Finite dimensional spaces with a 1-symmetric basis are examples of spaces with enough symmetries. On the other hand, B(g~, g~) has enough symmetries but, as noted in Section 9, does not even have a "good" unconditional basis. In general, ideal norms for the operators on a space with enough symmetries are better behaved than on general finite dimensional spaces. For example, if X has enough symmetries, the operator one obtains from applying Lewis' lemma to an ideal norm on B ( X , X) is a scalar multiple of Ix. This is a consequence of the following: If dim X = n and X has enough symmetries and ~ is an ideal norm, then ot(Ix)ot*(Ix) = n. The inequality ~(Ix)c~*(Ix) >~ trace Ix = n is clear. To see the reverse inequality, take T E B ( X , X) so that trace T = n and ~ ( I x ) ~ * ( T ) = n. Let G be the (compact) group of isometries of X and/z normalized Haar measure on G. For each S in G, trace S - 1T S = trace T = n and, by the ideal property of c~*, or* (S - 1 T S) = or* (T). The operator

To "= fG S - I T S d l z ( S ) satisfies trace To = trace T, ot*(T0)~< ot*(T), and T commutes with all elements of G. Hence To = )~Ix and since trace T o - n we conclude that )~-- 1. It follows that a ( I x ) a * ( I x ) <<,n and thus we have equality. A particular case of this last result is worth singling out: If dim X = n and X has enough symmetries then the projection constant of X is equal to n/zrl (Ix).

11. Interpolation In this section we give a very sketchy introduction to modem interpolation theory and mention a few applications to Banach space theory. The basic theory is exposed in the books [4] as well as [13] and [5]. The article [31] gives an overview of the subject and further connections to Banach space theory. We begin by describing the abstract framework of interpolation theory. A Banach couple is a pair X = (X0, X1) of Banach spaces which are contained in a Hausdorff topological vector space Z such that both injections Ji : Xi --+ Z, i - 0, 1, are continuous. The ambient

Basic concepts in the geometry of Banach spaces

75

space Z can be replaced with the algebraic sum Z ( X ) "-- Xo + X1 C Z topologized with the (complete) norm

Ilxll - inf[ Ilxollto] -+- Ilxl Ill1]" x - xo -4- Xl, xo E Xo, xl ~ x l }, where I1" I1[i1 is the norm of X i . We always take for Z the space Z ( X ) . The space A ( X ) "-Xo N X1 also plays a role in the theory; it is naturally normed by the (complete) norm IIx II - IIx Ilto] v IIx lit 1]. Thus we have for i -- 0, 1 inclusions

Z(X) with Ilni II ~ 1, IIJi II ~ 1. Any Banach space X satisfying A ( X ) C X C Z ( X ) with both inclusions continuous is called an intermediate space between Xo and X1 (or an intermediate space with respect to X). The spaces Xo, X1, A ( X ) , and Z ( X ) are all intermediate spaces with respect to X. Given Banach couples X -- (Xo, X1), Y -- (Yo, Y1 ), and a linear mapping T" Z (X) ----> Z ( Y ) , we write T E B ( X , Y) provided Tix i E B(Xi, Yi) for i - - 0 , 1. If X and Y are intermediate spaces with respect to X and Y, respectively, we say that X and Y are an interpolation pair for X and Y provided that if T E B ( X , Y), then TIx B(X, Y). If always IIT IIx, Y ~ IIT IIgo, Yo v IIT IIx,, Y, X and Y are said to be an exact interpolation pair for X and Y, where IlSlJx, Y :-][Slxl]B(X,y ) (if S X r Y then I]Sllx, Y "- oc). Finally, if 0 ~< 0 ~< 1 is such that the inequality 1-0

0

IIT IIx, Y ~ cII TIIxo,Yo. IIT IIx,,y, always holds, then the interpolation pair X and Y are said to be of exponent 0 (and exact of exponent 0 if C -- 1). W h e n X -- Y and X -- Y, we abbreviate by saying, e.g., that X is an interpolation space with respect to X provided X and X are an interpolation pair with respect to X and X. The most classical realization of this abstract set-up occurs in the scale of L p ( # ) spaces. For 1 ~< p0, pl ~< oc, the pair (Lpo(#), Lpl (#)) is a Banach couple, and for p0/x pl ~< p ~< P0 v pl the space Lp(l z) is an intermediate space between Lpo (/z) and Lpj (#). In the language of interpolation theory the Riesz-Thorin interpolation theorem can be stated as follows. Let 1 <. Po, Pl <<.ec, 1 <. qo, ql <<.oc, and 0 < 0 < 1. Define p and q by 1 -= p

1 -0

p0

+

0

pl

,

1 -= q

1 -0

q0

§

0

ql

76

W.B. Johnson and J. Lindenstrauss

Then for all measures tx and v, (complex) Lp(Ix) and Lq(v) are an exact interpolation pair of exponent 0 with respect to (Lpo(ix), Lpl (Ix)) and (Lqo(V), Lq, (v)). It is formal to derive from the Riesz-Thorin theorem that its statement remains true in the setting of real scalars as long as P0 ~< pl and q0 ~< ql (or, what is the same, if pl ~ p0 and ql <~ q0). In other cases the word "exact" must be deleted from the statement of the Riesz-Thorin theorem when the scalars are real. The modern approach to the Riesz-Thorin theorem proceeds via a construction called the complex method. Let X = (X0, X1) be a Banach couple of complex Banach spaces and let 7-/(X) denote the space of functions f from the strip S "= {z 6 C: 0 ~< Re z ~< 1 } into 27 (X) which are bounded, continuous, and analytic in the interior So of S, and so that for j = 0, 1 the restriction of f to the line j + iN is a bounded, continuous function into X[j]. The space 7-/(X) is normed by IIf 1IT-t(N) -

sup[I f(it)[1 t~IR

[o1

v Ilf(1 § it) lit,

The norm II 9 I1~(~) is a complete norm on 7-/(X) [4, 4.1.1]. Indeed, by the maximum modulus principle, for z 6 S

[I

supl[f(it)[lz(~)v t6R

[If(l

IlfllT-t(y).

Using the completeness of 27(X), it is easy to check that if ~n~=l Ilfn II~(~) < ~

then

l fn converges in 7-/(X) For 0 < 0 < 1 define X[ol (or simply X[0]) to be all x in S ( X ) for which x = f(O) for some f 6 7-/(X) with the norm

Ilxll[0] " - inf{ Ilfll~(~)"

f (O) -- x} < ec. m

The space (X[0], II" lit0]) is in a natural way isometric to the quotient 7-[(X)/No(X), where No(X) "= { f 6 ~ ( X ) : f(O) = 0 } is the kernel ofthe operator 7-[(X) --+ ~ ( X ) defined by f w-~ f(O), which as noted above has norm at most one. Consequently X[o] C 27(X) with the inclusion of norm ~ 1. To see that A (X) C X[o] with inclusion of norm ~< 1, take any x in A(X) and define f 6 7-/(X) by f ( z ) - x. Then f(O) - x and for t 6 R we have I

IIf(it)ll[o]- Ilxll[o],

IIf(1 + i t ) l l t m ] - Ilxll[1],

so that

Ilxil[e] ~ Ilfll~(~) ~ Ilxil[o] v lixli[1]- IlxllA(~). Thus X[o] is an intermediate space with respect to X. Moreover, if X and Y are Banach couples and 0 < 0 < 1, then X[o] and I1[o] are an exact interpolation pair of exponent

77

Basic concepts in the geometry o f Banach spaces m

0 with respect to X and Y. To see this, let T 9 B(X, Y), x 9 X[ol, e > 0, and choose f 9 7-/(X) so that f(O) -- x with IlfllT-t(~) ~< Ilxll[0] + e. Define g ' S --+ S,(Y) by g(z)--

IlZll%olyollrll-=X1, Y1 T f ( z )

--

0-1

-0

Then g is in 7-/(Y), IlgllTt(F) ~< IIf llT-t(N), and g(O) -IlTllxo,vollTll x,, Y, Tx so that 1-0

IITxll[ol ~ IITIIxo,YollTIl~ 1-0

<~ IITIIxo,YollTIl~

Ilgll~(y) (llxll[o] + e).

In order to derive the Riesz-Thorin theorem from this last result, it suffices to check that (Lpo (/z), Lpl (/z))[0] can be identified with Lp(/z), where ! -- 1-0 + 0 First suppose that t, m P0 P-I-" x 9 Lp(/z) with IlXllp - 1. Define f 9 7-/:= 7-[(Lpo (/z), Lpj (/3.)) by

f(z)-lxl

pIp(z)

where

1 ._ (l-z) p(z) Po

4- Z and 0 " - 0. x pl 0 Ixl'

Evidently Ilfll~

~ 1 and f ( O ) - x, so x 9 (Lpo(/z) , Lpl (/Z))[0] and Ilxll[0] ~ 1 - Ilxllp. Next suppose that x 9 (Lpo(/z), Lpi (/z))[0], Ilxll[0] < 1, and f 9 7-[ satisfies IlfllT-t < l, f (0) - x. To see that IIx IIp ~< 1 it is sufficient to check that if y is a/z-measurable function with/z[y :/= 0] < oc and Ilyllp* ~< 1, then I(x, Y)I "- f Ixyl d/z ~< 1. For z 9 S set 8tJ"rZ' - - I I'Y'P*/P*(Z) Y lyl

where

1 ._ 1 - z t z p*(z) Po Pl

and define h(z) - ( f ( z ) , g(z)). Then since IlfllT~ < 1, for t 9 R we have Ih(it)l v Ih(1 + it)l ~< 1. By the maximum modulus principle we conclude that I(x, Y ) I - ]h(0)l ~< 1 and hence I(x, Y)I ~< 1. Of the many classical applications of the Riesz-Thorin theorem we mention only one, the Hausdorff-Young theorem: Let .U be the Fourier transform for functions on R n, defined by .U f (s) - fR,, f (t)e-i(t's) dt. Then ~ maps L p ( R n) into L p , ( N n) for 1 ~< p ~< 2. To prove the Hausdorff-Young theorem, just observe that IlgClILj,L~ -- 1 and IIgClIL2,L2 -27r n/2, so that for 1 ~< p ~< 2, IlU]lL/,,Lp, <. 22/P*rc n/p* by the Riesz-Thorin theorem. There are also useful vector-valued versions of the Riesz-Thorin theorem. These can be deducedfrom the following. Let (Xo, XI) beaBanachcouple, 1 <~Po, Pl < oc, 0 < 0 < 1, and set • := 1-0 4- 0__. Then P

Po

P~

(Lpo (#, Xo), Lp, (#, X1))[o I -- L p ( # , (Xo, X1)[o]).

If Po < oc, then also (Lpo(/Z, Xo), L ~

X1))[ol -- Lp(/Z, XO, XI)[O],

where L~ X) is the closure in Lec(/z, X) of the simple functions. The proof of this theorem (see [4, 5.1.2]) is only a bit more complicated than the argument given above that

W.B. Johnson and J. Lindenstrauss

78

(Lpo (/Z), Lpl ( / z ) ) [ 0 ] --- Lp(/Z). The argument does use the following general fact about the complex method. For any Banach couple X and 0 < 0 < 1, A(X) is dense in X[o] (see [4, 4.2.2]). 1-0 _z- ~-~-, 0 has a generalization to Banach The identification Lp - (Lpo, Lpl)[O] , ~1 = -~-~-lattices. Assume that X0 and X1 are complex Banach lattices of/z-measurable functions which are ideals in the space of all/z-measurable functions, 0 < 0 < 1, and set S0

1 - 0

XOl ._ {(sign xoxl)lxoll-~

If Xo is order continuous then (Xo, Xl)[O]

~ xo ~ S0, Xl E Xl} .

=

.

X~ -0 XO1 and

1-0 Ilxl II~l] 9 Ixl-Ix011-~ Ix1 I~ }. Ilxll[0] -inf{llx011[0]

(38)

For a proof see Section iv.l.11 in [13]. The inequality <~ in (38); namely,

[llxoll-~176

1--0 Ilxl II~l], <. IIx011t01

does not require order continuity of X0. It reduces to the lattice H61der inequality (4) when XO -- X1.

Since X~ - ~ (#)o = XI-O = X o(~-o) (the l_--~-convexification of X0), it follows that the p-convexification, 1 < p < cx~, of an order continuous Banach lattice X0 can be obtained by using the complex method to interpolate X0 with an appropriate abstract M-space (here there is no problem in passing to the complexification of a real Banach lattice and returning at the end to the real setting). We turn now to a discussion of another interpolation method, which is a special case of what is usually called the 1C-method. Given a Banach couple X = (Xo, X1) and 0 < a, b < cx~, let/C(., a, b) be the norm of the algebraic sum of (X0, all" lit01) and (Xj, bll" Iltl~); that is,

~C(x,a,b)-inf{allxoll[o] +bllxllltl]" x - - x o + x l , x o E X0; Xl ~ Xl}.

(39)

Here we follow the notation of [15, 2.g] because there applications of the/C-method to Banach space theory are described. Usually in interpolation theory one uses only/C(-, t) "-/C(., 1, t); in this notation/C(., a, b) is just alC(., b/a). Let X - (X0, X1) be a Banach couple. We mention a method for using the/C-functional to build interpolation spaces with respect to X. Let E be a space with a normalized 1-unconditional basis {en}~__l and let a - {an}~:l, b - {bn}~--1, be sequences of positive scalars such that ~n~=l an /x bn < c~. The space/C(X, E, a, b) is defined to be all x 6 Z:(X) for which )--~n=l/C(x, an, bn)en converges in E, normed by O0

IIx IIK:~,a,b) -

(4o)

E 1C(x, an, bn)en n----1

E m

It is simple to check that if Y is another Banach couple, then /C(X, E , a , b) and /C(Y, E, a, b) are an exact interpolation pair with respect to X and Y.

Basic concepts in the geometry of Banach spaces

79

Earlier we saw how the complex method can be used to construct the p-convexification of a Banach lattice. The /C-method can also be used to construct Banach spaces with interesting properties. Suppose, for example, that both X0 and X t have a normalized 1-unconditional basis, each of which we identify with the unit vectors {en}n~=l so that X0 and X1 are both contained in co and thus X - (X0, X1) is a Banach couple. Let E be another space with a normalized 1-unconditional basis, also denoted by {en}n~ Since /C(X, E, a, b) is an exact interpolation space with respect to X, it follows that {en}n~__l is a 1-unconditional basis for/C(X, E, a, b) which moreover is even 1-symmetric if {en}~__l is 1-symmetric in both X0 and in X1 (one need only check that the span of {en }n~ is dense in /C(X, E, a, b)). Assume now that {en}ncC__lis 1-symmetric in both X0 and X1, X0 C X1, and Y~k-1 ek [l[0] II Ek-1 ekll[l] --+ 0 as n --+ oc. It turns out (see [14, 3.b.4]) that if an and the weight sequence b satisfies bn 1' ec sufficiently quickly, then there are disjoint finite subsets Cn of N so that the mapping en ~ II lc~ IlK;(N,a,b)lcn extends to an isomorphism from E onto a complemented subspace of/C(X, E, a, b). This is how one proves the result mentioned in Section 3 that a space with an unconditional basis is isomorphic to a complemented subspace of a space which has a symmetric basis. "Good" properties (such as reflexivity and superreflexivity) possessed by the Banach spaces X0 and X1 (often possessed by just one of the spaces) generally pass to interpolation spaces between X0 and X l which are obtained by the complex method or by the E-method (at least when the weight sequences a, b satisfy some growth conditions and the space E is "nice"). Sometimes interpolation spaces even have a good property which neither X0 nor X1 possesses. For example, suppose that X0 C X1 (this can be relaxed but is good enough for applications). If E is reflexive, the inclusion J" Xo ~ X1 is weakly compact, O0 }~,n=l an < oc, and bn "~ oc, t h e n / C ( X , E, a, b) is reflexive. For a proof when E - ~2, which is easily modified to cover the general case, see [ 15, 2.g. 11 ]. A consequence of this result is the following factorization theorem for weakly compact operators. If T" X ~ X1, is weakly compact then T factors through a reflexive space; that is, there exists a reflexive Banach space Y and operators A ~ B ( X , Y), B ~ B(Y, X1) so that T -- B A . To derive this factorization theorem from the interpolation result mentioned above, it suffices to take Y - / C ( X 0 X1 ~2 { 2 - n }n= ec 1 {2n }oc 1) where X0 is the span in X of T Bx with T Bx as the unit ball of X0. Let A be T, considered as an operator into Y, and let B be the formal inclusion from Y into X l. Then A and B are operators and T -- BA. We turn now to the realization of the /C-method which is used most often in analysis and is discussed extensively in the books on interpolation theory we have mentioned. Given a Banach couple X - (Xo, X1), 0 < 0 < 1, and 1 ~< p < ec, Xo,p denotes the space /C(X, ~p, a, b) where a2n "- e On, bzn "- e -(1-0)n, azn+l " - e -On, b2n+l " - e (1-0)n. Instead of using [I 9II/c(N,a,b~ on Xo,p, it is customary to use

Ilxll0,p - - i n f

[le~

v

IIe -

-~

where the infimum is over all x0(t), Xl (t) for which e ~ xo(t) E L p ( R , X0), e-(1-~ (t) c L p ( R , X1), and x - xo(t) + xl (t) for every t in R. The expression II 9I]0,p is a norm on Xo,p which is equivalent to II 9II~c~,a,b> but is better behaved. For example, (Xo,p, I1" II0,p)

W.B. Johnson and J. Lindenstrauss

80

is uniformly convex (rather than just superreflexive) if either X0 or X 1 is uniformly convex and 1 < p < cx~ [15, 2.g.21]. Moreover, I]" IIO,p is very good for interpolation purposes, for if Y is another Banach couple then (Xo,p, ]]. ]]0,p) and (Yo,p, ][" ]]0,p) are an exact interpolation pair of exponent 0 with respect to X and Y. It is of course important to identify Xo,p when X is a concrete Banach couple. The spaces that arise in this connection when X is a couple of L p ( # ) spaces are the L p,q spaces. Given 0 < p < ~ , 0 < q < cx~, a measure #, and a Banach space X, Lp,q (#, X) is the space of X valued strongly measurable functions x for which I]Xllpq "--

( f0 q/p

[tl/px.(t)]q dt t

)

1/q < o~,

(42)

where x*(t) is the decreasing rearrangement of IIx(t)llx. For 0 0

W h e n X is the scalar field we write Lp,q(#). E v i d e n t l y Lp,p(#, X) = Lp(#, X). Note that for p > q ~> 1, the space Lp,q(#) is the Lorentz function space Lw, q(#) defined in Section 5 with W(t) - - q t q/p-I and hence is a Banach space. W h e n q > p ~> 1 the expression II 9I]p,q does not satisfy the triangle inequality, although II 9Ilp,q is equivalent to a norm when p > 1. In any case Lp,q (#, X) is a metrizable topological vector space. The main result about spaces obtained from L p and L p,q spaces via t h e / C - m e t h o d is the following (see [4, 5.3.1]): Let 0 < Po, Pl, qo, ql <~ ~ , 0 < 0 < 1, and define p by 1 / p = (1 - O ) / p o + O / p l . If po =/=pl, o r i f po = pl and 1/q = (1 - O ) / q o + O / q l , then

(Lpo,qo(~, X), Lpl,q 1(#, X))O, q -- Lp,q(#, X)

(44)

up to equivalence of the "norms"; i.e., for some constant C > 0, C - 1 ] ] 9 ]]p,q ~ ]] 9 ][O,q C II 9 II p,q. W h e n the expressions II 9 II Pi ,qi' i -- 0, 1, are equivalent to norms one can deduce from (44) and earlier comments an interpolation theorem for operators. In fact, there are cases where interpolation is valid even when the spaces are not all Banach spaces. In particular, assume that T" Lpi,r i (l~, X) -+ Lqi,s i (19, X ) is continuous for i -- 0, 1, with P0 ~: Pl, q0 ~ ql, 0 < 0 < 1, and define p, q by 1 / p - (1 - O ) / p o + O/pl, 1/q - (1 - O ) / q o -O/ql. If p ~< q, then T ' L p ( l Z , X) ~ Lq(19, Y) is continuous and for 0 <~ r ~< s ~< c~, T "Lp,r(lZ, X) ~ Lq,s(19, Y) is continuous (see [4, 5.3.1]). In some cases it is possible to characterize the interpolation spaces with respect to a concrete Banach couple. The Banach couple X - (X0, X1) is called a Calder6n couple if whenever x, y in E (X) satisfy/C(y, 1, t) <~/C(x, 1, t) for all t, then there is T in B ( X , X) so that T x -- y. If always T can be chosen so that [[Tllxo,Xo v [[Tl[xl,x~ ~< 1, then X is called an exact Calder6n couple. It is easy to see that if X is an intermediate space with respect to a Calder6n couple X, then X is an interpolation space with respect to X if and only if it is E-monotone for X, which means that if x E X, y E A ( X ) , and /C(y, 1, t) <~/C(x, 1, t) for all t, then y E X. It then follows from the E-divisibility theorem

Basic concepts in the geometry of Banach spaces

81

that X is obtainable from X via the/C-method. This theorem says the following. If X is ~Cmonotone for X, then there are E, a, and b so that X = / C ( X , E, a, b), up to an equivalent renorming. The pair (Lp(O, 1), Lq(O, 1)), 1 ~< p < q ~< ~ , is a Calder6n couple, and many other examples are known (see [31]). One interesting problem which is not completely solved is to determine a necessary and sufficient condition for a Banach couple of symmetric lattice ideals on [0, 1] (see Section 5) to be a Calder6n couple. Much is known about the interpolation spaces for such a couple. For example, an exact interpolation space with respect to such a couple must itself be a symmetric lattice ideal (the first step is to observe that if r ' [ 0 , 1] --+ [0, 1] is a measure preserving automorphism, then x w-> x ( r ) defines an isometric automorphism of any symmetric lattice on [0, 1]). When the couple is (L1 (0, 1), L ~ ( 0 , 1)), it is reasonable to guess that the converse is true. It is not, but: If L~(O, 1) is dense in the symmetric lattice ideal X, then X is an exact interpolation space with respect to (Ll (0, 1), L ~ ( 0 , 1)) [13, Theorem 4.10]. Most natural symmetric lattice ideals on [0, 1], including all the separable ones, satisfy the hypothesis of this theorem.

12. List of symbols Here is a list of symbols used in this introductory article and, where appropriate, a reference to where they are defined. N R C T IP E

Ap, Bp KG C(K;X)

C(K) L p (l,t, X)

Lp(#) L~(#,X) Lp(O, 1)

Cp(~r) ep(r)

The natural numbers. The real numbers. The complex numbers. The complement of the set S. The unit circle in the complex plane. A probability measure (Section 2). f . dip (Section 2). The constants in Khintchine's inequality (1); also the constants in the Kahane-Khintchine inequality (24). The constant in Grothendieck's inequality (Section 10). Continuous functions f on the (usually) compact Hausdorff space K taking values in the (usually) normed space X, normed by ]lfll = suPtcK IIf ( t ) II. C (K; X) when X is the scalar field. The #-measurable X-valued functions f for which Ilfllp := ( f IIf[I F d#) 1/p < ~ (Section 7). Here 0 < p < ec. L p (#, X) when X = R. The #-measurable essentially bounded X-valued functions, with norm 11f II ~ L p (#) Lp(#) Lp(#)

:= inf~A=O sup Ifl A I" when # is Lebesgue measure on the unit interval. when # is normalized Lebesgue measure on the unit circle. when g is counting measure on the set F.

W.B. Johnson and J. Lindenstrauss

82

~.p ~p17

g~p(F) when F = N. g~p(F) when F -- {1, 2 . . . . . n}.

C

The subspace of s of scalar sequences which have a limit. The closure in s (F) of the scalar sequences which have finite support. co (F) when F -- N. The /z-measurable X-valued functions f for which ][f][p,q "= (q f o ][tl/pf *(t)][q ~)l/q < co(Section 11). Here 0 < p,q < co and

co(F) CO L p,q

(/z, X )

L p, oc (/z , X)

L p,q (/z )

Bx Bx(x,r) s-L S_L

d(X, Y) X ,~ Y ~x(.) px(.) Xu

Xu,

Ix Jx Ip Ip,q

B(X,Y) K(X, Y) WK(X, Y) SS(X, Y) Fr(X, Y) .N'(X, Y) A/'(T) M(P)(T) M(p)(T) Tp(X) Cp(X)

f * is the decreasing rearrangement of []fllx. The /z-measurable X-valued functions f for which [1f ]]p,oc "= supt>otl/Pl[f*(t)l[x < co (Section 11). Here 0 < p ~< co and f * is the decreasing rearrangement of 11f ]Ix. Lp,q (/z, X) when X -- ]R. The closed unit ball of the Banach space X. The closed ball of radius r with center x in the Banach space X; denoted also B(x, r) when X is understood. All linear functionals which vanish on S (when S is a subset of a Banach space). The intersection of the kernels of all linear functionals in S (when S is a subset of the dual of a Banach space). The Banach-Mazur distance from X to Y (Section 2). The space X is isomorphic to the space Y. The modulus of convexity of the space X (Section 6). The modulus of smoothness of the space X (Section 6). When X is a lattice and u ~> 0, the abstract M-space which has the order interval [ - u , u] as the unit ball (Section 5). When X is a lattice and u* ~> 0 in X*, the abstract L 1-space which is the completion of X under the seminorm [[x ]]u. -- u* ([x[) (Section 5). A Rademacher sequence (Section 4). The identity operator on the space X. The canonical embedding of X into X**. The formal identity operator from C(K) to Lp(/z) (when/z is a finite measure on the compact Hausdorff space K). The formal identity mapping from Lp(/z) to Lq(/z). The bounded operators from X to Y. The compact operators from X to Y. The weakly compact operators from X to Y. The strictly singular operators from X to Y (Section 10). The Fredholm operators from X to Y (Section 10). The nuclear operators from X to Y (Section 8). The nuclear norm of the operator T (Section 8). The p-convexity constant of the operator T (Section 5). The p-concavity constant of the operator T (Section 5). The type p constant of the Banach space X (Section 8). The cotype p constant of the Banach space X (Section 8).

Basic concepts in the geometry of Banach spaces ITp(X, Y) 7rp(T) I n ( X , Y) ip(T)

The The The The

83

p - s u m m i n g o p e r a t o r s f r o m X to Y ( S e c t i o n 10). p - s u m m i n g n o r m o f the o p e r a t o r T ( S e c t i o n 10). p - i n t e g r a l o p e r a t o r s f r o m X to Y ( S e c t i o n 10). p - i n t e g r a l n o r m o f the o p e r a t o r T ( S e c t i o n 10).

References [1] C.D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York, 1985. [2] B. Beauzamy, Introduction to Banach Spaces and Their Geometry, Mathematics Studies 68, North-Holland, Amsterdam (1985). [3] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Coll. Pub. 48, Amer. Math. Soc., Providence, RI (2000). [4] J. Bergh and J. Lrfstrrm, Interpolation Spaces: An Introduction, Grundlehren der Mathematischen Wissenschaften 223, Springer-Verlag, New York (1976). [5] Yu.A. Brudnyi and N.Ya. Kruglyak, Interpolation Functors and Interpolation Spaces I, North-Holland, Amsterdam (1991). [6] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Longman Scientific & Technical, Essex (1993). [7] J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York (1984). [8] J. Diestel and J.J. Uhl, Vector Measures, Mathematical Surveys 15, Amer. Math. Soc., Providence, RI (1977). [9] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Math. 43, Cambridge University Press, Cambridge (1995). [10] R. Durrett, Probability: Theory and Examples, Duxbury Press (1996). [ 11] P. Habala, P. H~ijek and V. Zizler, Introduction to Banach Spaces I, Matfyzpress, Univerzity Karlovy (1996). [ 12] P. Habala, P. Hfijek and V. Zizler, Introduction to Banach Spaces II, Matfyzpress, Univerzity Karlovy (1996). [13] S.G. Krein, Yu.I. Petunin and E.M. Semenov, Interpolation of Linear Operators, Transl. Math. Monogr. 54, Amer. Math. Soc., Providence, RI (1982). [14] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Springer-Verlag, Berlin (1977). [15] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II: Function Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 97, Springer-Verlag, Berlin (1979). [16] V.D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer-Verlag, Berlin (1986). [17] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math. 94, Cambridge University Press, Cambridge (1989). [18] H.L. Royden, Real Analysis, 3rd edn., Macmillan, New York (1988). [19] W. Rudin, Functional Analysis, 2nd edn., McGraw-Hill, New York (1991). [20] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Longman Scientific & Technical, Essex (1989). [21 ] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics 25, Cambridge University Press, Cambridge (1991).

H a n d b o o k articles which are referenced in this introductory article [22] [23] [24] [25] [26]

D. Alspach and E. Odell, L p spaces, This Handbook. D.L. Burkholder, Martingales and singular integrals in Banach spaces, This Handbook. EG. Casazza, Approximation properties, This Handbook. R. Deville and N. Ghoussoub, Perturbed minimization principles and applications, This Handbook. V. Fonf, J. Lindenstrauss and R.R. Phelps, Infinite dimensional convexity, This Handbook.

84

W.B. Johnson and J. Lindenstrauss

[27] T. Gamelin and S.V. Kislyakov, Uniform algebras as Banach spaces, This Handbook. [281 A.A. Giannopoulos and V. Milman, Euclidean structure in finite dimensional normed spaces, This Handbook. [29] G. Godefroy, Renormings of Banach spaces, This Handbook. [30] W.T. Gowers, Ramsey theory methods in Banach spaces, This Handbook. [31] N.J. Kalton and S.J. Montgomery-Smith, Interpolation andfactorization theorems, This Handbook. [32] S.V. Kislyakov, Banach spaces and classical harmonic analysis, This Handbook. [33] A. Koldobsky and H. K6nig, Aspects of the isometric theory of Banach spaces, This Handbook. [34] M. Ledoux and J. Zinn, Probabilistic limit theorems in the setting of Banach spaces, This Handbook. [35] J. Lindenstrauss, Characterizations of Hilbert space, This Handbook. [36] E Mankiewicz and N. Tomczak-Jaegermann, Quotients offinite-dimensional Banach spaces; Random phenomena, This Handbook. [37] B. Maurey, Banach spaces with few operators, This Handbook. [381 B. Maurey, Type, cotype and K-convexity, This Handbook. [39] D. Preiss, Geometric measure theory in Banach spaces, This Handbook. [401 H.E Rosenthal, The Banach spaces C (K), This Handbook. [41] L. Tzafriri, Uniqueness of structure in Banach spaces, This Handbook. [421 P. Wojtaszczyk, Spaces of analytic functions with integral norm, This Handbook. [43] M. Zippin, Extension of bounded linear operators, This Handbook. [441 V. Zizler, Nonseparable Banach spaces, This Handbook.

CHAPTER

2

Positive Operators Y.A. Abramovich Department of Mathematical Sciences, Indiana University-Purdue University, Indianapolis, IN, USA E-mail: yabramovich @math. iupui, edu

C.D. Aliprantis Department of Economics and Department of Mathematics, Purdue University, West Lafayette, IN, USA E-mail: aliprantis @mgmt.purdue, edu

Contents 1. Ordered vector spaces and Banach lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Operators between Banach lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. W h e n is every continuous operator regular? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Domination, compactness, and factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Invariant subspaces of positive operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Compact-friendly operators and invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Integral operators and invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Added in Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H A N D B O O K OF T H E G E O M E T R Y OF B A N A C H SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All fights reserved 85

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The theory of positive operators is a distinguished and significant part of the field of general operator theory. The extra feature of this part of operator theory is the existence of an order on the spaces involved. This ingredient is of tremendous importance and lies in the core of many specific results valid for positive operators. The two main objectives of the present work are to discuss the relationships between general operators and positive operators and to demonstrate the effects the order structure has on general operators acting between Banach lattices. We list here several books devoted primarily to positive operators between ordered spaces: [15,22,29,41,75,78-80,86,87,96,100,123,126,135,144,145]. The size constraints of this article as well as the areas of interest of the authors have inevitably influenced the selection of the material for this survey and have precluded us from mentioning many existing directions within the theory of positive operators. We especially regret having to omit the ergodic and interpolation properties of positive operators. For these topics the reader is referred to [83] and [84] respectively. Since our survey is devoted to operators, we have reduced to a bare minimum the general theory of Banach lattices presented here. Most of the books cited above, the series of notes by Luxemburg and Zaanen [93,92] and the two surveys [39,40] are excellent sources for such a theory as well. Throughout this work the word "operator" is synonymous with "linear operator".

1. Ordered vector spaces and Banach lattices Though we will be dealing mostly with operators on Banach lattices, it is fruitful to start with the general framework of (partially) ordered vector spaces. We will introduce some necessary terminology and prove a few basic results; for a systematic presentation we refer the reader to [70,94,101,107,134,143]. Recall that a real vector space X equipped with a partial order ~> is said to be a (partially) ordered vector space whenever x ~ y imply otx ~> oty for all ol ~> 0 and x + z ~> y + z for all z E X. The set X+ = {x E X: x ~> 0} is called the positive cone of X and its elements are referred to as positive vectors. The cone X+ is said to be generating whenever X -X+ - X+, i.e., whenever every vector can be written as a difference of two positive vectors. An ordered vector space X is said to be Archimedean whenever n x <, y for each n and some x, y c X implies x ~< 0. In an Archimedean space if - e y <~ x <~ ey for all scalars e > 0, then x = 0. As the next lemma shows many ordered topological vector spaces are Archimedean. LEMMA 1. I f an ordered vector space X admits a H a u s d o r f f linear topology r f o r which X+ is closed, then X is Archimedean. PROOF. Assume that n x <, y for all n and some vectors x, y 6 X. Then ~1 y - x ~> 0 and 1

y - x

r

> - x imply - x ~> 0 or x ~< 0.

E]

THEOREM 2 (M. Krein-Smulian). Let X be a Banach space ordered by a closed generating cone. Then there is a constant M > 0 such that f o r each x ~ X there are x l, x2 E X+ satisfying x -- xl - x2 and Ilxi II ~< Mllxll f o r each i.

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PROOF. We present a sketch of the proof. For each n define the set En = {x ~ X" =tXl, X2 G_ X - t- with x - Xl - x2 and [Ixi II ~ n (i = 1, 2) }.

Clearly, each En is convex, symmetric, and 0 6 En. In addition, note that En cc_ Em whenever n ~< m. Since X+ is generating, we see that X - Un~=l En. So, by the Baire Category Theorem, some Ek contains a closed ball B(xo, r) = {x 6 X: Ilx0-xll ~< r}. The properties ofthe sets En imply that B(0, r) c_ E~. Now by imitating the proof of the open mapping theorem, we can show that B(O, r) c_ Ezk holds and the proof is done. The details can be found in [135]; see also [4]. D C O R O L L A R Y 3. Let X be an ordered Banach space whose positive cone is closed and generating. I f xn --+ x holds in X, then there exist Yn, Zn, Y, z ~ X + such that Xn - - Yn -- Zn, x = y -- Z, Yn --+ Y and Zn --+ z. C O R O L L A R Y 4. Let X be a Banach space partially ordered by a closed generating cone, and let Y be a topological vector space. Then an operator T : X --+ Y is continuous if and only if the restriction o f T to X + is continuous.

And now we introduce the central concept of this work. D E F I N I T I O N 5. An operator T : X ~ Y between two ordered vector spaces is said to be positive if T (X+) _ Y+, i.e., if x ~> 0 implies T x >~ O.

It is a remarkable fact that quite often positive operators are automatically continuous. This was first proven by M. Krein for positive linear functionals [81 ] and later was generalized in several contexts by various authors; see, for instance, [32,99,101,121]. The next result, due to Lozanovsky, is the strongest in this direction and appeared in [136]. COROLLARY 6 (Lozanovsky). Let X and Y be two ordered Banach spaces whose cones are closed. I f additionally, the cone o f X is generating, then every positive operator T : X --+ Y is continuous. PROOF. It suffices to show that a positive operator T has a closed graph. So, assume Xn --+ 0 in X and Txn --+ y in Y. By passing to a subsequence, we can also assume oo Y~n=l n ]lXn I] < cx~. By Theorem 2 there exist some M > 0 and two sequences {Yn } and {Zn} of X+ satisfying Xn = Yn - Zn, Ilynl] ~< m[Ixnll, and I[Zn [] ~< mllxnl[ for each n. Since X+ is closed, the vector z = ~-~n~=l n(yn + Zn) of X belongs to X+ and - z <<.nxn <<.z holds

for each n. The positivity of T implies that _in T z <<.Txn <~ n1 TZ. Taking into account that Y+ is also closed and n1 T Z --+ O, we conclude that 0 <~ y ~< 0, that is, y -- 0.

U]

From now on we consider only Archimedean vector lattices. For the definition and basic properties of Banach lattices the reader should consult the article Basic concepts in the geometry o f B a n a c h spaces by W. Johnson and J. Lindenstrauss, starting off this volume [72]. Below we will introduce a few necessary definitions.

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A n o r m I1" II on a vector lattice is said to be a lattice norm if Ixl ~ lyl implies Ilxll ~ Ilyll. A n o r m e d vector lattice is a vector lattice e q u i p p e d with a lattice n o r m . A n o r m c o m p l e t e n o r m e d vector lattice is called a Banach lattice. H e r e is a list of the m o s t i m p o r t a n t properties b e t w e e n the t o p o l o g i c a l and order structures of a B a n a c h lattice that will be e m p l o y e d in this work. T h e first one, the order continuity, is discussed at length in [72]. DEFINITION 7. A lattice n o r m on a vector lattice X is said to be:

(1) (2) (3) (4)

order continuous, if xc~ $ 0 implies ]]x~ 11 $ 0; a Levi n o r m , if 0 ~< x~ 1" and ]lx~ 11 ~< 1 i m p l y that sup~ x~ exists in X; a Fatou n o r m , if 0 ~< xc~ 1" x implies Ilx~ II 1" Ilxll; a weak Fatou n o r m , if there exists s o m e constant K ~> 1 such that

0 ~ xo~ 1" x ------->,IIx II ~ K lim Ilxo~ II. Of A B a n a c h lattice with an order c o n t i n u o u s Levi n o r m is u s u a l l y referred to as a K a n t o r o v i c h - B a n a c h space, or as a KB-space in short. T h e m e a n i n g s of the e x p r e s s i o n s " X is a Levi B a n a c h lattice", or " X has the Levi p r o p e r t y " , or " X has the w e a k Fatou property", etc. should be clear and they will be used t h r o u g h o u t this work. If X has a Levi n o r m , then o b v i o u s l y X is D e d e k i n d c o m p l e t e . N o t e that two equivalent (lattice) n o r m s are either both Levi or both fail to be Levi. T h e same is true for the w e a k Fatou property. It is well k n o w n [30] that if a n o r m is Levi, then necessarily it is also w e a k Fatou. P r o p e r t y (3) is clearly an isometric property. If, in the definition of an order c o n t i n u o u s , or a Levi, or a (weak) F a t o u n o r m one replaces arbitrary nets by sequences, then one arrives at the definition of a sequentially order continuous norm, or a sequentially Levi norm, or respectively, of a sequentially (weak) Fatou norm. 1 Thus, for e x a m p l e , a n o r m on a B a n a c h lattice X is s e q u e n t i a l l y F a t o u (or simply ~ - F a t o u ) if for each s e q u e n c e 0 ~< xn t x we have ]Ix ]1 = lim ]]xn 11. T h e r e is a very rich structural t h e o r y of B a n a c h lattices (reflecting partly the c o r r e s p o n d ing theory of B a n a c h spaces) in w h i c h various properties of B a n a c h lattices are characterized by the a b s e n c e or p r e s e n c e of subspaces or vector sublattices w h i c h are i s o m o r p h i c to s o m e classical spaces ~p (1 ~< p ~< ec) or co. We refer to [29,39,40,96] for these results. As an illustration, we m e n t i o n only the f a m o u s t h e o r e m stating that a B a n a c h lattice X is a K B - s p a c e if and only if X does not contain a vector sublattice order i s o m o r p h i c to co. 1 These properties appear in the literature under many different names. Fremlin [58] was the first to associate the Levi and Fatou names with these properties. Nakano [100, pp. 129-130] used the term monotone complete norm for a sequentially Levi norm and universally monotone complete norm for a Levi norm. Zaanen [144] considers sequentially Levi norms in two places under different names. First on p. 305 he refers to them, like Nakano, as monotone complete norms, and then on p. 421 as norms with the weak Fatou property for monotone sequences. The term weak Fatou property for directed sets (p. 390) is used by Zaanen for what we refer to as a Levi norm. Meyer-Nieberg [96, p. 96], uses the term "monotonically complete norm" for a Levi norm. The Soviet school on Banach lattices used the symbols (A) and (A f) to denote the sequential order continuous and order continuous norms, respectively; they also used the symbols (B) and (B r) to denote the sequential Levi and the Levi properties respectively, and finally they used the symbols (C) and (C f) to denote the sequential Fatou and the Fatou properties respectively.

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A vector subspace J of a vector lattice X is said to be an (order) ideal if Ixl ~ lyl and y 6 J imply that x 6 J. For each 0 <~ u 6 X the principal ideal Xu generated by u is the ideal Xu = {x 6 E: 3 )~ > 0 such that Ix[ ~< )~u}. A positive element u 6 X is called a (strong) unit if Xu = X. Each unit generates the so called unit-norm II" Ilu via IIxllu = inf{)~ >~ 0: Ix[ ~< ~.u}. Clearly this norm is always Fatou; if X is a-Dedekind complete, then I]" Ilu is complete.

2. Operators between Banach lattices If X and Y are Banach spaces, then the symbol s Y) denotes the Banach space of all continuous operators from X to Y; we let s = E ( X , X). 2 If X and Y are in addition Banach lattices, then there are several very important classes of operators associated with the order structure on these spaces. We denote by 12+ (X, Y) the collection of all positive operators. This collection is a cone and it induces a natural (partial) order on the Banach space C ( X , Y). The subspace s Y) - E + ( X , Y) of/2(X, Y) generated by the c o n e / 2 + ( X , Y) is denoted b y / 2 r (X, Y), or simply/2 r, and is referred to as the space of regular operators. In other words, an operator is said to be regular if it can be written as a difference of two positive operators. An operator T" X --+ Y is order bounded if T maps order bounded sets in X to order bounded sets in Y. The symbol/2 b (X, Y) denotes the space of all order bounded operators. Clearly each regular operator is order bounded, that is, s c_ E b. The converse is not true in general (see [29] for several counterexamples, the first of which was found by S. Kaplan). Here we can ask the question: when does the equality s = F b hold? The next famous theorem of E Riesz and L. Kantorovich describes a rather general situation when this happens. THEOREM 8 (Riesz-Kantorovich). Let X, Y be vector lattices with Y Dedekind complete. Then E r (X, Y) = 12b (X, Y), i.e., each order bounded operator T : X --+ Y is regular. Moreover, in this case, Er (X, Y) is itself a Dedekind complete vector lattice whose lattice operations satisfy the Riesz-Kantorovich formulas, according to which f o r each T ~ E r (X, Y) and each x ~ X+ we have T + ( x ) -- sup{Tu" 0 <~ u <<,x}

and

I T l ( x ) - sup{Tu" ]u[ <~ x}.

The operator ITI = T x / ( - T ) (if it exists in E r (X, Y)) is called the modulus of T. There are several interesting questions and directions motivated by the Riesz-Kantorovich theorem. (1) Is there a non-Dedekind complete vector lattice Y such t h a t / U (X, Y) --/2 b (X, Y) for each vector lattice X ? This question was asked by Fremlin and was answered in the affirmative by Abramovich and Geyler in [16]. 2 We use the symbol 12(X, Y) instead of B(X, Y) utilized in [72] to be consistent with the standard notation for many other classes of operators between Banach lattices.

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(2) Is there a non-Dedekind complete vector lattice Y such that s (X, Y) is a vector lattice for each (some) vector lattice X? The answer is yes, and we refer to [18] for such examples and related discussions. (3) Assume that E r (X, Y) is a vector lattice, and so for each T E E r (X, Y) its modulus ITI exists. Is it true that this operator is necessarily given by the Riesz-Kantorovich formula? This is a long standing open problem. (4) Assume that for a fixed vector lattice Y the vector space s (X, Y) is a vector lattice for all vector lattices X in some special class of vector lattices. What impact does this have on the order structure of Y? In connection with (3) note that even when s (X, Y) is not assumed to be a vector lattice, there is no single example in the literature of a regular operator T : X --+ Y for which its modulus ITI exists and cannot be obtained by the Riesz-Kantorovich formula. Abramovich and Wickstead have studied this question rather systematically and described many cases covered by the Riesz-Kantorovich formula [ 18,19,140]. For the special domain of C(K)-vector lattices, this problem was also studied by van Rooij [118,119]. Now let X and Y be Banach lattices. On the space E ( X , Y) we will consider the usual operator norm. Under this norm the subspace s (X, Y) is rarely closed. However, there is a natural norm on s (X, Y), the so-called regular norm, which is closely related to the structure of the regular operators. The regular norm on s (X, Y) is defined by IlTllr - inf{ IISII" s E s

Y) and S ~> i T } .

Clearly IIT II ~ IIT IIr and for positive operators both norms coincide. It is well known that (Z;r (X, Y), II" IIr) is a Banach lattice. If Y is Dedekind complete, then the modulus Irl exists by Theorem 8, and in this case IIr lit = IlITI II.

3. When is every continuous operator regular? As the title of this section indicates, we shall discuss here the pairs of Banach lattices X and Y for which the space of all continuous operators E(X, Y) coincides with the space of all regular operators s (X, Y). We should distinguish here between two closely related but different questions. The first one requires simply the set-theoretic equality IZ(X, Y) = ~,r ( x , Y),

(*)

while the second requires (,) and, additionally, the norm equality IIT II = IIT Ilr

for each T E L;(X, Y).

(**)

We will refer to these problems as the isomorphic and isometric problems, respectively. The following two theorems, which are essentially due to Kantorovich and Vulikh [76], describe the two most important classes of Banach lattices for which each continuous operator is regular. These are the familiar AM- and AL-spaces, also known under the names abstract M spaces and abstract L l spaces, respectively, and defined in [72, Section 5].

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THEOREM 9. Let Y be a Dedekind complete Banach lattice with a strong unit. Then f o r each Banach lattice X equality (,) holds, that is, s Y) = s (X, Y). If the norm on Y is Fatou (in particular, if Y is equipped with a unit norm and, thus, is an AM-space), then (**) also holds. THEOREM 10. If X is an AL-space and Y is an arbitrary Banach lattice with a Levi norm, then equality (,) holds. If the norm on Y is Fatou, then we also have (**). To be precise, it was assumed in [76] that Y in Theorem 10 was a KB-space, and it was noticed in [ 129] that the original proof could be easily carried over to an arbitrary Banach lattice with a Levi norm. For a KB-space Y the proofs can be found in [29, Theorem 15.3], and [ 134, Theorem 8.7.2]. Under the assumption that Y is positively complemented in Y** (which is somewhat stronger than the Levi property) a proof of Theorem 10 is presented in [96, Theorem 1.5.11]. Two principal questions can be asked in connection with the previous theorems. (1) Do Theorems 9 and 10 characterize the AL- and AM-spaces? (2) To what extend is the Levi condition essential f o r the validity o f (,)? The first question is quite old and has various interpretations depending on whether we deal with the isometric or the isomorphic version. We refer to [3] for a brief survey of major results in this direction. Here we mention a few of them, starting with the isometric version of this question. THEOREM 1 1. Suppose that the Banach lattices X and Y satisfy both (,) and (**). Then either X is an AL-space or Y is an AM-space. The isomorphic version is much deeper and has, in general, a negative solution [2]. THEOREM 12. There exist two Dedekind complete Banach lattices X and Y satisfying (,) such that X is not order isomorphic to any AL-space and Y is not order isomorphic to any AM-space. Moreover, f o r any given e > 0 the Dedekind complete Banach lattices X and Y can be constructed in such a way that IIT IIr ~< (1 + e)II T ]lf o r each T ~ 12(X, Y). The strongest affirmative isomorphic result is also due to Cartwright and Lotz [44] and is stated next. A simple proof of this result is given in [ 17], where the technique introduced by Tzafriri in [133] of abstract Rademacher functions is utilized. The works of Fremlin [59], ~rno [ 105] and Schaefer [ 122] contained some versions of the next result. THEOREM 13. Let two Banach lattices X and Y satisfy (,), and assume that either X* or Y contains vector sublattices uniformly in n isomorphic to g np f o r some p E [1 cx~). In the f o r m e r case Y is order isomorphic to an AM-space and in the latter case X is order isomorphic to an AL-space. The case when X = Y is exceptionally nice and was studied in [2]. THEOREM 14. I f a Banach lattice X satisfies s either to an AL- or AM-space.

= s

(X), then X is order isomorphic

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PROOF. To sketch a proof, note that the following alternative is true: X either contains uniformly in n the vector sublattices order isomorphic to g ~ or it does not. In the former case, X* contains uniformly in n the vector sublattices g~ and so, by Theorem 13, X is isomorphic to an AM-space. In the latter case, X is isomorphic to an AL-space by Drno's theorem. [] Questions (1) and (2) formulated above, though quite natural, have been addressed by Abramovich and Wickstead only recently in [21 ]. THEOREM 15. A Dedekind complete Banach lattice Y has a Levi norm if and only if for every AL-space X we have s Y) = s (X, Y). This converse to Theorem 10 is somewhat partial since we assume Y to be Dedekind complete. On the other hand, as the next two theorems in [21 ] show, it is the best one can get. Note also that there are non-Dedekind complete Banach lattices (hence, without a Levi norm) which, nevertheless, satisfy (,). THEOREM 16. For a Banach lattice Y the following statements are equivalent:

(a) Y is Dedekind complete. (b) For all Banach lattices X, the space f_r (X, Y) is a Dedekind complete vector lattice. (c) For all AL-spaces X, the space s (X, Y) is a vector lattice. THEOREM 17. For a Banach lattice Y the following statements are equivalent:

(1) (2) (3) (4)

Y is ~-Dedekind complete. s (X, Y) is ~-Dedekind complete for each separable Banach lattice X. s (c, Y) is a vector lattice, where c is the space of all convergent sequences. /:r (LI [0, 27r], Y) is a vector lattice.

We close this section by mentioning one more direction of research devoted to the connections between the order and topological properties of compact operators. Namely, what can be said about the modulus of a compact operator T :X --+ Y between Banach lattices? Does this modulus exist? When is it compact or weakly compact? A well known example by Krengel [29, Example 16.6] shows that even on ~2 the modulus of a compact operator may fail to exist. We refer to [20] for some history, further references and concrete results concerning the properties of the modulus of a compact operator.

4. Domination, compactness, and factorization DEFINITION 18. Let T, B : X --+ Y be two operators between vector lattices with B positive. We say that the operator T is dominated by the operator B (or that B dominates T) provided ]T(x)l ~< B(Ixl) for each x 6 X.

It should be clear that every operator between Banach lattices dominated by a positive operator is automatically continuous, and that a positive operator T is dominated by another positive operator B if and only if 0 ~< T ~< B. When Y is Dedekind complete, an

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operator T is dominated by a positive operator B if and only if T is regular and ITI ~< B holds (use Theorem 8 to check this). We are now ready to discuss a new and interesting direction dealing with dominated operators on Banach lattices. This direction can be described collectively as the domination problem. THE DOMINATION PROBLEM. Assume that B : X --+ Y is a positive operator between Banach lattices and assume that B satisfies some property ( P ). What effect does this property have on an operator T dominated by B ? Of course, the best one can expect is that T also satisfies (P). Here are a few examples of the properties that have been studied in connection with the domination problem: compactness, weak compactness, Dunford-Pettis, and Radon-Nikodym. A first general domination result was established in 1972 by Abramovich [ 1] for weakly compact operators. He proved that if a linear operator S : X --+ Y from a Banach lattice to a KB-space is dominated by a weakly compact operator, then S is also weakly compact. Several years later Pitt [ 109] proved a domination result for compact operators, namely he s h o w e d t h a t i f B isacompactpositive operatoron L p ( # ) with 1 < p < oo andO <<,T <~ B, then T is also compact. This was an individual result about the properties of positive compact operators on a special Banach function space. At the same time Dodds and Fremlin [52] found a large class of Banach lattices, including Lp(#)-spaces (1 < p < oo), with the affirmative solution of the domination problem for compact operators. This important theorem is given next. THEOREM 19. If E and F are Banach lattices such that both E* and F have order continuous norms, then each operator from E to F which is dominated by a compact positive operator is compact. We proceed with a general result that does not require any additional assumptions regarding the spaces. It was proved by Aliprantis and Burkinshaw in [25] for real Banach lattices; the proof for the complex case can be derived easily from the real one. THEOREM 20. Suppose that in the scheme E Ml F M2 G M~ H of continuous operators between (real or complex) Banach lattices each operator Mi is dominated by a compact positive operator. Then M3M2 M1 is a compact operator. In particular, if an operator T : E --+ E on a Banach lattice is dominated by a compact operator, then T 3 is compact. THEOREM 21. Suppose that in the scheme E ml F M2 G of continuous operators between (real or complex) Banach lattices each operator Mi is dominated by a compact positive operator and either the norm of E* or the norm of G is order continuous. Then M2 M1 is a compact operator. In particular, if an operator T : E ~ E on a Banach lattice is dominated by a compact operator and either E or E* has order continuous norm, then T 2 is compact.

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The exponents three and two in the previous two theorems are the best possible. Some historical comments regarding this theorem and its predecessors can be found in [29, p. 279]. Are there any other cases (not covered by the Dodds-Fremlin Theorem) where one can deduce the compactness of the dominated operator itself? It turns out that there are. This was shown by Wickstead in [142]. Before stating this result, let us remind the reader that by a well known theorem of Walsh [ 137] the atomic Banach lattices with order continuous norm are precisely those Banach lattices in which every order interval is norm compact, and that in such spaces the solid hull of any norm compact set is again norm compact [138, Theorem 5]. THEOREM 22. For a pair of Banach lattices X and Y the following two statements are equivalent: (1) Every operator from X to Y which is dominated by a compact operator is itself compact. (2) One of the following three (non-exclusive) alternatives holds: (a) Both X* and Y have order continuous norm. (b) Y is an atomic Banach lattice with order continuous norm. (c) X* is an atomic Banach lattice with order continuous norm. A complete solution of the domination problem for weakly compact operators preceded the solution of the compact case and was found also by Wickstead [ 139]. THEOREM 23. For a pair of Banach lattices X and Y the following two statements are equivalent: (1) Each operator from X to Y which is dominated by a weakly compact operator is weakly compact. (2) Either X* or Y has order continuous norm. The following analogue of Theorem 21 for weakly compact operators (that is, when no conditions on Banach lattices is imposed) was established in [26]. THEOREM 24. Suppose that in the scheme X M~ y M2 Z of continuous operators between (real or complex) Banach lattices each operator Mi is dominated by a weakly compact positive operator, then M2 Ml is a weakly compact operator. In particular, if an operator T : X --+ X on a Banach lattice is dominated by a weakly compact operator, then T 2 is weakly compact. Recall that a continuous operator T : X --+ Y between Banach spaces is said to be a Dunford-Pettis operator if T carries weakly precompact subsets of X onto norm precompact subsets of Y. The domination problem for Dunford-Pettis operators was originally considered by Aliprantis and Burkinshaw [27]. Improving the main result in [27], Kalton and Saab [73] proved the following theorem. THEOREM 25. Let B : X --+ Y be a Dunford-Pettis operator between Banach lattices and assume that Y has order continuous norm. Then any operator T dominated by B is also Dunford-Pettis.

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Wickstead [ 142] showed that the only other case when the Dunford-Pettis property is inherited by a dominated operator is when X has weakly sequentially continuous lattice operations. The atomic Banach lattices with order continuous norm and the AM-spaces have this property. For such X the Banach lattice Y may be arbitrary. Some interesting results on the domination problem for nonlinear operators can be found in [38]. Now we turn our attention to the factorization problem. Recall that a continuous operator T : X --+ Y between Banach spaces factors through a Banach space Z if there exist continuous operators U : X --+ Z and V : Z --+ Y such that T -- V U. Depending on the properties of the space Z and of the operators U and V, a lot can be learned about the properties of the operator T. If the spaces under consideration are Banach lattices, then we can immediately look at positive or regular operators between them and demand more of the factorization space Z and of the factor operators U and V. A classical example to illustrate this topic is the following. If a continuous operator T : X --+ Y between Banach spaces factors through a reflexive Banach space, then T is weakly compact. Remarkably, as shown by Davis, Figiel, Johnson, and Petczyfiski [50], this is a characterization of weakly compact operators. THEOREM 26. An operator between two Banach spaces is weakly compact if and only if factors through a reflexive Banach space. As a matter of fact, Theorem 26 can be considered as the starting point for the subject of factorization. Without trying to be complete on this topic, we will present here only a few results on factorization of operators between Banach lattices. Additional material on factorization of positive operators may be found in [28] and [96]. Figiel [57] and Johnson [71 ] found which extra conditions on the factors and the transient space can be deduced for compact operators. THEOREM 27. Every compact operator between Banach spaces factors with compact factors through a separable reflexive Banach space. Inspired by these results, Aliprantis and Burkinshaw [28] studied the factorization problem under the presence of order structures. The following question is central for this topic. THE FACTORIZATION PROBLEM. Does every positive compact operator acting between Banach lattices factor (with positive compact factors if possible) through a reflexive Banach lattice? This problem (reiterated in [68] as Problem 9) is still open. We will mention some recent affirmative results below. But first, we should point out that a slightly weaker problem, when the given operator is assumed to be only weakly compact, is very difficult, too. It has been solved in the negative not a long time ago by Talagrand [130]. EXAMPLE 28. There exists a positive weakly compact operator T:s cannot be factored through a reflexive Banach lattice.

--~ C[0, 1] which

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The role of the space C[0, 1], or more generally, of the spaces C(K), for the factorization problem was studied in great detail by Ghoussoub [61 ] and Ghoussoub and Johnson in [62, 63]. On a positive note regarding the factorization problem, we will mention the following theorem obtained in [28]. THEOREM 29. The square o f a positive compact (respectively weakly compact) operator on a Banach lattice factors through a reflexive Banach lattice with positive compact (respectively weakly compact) factors. From this theorem and Example 28 it follows that not each weakly compact operator is the square of a weakly compact operator. This, of course, follows from other known results as well. (For instance, there are cases when the square of a weakly compact operator is necessarily compact.) The next result by Aliprantis and Burkinshaw is a special case of Theorem 17.13 in [29]; it demonstrates a nice relation between factorization and domination. THEOREM 30. Assume that T : X --+ X is a positive compact operator on a Banach lattice whose dual X* has order continuous norm. Then there exist a reflexive Banach lattice Z and positive operators U : X --+ Z, V : Z --+ X such that T = V U. Moreover, any operator S : X --+ X which is dominated by the operator T also factors through Z and S -- V1U f o r some O <, V1 ~ V.

Now let T : X ~ Y be a compact positive operator between Banach lattices. For the special (but important) case when either X* or Y satisfies the approximation property, Chen [47] made an advance toward answering the factorization problem by proving the existence o f a factorization with at least one factorpositive. As was pointed out by Johnson (see [28, p. 380]) the second case, i.e., when Y satisfies the approximation property, can be deduced from the results in [71 ]. It is interesting to observe that it is not for the first time (see, for instance, [64] and Theorem 31 below) that the approximation property makes its appearance for operators between Banach lattices. If we demand additionally that one of the factors is an interval preserving lattice homomorphism then, as shown by Chen in [47], the required factorization may fail to exist. Finally, we formulate an interesting result of Popovici and Vuza [ 110], where the factorization is intertwined with the order structure more subtly. For Banach lattices X and Y we denote b y / ~ (X, Y) the subspace of/2 r (X, Y) generated by the cone K~+(X, Y) of all compact positive operators from X to Y. THEOREM 31. Let X, Y and Z be Banach lattices with Z reflexive and discrete. For an operator T ~ s (X, Y) the following three conditions are equivalent. (1) T is regularly approximable, i.e., T belongs to the I]" lit-closure o f the space o f finite rank operators. (2) T = V U, where U : X --+ Z and V : Z -+ Y are regular and at least one o f them belongs to )Ur (X, Z) or )Ur (Z, Y). (3) T = VU, where U : X --+ Z and V : Z --+ Y are regularly approximable, a n d a t least one o f them (whichever one chooses) is positive.

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An example in [110] shows that one cannot improve (3) by getting, for a positive regularly approximable T, both factors positive. A detailed study of when, in that example, both factors can be chosen positive is presented in [ 111 ].

5. Invariant subspaces of positive operators In this section we will discuss the Invariant Subspace Problem for operators that are either positive or closely associated with positive operators. The general theory concerning the invariant subspace problem will be presented in a separate article prepared for this volume by Enflo and Lomonosov. THE INVARIANT SUBSPACE PROBLEM. Does a continuous linear operator T : X --+ X on a Banach space have a non-trivial closed invariant subspace? A vector subspace is "non-trivial" if it is different from {0} and X. A subspace V of X is T-invariant if T (V) _c V. If V is invariant under every continuous operator that commutes with T, then V is called T-hyperinvariant. If X is a finite dimensional complex Banach space of dimension greater than one, then each non-zero operator T has a non-trivial closed invariant subspace. On the other hand, if X is non-separable, then the closed vector subspace generated by the orbit {x, T x , TZx . . . . } of any non-zero vector x is a non-trivial closed T-invariant subspace. Thus, the "invariant subspace problem" is of substance only when X is an infinite dimensional separable Banach space. Accordingly, without any further mentioning, all Banach spaces under consideration in this section will be assumed to be infinite dimensional separable real or complex Banach spaces. The only exception will be made while discussing the Perron-Frobenius Theorem. In 1976, Enflo [56] was the first to construct an example of a continuous operator on a separable Banach space without a non-trivial closed invariant subspace, and thus he demonstrated that in this general form the invariant subspace problem has a negative answer. Subsequently, Read [ 115-117] has constructed a class of bounded operators on el without invariant subspaces. For operators on a Hilbert space, the existence of an invariant subspace is still unknown and is one of the famous unsolved problems of modern mathematics. Due to the above counterexamples, the present study of the invariant subspace problem for operators on Banach spaces has been focused on various classes of operators for which one can expect the existence of an invariant subspace. We start our invariant subspace results with a version of the classical Perron-Frobenius theorem for positive matrices. As usual, we denote by r ( T ) the spectral radius of an operator T. THEOREM 3 2 (Perron-Frobenius). I f A is a non-negative n x n matrix such that f o r some k >~ 1 the matrix A k has strictly positive entries, then the spectral radius of A is a strictly positive eigenvalue of multiplicity one having a strictly positive eigenvector. The proof of this theorem, discovered by Frobenius [60] and Perron [108], is available in practically every book treating non-negative matrices, for instance in [33,35,98]. One

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more proof of the Perron-Frobenius theorem as well as many interesting generalizations 1 can be found in [ 112]. If all entries of A are strictly positive, then the sequence { [r(a)] k Aku} converges to a unique strictly positive eigenvector corresponding to the eigenvalue r (A), no matter which initial vector u > 0 is chosen. This fact has numerous applications. A major step in extending the Perron-Frobenius Theorem to infinite dimensional settings was done by Krein and Rutman [82] who proved the following theorem. THEOREM 3 3 (Krein-Rutman). For any positive operator T" X --~ X on a Banach lattice r ( T ) E or(T), i.e., the spectral radius of T belongs to the spectrum o f T. Furthermore, if T is also compact and r ( T ) > O, then there exists some x > 0 such that T x -- r ( T ) x . PROOF. We will sketch a proof. The inclusion r ( T ) E cr(T) is caused merely by the positivity of T. Indeed, if we denote by R()~) the resolvent operator of T, then clearly R0~) > 0 for each )~ > r ( T ) , Also for each )~ with 1)~1 > r ( T ) the inequality ]]R(])~[)]] ~> ]]R()~)]] 1 holds. Therefore, for )~, -- r ( T ) + -d we have []R()~n)l[ ~ oc, whence r ( T ) ~ or(T). Assume further that T is compact and r ( T ) > 0. There exist unit vectors Yn E X+ such that ]lRO~,)yn I[--~ oc. Using the vectors y,, we introduce the unit vectors xn = R()~n)yn/llR()~n)Yn][ ~ X+. Since T is compact we can assume that Txn --+ x ~ X+. Finally, using the identities r ( T ) x n - Txn -

[ r ( T ) - Xn]xn + ()~n - T)xn

= [r(T) - ~ , ] x , + y~/IIRO~)y~II and that r ( T ) > 0 we infer that x 5~ 0 and that T x -- r ( T ) x . The conclusion of the previous theorem remains valid if we replace the compactness of T by the compactness of some power of T. Indeed, assume that T k is compact for some k. Since r ( T k) = [r(T)] k > 0 the previous theorem implies that there is a vector x > 0 such that T k x = [r(T)]kx. It remains to verify that the non-zero positive vector k - I r i T k - l - i x is an eigenvector of T corresponding to the eigenvalue r ( T ) . Y -- Y~i=0 The reader is referred to [121,123,144] for complete proofs and many pertinent results concerning the Krein-Rutman theorem. Some relevant results can be found in [4]. Note that the Krein-Rutman theorem holds not only for Banach lattices but for ordered Banach spaces as well. There is an interesting approach allowing to relax the compactness assumption. Namely, as shown by Zabrel3cO and Smickih [146] and independently by Nussbaum [ 102], instead of the compactness of T it is enough to assume only that the essential spectral radius re(T) is strictly less than the spectral radius r ( T ) . A different type of relaxation is considered in [121], where the restriction of T to X+ is assumed to be compact, that is, T maps the positive part of the unit ball into a precompact set. A version of this result, given in terms of re(T), can be found in [ 102]. Another classical result by M. Krein [82, Theorem 6.3] is the following. THEOREM 34 (Krein). Let T" C(I-2) --+ C(I-2) be a positive operator, where S72 is a compact Hausdorff space. Then T*, the adjoint o f T, has a positive eigenvector corresponding to a non-negative eigenvalue.

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PROOF. Consider the set G -- { f E C(t'2)+ f ( 1 ) = 1}, where 1 denotes the constant function one on t'2. Clearly, G is a nonempty, convex, and w*-compact subset of C ( ~ ) * . Next, define the mapping F ' G --+ G by

F ( f ) --

f+T*f f+T*f = [f + T'f](1) 1 + f(rl)

A straightforward verification shows that F indeed maps the set G into itself and that F: (G, w*) --+ (G, w*) is a continuous function. So, by Tychonoff's fixed point theorem (see, for instance [22, Corollary 16.52]) there exists some q~ E G such that F(4~) = 4~. That is, q~ + T*4~ -- [1 + 4~(T1)]4), or T*4) -- 4)(T1)4~, establishing that 0 < 4~ E C(12)~_ is an eigenvector for T* having the non-negative eigenvalue 4~(T1). M A proof of Theorem 34 that does not use fixed point theorems can be found in [77]. COROLLARY 3 5. Every positive operator on a C (Y2)-space (where Y2 is Hausdorff, compact and not a singleton) which is not a multiple of the identity has a non-trivial hyperinvariant closed subspace. PROOF. Let T ' C ( S 2 ) --+ C(Y2) be a positive operator which is not a multiple of the identity. By Theorem 34 the adjoint operator T* has a positive eigenvector. If )~ denotes the corresponding eigenvalue, then the subspace (T - )~I)(X) has the desired properties. V] Recall that a continuous operator T : X ~ X on a Banach space is said to be quasinilpotent if its spectral radius is zero. It is well known that T is quasinilpotent if and only if l i m n ~ ]lTnxl] 1/n = 0 for eachx c X. It can happen thata continuous operator T: X ~ X is not quasinilpotent but, nevertheless, l i m n ~ IlTnx ]ll/n = 0 for some x ~ 0. In this case we say that T is locally quasinilpotent at x. This property was introduced in [6], where it was found to be useful in the study of the invariant subspace problem. The set of points at which T is quasinilpotent is denoted by QT, i.e., QT = {x 6 X: l i m n ~ IlTnxll 1/n = 0 } . It is easy to prove that the set QT is a T-hyperinvariant vector subspace. We formulate below a few simple properties of the vector space QT. 9 The operator T is quasinilpotent if and only if QT = X. 9 QT = {0} is possible every isometry T satisfies QT = {0}. Notice also that even a compact positive operator can fail to be locally quasinilpotent at every non-zero vector 9 For instance, consider the compact positive operator T:g2 -+ ~2 defined by T(xl, X 2 , 9 9.) - - (X l , x22 ' x33 . . . . ). For each non-zero x E ~2 pick some k for which Xk r 0 and note that IlTnxl] 1/n ~ ~]xk[ 1/n for each n, from which it follows that T is not quasinilpotent at x. 9 QT can be dense without being equal to X. For instance, the left shift S: g2 -+ g2, defined by S(xl, x2, x3 . . . . ) = (x2, x3 . . . . ), has this property. 9 If QT ~: {0} and QT ~ X, then QT is a non-trivial closed T-hyperinvariant subspace of X. The above properties show that as far as the invariant subspace problem is concerned, we need only consider the two extreme cases: QT = {0} and QT = X.

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We are now ready to state the main result about the existence of invariant subspaces of positive operators on gp-spaces. It implies, in particular, that if a positive operator is quasinilpotent at a non-zero positive vector, then the operator has an invariant subspace. This is an improvement of the main result in [6]. THEOREM 36. Let T : g~p --+ g~p (1 <~ p < ~ ) be a continuous operator with modulus. If there exists a non-zero positive operator S : g~p --+ g~p which is quasinilpotent at a non-zero positive vector and SITI <. ITIS, then T has a non-trivial closed invariant ideal. We do not know presently if an analogue of the previous result is true if the inequality SITI <<. ITIS is replaced by SITI ~ ITIS. However, if we assume additionally that S is quasinilpotent (rather than just being locally quasinilpotent), then such an analogue is true. We now state several immediate consequences of the preceding results.

COROLLARY 37. Assume that a positive matrix A = [aij] defines an operator on an g~pspace (1 ~< p < ~ ) which is locally quasinilpotent at some non-zero positive vector. If ll3 "~ {llOij : i, j = 1, 2 . . . . } is an arbitrary bounded double sequence o f complex numbers, then the continuous operator defined by the weighted matrix Ato = [lloijaij] has a nontrivial closed invariant subspace. Moreover, all these operators A to have a common nontrivial closed invariant subspace. PROOF. By Theorem 36, the operator A has a non-trivial closed invariant ideal J. Now if B = [bij] is a matrix such that IBI ~< cA, then from IBxl <. IBl(Ixl) ~< c A ( l x l ) it follows that B x E J for each x E J, i.e., J is B-invariant. It remains to let B = Ato. IS] It is worth mentioning that in the preceding corollary our assumption that the weights are bounded is not necessary. It suffices to assume only that the modulus of the matrix A to defines an operator on g p. COROLLARY 38. l f the modulus o f a bounded operator T : g~p --+ g~p (1 ~< p < cx~) exists and is locally quasinilpotent at a non-zero positive vector, then T has a non-trivial closed invariant subspace. COROLLARY 39. Every positive operator on an g~p-space (1 <~ p < cx~) which is locally quasinilpotent at a non-zero positive vector has a non-trivial closed invariant subspace. For quasinilpotent positive operators on/~2, Corollary 39 was also obtained in [48]. Although Theorem 36 and its corollaries are new even for a quasinilpotent operator on gp, their main attractiveness is due to the fact that we do not really need to know that a positive operator T :gp --+ g p is quasinilpotent. The only thing needed is the existence of a single vector x0 > 0 for which IlT'xoll 1In --+ O. This alone implies that T has a non-trivial closed invariant subspace of a simple geometric form. In view of this, the following important question arises. How can we recognize by "looking at" a matrix [tij] defining a positive operator T:~p ~ g~p if the set of positive vectors at which T is locally quasinilpotent is non-trivial? This question is addressed in [9].

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A very interesting open problem related to Corollary 39 is whether or not each positive operator on el has a nontrivial closed invariant subspace. Since each continuous operator on el has a modulus (see Theorem 10), a natural candidate to test this problem is the modulus of any operator on g l without a nontrivial closed invariant subspace. In particular, each operator on g 1 without invariant subspace constructed by Read [ 115,117] is such a candidate. Troitsky [ 131 ] has recently managed to handle the case of the quasinilpotent operator T constructed by Read in [ 117]. Not only does ITI have a nontrivial closed invariant subspace, but ITI also has a positive eigenvector. In our previous discussion, we were considering only operators on gp-spaces. However, we only used the discreteness of g p-spaces, the above results remain true for operators on arbitrary discrete Banach lattices, 3 in particular, for operators on Lorentz and Orlicz sequence spaces. For instance, the following analogue of Theorem 36 is true. THEOREM 40. Let T : X --+ X be a continuous operator with modulus, where X is a discrete Banach lattice. I f there exists a non-zero positive operator S : X -+ X which is locally quasinilpotent at a non-zero positive vector and SITI <~ [TIS (in particular this holds if S commutes with ITI), then T has a non-trivial closed invariant subspace. Generalizing the approach developed in [5-7] for individual operators, Drnovgek [53] and Jahandideh [69] considered various collections of positive operators (for instance semigroups of operators) and proved the existence of common invariant subspaces for such collections. The main result in [53] is given next. THEOREM 41. Let S be a (multiplicative or additive) semigroup o f positive operators on a Banach lattice X. I f there exists a discrete element xo ~ X such that each operator in S is locally quasinilpotent at xo, then S has a common non-trivial closed invariant ideal. We refer to [8] for generalizations of some of the results in this section to Banach spaces with a Schauder basis. The situation with non-discrete spaces is considerably more complicated and will be discussed in the next section.

6. Compact-friendly operators and invariant subspaces In the Banach space setting, when we are looking for invariant subspaces of an operator T, the famous Lomonosov invariant subspace theorem (proved in [88]) asserts the following: if T commutes with an operator S that is not a multiple of the identity and S commutes with a non-zero compact operator K, then T has a non-trivial closed invariant subspace. 4 In the Banach lattice setting, when we are looking for invariant subspaces of a positive operator, the order structure provides new possibilities of bringing compactness into the picture. This was discovered by Abramovich, Aliprantis and Burkinshaw in [7] who proved 3 In actuality it is enough to assume only the discreteness of an element at which the operator in question is locally quasinilpotent. 4 It is interesting to note that Troitsky [132] has recently shown that this theorem cannot be generalized to "chains" of more than three operators.

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the next two results. The proof of the first one was inspired by Hilden's proof (included in [97]) of a simplified version of Lomonosov's invariant subspace theorem mentioned above. THEOREM 42. Let B : X --+ X be a positive operator on a Banach lattice. A s s u m e that there exists a positive operator S : X --+ X such that S B <<, B S, the operator S is locally quasinilpotent at some xo > O, and S dominates a non-zero c o m p a c t operator. Then the operator B has a non-trivial closed invariant subspace. Moreover, we can choose this invariant subspace to be the closure o f a principal ideal in X. THEOREM 43. Let B, S : X --+ X be two commuting non-zero positive operators on a Banach lattice. I f one o f them is locally quasinilpotent at a non-zero positive vector and the other dominates a non-zero c o m p a c t operator, then B and S have a c o m m o n non-trivial closed invariant ideal.

In the preceding two theorems, the compactness appears by means of the inequality IKxl <<, B(Ixl). But there are several other ways for a compact operator to be associated with an operator B. Here are three of them: (1) B may be dominated by a compact operator, (2) B may commute with a positive operator which dominates a compact operator, and (3) B may commute with a positive operator which in turn is dominated by a compact operator. As shown in [7] each one of these cases is "good enough" to guarantee the existence of a (non-trivial) closed invariant subspace. To accomplish this, a new notion was introduced in [7]. This notion is related to compactness in terms of the order structure and, in spite of some cumbersomeness, is much weaker than compactness and includes all the cases mentioned above. This will allow us to extend Theorem 42 considerably. DEFINITION 44. A positive operator B : X --+ X on a Banach lattice is called compactfriendly if there exists a positive operator in the commutant of B that dominates a non-zero

operator which in turn is dominated by a positive compact operator. That is, B is compactfriendly if and only if there exist three non-zero operators R, K, C :X ~ X with R, K positive and K compact such that RB = BR,

IC(x)l ~ R(IxI),

and

IC(x)l ~ K(IxI)

for eachx ~ X.

Clearly, every power of a compact-friendly operator is itself compact-friendly. Here are some classes of compact-friendly operators: positive operators with a compact power, in particular all positive compact operators; positive operators commuting with non-zero positive compact operators; positive operators that dominate non-zero compact positive operators; positive operators that are dominated by compact operators; positive integral operators; and composition operators. The last four cases are not obvious. Let us discuss them. Assume first that 0 < K0 ~< B, where K0 is compact. Then for the "triplet" (R, C, K), which is needed to conclude that B is compact-friendly, take R -- B, C -- K0 and K = K0. Suppose now that 0 < B ~< K0,

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where K0 is compact. Then the required "triplet" (R, C, K) is the following: R = B, C = B and K = K0. The proof that each positive integral operator is compact-friendly will be given in Theorem 52, and the proof for composition operators can be found in [ 11 ]. We will also need the following refinement of the concept of the strong unit introduced at the end of Section 1. A positive element u in a Banach lattice X is called a quasi-interior point whenever Xu is norm dense in X, i.e., Xu -- X. LEMMA 45. I f u is quasi-interior point in a Banach lattice X, then (1) For every non-zero element y ~ Xu there exists an operator V : X ~ X dominated by the identity operator and such that V y > O. (2) For every element v satisfying 0 <<.v <~ u there is an operator U: X --+ X dominated by the identity operator and such that Uu = v. The next result proved in [7] and [12] is the strongest invariant subspace theorem for positive operators proved so far. Its proof demonstrates the importance of the lattice structure. THEOREM 46. I f a non-zero positive operator B" X --+ X on a Banach lattice is compactfriendly and is locally quasinilpotent at some xo > O, then B has a non-trivial closed invariant ideal. Moreover, if another positive operator T" X --+ X commutes with B, then T and B have a common non-trivial closed invariant ideal.

PROOF. Let B be quasinilpotent at some x0 > 0. Also, let T" X --+ X be another positive operator that commutes with B. Fix three non-zero operators R, C, K ' X ~ X with K compact such that RB-

BR,

ICxl ~ R(IxI),

and

ICxl ~ K(IxI)

for each x 6 X.

We can suppose that l[B + T II < 1 and define A -- ~ n = 0 ( B + T) n. Clearly, the positive operator A commutes with both B and T and satisfies A x >~x for each x >~ 0. Also, for each x > 0, let Jx denote the non-zero principal ideal generated by A x in X, that is, Jx -- {Y ~ X: lyl ~< )~Ax for some )~ > 0}. If Jx ~ X for some x > 0, then the ideal Jx is a non-trivial closed (B + T)-invariant ideal. This ideal Jx is, of course, also invariant under both B and T. So, we can assume that Jx - X for each x > 0, i.e., A x is a quasi-interior point in X for each x > 0. Since C r 0, there exists some Xl > 0 such that CXl ~ O. Since A l C x l l is a quasiinterior point and ICxl[ <~ AICxl[ holds, it follows from Lemma 45(1), that there exists an operator V1 "X ~ X dominated by the identity operator such that x2 - Vl Cxl > 0. Put M1 -- VI C, and note that M1 is dominated both by the compact positive operator K and by the operator R. From Jx2 = X and C ~-0, we see that there exists some 0 < y <<. Ax2 such that Cy ~ O. Since Ax2 is a quasi-interior point, it follows from Lemma 45(1) that there exists an operator U ' X --+ X dominated by the identity operator such that U Ax2 - y. Now note that the element AICy[ is (again by our hypothesis) a quasi-interior point. Since

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ICyl <<,AICyl, it follows from Lemma 45(2) that there exists another operator V2 : X --+ X dominated by the identity operator such that x3 = VzCy -- V2CUAx2 > 0. Let M2 = V z C U A and note that M2 is dominated both by the positive compact operator K A and by RA. If we repeat the preceding arguments with the vector x2 replaced by x3, then we obtain one more operator M3 :X --+ X which satisfies M3x3 > 0 and which is dominated by both the positive compact operator K A and by RA. From M3 M2 M1 x 1 : M3x3 > 0, we see that M3 M2 M1 is a non-zero operator which (by Theorem 20) is also compact. Moreover, an easy argument shows that for each x 6 X IM3M2M1 (x)[ ~ R A R A R ( I x l ) ~ [ R A R A R + T](Ixl). Now consider the non-zero positive operator S -- R A R A R + T. Then B and S commute, S dominates the non-zero compact operator M3MzM1, and B is quasinilpotent at x0. By Theorem 43, S and B have a common non-trivial closed invariant ideal. This ideal is invariant under both B and T. [2 We will illustrate this theorem while studying integral operators in the next section. Theorem 46 may suggest that the compact-friendly operators characterize the positive operators with an invariant subspace. However, this is not true. An example of a positive operator that is not compact-friendly but has a non-trivial invariant subspace is given in [ 11 ]. The operator itself is very simple, just a multiplication operator M~; but the fact that for some multipliers the operator is not compact-friendly is far from being trivial. As shown in [ 13], a multiplication operator M~o on an L p ( # ) is compact-friendly if and only if the multiplier q) is constant on a set of positive measure. Nevertheless, in the last section of [ 11], a program is outlined for how the compact-friendly operators may be used to include the exceptional cases as well. We conclude this section by mentioning the following interesting conjecture of Lomonosov [89]: The adjoint of an arbitrary operator on a Banach space has a non-trivial closed invariant subspace. For operators on Banach lattices this conjecture is also open. We refer to the papers by de Branges [36], Abramovich, Aliprantis and Burkinshaw [10] and Simoni6 [127,128] that deal with this and related problems.

7. Integral operators and invariant subspaces After a brief introduction to the theory of integral operators, we will prove in Theorem 52 that each positive integral operator is compact-friendly. After that, applying the theory of compact-friendly operators, we will establish in Theorem 54 the existence of a nontrivial invariant subspace for each locally quasinilpotent positive integral operator. The rest of the section will be devoted to a discussion of the And6-Krieger theorem and its generalizations. We begin by defining the classes of integral (= kernel) and almost integral operators. 5 Throughout this section X and Y will denote two Dedekind complete Banach lattices which 5 To avoid a possible ambiguity let us point out that the integral operators considered in this section should not be confused with the integral operators in the sense of Grothendieck discussed in [72].

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are order dense ideals in the spaces (of equivalence classes) of all measurable functions L0(I2j, E l , #1) and L0(f22,272, #2) respectively, where (I21,271, #1) and (Y22, I72, #2) are two arbitrary "non-pathological" measure spaces, for instance, a-finite measure spaces. DEFINITION 47. An operator T" X --+ Y is said to be an integral operator if there exists a #1 x #z-measurable function T (., .) on s x ~(22 such that for each x e X we have

Tx(t) -- f ~ T(s,t)x(s)dlzl(s) 1

for #2-almost all t e S22. The function T(., .) is usually referred to as the kernel of the operator T. It is easy to prove that an integral operator is positive if and only if its kernel is nonnegative almost everywhere on S21 x I22. Furthermore, it is well known that an integral operator T with kernel T (., .) is a regular operator from X to Y if and only if for each x e X the function f IT(s, .)llx(s)l d#l (s) belongs to Y. Moreover, under this condition the modulus [TI:X --+ Y is also an integral operator, and its kernel is given by the absolute value IT (s, t)[ of the initial kernel. It should be pointed out that many authors include the last condition in the definition of the integral operator, thus guaranteeing that the modulus ITI of the operator T also acts from X to Y. To stress this property, such operators are sometimes referred to as absolute integral operators. We refer to [41], [74, Chapter 11] and [144, Chapter 13] for a systematic study of integral operators with emphasis on the order structure of the spaces involved. Apart from these three books there are, of course, many other books devoted to integral operators; see for instance [51,54,67,75,78]. Whenever convenient we will identify an integral operator with its kernel. The integral operators are closely related to finite-rank operators. A rank-one operator from a Banach lattice E to a Banach lattice F is any operator of the form 4~@ Y, where 4~ e E*, y e F and ~b | y (x) -- 4~(x)y for each x e E. Any operator of the form Y~i=ln ~bi | Yi is called a finite-rank operator. We denote by E* | F the vector space of all finite rank operators from E to F. The finite-rank operators are precisely the continuous operators whose ranges are finite dimensional. Recall that the subspace of all order continuous linear functionals on E is denoted by E*. DEFINITION 48. A bounded operator T ' E ~ F between Banach lattices, with F Dedekind complete, is said to be an almost integral operator if T belongs to the band generated in Z2r (E, F) by the collection E* | F of all finite-rank operators. The term "almost integral operator" has its origin in Theorem 50 below that is due to Lozanovsky [90], who introduced both this class of operators and the term itself. According to this theorem the classes of almost integral and integral operators are essentially the same. Let us mention also the paper by Synnatzschke [129], where a refined version of almost integral operators is studied. Namely, one can fix arbitrary subsets N _ E* and M c_ F, and then define the band of almost integral operators (which depends on N and M) as the

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band generated in/~r (E, F) by {q) | y" q9 E N, y E M}. The case of N = E* and M -- F considered in Definition 48 is the most important. A key element to Lozanovsky's characterization of almost integral operators is the famous Dunford theorem (see Theorem 49) regarding the integral representation of operators from L1 into Lp [55]. There are many proofs of this important theorem. A simple proof, utilizing the work of Benedek and Panzone [34] on spaces with mixed norm, was also found by Lozanovsky [91 ]. To outline the proof we need some terminology and notation. For p E (1, ec), let q denote its conjugate exponent, that is, p-1 + q-1 _ 1. We denote by L(p, oc) and L(q, 1) the usual Banach spaces with mixed norm [34], which consist of all real-valued #l x #2-measurable functions f ( s , t) on I21 x I22 satisfying respectively the following conditions:

If(s,t)lPd#2(t)

Ilfllp,oc " - I l l l f l l p l l ~ - esssup s ES22

]'J'

1

and

If(s, t)[ q d#2(t)

Ilfllq,1 "-IIIIfllqlll 1

d # | (s) < cx:~.

2

It is proved in [34] that [L(q, 1)]* = L(p, cx~). Notice that the definition of the space L(p, cx~) remains meaningful for p = 1. The introduced spaces L(p, cx~) are very convenient to describe all integral operators acting from L l ( # l ) into Lp(#2). Namely, a measurable function T(s, t) on S-21 x S-22 belongs to L(p, cx~) if and only if the integral operator determined by the kernel function T (s, t) is a regular integral operator from L l (#1) to Lp(#2). Under this condition IITIIIITIIp,~. This result can be found in many books (see [51,54,74,75]) and it will be used below to prove Dunford's classical theorem. THEOREM 49 (Dunford). If p > 1, then each continuous operator from Ll(lzl) into L p (#2) is a regular integral operator. PROOF. We will sketch a proof. Take an operator T:L1 (#1) --+ Lp(#2) with IITII ~ 1. Without loss of generality (see Theorem 10) we can assume that T 7> 0. We denote by b/the set of all finite collections U = (XA~, XA2 . . . . . XAn) of pairwise disjoint characteristic functions, where the sets Aj E SI (j -- 1, 2 . . . . . n) and have finite measure. Further, for each U we denote by R(U) the collection of all nonnegative functions K in L(p, cx~) such that IIg(s,t)llp,~ ~< 1 and fs2~ K ( s , t ) X z ; ( s ) d # l ( s ) - TXAj(t) for each j = 1, 2 . . . . n. n 1 The set R(U) is nonempty since the function K(s, t) -- ~ i = l lz(zj)Xzi (s)TXAi(t) belongs to it. It is a routine exercise to verify that R(U) is weak* closed and satisfies the finite intersection property. Therefore, (-]{R(U): U E b/} r 0. Since the linear combinations of the characteristic functions are dense in L 1(#l), it is obvious that any function T (s, t) from ~ { R ( U ) : U E H} is the desired kernel. D

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THEOREM 50 (Lozanovsky). An operator T : X--+ Y between two Banach function spaces is an almost integral operator if and only if T is a regular integral operator Furthermore, for this operator there are measurable functions wi ~ Lo(#i) (i = 1, 2) and a unique (up to an equivalence) integrable function T(s, t) on 121 x ~C22 such that for each x ~ X and for lzz-almost all t ~ ~-22 the following representation holds:

Tx(t) = w2(t) fs2 T(s, t)x(s)wl (s) d/zl (s). 1

The idea of the proof lies in consecutive approximations obtained by "squeezing" X and Y between weighted L ~ and L1 spaces (this procedure is described in [72, Section 5]) and then applying the Dunford theorem. The previous theorem is a global order-theoretic characterization of the space of integral operators. An important internal characterization of an individual integral operator is due to A. Bukhvalov [37,41], see also [125], and it is given next. Many applications of this criterion can be found in [41 ]. THEOREM 51. An operator T : X --+ Y between two Banach function spaces is an integral operator if and only iffor every order bounded sequence {Xn } C X which converges to zero in measure, Txn ~ 0 holds #2-almost everywhere. THEOREM 5 2. Every positive integral operator T is compact-friendly. PROOF. As we already know T is an almost integral operator. Now note that for each almost integral operator there exists a net {T~ } of positive operators such that 0 ~< T~x t Tx holds for each x ) 0, and each Tc~ is dominated by a positive finite-rank operator. This implies that T is compact-friendly. D LEMMA 5 3. If S : X --+ X is a strictly positive integral operator on an order complete Banach lattice, then for each element xo > 0 there exists a compact positive operator K: X --+ X satisfying 0 ~ K ~ S 3 and K xo > O. PROOF. Let S : X ---> X be a strictly positive integral operator and fix x0 > 0. So, there exists a net {S~ } of positive operators such that 0 <~ S~ t S and each S~ is dominated by a positive finite rank operator. Again, by Theorem 20, each S3 is a compact operator. From S~xo ~ Sxo and Sxo > 0, we see that there exists some index ~l such that S~x0 > 0 for each c~/> ~l. Similarly, from S~(S~,xo) ~ S(S~,xo) and S(S~,xo) > 0, we see that S~(S~xo) > 0 for all ~ t> Ot2 ) Otl. Finally, from the facts that S(S~2S~lXO) > 0 and S~(S~2S~lXO) t~ S(S~2S~,xo), it follows that S~(S~2S~,xo) > 0 for all possible indices O~ /) O~3 /) Ot2 )/Otl. It remains to note that if we let K -- $33 , then K" X ~ X is a positive compact operator satisfying 0 ~< K ~< S 3 and K xo > O. D And now we are ready to present one of the main invariant subspace theorems for integral operators obtained in [7]. The proof illustrates the role of compact-friendly operators.

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THEOREM 54. Let S : X --+ X be a non-zero positive integral operator on a Dedekind complete Banach lattice and let B : X --+ X be another non-zero positive operator commuting with S. I f either S or B is locally quasinilpotent at a non-zero positive vector, then the operators S and B have a common non-trivial closed invariant ideal. PROOF. If the operator S is itself locally quasinilpotent at a non-zero positive vector, then S is a compact-friendly operator that is locally quasinilpotent. So Theorem 46 is applicable and guarantees the existence of the desired invariant subspace. Assume now that B is locally quasinilpotent at a non-zero positive vector. Without loss of generality we can assume that S is strictly positive (since otherwise the null ideal Ns provides at once a common invariant subspace). Therefore, by Lemma 53, the operator S 3 dominates a non-zero compact positive operator K. Consider now the operator B 4- S 4- S 3 . Clearly, it dominates K and commutes with both S and B. It follows that B is compactfriendly and by hypothesis B is locally quasinilpotent at a non-zero positive vector. Hence, Theorem 46, applied to B and B 4- S 4- S 3, guarantees the existence of a common nontrivial closed invariant ideal for these two operators. Since both S and B are dominated by B 4- S + S 3, it follows that this ideal remains invariant under both S and B. D COROLLARY 55. Every positive integral operator that is locally quasinilpotent at a nonzero positive vector has a non-trivial closed invariant ideal.

For the classical Lp-spaces (and, as a matter of fact, for any Banach function space), the preceding corollary yields the following. COROLLARY 56. Let B : L p ( # ) --+ L p ( ~ ) (where # is a-finite and 1 <~ p < r be a positive integral operator with kernel B(., .), and let B be locally quasinilpotent at some non-zero positive function. Then the kernel operators T : L p ( # ) -+ L p ( # ) defined by T f (s) --

f

w(s, t) B(s, t) f (t) d # ( t ) ,

f E Lp(lZ),

where w(., .) is an arbitrary bounded lZ x #-measurable function, have a common nontrivial closed invariant ideal.

The proof is immediate since any operator T defined by a kernel w(.,-)B(., .) is dominated by a multiple of B, since w is a bounded function, and so our conclusion follows from Corollary 55. A positive operator B : X ~ X on an Dedekind complete Banach lattice is called a Harris operator if some power of B is not disjoint from the band of almost integral operators on X. It is easy to see that since some power of B dominates a non-zero positive almost integral operator, B is compact-friendly, and therefore the next corollary is an immediate consequence of Theorem 46. COROLLARY 57. Every Harris operator which is locally quasinilpotent at a non-zero positive vector has a non-trivial closed invariant ideal.

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We conclude this section by discussing the well known And6-Krieger theorem and its generalizations. DEFINITION 58. A positive operator T :X --+ X on a Banach lattice is ideal irreducible, if T does not have a non-trivial invariant closed ideal, i.e., if T (J) ___J and J is a closed ideal, then either J = 0 or J = E. Similarly, T is band irreducible, if Tdoes not have a non-trivial invariant band.

In 1957, And6 [31 ] proved that every band irreducible compact positive integral operator has a strictly positive spectral radius. Twelve years later, in 1969, Krieger [85] removed the compactness condition from And6's theorem, and the remarkable result asserting that a band irreducible positive integral operator has a strictly positive spectral radius, is today known as the And6-Krieger theorem. It is obvious that this result is an invariant subspace result as soon as it is reformulated as follows: each quasinilpotent positive integral operator has a non-trivial invariant band. The proofs presented by And6 and Krieger were not at all transparent and several attempts were made to clarify the intrinsic properties behind the structure of kernel operators. In addition to these efforts, many investigators looked at the possibility of generalizing the And6-Krieger theorem. In spite of all these research activities, it was not until somewhat recently that the And6-Krieger theorem was understood relatively well and generalized in various directions. Several people have contributed to the understanding of this subject; among them Caselles [45,46], Grobler [65,66], de Pagter [106], Schaefer [124,120], and Zaanen [ 144, Section 136]. The early attempts to generalize the And6-Krieger theorem to operators more general than integral were fraught with many obstacles from the outset. Schaefer [ 120] constructed an example of an ideal irreducible operator on Lp(#) (1 < p < e~) which was quasinilpotent. This showed that even for the class of ideal irreducible operators (which is larger than that of band irreducible operators) some extra condition is necessary. For a long time it was not clear whether the compactness would be such a condition. This problem was solved by de Pagter [ 106] who proved that every ideal irreducible compact positive operator on a Banach lattice has a strictly positive spectral radius. This remarkable theorem was the starting point in [5], where, instead of imposing all conditions on one operator, some of the properties were shifted to the commutant of the operator. The next two results not only demonstrate the difference between the ideal and band irreducible operators but also are the strongest generalizations of the And6-Krieger theorem for ideal and band irreducible operators obtained so far. We refer to [5] for these and many relevant results. THEOREM 59. If an ideal irreducible positive operator U : X ~ X commutes with a compact positive operator V : X --+ X, then r ( U V ) > O, and in particular r(U) > 0 and r ( V ) > O. PROOF. We will outline the main points of the proof. First of all, let us show that every non-zero compact quasinilpotent positive operator T has a non-trivial closed positively hyperinvariant ideal J, that is, J is invariant under each positive operator in the commutant {T}' of T.

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Let F be the closed ideal generated by the range of T. We claim that F is the closure of a principal ideal. The compactness of T implies that the range of T is separable. So, there exists a countable subset {yl, y2 . . . . } of T (X) consisting of non-zero vectors that is norm dense in T ( X ) . Consider u -- y~'n~__l 2,I~1'~I,ii which is in F. For the principal ideal A, generated by u we clearly have A, _ F. Conversely, the inclusion {yl, y2 . . . . } _ A, implies that T (X) ___A,. Hence, F c_ Au and so A, = F. Clearly, F 5~ {0} and it is easy to verify that F is indeed a positively hyperinvariant ideal for T. We are done if F is proper. So assume that F -- A, = X, which implies in particular that X has quasi-interior points. In this case, the existence proof of a non-trivial closed positively hyperinvariant ideal is just the second part of the proof of de Pagter's result mentioned above [106, Proposition 2]. The basic steps of this proof follow. Let J + = {0~< S E s 3 R E {T}'with0~< S ~< R}. Also, consider the vector subspace ,7 = {S1 - $2: Sl, $2 E ,.7+ } of s In addition, for each x >~ 0, let J [ x ] = {Sx: S E J } . It is shown in [106] that: (1) For each x > 0 the closure J [ x ] is a non-zero T-invariant closed ideal; and (2) For some x > 0 the closed ideal J [ x ] is non-trivial. Next, let C be a positive operator that commutes with T. If S E J , then there exist S1,82 E , J + and RI, R2 E {T}' satisfying S = Sl - 82 and 0 ~< Si ~ Ri (i -- 1,2). Then, 0 <~ CSi <~ CRi and CRi E {T}' for each i and so C S = CS1 - CS2 E ,J. It follows that C ( J [ x ] ) c_ J [ x ] for each x > 0. Consequently, T has a non-trivial closed positively hyperinvariant ideal. Now we are ready to finish the proof. Let U and V be two operators on X satisfying the hypotheses of the theorem. Assume that r ( U V ) = 0. Then, by the first part of the proof, the positive operator T -- U V has a non-trivial closed hyperinvariant ideal J. Since the operator U commutes with U V, we see that U (J) ___ J, which contradicts the ideal irreducibility of U. Hence, r ( U V ) > 0 holds. It remains to observe that r ( U V ) = r ( U ) r ( V ) since U and V commute. 71 For an order continuous operator U we can replace an ideal irreducible operator V in the commutant of U by a band irreducible operator. THEOREM 60. Every order continuous compact positive operator that commutes with a band irreducible positive operator has a positive spectral radius.

The two previous theorems imply immediately the corresponding results by Grobler [66] and Schaefer [ 124]. It is worth pointing out that Corollaries 55 and 56 together with Theorems 59 and 60 improve the And6-Krieger theorem considerably by replacing the quasinilpotence assumption with local quasinilpotence at a single positive vector, and by removing the assumption of the order continuity of the operator; see also [5,106].

8. Applications In this concluding section we will discuss briefly several areas of applications of positive operators. The interested reader is referred to the original works for necessary background

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and details. Each of these applications requires the development of fairly sophisticated machinery that cannot be presented here because of space limitations. More areas of applications can be found in the interesting surveys by Dancer [49] (devoted to some recent work on maps, both linear and nonlinear, between cones in Banach spaces) and by Nussbaum [103,104] (devoted to the fixed point index and some of its applications). Biology. We begin with some interesting and surprising applications of the abstract resuits on domination described in Section 4. The first example of such an application was discovered by Btirger [42] in biology. In [42] a very general discrete-time model that describes the evolution of type densities under the action of mutation and selection in an asexually reproducing population is developed and analyzed. As a result of this analysis, the author proves the existence and uniqueness of a strictly positive equilibrium. The analysis unfolds in some L1 (#)-space, that is, in a special Banach lattice with order continuous norm and so the Aliprantis-Burkinshaw Theorem 21 is applicable. This allows the author to conclude that for some positive compactly dominated integral operator U its second power U 2 is compact and then to apply to U the improvement of the Krein-Rutman theorem mentioned after Theorem 33, according to which r(U) > 0 is an eigenvalue, and U has a unique strictly positive normalized eigenvector corresponding to r(U). This model and its analysis have been recently generalized by BUrger and Bomze [43]. A similar application, but for a completely different model, has been obtained by Rhandi [ 113]. He has also applied Theorem 21 in his investigation of a population equation to obtain compactness of the remainder of order 3 of the Dyson-Philips expansion. In his recent work with Schnaubelt [114], they deal with a more complicated model describing a non-autonomous population equation with diffusion in L1, and again apply Theorem 21 to deduce the compactness of some operators. This allows them to obtain existence and uniqueness of positive solutions to this equation. A beautiful application of the Perron-Frobenius Theorem 32 to simple growth processes is given next. Suppose that there are N different types of biological objects. Assume also that at the times t = 0, 1, 2 . . . . each individual object produces a certain number of offspring among which every type is present. To put it mathematically, we denote by aij the number of offspring of type i produced by a single object of type j. If we denote by xi(n) the number of the ith type present at time t = n, then the size of the system at this time is determined by N numbers xl (n) . . . . . xN(n). Hence, the size of the i th type at time t - n + 1 will be given by Y~Y-1 aijxj (n). In vector form we have x (n + 1) = A x (n), where A = [aij ] and the vector x (n) = (x 1(n) . . . . . XN (n)). We are interested in the asymptotics of x (n) as time n tends to infinity, that is, we are interested in the behavior of the sequence A n (x (0)), where x(0) is the initial distribution of the system. The Perron-Frobenius Theorem allows us to solve this problem. 1 A nu}, where u is an Indeed, as we mentioned after Theorem 32, the sequence { [r(a)]~ arbitrary positive vector, converges to a unique (up to a multiplier) vector that is collinear with the unit eigenvector v corresponding to the eigenvalue r(A) > 0. This multiplier depends on our choice of the vector u. In other words, the sequence {x(n)} asymptotically behaves as [r (A) ]n v.

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Economics: Leontief's model. A very similar application of the Perron-Frobenius Theorem 32 to economics is produced by the Leontief model. Assume that we have N factories each of which produces a single commodity (or good). To produce each commodity other commodities must be employed. We denote by aij the amount of the j th good that is needed to produce one unit of the i th good. These numbers are called the input coefficients and they are usually assumed to be constant and nonnegative. Let us denote by di the demand for the i th good and by xi the total output of the i th good over a fixed period. Then the standard equilibrium equation "Supply -- Demand" yields the following system of equations: N

Xi -- Z a i j x j -- di, j-1

i - - 1 . . . . . N,

or (I - A ) x -- d. It is clear from the setup that A >~ 0 and also that the off-diagonal entries of I - A are nonpositive numbers. The economic situation is "feasible" if and only if I - A is nonsingular and r ( A ) ~< 1. Under these conditions the system can be uniquely solved for each output vector d ~> 0 and x - (I - A ) - l d ~> 0. For details and various generalizations see [33,35].

Economics: general equilibrium. Here we shall describe the basic framework of general equilibrium theory in economics" for details see [23]. The principal mathematical tools needed in this area are functional analysis and positive functionals rather than operators. Nevertheless, it seems very appropriate to mention this area in order to demonstrate the extent to which mathematical sophistication has penetrated work in economics and finance. The economic intuition of commodities and prices is understood by means of a dual system (X, X'). The vector space X is the commodity space and X I is the price space. The evaluation (x, x I) is interpreted as the value of the "bundle" x at prices x ~. For simplicity, we shall assume here that X is a (finite- or infinite-dimensional) Banach lattice and that X ~ is its norm dual. An exchange economy consists of a finite number of consumers, say m, indexed by i (i -- 1 . . . . . m) whose commodity-price duality is described by (X, X'). Every consumer i has a preference relation >-i, i.e., a binary relation on X+, which allows the consumer to distinguish out of any two bundles x and y the one which is better for him. The relation x ___iY is read "the bundle x is preferred to y " T h e preference ___.iis assumed to be complete (i.e., any two vectors in X+ are comparable), transitive, convex (i.e., the better-than-x set {y E X+" y ___iX} is a convex set for each x E X+), continuous (i.e., for each x E X+ the sets {y E X+" y ~i X} and {y E X+" x ___iY} are both closed), and monotone (i.e., "more is better" in the sense that y > x ~> 0 implies y ___iX). Each consumer i has also an initial endowment COi > 0. The total or the social endowment (or the resources) of the economy is the vector c o - Ziml coi. An allocation (redistribution) is any m-tuple (xl, x2 . . . . . Xm) such that xi >~ 0 for each i and Ziml x i - co. The objective of the economic activity in the economy is to redistribute the resources among the consumers in such a way that each consumer becomes better off. The efficient ways of redistributing the goods among the consumers are achieved by means

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of Pareto optimal allocations. An allocation (x l, X 2 . . . . . Xm) is said to be Pareto optimal if there is no other allocation (yl, y2 . . . . . Ym) such that Yi ~-i Xi for each consumer i and Yi ~ i xi for s o m e / ( w h e r e yi ~i Xi means yi ~i Xi and xi ~ i Yi). In other words, a distribution of the resources is Pareto optimal if it is impossible to make some consumer better off without making some other one worse off. Another important property of allocations is that of individual rationality: an allocation (x l, x2 . . . . . Xm) is individually rational if Xi ~ i (-t)i holds for each consumer i; that is, the redistribution does not give any consumer a bundle which is worse than his initial endowment. If the order interval [0, co] is weakly compact (in particular, if X has order continuous norm), then an easy application of Zorn's lemma guarantees that Pareto optimal individually rational allocations exist. The above efficiency notion requires that the acceptable redistributions of the goods are Pareto optimal and individually rational. This brings us to the important decentralization problem in economics, which in mathematical terminology can be stated as follows: Is there a price system (i.e., some linear functional x ~ E X ~) that can be used to enforce a given Pareto efficient allocation? Put another way: Can we use a system of prices - like the one we see in every day life - to redistribute the resources of the economy? This problem is related to the mathematical notion of supportability of convex sets by continuous linear functionals. Recall that a subset A of a vector space is said to be supported by a non-zero linear functional f at some point a E A if f (a) ~< f (x) holds for all x 6 A. An allocation (x l, x2 . . . . . Xm) is said to be supported by a non-zero price x ~ E X ~ if x' supports the better-than-x/set of each consumer i (i.e., the set {x E X+: x ~i Xi }) at the point xi. That is, x ~ E X~supports the allocation (Xl, x2 . . . . . Xm) if and only if

x •

xi---> (x, x') >>,(xi, x').

Now the economic concept of the decentralization of the allocations finds its best interpretation in the two famous theorems of welfare economics. These theorems were first proved for finite dimensional commodity spaces by K. Arrow and G. Debreu and are stated next. 1 S T WELFARE THEOREM. Every allocation supported by prices is Pareto optimal. 2ND WELFARE THEOREM. Every Pareto optimal allocation can be supported by prices. In particular, the second welfare theorem is of fundamental importance and its validity is a "must" for any economic system. To establish the validity of this theorem is, in general, a difficult problem, especially when one deals with an infinite dimensional commodity space. Supporting a convex set at a given (boundary) point by a non-zero continuous linear functional is not always feasible and supporting m convex sets each at a specified point by the same non-zero continuous linear functional is, if not impossible, a very serious matter. To achieve this goal, we must impose some extra conditions. The existence of a supporting price for a given Pareto optimal allocation (x 1, x2 . . . . . Xm) is proved by utilizing the lattice structure of the price space X ~. It usually consists of two steps. In the first step one proves the existence of a non-zero price x i that supports the

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better-than-x/ set at Xi. Once this is done, then the second step requires a sophisticated argument to establish that the supremum linear functional

x' -- x'l v x~ v . . . vx~ is non-zero and supports the allocation (x l, x2 Xm). The lattice structure of X' is of fundamental importance. As a matter of fact, if X' is not a vector lattice, there are examples of Pareto optimal allocations that cannot be supported by prices. . . . . .

Finance: choosing the optimal portfolio.

We shall discuss here how positive operators appear in the theory of finance in the context of choosing an optimal portfolio under uncertainty. For a systematic presentation the reader can consult the excellent book [95]. All investment decisions are made under uncertainty regarding future financial conditions. To analyze the decision process under uncertainty in the theory of finance, one usually introduces a two-period model (the simplest possible model), where at period 0 (today) everything is known and at period 1 (tomorrow) everything is unknown and uncertain. To capture the notion of uncertainty tomorrow one usually introduces the set S of all possible states of the world tomorrow. The usual assumptions about S is that it is either a finite set S -- {1, 2 . . . . . S }, or a more general compact Hausdorff topological space, or a probability space (S, P). By the term security one should understand any object or property that can be traded in the markets and that has a value today. While any such object (security) has a specific value today, its value tomorrow is uncertain and depends, in general, upon the prevailing state of the world tomorrow. Thus, a security is characterized by a value (or a payoff) at each possible state of the world tomorrow. Mathematically, this naturally leads to the interpretation of a security simply as a function x : S --+ R, where x(s) is interpreted as the value of the security x tomorrow when the state of the world s has been realized. Obvious economic considerations require that the collection of all securities X is a vector space under the usual pointwise operations. This space is called the space of securities. The most common spaces of securities used in finance are C (S) and L2 (S, P); the spaces L p (S, P) also are used quite often. In sum: the vector space of securities represents every object or property available that can be possibly traded in the market today. However, in today's market not every security is available for trade. For instance, every house is a security, but not everyone's house is in the market for sale today. The collection M of all securities that are available for trading today is assumed to be a vector subspace of X, and this space M is called the marketed space. If M -- X, then the markets are called complete and if M -r X the markets are incomplete. From the practical point of view the markets are always incomplete. Clearly, the structure of M should play an important role in making decisions today given the uncertainty of tomorrow. Now let us discuss the hedging problem. In order to protect their securities, given the uncertainty of tomorrow, people are forced to buy insurance. An insurance is simply a non-negative marketed security. If a person owns a security x and wishes the value of this security tomorrow never to fall below a certain level k (called the striking price or the floor), then he must purchase some insurance a c M+ = M A X+ so that x + a ~> k, or x(s) + a(s) >7k for each state s of the world tomorrow. This implies a ~ k - x, and hence a ~> (k - x) +, where (k - x)+(s) =

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max{k - x(s), 0}. Therefore, the "smallest" desired insurance seems to be (k - x) + and it is called the put option of the security x with striking price k. Since M need not be a vector sublattice of X, the put option (k - x) + need not be available in the market today, i.e., (k - x) + need not belong to M. This is why the theory of lattice-subspaces plays an important role here along with the theory of Banach and vector lattices (see for instance [ 14,24]). And now we can state the hedging problem of the theory of finance. THE HEDGING PROBLEM. Given a marketed security x, what is the "cheapest insurance" that one can purchase in order to guarantee at least the value k for the security tomorrow no matter what the prevailing state will be? Of course, if M is a vector sublattice of X and x, k 6 M, then a = (k - x) + should be the desired answer. The general solution of the Hedging Problem is non-trivial and is related to the meaning of the "cheapest insurance" 9This is the place where positive operators and their properties make their appearance. To simplify matters further, let us assume that there is either a finite or a countable number of linearly independent securities whose linear span is M. Considering the simplest case of the two, we will suppose that there are N securities x l, x2 . . . . . XN in M+ that span M and that S = {1, 2 . . . . . S}. In this case we have N <~ S (and so we have complete markets if N -- S and incomplete if N < S), and we can write

xl x~ v X l

D

.

,

X

2

D

.

,...,

.

XN

A portfolio is any vector 0 - - ( 0 1 , 0 2 . . . . . ON), where Oi represents the number of shares of security i. The payoff of a portfolio 0 is the marketed security R(O) -- y ] N 10iXi; notice that if s is the state of the world tomorrow, then the payoff of the portfolio will be R(O)(s) = ~-]U 10ix~. Clearly, this formula defines a positive operator R" ]~N ~ M, referred to as the payoff operator. In this case, the payoff operator can be represented by the following S x N matrix:

R

Now let us recall that a price p 6 M* (i.e., a continuous linear functional on M) is said to be arbitrage free if 0 < x 6 M implies p(x) > 0, i.e., if there are no "free lunches" in the sense that if one obtains a non-negative security, then he must pay a positive amount 9 The payoff operator by means of its adjoint operator allows us to describe the open cone of arbitrage free prices 9First of all, let

K -- {O 6 ]RN" RO > O}.

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Then a price p 6 M* is arbitrage free if and only if (p, RO) = (R* p, 0) > 0 for each 0 6 K, where R* :M* --+ ~I~u is the adjoint operator of R. This duality property is very useful in determining the arbitrage free prices. From economic considerations only arbitrage free prices are acceptable prices and the Hedging Problem can be stated now as follows: Given an arbitrage free price p 6 M*, a security x and a striking price k, find the portfolio 0 E R x that minimizes the expression (R* p, O) = (p, RO) subject to RO ~ x x/ k. The Hedging Problem is also related to the notion of a derivative in the theory of finance and has been studied in m a n y contexts. A derivative is a security of the form x ( s ) = f ( y l ( s ) , y2(s) . . . . . yk(s)), where (in the case X = C ( S ) ) f : R k --+ R is a continuous function and yl, y2 . . . . . yk are k given securities; that is a derivative is a security whose payoff depends on the payoffs of a finite number of other securities.

Acknowledgment The authors would like to express their thanks to A. Kitover and V. Troitsky for many useful suggestions.

Added in Proof Detailed proofs of the results discussed in this survey as well as an extensive treatment of many related topics can be found in the forthcoming book "An invitation to Operator Theory" by the authors that will be published shortly by the American Mathematical Society.

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CHAPTER

3

Lp Spaces Dale Alspach Department of Mathematics, Oklahoma State University, Stillwater, OK, USA E-mail: alspach @math. okstate, edu

Edward Odell* Department of Mathematics, The University of Texas, Austin, TX, USA E-mail: odell@ math. utexas, edu

Contents 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Global structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Sequences in ~ p and L p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Subspaces of L p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. /2p-spaces, 1 < p < cx~, p-f: 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

*Research supported by N S E H A N D B O O K OF T H E G E O M E T R Y OF B A N A C H SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved 123

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In this chapter we will discuss the structure of the L p-spaces and their subspaces. We are concerned mainly with the reflexive case (1 < p < ~ ) . The space L ~ and the /21-spaces are considered elsewhere [90]. A theory requires and feeds on its examples. The L p-spaces have provided much fodder for the general theory of Banach spaces because they appeared early in the theory and the study of these spaces has motivated the definitions of many properties of more general Banach spaces. For example, with its usual norm L p is a Banach lattice under the pointwise almost everywhere ordering. In the reflexive case it also has an unconditional basis and thus has another lattice structure as a sequence space (and is a Banach lattice under a different norm except in the case p - 2). These spaces naturally occur as interpolation spaces and are the simplest of the rearrangement invariant spaces. The Hardy spaces, Hp(D), of analytic functions on the unit disk with Lp boundary values are isomorphic to Lp, 1 < p < cxz, and the Bergman spaces A p ( D ) of analytic functions on the unit disk which are in L p ( D ) are isomorphic to ~p, 1 ~< p < e~, [67]. The study of the structure of the finite dimensional subspaces of L p paved the way for much of the extraordinary development of the local theory of Banach spaces in the 1980's [77]. In investigations of other Banach spaces and operators the existence and classification of operators from, into or factoring through L p-spaces provide fundamental information on the structure. We will concern ourselves primarily with the infinite dimensional isomorphic structure of the Lp-spaces. In particular we shall concentrate on separable Lp-spaces and, as noted in the basic concepts chapter, [49, Section 4], this reduces essentially to studying g p and L p(O, 1) (which we shall denote by L p). Functions are assumed to be real-valued as in general there are only minor adjustments needed for the complex case. As the theory of Lp-spaces was developed some natural questions arose. What are the complemented subspaces of Lp (or ~p).9 When does a given Banach space X embed into Lp (or gp)? If X c_ Lp, what subspaces must live inside of X? Does every Banach space contain some g p or co? What can be said about the structure of an unconditional basic sequence in L p ? We address these problems and others below. With apologies to the experts we choose to include some of the well known (to them) structural results that were mentioned in the basic concepts chapter or appear in books such as [70]. In Section 1 we review certain inequalities for sequences in L p. For example we show that by integrating against the Rademacher functions one can deduce that a normalized unconditional basic sequence in Lp (2 < p < CX~)admits upper ~2 and lower gp estimates (respectively, for 1 < p < 2, upper gp and lower ~2 estimates). In Section 2 we study the global structure of L p and in particular the Haar basis. Among the results given we examine the span of subsequences of the Haar basis and Schechtman's result that every complemented subspace of L p with an unconditional basis is isomorphic to a complemented subspace spanned by a block basis of the Haar basis. Section 3 begins with some properties and characterizations of the unit vector basis of p. The details of the argument using the Pelczyfiski decomposition method that a complemented subspace of g p is isomorphic to g p are presented to help give context and meaning to the notion of (p, 2) bounded operators discussed later in Section 5. We also discuss spreading models and types and the Krivine-Maurey theorem. This latter result gives a sufficient condition, in terms of types, for a Banach space X to contain almost isometric copies of g p for some p. Section 4 deals with subspaces X of L p. For example if

D.AlspachandE. Odell

126

C Lp

X (2 < p < cx~) then X must contain an isomorph of s or s and if X does not contain s it must embed into s We also consider a necessary and sufficient condition for a reflexive space X to embed into the s sum (1 < p < cx~) of finite dimensional spaces. The last section c o n c e r n s / 2 p - s p a c e s - those complemented subspaces of L p which are not Hilbert spaces. Here the known examples and their isomorphic classification are discussed. In particular Rosenthal's space X p and its generalizations are presented in detail. The role of (p, 2)-bounded operators is also explained. Finally we discuss some of the results for s with )~ near one and their relation to the isometric theory of Lp-spaces.

1. Preliminaries We first recall a few key properties of L p and s which are discussed throughout the basic concepts chapter. The unit vector basis for s is a 1-symmetric basis [49, Section 3]. The Haar basis (hi)~ is an unconditional basis of Lp for 1 < p < c~ [49, Section 3], [24]. It is also a monotone basis for for 1 ~< p < c~. The Rademacher functions (rn)n~=l , [49, Section 4], are equivalent to the unit vector basis of s for p < ~ , (and the unit vector basis of el for p = ~ ) . Thus for 0 < p < oc there exist constants A p, B p with

Lp

Ap(~lanl2)l/2 <. (fo I I}2a,

)l/p <~Bp(~lan] 2)1/2

(1.1)

for all scalars (an). Ap -- 1 if 2 ~< p < ~ and Bp -- 1 if p ~< 2. If (xi)~ is a normalized sequence of disjointly supported functions on [0, 1] in Lp with (1 ~< p < ~ ) then (xi) is 1-equivalent to the unit vector basis of s and [(xi)] is 1-complemented via the projection

P(x) -- ilcx~(fo1 ~ signxi(t)lxi(t)lP-lx(t)dt

) Xi.

For 1 < p < cx~, L p is uniformly convex and uniformly smooth with modulus of convexity (respectively, of smoothness) of power type p (respectively, of power type q with 1 + ~1 -- 1), [49, Section 6], Lp is of type min(2, p) and cotype max(2, p) for 1 ~< p < c~ [49, Section 8]. For (xi)~

~ Lp

1/p Zri(t)xi

,

1

at

I1' p

~

Ilxill p

(1.2)

Lp spaces

127

if 1 ~ I1" lie2 to obtain

~ri(t)xi

d,-f'Ji' ~ri(t)xi(s)

p dt ds

1

1

io[So = fo

2 ]p/2 ~ri(t)xi(s) dt ds 1

ds

Ixi (s)I 2

fo ~n Ixi(s)l pds

(1.4)

1

which yields the right hand inequality of (1.2). Also by (1.1),

io'lo'

n

~ri(t)xi(s)

dt ds >~APpfo

]xi(s)] 2

ds.

1

Now

Ilxi I12 p

p

/'/

= II(iix, li~), II~2t. /7

= ~ [Ixi []Ppai

for some

(ai)~z ~ e2/(2-p)

of n o r m 1

1

~ Js

[Xi(S)I2)P/2(~I [oi12/(2-P))(2-p)/2ds

by H61der's inequality

"-- fo which completes the proof of (1.2).

iXi(S)12

ds,

D. Alspach and E. Odell

128

(1.2) and (1.3) can be viewed as generalizations of Clarkson's inequalities [29]. Since II 9IILp ~< I1" IlL2 for p ~< 2 we also have using (1.2) for p = 2 that

(So

1/2

~ri(t)xi 1

p

dt

~<

IIx/ll 2

for 1 ~ p < 2,

(1.5)

for 2 < p.

(1.6)

and similarly

(So

1/2

~-~ ri (t)xi 1

p

at

1>

ilxill 2

The technique of integrating against the Rademacher yields some useful inequalities for unconditional basic sequences in L p. If (Xn) is a A-unconditional basic sequence in L p then

A-I [f01 (Elanl2iXn(S)l

<~ABp

(AAp) -1

2)p/2as ]l/p I[Ea, x, II"

[f0'(z lanlZlxn(S)l2t P/Zds], (Elani21Xn(S)l

i f 2 ~< p < oo,

<~IIEa.x

as

A If01( E lanlZlXn(S)l 2) p/2] 1/p

if 1

(1.7)

ll '

~

(1.8)

which implies that (Xn) and (Ixnl) are equivalent. If (Xn) is also normalized,

A-l(ElanlP) 1/p

<. ABp(Elanl2) 1/2

i f 2 <~ p < ex~,

(AAp)-l ( ~ ianl2) 1/2 i f l ~
(1.9)

IIEanXn

A(~lanlP) 1/p (1.10)

These last two inequalities are immediate consequences (1.2) and (1.3). Any martingale difference sequence in L p is unconditional [25] which generalizes the fact that the Haar basis is unconditional. In particular any sequence of mean zero independent random variables in L p is unconditional. Rosenthal's inequality [91] gives us some information on such sequences. Let 2 < p < ex~. There exists Kp < cx) so that if (Xi)~ are

129

L p spaces

independent mean zero random variables in L p then

max

Ilxi IlPp

,

i:1

~Xi i:1

Ilxi II2 i--1

~ p

Kpmax

Ilxi lipp i=1

t

,

IIx/ll

9

(1.11)

It is shown in [55] that Kp ~ p~ In p. A Banach space X is a L;p,Z-space if for all finite dimensional spaces F ___X there exists a finite dimensional E with F c E cm X so that d ( E ~ fdimE) ~ )~, It ultimately turns out vp (see Section 5) that a separable X is s for some )~ and 1 < p < ec iff X is isomorphic to a complemented subspace of L p which is not isomorphic to Hilbert space [66,68]. The situation for L 1 is more complicated. It is conjectured that every infinite dimensional complemented subspace X of L1 is isomorphic to L1 or el. It is known that if X contains an isomorph of L1 then X is isomorphic to L1 [36] and if X embeds into el then X is isomorphic to el [65]. Various characterizations of/21- (and L;oc-) spaces are given in [68]. Much work was done to study and attempt to classify the L;p-spaces up to isomorphism and this is discussed in Section 5 below. We begin with some results on the global structure of L p and in particular those involving the Haar basis. All Banach spaces are presumed to be separable unless otherwise stated.

2. Global structure

In Hilbert space strong geometric properties follow from the properties of the inner product. Early in the development of operator theory on more general spaces Murray [81 ] recognized the critical role that the orthogonal projection played for operators on L2 and attempted to find generalizations for the L p-spaces. While we know now (and as Murray himself discovered in part) that there are far fewer bounded projections on the typical Banach spaces than on Hilbert space, nevertheless, spaces such as L p possess a rich family of projections and of corresponding complemented subspaces. Given the unconditionality of the Haar basis the simplest way to get a complemented subspace of Lp (1 < p < oc) is to take [(hji)]~__l where is a subsequence of (hj). There are however only two isomorphism types of such subspaces.

(hji)

THEOREM 1 ([39]). Let 1 < p < oc and let X -- [(h j;)] f o r some subsequence (hji) o f

(h j). Then X is isomorphic to e p or L p. PROOF. By the decomposition method [49, Section 4], (or see the proof of Theorem 15 c

c

below) it suffices to prove that either X ~ e p or L p ~ isomorphic to a complemented subspace of Y.) Let

c

X. ( X ~

A = {t c [0, 1]" t c supp hji for infinitely many i's}.

Y means that X is

130

D. Alspach and E. Odell C

We shall show that X ~ s if m (A) -- 0 and otherwise L p ~ X. So assume m(A) = 0. We shall produce a sequence (Sn) of disjoint measurable subsets of [0, 1] with m ([0, 1] \ Un Sn) = 0 and such that every element of X is constant on each Sn. It then follows that C

X ~-+ [(l&)]~ g.p. To do this it suffices to show that if n E N and

An

-

-

{t ~ [0, 1]" t ~ supphji for exactly n i's},

then An may be split into such a sequence of sets. Let (Xi) be the subsequence of those h ji's involved in the definition of An and let (yi) be the (possibly finite) subsequence of (xi) determined by U i s u p p yi ~ An and if Xn :/= Yi then suppxn g supp Yi. The yi's a r e disjointly supported and

An

-

-

U(Ann suppy+) U U(Ann supp y?). i

i

This gives the desired splitting of An. If m(A) > 0 we partition (hji) into a countable number of subsequences (hji)iEN k so that supp hji A supp h je = 0 if i -J= e 6 Ark and so that if k >~2, i ~ Nk then there exists ~ Nk-1 with supphji c_ supphje. Set Bk = UicNk supphji" Clearly B1 ___ B2 ___ " "

and

(-]Bk - - A .

One now chooses a subsequence (kn) of N with m (Bk, \ A) < en, en .[. 0 rapidly. Define

do- L icNk o

d,- L

hji

i~Nk 1

and inductively for m odd

+ }, d2"+m -- Z hji summed over {i 6 Ark," supphji ___suppd2n_l+~(m+l)/2]] and for m even

d2n+m-Zhjisummed over {i ~ Nk," supphji

c_ d~_,+~(m+,)/2~}.

A perturbation argument yields that (dn) is equivalent to the Haar basis and it is not hard to see that [(dn)] is complemented in X, [10]. D A vector-valued version of this theorem exists [78]. There also exists a finite dimensional version of this theorem.

L p spaces

131

THEOREM 2 ([80]). Let 1 < p < cx~. There exists Kp < ~X) so that if (hji)in___l is anyfinite subsequence o f (hj) and E -- ((hji)~) then d ( E , ~np) <~ Kp. ((.) denotes the linear span.) The Rademacher functions (ri) form a block basis of (hi) and so one can find the unit vector basis of ~ 'S uniformly as a block basis of a suitably large initial segment of the Haar basis. This is true more generally if we replace "block basis" by block basis of a permutation and false without this change. THEOREM 3. (a) ([31]) Let 2 < p < cx~ and let ( X i ) be an unconditional basic sequence in L p which is not equivalent to the unit vector basis o f g,p. Then there exists K so that f o r all n there exists a block basis (yi )~ o f a permutation o f (xi ) which is K-equivalent to the unit vector basis o f g,~. (By Krivine's theorem this holds f o r all K > 1.) (b) ([79]) Let 1 / 1 and f o r all n a rearrangement (hn~2 n

2n

n 2n

~'i 'i=0 ~ (hi)i=o so that if M E N and (zi) M is a normalized block basis o f (b i )i=0 then there exists A c {1 . . . . . M} with [A[ ~> M / 2 so that ( Z i ) i E A i s Kp equivalent to the unit Ial vector basis o f g~p .

The rearrangement is defined in terms of the supports of the Haar functions. For i < j either supp b n and supp b jn are disjointwithsuppbi n lying to the left of supp bjn orsuppbi n _ suppbj. Every s has a basis ([54], see Section 5 below) but it is still open as to whether or not it must have an unconditional basis. However those that do can be realized isomorphically as being spanned by block bases of (hn). THEOREM 4 ([95]). Let 1 < p < cx~ and let X be a complemented subspace o f Lp with an unconditional basis (Xn). Then there exists a block basis (Yn) o f the Haar basis with [(Yn)] complemented in L p and such that (Xn) is equivalent to (Yn). The first part of this theorem follows from the following proposition. PROPOSITION 5 ([95]). Let (Xn) be an unconditional basic sequence in Lp (1 < p < ~ ) . Then there exists an equivalent block basis (Yn) o f (hn). PROOF. From (1.7) and (1.8) it follows that if (yi) is another unconditional basic sequence in Lp with [yi[ -- Ixi[ then (yi) is equivalent to (xi). By a perturbation argument we may assume that there exist integers m n ~ ~ and scalars (ai) so that 2 mn - l

Xn--

Z azmn+klh2mn+k[" k=0

One defines the yn'S via this formula by removing the absolute values.

(2.1)

[3

D. Alspach and E. Odell

132

Before completing the proof of Theorem 4 we recall a useful isomorphic representation of Lp as Lp(~2) for 1 < p < cxz. The latter is the Banach space of all sequences (fi) of measurable functions on [0, 1] with

II(fi)llL,r

fi2 (t)

-

dt

Lp embeds as a complemented subspace of Lp(g2) via the map f ~ (f, 0, 0 . . . . ). Also it is not hard to show that the map Lp(s

9 (f/.) i

~Z

fl(t)ri(s)

embeds Lp(s as a complemented subspace of Lp([0, 1] 2) ~ Lp. Thus by the decomposition method, Lp ~ Lp(g2). PROOF OF THEOREM 4 (Sketch). As in the proof of Proposition 5 we may assume that (2.1) holds. For each i, j E N set f i j = (0 . . . . . O, xi, 0 .... ) with xi in the j th coordinate and let f i __ fii. For each n E N and 0 ~< k < 2 m~ let

h 2mn+k - - (0 . . . . . 0, Ih2mn+kl, 0 . . . . ) with ]h2mn§ in the nth coordinate. Let P be a projection of Lp onto X. Then P ( f i ) - ( P f / ) is a projection of Lp(g2) onto [(fij)]. Moreover if p > 2 it can be shown that (fij) is unconditional and since ( f i ) is a subsequence of (fij) we obtain a projection of Lp(s onto [(fi)]. ( f i ) is equivalent ~'2mn--1 F u r t h e r m o r e (h 2mn+k) is equivalent to to (xi) and is a block basis o f (h 2mn+e)n ~ 1,k=0 (h2mn+k) and this completes the prooffor p > 2. For p < 2 a duality argument is necessary [951. D Some Banach spaces have the property that whenever they are isomorphic to a subspace of another nice enough Banach space that there is another isomorphic embedding with better properties. One property of this type is reproducibility. A basis (xi) for a Banach space X is reproducible if whenever X ___Y and Y has a basis (yi) then some block basis of (yi) is equivalent to (xi). It is easy to see that because they are weakly null and subsymmetric that the unit vector bases of s (1 < p < cxz) or co are reproducible. It is also not hard to show that the unit vector basis of s is reproducible. THEOREM 6 ([67]). The Haar basis for Lp (1 ~< p < cx~) is a reproducible basis. Moreover the same is true for any unconditional basis of Lp (1 < p < cx~). PROOF. By virtue of Proposition 5 it suffices to prove that (hi) is a reproducible basis for L p for 1 ~ 0 there exists a measurable set B c_ A with m(B) -- m(A \ B) and x*(1B) -x*(1A\B) for x* E F. Let en ,~ 0. We can use the above to produce a dyadic tree of sets (An,j a~'2n +n=0, j= 1 as in the supports of the Haar basis and a sequence (fn) as follows. Set f0 -- 1 a0,1 = 1, fl

-- 1Al,l

-- 1Al,2,

f2 -- 1A2,1 - - 1A2,2

and so on. Then ( f , ) is 1-equivalent to (hn) and for some subsequence (Pk) of N cx~

O<3

fn-- Z x;(fn)xi i: Pn

with

Z X* (fn)Xi i=pn+l

~n.

If fn has been defined and pn, pn+l are as above we apply the fact that for F = (x.]Pn+l ~ i J i = l --1 and the appropriate set Am,j (for which I f , + l [ -- 1am,j). T h u s (fn) is a perturbation of a block basis of (Xi). [-] The proof also works in the setting of rearrangement invariant function spaces on [0, 1], [72]. Above we saw that subsequences of the Haar basis span either a subspace isomorphic to p or to L p and thus if the Haar basis is split into two sets at least one of the two must span L p. Actually L p has a stronger property. Recall that X is primary if whenever X -- Y 9 Z then Y or Z is isomorphic to X. s is primary for 1 ~< p < cx~ by the results of Section 3. THEOREM 7 ([2,74], [72, T h e o r e m 2.d.11]). Lp is primary for 1 <<.p <<.cx~. PROOF. We confine ourselves to the case 1 < p < cx~. We use Lp(~2) as a representation of Lp and recall that (h ij) is an unconditional basis for Lp(~2) where h ij = (0 . . . . . O, hi, 0 . . . . ) and hi is in the jth-coordinate. Let Lp(~2) -- X • Y and Px, PY be the corresponding projections. N o w h ij -- PX hij + Pyh ij and so either h ij* (Px hij) ~ 1/2 or h ij* (Pyh ij) ~ 1/2, w h e r e (h ij*) are the biorthogonal functionals to the basis (h tj). Let ..

I = {i" h iJ*(Pxh ij) >~ 89for an infinite n u m b e r of j ' s } and

J -- {i" hi J* (pyhiJ) ~ 1 for an infinite n u m b e r of j ' s }. By T h e o r e m 1 either [hi]iEI or [hi]iEJ is isomorphic to Lp. A s s u m e the former. Let I (in). Since (Pxh ij) c o n v e r g e s weakly to 0 we m a y assume there are integers jn so that (Px hi''j') is a block basis of (h ij) and h inj'* (Pxh in'j~) ~ 89 LEMMA 8 ([27]). Let (Xn) be a bounded unconditional basis for a Banach space X with

biorthogonal functionals (x*). Let T be an operator on X such that (Txi,) is a block basis

134

D. Alspach and E. Odell

of (Xn) f o r some subsequence (Xin) and Ixi*n (Txi~)l ~ e f o r all n and some fixed e > O. Then (Txi~) is equivalent to (xi~) and [(Txin)] is complemented in X. By the lemma ( P x ( h in,jn)) is equivalent to (h in'jn) and [Px(h in,j~ )] is complemented in Lp(~2). Also (h i"'j~) is equivalent to (hi~) and this completes the proof. D The nonseparable Lp-spaces are built on the spaces gp(F) and Lp{O, 1}~. If one is 1

considering only separable subspaces X of such spaces then X ~ L p. But the global structure of nonseparable L p-spaces can be somewhat different. These spaces need not have an unconditional (uncountable) basis. Recall dim X = inf{ot: there exists S C X with S dense in X and card S = o~}. THEOREM 9 ([37]). Let 1 l'~co. Then L p (#) does not embed into a space with an unconditional basis.

3. Sequences in gp and Lp In this section we look at some special subspaces of ~p and L p which are isomorphic to gr, for some r. Of course for ~p, the only possible value of r is p itself, but for Lp there is always r = 2 and for p < 2 an entire interval of possibilities. General sequences and even unconditional basic sequences in L p are probably too varied to ever be described (see Section 10, [50], for a particularly peculiar subspace with unconditional basis), however, near the end of this section we note some results for Orlicz sequences and other spaces with a symmetric basis. In the previous section we looked at the span of subsequences of the Haar system. For the unit vector basis of gp, subsequences are not mysterious, and in this simpler situation we can successfully describe the span of block bases. Let (ei) be the unit vector basis of ~p. We begin by recalling the a simple fact about block bases of (ei) which was mentioned in [49, Section 64], [71, Proposition 2.a. 1], [44, Proposition 1.5.3]. PROPOSITION 10. Suppose that (Xn) is a block basis of the unit vector basis of g.p. [Xn: n E IN] is isometric to g.p with the isometry which is induced by the basis map Xn ----> ]]Xn lien, n = 1, 2 . . . . . A normalized basic sequence is said to be perfectly homogeneous if it is equivalent to all of its normalized block bases. We have thus shown that the unit vector basis of g p has this property. It is easy to see that the unit vector basis for co is also perfectly homogeneous. It turns out that this is a very special property. THEOREM 11 ([101]). Suppose that X is a Banach space with a normalized perfectly homogeneous basis (Xn), then (Xn) is equivalent to the unit vector basis of g p or co. We refer the reader to Zippin's paper or the book [71, p. 60], for the proof. There have been some generalizations of this result in the context of symmetric spaces [9].

Lp spaces

135

Another interesting property of block bases in ~ p mentioned in [49, Section 4] is the fact that the closed span is always complemented. Indeed, let Xn = ~iEFn aiei be a normalized block basis of (ei) (here the Fn are finite subsets of I~t and F1 < F2 < ... in the sense that maxFn < min Fn+l)and define x n = IlXnll- p Y]icFn lai]P-2aie* for each n, where, as usual, e* denotes the biorthogonal functional to ei. Then (x*) is biorthogonal to (Xn). The formula OO

P x -- E

x* (x)xn

forxeep

n=l

then defines a norm one projection from g p onto the closed span of (Xn), [71, Proposition 2.a. 1], [44, Proposition 1.5.3]. This gives THEOREM 12. If (xn) is a block basis of g.p, then there is a norm one projection from g.p onto [Xn : n ~ H]. As noted in Section 2, a very similar argument yields that if (Xi) ~ Lp is a nonzero disjointly supported normalized sequence then (xi) is 1-equivalent to the unit vector basis of ep and [(xi)] is 1-complemented in Lp. The analog of Theorem 12 holds for co and, as it turns out, is peculiar to these unconditional bases. THEOREM 13 ([69]). Suppose that X is a Banach space with an unconditional bases such that f o r every permutation rc of H and every block basis (yj) of (Xjr(n)), [yj: j ~ H] is complemented, then (Xn) is equivalent to the unit vector basis of g.p or co. As was noted in [49, Section 4], it is an immediate consequence of Theorem 12 and standard perturbation arguments that COROLLARY 14. If Y is an infinite dimensional subspace of g.p, then f o r every e > O, Y has a (1 + E)-complemented subspace (1 + e)-isomorphic to g.p. We turn to some results of Petczyfiski [86] which were important in stimulating interest in classification problems for complemented subspaces. THEOREM 15. If X is an infinite dimensional complemented subspace of g.p, then X is isomorphic to ( p. A proof using Pe{czyfiski's decomposition method, [86] is given in [49, Section 4]. A more effective but more complex proof is given in [7]. That proof proceeds by directly showing that a complemented subspace of g p actually decomposes into a sum of finite dimensional subspaces each isomorphic to gnp for an appropriate n and then deducing that the space is isomorphic to g p from the decomposition. The proof using the decomposition method depends rather heavily on the ability to estimate the norm of an operator on a g p sum by norms of the restrictions to the summands. In

D. Alspach and E. Odell

136

Section 5 we will see that the fact that this does not work for other types of unconditional sums leads to a rather delicate approach to understanding the complemented subspaces of C

Lp. It is not true in general that if X ~ Y and Y ~ X then X is isomorphic to Y [42]. One problem that was open until the 1970's was whether every Banach space contained g p for some p or co. If we are satisfied with finite dimensional subspaces Dvoretzky's theorem [33] tells us that for every Banach space X, ~2 is finitely represented in X. Krivine's theorem [61 ] extends this to basic sequences: Every basic sequence admits a p so that g p is block finitely represented therein. Going the other way if X is such that for some ~. < cx~, 1 ~< p < ~ , every finite dimensional subspace of X X-embeds into ~pm for some m then X embeds into L p. This is easy via an ultraproduct argument. In any event if we are seeking some collection of infinite dimensional "atoms" which must live inside of every X, these must include more than just co and g p, 1 ~< p < cxz, as witnessed by Tsirelson's example [99,28]. However we do get a positive result for subspaces X of Lp (1 ~< p < c~). In fact within a certain class of "stable" spaces the conjecture is true in a strong sense. Before we discuss these results let us define a notion that is a little stronger than finite representability. DEFINITION 16. Suppose that (Xn) is a sequence in a Banach space X and that Y is a Banach space with a basis (Yn). We say (Xn) generates the spreading model (Y, (Yn)) if for every s > 0 and k E N, there is an N such that

(1 + / 3 ) -1 ~ a i Y i k=l

aiXni ~ (1 + s) i--1

ai Yi k--1

for all N < ni < n2 < ... < nk and finite sequences of scalars (ai). (Y, (Yn)) is said to be a spreading model over X if there is a sequence (xn) in X generating the spreading model (Y, (y~)) and such that limnl
L p spaces

137

DEFINITION 17. Suppose X is a separable Banach space. We say that X is stable if for any two bounded sequences (xn) and (yn) in X and free ultrafilters b / a n d 12 on N, lim

lim Ilxn -+- Ym II -

(n)E/~ (m)EV

lim

lim Ilxn -+- Ym II.

(m) E'I2 (n) E/~

It is easy to see that all finite dimensional spaces are stable. In the case of Hilbert space a simple computation with the inner product establishes stability. Indeed, without loss of generality let (xn) and (Yn) be weakly convergent sequences in a Hilbert space X with limits x and y, respectively. (We denote the inner product by (., .).) Noting that lim(n)cU Ilxn II and lim(m)~V [lYm I[ exist and lim

lim IIx~ -+- Ym II2 -- IIx~ II2 - Ilym II2 - 2 lim

(n)c/g (m)E]2

= 2(x, y ) -

lim (xn, Ym)

(n)ELr (m)EV

lim

lim Ilxn + Ym II2 -[IXnll 2 --IlYm 112,

(m)EV (n)EU

(3.1)

we see that X is stable. Clearly any subspace of a stable space is stable. Maurey and Krivine proved that if X is a stable Banach space and 1 <~ p < oc, then L p ( X ) is also stable. In particular, Lp is stable. Several other standard function spaces are known to be stable, [89,87] and [40]. Also stability is an isometric rather than isomorphic property, co is easily seen to be not stable by letting xn -- ~in_=l ei and Yn -- en. For then lim

lim Ilxn -+- Ym II- 1

(n) Eb/(m)E])

and

lim

lim Ilxn -+- Ym I I - 2.

(m)EV (n) E/A'

At this point stability would seem to be a curiosity, but the interest stems from the idea of a type on a Banach space. DEFINITION 18. A type on a Banach space X is a non-negative real-valued function r on X such that there exists a sequence (Xn) in X and an ultrafilter b / o n N such that r (x) = lim(n)eu I[Xn + xll for all x 6 X. The type r is said to be realized by (Yn) and 12 if r ( x ) = lim(n)ev IlYn + x II for all x 6 X. Typically types are viewed as a (topological) subspace of the space of non-negative realvalued functions on X in the topology of pointwise convergence. Note that the types are actually Lipschitz with constant 1. Types have a natural scalar multiplication since for a scalar c ~ 0 1 lim Ilcxn + x l l - Icl lim Xn + - x C (n)eU (n)cU

Thus if r is a type which is realized by (Xn) and b/, then we can define c r to be the type realized by (cxn) and b/. Thus c r ( x ) - I c l r ( ( 1 / c ) x ) if c r 0. (Define Or - 0 . )

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The convolution of two types r and or, realized by (Xn), bl and (Yn), 12 respectively, is defined by lim

r * cr ( x ) =

lim IIx + Xn + Ym II,

(n) e/.d ( m ) c V

where r and o- are realized by (Xn) and L / a n d (Ym) and V, respectively. If the space X is stable the convolution is commutative. Furthermore the definition then does not depend on the particular realizations of r and o- and r 9 cr can be easily shown to itself be a type on X. A type r is said to be symmetric if r = - r , i.e., r (x) -- r ( - x ) for all x. Observe that if (Xn) is the fundamental sequence of a spreading model which is isometric to g p over X then the type realized by (Xn) and any ultrafilter/,4 is symmetric. DEFINITION

19.

A

scalars a, b, a r * b r

symmetric type r is said to be an s =

if for any non-negative

(a p + b p ) I / P r .

Notice that if (Xn) is a weakly null sequence in s and limn---,~ Ilxn II = P , 1 < p < cxz, then r ( x ) = limn~oc Ilxn + xll = (PP + IlxllP) 1/p, for all x e ep, is a symmetric type. Also (ar,br)(x)

= =

lim Ilx + Xn +Xmll

lim

n--+ oc m--+ oo

(llxll p

+ a P p p -+-bPpP) 1 / p -

(a p n t - b p ) l / P r ( x ) .

(3.2)

Thus r is an gp-type. Now suppose that (Xn) is a normalized sequence and L/is an ultrafilter that realize an p-type r on X. Then for any x e X,

lim lim . . . (nl)Eb/(n2) cb/

-

a i=1

lim

(nk)cld

r(x)-

ql

x nt-

aixni i--1

lim

(n)cld

x+

II

-- a l ' r . a2"c -k . . . . a k r ( x )

a

Xn 9

(3.3)

(3.4)

Thus (Xn) is the generating sequence of a spreading model over X which is isometric to (~p, (e,,)) over X. As noted above Krivine, [61], had earlier shown that in any basic sequence the space p, for some p, or co is block finitely representable. Usually finite representability tells us little about the infinite dimensional structure of the space. Maurey and Krivine show that in a stable Banach space Krivine's theorem can be used to obtain much more. THEOREM 20. Suppose that X is a separable stable Banach space. Then f o r some 1 <. p < cx~ there is a non-trivial g.p-type on X.

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THEOREM 21. Suppose that (Xn) is a sequence in a separable stable Banach space X which generates a spreading model which is isometric to (s p, (en)) over X. Then f o r every e > 0 there is a subsequence of (Xn) which is (1 + e)-equivalent to the basis of s The proof of Theorem 21 is quite easy. Given e > 0 we choose suitable ek $ 0 and using a compactness argument inductively choose a subsequence (x/7i) of (x/7) satisfying" for all k and (/~i)'l __V_[ - 1 , 1],

k-2 ~~j2~ j--I k-2

+ (l~k_l Ip -+-I~klP)l/Pxn~_,

-- ~

fljXnj + ([flk-I Ip + ]flkl p) l/pel

< 6k

j=l and

Z fljXnj

k-2 Z f l j X n j + flk-lel + flke2

j=l

j--I

k

< 6k.

Using these two inequalities one can iteratively keep combining the last two terms in y~] ~ j x j to obtain the result. The proof of Theorem 20 is more difficult. One first shows that a symmetric type cr on a stable space X is an s type for some 1 ~ 0 there exists 15 ~> 0 with ~r, (otcr) --/3cr. To get the existence of such acr is fairly involved and employs the notion of conic classes of types. See the original paper of Maurey and Krivine, [62,44,40] or [23] for proofs. The last reference proves the existence of s by using an ordinal index approach which is quite different from any of the others and is somewhat simpler. One uses Krivine's theorem in a certain manner to obtain a p. Then given e > 0 one considers

He,p(X) -- { (xi)~ ~ X: (xi)~ is (1 + e)-equivalent to the unit vector basis /7 Of~p}.

This is naturally a closed tree and can be given an ordinal index [ 16]. One shows that the index is Wl and so the tree has an infinite branch. In general, even for very well-behaved sequences in L p for 1 ~< p < 2, one cannot find a nice s basis by passing to subsequences. (See [43].) These results have been extended to weakly stable Banach spaces (defined as in Definition 17 except one also requires (Xn) and (Yn) to be weakly null) which allows for the case of co [12]. Thus every subspace of a stable Banach space contains a subspace (1 + e)-isomorphic to s One way to look at these results is to say that every subspace of L p, 1 <<,p < oe, (or any stable Banach space) contains s for some r, almost isometrically. One cannot find a distorted norm (see [85]) for s by looking at subspaces of stable spaces. It is natural to

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140

inquire if there are stronger results than Theorem 20. One such possibility is to replace X is stable by a condition implied by stability: the set of types on X, r (X), is separable in the strong topology (that generated by the pseudometrics dM(a, r) = sup{la(x) -- r(x)l : ~ X, Ilxll ~< M}). Unfortunately the dual of Tsirelson's space [99] satisfies this condition so the attempt fails. However Haydon and Maurey [45] have shown that if r(X) is strongly separable then X contains either a reflexive subspace or an isomorph of ~1. One might also ask whether there are other nice sequence spaces which embed isomorphically into L p. The approach of types using convolution cannot be used in these cases because they depend on formulas which must give rise to an ~p basis, [15,72]. The situation for p > 2 is constrained by the gp - ~2 dichotomy (see Section 4 or [49, Section 4]), but for 1 ~< p < 2 there is some room to maneuver. Dacunha-Castelle [30], showed that in order for a symmetric sequence space to embed in L1, the space must be isomorphic to an average of Orlicz sequence spaces. Additional work was done by Kwapi6n and Schtitt [64,63], and Raynaud and Schtitt [88]. The proofs of these results are too technical for us to delve into here, so we will content ourselves with stating a few of them. THEOREM 22. There is a universal constant K such that if Y is a subspace of L1 with

C-symmetric basis, then there is a probability space (I2, P) and a family of 2-concave normalized Orlicz functions F~o,o9 ~ I2 such that (KC)-If~

IlfllF~o dP ~< ]lfll ~< K C f s 2 [ifl[Fo~dP

for all f e Y. Bretagnolle and Dacunha-Castelle [22] showed that each Orlicz space with a 2-concave Orlicz function embeds into L1. Thus these two results give a pretty clear picture of the symmetric sequence spaces which can embed in L1 and hence in Lp, for 1 < p < 2, [93]. (See [53] for some additional material on symmetric and unconditional sequences in L1.) An example of a more general result from [88] is THEOREM 23. Lr

is isomorphic to a subspace of L1 if and only if there is a p such that Lr is order isomorphic to a sublattice of L1 (Lp) and X is isomorphic to a subspace

of Lp. 4. Subspaces of

Lp

As noted in Section 3, if X c s then for all e > 0, X contains Y with d (Y, s < 1 + e and Y is (1 + e)-complemented in s Not much more can be said about an arbitrary X c s X need not even have the approximation property, let alone a basis if p ~: 2, [26]. But we can say something if X has an FDD. THEOREM 24 ([57]). Let (En) be an FDD for X c_ g~p. Then (En) can be blocked into an g~p decomposition (Fn). Thus X ~ (Y~ Fn ) p.

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We shall sketch the proof which illustrates the blocking techniques developed by Johnson and Zippin in several papers [57,56] and later further developed [52]. This technique will have application for subspaces of L p. An extension of this theorem is given in Proposition 31. If A, B and C are positive numbers we write A c B if A ~< C B and B <<,C A . (Below (-) denotes the span.) PROOF. Let 6n $ 0 rapidly. We let (ei) be the unit vector basis of ~p and let Pn be the basis projection of gp onto (ei)~. Choose nl so that [l(l - Pn~)xll < 61 if x E BE~. Set m0 : 0 and m 1 = 1. The F D D (En) is necessarily shrinking so there exists m2 > m 1 with l] Pn~ x II < 61 if x E B[En]~ 9Then choose n2 > n 1 so that 2-1-1

I1(I - P~2)xll < 62

if X E B(Ei)I2.

Continue in this manner to obtain subsequences (m i) and (n i) of 1~ satisfying the following mn

for Gn -- (Ei)i=mn_l+l (a) IIP~.xll < 6k if x E B[ci]i>~+:, k >~ 1, and (b) I1(I - P ~ ) x l l < ek i f x E B(Gz)~: ' , k ~ > l . A (possibly finite) sequence (x~) is a skipped block sequence with respect to (Gi) if there exist integers r l <~ s l < s l + 1 < r2 ~< s2 < . . . so that x~ E (Gi)i=~ for all n. The conditions a) and b) yield that for such a sequence (x~)

provided the 6i'S decrease rapidly enough to 0. This nearly gives what we want but we must overcome the skipped requirement. We do this by a further blocking of (Gn) using LEMMA 25. Given e > 0 and n E 1~ there exists m E 1~ so that if x = ~ Xi E X with xi E Gi f o r a l l i, then there exists k E ( n , m ] with Ilx/~l[ ~< ellxll. The l e m m a is proved by contradiction. It is valid for any F D D for a reflexive space X. If not then for all m there exists x m - y~ x im E S x with X .m E Gi for all i and IIx m II/> e for all n ~ e for all i, a contradiction. Using the l e m m a we can find integers 0 = C0 < ~1 < g2 < " " so that if x : Y~.xi with xi E Gi for all i then for all n ~> 1 there exists Pn E ( g n - l , g n ] with IIxp~ll ~<

enllxll. Set Fn

-- (Gi)iE(~n_l,s

and let x be as above. Write yl - Y - ~ l - l x i

and Yn -

_l+lXi for n > 1. Then (Yn) is skipped so [[xll 2 ][ Y~Yn[] 2 (~-.~ ]]ynllp)l/p" Furthermore if x -- ~ zn, Zn E Fn then by the triangle inequality for some constant C,

(Y-~ IIznIIp)I/PL(~_. Ilyn[lP) 1/p. Thus (Fn) is the desired blocking.

7q

What can be said about a general subspace X of Lp ? What subspaces must it contain? We begin by summarizing some results from the basic concepts chapter which describe which L p- or s p-spaces e m b e d into others, [49, Section 4].

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THEOREM 26. Let 1 <<.p, q < cx~. (a) g p ~ eq if p ~ q. (b) Lp ~ Lq (and in fact isometrically) iff q <<.p <<.2 or if p = 2. (c) g.p ~-+ Lq (and in fact isometrically) iff q <<.p <<.2 or p = 2. If X c_ Lp, then X contains an isomorph of s for some q. This follows directly from the results in Section 3 but as noted in the basic concepts chapter is much easier to prove directly in the case 2 < p < ~ . Our next theorem addresses this and also the case of when a subspace of L p for 1 ~< p < 2 must contain a copy of gp. Recall [49, Section 4] that a subspace X of L1 is reflexive if and only if B x is uniformly integrable. And if X is not reflexive then X contains a copy of el. Part (c) extends this to Lp-spaces.

THEOREM 27 ([58]). (a) Let 2 < p < cxz and let (Xi) be a normalized weakly null sequence in Lp. Then there exists a subsequence (Yi) which is equivalent to either the unit vector basis of g.p or to the unit vector basis of g2. (b) Let 2 O, X contains a subspace Y which is (1 + e)-complemented in Lp and satisfies d(Y, g~p) < 1 + e. (c) Let 1 <~ p < 2 and let X be a subspace of Lp. If {Ixl p" x E Bx} is not uniformly integrable then f o r all e > 0 there exists Y c X with d(Y, g.p) < 1 + e and Y is (1 + e)-complemented in Lp. PROOF (Sketch). (a) is proved by noting that since Lp has an unconditional basis we may assume by passing to a subsequence and perturbing that (xi) is a block basis of the Haar basis and hence is unconditional. Thus by (1.9) we automatically have an upper ~2 and lower g p estimate. If Ilxi 112~ 0, then one shows that some subsequence is a perturbation of a disjointly supported sequence, the latter spanning a subspace of L p which is 1-complemented and isometric to e p. Otherwise (xi) may be presumed to be a semi-normalized unconditional basic sequence in L2 and since II" IIL2 ~ II" IIL p we obtain a lower/~2 estimate and thus the ~2 alternative. The extrapolation principle [49, Section 4] yields the complementation. (b) follows by similar considerations. To prove (c) for p > 1 we note that from the hypothesis one can obtain a normalized sequence (Xi) ~ BX, 6 > 0 and disjoint measurable sets (Ei) with IlXi[Ei lip > ~ for all i. By passing to a weakly convergent subsequence, taking differences and renormalizing we may assume that (xi) is in addition weakly null and moreover unconditional. LEMMA 28. Let (Xn) be a normalized h-unconditional basic sequence in Lp, 1 <<.p < 2. Assume there exists disjoint measurable sets (Ei) with [[xi IEi ]lp >/~ f o r all i. Then there exists K -- K()~, p, 3) so that (xi) is K-equivalent to the unitvector basis of g.p. PROOF. By (1.10) it suffices to show

IlaixilEi

I1)

llZaixill

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143

The mapping T ' [ ( X n ) ] --+ (Y~ L p ( E n ) ) p given by T x -- (x[En) is norm 1. The diagonal map D ( y ~ a~x~) -- (a~xn IEn) is bounded with ]]D[] ~< )~ [71, p. 20]. This yields the result. E3 To obtain the "1 § e" conclusion of (c) we need to pass to a normalized block basis (yi) of (xi) which is a perturbation of a disjointly supported sequence. For example if p = 1 we may assume that xi = ui § di where (di) is disjointly supported and bounded away from 0 in norm (hence equivalent to the unit vector basis of g l) and (u i) is uniformly integrable. This follows from the subsequence splitting lemma. THEOREM 29. I f ( f i ) CC_BLi then there exists a subsequence f / -- ui § di where ]Ui] /~ ]di] -- 0 f o r all i, (u i) is uniformly integrable and (di) is a disjointly supported sequence. Since a uniformly integrable subset of L l is relatively weakly compact [49, Section 4], we may assume that (u i) is weakly convergent and hence cannot be equivalent to the unit vector basis of el. Thus some normalized block basis (yi) of (Xi) is a perturbation of a block basis of (di) which yields (c). D REMARK. While Hilbert subspaces of Lp (2 < p < cx~) are automatically complemented, for 1 < p < 2 every Hilbert subspace of L p contains a subspace which is complemented in Lp. Indeed let (xi) be a basis for X equivalent to the unit vector basis of ~2 with biorthogonal functionals (x*). Let Q ' L q --+ X* be the natural quotient map (p1 + q1 - 1). Let Qyi - x* with ([[yi [[) bounded. By Theorem 27(a) it follows that some subsequence of (y~) must be equivalent to the unit vector basis of ~2 and [(y~)] is complemented in Lq by the orthogonal projection. It follows that [(x~)] is complemented in L p. A subspace of L p which is "small" embeds into g p. THEOREM 30 ([51,46]). Let X c_ L p, 1 < p :fi 2 < cx~. (a) I f 2 < p < cxz and ~2 5/-+ X then X ~ gp. (b) I f 1 < p < 2 and there exists K < ~ so that every normalized weakly null sequence in X has a subsequence K - e q u i v a l e n t to the unit vector basis o f g.p then X ~ g.p. Part (a) has been strengthened [60] to yield that for all ~ > 0, X l_+~s We shall sketch a unified approach to proving both parts. Let T~o -- {(nl . . . . . nk)" k E N and ni E • for 1 ~< i ~< k} be the countably branching tree orderedby (nl . . . . . nk) <~ (ml . . . . . me) ifk ~< andni - - m i for/~< k. W e s a y T E T~o(X) i f T - - {X(nj ..... nk)" (ni . . . . . nk) E Too} c_ S x . T is O9

O)

a weakly null tree if x(n) ~ 0 a s n - - + c x ~ a n d f o r a l l (nl . . . . . nk) E T~o, x(nt ..... nk,n) ~0 cx~ as n --+ ~ . If X has an FDD (En) then T is a block tree with respect to (En) if (X (n))n=l and (x(ni ....9 nk ,n~) n=l ~ are all block bases of (En ) forall (nl . . . . . nk) ET~o.

PROPOSITION 31 ([84]). Let 1 <<.p < cx~ and let (En) be a boundedly complete F D D f o r a Banach space X. A s s u m e there exists K < cx~ so that every block tree T E T~o(X) with respect to (En) admits a branch K - e q u i v a l e n t to the u n i t v e c t o r basis o f g.p. Then (En) can be blocked into an g p decomposition.

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PROOF. By the proof of Theorem 24 (using Lemma 25) it suffices to show that (En) can be blocked into a skipped gp decomposition, i.e., there exists a blocking (Fn) of (En) and C < ec so that if (xn) is any normalized skipped block sequence of (Fn), then (Xn) is Cequivalent to the unit vector basis of g p. C may be taken to be K + 1. By a compactness argument it suffices to prove the following. (A) 3nl 'v'xl E S(Ei)~ 3n2 Vx2 G_ S(Ei)~ 3n3 Vx3 E S(Ei)~ ... s o that ( x i ) ~ is Kequivalent to the unit vector basis of gp. The formal negation of (A) is (B) Vnl 3Xl E S(Ei),,1 Vn2 3x2 E S(Ei),~2 Vn3 3x3 E S(Ei),,~... so that (xi) is not Kequivalent to the unit vector basis of gp. Since these are infinite sentences we cannot automatically assume that either (A) or (B) must be true. However by Martin's theorem that Borel games are determined [73] it does follow that (A) or (B) is true. Indeed consider a two player game where player (I) chooses n l E N and player (II) chooses X l E S(Ei)~. Then (I) chooses n2 > max(suppxl), the support being with respect to the Ei's. (II) chooses x2 E S/E i )n~ and so on. An outcome of this game is a sequence in

~ - - { ( n l , x l , n 2 , x2 . . . . )" Xj E S(Ei)nj and nj+l > max(suppxj) for j EN}. Player (I) wins if the outcome belongs to r -- { (n 1, x l, n2, X2 . . . . ) E ~" (Xi)C~ is K-equivalent to the unit vector basis of g p }. The precise meaning of (A) is that player (I) has a winning strategy while (B) means that (II) has a winning strategy. If ~ is given the relative product topology then r is a closed subset of ~. Martin's theorem yields then that this game is determined, i.e., either (I) or (II) has a winning strategy. But if (B) holds one easily constructs a block tree T E To(X) with respect to (En) so that no branch is K-equivalent to the unit vector basis of ~ p. D By employing some further blocking and perturbation arguments one can extend this result to PROPOSITION 32 ([84]). Let 1 < p < oc and let X be a reflexive space. Assume there exists K < oo so that whenever T E T~o(X) is a weakly null tree then some branch of T is K-equivalent to the unit vector basis of g.p. Then there exists a sequence of finite dimensional spaces (Fn) so that X ~ ( ~ . Fn)ep. PROOF OF THEOREM 30. One first shows that the hypotheses of both (a) and (b) imply the hypothesis of Proposition 32. This depends upon the fact that X c L p and arguments like those used to prove Theorem 27. For example (b) yields that there exists 3 -- 3(K) > 0 so that if (xi) is a normalized weakly null sequence in X then some subsequence can be written as x if -- Yi -Jr-di where

L p spaces

145

Yi /X di

- - 0 , the di's a r e disjointly supported and Ildi I1 ~> 6 for all i while (a) yields in addition that one can take []di ]1 --+ 1. Using this given T ~ T~o(X) one can select a branch of the tree exhibiting similar behavior and hence is C (K)-equivalent to the unit vector basis

Of~p.

The proof of Proposition 32 yields that if X c Y, where Y is reflexive with an FDD (En) then one can block (En) into (Fn) so that X ~ ( ~ Fn)ep. Thus one can block the Haar basis into an FDD (Fn) so that X embeds into ( ~ Fn)ep. Since each Fn ~ gp" for some m n ~ N, the latter space is a subspace of ~p. ff] From Theorems 30 and 27 it follows that a subspace X of L p for 2 < p < oc which is not isomorphic to ~2 nor to a subspace of s must contain ~2 | g p. What additional conditions on X ensure that X ~ g p ff~ g2 ? The following gives one such result. THEOREM 33 ([52]). Let 2 < p < oc. Let X be a subspace of Lp which is isomorphic to a quotient o f a subspace ofg~p • ~2. Then X embeds into g~p 9 g~2. The method of proof is a more complicated blocking argument which ultimately produces a blocking (Hn) of the Haar basis so that the natural mapping

P

is an into isomorphism. The following is open. It concerns the next layer of small subspaces of Lp. PROBLEM 34. Let X be a subspace of Lp, 2 < p < oc and assume that ( ~ s Does X ~ s 9 s

~

X.

The situation for subspaces of L p with 1 < p < 2 is of course more complicated (cf. Theorem 24). But we can say the following. THEOREM 35 ([93]). Let 1 <. p < 2. Let X be a subspace of Lp. Then either g~p ~-+ X or X ~ Lq f o r some p < q <. 2. In particular if X is a reflexive subspace of L1 then X embeds into Lq f o r some q > 1. PROOF (Sketch). We shall briefly sketch the technical steps of the proof in the case p -- 1. c

By Theorem 26 if X c_ L1 is not reflexive then el ~ X. Let Is(X) be the r-absolutely summing norm [49, Section 10] of the canonical map from Loc into X* (where 71 4- 71 -- 1). Step 1" Let 1 < q ~< 2. lq (X) < e~ implies that X ~ Lq. One shows that there exists an integrable function q~ on [0, 1] with ~b(t) > 0 for all t 6 [0, 1] and K < oc so that for all x 6 X X

~K Lq(v)

X LI(v)

D. Alspach and E. Odell

146

where d v = ~bdm. Thus since x --+ x is an isometry of L1 with L l(v), the theorem will follow provided we have Step 2: Iq (X) < cx~ for some 1 < q ~< 2. If not one shows that s is finitely representable in X. This yields that Bx is not uniformly integrable and hence X is not reflexive. D

5. s

1 < p < cx~, p 7~ 2

Recall that X is a s if and only if there is a constant )~ < ~ such that for each finite dimensional subspace Z of X there is a finite dimensional subspace Y of X such that d(Y, s /7p) <~ )~, where n -- dim Y, and Z C Y. For the range of p, 1 < p < c~, this is precisely equivalent to the statement that X is isomorphic to a complemented subspace of L p ( # ) for some # but X is not isomorphic to s [66,68]. This latter description has turned out to be the most useful for determining whether a space is a s The local description has been useful for a few results concerning isomorphic properties of these spaces and for some results in the local theory and the related area of operator ideals. One consequence of the local description is that THEOREM 36. If l < p < oc and X is aseparable ~p-space then X has aSchauderbasis. We will not give the complete proof, but we can outline the steps. It has already been shown in [49, Section 9] that because X is a s there exists a )~ < cx~ and a sequence of finite rank projections (Pn)n~ 1 defined on X such that, IIe~ll ~< x for all n, d(Pn(X),s

kn

) ~ )~, where kn = d i m P n ( X ) , Pn(X) C Pn+l(X) for all n, and Un~ l Pn (X) -- X. The next step is to replace the projections Pn by a family of commuting projections. In order to accomplish this we need to have Qn*(X*)• - ker Qn c ker Qn-1 = On*_l(X*)• In other words O*(X*) 3 On*_l(X*). Assume that we have (Oi)n-~. We have that Pk*x* ~ x* weakly for each x* E X*. Therefore there are convex combinations R~ of (P~) which are essentially the identity on P*n(X) U O*-i (x). Fix a large j and let R -- R j . Because R is finite rank, we can choose k such that Pk is essentially the identity on R(X). By a perturbation argument we can replace Pk by a projection P which is the identity on R(X). Let Qn = R + P - R P. kn We now have an FDD for X. The last step is to use the fact that d(Qn(X), s ) is uniformly bounded and the fact that X contains a complemented subspace isomorphic to s to create a basis. A little care must be used here since it is not true that a space with an FDD (Fn) in which each summand (Y~nm__lFn) has a basis with good constant has a basis, [98]. The idea in this case is to make sure that the basis is chosen so that the basis projections are similar to projections of known uniformly bounded norm in s m W e direct the reader to [70, p. 203], [54] or [82] for the complete argument. The construction of the basis for X is highly conditional so it is unlikely that this approach could be used to show that/2 p-spaces have unconditional bases. Next let us consider the isomorphic types of complemented subspaces of Lp. The two obvious examples are Lp and s Clearly Lp is isomorphic to (Y-~n~__lLp)p since we can

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147

realize this space as (y-~n~ 1 Lp (2 -n, 2 -n+l) p. Thus by the decomposition method any complemented subspace of L p which contains L p complemented is isomorphic to Lp. In particular L p G g . p ( F ) , where F is a finite or countably infinite set, is isomorphic to L p. It follows that these two spaces are all of the isomorphic types that we get as the range of a conditional expectation onto a (non-finite) sub-cr-algebra of the Lebesgue sets. A slightly more general class of projections are those of the form P f -- g E ( f g -1 lsuppg I~) where g E L p and/3 is a sub-a-algebra. This is the general form of a contractive projection, [10, 100]. These do not give us any new spaces since the range of such a projection is isometric to L p (#) for some measure #. We have already seen that the sequence of Rademachers or Gaussians span complemented subspaces isomorphic to g2. It is easy to see then that there are complemented subspaces of L p which are isomorphic to ~p @ ~2 and (y~ ~2)p. A natural question is whether L p can be split into two spaces which are not isomorphic to L p, i.e., are there complemented subspaces X and Y of L p such that L p -- X 9 Y but neither X nor Y is isomorphic to L p? As we have seen in the case of L p (and s (Theorem 7 and Theorem 15) the answer is no. The fact that L p is primary tell us "half" of the s but little about the far more interesting other "half". Let us now consider the one complemented subspace of L p which is not a s Suppose that X is a subspace of L p which is isomorphic to ~2. By [58] if p > 2, the L p and L2 norms are equivalent and the orthogonal projection is bounded. If p < 2 it is no longer the case that a subspace isomorphic to ~2 is complemented. This was first proved for certain values of p by using some results from harmonic analysis, [13, p. 52], [94,20]. Later a completely different approach yielded the existence of uncomplemented subspaces of L p , for each p, 1 < p < 2, which are isomorphic to g2, [13]. This immediately implies the existence of subspaces of L p which are isomorphic to e p and not complemented, since g p is isomorphic to ( ~ g~)p. For 2 < p < oc, Rosenthal showed that there are uncomplemented subspaces of g p which are isomorphic to g p, [91, Theorem 6]. Curiously, such a subspace cannot be the span of a sequence of independent random variables, [31,17]. For some time it was thought that the situation might be different for p - 1, but in [ 17] Bourgain shows that there are subspaces of L1 isomorphic to el which are uncomplemented. If we look at projections there is one very special property of g2 subspaces of L p, 1 < p < 2. While it may not be true that the subspace itself is complemented, as we have seen, for any subspace X of L p which is isomorphic to g z, there is an infinite dimensional subspace Y of X which is complemented. If X is a subspace of L p which is isomorphic to p, then as was shown in Section 3 it must contain a subspace which is complemented (and necessarily isomorphic to gp). It follows that for a subspace of L p which is isomorphic to ~p (~ ~2 there is a subspace isomorphic to ~p • ~2 which is complemented in L p . This sort of hereditary complementation property does not hold for general complemented subspaces of L p. Consequently, a special class of operators has been introduced which have somewhat better properties with respect to g2 subspaces. This concept occurred implicitly in [91] and then was developed further in several papers, [7,5,6,38,52]. In [58] it was shown that subspaces of L p, 2 < p < oc, which are isomorphic to/~2 are complemented by showing that the orthogonal projection acts as a bounded operator on L p. This is an example of an operator which is simultaneously bounded in both norms on L p. This, as it turns out, is not really an unusual situation.

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THEOREM 37 ([47,53]). Suppose that 1 <. p < oo and that T is a bounded linear operator on L p. Then there is a constant K and a function g such that

f lg-1/,,r(f . gl/p)12g <~ K 2 IITII2

f

Ifl 2 g

for all f ~ Lp(g d)~) A L2(g d)~). Moreover K is independent ofT. (Here)~ is Lebesgue measure on [0, 1].) Thus the operator S on Lp(g d~.) N L2(g d~.) defined by S f -- g - 1 / p T ( f g l / p ) is bounded in the L2 and Lp norms. S is the operator on Lp(g dl.) induced from the operator T by the change of measure isometry ~b(f) -- f g - 1 / p from Lp onto Lp(g d~.). The proof of this result uses change of density arguments due to Maurey, [76]. Rosenthal [91] used such operators on subspaces of Lp as well and in cases where the subspaces carried two norms which essentially defined the subspace. An important construction which has its roots in this paper is that of (p, 2)-sum of subspaces of Lp. DEFINITION 38. Let (Xn) be a sequence of subspaces of Lp(S-2, #) for some probability measure #, and let (Wn) be a sequence of real numbers, 0 < Wn ~< 1. For any sequence (Xn) such that Xn ~ Xn for all n, let II(Xn)IIp,2,(wn)= max{ ( ~

Ilxn lip p) 1/p,

tz ilXnll]w 2t

and let = {(Xn)" Xn ~ Xn for all n and II(Xn)llp,2,(Wn) < CX~}. We will say that X is the (p, 2, (Wn))-sum of {Xn}. In the special case that Wn = 1 for all n, we will sometimes write ( ~ Xn)p,2 instead of ( ~ Xn)(p,2,(1)). Also, in this context, and

llx.)ll2 (E IIx.ll .) _

Notice that if we take the sequence (Wn) such that ~ W2p/(p-2) inequality

2

<

OO then by H61der's

Thus (y~ Xn)(p,2,(Wn)) is isomorphic to the gp sum of (Xn). (p, 2)-sums of spaces give us a very general method of producing new spaces. We shall see below that these sums and a closely related tensor product are the basic tools for constructing all known/2p-spaces.

L p spaces

149

In order to deal with spaces with a norm defined in terms of an L2 norm and an Lp norm and with similar subspaces of Lp, we will use the terms (p, 2)-bounded, (p, 2)isomorphism, etc., to indicate that the map is bounded, is an isomorphism, etc., in both norms. For example, if T is a map from ( ~ Xn)(p,2,(w,,)) into Lp, we would say that T is (p, 2)-bounded with IlTllp,2 ~< K, if there exists a constant K such that IlT(xn)ll2 ~< 2 1/2 and IIT(x~)llp <~KIl(x~)llp,2,~,,~ for all (Xn) ~ (Y~ Xn)(p,2,(w,,)). In the K(y~ Ilxnll~w~) second inequality we have used II(x~)II p,2,~,,) which is equivalent to the Lp-norm on some subspace of Lp. For an operator defined on a general subspace Y of L p, the inequalities would be IITylI2 ~< KllYlI2 and IITylIp <. KllYlI/,. At times using the (p, 2, (wn))-norm rather than the Ln-norm gives better estimates. It may appear that in general these (p, 2)-sums have little to do with subspaces of Lp but it follows from Rosenthal's inequality, (1.11), that such a sum can be realized inside of L p. Indeed, first observe that if X is a subspace of L p[O, 1] and 0 < w ~< 1, then the subspace

Y - {ll)-2/(p-2)x(s)(l[o, w2p/fp-2)/2] - l(w2p/{p-2)/2,w2,,/t,,-2)])(t): x E X } of Lp([0, 1] x [0, 1]) is isometric to X in the p-norm and ]lTxl]2 = wI]xII2, where T is the L p isometry. Now given a sequence of subspaces of L p[O, 1], (Xn), and a sequence (wn) of numbers in (0, 1], we can isomorphically embed (y~ Xn)~p,2,(w,,)) into L p ( 1-[n=l [0, 1] x [0, 1]) by embedding Xn isometrically into the subspace of functions which depend on the 2n and 2n + 1 coordinates of the infinite product as above. Let Tn denote this isometry. If (x~) is in (Y~ X~)~p,2,(w,,)), then (Tnxn) is a sequence of independent, mean-zero elements of L p I-In=l [0, 1] x [0, 1]). It is now immediate from Rosenthal's inequality that II ~ T~ x~ II is equivalent to II(x~)II. Now observe that if X = (Y~ X~)~p,2) and Y = (Y~ Y,)(p,2) and T~ is a (p, 2)-bounded operator from X,, to Y, for each n, then the operator T from X into Y defined by T (xn) = (Tnxn) is (p, 2)-bounded with norm at most max IIT~lip,2. This makes it possible to use the decomposition method in spaces with (p, 2) structure. We now turn to a description of the known s We have already encountered the (mutually non-isomorphic) basic spaces Lp, g.p, g.p 9 g~2and (y~ g2)p. A basic question is whether any of these spaces has any other complemented subspaces. Of course, we know that every infinite dimensional complemented subspace of g p is isomorphic to g p. It is not hard to see that for p > 2 that a complemented subspace X of g p 9 g2 is isomorphic to g2 or contains g p. If the subspace does not contain g2 it is isomorphic to a subspace of g p by [51]. It now follows from [57] that X is isomorphic to g p. If a complemented subspace of gp | g2 contains both g2 and gp then it contains a complemented subspace isomorphic to gp | ~2. At first it might seem that it is possible to apply a (p, 2) decomposition method in this case to conclude that the subspace is isomorphic to g p | g2, however no such proof has been given. Instead, Wojtaszczyk and Edelstein [34] used a functional calculus approach to show that a complemented subspace of gp @ ~2 which contains a complemented subspace isomorphic to s | ~2 is itself isomorphic to s ~3 ~2- An important tool in this proof is the fact that every operator from ~2 to s or s to ~2 is strictly singular and their proof generalizes to other direct sums of spaces which are incomparable in this sense.

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The complemented subspaces of (Y]~~2)p have also been classified, [83]. It turns out that only the obvious spaces occur, g2, gp, gp (9 ~2 and (}-~'g2)p. The proof is rather technical and mainly consists of carefully distinguishing between complemented subspaces which are isomorphic to complemented subspaces of gp | ~2 and those that contain (y]~g2)nIn 1972 Rosenthal proved his inequality and used it to show that there were several other s In particular he defined the space X p which had some surprising properties and led to the construction [21] of uncountably many isomorphically distinct s Because X p has proved to be a fundamental space we will present some of Rosenthal's results in detail. To construct the space X p for p > 2, we let Xn be the one-dimensional space spanned by 1[0,1] for all n ~ N, and let (wn) be a sequence of numbers in (0, 1] such that for every e > 0, ~n:wn<e w2p/(p-2) : oo. Define X p to be (y]~Xn)(p,Z,(wn)). In the sequel we will refer to a sequence (Wn) satisfying this condition as one which is thick near O. A priori this is not one space but a family of spaces, however they are all isomorphic. Before pursuing this and some other questions, note that the restriction on the sequence (Wn) is made to avoid getting one of the three spaces g2, gp, and gp ~3 g2. As we have seen above if ~ W2p/(p-2) < OO the (p, 2, (Wn))-sum becomes an gp-SUm. If infwn > 0 then ( ~ Ixn [2 Wn) 2 1/2 >~ (infwn)(y]~[xnlP) lip so that the norm is equivalent to the g2 norm. There is also a combined case where the sequence breaks into two pieces and thus yields ~p (~ ~2.

We have two issues to resolve with X p at this point. How do we see that it is a s space and how do we see that we really have only one isomorphism class? The solution to both involves the construction of (p, 2)-bounded projections. (Also note that we can then define Xq for q < 2 to be the dual of X p where p and q are conjugate indices.) We will , ( OO use the representation above for (y]' Xn)(p,2 (w,,)) in Lp 1-1,,=1[0, 1]), where the X,, are one-dimensional spaces spanned by a function Xn whose modulus is an indicator function. For each n E N let x n = ]lxn 1122Xn and notice that P x - y]~n~ ( f x n X)Xn is the orthogonal projection onto the closed span of (Xn) in L2. Thus P is bounded in the Lz-norm. Moreover by Rosenthal's inequality (1.11)

n=l

~< C max {

, n=l

Ix~,xl n=l

1"

IIx. II p

9

Because P is bounded in the L2-norm and p > 2, the first expression in the max is bounded by Ilx l]p. To see that the second is also notice that

x*~x n=l

IIx. llpp

L p spaces

151

Because Ixn I is an indicator function, Ilxn II2 - IIx~ IIp and thus (Xn Ilxn 1122Ilxn IIp) is a norm one sequence of mean zero symmetric independent random variables in Lq, where q -p / ( p - 1). Consequently, the sequence is unconditional and has an upper s estimate. Therefore oo

f x, x IIx, 1122 IIx, II p n--I

- fx~b~x~llx~l12211x~llp n--

1

~< IIx IlpC

Ib, Iq n--I

for some constant C. This proves that the second expression is also bounded. The argument just given has been generalized to replace the one-dimensional spaces by the range of an orthogonal projection which is bounded on Lp. See [38, Theorem 4.8]. THEOREM 39. Let 2 < p < cx~ and let (wn) be a sequence of scalars in (0, 1]. For each n let Xn be a subspace of mean zero functions in L p such that the orthogonal projection from Lp onto X , , Pn, is boundedand supllPnll < cx~. Then (~-~ Xn)(p,2,(w,,)) is isomorphic to a complemented subspace of L p. The proof of this generalization is very similar to the proof given above. The main point is to realize that the pairs (x,, x*) can be replaced by orthogonal projections and their adjoints. Next we will use a version of the decomposition method to show that there is really only one isomorphism class for the spaces X(p,Z,(w,,) ) with (w,) thick near 0. To use the decomposition method we need a suitable infinite sum of spaces. Rosenthal, [91 ], called this a symmetric sum; in our terminology this is a (p, 2, (1)) sum, i.e., the sequence (wn) is just the constantly 1 sequence. As we noted earlier a key feature of the s that is used in the decomposition method is that one can combine a sequence of uniformly bounded operators on the individual summands to create an operator on the sum. For the (p, 2, (1))-sum we have a similar property. LEMMA 40 ([6, Lemma 2.51). If {Xn} and {Yn} are two sequences of subspaces of Lp(S2, #) for some probability measure #, and there is a constant K and (p, 2)continuous operators {Tn} such that I]Tnllp,2 <~ K for all n E N then the operator T = ~ OTn is a K (p, 2)-bounded operator from ( ~ Xn)(p,Z,(w,,)) into (~-~ Yn)(p,Z,(w,,)). Consequently, (1) if each Tn is a (p, 2)-isomorphism and IITn-11]p,2 ~ K as well then T is a (p, 2)isomorphism; (2) if each Tn is a projection, then y~ G Tn is a (p, 2)-bounded projection onto (ZTnXn)(p,Z,(w,,))

9

With this lemma it is easy to verify that the decomposition method works for subspaces of L p, X and Y, provided X and Y are (p, 2)-isomorphic to (p, 2)-complemented subspaces of Y and X, respectively, and X is (p, 2)-isomorphic to the (p, 2, (1)) sum of infinitely many copies of itself. Notice that the condition that (wn) is thick near 0 is not

D. Alspach and E. Odell

152

affected by redundancy. Thus we can take a thick sequence (Wn) and define Wm,n = Wn for all n, m in N, to get a space X = X(p,Z,(wm,n)) which is obviously (p, 2)-isomorphic to the (p, 2, (1)) sum of infinitely many copies of itself. Now we show how to construct projections which are (p, 2)-bounded. As Rosenthal discovered one can take a particular combination of any finite set of weights (Wn)ncF and get a block of the natural basis (en) of X(p,2,(Wn)) and a corresponding functional which is (p, 2)-continuous. If

Y -- E'wn2/(p-2)D~n and

y* -

Zn6F It~2p/(p-2) ~ 1, let

(E

ncF

w2P/(P-2)

)-'

n~F

E l l ) n2(p-1)/(p-2)e,n. nEF

Now notice that the map

Pn(Eanen)

-

-

y*(Eanen)y E w2(p-1)/(P-2)an) y n~F

is (p, 2)-continuous. Indeed,

(~

w2P/(P -2)

)'(z

)

ll02(p-1)/(P-Z)any

n6F _

( z ll)2p/(P ) -2) '1~ Wn2~ "~ 2'a~( Z -113n2~-2' ncF

<~

n~F

,j2

n6F

lan I2 1/3 n2 n~F

by the Cauchy-Schwarz inequality and

(~ Wn2p/(p-2) )'(z

Wn2 ( p - 1 ) / ( p

2,an) y

n6F

-

~~'1;~: w,

ncFk

110- p / ( p - 2 )

n6F

It is easy to see that a block basis (yk) of

(E

a~ Zl(~m:~~) '~ E

n6F

Yk --

P

lakl p

)-'E anen n6Fk

X(p,2,(Wn)) with

(p-1)/p

Ela.l" n6F

1/p

L p spaces

153

for all k is (p, 2)-equivalent to the natural basis of X(p,2,(vn)) where

v.-(

~ , a n , 2 w.2 ) ' / 2 / ( ~_~ la.I p n~Fk n6Fk

Notice that for the special case an --

2/(p-2)

ton

),/p

the weight vk is

)(p-2)/(2p) Z to2p/(p-2) n~Fk Also if K1 and K2 are positive constants and (Wn) and (W~n) are two sequences in (0, 1] such that KlWn <~ w n <~ KZWn for all n E N then the natural map from X(p,2 (Wn)) to X(p,2,(w.)) is a (p, 2)-isomorphism. An important observation is that given a sequence (wn) that is thick near 0 and any )~ 6 (0, 1] there exists disjoint finite sets (Fn)n~N such .2p/(p-2)-~(p-2)/(2p)/)~ < 2, for all n. that 1/2 < (ZjEFn "wj ] Assembling these observations we see that we can apply the decomposition method to X(p,Z,(wm,n)) and any X(p,Z,(w~n)) where (W~n)is thick near 0. Therefore

For any two sequences of numbers (Wn) and (wln) in (0, 1] which are thick near 0, X(p,Z,(wn)) and X(p,Z,(w~7)) are (p, 2)-isomorphic. THEOREM 41.

In the early 1970s research concentrated on the isomorphic properties of the known /3p-spaces, but because it was known that there were uncountably many isomorphically distinct separable C(K)-spaces (and thus s it was suspected that there might be infinitely many isomorphically distinct s for every p. The first construction was given by Schechtman, [96]. The basic tool is a tensor product which is specific to subspaces of Lp. DEFINITION 42. Suppose that X1 and X2 are subspaces of Lp(S-2i, ~V'i, # i ) , i = 1, 2, respectively. Let

XI Q X2 -- [ f (col)g(co2)" f E Xl, g 6 X2, f lf (col)g(co2)lP d#l dlz2 < cx~] C Lp(~21 x ~-22, r l @ r 2 , #1 x #2). The definition is most useful in the case that the measure spaces are both separable probability spaces and we will restrict to that case. It is not difficult to see that if X1 and X2 are complemented with projections P1 and P2, respectively, then X1 | X2 is complemented by the tensor product ofthe projections, i.e., (P1 | P z ) ( f | g) (col, co2) = P1 ( f ) (COl)P2 (g(co2) for pure tensors f | g and then extend linearly. It is also not difficult to see that Lp(S'21, ~W'l,#1) | Lp(S22, r 2 , #2) -- Lp(S-21 x S'22, r l @ r 2 , #1 x #2), and thus the tensor product of/2p-spaces is again a/2p-space. k Schechtman showed that the spaces @n= 1 X p are isomorphically distinct. His argument is based on the fact [92] that for p < 2, g~, p < r ~< 2, is isomorphic to a subspace of X p

154

D. Alspach and E. Odell

and is uniformly p-integrable. Using this Schechtman showed that @kn= 1 X p contains the k-fold iterated sum

(Z(Z (Z'r')r2 where 2 ~> rl > r2 > ... > rk ~> p, but not a (k + 1)-fold iterated sum. Thus the spaces 1 Xp, k - , 1 2,.. . , are different. The tensor product has one serious drawback in that the norm of the tensor product of the projections grows rapidly. (See [6, Chapter 1] for some details.) Thus there was no obvious way to use this to produce uncountably many different separable/2p-spaces. In [21,19] a completely different approach using independent sums was used to produce COl distinct/2p-spaces. In order to construct/2p-spaces based on independent sums without having difficulties with constants, Bourgain, Rosenthal and Schechtman introduce an isomorph of L p which is convenient for the construction. Let T denote the usual dyadic tree and for each node N of 7- we consider a copy of {0, 1}, X2N, with the measure of each point one half. Now we define a certain subspace L of L p on the natural product algebra and probability measure on I-IN~'T ~(2N, which is generated by those functions which are measurable with respect to any branch sub-a-algebra. That is, if B is a sequence of pairwise comparable nodes in the tree, then we consider the sub-o'-algebra generated by sets which depend only on the coordinates in B and L contains the functions which are measurable with respect to each such sub-o'-algebras. It is obvious that this space contains complemented subspaces isomorphic to L p of the Cantor set (with the usual product measure) because the o'-algebra generated by one branch is isomorphic to the Cantor a-algebra. Less obvious is the fact that this subspace L is complemented and thus isomorphic to L p. By considering subspaces of L, Rp, which are generated by functions which depend on certain subtrees of 7- of order c~, Bourgain, Rosenthal and Schechtman show that there are uncountably many s Their argument is actually to some extent an existence argument. The proof depends on an ordinal Lp-index which they cannot compute precisely but are able to show grows with the ordinal index of the subtree. In [6] the precise isomorphic classification of these spaces is given. It is also shown there that these spaces are not isomorphic to any of those constructed by Schechtman. More details about the ordinal L p index can be found in [11 ]. For the case p = 1, Bourgain [ 18,19] adapted some of the ideas from [21 ] to produce an uncountable family of s with the Radon-Nikodym property. The difficulty was that the analogous spaces for p = 1 are not complemented so Bourgain showed that there are enveloping/21-spaces which are complemented. Earlier Johnson and Lindenstrauss, [48], showed by a completely different approach that there are uncountably (actually 2 s~ many separable non-isomorphic s which have the Radon-Nikodym and Schur properties but do not embed in separable dual spaces. At this point a complete isomorphic classification of the separable s seems out of reach. Most of the known relations among the smaller spaces can be found in [38]. The space X p itself has resisted study, [5]. It is not known whether there are any complemented subspaces of X p other than gp, ~2, ~p 9 ~2, and X p or any other s except under additional assumptions such as having an unconditional basis, [52].

L p spaces

155

As we have noted above the general s are mysterious but the contractively complemented ones are just Lp(l z) spaces. This raises the question as to what is the size of the largest constant 9/such that if P is a projection of norm less than y, then the range of P is isomorphic to L p (#) for some #. Because the norms of projections are linked to the parameter )~ in the original definition of the s there is a companion question as to the largest constant )~ so that if X is a s then X is isomorphic to L p ( # ) . One can also consider the question of the largest constant )~ such that if X is a s of L p then X is complemented in Lp. In [32] it was shown that there is a constant )~p such that if X is a finite dimensional subspace of L p and T is an isomorphism from s /7 with standard basis (e j) onto X with ][TI[II T-1 l [ - )~ < )~p then there are disjoint sets and e > 0 such that IlTejla j II > e for each j. Moreover e converges to 1 as )~ converges to 1. For the case p = 1 this was sufficient to imply that X is complemented. The main lemma in Dor's proof is the following. LEMMA 43. Let g l , g 2 . . . . O
/o'

be a sequence of non-negative functions in L1 and let

m a x oli gi ~ c

i <~n

oti i=1

for all n E 1~ and all (Oli ), Oti ~ 0 f o r all i, then there exist disjoint measurable subsets A1, A2 . . . . of [0, 1], satisfying

fA gi >/ c, i

f o r i = 1,2 . . . . . The proof of this uses an extreme point argument. From this lemma Dor deduces THEOREM 44. Let 1 ~ p < cx~, p =/: 2, 0 < 0 ~< 1, and let (fn) be a sequence in Lp such that f o r all sequences of scalars (Otn) either

1 ~ p < 2,

II in II ~ 1 f o r all n

and

~i f/ i--1

I]

~ 0

Ioti Ip

or

2 1 f o r all n

and

• i--1

O/i f/

<~ 0 -1

I~i lp i--1

Then there exist disjoint measurable subsets A1, A2 . . . . of [0, 1] such that f o r each i

[[filAi [[ --

(fA i

If/(t)[ p dt

~ 0 2/(p-2)

156

D. Alspach and E. Odell

In the case p = 1 it is sufficient to have a uniform bound on the individual errors

[Ifnl[O, 1]\ai II in order to show that the obvious candidate for a projection is bounded. For p > 1 Schechtman [97] was able to obtain an error estimate of the form

II~ajTejlAj

II

a()~, e ) ( ~

lajlP) '/p

with a0~, e) --+ 0 as )~ ~ 1 and ~ --+ 1 which gave the complementation result in this case also. Thus if X is a/2p,Z subspace of L p, 1 < p < c~, with )~ sufficiently close to 1, by passing to a limit of projections on an increasing family of subspaces Xn we get that X is complemented. (See, for example, [67].) For the case p = 1 passing to a limit is more problematical because the spaces are not reflexive. Nonetheless it was shown in [8] that a/~l,l+e subspace of L1 is complemented if ~ is sufficiently small. Recently [41] a somewhat different approach to these problems for p = 1 has been given which makes use of a representation theorem of Kalton for operators on Lp, 0 < p <~ 1, [59]. The determination of the constants in the results of Dor and Schechtman, the Lp versions of their results and the relationship between the norm of a projection and the distance to L p ( # ) have proved illusive. Some partial results were obtained in [4]. The results of Dor and Schechtman have also proved useful for proving that small isomorphisms on L p are in fact close to isometries. An isomorphism is said to be small if IITII lIT -1 II < 1 + ~ and e is close to O. The results of Dor and Schechtman give that small isomorphisms from I pn into L p are close to isometries. In [3] it is shown that by applying an appropriate limiting argument the same is true for small into isomorphisms from L p ( # ) into L p ( v ) . Thus there is a function f ( p , E) which approaches 0 as ~ approaches 0, so that if T is an isomorphism from Lp(lZ) into Lp(1)) with IITIIIIT -111 < 1 + E then there is an isometry S from Lp(#) into Lp(v) such that liT - S[] < f ( p , E). This is an isometric property of the L p - n o r m because H p ( D ) is isomorphic to L p, [ 14], and it is known that for small isomorphisms on lip, there cannot be a corresponding result, [35].

References

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[43] S. Guerre-Delabribre and Y. Raynaud, On sequences with no almost symmetric subsequence, Texas Functional Analysis Seminar 1985-1986 (Austin, TX, 1985-1986), Longhorn Notes, Univ. Texas, Austin, TX (1986), 83-93. [44] S. Guerre-Delabri~re, Classical Sequences in Banach Spaces, Monographs and Textbooks in Pure and Applied Mathematics 166, Marcel Dekker, New York (1992). With a foreword by Haskell P. Rosenthal. [45] R. Haydon and B. Maurey, On Banach spaces with strongly separable types, J. London Math. Soc. 2 (1986), 484-498. [46] W.B. Johnson, On quotients of Lp which are quotients oflp, Compositio Math. 34 (1) (1977), 69-89. [47] W.B. Johnson and L. Jones, Every Lp operator is an L 2 operator, Proc. Amer. Math. Soc. 72 (2) (1978), 309-312. [48] W.B. Johnson and J. Lindenstrauss, Examples of 121 spaces, Ark. Mat. 18 (1) (1980), 101-106. [49] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1-84. [50] W.B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19 (217) (1979), v+298. [51] W.B. Johnson and E. Odell, Subspaces of Lp which embed into lp, Compositio Math. 28 (1974), 37-49. [52] W.B. Johnson and E. Odell, Subspaces and quotients of lp @ 12 and Xp, Acta Math. 147 (1-2) (1981), 117-147. [53] W.B. Johnson and G. Schechtman, Finite dimensional subspaces of Lp, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 837-870. [54] W.B. Johnson, H.E Rosenthal and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488-506. [55] W.B. Johnson, G. Schechtman and J. Zinn, Best constants in moment inequalities for linear combinations of independent and exchangeable random variables, Ann. Probab. 13 (1) (1985), 234-253. [56] W.B. Johnson and M. Zippin, On subspaces of quotients of (y~ Gn)Ip and (y~ Gn)c 0 , Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), Israel J. Math. 13 (1972), 311-316. [57] W.B. Johnson and M. Zippin, Subspaces and quotient spaces of (y~ Gn)lp and ( ~ Gn)co, Israel J. Math. 17 (1974), 50--55. [58] M.I. Kadets and A. Pe~czyfiski, Bases, lacunary sequences and complemented subspaces in the spaces Lp, Studia Math. 21 (1961/1962), 161-176. [59] N.J. Kalton, The endomorphisms of Lp (0 <<,p <<,1), Indiana Univ. Math. J. 27 (3) (1978), 353-381. [60] N.J. Kalton and D. Werner, Property (M), M-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137-178. [61] J.L. Krivine, Sous-espaces de dimension finie des espaces de Banach r~ticul~s, Ann. of Math. (2) 104 (1) (1976), 1-29. [62] J.L. Krivine and B. Maurey, Espaces de Banach stables, Israel J. Math. 39 (4) (1981), 273-295. [63] S. Kwapiefi and C. Schtitt, Some combinatorial and probabilistic inequalities and their application to Banach space theory, Studia Math. 82 (1) (1985), 91-106. [64] S. Kwapiefi and C. Schiitt, Some combinatorial and probabilistic inequalities and their application to Banach space theory. II, Studia Math. 95 (2) (1989), 141-154. [65] D.R. Lewis and C. Stegall, Banach spaces whose duals are isomorphic to ~1 (F), J. Funct. Anal. 12 (1973), 177-187. [66] J. Lindenstrauss and A. Petczyfiski, Absolutely summing operators in 12p spaces and their applications, Studia Math. 29 (1968), 275-321. [67] J. Lindenstrauss and A. Petczyfiski, Contributions to the theory of the classical Banach spaces, J. Funct. Anal. 8 (1971), 225-249. [68] J. Lindenstrauss and H.E Rosenthal, The 12p spaces, Israel J. Math. 7 (1969), 325-349. [69] J. Lindenstrauss and L. Tzafriri, On the complemented subspaces problem, Israel J. Math. 9 (1971), 263269. [70] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Lecture Notes in Math. 338, Springer-Verlag, Berlin (1973).

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[71] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I, Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Springer-Verlag, Berlin (1977). [72] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete 97, Springer-Verlag, Berlin (1979). [73] D.A. Martin, Borel determinancy, Ann. of Math. 102 (1975), 363-371. [74] B. Maurey, Sous-espaces compl~ment~s de L p, d'aprks P. Enflo, S6minaire Maurey-Schwartz 19741975: Espaces LP, Applications Radonifiantes et G6om6trie des Espaces de Banach, Exp. No. III, Centre Math., l~cole Polytech., Paris (1975), 15 (erratum, p. 1). [75] B. Maurey, Tout sous-espace de L 1 contient un lp (d'aprOs D. Aldous), Seminar on Functional Analysis, 1979-1980 (French), Exp. 1-2, t~cole Polytech., Palaiseau (1980), 13. [76] B. Maurey, Th~orkmes de factorisation pour les op~rateurs lin~aires gt valeurs dans les espaces L p, Soci6t6 Math6matique de France, Paris (1974). With an English summary, Ast6risque 11. [77] B. Maurey, Local theory: history and impact, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), to appear. [78] P. Mtiller and G. Schechtman, Several results concerning unconditionality in vector valued LP and H 1(Un) spaces, Illinois J. Math. 35 (1991), 220-233. [79] P. Miiller and G. Schechtman, A remarkable rearrangement of the Haar system in Lp, Proc. Amer. Math. Soc. 25 (1997), 2363-2371. [80] P.F.X. Miiller, A local version of a result of Gamlen and Gaudet, Israel J. Math. 63 (2) (1988), 212-222. [81] EJ. Murray, On complementary manifolds and projections in spaces Lp and g.p, Trans. Amer. Math. Soc. 41 (1937), 138-152. [82] N.J. Nielsen and E Wojtaszczyk, A remark on bases in s with an application to complementably universal 12~-spaces, Bull. Acad. Polon. Sci. S6r. Sci. Math. Astronom. Phys. 21 (1973), 249-254. [83] E. Odell, On complemented subspaces of (~-~12)lp, Israel J. Math. 23 (3-4) (1976), 353-367. [84] E. Odell and Th. Schlumprecht, Trees and branches in Banach spaces, Preprint. [85] E. Odell and Th. Schlumprecht, Distortion and asymptotic structure, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), to appear. [86] A. Petczyfiski, Projections in certain Banach spaces, Studia Math. 19 (1960), 209-228. [87] Y. Raynaud, Stabilit~ des espaces d'op~rateurs CE, Seminar on the Geometry of Banach Spaces, Vol. I, II (Paris, 1983), Publ. Math. Univ. Paris VII, 18, Univ. Paris VII, Paris (1984), 1-12. [88] Y. Raynaud and C. Schtitt, Some results on symmetric subspaces of L 1, Studia Math. 89 (1) (1988), 27-35. [89] Y. Raynaud, Sur les sous-espaces de L P (Lq), S6minaire d'Analyse Fonctionelle 1984/1985, Publ. Math. Univ. Paris VII, 26, Univ. Paris VII, Paris (1986), 49-71. [90] H.P. Rosenthal, The Banach spaces C(K), Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), to appear. [91 ] H.P. Rosenthal, On the subspaces of L P (p > 2) spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273-303. [92] H.P. Rosenthal, On the span in L P of sequences of independent random variables. II, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, CA, 1970/1971), Vol. II: Probability Theory, Univ. California Press, Berkeley, CA (1972), 149-167. [93] H.P. Rosenthal, On subspaces of LP, Ann. of Math. (2) 97 (1973), 344-373. [94] W. Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203-227. [95] G. Schechtman, A remark on unconditional basic sequences in Lp (1 < p < ~ ) , Israel J. Math. 19 (1974), 220-224. [96] G. Schechtman, Examples o f s spaces (1 < p ~ 2 < cx~), Israel J. Math. 22 (1975), 138-147. [97] G. Schechtman, Almost isometric Lp subspaces of Lp(O, 1), J. London Math. Soc. (2) 20 (3) (1979), 516-528. [98] S.J. Szarek, A Banach space without a basis which has the bounded approximation property, Acta Math. 159 (1-2) (1987), 81-98. [99] B.S. Tsirelson, It is impossible to imbed l p or c O into an arbitrary Banach space, Funkcional. Anal. i Prilo2en. 8 (2) (1974), 57-60. (In Russian.) [100] L. Tzafriri, Remarks on contractive projections in Lp-spaces, Israel J. Math. 7 (1969), 9-15. [ 101 ] M. Zippin, On perfectly homogeneous bases in Banach spaces, Israel J. Math. 4 (1966), 265-272.

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CHAPTER

4

Convex Geometry and Functional Analysis Keith Ball Department of Mathematics, University College London, London, England E-mail: kmb @math. ucl. ac. uk

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163

1. Convolution inequalities in convex geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. The Brascamp-Lieb inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 164

1.2. The reverse isoperimetric inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169

1.3. The reverse Brascamp-Lieb inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Higher dimensional projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Volumes of sections of convex bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Vaaler's Theorem and its relatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Upper bounds for the volumes of sections of cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Plank problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Bang's lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The affine plank theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Nazarov's solution to the coefficient problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Related results and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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171 173 174 177 181 182

185 185 189 192 193

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Introduction This article describes three topics that lie at the intersection of functional analysis, harmonic analysis, probability theory and convex geometry. The first section consists principally of applications of harmonic analysis to convex geometry. The second section uses probabilistic inequalities to estimate volumes of convex bodies. The last section links an old problem in geometry, the plank problem, with some of the fundamental principles in functional analysis, and with the coefficient problem in classical harmonic analysis.

1. Convolution inequalities in convex geometry The classical isoperimetric inequality states that among bodies of a given volume in R n, the Euclidean balls have least surface area. This principle seems to have been recognised, at least in two dimensions, by the ancients, and by the end of the last century, there were a number of proofs which worked in arbitrary dimension. Some time in the 1930's, a reverse isoperimetric problem was formulated for convex bodies. Some care is needed in the formulation, since even convex bodies can have large surface area and small volume. The most natural way to pose the reverse problem is to consider affine-equivalence classes of convex bodies. The following solution to the problem appeared in [2] and [4]. THEOREM 1 (Reverse isoperimetric inequality). If C is a convex body in R n and T is a regular solid simplex in R n then C has an affine image C f o r which vol(C) - vol(T),

but

vol(0C) = vol(0T).

If C is centrally symmetric, T may be replaced by a cube Q. The proof of Theorem 1 uses a sharp convolution inequality coming from harmonic analysis. There has been quite a lot of recent work dealing with other links between harmonic analysis/PDE and convex geometry: some years ago Jerison found an analogue of Minkowski's existence theorem for harmonic measure and later he, Cafarelli and Lieb found a way to extend much of the classical Brunn-Minkowski theory, to capacity: but these developments will not be described in the present article. It may not come as a surprise that harmonic analysis is linked with isoperimetric problems: after all, the classical isoperimetric inequality in R" has an extension to compactly supported functions on R n as the simplest Sobolev inequality: cn ll f l[~

~
for an appropriate constant Cn. This inequality belongs squarely within harmonic analysis. However, the relationship between the reverse isoperimetric problem and harmonic analysis is a bit more mysterious: in particular, a crucial role is played in the proofs by Gaussian densities.

164

K. Ball

It makes sense at this point to explain which affine image is used in the proofs of the assertions in Theorem 1. Of course, for any convex body, there is an affine image which is optimal: an affine image of the given volume, whose surface area is least. This affine image was characterised by Petty in the 60s, [35]; (some consequences of this characterisation were recently studied by Giannopoulos and Papadimitrakis, [22]). But the characterisation didn't seem to be of much use for tackling the reverse isoperimetric problem. However, as explained in the article [26, Section 8], each convex body includes an unique ellipsoid of largest volume; this, maximal ellipsoid, was characterised by Fritz John. The affine image of a body C, that will be used to prove the theorem, is the one whose ellipsoid of largest volume is the standard Euclidean unit ball. If C has the Euclidean ball as its maximal ellipsoid, then, according to John's theorem, there are Euclidean unit vectors (ui)T on the boundary of C and positive numbers (ci)~, for which

E C i U i --0

(1)

i

and

E

Ci lgi (~ Ui = In,

(2)

i where In is the identity on R n. The principal tool used in the proof of Theorem 1 is an inequality of Brascamp and Lieb [ 10] which among other things, is a sharp form of Young's inequality for convolutions. Over the last 20 years there has been a steady growth in understanding of the best constants in classical inequalities like the Hausdorff-Young and Young inequalities. The paper of Lieb [30] contains very general results of this kind and describes the developments leading up to them. Recently, a delightful (and delightfully short), new proof of the Brascamp-Lieb inequality was found by Barthe [9]. He also proved a reverse analogue of the inequality which had been conjectured by the present author and which, like the forward inequality, has applications to the study of convex bodies. It should be remarked that Barthe's new proof is sufficiently simple that he was able to determine cases of equality in his Theorems and in the applications to geometry. The body of this chapter is divided up as follows. The Brascamp-Lieb inequality is discussed in Section 1.1 together with Barthe's new proof. The proof of the reverse isoperimetric inequality is given in Section 1.2. Section 1.3 contains the reverse inequality and a brief example of how it may be used and Section 1.4 discusses a higher-dimensional version of these inequalities.

1.1. The Brascamp-Lieb inequality Young's inequality for convolution on a locally-compact group states that if ~l_q_ ~1 _ 1 + s1 then for functions f and g in the appropriate spaces

IIf * glls ~ Ilfl[p[]g[lq.

Convex geometry and functional analysis

165

On compact groups, where constant functions belong to L p spaces, the inequality is sharp: but on R it is not (except for special values of p and q). In the 70's Beckner and Brascamp and Lieb independently showed that the extremal functions in Young's inequality are Gaussian densities" x w-~ e -ax2 . Using the duality between L p spaces, it is possible restate Young's inequality as follows" if ~1 + ~1 + r1 -- 2 then

f 9 g 9 h(O) <~ Ilfllp]lg]lq Ilhllr. (The Ls norm of f 9 g is realised as an integral against a function in Lr.) The left hand side of this inequality is

ff

(3)

f ( x ) g ( y - x ) h ( - y ) dx dy.

This can be rewritten as

fR f ((x, el))g((x, e2))h((x, e3))dx, 2

where el = (1,0), e2 -- ( - 1 , 1) and e3 = (0, - 1 ) . The inequality of Brascamp and Lieb states that if v l, v2 . . . . . Vm are vectors in R n, f l , f2 . . . . , fm are functions on the line and 1 = n then the ratio (pi) are numbers between 1 and cx~ satisfying y~ p--?

fR ~ I--Ir~fi((X, vi))dx

I-I~' IIf, IIpi is "maximised" by appropriate Gaussian densities. (There are degenerate cases in which the maximum is not attained.) The value of the maximum is not easily computed since the ai are the solutions of nonlinear equations in the pi and vi. This apparently unpleasant problem evaporates in the context of convex geometry: the inequality has a normalised form, introduced in [2], which "matches" Fritz John's Theorem. THEOREM 2 (Brascamp-Lieb). Suppose (Ui) is a sequence of unit vectors in R n and (Ci)

is a sequence of positive numbers satisfying the John condition Z

CiUi @ Ui = In.

If fi : R --+ [0, oc) are measurable then

f i ( ( X , Ui)) ci dx ~ n

1

fi

9

1

In this version, the weights Ci play the role of 1/Pi. If one takes traces of the operators in the John condition one obtains, ~ ci = n. The use of the John condition has magically

K. Ball

166

guaranteed that the best constant is 1. Moreover, the replacing of the extremal functions are identical Gaussian densities;

f i pi ,

by ft., ensures that

fi(t) - e -t2 for example. It is perhaps instructive to see why this is. The Fritz John condition, can be written

Z Ci(X, Ui)2 --

Ixl 2

for each vector x in R n. So with ft. (t)

fR Hm ft" ((X, Ui))ci dx - fR n

--

e -t2 for each i,

e -Ix12 d x - n

(f~

e -t2 dt

1

)~

--

vI(f

f/"

)ci

1

The rest of this section is devoted to the recent proof of Theorem 2 found by E Barthe. First a simple lemma.

(Ui) and (Ci) be vectors and weights satisfying

LEMMA 1. Let

the Fritz John condition (2)

Z CiUi (~ Ui -- In. i (1) If x -- y~ CiOiUi, then [xl 2 ~ ~ ciO2. (2) For any operator T on R n, det T ~< H IITuill Ci. i

(3) For any sequence

(Oti) of positive

numbers

det(~--~ciotiui Qui) ~ Hot~ "i. REMARK. The third statement is exactly what is needed to check that identical Gaussian densities are extremal among Gaussian densities, for the inequality of Theorem 2. PROOF. The first statement is just the Cauchy-Schwarz inequality. For the second, it may be assumed that T is positive semi-definite and symmetric" say,

T -- ~

~.j ej | ej

J for some orthonormal basis (e j). Now, for each i,

IITuill 2 = ~ ) ~ j 2 (Ui , ej) 2 J

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167

and since ~ j (ui, ej ) 2 __ 1, this is at least

E )~(ui,ej) 2

J Hence

I[Zbli [I~

E i~ i'ej)2 J

Raise this inequality to the power r and take the product. The third inequality is just the dual of the second. For each sequence of oti, there is an operator T of determinant 1 for which

(

cioliui(~ lgi

det

'(E

)

CiOliTui | Tui 17 __-- _1~_~ CiOliIITui II2 n i (~i Oli)Ci/n(I--I IITui ll2 )Ci/17 i --

--tr

and this is at least

(~i Oti)ci/n by part (2).

I-1

Now for Theorem 2. PROOF. Let ( U i ) and (Ci) be vectors and weights satisfying the John condition, and (fi) a sequence of functions on R with

for each i. The aim is to show that

m fR n E fi((lgi,X)) ci dx ~ 1. 1

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Let g be the function given by g(s) -- e -zrs2, chosen so that f g = 1. Assume that each fi is strictly positive and smooth. For each i, define T/:R ~ R by

fi (s) ds =

g(s)ds.

oo

Then, for each t and i,

fi(t) -- g(Ji(t)).Ti'(t). So

f I-I fi((Ui'x))Cidx

f I-Ig(Ti((tti'x)))cil-'I

Tit((bli'x))Ci dx

f Hg(Ti((ui,x))) cidet(~CiTl.t((ui,x))ui @ui)dx --

f exp(-Zci(Ti((ui,x))) 2) det(~ciTit((ui,x))ui @ui)dx

where the inequality follows from part (3) of the lemma. The determinant in the last expression is the Jacobian J (y, x) for the n-dimensional change of variables given by

y-- ~ciTi((ui,x)).ui. With y given by this equation, the first part of the lemma shows that lYl2 ~<

~ci(Ti((ui,x))) 2,

and hence e -zrlyl2 ~> exp ( - J r ~

ci(Ti((ui,x))) 2 ) .

So the integral is at most

f e-~lyl2j(y,x)dx- 1. As was mentioned in the introduction, Barthe obtained conditions for equality in the above result, but these will not be discussed here.

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1.2. The reverse isoperimetric inequality The main purpose in this section is to describe the proofs of the two estimates in Theorem 1. Both the symmetric and the general case of the theorem are deduced from volume-ratio estimates. The volume-ratio of a convex body K in R n, whose ellipsoid of maximal volume is C, is the number vol(K)

1/n

vol(s ) (See the article [21 ] in this collection.) Most of the effort goes into proving the following. THEOREM 3 (Volume ratios). The simplex has largest volume ratio among convex bodies of a given dimension, while among symmetric bodies, the cube is extremal. Let's see how the volume-ratio estimate implies the reverse isoperimetric inequality, for example, in the case of the cube. The aim is to show that if the symmetric body C has the standard Euclidean ball, B~, as its maximal ellipsoid, then, vol(0C) <~ 2n vol(C) (n-1)/n. The volume ratio estimate guarantees that vol(C) ~< 2 n since this is the volume of the cube whose maximal ellipsoid is B~. Now, from the fact that C contains B 2n , it follows that

vol(0C)-

lim

vol(C + eBb) - vol(C)

e--+0

~< lim

8

vol(C + eC) - vol(C)

e--+0

8

= vol(C) lim e--+0 ~.

vol(C

m

(l+e)n-

1

8

(vol(C )

F/

(vol(C )

2 (vol(C ) as required. The argument for the simplex is similar. The problem, then, is to prove Theorem 3. PROOF. For the symmetric case, the result is almost immediate from the normalised Brascamp-Lieb inequality and John's Theorem. If the maximal volume ellipsoid of C is

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K. Ball

B 2/7 , then, since the vectors U i given by John's Theorem are points of contact between the boundaries of C and the Euclidean ball, the set C is included into the set K = K (C).

{x 6 R n" [(x, ui)l ~ 1, forall i}. So it suffices to estimate the volume of K. Now if fi is the indicator of the interval [ - 1, 1], for each i, then the indicator 1K of K is exactly I-I f i ( ( x , ui)) ci . By Theorem 2,

m(f f/ )ci

vol(K) ~< I-[

m

-~ H 2ci -- 2n"

1

1

The volume ratio estimate for general bodies needs a bit more. A simplex whose maximal ellipsoid is B~, has volume

nn/2(n + 1)(n+l)/2 n! So the aim is to show that if B~ is the maximal ellipsoid in C, then the volume of C is at most this number. As usual, let ui and ci be the unit vectors and the weights guaranteed by John's Theorem. In this case, the most we can say is that C is included into

K = {x ~ R n" (x, ui) ~< 1, for all/}. The absence of absolute values in the definition of this set makes it impossible to apply the Brascamp-Lieb inequality directly. The clue as to what to do is provided by the fact that a regular simplex of dimension n sits more naturally (as a section of an orthant) in R n+l than it does in R n . Regard R n+l as R n x R. For each i define a unit vector vi in R n+l by

(1) l)i =

x/n + 1

Ui

~/n + 1

and a weight di -~ ~ci.n+l _ Using the two conditions from John's Theorem, it is not too hard to check that E

di l)i Q 13i -- In+l.

For each i, let

36 (t) -

e t 0

i f t >~0, otherwise,

and let F be the function on R n+l

F ( y ) -- l--I f i ( ( y , vi)) di 9

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Convex geometry and functional analysis

By Theorem 2 the integral of F, over R n+l is at most 1. Now, let y = (x, r) be a point of R n+l . For each i, x/'ff (y, Wi) = - - ~ ( X , x/n + 1

Ui) %"

1 x/n + 1

r.

Since Y~ CiUi = O, Z Ci (X, lgi) ~-- 0 as well. Hence, if r < 0 there is at least one i for which (y, vi) is negative. This will ensure that F (y) = 0. On the other hand, if r ~> 0, the inner products (y, vi) are all non-negative, precisely if 1 (X, Ui) ~ ~ r

for every i. The function F is thus supported on a cone whose vertex is 0 and whose crosssection at distance r from the vertex is ~E K. At a point (x, r), on this cross-section, the function's value is

exp - Z

W/n

di - ~ / n + 1

(X, Ui ) +

~/n + 1

r

and since ~ Ci bli -- 0 and y~ di -- n + 1, this reduces to e - ~ therefore e_ n,/h-~r

r

vol(K) dr -

r. The integral of F is

n! vol(K) nn/2(n -+- 1)(n+l)/2"

This completes the proof. The article [4] includes a number of other applications of the Brascamp-Lieb inequality. In particular, it is shown that if 1 ~ p < ec, the space g np has largest volume-ratio among all n-dimensional subspaces of L p. This fact is of most interest for p = 1.

1.3. The reverse Brascamp-Lieb inequality At the time that the article [4] was written, the author conjectured a reverse form of the Brascamp-Lieb inequality. This conjecture was proved by Barthe [9]. Indeed, his proof of the forward inequality, described above, also gives the reverse inequality, if the roles of the j5 and g are interchanged. In fact, as Barthe pointed out, the two arguments can even be combined into one. The reverse inequality is the following. THEOREM 4 (Barthe). Suppose (Ui) is a sequence of unit vectors in R n and sequence of positive numbers satisfying the John condition ~

CiUi (~ Ui = In.

(Ci) is a

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K. Ball

Suppose fi :R --+ [0, oc) and F : R --+ [0, r

are measurable, and that

F(x) ) I--I fi(Oi)ci'

wheneverx --" Z c i O i u i .

fa

fi

Then F(x) dx >1

)ci

n

1

The reverse inequality also has applications to convex geometry. Barthe used it to prove an extremal property of the simplex with respect to mean width. The following estimate for "outer volume ratio" provides a simpler illustration of its use. THEOREM 5. If C is a symmetric convex body in R n, then K is included in an ellipsoid,

s with vol(g) vol(K)

no more than the corresponding ratio for the s ball. PROOF. The dual form of John's Theorem guarantees that if the ellipsoid of minimal volume containing C is the ball B~, then there are unit vectors (ui) in C, and weights ci satisfying the John condition (2). The problem is to show that the absolutely convex hull of these (ui) has volume at least that of the s ball; namely 2n/n!. Let K be the absolutely convex hull and let II. II be the norm corresponding to K. By using polar coordinates it is easy to check that vol(K) = ~1

fR,, e-llxlldx

For any x,

]lxll-min{~l)~il" x-=-Z)~iui ] -min[~cilOil" i i i

x---ZciOiui }. i

Hence, if fi (t) = e -Itl for each i, e -Ilxll = m a x [ H f i ( O i ) ci"

i

X-~~CiOiUi]. i

By Barthe's Theorem, fa

e-IIx IIdx ~ 2 n n

as required.

[~

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173

1.4. Higher dimensional projections In his paper [30], Lieb produced a greatly generalised version of the Brascamp-Lieb inequality, which, in the normalised form, reads: THEOREM 6 (Lieb). If (Pi) are orthogonal projections on R n and (Ci) are positive num-

bers satisfying ~ ci Pi = In, and for each i, f i is a non-negative function on the image of Pi, then

fR

) Ci

n

1

1

This higher-dimensional version also has a simpler proof, recently found by Barthe [9]. The basic idea is the same as for the 1-dimensional case: to use a homeomorphism of Euclidean space to represent a given density in terms of the Gaussian. But on higher dimensional spaces, there is a wide choice of such homeomorphisms and the extra problem is to find one which is appropriate to the inequality. Barthe does this by invoking a theorem of Brenier [11 ] (see also McCann [32]). The results of Brenier, McCann and others are part of an active current investigation of the transportation of measures. The special case in which m = n and the Pi are the 1-codimensional projections onto the coordinate hyperplanes in R n, goes back (essentially) to Loomis and Whitney. They proved that if K is compact in R n then

vol(K) ~< [-I vol(Pi K) 1~(n-l) 1

This follows from Lieb's Theorem applied to the functions j~(x) = 1PiK(X ),

whose product on R n is equal to 1 at each point of K. Although their result is specific to indicator functions, their argument also gives the case of general functions. The reason that Lieb's theorem is much more difficult, lies not in the passage from sets to functions, but in the fact that his projections Pi do not commute with one another. The Loomis-Whitney inequality already looks a bit like an isoperimetric inequality since it estimates the volume of K in terms of the volumes of 1-codimensional sets derived from K. In fact, a simple generalisation of the Loomis-Whitney inequality is the main technical tool in Gagliardo's proof of the Sobolev Embedding Theorem, which goes back to the late 50s. One can begin to see how Loomis-Whitney could be used, by noticing that it immediately implies a version of the isoperimetric inequality (without the right constant). If K is compact and convex, then by Loomis-Whitney,

vol(K) (n-1)/n <~V i v o l ( P i K ) 1/n <~1

n

vol(PiK). l

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K. Ball

Now average the inequality over all choices of coordinate systems. The result is vol(K) (n-1)/n <, Average(vol(PK)),

where the average is taken over all orthogonal projections of codimension 1. The last expression is proportional to the surface area of K (with a constant of proportionality depending upon n, but not upon K).

2. Volumes of sections of convex bodies

In the previous section it was explained that the simplex has the largest volume ratio of any convex body, and that the cube has the largest of any symmetric body. For a convex body whose maximal ellipsoid is known, these volume ratio estimates automatically provide upper bounds for the volume of the body. It turns out that such upper bounds are pretty accurate, as long as the body has "few" faces, relative to the dimension. (The precise meaning of "few" will become clear later.) This section of the article describes some lower bounds for the volumes of bodies with few faces, which complement the upper bounds of the last section and indicate how sharp they are. The oldest discovery in the Geometry of Numbers is Minkowski's Box Theorem which guarantees that every symmetric convex body in R n, of volume at least 2 n, contains a nonzero point of the integer lattice Z n. An immediate consequence is that if A (aij) is an n • n matrix of determinant at most 1, then there is a sequence (zj)~ of integers, not all 0, so that for each i, -

-

~i aij z j A natural question, which arises in applications to number theory, asks what happens if the matrix is not square. A very natural conjecture, which was publicised by A. Good, went as follows. CONJECTURE 1. Let A

= (aij ) be an m

• n matrix satisfying

det(A*.A) ~ 1. Then there is a

(4)

sequence (zj)~ of integers, not all O, so that for each i,

~i aij zj

~<1.

The problem is to show that if A is an m • n matrix satisfying (4) then the set

{x

I

~< 1, for all i }

Convex geometry and functional analysis

175

has volume at least 2 n. The inequality (4) states precisely that the map from R n to R m given by A does not increase n-dimensional volume. The image of K under this map, is the n-dimensional slice of the cube [ - 1 , 1] m by the subspace A(Rn). Thus, as Good himself pointed out, the conjecture can be restated: for each m and each n ~< m, every n-dimensional slice through the centre of the cube [ - 1 , 1]m has volume at least 2 n. For n = m - 1, this conjecture was proved by Hensley [24] and for arbitrary n, by Vaaler [38]. Yet another formulation of Vaaler's Theorem reads as follows: THEOREM 7 (Vaaler). Suppose (l)i)r~ is a sequence of vectors in R n and we set

r2

_

[vii 2

_

n

1

Then

2 vol({x" I(x, vi)] ~ 1, f o r a l l i } ) 1/n >~ - . r

Some years after Vaaler's result, several authors, independently, found an analogue, in which the maximum length of the vi is controlled, instead of the sum of the squares of the lengths. It states: THEOREM 8. There is a constant ~ > 0 (independent of everything) so that if (l) i )r~ is a sequence of unit vectors in R n, then vol({x" I(x, vi)[ ~ 1, forall i}) 1/n ~log(1 + m) This statement can be extracted from some results of Carl, and this was done by Carl and Pajor [ 14]: it was proved independently by Gluskin [23] and by Barany and FurEdi [7]: and it was rediscovered by Bourgain, Lindenstrauss and Milman [ 12]. In fact Gluskin's method gives something a bit stronger: it is not necessary to assume a "uniform" estimate on the lengths of the vectors. THEOREM 9 (Gluskin). If (vk) is a sequence of vectors in R n satisfying

Ivkl ~< 2v/log(1 + ~) f o r each k, then

vol({x" I(x, vi)l ~ 1, f o r a l l i}) 1/n

1

>~.

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K. Ball

It should be remarked immediately, that if one replaces the expressions log(1 + ~) by log(1 + k), then the statement becomes almost trivial. The improvement given by the theorem is of interest if the number of vectors is fairly small compared to the dimension: especially if it is no more than a fixed multiple of the dimension. This is also the range in which Vaaler's Theorem is most useful (and is typically used). Gluskin's result (in the form stated above) includes Vaaler's, apart from the value of the "constant" involved. To see why, observe that if 1~

Ivkl 2 ~ 1

n

and the vk are in decreasing order of norm, then

~log(1 + ~) Strangely enough, the proofs of Theorems 7 and 9, like the proofs of the volume ratio estimates of the last chapter, use comparisons between Lebesgue measure and an appropriate Gaussian measure. In the case of Theorem 9, the crucial property of Gaussian measure is captured by Sidak's Lemma. A symmetric slab in R n is a set of the form {x 6 R n" I(x, u)l ~< t} for some vector u and positive number t. LEMMA 2 (Sidak). If y is standard Gaussian measure on R n and (Si) is a sequence of symmetric slabs then

z(nsi)

>~l--I z(si).

Sidak's Lemma (and more especially its proof) solves a special case of a well-known and rather intriguing problem: the correlation problem for Gaussian measure. QUESTION 1. Is it true that if y is standard Gaussian measure on R n and K and L are symmetric convex bodies in R n, then

y ( K A L) >~y ( K ) y ( L ) ? For a discussion of this problem see, e.g., [36]. In his article on the central sections of the cube, Hensley not only showed that the volumes of the 1-codimensional sections are at least 1, but also that they are at most 5, independent of the dimension. At first sight this seems very surprising; but in a second article [25] Hensley showed that all convex bodies have a similar property; each convex body (after an appropriate linear map), has 1-codimensional sections with "almost constant"

Convex geometry and functional analysis

177

volume. This fact is of considerable importance in a number of situations. Hensley also asked for the sharp upper bound for the volumes of 1-codimensional sections of the cube, conjecturing the value x/2. The present author proved this in [1 ], and in a later paper [2] found the optimal upper bound for k-codimensional sections, (x/2) k . These results will not be proved here, but in the short Section 2.2, a general upper bound for volumes of sections of the cube will be deduced from the Brascamp-Lieb inequality of the last chapter. An unexpected byproduct of the upper bound, x/2, for 1-codimensional sections of the cube, was a "concrete" solution of the so-called Busemann-Petty problem. In [13], these authors asked the following question. QUESTION 2. Suppose that K and L are symmetric convex bodies in some Euclidean space with the property that, for each 1-codimensional subspace H, vol(H n K) ~< vol(H n L). Does it follow that vol(K) ~< vol(L)? At the time, there was a widespread belief that the answer should be yes: among other things, it was known that if two such bodies have sections of equal volume, then they are the same body. So it came as something of a surprise, when Larman and Rogers [29] constructed a random symmetric perturbation of the Euclidean ball in R 12, whose slices all had smaller volume than those of a ball of equal volume. However, with the knowledge that the unit cube has sections of volume at most x/2, the problem loses some if its mystique. When the dimension is large, the Euclidean ball of volume 1 has 1-codimensional sections of volume about x/-e. Thus, for large enough dimension, a cube and an Euclidean ball of slightly smaller volume, provide a counterexample for the Busemann-Petty problem. Following this observation, Giannopoulos [ 19] pointed out that, in fact, there are extremely simple concrete counterexamples, in dimensions as low as 7. It is now known that the Busemann-Petty problem has a negative answer if and only if the dimension is at least 5. The positive answer in dimension 3 was provided by Gardner [ 16] and that in dimension 4, by Zhang [39]. Recently Koldobsky [28] has developed a new approach to the problem which greatly simplifies and clarifies matters. This led to a definitive unified solution to the problem in all dimensions, which will appear in [ 18]. (The topic was also dealt with at some length, in the book by Gardner [17].) This chapter is organised as follows. Vaaler's Theorem, Sidak's Lemma and Gluskin's Theorem are proved in the first section. The second section gives a brief account of some upper bounds for volumes of sections of cubes.

2.1. Vaaler's Theorem and its relatives As explained in the introduction all the results in this section depend upon comparisons between Lebesgue and Gaussian measures. The comparison method was studied in detail by Kanter [27]. It begins with the following definition.

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K. Ball

If # and v are two probability measures on some Euclidean space R n, say that # is more peaked than v if for every symmetric convex set K in R n ,

# ( K ) >~ v(K). In order to prove Vaaler's Theorem, the aim will be to show that Lebesgue measure on the cube is more peaked than an appropriate Gaussian measure. To begin with, l e t / z be Lebesgue measure on the interval [- 89 89 and let v be the unique Gaussian probability on R whose density

f ( t ) - - e -Jrt2 satisfies f (0) = 1. Clearly in this case, # is more peaked than v. Kanter's basic lemma, shows that peaking-order is preserved under the formation of product measures, at least in the presence of some restriction. This restriction involves a notion of unimodality: several such notions have been considered. The most versatile seems to be the one used here, which was introduced by Barthe [8]. A probability measure on R n will be called unimodal if it has a density f , which can be expressed as the increasing limit of a sequence of functions, each of which is a positively weighted sum

~jpilKi of characteristic functions, 1 gi, of symmetric convex sets. Clearly, Lebesgue measure on the cube and Gaussian measures are unimodal. LEMMA 3 (Kanter). If lZl and 1~2 are probabilities on R n, with #1 being more peaked than #2, and v is a unimodal probability on R m, then lZ l | v is more peaked than lz2 Q V. PROOF. The problem is to show that for every symmetric convex K in R n+m ,

# l | v ( K ) >~# 2 (~ v(K). By the unimodality of v, it may be assumed that the density of v is the characteristic function of a symmetric convex set C. If K is the intersection of K with the cylinder R n x C, then for any probability # on R n,

m

m

It is a standard consequence of the Brunn-Minkowski inequality that the function g, defined on R n by

g(x) -- fRm 1 ~-(X, y) dy

Convex geometry and functional analysis

179

has a concave logarithm. Also, g is an even function, because the set K is symmetric. From this it follows that g can be approximated by positive combinations of the characteristic functions of symmetric convex sets in R n. The fact that #l is more peaked than #2 now ensures that

fR n g(x) d#l (x) ~> fR n g(x) d#2(x), as required.

D

By applying Kanter's Lemma repeatedly it is easy to see that if # and v are the uniform and Gaussian probabilities on R, described above, then #|174174 is more peaked on R n than

v|174174 From this one gets Vaaler's Theorem as follows. PROOF. The task is to show that if H is a k-dimensional subspace of R n and Q is the unit cube [- 89 89 ]n then the k-dimensional volume of H n Q is at least 1. As explained above, if K is any symmetric convex body in R n, then K n Q has volume at least as large as Ke Trlxl2dx. By approximating the subspace H by very thin convex "tubes", one obtains that the k-dimensional volume of H n Q is at least the integral over H (with respect to k-dimensional measure) of e -Jrlxl2", and this is 1. [3 The reader is invited to check that the two other formulations of Vaaler's Theorem, described in the introduction to this chapter, follow from the statement just proved. An application of Kanter's Lemma, to the volumes of sections of g np balls other than cubes appears in the article of Meyer and Pajor [33]. It was mentioned in the introduction, that the proof of Sidak's Lemma is closely related to the Gaussian correlation problem. Sidak's Lemma says that the Gaussian measure of an intersection of symmetric slabs, is at least the product of the Gaussian measures of the slabs. It is clear that this statement can be deduced by induction from the following lemma. LEMMA 4. If K is a symmetric convex body in R n, S is a symmetric slab and y is standard Gaussian measure on R n, then

•

n S) >1•215

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K. Ball

It was pointed out by A. Giannopoulos [20] that this lemma can readily be recast so as to fit into the framework described earlier. Define a probability/z on R n by #(A)

=

y(A AS) •

The lemma states that # is more peaked than 9/. By writing R n as the direct sum of the 1-dimensional space spanned by the vector perpendicular to the slab S and the (n - 1)dimensional space parallel to S, the measure # can be regarded as the product of an (n - 1)dimensional Gaussian measure and a 1-dimensional measure. The latter has a density of the form s w+ (constant) e - s 2 / 2

l[-t,t](s),

obtained by restricting the Gaussian density to a symmetric interval, and then renormalising. Plainly, this 1-dimensional probability is more peaked than the standard Gaussian. Therefore, by Kanter's Lemma, # is more peaked than y. Sidak's Lemma gives Gluskin's Theorem in the following way. k -1/2 and let PROOF. Assume that for each k, v~ has norm at most (4 log(1 4- n))

K = {x" I(x, vk)l <~ 1, for all k} - A Sk, where for each k, Sk is the slab

{x ~ R "" I(x, v~)l <~ 1}. Clearly vol(K) ~> ( ~ / ~ ) n y ( K ) , and by Sidak's Lemma, this is at least

k

For each k the slab Sk has width 2t where

t --

Iluk II ~

log

1 +

.

Now a slab of width 2t has Gaussian measure

'i

e - S 2/2

ds

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Convex geometry and functional analysis

and it is well-known that this is always at least 1 - e -t2/2. Hence vol(K) ~> (v"2-~-~)n VI(1

--e -21~

--(%/~)n

k

Vi(1 _ k

1) k)2

(1+ n

"

The product can easily be estimated since

Z

' k)2 ) ~>

log (1 -

(1+~

(

log 1 -

1 x

) 2 ) dx -- - 2n log 2.

D

(1+~

REMARK. It is clear that the preceding argument can be adapted to show that if the vk satisfy the stronger estimate

Ivkl

v/log(1 + k)

then the Gaussian measure of the intersection of the corresponding slabs is at least 1/2 (not just 1/2n). This implies that if I[. II is the norm whose unit ball is the intersection of these slabs, then the random variable II. II on the probability space (R n , V) has a median which is at most 1. In fact, it is not too hard to show that under this assumption on Irk l, even the mean

fR" Ilxlldy(x) is at most a constant. Using his remarkable majorising measure theorem, [37] Talagrand has shown that the "converse" is true: if the mean of ]].[[ is at most 1, then the unit ball contains a convex body which is the intersection of slabs, (Sk), whose widths are at least 3v/log(1 + k).

2.2. Upper bounds for the volumes of sections of cubes The purpose of this section is to explain how the Brascamp-Lieb inequality, quickly provides upper estimates which are complementary to those of the previous section. In fact, the argument here is really no more than a disguised form of the volume ratio estimate for symmetric bodies; but the statement looks a bit different. THEOREM 10 (Sections of the cube). Every k-dimensional section of an n-dimensional unit cube has volume at most kj2

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K. Ball

The estimate is sharp whenever k divides n; but even in the cases where it is not sharp, the estimate has roughly the right "shape". As mentioned above, the alternative estimate (~/-~)n-k is known, and this is sharp whenever k >~ n/2. PROOF. Let Q be the unit cube, [ - l , 89 ]n. Let H be a k-dimensional subspace of R n and let P be the orthogonal projection from R n onto H. For each i, 1 ~< i ~< n, let Vi m P e i ,

where el, e2 . . . . . en is the standard basis of R n, and let [2.

Ci - - [Vi

Since P acts as the identity I/4, on H, Citti (~ Ui

I H m Z V i @ V i : ~ i

i

where the U i are unit vectors. If x 6 H, then x 6 Q if and only if, I(x, vi)[ ~ 89for every i. In terms of the u i, H A Q is the set

{

1}

x 6 H" I(x, ui)l ~ 2x/-6-/

9

An application of Theorem 2 in the space H gives,

vol(n['~a)~H(~i) 1

(5)

Since ~ Ci k, the convexity of the function c w-~ c log c shows that the right-hand side of (5) is maximised when all Ci are equal: and hence equal to k / n . [] :

3. Plank problems In the early 1930's, Tarski asked the following question (in connection with the BanachTarski paradox). Let us call the region between two parallel hyperplanes in R n , a plank. QUESTION 3. If a convex body of minimum width 1, is covered by a union of planks, must the widths of the planks add up to at least 1? Clearly it is possible to cover with planks of total width 1, by aligning them perpendicular to the direction in which the body has minimum width. Tarski himself showed that the answer to the question is yes, if the body is an Euclidean ball in 2 or 3 dimensions. During the 40s, the problem gained a certain notoriety. It was

Convex geometry and functional analysis

183

finally solved in the affirmative by T. Bang in 1951 [6]. The basic lemma used by Bang, is proved below. Though apparently simple, this lemma turns out to be the starting point for several surprising developments. At the end of his paper, Bang asked a further question, which is a good deal more natural than Tarski's original one, since it is affine invariant. Given a convex body K, the relative width of a plank, S, is the width of S, divided by the width of K in the direction perpendicular to S. Bang's affine plank problem asks:

QUESTION4. If a convex body is covered by a union of planks, must the relative widths of the planks add up to at least 1? Plainly, an affirmative answer to Bang's question, strengthens Bang's result. The general case of this affine plank problem is still open, but the most important case, in which the body is assumed to be centrally symmetric, was proved in [3]. If K is such a body, then it may be regarded as the unit ball of some finite-dimensional normed space. A plank in a normed space is a set of the form {x c R ~" lOS(x)- ml <~ w] for some continuous linear functional 4~ on the space. If the norm of 4~ is 1, then the relative width of the plank is w. The result in [3] applies to infinite-dimensional spaces as well as finite: in the language of functional-analysis, it states: THEOREM 1 1 (Ball). I f (~i ) is a sequence of linear functionals, o f norm 1, on a Banach space X, mi is a sequence o f real numbers and (wi) is a sequence o f positive numbers satisfying 0(3

Z

tOi < 1,

1

then there is a point x in the unit ball o f X f o r which Iqbi(x) - mil > llOi f o r every i.

This formulation of the theorem makes it clear that the result simultaneously extends the Hahn-Banach Theorem and sharpens the Uniform Boundedness Principle. It also suggests a similarity between the plank problem and the "Coefficient Problem" in harmonic analysis. For any sequence of numbers (Ck) satisfying Ick[ 2 < ~ , the function t w+ ~

Ck e ikt

(6)

184

K. Ball

is square-integrable on the circle. For an integrable function f on the circle, let f ( k ) denote the kth Fourier coefficient of f . The de Leeuw-Kahane-Katznelson Theorem [ 15] states that for any sequence (Ck) of positive numbers satisfying (6), there is a bounded function f on the circle for which If(k)l > ck for every k. Thus, it is impossible to distinguish bounded functions, from other squareintegrable functions, just by looking at the sizes of their Fourier coefficients. It is natural to wonder just what conditions on a sequence, (~pk), of functions will ensure that for any square-summable sequence (Ck) of coefficients, there is a bounded function f for which I(f, ~P~)l > c~ for every k. The de Leeuw-Kahane-Katznelson Theorem is proved using modifications of random functions such as

Z ekCkgrk, where (ek) is a sequence of independent choices of sign. The method requires at least some control of the functions ~Pk in a norm "bigger than 2": a uniform estimate on their Lz+e norms, for example. More crucially, it depends heavily upon the orthogonality of the characters. The method, therefore, leaves open several natural questions; for example, what happens when the 7tk are the trigonometric characters, but it is required that the function f be supported on some restricted part of the circle? The situation was resolved in a rather startling way by the following theorem of Nazarov [34]. (All functions and coefficients are, from now on, assumed to be real, since this results in no loss of generality.) THEOREM 12 (Nazarov). Let (Tzk) be a sequence of unit functions in L1 (of a measure space) which satisfy an upper-2-estimate: that is, there is a constant M so that

<~M ( Z a 2 ) ' / 2 for any sequence (ak). Then, for any square-summable sequence (ck), there is a bounded function f satisfying f f~Pk

> Ck

for every k. Nazarov's Theorem is proved in Section 3.3. Although the proof of this theorem does not formally depend upon the other results in this chapter, it will be clear that the methods used

Convex geometry and functional analysis

185

are closely related. The basic lemma used by Bang, is proved in Section 3.1. Section 3.2 contains a proof of Theorem 11, which uses Bang's Lemma in its original form. Finally, the chapter closes by mentioning some related results and open problems.

3.1. Bang's lemma The crucial lemma used by Bang in his solution of the original plank problem is the following. This lemma will be used in Section 3.2. LEMMA 5. Let (Ui) be a sequence of unit vectors in an Euclidean space, ( m i ) a sequence of reals and (wi) a sequence of positive numbers. Then there is a choice of signs (ei) f o r which the vector X ~ ~6ill)iUi

satisfies [(X, Ui ) -- mi l >~ tOi

f o r every i.

PROOF. The following argument deals with the special case in which all the mi are 0. The general case is only marginally more complicated: see, e.g., [3] for a proof. Choose the sequence of signs for which the vector x is longest. Now, suppose k is between 1 and n and let y

-

-

~EiWiUi iCk

so that x -- y + ek w~uk. By the choice of e~, x has norm at least as large as that of y ek wk uk. This means that

e~:(y, u~:) >~O. This in turn guarantees that the number

(x, uk) = (y, uk) + ekwk = ek(ek(y, uk) + w~:) is at least w~ in size. This is what was wanted.

D

3.2. The affine plank theorem The aim in this section is to prove Theorem 11, at least in the special case in which all m i are zero. The argument in the general case is no different; it merely uses the more general case of Bang's lemma.

186

K. Ball

Most of the work goes into solving the problem on a finite-dimensional space. In this case it may be assumed that there are only finitely many functionals (0i)~, and it suffices to prove that if

Zwi = l , then there is a point x in the unit ball for which Iqbi(x)l ~ Wi for each i. By slicing the planks into thin "sheets" it may also be assumed that all the tOi are the same, i.e., that each is equal to 1In. For each i, let xi be a vector of norm 1 for which

~i(Xi) = 1. Let A

=

(aij) be the matrix given by aij -- q~i (x j)

so that each diagonal of A is 1. It suffices to find a sequence (~.j) for which ~ I)~j[ ~< 1 and 1

~j aij )~j / > -

n

for each i; for, given such a sequence, the vector

X = Z ~.jXj, J has the required properties. The main task will therefore be to prove the following combinatorial statement: THEOREM 13. Let A = (aij ) be an n x n real matrix with 1 's on the diagonal. Then there is a sequence (~.j ) f o r which Z

I)~jl ~< 1,

(7)

but

ZJ aij )~j f o r each i.

>>-1

n

Convex geometry andfunctional analysis The argument below will guarantee rather more; namely, that the sequence 2 1 ~j ~< - .

187

()~j)

satisfies (8)

n

This implies (7) by Cauchy-Schwarz. The advantage of aiming for (8) is that one can apply Hilbert space methods to modify the matrix A. From now on, a matrix will be called positive, if it is symmetric and positive semi-definite. The modification of a matrix needed here, is described in the following lemma. LEMMA 6. Let A be an n x n matrix, o f which each row contains a non-zero entry. Then there is a diagonal matrix 69, with non-negative entries, and an orthogonal matrix U, so that the matrix H--OAU is positive and has all diagonal entries equal to 1.

Lemma 6 can be proved, either from Brouwer's Fixed Point Theorem or, more directly, by a variational argument: see [3]. The entries in the diagonal matrix, 69, whose existence is guaranteed by this lemma, are not easily expressed in terms of the original matrix A. For the present purpose it is necessary to understand how large these entries can be. The next lemma will provide an estimate for them. LEMMA 7. Let H be a positive matrix with 1 's on the diagonal and U an orthogonal matrix. Then the diagonal entries o f the product H U satisfy

~-~(HU)2i <~ n.

i

REMARK. The fact that the trace of H U is at most n, is standard. The above, quadratic, estimate looks somewhat unusual. The elegant proof below was shown to me by E LustPiquard. PROOF. For each i, let Yi -- ( H U ) i i , and let F be the diagonal matrix with diagonal entries, Y1. . . . . Yn- Let T be the positive square root of H. Then Z(HU)2

i -- t r a c e ( F H U ) -- t r a c e ( F T T U )

i <, t r a c e ( F T T F ) l / 2 t r a c e ( T U U * T ) l / 2

= trace(F2H) 1/2 trace(H)1/2

-- (t~. yi2) 1/2~ -

( ~ i (oe)2i) 1/2

188

K. Ball

the inequality in the middle following from the "matrix Cauchy-Schwarz" inequality. This proves the lemma. D Theorem 13 is proved by combining these lemmas with Bang's lemma. PROOF. Given the matrix A, choose 69 and U as guaranteed by Lemma 6 and put H = 69A U. Let 01 . . . . . On be the diagonal entries of 69. Since H is positive, and has l's on the diagonal it is the Gram matrix of a sequence (u i) of unit vectors in Euclidean space. By Bang's Lemma there is a sequence of signs (8i) for which the vector

X -- Z

8jOjUj

satisfies [(x,

ui)l ~ Oi

for each i. This is equivalent to

Z(H)ijOjsj J

Oi

for each i. In view of the definition of H this is in turn equivalent to

Z Oi(AU)ijOjsj

Oi,

J

and hence to 1

! Z(AU)ijOj8 j />?/ rt J for each i. The sequence given by 1

n

J

will therefore solve the problem, provided that 1 ?/

Now, since U is an orthogonal matrix, this is the same as asking that

Z O j 2 <~n.

Convex geometry and functional analysis

189

But O A = H U* and since A has l's on its diagonal this implies, in particular, that for each i,

Oi = ( H U ) i i . The desired estimate now follows from Lemma 3.2. This solves the finite-dimensional version of the plank problem. The infinite-dimensional version does not follow formally from the finite-dimensional, except in reflexive spaces where there is weak-compactness. However, the "quadratic" estimate obtained in the special case in which all the plank widths are equal, can be transferred back to the more general situation: the result that has really been proved is this. THEOREM 14. Let A -- (aij ) be an n x n real matrix with 1 's on the diagonal and (Wi ) a sequence of positive numbers. Then there is a sequence O~j ) for which

Z ~y;1 )~J2 <~Z

w j,

(9)

but

Z aij)~j

wi

J

for each i. This quadratic control is sufficient to handle the infinite-dimensional situation. Given

Z//3i <

1,

use the theorem, for each n, to construct a sequence (/~j)~, which corresponds to a point where the first n functionals are large. The quadratic control on these sequences ensures that they are uniformly summable; so it is possible to find an accumulation point.

3.3. Nazarov's solution to the coefficient problem The purpose of this section is to prove Theorem 12, stated earlier. Nazarov's argument begins with slightly stronger hypotheses than those above. It initially gives: THEOREM 15. Let (l[rj) be a sequence of unitvectors in L1 of aprobability space, which satisfy an upper-2-estimate in L2: i.e., for any sequence of coefficients (a j),

IIaJJ[L2 M(ZaJ)'J2

190

K. Ball

Then for any sequence

~Cj

2

(Cj) satisfying

< O0

there is a bounded function f satisfying I E f ~ j l / > ck for each j. Theorem 12 reduces to Theorem 15 via the following weak form of Grothendieck's inequality (see [26, Section 10]). THEOREM 16. If (~k) is a sequence of unit vectors in L 1 (of a measure space (S'2, .T, #)) which satisfy an upper-2-estimate (in L1) then there is a non-negative function ~ for which f ~bd/z-- 1 and for which the sequence (~k/qb) satisfies an upper-2-estimate in L2 of the probability space whose measure is dpdlz. The details of the reduction are left to the reader. The proof of Theorem 15 follows. PROOF. Consider the functions of the form oo

fe -- Z 6jCj~ltj 1

for all possible choices of sign e L1. Moreover, the map

-

(6j).

Each of these sums converges in L2 and hence in

~f~ is continuous from the compact space {-1, 1}o~into L1. The functions fe certainly need not be bounded: but the functions ge given by g~ -- tan- 1 fe are bounded. The aim will be to show that for an appropriate choice of e, and every k,

I ac~, (or being a constant depending only upon M).

Convex geometry and functional analysis

191

Define s : R --+ R by

fo x tan -1 t dt.

s ( x ) --

s is Lipschitz, with constant re/2. This ensures continuity of the function S : L 1 ~ R given by S(f) = Es(f).

There is, therefore, a choice of signs s = (8j) for which S ( f s ) is maximum. By altering the signs of the 7tj, it may be assumed that the maximum occurs for e:(1,1,1

. . . . ).

Use f to denote the function f(1,1,1 ....) integer k, take

A

-- ~

--

CjI[rj and g to denote tan -1 f . For a fixed

E

cj ~ j - ck 7tk -- f - 2Ck ftk.

j~k Then, by the maximality of the choice of signs,

(lO)

0 <~ S ( f ) - S(fk) = E ( s ( f ) - s ( f k ) ) .

Apply the second mean value theorem to the function s (at each point of the probability space) to find a function h, so that at each point

s ( f k ) -- s ( f ) -+- s ' ( f ) ( f k

1 st '

-- f ) + -~

= s ( f ) -- 2s' ( f ) c k ~

( h ) ( f k -- f ) 2

+ 2s" (h)c 2 ft 2

and h lies between f and fk. Combine this with Equation (10) to get 0 <. E ( 2 s ' ( f ) c k C / k -- 2 s " ( h ) c 2 f t 2 ) .

This implies that EgTtk -- Es' ( f ) ~ k ~ ckEs" (h)~ k" 2

To complete the proof it remains to show that Es"(h)O~

192

K. Ball

cannot be too small. By virtue of the fact that s' = tan-1 this integral is

1 1 4-h 2~2" By the Cauchy-Schwarz inequality ~2

1 = El~k[ ~< E 1 4- h2

)1/2

(E(1 4- h2))1/2.

So it is enough to check that E(1 4- h 2) cannot be too large. But since h is pointwise between f and fk, the latter is no more than g(1 4- f 2 4- (fk)2) ~< 1 4- 2M 2.

[q

Nazarov's article [34] includes several extensions of the above result, to spaces other than Loo.

3.4. Related results and open problems An old problem on the boundary between convex geometry and number theory is that of estimating the maximum density of lattice packings of spheres, in Euclidean space. The densest packings found to date, were obtained by this author in [5], using Bang's Lemma. A delightful analogue of Nazarov's Theorem, in a non-commutative setting, was found by Lust-Piquard [31]. She proves that if (aij) is a matrix of reals, for which there is a uniform bound upon the sums of the squares of the entries in any row or column, then there is a bounded operator B on Hilbert space, whose matrix (bij) satisfies,

Ibij l > aij for all i and j. Nazarov's Theorem and Bang's Lemma show, respectively, that if X is either L1 or L2, and (xi) is a sequence of unit vectors in X, satisfying an upper-2-estimate, then for any square-summable sequence (wi), there is a functional 4~ in X* for which I~b(xi)l > tOi for every i. The obvious question is "For which X is this true?" A natural guess would be "Spaces of cotype 2". The adventurous might like to conjecture that a still more general statement is true; a statement that also includes a version of Theorem 11 (up to some constant).

Convex geometry and functional analysis

193

References [1] K.M. Ball, Cube slicing in R n , Proc. Amer. Math. Soc. 97 (1986), 465-473. [2] K.M. Ball, Volumes of sections of cubes and related problems, Israel Seminar on G.A.EA., 1987-1988, Lecture Notes in Math. 1376, Lindenstrauss and Milman, eds, Springer-Verlag (1989), 251-260. [3] K.M. Ball, The plank problem for symmetric bodies, Invent. Math. 104 (1991), 535-543. [4] K.M. Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. 44 (1991), 351-359. [5] K.M. Ball, A lower bound for the optimal density of lattice packings, Duke Math. J. 68 (1992), 217-221. [6] T. Bang, A solution ofthe "Plank problem", Proc. Amer. Math. Soc. 2 (1951), 990-993. [7] I. Barany and Z. Ftiredi, Computing the volume is difficult, Discrete Comput. Geom. 2 (1987), 319-326. n C.R. Acad. Scis. Paris 321 (1995), 865-868. [8] E Barthe, Mesures unimodales et sections des boules B p, [9] F. Barthe, Indgalitds de Brascamp-Lieb et convexitd, C. R. Acad. Scis. Paris 324 (1997), 885-888. [10] H.J. Brascamp and E.H. Lieb, Best constants in Young's inequality, its converse, and its generalization to more than three functions, Adv. Math. 20 (1976), 151-173. [11] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), 375-417. [12] J. Bourgain, J. Lindenstrauss and V. Milman, Approximation of zonoids by zonotopes, Acta Math. 162 (1989), 73-141. [13] H. Busemann and C. Petty, Problems on convex bodies, Math. Scand. 4 (1956), 88-94. [14] B. Carl and A. Pajor, Gelfand numbers of operators with values in a Hilbert space, Invent. Math. 94 (1988), 479-504. [ 15] K. de Leeuw, J.P. Kahane and Y. Katznelson, Sur les coefficients de Fourier des fonctions continues, C. R. Acad. Sci. Paris 285 (1977), 1001-1003. [16] R.J. Gardner, A positive answer to the Busemann-Petty problem in three dimensions, Ann. of Math. 140 (1994), 435-447. [ 17] R.J. Gardner, Geometric Tomography, Cambridge University Press, New York (1995). [ 18] R.J. Gardner, A. Koldobsky and Th. Schlumprecht, An analytic solution to the Busemann-Petty problem on sections of convex bodies, Ann. of Math. 149 (1999), 691-703 [19] A. Giannopoulos, A note on a problem of H. Busemann and C.M. Petty concerning sections of symmetric convex bodies, Mathematika 37 (1990), 239-244. [20] A. Giannopoulos, Private communication. [21] A.A. Giannopoulos and V.D. Milman, Euclidean structure in finite dimensional normed spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 707-779. [22] A. Giannopoulos and M. Papadimitrakis, Isotropic surface area measures, Mathematika, to appear. [23] E. Gluskin, Extremal properties of rectangular parallelepipeds and their applications to the geometry of Banach spaces, Mat. Sb. (N. S.) 136 (1988), 85-95. [24] D. Hensley, Slicing the cube in R n and probability, Proc. Amer. Math. Soc. 73 (1979), 95-100. [25] D. Hensley, Slicing convex bodies-bounds for slice area in terms of the body's covariance, Proc. Amer. Math. Soc. 79 (1980), 619-625. [26] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1-84. [27] M. Kanter, Unimodality and dominance for symmetric random vectors, Trans. Amer. Math. Soc. 229 (1977), 65-85. [28] A. Koldobsky, Intersection bodies, positive definite distributions and the Busemann-Petty problem, Amer. J. Math. 120 (1998), 827-840 [29] D. Larman and C.A. Rogers, The existence of a centrally symmetric convex body with central sections that are unexpectedly small, Mathematika 22 (1975), 164-175. [30] E.H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), 179-208. [31] E Lust-Piquard, On the coefficient problem: a version of the Kahane-Katznelson-DeLeeuw Theorem for spaces of matrices, J. Funct. Anal. 149 (1997), 352-376. [32] R.J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80 (1995), 309-323.

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[33] M. Meyer and A. Pajor, Sections ofthe unit ball of~np, J. Funct. Anal. 80 (1988), 109-123. [34] E Nazarov, The Bang solution of the coefficient problem, Algebra i Analiz 9 (1997), 272-287. [35] C. Petty, Surface area of a convex body under affine transformations, Proc. Amer. Math. Soc. 12 (1961), 824-828. [36] G. Schechtman, Th. Schlumprecht and J. Zinn, On the Gaussian measure of the intersection of symmetric convex sets, Ann. Probab. (1998), 346-357. [37] M. Talagrand, Regularity of Gaussian processes, Acta Math. 159 (1987), 99-149. [38] J.D. Vaaler, A geometric inequality with applications to linear forms, Pacific J. Math. 83 (1979), 543-553. [39] G. Zhang, A positive solution to the Busemann-Petty problem in R 4, Ann. Math. (1999).

CHAPTER

5

A p-sets in Analysis: Results, Problems and Related Aspects Jean Bourgain School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA E-mail: bourgain @math. ias. edu

Contents 1. 2. 3. 4. 5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Existence and construction of A p - s e t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Kolmogorov rearrangement problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lattice points on spheres; Quantum limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Restriction of the Fourier transform to spheres and the Hausdorff dimension of Besicovitch sets 6.1. Further comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. On the distribution of Dirichlet sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H A N D B O O K OF T H E G E O M E T R Y OF B A N A C H SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved 195

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197 199 204 208 213 215 222 223 230

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A p-sets in analysis: Results, problems and related aspects

197

1. Introduction

A subset S of the integers Z is called a Ap-set (p ~> 1) provided for some constant C and all scalar sequences (an)n~S we have the inequality

n~S

LP(T)~< C (n~Sia~ 12)1/2

E an e inx

L P (T)

E

a n e inx

nES

<~c

E

if p > 2,

a n e inx

if p < 2,

(1.1)

(1.2)

L 1(T)

nES

where qr refers to the circle group. The smallest constant C satisfying (1.1), respectively (1.2), will be naturally called the Ap-constant K p ( S ) of the set S. This notion may obviously be extended to the setting of an arbitrary compact Abelian group. The main purpose of this paper is to bring together a number of items directly related to this concept. Those play an often important role in diverse mathematical areas, including harmonic analysis, of course, but also PDE's and analytic number theory. Without intention to survey these topics in any detail, we will mention some of the basic facts and elaborate a bit more on the related problems relevant to this expos6. Some of these generated in fact active research areas. Thus, we will try to change somewhat its general perception as a perhaps interesting, but certainly narrow subject. One of the major papers on "thin sets" in harmonic analysis, in particular on Ap-sets, is that of W. Rudin [38]. It raises in particular the natural question on the existence of A p sets that are not Aq for any q > p. We will discuss related results and problems in the next section. Probabilistic methods turn out to play an important role here. There have been also some byproducts of the methods developed for the purpose of this research, leading for instance to progress on Kolmogorov's rearrangement problem. On the other hand, there are several instances where one is led to estimate Ap-constants for specific finite sets of integers or real numbers. We mention two of them. (i) A basic tool in the theory of dispersive PDE's are Strichartz' type inequalities. For instance, the Cauchy problem for the nonlinear Schr6dinger equation (NLS) depends essentially on the following inequality for the linear Schr6dinger group (eitA)t~R 2(d + 2)

(1.3)

where A denotes the usual Laplacian. Considering the corresponding problem on bounded domains, for instance with periodic boundary conditions, the expression (eitAr ( x ) -

f ~"(se)ei<xse+t~2)

dse

(1.4)

is replaced by an exponential sum of the form

E an e i(nx+lnl2t)

nEZ d

(1.5)

J. Bourgain

198

considered as a function on the (d + 1)-torus q[,d+1. In order to obtain an analogue of (1.3), the issue becomes clearly to evaluate

Kp({(n, It/] 2) E ~d+l ] ]r/] ~ N / ) .

(1.6)

This question is largely unresolved, especially in higher dimensions. (ii) One of the major themes in analytic number theory is to evaluate moments of the ~'-function and more generally Dirichlet series and sums of the form

E annit' n,,~N

(1.7)

where t is restricted to some interval [0, T]. The problem of bounding expressions

fo T

P

ann it dt n,~,N

E

(1.8)

in various ranges of N and T has been extensively studied. Estimates for (1.6) with p an even integer play an important role in zero-density estimates of the ~'-function and L-functions (cf. [32]). On the other hand, several most significant conjectures (known as Montgomery's conjectures) remain essentially unresolved. Some of these conjectures, if true, would also have interesting implications in geometric measure theory, such as maximal Hausdorff dimension of any Besicovitch set in IKd, presently only known in dimension d = 2. Recall that a Besicovitch set in •d is a measurable subset ,4 of ]Kd containing a line or line segment in every direction. A well-known construction in the plane (going back to Besicovitch) shows that such sets may be of zero measure. But it seems reasonable to conjecture that they always have maximal dimension, thus dim,/[ = d .

(1.9)

This problem is also of primary importance in Fourier Analysis related to restriction theory of the Fourier transform to surfaces. What we have in mind here are improvements of the well-known Tomas-Stein theorem for the sphere S d-l, do"

I1~11 L 2(~+,~ d-1 (~d)

~< c

d#

L2(Sd-l)

(1.1o)

when one makes the stronger assumption of a measure # on S d-1 with bounded density dlz/da rather than assuming d#/da E LZ(a). As mentioned, there is an amount of recent research and results on the subject and part of it will be described briefly here. We made up the reference list strictly for the purpose of this short expos6 and even as such there are surely serious omissions. It should certainly not be viewed as a measure for the significance of the work of the various contributors to the subject.

A p-sets in analysis: Results, problems and related aspects

199

2. Existence and construction of Ap-sets Consider first the case p > 2. The methods involved to produce Ap-sets are either of combinatorial (when p is an even integer) or probabilistic nature. THEOREM 2.1. For every p > 2, there exists a subset S ~ Z which is a A p-set and not a

Aq-set for any q > p. Denote

S N - - S A [0, N]. One has

Kp(S)[SN[ 1/2 >/

einX

~]SNI N-1/p,

(2.1)

hence

[SNI ~ Kp(S)2N 2/p.

(2.2)

Thus the intersection of a Ap-set S with [0, N] (hence also any arithmetic progression of length N) has cardinality at most C N 2/p. PROPOSITION 2.2. Fix p > 2. Then for any integer N > O, there exists a subset S C

{ 1 . . . . . N} satisfying I S [ - [N2/p],

(2.3)

Kp(S) < Cp.

(2.4)

In order to derive Theorem 2.1 from Proposition 2.2, recall the Littlewood-Paley theorem (2.5) for 1 1 2/(-1~
Proceed then as follows. Apply Proposition 2.2 with N -- 2 k in order to produce a set S(k) C {2k-1 . . . . . 2 k - 1} satisfying IS(k) [ "~ 4 k/p,

(2.7)

Kp(S(k)) < Cp.

(2.8)

J. Bourgain

200 Define

S = U S(k).

(2.9)

k From (2.5), (2.8) ~j

an e inx

E a. ei" 2) 1/2 P

<~ Cp

n6S(~)

ncS

c ( ll a ei x n~S(~:)

<~ Cp'

Z

2)1/2 P -Cp'

lanl2

ncS(k)

y~la~l 2

(2.10)

ncS

proving that S is a A p-set. In case p is an even integer p = 2h, sets satisfying the conclusion of Proposition 2.2 may be produced by combinatorial means. Observe first that by the Cauchy-Schwartz inequality

Z

=

an e inx

ncS

L2h(T)

an e inx n~S

2

E

--

anl " " "anh

12t

m = n 1%-" .-q-nh

njES

lanl2) h/2

<~ (mmaxRm(h; S)1/2)(

(2.11)

nES

where Rm (h; S) denote the number of representations of m in the form

m--n1 -+-...-+-nh;

nl . . . . . nh ES.

(2.12)

S ) 1/2h.

(2.13)

From (2.11), we get thus K2h(S)

<~ m a x R m ( h ; m

Call S C Z a Bh-sequence if for every integer m (2.12) has at most 1 solution such that nl <<,n2 <~ ... <<,n h . By (2.13), a Bh-sequence is thus a A2h-set. Density estimates on Bh-sequences are sometimes referred to as Sidon's problems. As a general reference on this subject, the reader may consult the book by Halberstam and Roth [25] (in particular Chapter II).

A p-sets in analysis: Results, problems and related aspects

201

PROPOSITION 2.3 (Bose-Chowla). For every h ~ 2 and prime power m, there is a Bhsequence S consisting of m elements and contained in [ 1, m h - 1]. By (2.13), the sets S C {1 . . . . . N = m h } produced by Proposition 2.3 have bounded K2h (S) constant and are of maximal density. This proves Proposition 2.2 when p is an even integer. Let us mention that for h - 2, there is the following more precise density bound on Bz-sequences due to Erd6s and Turan. PROPOSITION 2.4. If S is a B2-set, then [SN[ < N 1/2 -1- O(N1/4). If p is not an even integer, there is no constructive approach to Proposition 2.2 known. Its verification is done probabilistically and in fact in the generality of any bounded orthonormal system. PROPOSITION 2.5. Let {~bl. . . . . ~bN } be an arbitrary orthonormal system on a probability space (Y2, d/z). Assume II~bjll~ < C

(1 ~ j ~< N)

(2.14)

f o r some constant C. Let p > 2. Then there exists a subset S of {1 . . . . . N} such that

I S I - [N 2/p]

(2.15)

and 1/2

(2.16) LP(dlz)

f o r all scalar sequences {aj }j6S. In fact, the property is valid f o r "most" subsets S of {1 . . . . . N} of size ~ N 2/p.

Proposition 2.5 is due to the author [7]. The argument is rather technical, but based on elementary probabilistic considerations. Another proof has been given by Talagrand [42]. It avoids certain technicalities from [7] but uses a more sophisticated probabilistic apparatus (known as the majorant measure theorem). Both arguments are purely probabilistic and hence do not produce explicit examples. Let us give an idea how the proof of Proposition 2.5 given in [7] works. First, there is the following general estimate 2.6. Let s be a subset oflK~ and B -- SUPx~g Ixl. Let 0 < ~ < 1 and consider (~j ) I <<.j<.N independent O, 1 valued random variables of

PROPOSITION mean

f ~j (co)dco-

(1 ~<j ~
J. Bourgain

202

Let 1 <. rn <. N. Then

~j (O))Xj

sup

do)

<. C(6m + (log 1/6) -1)1/2B +

fo

log

B v/log N2(g, t) dt.

(2.17)

Here Ixl - ( ~ X2) 1/2 and for t > 0, we denote N2(s t) the metrical entropy numbers of the set g with respect to the g2-distance, i.e., the minimum number of g2-balls of radius t needed to to cover g. Choose 6 = N (2/p)-I and generate the random set So) of {1 . . . . . N} of (expected) size m = [N 2/p] defining

S-Sco=

{1 ~<j ~< N I ~j(o)) -- 1}.

(2.18)

Write

p-2 P

j'6So)

Z

j'~So)

aj,qbj,

.

(2.19)

Assuming (2.20)

[aj l <~ it follows that (2.19) ~<

1

sup

]

~j (O))Xj ,

(2.21)

xEEo), IAl<~m where we denoted

(2.22)

Assume that we succeeded in "decoupling" (2.21) as

sup X~go~,, Ial<~m

~j (o))xj ,

--~

(2.23)

A p-sets in analysis: Results, problems and related aspects

203

where co, co' refer now to independent random variables. We then apply (2.17) to the co average with fixed o91. Denote further

Kp(co) --

sup

1 lajl<~

ajdpj

Z

(2.24)

jr

and

-

-

m

<<.1.

]

(2.25)

jEA Assume 2
(2.26)

By Bessel's inequality, the number B -- SUPxcEo/Ixl is clearly bounded by p-2

E

sup

Ibjl<~l

j ~c

bj,~j,

- [~'2~_~o'~] ~-l

So/ p-2

<"C[Kp(~')]P/2 (\ 4IS~, ~ I)

[Kp(co')]P/2m

p-2 4

(2.27)

recalling (2.14). Writing for g, h ~ 79m

Iglgl p-2 - hlhlp-2l ~ Clg - hl(lg] p-2 -+-Ihl p-2) and using H61der's inequality (2 < p < 3), it follows

I[glgl p - 2 - h l h l p - 2 l l 2 ~ C l l g - h l l

2 .

(2.28)

3-p

Another application of Bessel's inequality together with an entropy estimate of 79m in L q , q -- 3--~, gives (for some v -- v(q) > 2) log N2 (Co,,, t) ~< log NLq (JQm, ct)

{ log(1 + ~-) + log 71

~ Cm.

log(1 + ~),-~

fort 1.

Consequently, from (2.19), (2.21), (2.17), (2.27), (2.29), we get

J. Bourgain

204

f Kp(o)) p dw ~ S1/2m p-~

(f

)

Kp(w')p/2 dw' + m -1/2 log ~'p (O)) p dw

m 4P 91-

~

-+- log

v/(2.29) dt l o g -m

.

(2.30)

Hence, since m ~ N 2/p, Kp(cO) has bounded expectation. This means that the random set S - S~o of size m will satisfy ~< C

(2.31)

j~S aj~j p whenever lajl ~ 1/~f-m. Extending this conclusion to general coefficient sequences {aj I j E S}, lal ~< 1 is a bit more technical and we refer to [7] for details. We conclude here our discussion of Theorem 2.1 for p > 2. Our understanding of Ap-sets for p < 2 is much less satisfactory. The following result is due to Bachelis and Ebenstein [3] based on a theorem of Rosenthal [37] as main ingredient. THEOREM 2.7. Let S C Z. Then {p E ]1, 2 [ I S is a Ap-set}

is an open interval. Thus, Theorem 2.7 exhibits a different behaviour from the p > 2 case. Remark that for p > 2, the interval {p E ]2, c~[I S is a Ap-set } may or may not contain its endpoint. On the other hand, the only known examples of Ap-sets for p < 2 are Aq-sets for some q > 2. It may be verified that for a random set SN C {1 . . . . . N} of size 6(N)N with density 6(N) satisfying lim log N-+cx~

N~ log 6 ( N )

-1

--

~

(2.32)

one gets for all p > 1 lim N--+~

Kp(SN)

-

cx~.

(2.33)

In fact, for p < 2, probabilistic techniques alone do not seem to suffice to produce nontrivial examples.

3. Further comments (1) Theorems 2.1 and 2.7 from the previous section remain valid if q[' is replaced by any (infinite) compact Abelian group.

A p-sets in analysis: Results, problems and related aspects

205

(2) B2-sets are sometimes also called Sidon sets. Thus a set S C Z is a Sidon sequence if all the sums a + b, a, b E S are distinct. Remark that the name "Sidon set" is also attached to another related but different concept that will be discussed later. As mentioned, Sidon (= B2) sets are A4-sets. Although it is easy (as explained earlier) to construct an infinite A4-set S satisfying

I S N I - IS A [1,

N] I >

N 1/2

(3.1)

for all N

the situation for Sidon sequences is different. Erd6s showed that if S is an infinite Sidon set, then lim ISNl~/log N N

~/N

~< 1.

(3.2)

In the other direction, the best known result remained for a long time a construction of Ajtai, Komlos and Szemeredi who produced a Sidon sequence satisfying the density property lim ISNI >0. N (N log N) 1/3

(3.3)

This was only recently improved by I. Ruzsa [39], who constructed an example of a Sidon sequence S for which

ISNI > N •176

9 / - ~ / 2 - 1.

(3.4)

The reader is referred to the paper [39] for further references. (3) The other meaning of "Sidon set" is that of a subset S C Z such that for some constant C, the inequality

~la~l ~ C nES

Z an e inx

(3.5)

nES

holds for all sequences {a, In E AS}. Equivalently, any functionAf E C(T) for which the support of the Fourier transform f is contained in S, thus supp f C S, satisfies f -- A (T) ( = the space of absolutely convergent Fourier series). There has been an extensive study of the structure of those sets over the past years. Answering a problem raised by Rudin in [38], Pisier showed that S C Z is a Sidon set r S is a Ap-set for all p < cx~ and sup

Kp(S)

p>2 v/P

< c~.

In particular, finite unions of Sidon sets are still Sidon sets (Drury's theorem).

(3.6)

J. Bourgain

206

Call a subset S C Z "independent" if any relation elnl

nt-e2n2-+-...nt-ehnh--O

(es -- 0,1, - 1 )

(3.7)

between distinct elements of S is necessarily trivial, i.e. (3.7) =r

e= = 0

(1 ~< s ~< h).

(3.8)

Independent sets are Sidon sets and the interpolating measures are produced by Rieszproducts H(1 +

bncosnx)

( - 1 ~< bn

<~1).

(3.9)

nES Observe indeed that by (3.7)-(3.9) defines a probability measure on ~'. It was shown by Pisier [35] that S C Z is a Sidon set r there is a constant 6 > 0 such that any finite subset S1 C S contains an independent subset $2 C $1 satisfying (3.10)

IS21 > ~lS~l.

Another proof is given in [6], where it is also shown that moreover for an arbitrary Sidon set S, interpolating measures may always be obtained form averages of Riesz products on independent sets, of the form (3.9). Equivalently IlfllA(•) ~

sup

I(f, #)1

(3.11)

# of the form (3.9) A

if supp f c S. The natural question whether S is always a finite union of independent sets is still open. This is known to be the case if the circle group ~' is replaced by an infinite product group 1--I~ Zp where Zp = Z/pZ with p prime (the Rado-Horn theorem) or p a simple product of distinct prime numbers (the case p - 4 seems still open). See [35,36] as general reference on these matters. (4) Since Rudin's paper [38], there has also been interest in the harmonic analysis and related arithmetical properties of the sequence of squares

S-{n21n-l,2

.... }.

(3.12)

Elementary considerations show that logN (log N) 1/4 ~< K4(SN) ~ e x p - loglogN The problem whether

Kp(S) < oo for some p > 2 is open (and seems difficult).

(3.13)

A p-sets in analysis: Results, problems and related aspects

207

Rudin raised the question of density of S in an arbitrary arithmetic progression P of length N and conjectured that always

IS A PI < CN1/2.

(3.14)

The best known bound here is due to Bombieri, Granville and Pintz [5]

IS A PI < CN2/3

(3.15)

based on effective versions of Falting's theorem on the number of rational points on elliptic curves. Already the fact that IS A P] = o(N) is nontrivial. This follows for instance by combining Szemeredi's theorem on the existence of progressions in subsets A of {1 . . . . . N} of positive density iA] > 6N (we only need length 4 progressions for which a new and very recent proof of T. Gowers asserts that ]AI > C N / l o g l o g N suffices) and the fact that indeed S does not contain 4 elements in arithmetic progression (an old result published first by Euler). The reader is referred to [5] for details. One may also ask for nontrivial estimates on K4 (A) if A is an arbitrary set of m squares. The fact that sup K4(A) -- o ( m ACS IAl=m

1/4)

(3.16)

(and more precise versions of it) may be seen as follows. Assume that K4(A) "-~m 1/4, then necessarily 4 ~,m 3

einX

(3.17)

4 implying that the equation

a+b-c+d

(a,b,c, dEA)

(3.18)

has a number of solutions of the (essentially maximal) order of m 3. A general combinatorial result of Balog and Szemeredi (see [33]) implies then the existence of a subset A1 C A, IAll ~ m with small subset A 1 + A1, i.e. IA1 + All ~< m.

(3.19)

Finally, Freiman's theorem (see [33]) asserts that A 1 is contained in a generalized (= multidimensional) arithmetic progression 79 satisfying 1791 < m. This last fact then in turn implies that A1 (hence S) has large intersections with 1-dimensional progressions P, i.e. IA~ n PI ~ IPI contradicting in particular (3.15).

(3.20)

J. Bourgain

208

(5) Proposition 2.6 is in spirit closely related to Dvoretzsky's theorem on almost spherical sections of convex bodies in R n. Recall that a normed space X, II II is said to be of cotype p ~> 2 provided there is an inequality

c f l , - 1}n

~-~SjXj

de >~

Ilxj IIp

(3.21)

j=l

valid for some fixed constant C and any finite sequence of v e c t o r s {xj I j - 1 . . . . . n} in X. In the context of Proposition 2.6, the relevant spaces X are of course subspaces of L p (p > 2). A result from [21] states that if •n is endowed with a norm II II such that X -- ~n, II II has bounded cotype p constant and moreover the standard euclidean ball

Bn-{xERnlIxI~I}

(3.22)

coincides with the maximal volume ellipsoid ( - - t h e E John ellipsoid) contained in the II II-unit ball {x 6 Rn I IIx II ~ 1}

Kthen for some X

F

~

Gn,m, m

(3.23)

~ n 2/p

Ixl % Ilxll ~ 2~lxl

for all x E F,

(3.24)

where X denotes the mean

- fs

IIx II~(dx).

(3.25)

n--1

Here Gn,m is the Grassmannian of m-dimensional subspaces of ]1{n. Again property (3.24) holds for the "random" section, i.e. for "most" F ~ Gn,m endowed with its invariant measure. Thus here we consider random subspaces of certain dimension rather than random subsets of certain size of a given basis as in Proposition 2.6. Results along this line constitute an important part of the "Geometry of Banach spaces" which is however not our primary line of interest here. The reader may consult [31] for instance for the basics of this theory.

4. The K o l m o g o r o v r e a r r a n g e m e n t p r o b l e m

The question is whether any ONS q~ - ((/9n ) n - - 1,2 .... may be rearranged to become a system of convergence. This means that there is a permutation :r of Z+ such that for any scalar sequence ~ = {an}, with lal --

(~--] ]anl2) 1/2 <

oo

(4.1)

A p-sets in analysis: Results, problems and related aspects

209

the partial sums N

(4.2)

~ anq97v(n) n--1

converges almost surely (a.s.). The problem is open, also for uniformly bounded systems. It is shown in [8] that a positive answer implies the classical (deep) fact that the Walshsystem (ws) on {1, - 1}U in its natural ordering is a system of convergence (a result due to Billard - proven first by Carleson for the trigonometric system). But proofs of those facts seem to depend heavily on the additional structure of these classical systems, not present in the general context of Kolmogorov's problem. This may be an indication that the answer may well be negative. A result of Garsia [22] asserts that if the coefficients {an} satisfying (4.1) are given, then (4.2) will converge a.s. for most permutations Jr. Garsia also raised the question (sometimes referred to as "Garsia's conjecture") whether given any (uniformly bounded) finite orthonormal sequence ~01. . . . . qgn of functions, there is a permutation Jr of {1. . . . . n } such that the L2-maximal inequality holds, i.e.

k ~-~ a j q)rr(j ) max k <~n

j=l

Clal

(4.3)

2

for all scalar sequences {aj [ j = 1 . . . . . n} and where C is an absolute constant. It may be shown that this statement for finite sequences implies the same statement for infinite sequences and hence an affirmative answer to Kolmogorov's problem. The discussion that follows is related to (4.3). Recall first the classical (and optimal) inequality of Menshov stating that always

max

k <~n

•

ajqgj

(logn)lal.

(4.4)

j=l

A byproduct of the probabilistic techniques involved in the solution of the Ap-problem, in particular Proposition 2.5, is the following improvement of (4.4), if we allow a rearrangement. PROPOSITION 4.1. Assume that ci9 -- (qgj) 1 <~j<~n is a uniformly bounded OS. Then the rearranged system (qgzr(j) ) l <~j<~n, where Jr E S y m ( n ) is a random permutation, will satisfy with high probability max k k<~n Z aJqgzr(J)

~< (log log n)lal

j=l

f o r all scalar sequences a -- {aj}j= 1.

(4.5)

210

J. Bourgain

Again this statement may be shown to be the best possible in the sense that the (log log n)-factor may be necessary if one considers random rearrangements. Thus a random rearrangement only does not suffice to solve Garsia's conjecture (assuming the solution affirmative). REMARK. C. Fefferman observed that the double trigonometric system {ei(mx+ny) I m 9 Z , n e Z }

(4.6)

is not a system of convergence for rectangular summation and the Lebesgue constant

Z

amn e i(mx+ny) sup m a x lal~l k,g~N Iml~
~> v/log N.

(4.7)

2

It is remarkable that for a random rearrangement 7r of Z, one gets an inequality

suP[I

E

max

amn e i(rc(m)x+zr(n)y)

lal~< 1 k,e<~g Iml~k, Inl~e

~< (log log N) 2,

(4.8)

2

hence improving the Lebesgue constant of the product system (4.6) in its natural order. PROOF OF PROPOSITION 4.1 (Sketch). The first observation is the following. Denote the normalized invariant measure on the symmetric group Sym(n). Then the image measure of the map Sym(n_),P

> 79m "7r

> (7r(1) . . . . . 7r(m))

(4.9)

is the normalized counting measure on 7)m. Next, for given 1 ~< K < oo, define the convex function

fg(t)-

{

Itl5/2

if Itl ~< K,

5K1/2t2 _ lK5/2

if Itl > K

(4.10)

and denote by II IIFK the corresponding Orlicz function norm, i.e.

IlfllLr~ -inf{)~

> 0 f F/((f)

~< 1}.

(4.11)

LEMMA 4.2. Formostpermutations yr, the system ~ j = GTr(j) (1 <. j <. n) satisfies for all

intervals I C { 1..... n} the inequality ) 1/2

<" C(l~ ~ei ajPj

for all scalars

{aj}

E laJ 12

Lv

and where F

I --

fn/m, m -- III.

(4.12)

A p-sets in analysis: Results, problems and related aspects In view of (4.9), L e m m a 4.2 is a consequence of the fact that if 6 =

211

m/n

and

{~j I j

=

1 . . . . . n } independent selectors of mean 6, then

sup Z lal~l j=l

aj ~j (c~

L

r~_

with high probability. Decomposing lowing one sup Z [Bl=r, [bjl~-~1 j~B

<. C(logn) 3/4

(4.13)

1

{aj } in

bj ~j (co)q)j

level sets, this statement results from the fol-

<<.C(6K + qo + logn)

1/4

(4.14)

LFK Lq0(dco)

Inequality (4.14) is proven using the same methods as in Proposition 2.5. The use of Orlicz norms L rx for variable K seems here more appropriate than L P-norms. The role of q0 ( = log n for instance) is to insure the random event to occur with high probability. Assuming L e m m a 4.2, we explain how to proceed further. First, (4.2) implies the following distributional inequality if g - Z j 6 I a j ~ j , I C { 1 . . . . . n } an interval of size m ~< n

L

Igl 2 ~< A-1/2(logn)15/allgll 2 [>/llgl12

// for A < --. m

(4.15)

Take

f--

~aj~j j=l

with coefficient sequence a -- {aj}j= 1 satisfying lal ~< 1. Our aim is to estimate the partial sums

k

S x f - - Z a j r j.

(4.16)

j=l Construct subsequent refining partitions of 11,1 - {1 . . . . , n} in intervals (/r,s)s=l the following way

aj2 <<.2 - - r .

..... 2 r - 1

in

(4.17)

jEIr, s If I/r,s I > 1, split/r,s as

Ir, s = / r + l , 2 s - 1 U {jr, s} U/r+l,2s,

(4.18)

212

J.

Bourgain

where

~_~ aj2

~ -21Z aj2

j E I r+ 1,s'

for s ' -

2s - 1, 2s

(4.19)

j E Ir,s

and

lajr, s I <~ 2 -r/2

(4.20)

(we introduce an extra index jr, s in (4.18) to take care of a possible large coefficient). Considering (4.16), one may write [1,k] C

(4.21)

U ( I r , sr U {jr, srt} ) r

for appropriate sequences {Sr }, {Sr~}. Thus in particular

r[msXZa, , +msXa,rs] j EIr, s

aj~j

~< EmsaX I Z r

jEIr, s

(4.22)

+ ~ _ . 2 -r/2 r

by (4.20). Observe that the right side of (4.22) does not depend on k. For fixed r, distinguish then intervals Ir, s such that IIr, sl < 2 - r / 2 n

(s is of type (I)),

IIr, sl ~ 2-r/2n

(s is of type (H)).

(4.23)

The number of type (H) intervals is at m o s t 2 r/2 and, by (4.17), their contribution is bounded by (Y~s(ii) lY~dEir, s ajTtjl2) 1/2 of L2-norm at most ( 2 r / 2 2 _ r ) 1/2 _ 2_r/4.

For type (I) intervals, we apply inequality (4.15). Thus write, again using (4.17)

/maxl s(I) I

2

j E Ir,s

<~ ( 2 r / l O 2 - r / 2 ) 2 -+- Z

f

2

(4.24)

s(I) {[ ~jelr, s aj~ji>2r/lO(y~-jeIr, s a2)1/2} j-

213

A p-sets in analysis: Results, problems and related aspects Apply (4.15) to bound the second term in (4.24). Thus

g --

EjGIr, s aj~j,

A --

2 r/10 <

2r/2 < n/llr,sl by (4.23). Hence f{

igl2< 2-r/20(logn)15/8(j~Ei~ ' a 2) Igl> 2r/10 Ilg1121 s

(4.25)

and substituting in (4.24) gives the bound (4.24) ~< (2r/lO2-r/2) 2 + (logn)15/82 -r/20 Z

~ aj2

s(I) jEl .... < (logn)

15/82-r/2~

(4.26)

Collecting estimates (4.23), (4.24), (4.26) gives

aj~j

msaX

< (logn) |5/8 2-r/20.

(4.27)

j E I.... Observe also the obvious bound

2) 1/2 aj ~rj

max j EIr,s

<(~

Z ajoj

9 j EIr,s

~< 1.

(4.28)

2

Coming back to (4.22), write Y~r Er<~ro -~- ~r>ro and apply (4.28) for r ~< r0, (4.27) for r > r0. Hence, choosing r0 ~ log log n, we conclude that -

~< [(4.22)12 ~ r0 + (logn) 15/8. Z < log log n

r >r0

2-r/20 (4.29)

proving Proposition 4.1. REMARK. The uniform boundedness hypothesis SUPn IlgOnI1~ ~ c may be weakened to supn II~0nIIp < ec for some p > 2. For details on the preceding and further results on systems of convergence, the reader may consult [8]. As a general reference work on the subject, see Olevskii's book [34].

5. Lattice points on spheres; Quantum limits Our interest in this section is the behaviour of eigenfunctions q9 of the Laplacian on the d-torus qFd - R d / Z d of eigenvalue )~ when X ~ cx~. Equivalently, these are trigonometric polynomials q9 on qFd such that supp ~" C Z d A x/X Sd-1 -- {r C Z d I l~12 _ A }.

(5.1)

214

J. Bourgain

Thus q9 is of the form ~o

-

Z

-

a~ e ix'~ .

(5.2)

~ egd Nv/-~Sd_ l For d - 2, there is the following result of Cook (see [48,17]). PROPOSITION 5.1. Let q) be an eigenfunction of the Laplacian on qF2. Then

11~o114 11~o112

51/4 .

(5.3)

Thus uniformly for all )v > 0

K4 (Z 2 N v/~S1 ) < 51/4.

(5.4)

The question w h e t h e r K p ( Z 2 N x/~S1) < Cp for all p < oo is open. In general, it seems reasonable to conjecture that

Kp(Z d N

V/-~Sd_l) < Cp

for p <

2d

(5.5)

d-2

and d-2 K p ( ~_J n %li~ Sd_ l ) ~ ~ 4

2d 2d--p-q-8 for p ~> ~ . d-2

(5.6)

In [7], the following partial verification of (5.6) is obtained. PROPOSITION 5 2 9

"

(5.6) holds if d >>.4 and p ~ 2(d+l) d - 3

"

For d ~> 3, no estimate of the form (5.5) for some p > 2 seems to be known 9 Next, we discuss the notion of a quantum limit. These are the weak*-limits of sequences of m e a s u r e s ~ j on ,-~d of the form d#j dx

- Igoj12,

(5.7)

where the qgj are L2-normalized eigenfunctions of eigenvalue ~,j with limj_+oc ~,j -----OO. For details on the following results, the reader should consult [28]. PROPOSITION 5.3. The density of any quantum limit v on 7~2 is a trigonometric polynomial whose Fourier transform is supported by at most 2 circles centered at the origin. PROPOSITION 5.4. For 3 <. d ~ 5, every quantum limit v on 72d satisfies

~ kEZ d

I~(k)l d-2 < ~ ,

(5,8)

A p-sets in analysis: Results, problems and related aspects PROPOSITION 5.5.

215

Quantum limits are absolutely continuous in any dimension.

The proof of Proposition 5.5 depends on the following geometric result due to B. Connes. We first introduce certain graphs on spheres. For an r-dimensional sphere S in R d and p > 0, define G p (S) as the graph with S A Z d as vertex set and connecting 2 points provided they are at distance < p. PROPOSITION 5.6 (Connes [16]). There is a function gr(R), l i m n ~ gr(R) --cx~, such that if S is an r-dimensional sphere in R d of radius R, then for O < p < gr(R), the connected components of Gp(S) are contained in one of the ( r - 1)-spheres I21 . . . . . f2m formed by intersection of S with a subspace of R d of dimension r. Proposition 5.6 as well as Proposition 5.1 have been motivated by problems on uniqueness of trigonometric series. More precisely, the uniqueness question for spherical summation: Let )-~ czJ a~ e ix~ be a trigonometric series such that

Z

a~ e ix~ R ~ >

0

pointwise.

(5.9)

I~I~ 2 is a more recent result due to the author. The starting point is the following fact (due to Connes) and which is a consequence of Proposition 5.6. PROPOSITION 5.7. Let qgj be a sequence of eigenfunctions of the Laplacian on 7fd such that l i m j ~ q g j ( x ) - 0 pointwise for x in a nonempty open subset of qFcl (we do not assume any normalization). Then lim IIgoj112 -

j--+ cx:~

o.

(5.10)

The reader interested in details and references on uniqueness problems for summation of trigonometric series may consult [ 1,2] which is a recent survey on the subject.

6. Restriction of the Fourier transform to spheres and the Hausdorff dimension of Besicovitch sets The first result we mention is the Tomas-Stein theorem (see [45,40]), which plays an important role in harmonic analysis.

216

J. Bourgain

PROPOSITION 6.1. Let S d-l, o" be the unit sphere with its surface measure or. Let lZ M (S) be a measure such that /z << cr

and

d# d~

- -

e

L2 (~).

Then the Fourier transform ~'(x) -- f s eix~/z(d~)

is an L q (IRd)-function for 2(d + 1) d-1

~< q ~< cx~.

(6.1)

The restriction (6.1) on q is optimal. There is also the following interpretation of Proposition 6.1. Choose R a large number and let {~ [ k = 1 . . . . . K} be a system of frequencies on RS that are 1-separated, i.e. I ~ 1 - R,

i~e~ _ ~k'l > 1

for k r k'.

(6.2)

Thus

K <~R d-1.

(6.3)

For each k, denote [ ~ ] 6 Z d the vector obtained by replacing each of its coordinates by its integer part. Consider {ei[~k]'x [ k -- 1 . . . . . K} as exponentials on ~,d. Then, by rescaling and discretization of Proposition 6.1, we get that the Aq-constant d-1

Kq {[~11 k = 1 . . . . . K} < CR 2

d q

for

2(d + 1) d-1

~ q ~< oe.

(6.4)

Observe that this statement differs from the results in Section 5 because the set of lattice points {[~] [k - 1 . . . . . K} is not contained in a sphere anymore. Proposition 6.1 has the following version. PROPOSITION 6.2. Let f E L p(IRd), 2 ( d + 1) 1 ~< p ~< ~ . d+3

(6.5)

Then the restriction of the Fourier transform f~" of f to the sphere Sd-1 is an L 2 -function and

(f,

I~12 d~

)l j2

~ ell flip.

(6.6)

A p-sets in analysis: Results, problems and related aspects

217

Let us mention that in these statements, the sphere may be replaced by any compact smooth hypersurface of nonvanishing curvature. In particular, replacing the sphere by the paraboloid, one obtains Strichartz' inequality [41] for the linear Schr6dinger group (eitA)teR. PROPOSITION 6.3.

Let cb E L2(R d) and define

(eitA~b) (x) --

f x()ei(x'~+tl~12)dse.

(6.7)

Then I eitA~bl Lq(P#+i) <~CIl~bll2 for q --

2(d 4- 2)

(6.8)

Observe that here the integral (6.7) is unrestricted because of the scaling property when replacing 4>by 4~z (6.9)

dt))~(x) = )~d/2~)()~x).

PROOF OF (6.8) (Sketch). By stationary phase, one has the decay estimate

Ilei'A4,l[~ <

Cltl-d/2114~ll~ for t ~ 0.

(6.10)

Hence, interpolating with the unitary property [ eit/x~bil2 -- 11q5112

(6.11)

we get IleitA~b q <~Cltl-( 89

1

(6.12)

In order to prove (6.8), proceed by duality and estimate

f e-i'Af(.,t)dt

(6.13)

Ilfll Lq'(Rd+')~ 1.

(6.14)

with

Squaring (6.13) yields by (6.14)

f f (e-itA f (., t), e-iSA f (., s))ds dt

f ei(S-t)Af( , t) dt

(6.15) Lxq,s

218

J. Bourgain

First, from (6.12), estimate --

f e i{s-t)A f ( . , t) dt

q' dt.

(6.16)

L~ Apply then the Hardy-Littlewood (or Young) inequality to bound ll(6.16)llLq, observing that by (6.8)

1

2 q

This gives the required result. Strichartz' inequalities for Schr6dinger and wave equations are an essential ingredient in the solution of the initial value problems of nonlinear equations. In the context of Proposition 6.3, especially the IVP for the NLS iut -+- A u i ulul p-2 = O, u(O) - ck ~ H s (R d)

(6.17)

has been studied in detail. It may be rewritten as follows (Duhamel's integral formula) u(t) -- eitAq~ 4- i f0 t ei(t-r)A(ululP-2)(r ) dr

(6.18)

and the mapping properties of e itA play a key role when estimating (6.18). Interesting new problems arise if one considers the case of periodic boundary conditions, thus replacing •d by qI'd. The linear evolution operator (6.7) is now given by (eitA~b) (x) -- Z

~'(n)e i(nx+lnl2t)

(6.19)

nCZ d

and should be considered as a function of (x, t) 6 ,~d§ The classical analogue of Strichartz' inequality, thus

]leitA~[ L 2(~+2______~
(6.20)

fails. For instance N Z ei(nx+n2t) n=l

>~ N 1/2 (log N) 1/6.

(6.21)

L6(• 2)

However, it seems reasonable to conjecture the following

Ile~'Ar

~ CI1r

ifq <

2(d + 2)

(6.22)

219

A p-sets in analysis: Results, problems and related aspects

and d d+2

UeitA~HLq(Ta+,) << N2

q +~114~112 ifq ~>

2(d + 2) d

(6.23)

A

where we assume supp 4~ C B(0, N). The following partial results are known (see [ 11 ]). PROPOSITION 6.4. For d -- 1, (6.22) holds with q - 4.

(6.24)

For d - 1, 2, (6.23) holds.

(6.25)

For d ~> 3, (6.23) holds when q >~ 4.

(6.26) A

PROPOSITION 6.5. There is the following distributional inequality, assuming suppq~ C B(0, N), ll4'll2 ~< 1 mes[(x, t) ~ Td+l I ]eitZx~] > X] << Ne~. -2(d+2)/d

(6.27)

provided we assume in (6.27)

(6.28)

)~ > N d/4.

The proof of Proposition 6.4 is based on simple arithmetic considerations. Proposition 6.5 is related to Proposition 5.2. Both are an application of the circle method technique. Applications of Propositions 6.4 and 6.5 to the Cauchy problem (6.17) for NLS with periodic boundary conditions, thus 4~ ~ HS (Td), are discussed in [11 ]. See also [23]. Next, coming back to Propositions 6.1 and 6.2, we discuss Stein's restriction conjecture. The conjecture is that A

IIf IxllLr(o)~ CIIflILP(Rd)

(6.29)

provided p <

2d d+l

and

d-1 d+l

1 1 + - ~> 1. r p

(6.30)

Thus Proposition 6.2 corresponds to the case r - 2, p ~< 2(d + 1)/(d + 3). Observe that if one disposes of the inequality A

(6.31)

Ilf ]SilL'(o-) ~< Cllfllp, then also the following one holds A

I]flsllLr(,~)~Cllfllp

if r < p .

(6.32)

J. Bourgain

220

Indeed, (6.31) corresponds to boundedness of the map

T" L p (IRd) --~ L ' (o-) " f ~ f ' l s

(6.33)

hence

T* " L cx~(o-)____~ L pt (IRd). Thus, invoking the theory of summing operators, T* is rt-summing, for any r < p. This means that there is a probability measure v on S d- 1 satisfying (6.34)

IIT*gollp, ~ Crllqglltr'(dv).

But because T is rotational invariant, a standard averaging argument permits us to replace dv by do-. Hence we get (6.32). Consequently, Stein's conjecture essentially amounts to the inequality ~IIf IsIItlfo~ ~< Cpllfllp

2d for p < d + 1"

(6.35)

(This argument is only applicable for the sphere.) What is known? For d = 2, the restriction conjecture is correct and due to Stein. For d ~> 3, the problem is open and much harder. The truth of inequality (6.35) has deep consequences in geometric measure theory, which we explain next. DEFINITION 6.6. A measurable subset .,4 of R d is called a Besicovitch set provided .,4 contains a line or line segment in every direction. An old construction (see [18]), going back to Besicovitch, permits us to produce a measure-zero Besicovitch set in the plane (d -- 2). Then, straightforward product techniques show that there exist Besicovitch sets of measure-zero in any dimension d ~> 2. The following conjecture is verified for d = 2 but open for d ~> 3. It can be shown to be a formal consequence of (6.35) (see [9]). CONJECTURE 6.7. A Besicovitch set in R d has always full Hausdorff dimension. Thus dim,,4 = d.

(6.36)

REMARK. As we will see in the next section, there is another problem on the distribution of Dirichlet sums, known as Montgomery's conjecture, that implies Conjecture 6.7 equally well. It should be mentioned that the role of Besicovitch sets in harmonic analysis became first apparent from C. Fefferman's work on the ball multiplier. He proved the following: For d ~> 2, the map f --+ I

ff(~)e ix~ d~ I~<1

is only bounded on LP(• d) for p = 2 (see [19]).

(6.37)

A p-sets in analysis: Results, problems and related aspects

221

A key ingredient in his argument is indeed the existence of measure-zero Besicovitch sets. From Proposition 6.2, (6.35) holds clearly if p ~ 2(d + 1)/(d + 3). The first improvement (d ~> 3) was obtained in [9]. More precisely, for d = 3, (6.35) was established for p < 58/43. The first step consists in obtaining certain "nontrivial" geometric information related to Conjecture 6.7. We introduce the following maximal function f~*, for given 6 > 0 (which we refer to as "Kakeya maximal function"). Let f be a function on R d. Define for ~ E S d - l

f~*(~)

-

supWi Ifo<)l

(6.38)

In (6.38), the supremum is taken over all tubes r of unit length with given direction ~ and width 6 (thus there is a translation freedom). Also Irl refers to the measure of Ir I, thus Irl ~, 8d-1. CONJECTURE 6.8. For 1 <~ p <~d, one has the following inequality d

[IflltP(Rd).

(6.39)

The optimality of this statement may be seen by simply choosing f = A's(0,~). Conjecture 6.8 implies Conjecture 6.7. Furthermore, if (6.39) holds for some 1 ~ p

(6.40)

holds for any Besicovitch set A in R d. Again Conjecture 6.8 holds for d = 2. In [9] (6.39) was proven for p ~< 7/3. The second step is then to deduce from this geometric fact new harmonic analysis results on the restriction problem and we succeeded in obtaining (6.35) for p < 58/43. This process is thus essentially converse to the line of thought in Fefferman's result on the ball multiplier (6.37) mentioned above. Since [9,12] appeared, there has been further progress that however is still far from solving the problem. Concerning (6.39), T. Wolff obtained the following [46]. PROPOSITION 6.9. Inequality (6.39) holds f o r p <~ d + 1 in any dimension. Remark that (6.39) with p -- (d + 1)/2 may be considered as the "trivial" inequality that corresponds to the condition p ~< 2(d + 1)/(d + 3) in (6.35), implied by the LZ-theory (i.e. Proposition 6.2). COROLLARY 6.10. If A is a Besicovitch set in N d, then d d i m A ~> ~ + 1.

(6.41)

222

J. Bourgain

In particular, for d = 3, we get 5 d i m A ~> ~

(6.42)

which is still the best result for Hausdorff dimension at the time of this writing. Very recently, the author showed that for a Besicovitch set A in dimension R d, one has 13 12 d i m A >> ~-~ d -+- 2--5 - e

(6.43)

which improves on (6.41) for large d. The argument is mainly based on T. Gowers' proof (contained in the paper [24]) of the Balog-Szemeredi theorem mentioned in Section 3. Some further progress in this direction, appears in the recent papers [30,29]. About conjecture (6.35) on restrictions, the best result in d = 3 at this point is due to Tao, Vargas and Vega (see [44] and forthcoming paper). PROPOSITION 6.1 1. Inequality (6.35) holdswith d - 3for p < 26/19.

6.1. Further comments (1) Related to the restriction conjecture is another problem known as the Bochner-Riesz summation conjecture in harmonic analysis. We first introduce the Bochner-Riesz multiplier m~, 0 ~< )v ~< (d - 1)/2,

m)~(~)

[ (1 -l~12) )~

/0

if I~1 ~

1,

(6.44)

ifl~l > 1.

Thus

cos Ix l

~(x)

(6.45)

ix ld-@+~

CONJECTURE 6.12. mx is a bounded Fouriermultiplier on LP(•d), p 7~ 2, i.e.

f f"(~)m x (~) e ix'$ d~

Cp,)~llfllp

(6.46)

p

if and only if (6.47)

)v>0

and 2d d-k- 1 +2)~

2d

1-2X

(6.48)

A p-sets in analysis: Results, problems and related aspects

223

Recall indeed that (6.47) is necessary by Fefferman's ball multiplier result. Condition (6.48) is easily implied by (6.45). It is known [43] that the Bochner-Riesz conjecture implies Stein's restriction conjecture. Again for d = 2, it is correct and was proven by Carleson and Sj61in [ 15] (another proof is due to H6rmander [27]). In arbitrary dimension, C. Fefferman [20] proved (6.12) under the restriction max(p, p') ~>

2 ( d + 1)

(6.49)

d-1

using Proposition 6.1. Again using a geometric input related to (6.39), progress was made in [9,12] where (6.12) is verified under the condition 58 max(p, p') > n . 15

(6.50)

The result has been subsequently subject to further small improvements. (2) The reader may wish to consult [ 12] for a survey on oscillatory integral operators and related aspects in geometric measure theory, including in particular the topics discussed here. Most of these results have been improved later however. (3) For a recent survey on results and problems related to Besicovitch sets, see [47].

7. On the distribution of Dirichlet sums

A Dirichlet series is an expression of the form

•

ann it ,

(7.1)

n--1 where the coefficients {an } are apriori arbitrary (thus not necessarily of arithmetic nature). Our purpose here is to mention some results and problems (known as Montgomery's conjectures) on the distribution of Dirichlet sums

2N ~

ann it.

(7.2)

n=N

The most relevant case here is the situation where

2~)1/2 max

lanl

<< N - l + ~

N<~n<~2N

(i.e., "flat" coefficients).

lanl 2 n=N

(7.3)

J. Bourgain

224

Bounds on Dirichlet sums play an essential role in density estimates on the zero's of the zeta function ((s) and systems of L-functions L(s, X) where X denotes a Dirichlet character modulo q. The main reference for the material discussed in this section is H. Montgomery's book [32]. Denote 2N

S(s)- Z

annS'

s 6 C.

(7.4)

n=N First, recall the following classical mean value theorem. PROPOSITION 7.1. For any real To and T > O, we have

fTo+T

( 47r ) N [S(it)[ 2dt ~< T + U Z [an[2"

dTo

(7.5)

n--1

REMARK. The ~3-coefficient in (7.5) is the best possible. Observe that multiplying Dirichlet sums produces another one according to the rule

( ~annit) n
( Z a t n n i t ) - Z bnnit n
(7.6)

bn m

an1 an2"

(7.7)

with

nl,n2
Thus

(Z )' Z ann it

--

n
an(k)n it,

(7.8)

n
where

an(k) -

Z

anlan2 """ank.

(7.9)

n--nln2...nk A crude estimate on the number of divisors yields

lan(

l 2 <
As a consequence of Proposition 7.1 and (7.10), we get.

(7.10)

A p-sets in analysis: Results, problems and related aspects

225

PROPOSITION 7.2. For all k -- 1, 2 . . . .

f0 IS(it)12kdt << U ~(T + N k)

[anl 2

(7.11)

.

Again Proposition 7.2 may be reinterpreted as a statement about A p-sets. Write for O<.t<.T t--TO

(0~<0<~1)

(7.12)

and n it - -

exp(iT (logn)0).

(7.13)

Observe that for T > N, the frequencies {T log n I 1 ~< n ~< N} are at least 1-separated. From Proposition 7.2, we deduce easily PROPOSITION 7.3. K2k({[N k logn] In -- 1 . . . . . N}) << N Cfor all N and k. Extending (7.11) to the (interpolating) statement involving fractional values of k would be most interesting. Montgomery conjectured the following. CONJECTURE

7.4.

fo

(

max lan I2v Is(it)[ 2v dt << N V T ~ (T + U v) N
)

(7.14)

f o r l <<.v <~ 2, T > N. From (7.11), (7.14) clearly holds for v -- 1 and v = 2. Denote for 1/2 < tr < 1 and T > 0 b y N(cr, T ) t h e number of zero's p - - / 3 + i y of ((s) satisfying > ~,

lyl < T.

(7.15)

PROPOSITION 7.5. Assume Conjecture 7.4 correct. Then the "density hypothesis" holds in the full range 1/2 < cr < 1, i.e. N(cr, T) << T 2(1-~r)+e.

(7.16)

See [32]. The density hypothesis implies in particular that Pn+l - Pn << n 89

(7.17)

for consecutive primes Pn, Pn+l. This upperbound on the gap size is essentially as strong as what one knows to deduce from the Lindel6f or Riemann hypothesis.

226

J. Bourgain

REMARKS. Conjecture 7.4 differs slightly from Conjecture 9.2 in [32] in the sense that estimate (7.14) involves an g~-norm of the coefficients rather than the g2-norm. The need of this slight modification appeared more recently. As mentioned in the beginning, cf. (7.3), this more restricted formulation has no effect on its implications. It shows however that the purely ~2 large sieve techniques are not sufficient. Conjecture 7.4 is a consequence of Montgomery's large values conjecture (the same remark applies). The known validity range for the density hypothesis is cr > 25/32, cf. [7]. C O N J E C T U R E 7.6. Let S(s) = Y~nN1 ann s and .T be a set of 1-separated reals in the interval [0, T], T > N. Then

E l s ( i t ) l 2 << T ~(N + 19t-I)N( max lanl2). l<.n<.N

(7.18)

t 6a~

Indeed, assume (7.18)valid. Take lan I << Ne N - 1/2

(1

~< n ~< N).

(7.19)

It follows then from (7.18) that mes[t 6 [0,

Z]lIs(it)l

> ~] << T e N ~ -2

(7.20)

provided )~ > T e.

(7.21)

Hence

fo

T ~]

[S(it)12<< T e N '

(7.22)

and for 1 ~< v ~< 2 T

N l+e

fo IS(it)I2V<<TI+e + NTe fT ~

)~2V-3d& << Tl+e'k- TeNV

(7.23)

which is (7.14). Thus Conjecture 7.6 implies Conjecture 7.4. If correct, Conjecture 7.6 is a remarkably deep statement about the behaviour of a broad class of functions. We wish to make some further comments on its meaning. The Dirichlet polynomial N

S(it) - Z n=l

annit

(7.24)

227

A p-sets in analysis: Results, problems and related aspects

with 0 ~< t ~< T may be substituted by the trigonometric polynomial on T N

F(O) -- Z

an e i[Tl~

(7.25)

n=l

for which supp F c { [T, log n] ] n -- 1 . . . . . N }.

(7.26)

Isupp F'I ~< N

(7.27)

supp F C [0, T. log N].

(7.28)

Thus

and

We claim that Conjecture 7.6 conjectures a similar behaviour for the random set (7.29)

I C [0, T ] N Z

of size III - N regarding the distribution of the associated trigonometric polynomials

F (0) -- ~

an e inO ,

1 lan [ < x/-N"

(7.30)

n6I

In fact, for random sets, there is the following precise result. PROPOSITION 7.7. Let I be the set of N random frequencies in [0, T] n Z, T > N. Then, with high probability, the following holds

Z IF(0)] 2 < C[N + (log T)Jf]]

(7.31)

06.f A

for any trigonometric polynomial F, supp F C I, of the form (7.30) and any subset ,T o f t whose elements are at least T1-separated, i.e.

1

IO - O'l > -~ for 0 ~ O' in ~ . PROOF (Sketch). Generate again the random set I as I = Ioj = {n c [0, T] n ZiOn(CO) - 1},

(7.32)

J. Bourgain

228

where the (~n) are independent 0-1 valued random variables of the required mean

f ~n(CO)dCO-- 8 = ~-. N

(7.33)

(7.31) is clearly equivalent to

C1N

Z IF(O)[ 2 < Oc~'N[IF[>C2 I~T]

(7.34)

for some constants C1, C2. Choosing C2 appropriately one may moreover assume in this statement that N I~1 < ~ . logT

(7.35)

Thus we need to estimate

f sup[o~7~a"~',.
2] 1/2 dco,

(7.36)

where the sup refers to coefficients 8 = (an)l <~n<~Tsatisfying (7.30), thus la~ I <

1

(7.37)

and subsets ,T of qI' consisting of -~-separated points for which (7.35) holds. Dualizing, (7.36) may be replaced by

~ ba ein~ ]] dco, I/~I--1,.TL n< T

(7.38)

Ol

where b = (bc~) and 9t- = {0c~}. Writing ~n = 8 + ~j, f ~n~ = 0, we get thus N1/2(7.38) ~< 8 sup

Z

~ b~ein~

]/~l=l, .T'n
(7.39)

c~

• f suplZ /~, )t-'ln < T

ZboteinO~

dco.

(7.40)

ot

From the separation hypothesis (7.32), (7.39) is easily seen to be bounded by CgT = Consider next (7.40). Since the ~nt are independent and of mean zero, we may write (7.40)---

ff

sup] Z /~, .Y'ln< T

where the

~fn(CO)gn(CO' ) E bo~einO~

{gn} is a system of independent

dco dco',

CN.

(7.41)

Ol

standard Gaussians, independent from {~n }.

229

A p-sets in analysis: Results, problems and related aspects

Applying Slepian's comparison lemma for Gaussian processes, we may drop the inner [ 9[ in (7.41). Thus (7.41) ~< C f f

supl~~n(CO)gn(CO')I~bc~ein~ ] do) t d(_o D, .T'In< T N~ log T

<-cffsupI ,_ ( Nlogr

),/2(ff

(N

~< C log T

211/2

]n~
~tn (W) gn (W') e inO

) 1/2

d~o~do) oo

n
(log T)l/2

dco' dco

If(

1/2 n
(7.42)

-CN.

CN 1/2.

Hence (7.36) < This gives (7.34) and proves Proposition 7.7. For details, see [13]. The range of T is Conjecture 7.6 that is of specific interest in the applications is N < T < N 2. Consider the more precise formulation Z I S ( i t ) l 2 ~< [ot(N)N + tc~"

fl(N)l.UI]N(maxlan[2)

(7.43)

n<~N

valid for all S(s) -- y~nUl ann it and 1-separated subsets ~ of [ - N 2, N2]. Here oe(N) and/3(N) denote some slow growing functions of N. The following combinatorial consequence may then be derived from (7.43), see [14]. PROPOSITION 7.8. Assume (7.43) holds for all N, S(s) - y~nN1 ann it and 1-separated subsets f C [ - N 2, N2]. Then the following is true. Let K, B > 1 and let 0 < f < 1 be a function on [0, B] such that If'l < K. Then sup

/2 r

o

(7.44)

b=l

Proposition 7.8 is then used further to show the following facts (cf. [26]). PROPOSITION 7.9. For each 0 0 such that (7.43) may only be valid if either or(N) > exp(log N) p

(7.45)

/3(N) > N ~(p).

(7.46)

OF

J. Bourgain

230

PROPOSITION 7.10. Assume f o r all e > O, one may take or(N) = C e N ~ = f l ( N ) , thus Conjecture 7.6 holds. Then the dimension Conjecture 6.7 f o r Besicovitch sets holds in any dimension.

See [ 14] for the proofs of Propositions 7.9 and 7.10. Consequently, there are the following two implications

(6.39) (6.41)

(7.23) It is not clear however how a knowledge (or partial knowledge) of Conjecture 6.7 may contribute to get new distributional results on Dirichlet sums. REMARK. From the discussions in Sections 6 and 7, it appears that some of the main problems in harmonic analysis and analytic number theory involve estimating A p-COnstants of specific finite sets S of exponentials, but in the more restricted sense of flat coefficients, i.e.

la~l ~ IS1-1/2

for r 6 S.

This involves thus bounding linear operators from s ~2(S) ~

(7.47) --+ L p rather than from

L p.

References [1] M. Ash and G. Wang, A survey of uniqueness questions in multiple trigonometric series, Harmonic Analysis and Nonlinear Differential Equations, in honor of V. Shapiro, Contemporary Math. 208, Amer. Math. Soc. (1997), 35-71. [2] M. Ash and G. Wang, Some spherical uniqueness theorems for multiple trigonometric series, Ann. of Math. 151 (2000), 1-33. [3] G.E Bachelis and S.E. Ebenstein, On A(p)-sets, Pacific J. Math. 54 (1974), 35-38. [4] G. Bennett, L. Dor, V. Goodman, W. Johnson and C. Newman, On uncomplemented subspace of LP, 1 < p < 2, Israel J. Math. 26 (1977), 178-187. [5] E. Bombieri, A. Granville and J. Pintz, Squares in arithmetic progressions, Duke Math. J. 66 (3) (1972), 369-385. [6] J. Bourgain, Sidon sets and Riesz products, Annales Fourier 35 (1985), 137-148. [7] J. Bourgain, Bounded orthogonal systems and the A(p)-setproblem, Acta Math. 162 (1989), 227-245. [8] J. Bourgain, On Kolmogorov's rearrangement problem for orthogonal systems and Garsia's conjecture, GAFA Seminar, Lecture Notes in Math. 1376, J. Lindenstrauss and V. Milman, eds, Springer (1989), 209250. [9] J. Bourgain, Besicovitch type maximal operators and applications to Fourier Analysis, GAFA 1 (2) (1991), 147-187. [10] J. Bourgain, Eigenfunction bounds for the Laplacian on the n-torus, Intern. Math. Res. Notices 3 (1993), 61-66. [11] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, GAFA 3 (2) (1993), 107-156 and (3) (1993), 209-262.

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[121 J. Bourgain, Some new estimates for oscillatory integrals, Essays on Fourier Analysis in honor of E. Stein, C. Fefferman, R. Fefferman and S. Waigner, eds, Princeton, UP (1994). [13] J. Bourgain, Remarks on Halasz-Montgomery type inequalities, Operator Theory, Advances and Applications 77 (1995), 25-39. [14] J. Bourgain, On the distribution of Dirichlet sums, J. Analyse Math. 60 (1993), 21-32. [151 L. Carleson and P. S61in, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287-299. [161 B. Connes, Sur les coefficients des sdries trigonometriques convergentes sphdriquement, C. R. Acad. Sci. Paris, S6r. A 283 (1976), 159-161. [171 R. Cooke, A Cantor-Lebesgue theorem in two dimensions, Proc. Amer. Math. Soc. 30 (1971), 547-550. [181 K. Falconer, The Geometry of Fractal Sets, Cambridge, UP (1985). [191 C. Fefferman, The multiplier problem for the ball, Ann. of Math. 94 (1971), 330-336. [201 C. Fefferman, A note on spherical summation multipliers, Israel J. Math. (1973), 44-52. [211 T. Figiel, J. Lindenstrauss and V. Milman, Almost spherical sections of convex bodies, Acta Math. 139 (1977), 53-94. [22] A. Garsia, Topics in Almost Everywhere Convergence, Lectures in Advanced Math., Maskham Publ. Co, Chicago (1970). [231 J. Ginibre, Seminar Bourbaki, March 96, Asterisque, Soc. Math. de France. [24] W.T. Gowers, A new proof of Szemeredi's theorem for arithmetic progressions of length four, Geometric and Functional Analysis 8 (1998), 529-551. [251 H. Halberstam and K. Roth, Sequences, Clarendon, London, Springer-Verlag (1983). [26] D.R. Heath-Brown, A large value estimate for Dirichlet polynomials, J. London Math. Soc. (2) 20 (1979), 8-18. [27] L. H6rmander, Oscillatory integrals and multipliers on FLP, Ark. Mat. II (1973), 1-11. [28] C. Jacobson, Quantum limits onflat tori, Ann. of Math. 145 (1997), 235-266. [29] N. Katz, I. Laba and T. Tao, An improved bound on the Minkowski dimension of Besicovitch sets in IR3, Ann. of Math., to appear. [301 N. Katz and T. Tao, Bounds on arithmetic projections, and applications to the Kakeya conjecture, Math. Res. Letters 6 (1999), 625-630. [311 V. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer (1986). [321 H. Montgomery, Topics in Multiplicative Number Theorem, Lecture Notes in Math. 227, Springer (1971). [33] M. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Springer-Verlag (1996). [341 A.M. Olevskii, Fourier Series with Respect to General Orthogonal Systems, Ergebnisse der Math., B. 86 Springer, Berlin (1975). [35] G. Pisier, De nouvelles charactirisations des ensembles de Sidon, Advances in Math. Suppl. Studies, Math. Analysis and Applications 7B (1981), 685-726. [361 G. Pisier, Arithmetic characterizations of Sidon sets, Bull. Amer. Math. Soc. 8 (1) (1993), 87-89. [371 H.R Rosenthal, On subspaces of L P, Ann. of Math. 97 (1973), 344-373. [381 W. Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 103-227. [39] I. Ruzsa, An infinite Sidon sequence, J. Number Th. 68 (1998), 63-71. [401 E. Stein, Oscillatory integrals in Fourier analysis, Beijing Lectures in Harmonic Analysis, E. Stein, ed., Annals of Math. Studies 112, 305-355. [411 R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705-714. [421 M. Talagrand, Sections ofsmooth convex bodies via majorizing measures, Acta Math. 175 (2) (1995), 273300. [43] T. Tao, The Bochner-Riesz conjecture implies the restriction conjecture, Duke Math. J. 96 (1999), 363-376. [441 T. Tao, A. Vargas and L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), 967-1000. [451 R Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 447-478. [461 T. Wolff, An improved bound for Kakeya type maximal functions, Revista Math. Iberoamericana 11 (1995), 651-674.

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[47] T. Wolff, Recent work connected with the Kakeya problem, Prospects in Mathematics (Princeton NJ, 1996), Amer. Math. Soc., Providence, RI (1999), 129-162. [48] A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189-201.

CHAPTER

6

Martingales and Singular Integrals in Banach Spaces Donald L. Burkholder Department of Mathematics, University of Illinois, Urbana, IL 61801, USA E-mail: donburk@math, uiuc. edu

Contents 1. 2. 3. 4. 5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric characterization of U M D spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Singular integral operators and U M D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The unconditional constant of the Haar basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential subordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. A sharp martingale inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Monotone bases and contractive projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. The complex unconditional constant of the Haar basis . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Inequalities for the martingale square function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Differential subordination of harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. The Beurling-Ahlfors transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Martingale characterization of spaces with the R N P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Martingale characterization of spaces with the A R N P . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Characterization of A U M D spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction Martingale theory provides insight into some of the classical Banach spaces such as the Lebesgue spaces, the Orlicz spaces, and the Hardy spaces, but it also plays a role in more abstract settings. One example, is Pisier's use of martingales to renorm superreflexive spaces; see Pisier [ 124], the earlier work of James [94] and Enflo [71], and the article on renormings by Godefroy [85] in this Handbook. Due to the size of the subject, the present article can treat only a small part of the fruitful interaction between Banach spaces and martingales, and the light this interaction sheds on other parts of analysis, including the theory of singular integrals. To study singular integral operators in a fairly general Banach space setting, one must somehow surmount the difficulties caused by the lack of classical orthogonality. The same can be said for martingales and some other mathematical creatures that do not always have an easy transition from their original and friendly environment to one not quite so friendly. But now and then the effort yields an unexpected byproduct. Certain kinds of biconvex functions designed to surmount these difficulties in the study of martingale transforms are just the fight tools to use in proving some of the most important sharp inequalities in the scalar-valued case, for example, the one giving the value of the unconditional constant of the Haar system. Also, the similar LP behavior of martingale transforms and singular integrals in the scalar-valued case carries over to those Banach spaces (the UMD spaces) for which, in an appropriate sense, martingale difference sequences are unconditional. In Section 2, we define ~'-convexity and show that a Banach space is UMD if and only if it is ~'-convex. In addition to the biconvex function ~', other biconvex and biconcave functions enter into the discussion. For a Banach space, the existence of any of these functions not only provides an additional equivalent to UMD but also leads to a sharp martingale inequality if the optimal such function can be found, as it often can be if the Banach space is R, C, or any other Hilbert space. The work of M. Riesz on the Hilbert transform in the 1920s and that of Calder6n and Zygmund on more general singular integral operators in the 1950s soon led to the natural question of how much of this work, so influential during the last fifty years, can be carried over to functions with values in a Banach space. As early as 1938, it could be seen as a consequence of the work of Bochner and Taylor that not all Banach spaces are well behaved even for the Hilbert transform. Later, it became clear that those Banach spaces that are well behaved for the Hilbert transform (the HT spaces) are also well behaved for those singular integral operators amenable to treatment by the Calder6n-Zygmund method of rotations, for example, those with odd kernels. Section 3 is focused on the following martingale characterization of the HT spaces: a Banach Space is an HT space if and only if it is UMD. Combining this result with that of Section 2, we have that a Banach space is an HT space if and only if it is ~'-convex. It is of interest that the only known proof of this geometrical characterization of the HT spaces is with the use of martingales. Figiel has shown recently how a remarkably large class of singular integral operators can be studied in the ~'-convex = UMD = HT context using a martingale approach. As a number of other authors have discovered, these spaces can also often provide the right setting for the study of evolution equations, the closedness of the sum of two closed operators, spectral

D.L. Burkholder

236

theory, multiplier theory, and other topics. Some of this work will be discussed briefly at the end of Section 3. In Section 4, we give a short derivation of the unconditional constant of the Haar basis of L~ (0, 1). One byproduct is a biconcave function on R x R that, with the substitution of the norm of a real or complex Hilbert space It for the absolute value function, becomes a biconcave function on H x It. With a simple transformation, the latter function becomes the special function U of Section 5, which is the key to the proof of a conjecture of Petczyfiski about the complex unconditional constant of the Haar system, and to the proofs of other results for martingales, harmonic functions, and the Beurling-Ahlfors transform (sometimes called the complex Hilbert transform). Section 6 provides an essentially self-contained proof of the Banach space version of Doob's theorem for the scalar case" A Banach space B has the Radon-Nikod~m property (RNP) if and only if every L 1 (0, 1)-bounded martingale converges almost everywhere. Section 7 treats the analogous characterization of Banach spaces with the analytic RadonNikod3)m property (ARNP) in which analytic martingales come into play. Section 8 focuses on the class AUMD, which consists of those Banach spaces for which, in an appropriate sense, the difference sequences of analytic martingales are unconditional. In this article, Banach space will mean nondegenerate Banach space with a similar convention for Hilbert space. For functions with values in an infinite-dimensional space, measurable will mean strongly measurable. Also, there will be no need to distinguish notationally between functions and their equivalence classes.

2. Geometric characterization of UMD spaces Let B be a real or complex Banach space, (Y2, .T, P) a probability space, and (.Tn)n>>O a nondecreasing sequence of sub-a-algebras of .T. A sequence (dn)n>~O of (Bochner) integrable B-valued functions is a martingale difference sequence relative to the filtration (3Cn)n>>Oif for all n >~ 0, the function dn is fin-measurable and E(dn+l ].Tn) - - 0 where the latter condition is simply that

fa

dn+l dP =O

forallA6f'n.

(1)

The Banach space B is UMD (unconditional for martingale differences) if tip(B) is finite for some p E (1, ec); equivalently, for all p 6 (1, oc). Here tip(B) denotes the least/3 [1, oc] such that

~-~dk k=0

p

(2)

k=0

for all B-valued martingale difference sequences d, all sequences e of numbers in {1, - 1}, and all nonnegative integers n. In this definition, the filtration (Un)n~>0 must vary, and the probability space must also vary unless it is nonatomic. By (1), if (an)n>~Ois a sequence of scalars, then (andn)n>~O is also a martingale difference sequence. So if B is UMD and the

Martingales and singular integrals in Banach spaces

237

series ~ - - 0 akdk converges in the Lebesgue-Bochner space L~ ($2, ~ , P), then the series converges unconditionally. That the finiteness of tip(B) for some p E (1, cx~) implies its finiteness for all p E (1, cx~) is due to Pisier; see Maurey [ 109]. It is also an immediate consequence of Theorem 1 below. Let fn Y~=0dk and Y~=0ekdk. The martingale g is the transform of the martingale f by e and this • 1-transform satisfies --

gn

--

[Igllp ~ ~p(B)llfllp.

(3)

Here IIf IIp - supn~>0 [Ifn IIp. Maximal functions will also play a role. The maximal function f * of f is defined on S2 by f*(co) = SUPn~>0 IIf~(co)ll. The inequality (3) holds with the same constant tip(B) if, in the definition of g, the number ek is replaced by an U(k- 1)v0measurable function that maps S-2 into the interval [ - 1 , 1]. If H is a real or complex Hilbert space, then tip(H) = p* - 1 for all p E (1, cx~) where p* = p x/q and 1/p + 1/q = 1; see Section 5. This includes R and C. Clearly, tip(B) >/ tip(R). Therefore, for all Banach spaces B,

tip(B)/> p * - 1.

(4)

Other results also follow from the scalar case such as tip (g P) = p* - 1 (integrate term-byterm). Indeed, flp(g~) - tip(B), which implies that if B is a UMD space, then so is ~ for all p E (1, cx~), and similarly for L~(0, 1). The spaces el, ~ , LI(0, 1), and L ~ ( 0 , 1) are not UMD. In fact, UMD spaces are reflexive, even superreflexive; see Remark 4 in Maurey [ 109] and the proof of this in Aldous [ 1]. However, this property does not characterize UMD spaces: there are superreflexive spaces that are not UMD [123]. Since every superreflexive space can be given an equivalent uniformly convex norm [71 ] and, by (2), every Banach space that is isomorphic to a UMD space is also UMD, the geometric property of uniform convexity does not imply UMD. As can be seen above, the classical reflexive Lebesgue spaces are UMD. Other examples include the reflexive Orlicz spaces [87], the reflexive trace-class spaces [87,20], and the reflexive noncommutative L p (M, r)-spaces associated with a v o n Neumann algebra M possessing a faithful, normal, semifinite trace r [ 12]. Is there a geometric property that these and other UMD spaces share that no other Banach space has? A real or complex Banach space B is ~'-convex if there is a biconvex function ~" : B x B R such that ~"(0, 0) > 0 and ~'(x, y) ~< [Ix + y [I if IIx II = Ily II = 1,

(5)

(A function ~" is biconvex on B • B if both ~'(x, .) and ~'(., y) are convex on B for all x, y E B.) The ~'-convexity property characterizes UMD spaces [30,36]. THEOREM 1. A Banach space B is UMD if and only if it is ~-convex. This is proved in [30] with a slightly different but equivalent notion of ~'-convexity. The equivalence is proved in [36].

238

D.L. Burkholder

A biconvex function ( that satisfies (5) must also satisfy ((0, 0) ~< 1" if IIx II = 1, then ( ( x , x ) <<.2, ( ( x , - x ) <~O, and ((0, 0) ~< [((x, 0) + ( ( - x , 0)]/2, so ((0, 0) ~< [((x, x) + ( ( x , - x )

+ ((-x,x)

+ ((-x,-x)]/4

~< 1.

If B -- I-l, a Hilbert space, and ( is defined by ((x, y) = 1 + (x, y) where (., .) is the real part of the inner product of H, then [((x, y)]2 ~< 1 -+- 2(x, y) -+-IIxII2llyll 2 - I I x + yll 2 + (1 -Ilxll2)(1 -Ilyll 2) so ( is a biconvex function satisfying not only (5) but also ((0, 0) = 1. This shows that the upper bound 1 can be attained. On the other hand, if B is not a Hilbert space, then the bound is not attained [31 ]. Also, see Lee [102,103]. Therefore, a Banach space is a Hilbert space if and only if there is a biconvex function ( satisfying (5) with ((0, 0) = 1. It is often convenient to consider biconvex functions that satisfy a stronger condition than (5). Let/2 be the family of biconvex functions u on B x B such that

u(x, y) <. IIx + yll

(6)

if Ilxll v Ilyll >/1,

and let (B be the greatest such function. Then B is (-convex if and only if (B (0, 0) > 0. The "if" part is clear and the "only if" part follows from Lemma 8.5 in [36] where it is also shown that there is a function u in E that satisfies

u ( x , - x ) <. u(0, 0)

ifx 6 B,

(7)

and

~B(0, 0) ~ 2u(0, 0).

(8)

The condition (7) is satisfied by u = (B for many Banach spaces, for example, by any space B that is not UMD, in which case (B(X, y) -- IIx + Y lI; by Hilbert spaces, in which case,

(H(X, y)

-

-

-

[llx + yll 2 + IIx + yll

( 1 - [[x[]2)(1- [[y[[2)],/2

if IIx II v Ily II ~< 1,

if [[x[[ v [[y[[ >~ 1;

and by some other UMD spaces such as s r where r 6 (1 oe) and E is UMD (see Remark 8.2 in [36]). THEOREM 2. If u" B x B --+ R is biconvex and satisfies (6) and (7) with u (0, 0) strictly positive, and g is a 4-1-transform of a B-valued martingale f , then

~P(g* >~~.) <~

2 u(O, O)

Ilflll,

)~ > O,

(9)

Martingales and singular integrals in Banach spaces

239

and

324 [[gl[P ~ u(O, O) ( P * - - 1)[[f[[p,

1 < p < o0.

(10)

The inequalities (8) and (10) imply that a (-convex space is UMD. If u is the greatest biconvex function satisfying (6) and (7), then inequality (9) is sharp; see Remark 4 below. Although (10) is not sharp, it does give, with (4), the correct order of magnitude of tip(B) as p --+ 1 and as p --+ ~ . In the proof of Theorem 2, it is enough to work with simple martingales. A martingale f is simple if f/7 is a simple function for all nonnegative integers n and there is a nonnegative integer n such that f/7 -- f/7+l . . . . . Then any + l - t r a n s f o r m g of f is also a simple martingale. Fix such a pair (f, g) and consider the martingales X and Y defined by /7

X/7 - f/7 + gn - L (1 + sk)dk k=0

and

Y/7 -- fn - gn -- ~--~(1 - ek)dk. k=O

Then Z, defined by Z/7 = (X/7, Y/7), is a simple martingale with the zigzag property: if n ~> 1, then either Xn - X , - 1 = 0

or

Yn - Y/7-, = 0 .

(11)

This implies that if U is a biconcave function on B • B, then U (Zn) is expectation decreasing: E U ( Z n ) <~ E U ( Z n - 1 ) <~... <~ E U ( Z o ) .

(12)

This is a consequence of the zigzag property of Z and Jensen's inequality for conditional expectations. If n ~> 1, then Zn = (Xn, I n - 1 ) or Zn -- ( X n - l , Yn). For example, assume the first possibility holds for some n. Then

E[u(z.) i f . _ , ] - E[u(x , v._,) i f . _ , ] <~ U(E[X~ I f~-~], r~_~) - V(Z~_~) and taking expectations gives E U (Zn) <~ E U (Zn_ 1). PROOF OF (9). It suffices to prove the inequality for )~ = 1. Let A denote the set {(x, y) B • B: IIx II v IlY II/> 1 }, and define U and V on B • B by U ( x , y) = u(O, O) - u(x, y), V ( x , y) = u(0, 0)IA(x, y) -- IIx + YlI,

D.L. Burkholder

240

where 1A is the indicator function of A and u satisfies the assumptions of Theorem 2. Then U is a biconcave majorant of V. The biconcavity of U follows from the biconvexity of u, majorization on A follows from (6), and majorization on its complement follows from

u(x, y) ~< u(0, 0) + Ilx + yll.

(13)

Note that (13) is a consequence of (7) and

u(x, y) - u ( x , - x )

<~ lim ) - l [ u ( x , - x + ~(x + y)) - u ( x , - x ) ] )~----->o o

~< IIx + Nil,

(14)

where, for x + y # 0, the convexity of u(x, .) and (6) come into play. Assume that u not only satisfies (6) and (7), but also u(x, y) = u ( - x , - y ) . (To see that this is permissible, consider u(x, y) v u ( - x , - y ) . ) Then, by the biconcavity of U,

U(x, O) - [U(x, O) + U ( - x , 0)]/2 ~< U(O, O) --0

(15)

with a similar inequality for U (0, y). Now use V ~< U and (12) to obtain

u(0, 0)P(llg~ll ~ 1) = u(0, 0)P(llSn - Ynll ~ 2) U(0, 0)P(llSnll V IIYnll ~ 1)

= EW(Zn) + EIISn + Y~II ~< EU(Zn) + 211fn Ill <~ EU(Zo) + 211fnlll ~ 211fnlll.

(16)

Here Z0 = (X0, 0) or Z0 = (0, Y0) and so, for example, if the first possibility holds, then U (Z0) ~< 0 by (15). Inequality (16) yields (9) for ~ -- 1. To see this, consider the simple martingale F with nth difference l{n 1}. Then apply (16) to F and G: on the right IlFn Ill ~< II/n Ill ~< Ilflll and on the left {g* ~> 1} = {llGn II I> 1} for all large n since g is simple. This completes the proof of (9). O Before proving (10), let us consider any function V : B • B ~ R and the family L/of biconcave functions U : B x B --+ R such that U/> V on B x B. If b / i s nonempty, let U0 denote the least U in L/. The function U0 has a representation based upon the simple zigzag martingales with values in B x B. Let Z(x, y) be the set of all such martingales Z with Z0 = (x, y). The filtration is allowed to vary but the probability space can be fixed if it is nonatomic. Since Z is simple, its pointwise limit Zoo exists and is a simple function with values in B • B. LEMMA 3. If lg is nonempty and (x, y) 6 B x B, then

Uo(x, y ) = s u p { E V ( Z ~ )

Z ~ Z(x, y)}.

(17)

241

Martingales and singular integrals in Banach spaces

PROOF. Denote the right side of (17) by W ( x , y). Let U 6 L/and Z ~ Z ( x , y). Then, by the inequality V ~< U and the monotonicity property (12), E V ( Z e ~ ) <<.E U ( Z c ~ ) <<.E U ( Z o ) = U(x, y).

Therefore, by its definition, the function W is majorized by U. This implies that W ~< U0. To prove the reverse inequality, note that W ~ V: consider the constant martingale with Zn = (x, y). That W is biconcave can be seen by using a splicing argument similar to that used in Section 5 in [36]. The conclusion is that W ~ L/so U0 ~ W and equality holds. D REMARK 4. Let/3 (B) denote the best constant in (9) and u0 the greatest function on B • B satisfying (6) and (7). Assume that B is ~'-convex so that, by (8), u0(0, 0) > 0. Then, by (9), fl(B) ~< 2/u0(0, 0). To see that equality holds, let B = {(x, y) E B • B: IIx - Yll/> 2} and V ( x , y) -- otlB(x, y) - I I x -+- YII, U(x, y) -- s u p { E V ( Z ~ ) " Z ~ Z ( x , y)},

where ot = 2/fl(B) and Z(x, y) is as in Lemma 3. Then, by the lemma, U is a biconcave majorant of V so U (x, - x ) ~> V (x, - x ) ~> 0. But, by the definition of or, E V (Zoo) ~< 0 for all Z 6 Z(0, 0). This implies that U(0, 0) ~< 0 so U(0, 0) = 0. Let u(x, y) = ~ - U(x, y). If IIx - Yll >~ 2, then u(x, y) <~c~ - V ( x , y) = Iix + yll. This implies, by an argument similar to that used to prove (14), that the inequality u(x, y) <~ IIx + Yll also holds if Ilxll v Ilyll ~> 1. Since U is biconcave, u is biconvex 9 Therefore, u E/2. Furthermore, u ( x , - x ) = - U ( x , - x ) ~< ot = u(0, 0) ~< u0(0, 0). So u satisfies (7) and fl(B) = 2/or ~> 2/u0(0, 0). This completes the proof that fi(B) = 2/u0(0, 0). REMARK 5. If ~" is any biconvex function satisfying (5) and g is a +l-transform of a B-valued martingale f such that P(g* 7> 1 ) = 1, then 211fllJ ~ C(0, 0)

(18)

and the inequality is sharp if ~" = ~'e. The main step in proving (18) is the application of the biconvex version of (12) and (15) to U = ~'. That (18) is sharp follows from a more general version of Lemma 3 in which both the starting point and the ending point of the zigzag martingale Z associated with f and g are taken into account; see Lemma 5.1 in [36]. LEMMA 6. Let 1 < p < cx~ and fl c [1, c~). Define V :B • B --+ R here by

V(x, y)=

x-y 2

p

x+y -tip

2

p 9

(19)

Then tip(B) ~
D.L. Burkholder

242

PROOF. Suppose that U ~ is such a biconcave function. Then U, defined by U(x, y ) infzr176 p, is also a biconcave majorant of V and satisfies U(otx, oty)= Iot[PU(x, y) for all ot 6 R. Therefore, U(0, 0) - 0 and, by the biconcavity, U(x, O) -U ( - x , 0) ~< 0. Similarly, U (0, y) ~< 0. Consequently, if g is a +l-transform of a simple martingale f and Z is the corresponding zigzag martingale, then Ilgn IlPp- flPllfn lipp - E V ( Z . ) ~ EU(Zn) <~EU(Zo) <~O. This implies the "if" part of Lemma 6. The "only if" part follows from Lemma 3.

D

REMARK 7. Assume as above that f is a B-valued martingale with difference sequence d relative to a filtration (Un)n>.O. Then f is a dyadic martingale if each a-algebra .Tn is generated by a finite partition/7n of s = {s }, and if A ~ Fin, then either A E/7n+l or A is the union of two sets in/7n+l of equal probability measure. So dn+l can take at most two values on A and, by (1), the sum of these two values is 0. This implies that Ildn+, II is constant on A and is therefore .Tn-measurable. By (1), the sequence of partial sums of a Haar series is a dyadic martingale. Lemma 6 holds if tip(B) is replaced by flo (B), the best constant in the inequality (3) under the condition that only dyadic martingales are considered. To see this, note that although the dyadic version of Lemma 3 gives only a midpoint biconcave majorant U of V, such a function U is locally bounded from below in L e m m a 6, since V is, and is therefore biconcave. Accordingly, t3o (B) ~
IIg*llp <~

(p 4- 1)2 "~llfllp, u (0, O) p -- 1 36

1 0, fl > 26 + 1, c~ - 86/[u(0, 0)(fl - 26 - 1)], and )v > 0, then

P(g* > fl~., f* <~~.) <~otP(g* > ~.).

(21)

If IIf011 > 6z, then the left side of (21) vanishes and the inequality holds. So assume that IIf011 ~< a~.. Define functions/z, v, a " s -+ [0, oo] by /z(o)) - inf{n" IIg,(og)ll >)v}, v(co) - inf{n" IIg,(co)ll >/~9~}, a(w) - inf{n" II/~(oo)ll > a)~ or IId~+l(O))ll > 26;.}.

Martingales and singular integrals in Banach spaces

243

1)n be the indicator function of the set {/z < n ~< v A a }, the empty set if n -- 0, and a set in f n - 1 if n >~ 1 (recall that Ildn+l I[ is .Tn-measurable). Therefore, Fn = Y ~ = 0 vkdk and Gn -- ~ = 0 ek vkdk define martingales F and G such that, by (9),

Let

P ( g * > fl ~. , f * <~ (3),.) = P ( # <~ v < cxz , cr = ~ ) <~ P ( G * > ill. - 261. - Z) 2

IIFII~

u(O, O)

fl)~ - 26)~ - )~

~<

<~ o~P ( g * > )0. So (21) holds for all Z > 0 and yields the inequality

EcI,(g*) <~ c E ~ ( f * ) , where r is any nondecreasing continuous function on [0, cr with r = 0 such that 4~(2)~) ~< V4~()0 for all ~. > 0; see L e m m a 7.1 in Burkholder [29]. Here the choice of fl and 6 depends on V and u(0, 0), so the choice of c needs to depend only on these two parameters. Application of this inequality to the map ~. w, )~P leads to (20); see Section 8 in [36] for the details. 71 PROOF OF THEOREM 1. We have already seen that T h e o r e m 2 implies that a ~'-convex Banach space is U M D for all p 9 (1, cx~). It remains to show that if B is U M D for at least one p 9 (1, cr then B is ~'-convex. To see this, fix p 9 (1, cr with tip(B) finite and let V be as in (19) with # = tip(B). Define ~" on B x B by

( ( x , y)

--

2

1 +tip

[1-

U(x

'

y)]

'

where U is a biconcave majorant of V with U (0, 0) -- 0. Such a U exists by L e m m a 6 and its proof. Then ~" is biconvex on B x B and satisfies ~"(0, 0) =

2

1-k-tiP

., 0.

(22)

Furthermore, (5) is satisfied as can be seen as follows. Let x and y satisfy Ilxll = Ilyll = 1 and set t = IIx + yll/2. Then IIx - yll/2 ~> Ilxll - t = 1 - t and 2

( ( x , y) <~ l + tiP 2 ~< 1 + # p [ 1 - ( l - t )

p + f l p t p]

~< 2t = [Ix + Y II, where the last inequality follows from 0 ~< t ~< 1, the inequality (4), and 0 ~< q~(t)"= t -- 1 + (1 -- t) p -Jr (p* -- 1)P(t -- tP).

D.L. Burkholder

244

If p = 2, then 45(t) = 0. To see that 4~ is also nonnegative on [0, 1] in the case p 7~ 2, note that q~ (0) = q~ (1) = 0 and 45'(0+) > 0, 4~' ( 1 - ) < 0. So there is a to E (0, 1) such that 4~'(to) = 0 and to is unique: if p < 2, then 45' is strictly convex on (0, 1); if p > 2, strictly concave. Therefore, 45 is increasing on [0, to] and decreasing on [to, 1] thus proving (5) and completing the proof that a UMD space is (-convex. 71

REMARK 8.

From (22) it follows that (B(0, 0) /> 2/[1 + tiP(B)]. A different method (see Section 6 in [36]) gives (a(0, 0) ~> 1~tip(B). I f B is a Hilbert space and p -- 2, then equality holds in both.

3. Singular integral operators and UMD Singular integral operators arise in many areas of mathematics, for example, in complex analysis, harmonic analysis, partial differential equations, and quasiconformal mappings. See Stein [134,136] and the many references therein for some of the history and much of the theory. Here we study an old problem of interest in many applications: How do such operators behave in a Banach space setting? Inequality (3) gives some information about the possible size of a + 1-transform g of a B-valued martingale f in comparison with the size of f . For example, if tip(B) is infinite, then Ilgllp can be exceedingly large even if ]lf l i p is small. Similar results hold for conjugates of B-valued harmonic functions. Let u and v be B-valued functions harmonic in the open unit disk of the complex plane with v(0) -- 0. Suppose that the Cauchy-Riemann equations, Ux = Vy and Uy = - V x , are satisfied. Let

Ilullpp =

sup

O
f0 7rIlu(re i~

II

dO

and define Otp(B) to be the least ot 6 [1, ~ ] such that Ilvllp ~ allullp

(23)

for all u and v as above. If 1 < p < cx~, then Otp (R) is finite. This is a classical result of M. Riesz [ 128], who used it to prove that the norm of the Hilbert transform on LP (R) is Otp (R) for all p E (1, cx~). If 1 ~e

f(x -y)

dy

Y

provided this limit exists for almost all x E R, in which case

IIHfllp ~ ~ p ( B ) l l f l l p

if 1 < p < c~.

(24)

The Banach space B is an H T space if Otp(B) is finite for some p E (1, cx~); equivalently, for all p E (1, ~ ) , an equivalence that will be proved below. If B is such a space then the

Martingales and singular integrals in Banach spaces

245

norm of H on LP(R) is C~p(B), which has a proof with the same pattern as the one for the scalar case; see p. 256 of volume II of Zygmund [ 141 ] or C16ment and de Pagter [51 ]. If B is an HT space, then, as in the scalar-valued case, the norm inequality for the Hilbert transform together with the method of rotations introduced by Calder6n and Zygmund [46] can be used to establish norm inequalities for a large class of singular integral operators on L~ (Rn). For example, let K(x)=

S2(x) ~ , Ixl ~

x E R n \ {O},

where I2 is an odd scalar-valued function with Y2(otx) = S2 (x) for all ot > 0, and the restriction of S-2 to the unit sphere is integrable. Let IIS2 II1 denote the integral of IS2l with respect to surface measure on the unit sphere. Then the limit f T f ( x ) -- l i m / f(x - y)K(y)dy e$O d[y I>e exists almost everywhere if f 6 L~ (R n) for some p satisfying 1 ~< p < ec. If 1 < p < ec, then (25)

IITfllz, ~ loep(B)ll~lll Ilfllp. THEOREM 9. A Banach space B is an H T space if and only if it is UMD.

That a UMD space is HT is due to Burkholder and McConnell; see Burkholder [33] where the proof, designed to be as accessible as possible, is based on the Rademacher functions rather than on the It6 stochastic calculus, the most natural setting. The question is also raised of whether an HT space is UMD. In response, Bourgain [18] proved that, indeed, HT implies UMD. PROOF THAT A U M D

SPACEIS HT.

C~p(B) ~< flZ(B),

This follows from the inequality

1 < p < ec.

(26)

It is enough to show that

f0 rr ]v( ei0 ) I1" dO ~< fi2p (B) f0 2rr ]lu(e i~

II~ dO

(27)

in the special case that the function u is harmonic and v is its conjugate on the entire complex plane with the ranges of both functions included in some finite-dimensional subspace of B. As usual, v(0) = 0. Let Z = (Zt)t>>o be a complex Brownian motion starting at zero with X and Y the real and imaginary parts of Z. The inequality (27) is equivalent to

D.L. Burkholder

246

where r (co) -- inf{t ) 0" IZt (co)l = 1} since Zr is uniformly distributed on the unit circle 9 Because u is harmonic, It6's formula gives

u ( Z r ) -- u ( 0 ) +

f0 Tux(Zt) dXt + uy(Zt) dYt

with a similar formula for v(Zr). The inequality (28) can now be proved with the use of the decoupling method. Let Z ~be another complex Brownian motion starting at zero. Assume as we can that Z and Z ~ are defined on the same probability space and that Z and Z f are independent. Then

II (z )ll

fo r vx(Zt) dXt -+-vy(Zt) dYt

=

/~p(B)

p

fo +v~(Zt)dX~ + vy(Zt)dY t

(29) p

Inequality (29) is the analogue of the decoupling lemma in Burkholder [33] and follows directly from the discussion in Garling [76]. A different proof is given in Burkholder [36]. By the Cauchy-Riemann equations and the fact that (r, X, Y, Y~, - X ~) and (r, X, Y, X ~, Y~) have the same distribution, the norm on the right of (29) is equal to

~0z Ux(Z,) dX~ + .y(Z,) dY/

fo r ux(Zt) dY t + uy(Zt) d(-X~) p

p

Here the norm on the right is less than or equal to

/7

tip(B) u(O) +

f0

ux(Zt) dXt + uy(Zt) dYt

-

and completes the proof of (26). Hence, a UMD space is HT. To prove Bourgain's result that an HT space is UMD, we shall use some of the properties of the periodic Hilbert transform H. Let f " R --+ B be defined by

f (0) -- E cveiV~ 1)

where the spectrum of f , the set {v 6 Z" cv ~ 0}, is finite 9Here, if B is a complex Banach space, then cv 6 B, but if B is a real Banach space, then cv = 89 (a~ - ibm) where av = a - v 6 B and by - - b - v 6 B. So for real Banach spaces, f can also be described as the trigonometric polynomial given by 1 ~ao + E v~>l

(av cos vO + by sin vO).

Martingales and singular integrals in Banach spaces

247

The periodic Hilbert transform is defined for such f by

H f (O) -- - i Z ( s g n v) cve ivO v#o Note that u(re i~ -- Y~v cvrlvleiv~ and v(re i~ = - i y~'v#0(sgn v) cvrlvle iv~ define Bvalued harmonic functions u and v on C that satisfy the C a u c h y - R i e m a n n equations, v(0) - 0 , u(e i0) - f(O), and v(e i0) - Hf(O). Just as for the classical case in which B - R, the integral in the definition of Ilu IIp is nondecreasing as a function of r. Therefore, by the continuity of u and v on the closed unit disk and the definition of Otp (B),

(30)

IIH fllp <~oep(B)llfllF

for all f as above, where 1[f [[~ -- f [ f (0)[P dO with the integration over any interval of length 2st. Inequality (30) is of course meaningful if and only if Otp (B) is finite, that is, if and only if B is an HT space for p. Note also t h a t / 4 f is an odd function if f is even and t h a t / ~ 2 f = _ f if the coefficient co = 0. The following observation is contained in the proof of L e m m a 1 in Bourgain [ 18]" LEMMA 10. Let f be as above. Then there is a positive integer M such that H ( f g ) --

f H g for every function g with scalar coefficients and finite spectrum defined by g(O) -- Z

~

##0 PROOF. Let M > N where N ~> ]v[ for all v in the spectrum of f . Then

H(fg)(O)- -i

~

~ (sgn(v + M/z))cv~ue i(v+Mu)~

[v[<~ N lZ=/=O

and sgn(v + M/z) -- sgnM/z. Therefore, H ( f g ) -- f H g . PROOF THAT AN H T SPACE IS U M D . This is due to Bourgain [18]. The proof consists of showing that

/~p(B) ~< %2 (B),

1 ~o be the Rademacher sequence: rn takes the values § 1 and - 1 alternately on the subintervals [0, 1/2"+1), [ 1/2 n+l , 2 / 2 n+l) . . . . of [0, 1). Consider the dyadic martingales (Fn)n~o and (G~)n~>o on [0, 1) such that F0 = Go -- 0, and for n ~> 1, /7

F, = ~ ~ok(ro. . . . . k--1

rk-1)rk,

Gn - Z k=l

ek~Ok(ro . . . . . rk-1 )rk,

D.L. Burkholder

248

where qg~" {17 - 1 }~ --+ B and ek 6 { 1, - 1 } for all k ~> 1. Then, to prove (31), it will suffice to show that (32)

2 (B) ll Fn llp [IGnllp ~ Olp

The restriction to martingales satisfying F0 = Go - 0 has no effect on tip(B); see, for example, Section 2 of Burkholder [34]. The restriction to dyadic martingales of the above form also has no effect as can be seen from the proof of tip(B) - t o (B) that is given in R e m a r k 7. Let 7r be the even function on R with !k (0) - 1 if cos 0 ~> 0 and ~p(0) - - 1 if cos 0 < 0. Then

F F• F F 7/"

7r

(pk(~(to) . . . . . ~ ( t k - 1 ) ) ~ ( t k )

k=l

dto

dtn

2rr

2Jr

n

iionll

.

.

.

.

7/"

7/"

ek~ok(~(to) . . . . . ~ ( t k - 1 ) ) ~ ( t k )

k=l

dto

dtn

2Jr

2Jr

since (rk)0~
F Y F F

I0 . . . .

jo . . . .

7/"

d t o . . , dtn,

~-~ ~k(tO, . . ., tk-1)~(tk)

7/"

k=l p

~-~ ek ~k (to . . . . . tk_ j ) qJ (tk )

7/"

dto. 99dtn,

k=l

where ~k(t) -- ~--~vcveiVt i n which t -- (to . . . . . tk-1) and v -- (vo . . . . . 1 3 k - l ) , Iffk(S) -ur c~ue ~us, both ~k and tPk have finite spectrum, ~k is even with real coefficients otu -c~_#, and ~k(eoto . . . . . ektk) -- ~k(to . . . . . tk) for all choices of the numbers ej ~ { 1 , - 1 } . Then the inequality J ~< Up2 (B)1 follows from j o ~
(33)

with the use of standard methods from classical Fourier analysis and the fact that 7r is an even function. If fk(O) -- clgk(to + MOO,..., tk-1 + Mk-lO) and gk(O) -- qJk(tk 4- MkO), then, by L e m m a 10, there exist integers M 0 , . . . , Mn such that H ( f k g k ) -- f k H g k . So, by the linearity of H ( n n and the inequality (30), H ~ k = l fkgk) = ~-~k=l f k H g k and

F ~A(O)~g~(O) 7/"

k-1

p

dO ~ ~ (a)

A(O)gk(O) dO. yr k = l

249

Martingales and singular integrals in Banach spaces

Using periodicity, integrate each side of this inequality with respect to tk over the interval [-7r, re), then with respect to tk-l, and so on. Finally, integrate with respect to 0. After the division of each side by 27r, the result is

F F Jr

Jr

cI)k (to . . . . . tk_ l ) H qJ (tk )

(34)

d t o . . , dtn <. u p (B) I ~

k=l

With the replacement of qs by H qJ, and 4~k by ekq~k, in both sides of this inequality, followed by the use of/-J21~ - - - - t p , the inequality becomes p

j o ~< ot~(B)

f F / ; s:l ...

--Jr

= Otp p (B)

Jr

~pP(B)

.

Jr

2p

Jr

.

dt0. 99dtn

n

9

Jr

-

ek q~k (to . . . . . tk- l) H q/(tk)

k=l P

cI)k(eoto . . . . . e k - l t k - 1 ) H O ( e k t k )

dt0.. "dtn

k=l

.

P

@k(~o . . . . . sk-1 ) HqJ (sk)

ds0. 99dsn

Jr k = l

0

~< Otp (B)I .

Equality in the second line of this array holds because q~k is an even function of each tj and the periodic Hilbert transform of an even function is an odd function, so ek H ( t k ) -H(ektk). Equality in the third line follows from the substitution sj -- e j t j . The inequality in the last line is an application of (34). This completes the proof that an HT space is a UMD space 9 D Asmar, Kelly, and Montgomery-Smith [6] have a proof resting partly on the work of Berkson, Gillespie, and Muhly [ 13], who show that HT spaces are well behaved for certain abstract conjugate function inequalities, and their own work, especially their observation that a particular martingale transform is the composition of two conjugate function operators. Here is one way to see that if Otp(B) is finite for some p 6 (1, ec), then it is finite for all p 6 (1, oc). As Bourgain has shown, the first condition implies that tip(B) is finite for the same p. Therefore, by Theorem 1, tip(B) is finite for all p E (1, ec). So, by (26), C~p(B) is finite for all p 6 (1, ec). It is an open question (the last two sentences in Burkholder [34] were prematurely optimistic) whether or not the inequality (26) can be sharpened to Otp(B) ~< tip(B), but this inequality is true for Hilbert spaces H as can be seen from (44) and (54) below. This can also be seen by comparing p* - 1 and Otp (H) = cot(rr/2p*), the Pichorides-Cole constant. On the other hand, if the exponent 2 is removed in (31), then the resulting statement is not true for p --/=2 in the case B = H as these sharp constants imply. Recent work on singular integrals of functions with values in a UMD space includes the important work of Figiel [75]. He introduces a martingale approach that is partly based on ideas from his earlier paper [74], to show that a large class of singular integral operators

D.L. Burkholder

250

are bounded on L~ (Rn), 1 < p < cx~, where B is UMD. Among other things, his work throws new light on the celebrated T1 theorem of David and Journ6 [54] and their work on generalized Calder6n-Zygmund operators in the scalar-valued case. McConnell [ 110] proved that if a Banach space B is UMD, then certain multiplier operators satisfying conditions slightly more restrictive than those of Mihlin-H6rmander are bounded on L~ (Rn), 1 < p < c~. Later, Zimmerman [140] showed this for the full Mihlin-H6rmander class. Since the Hilbert transform is such an operator, McConnell's theorem is an extension of part of Theorem 9: a UMD space is an HT space. The converse of McConnell's theorem is also true by the other half of Theorem 9: an HT space is a UMD space. For the special case of UMD Banach lattices, see Bourgain [19]. Related work includes that of Rubio de Francia [ 129,130], Rubio de Francia and Torrea [ 131 ], and Fernandez [73]. In the study of evolution equations and some other differential equations in an abstract setting, the Hilbert transform can play an important role. Coulhon and Lamberton [53] show in a simple example the relevance of the class UMD in the study of L p regularity. Dore and Venni [60] study a more general problem with an emphasis on one of its key components: what conditions on a Banach space and two closed operators A and B assure that the sum A 4- B is closed? Again, UMD comes into play via the Hilbert transform. Further work includes Dore and Venni [61,62], Prtiss and Sohr [127], Giga and Sohr [83], Giga, Giga, and Sohr [84], and Monniaux and Prtiss [113]. Also, see the survey by Dore [59], and the books by Prtiss [126] and Amann [2]. Berkson, Gillespie, and Muhly [12] show that a large part of classical spectral theory carries over to the UMD setting: the spectral theorem is proved for power-bounded invertible operators on UMD spaces, the Stone theorem for strongly continuous one-parameter groups of operators is also carried over, and more. Given its connection with singular integral operators, it is not surprising that UMD also enters into the study of various kinds of Hardy spaces H~: see, for example, Bourgain [20], Blasco [14,15], Hensgen [88,89], and Burkholder [36]. For some other properties of UMD and related spaces, see Pietsch and Wenzel [ 122] and the references given there.

4. The unconditional constant of the Haar basis One consequence of the main result of this section is the inequality (4) with equality holding for B = R. The proof leads to a special biconcave function on R • R that, in turn, leads in the next section to the function on H x H that we shall use to obtain, among other things, the complex unconditional constant of the Haar system. Here we consider the real case. Let h = (hn)n>~O be the sequence of Haar functions on [0, 1). By the work of Paley [ 116] and Marcinkiewicz [108], h is an unconditional basis of L p (0, 1) for all p E (1, cx~). The unconditional constant, here denoted by tip(h), satisfies

tip(d) <~ tip(h) <~ tip(e),

(35)

where tip(d) denotes the unconditional constant of a real-valued martingale difference sequence d and tip(e) denotes the unconditional constant of an unconditional basis e of

Martingales and singular integrals in Banach spaces

251

LP(0, 1). The inequality on the left is due to Maurey [109]. The inequality tip(d) <~tip(R) follows from the definition of tip(R), so the left side of (35) is a consequence of his result, mentioned in Remark 7, for the case B = R:

flp(h) = flp(R).

(36)

The inequality on the right of (35) is due to Olevskii [114,115]; see also Lindenstrauss and Petczyfiski [ 105]. So h is both an extremal martingale difference sequence and an extremal unconditional basis in L P (0, 1). These extremal properties suggest the question: What is the precise value of the unconditional constant of the Haar basis? The answer is given in the following theorem [32,35]. As above, p* denotes the maximum of p and its conjugate index q. THEOREM 1 1. The unconditional constant of the Haar basis of L p (0, 1) is

tip(h) = p* - 1 for all p ~ (1, oc).

(37)

Keeping (36) in mind, we search for the value of tip(R) and do this here without assuming that we already know the answer. The search is fairly simple thanks to the following reduction in the dimension of the problem. Define v on R by v(x) = V(x, 1) where

V(x, y) =

x+y 2

p

-tip

x-y 2

p 9

(38)

Note that v(x) = IxlPv(1/x) for all x --/=O. LEMMA 12. Suppose that fl ~ [1, oo). Then tip(R) <~ fl if and only if there is a concave function u majorizing v on R such that

u(x) = IxlPu(1/x)

for all x r O.

(39)

PROOF. By Lemma 6 and its proof, the condition tip(R) <~ fl implies the existence of a biconcave majorant U of V satisfying U(~x, ~y) = I~IPU(x, y) for all ot E R. Since the mapping (x, y) ~ U(x, y) A U(y, x) has these properties, we can assume that U is also symmetric. Therefore, u = U(., 1) is a concave majorant of v satisfying (39). The latter condition can be checked as follows. If x r 0, then u(x) = U(x, 1) = U ( 1 , x ) = IxlPU(1/x, 1) = IxlPu(1/x). To prove the "if" part, suppose there is a concave majorant u of v satisfying (39). Since u is locally bounded from below, the concavity implies that u is continuous. Also, u (0) ~< 0. For if u(0) were strictly positive, then, by (39) and continuity, both u(x) and u ( - x ) would be larger than u (0) for all large x, which would contradict the concavity of u. Define U on R 2 by U(x, O) = IxlPu(O) with U(x, y) = ly]Pu(x/y) if y r 0. It is easy to check that U is symmetric and majorizes V. By the concavity of u and the inequality u (0) ~< 0, the function U(., y) is concave so, by symmetry, U is biconcave. L e m m a 6 remains true if x + y and x - y are interchanged. Therefore, tip (R) <~ ft. D

252

D.L. B u r k h o l d e r

PROOF OF THEOREM 1 1. Let u be a concave majorant of v satisfying (39). Set x0 = (/3 - 1)/(/3 + 1). Then 0 ~< x0 < 1 and u(xo) >~ v(xo) = 0. Similarly, u(1) >~ v(1) = 1. Let L be the limit of ( u ( 1 ) - u ( x ) ) / ( 1 - x) as x 1' 1 and R the limit of (u(x) - u ( 1 ) ) / ( x - 1) as x $ 1. Because u is concave, both of these limits exist. If 0 < x < 1, then, by (39), u ( 1 ) -- u ( x )

=

u ( 1 ) -- x P u ( 1 / x )

1 -x

1 -- x p

= ~ u ( 1 ) 1 -x

1 -x

- x p-1

u(1/x)

(I/x)-

-- u ( 1 )

1

Letting x 1" 1, we see that L = p u ( 1 ) - R. Using the nonnegativity of u(xo) and the concavity of u, we obtain

p u ( 1 ) = L + R ~< 2L ~< 2

u(1) - u ( x o )

<~ (/~ + 1)u(1)

(40)

1 -- X0

and the strict positivity of u (1) implies that/3 ~> p - 1. An equally simple argument shows that/3 ~> q - 1. So the existence of a concave majorant u of v satisfying (39) implies that fl ~> p* - 1. Therefore, tip(R)/> p* - 1 by L e m m a 12. The argument giving q - 1 as a lower bound can of course be avoided by using duality: tip (R) = flq ( R ) . A natural question arises: Is this lower bound p* - 1 also an upper bound? Equivalently, does there exist a concave majorant u of v satisfying (39) if fl = p* - 1 in the definition of v ? For example, if p = 2, then the identity function has the desired properties. If p > 2, t h e n / 3 -- p - 1 and equality must hold throughout (40) for the desired function u. So u(xo) = 0 and L = R -- u ( 1 ) / ( 1 - x0). This implies that u must be of the form u(x) = ax - b on the interval [x0, 1]. Therefore, by (39), u(x) = a x p - 1 - bx p on the interval [1, l/x0] and L = R implies that b = a(1 - 2 / p ) . Both u and v vanish at x0 and in order for u to be concave and majorize v, it is necessary that u' ( x 0 + ) = v' (x0),

which yields a l p 2 ( 1 - 1/p)p-1. Thus, u is uniquely determined on the interval [x0, l/x0]. Because v is concave on the interval ( - ~ , x 0 ] and also on the interval [ l / x 0 , c~), the desired u can be defined on the complement of Ix0, l/x0] by u(x) = v(x) or in any other way such that the result is a concave majorant of v on R satisfying (39). Therefore, tip(R) ~< p* - 1 for all p ~> 2. The case 1 < p < 2 has a similar proof, as well as being an immediate consequence of duality. So /~p(R) = p* - 1

(41)

and this completes the proof of T h e o r e m 11. REMARK 13. If u is defined by u = U (., 1) where, for all x, y E R,

U(x, y) = ap {

x+y 2

-(p*-

1)

x-y 2

I{

x+y 2

x-y +

2

p-1

Martingales and singular integrals in Banach spaces

253

with Otp = p(1 - 1/p*) p - | , then u is also a concave majorant of v on R satisfying (39) for all p 6 (1, ~ ) . If p > 2, for example, then this function satisfies u ( x ) = ax - b for x E [x0, 1] just as uniqueness on this interval demands.

5. Differential subordination 5.1. A sharp martingale inequality As in Section 2, let B be a real or complex Banach space, f a B-valued martingale, and d its difference sequence" fn - Y~=0 dk for all nonnegative integers n. In this section, we relax the condition on the B-valued martingale g. As before, f and g are martingales relative to a common filtration (U,)n~>0 but here the only additional condition on the difference sequence e of g is that lien (co)II ~< IId~(~)I[

for all co E S2.

(42)

The martingale g is differentially subordinate to f . The condition of Section 2 that en = sndn, where sn E {1,--1}, is a special case in which tip(B) plays a key role. The corresponding quantity in the differentially subordinate case will be denoted by yp (B). So if (42)holds and 1 ?' II f IIp. The following theorem [30,38] compares the behavior of tip(B) and yp(B). THEOREM 14. Let 1 < p < oc. (i) The extended real number yp(B) is finite if and only if B is isomorphic to a Hilbert space. (ii) I f H is a Hilbert space, then /3p (H) = yp (H) = p* - 1.

(44)

For example, if r E (1,2) U (2, ec) and p E (1, ec), then ~p(~r) is finite but yp(gr) is infinite. By (i), this is of course true for every UMD Banach space that is not isomorphic to a Hilbert space. PROOF OF (i). The finiteness of yp(B) for some p E (1, cx~) implies that yp(B) is finite for all p E (1, ec); see Section 5 in [30]. So in the proof of (i) we can assume that p = 2. The "if" part then follows from the easy part of (ii): y2(H) = 1. For the "only if" part, let (rn)n~o be the sequence of Rademacher functions on [0, 1), (an)n~>0 a sequence in B, and b E B with Ilbll - 1. Define martingale difference sequences in L~(0, 1) by dn - anrn and

D.L. Burkholder

254

en = [[an[[rnb. Note that the associated martingales f and g are differentially subordinate to each other. Therefore, the finiteness of yz(B) implies that for all n,

•

/7

Ila~ Ilrkb

akrk

k=O

2

k=O

~ l l a ~ Ilrk 2

=

k=0

[[akl] 2 k=0

So, by a result of Kwapiefi [ 100], B is isomorphic to a Hilbert space.

El

PROOF OF (ii). By (41) and the obvious inequality tip(R) <~ tip(H) <~ yp(H), it suffices to show that if [If 11p is finite, then

Ilgnllp ~ (p* - 1)llfnllp

foralln/>0,

(45)

where f and g are H-valued martingales relative to a common filtration and g is differentially subordinate to f . The assumption that ]lf I]p is finite implies that []dn IIp is finite for all n, so by (42), both ]lfn lip and [Ignl[p are finite for all n. It is enough to prove that, under these assumptions,

E V ( f n , gn) <~O, where V" H x H --+ R is defined by V(x, y) = [lyl[p - (p* - 1)Pllxl[ p. The proof consists of showing that there is a majorant U of V satisfying both

E U ( f n , gn) < . ' " <<.EU(fo, go)

(46)

and EU(fo, go) <~O. One such function is U ' H x H --+ R defined by

U(x, y ) - Otp([[y[I - (p* -- 1)llxl[)(llx[[-I-Ilyl[) p-1 ,

(47)

where Otp = p(1 - 1/p*) p-1 . Note that U(x, y) is the value of the function U of Remark 13 at the point (llYll + llxi], llYll - Ilxll) in R 2 just as V(x, y) is the value of the function (38), with 13 replaced by p* - 1, at the same point. Therefore, U majorizes V, which can also be proved directly (see [38, p. 77]). The inequality EU(fo, go) <, 0 follows from U(fo, go) = U(do, e0) ~< Otp(2- p*)[ldoll p <~O. We now turn to the proof of (46), the final step of the proof. In the following, (., .) denotes the real part of the inner product of H. Fix x, y, h, k 6 H with llkll ~< ]lhll and suppose that llx + htll and ][Y + ktll are strictly positive for all t E R. Let x' = x/llxll and y ' = Y/llYll. Then the function U defined in (47) satisfies

U(x + h, y + k) <<.U(x, y) + (~o(x, y), h) + (7t(x, y), k), where

~o(x, y)

--

OtpXt[(p

-

p * ) l l y l l - p(p* - 1)llxll](llxll + Ilyll) p-2,

7t(x, y) - ~py'[pllyll + (p + p* - pp*)llx[I](l[xll + Ilyl[) p-2.

(48)

Martingales and singular integrals in Banach spaces

255

To prove (48), define M : R --+ R by M(t) = U(x + ht, y + kt) and notice that M is infinitely differentiable on R and that the inequality (48) is the same as M (1) ~< M (0) + M' (0).

(49)

This inequality follows from the concavity of M on R, the proof of which can be reduced by translation to showing that M I' (0) ~< 0. If p >~ 2, then M"(O) = - O t p ( A + B + C),

(50)

where A -- p ( p B -- p(p

-

C -

-

p(p

1)(llhll 2 -Ilkll2)(llxll + IlYll)p-2, 2 ) [ l l k l l 2 - (y',k)2]llYll -~ (llxll + Ilyl]) p-l, 1)(p - 2)[(x', h ) + ( y ' , k)]211xll(llxll + Ilyll) p-3

so M"(0) ~< 0: A is nonnegative since Ilkll ~ Ilhll, B is nonnegative by the CauchySchwarz inequality, and C is also clearly nonnegative. If 1 < p ~< 2, a similar expression for ( p - 1)M"(0) is obtained from (50) by interchanging x and y, h and k, and then multiplying the right-side by - 1 . This can be seen from (p-

1)U(x, y ) - -oep(llx I I - (p - 1)IlYlI)(IlYll-+-IIx II) p-~,

which rests on the identity (p - 1)(p* - 1) = 1, valid for 1 0. We can and do assume that H0 is a proper subspace (otherwise enlarge H slightly). Let a 6 H be in the orthogonal complement of H0 with 0 < Ila II < 1 and write Fn = a + fn and Gn = a + gn. Then, by (42) and (48), U(Fn+l, Gn+l) ~ U(Fn, Gn) + (9(Fn, Gn),dn+l) + (1/~(Fn, Gn),en+l). Each of these four terms is integrable and, by (1), the last two have zero expectation. Therefore, EU(Fn+I, Gn+l) <<,EU(Fn, Gn). Now let a --+ 0. By the continuity of U and the dominated convergence theorem, this yields the corresponding inequality for f and g and completes the proof of Theorem 14. D

5.2. Monotone bases and contractive projections Let 1 < p < oo. The Haar system is one example of an unconditional monotone basis of LP(0, 1). Dor and Odell [58] and, independently, Petczyfiski and Rosenthal [119] proved that every monotone basis of L p (0, 1) is unconditional. Their proofs are based on And6 [3], where contractive projections are shown to be isometrically equivalent to conditional

256

D.L. Burkholder

expectations, and Burkholder [28], where it is proved that tip(R), hence also/}p(C), is finite. Using instead the precise value of these constants obtained in Theorems 11 and 14 above, we see that the unconditional constant tip(e) of any monotone basis e of real or complex L p (0, 1) is given by tip(e) -- p* -- 1. Now suppose that 0 = P0, P1, P2 . . . . is a nondecreasing sequence of contractive projections in LP of any positive measure space. Thus, Pk :LP --+ LP is linear with a norm not exceeding 1 and Pj Pk = Pk Pj = Pj if 0 ~< j ~< k. If Ck 6 C and ]ck] ~< 1 for all k ~> 1, and f ~ LP, then O0

Z Ck(Pk -- P k - 1 ) f k=l

( P * - 1)llfllp.

This sharpens the contractive projection inequality of Dor and Odell [58] and throws further light on monotone Schauder decompositions. See Burkholder [34] for additional discussion. Doust [63] has used this sharp inequality in his study of spectral representation of operators on LP.

5.3. The complex unconditional constant of the Haar basis Let 1 < p < oo. Petczyfiski [118] conjectured that the complex unconditional constant of the Haar basis of L~[0, 1) is the same as the unconditional constant for the real case, equivalently, tip(h, C) = tip(h) where tip(h) is defined as in Section 4 and tip(h, C) is defined to be the least fl such that if n is a nonnegative integer, co . . . . . Cn are complex numbers, and 00 . . . . . On are real numbers, then

k=0

~Ckhk k=O

p

p

By Theorems 11 and 14, p* - 1 - tip(h) ~ tip(h, C) ~ p* - 1. Therefore, the complex unconditional constant of the Haar basis is the same as the unconditional constant for the real case.

5.4. Inequalities for the martingale square function The square function S ( f ) of a B-valued martingale f is defined by

o
)1/2

Z ]ldk 112 k=0

where d is the difference sequence of f . A simple application of Theorem 14 yields that if p 6 (1, oo) and f is an H-valued martingale, then

lls))

iilllp llp.

(51)

Martingales and singular integrals in Banach spaces

257

To see this, let K denote/72, the Hilbert space of sequences x - - ( x j ) j ~ / o with X j E I-I and IIx IlK -- (Z~--0 Ilxk 112) 1/2 finite. Let F and G be the K-valued martingales defined by

Fn=(fn,O,O .... )

and

G n - - ( d o . . . . . d,,O,O . . . . ).

Their respective difference sequences D and E satisfy lID, IlK - l I E , IlK. In addition,

IIF~ IlK - I I f ~ IIn and IIGn IlK -- (Y~'~=0 Ildk II2) i/2. So, by Theorem 14, IS(f ) I p-

IIGllp ~< (p* - 1)llFllp-

(p* - 1)llfllp.

This gives the left side of (51) and IIFIIp ~< (P* - 1)llGIIp gives the right side. Apart from the constants, special cases of (51) for real-valued martingales go back to the 1920s and 1930s with the work of Khintchine, Littlewood, Paley, Marcinkiewicz, and Zyground on Rademacher series, Haar series, and sums of independent random variables. Apart from the constants, the inequality was proved for all real-valued martingales by Burkholder [28]. Many other proofs are now known, including Gundy's proof [86] based on his martingale version of the Calder6n-Zygmund decomposition. Pittenger [125] proved that if p ~> 3, then the best constant on the right side of (51) is p - 1. The proof given above has the advantage of yielding the best constant for the right side if p ~> 2 and the best constant on the left if 1 < p ~< 2; in both of these cases the constant is p - 1. The best constants in the other cases are not known. Inequality (51) does not carry over with finite constants to any Banach space that is not isomorphic to a Hilbert space. This is clear from a result of Kwapiefi [100]; see the proof of the first part of Theorem 14 above. The following theorem provides a martingale square function inequality in the setting of rearrangement invariant function spaces; see Lindenstrauss and Tzafriri [ 107] for background on such spaces. If p ~ [1, oo), then L p (0, 1) and many other Orlicz spaces are examples. Here B is a rearrangement invariant function space on (0, 1) and the upper Boyd index qB of B plays an important role. It is the least q 6 [ 1, oo] such that the norm of the dilation operator Ds on B satisfies

IIDsII ~< s llq

for all s E (0, 1),

where ( D s x ) ( t ) = x ( t / s ) if t 6 (0, s], ( D ~ x ) ( t ) = 0 if t E (s, 1), and the function x : ( 0 , 1) ~ R belongs to B. The following theorem is due to Johnson and Schechtman [96]; also see Antipa [4] and Hitczenko [90]. THEOREM 15. Let B be a rearrangement invariant function space on (0, 1) with finite upper Boyd index qB. There exist positive real numbers c and C depending only on qB such that if f is a real-valued martingale on (0, 1) with respect to Lebesgue measure on the er-algebra of Borel subsets of (0, 1), then

clls(f)

B

IIf*llB

CIIs(z)IIB.

(52)

Conversely, if either side of this two-sided inequality holds for all such martingales (the filtrations must also vary), then the upper Boyd index qB is finite.

258

D.L. Burkholder

This contains the Burkholder-Davis-Gundy inequality [43], which is the inequality

c•

<~EcI)(f*) <~C• EcI)(S(f)),

where q~ is a nondecreasing continuous function on [0, oo], convex on [0, ~ ) , with q~(0) = 0, and satisfying r ~< yq~(,k) for all )~ > 0. The positive real numbers c• and C• can be chosen to depend only on the positive real number y. The condition of convexity can be omitted if the martingales f are assumed to be dyadic or otherwise satisfy the conditions assumed in the earlier paper of Burkholder and Gundy [42]; see Burkholder [29] for a short treatment of both cases. For the convex case, the associated rearrangement invariant function space B is an Orlicz space. Its upper Boyd index (see p. 139 of Lindenstrauss and Tzafriri [107] and the references given there) is qB = inf{q" sup

clgO~t)/c1900tq < ~}.

~.,t>~l

5.5.

Differential subordination of harmonic functions

The function U defined in (47) has other applications. Here is one of them. Let n be a positive integer, D a domain of R n, H a real or complex Hilbert space, K the Hilbert space I-In, u and v harmonic functions on D with values in H, and Vu = (OlU . . . . . OnU). Suppose that v is differentially subordinate to u: for all x E D,

IIv (x)ll

Ilvu(x)ll .

(53)

Fix a point ~ E D and suppose that Ilw(~)lln ~ Ilu(~)lln. Let Do be a bounded subdomain satisfying ~ Do C Do U ODo C D. Denote by/z the harmonic measure on 0 Do with respect to ~. If 1 ~< p < ec, let

IIu IIp = sup

Do

Do

IIu II~ d#

,

where the supremum is taken over all such Do. Then (54)

Ilollp ~ ( p * - 1)llullp.

To see this, recall that U majorizes the function V, where U and V are as in the proof of part (ii) of Theorem 14. It is not hard to show that U (u, v) is superharmonic on D. Therefore,

f0Do

V(u,

v)d#

<~fo

Do

U(u, v)d/z

~< U(u(~),

v(~)) <~O,

which gives (54). See Burkholder [39] for the details and an application to Riesz systems.

Martingales and singular integrals in Banach spaces

259

In the classical setting, n = 2, D is the open unit disk, ~ - - 0 , v(0) = 0, H = R, and the function v is conjugate to u so the Cauchy-Riemann equations give equality in (53). For (54), the conjugacy condition has been replaced by the weaker condition of differential subordination, a condition that makes sense for domains of R n . It is not yet known whether or not p* - 1 is the best constant in (54), as it is for the analogous martingale inequality proved above. For related work, see Burkholder [41], Wang [137], Bafiuelos and Wang [9,10], and Choi [49,50].

5.6. The Beurling-Ahlfors transform The function U of (47) is useful also in the study of the Beurling-Ahlfors transform, the singular integral operator B on L p (C), 1 < p < oc, that maps f to its convolution with - 1/re z 2. It is important in the study of quasiconformal mappings and some other parts of analysis; see, for example, the paper of Iwaniec and Martin [93] and the references given there. The norm liB lip of B is not yet known precisely, except for [IBII2 - 1, but such knowledge would be helpful in some applications. It is known that p*-l~
for a l l l < p < o o .

See Lehto [104] or Iwaniec [92] for the left side. The right side is due to Bafiuelos and Wang [9], where they use U in their proof. Iwaniec [92] conjectures that IIBIIp = P* - 1 and, indeed, this would follow if a conjecture of Bafiuelos and Wang could be shown to be true; see Bafiuelos and Lindeman [8]. The Bafiuelos-Wang conjecture is that

fc

U(Of, m Of) ~< 0

for all infinitely differentiable functions f :C ---> C with compact support, where

1 Of---~

Of i a--x- Oy

and

af--~

l(O

~xx+i-O-Ty .

Some numerical support for these and several closely related conjectures is contained in Baernstein and Montgomery-Smith [7], as well as a remarkable integral identity for the special function U.

6. Martingale characterization of spaces with the R N P

In the study of Banach spaces, the Radon-Nikod3)m property (RNP) has a long and rich history going back at least to the 1930s. Some of the early background is sketched in B.J. Pettis's foreword and the authors' introduction to Vector Measures by Diestel and Uhl [56]. In this influential book, the Radon-Nikod3)m property plays a major role, as it does in a number of articles in this Handbook. After recalling the definition of the RNR we

260

D.L. Burkholder

examine just one of the many necessary and sufficient conditions on a Banach space in order that it have the RNP. Let B be a Banach space and f" a a-algebra of subsets of S2. Here a function q9 : U ~ B is a vector measure if it is a-additive on U. Let I~01denote the measure (from U to [0, cx~]) of total variation. Then q) hasfinite variation if I~oI(s2) is finite. The Banach space B has the Radon-Nikodym property with respect to (I2, 3c, P) if for each vector measure q) of finite variation that is absolutely continuous with respect to the probability measure P there is an integrable function g: S2 --+ B such that ~0(A) = fAg dP for all A 6 U, and B has the RNP if it has the RNP with respect to every probability space. Reflexive spaces, hence UMD spaces, have the RNP, but some nonreflexive spaces such as s also have it; see pp. 217-219 of Diestel and Uhl [56] for three long and helpful lists: a list of some spaces that have the RNP, a list of some that do not, and a list of conditions on a Banach space, each equivalent to the RNP. After Doob's discovery [57] that L l - b o u n d e d martingales converge almost everywhere, it was natural to ask: Under what conditions on a Banach space B, do L l - b o u n d e d martingales converge almost everywhere? Partial results were obtained by Chatterji [47] and Scalora [133]. The complete answer is given in the following key theorem. THEOREM 16. A Banach space B has the RNP if and only if every L~-bounded martingale converges almost everywhere. The "only if" part of this theorem is due to A. Ionescu Tulcea and C. Ionescu Tulcea [91 ]; see also the note by Bellow [ 11 ]. For the "if" part, see Chatterji [48]. For much else that is related, see the books by Diestel and Uhl [56], Bourgin [23], Egghe [70], and Edgar and Sucheston [69]. PROOF (ONLYIF). In all of its essentials, the following proof is the same as Lamb's in the scalar-valued case; see Lamb [101]. Suppose that B has the RNP and f is a Bvalued martingale on (s U, P) with Ilflll finite (recall that Ilflll = SUPn~>0 Ilfn Ill)- Then f converges a.e. as can be seen as follows. Let )~ > 0 and define r:S-2 ~ [0, cx~] by r(co) = inf{n: Ilfn(co)ll > )~}. We shall see below that F = (frAn)n>~O converges a.e. This implies that f converges a.e. on the set {f* ~< )~} since f = F on this set. The complementary set satisfies P(f* > )~) ~< Ilflll/)~ by the Doob maximal inequality applied to the submartingale (llfnll)n>>O, so P(f* < ec) = 1 by the finiteness of Ilflll. Consequently, f converges a.e. The first step in the proof that F converges a.e. is to show that the expectation of F* is finite. Observe that F* ~< )~ on {r = ec} and F* = IIf~ II on {r < ec}. Therefore, E F* ~< )~ + limn--,oc fr<<,nIIf~ II where the integral is equal to n

~ f{r=k} k=0

Ilfkll ~ ~

k=0

f ~ =k/ IIfn II ~ IIfn II1 ~ IIf Ill

since (llfn II)n~0 is a submartingale and {r = k} 6 3ck. Here (Un)n~O is any filtration with respect to which f is a martingale. Thus, E F* ~< )~ + IIf Ill.

Martingales

and singular

i n t e g r a l s in B a n a c h

spaces

261

The next step is to show that there is an integrable function Foc such that

(55)

F, = E ( F ~ I 5t-n) for all n ~> 0.

Let . , 4 - ~n~=of'n and ~ be the least a-algebra containing the algebra A. Define r ".T'~ ~ B by qgn(A) -- fA Fn and the measure ~ ".T'~ ~ [0, cx~) by ~p(A) -- fA F*, and note that Ilqgn(A)ll ~< 7r(A). This leads to a vector measure qg~ : ~ ' ~ ~ B satisfying ~p~(A) = l i m n ~ ~pn(A) for all A E ~ as follows. By (1), F is a martingale: Fn -- Y]~=0 l{k~
A EA

where A A Aoc denotes the symmetric difference of A and Aoc. The limit q%c not only exists and but also satisfies IIq%c(A)ll ~< ~p(A) for all A in Uoc. So it is absolutely continuous with respect to ~k, hence with respect to P. By the RNP, there is an integrable 9coo-measurable function Foc such that

q)oc(A) -- f a F ~ dP

for all A 6 Foc.

In particular, this holds for all A E 9c, and gives

fA

in -- q?n(A) -- q)oc(A) -- fA F ~ .

This proves (55), which yields the a.e. convergence of F as follows. Let S be the family of all simple functions Goc : S2 --+ B that are measurable with respect to the algebra A. The martingale G defined by Gn -- E(Goc ] ,T'n) converges a.e. since Gn = G ~ for all large n. By the contraction property of conditional expectations, 11Fn - Gn I11 ~ ]]Foc - Goc ill. Therefore, lim sup IIFn m , n -+ oc

-

Fm I1~ ~< 2 inf IIFoc - Goc

I1, = 0.

G oc E S

Since (llFm+k - Fm II)k~>0 is a submartingale, Doob's inequality gives XP (

sup

IIFk -Fm It > k) ~< if Fn -Fm Ill

for all X > 0.

m <~k <~n

Accordingly, F converges a.e. so f does also. This completes the proof of the "only if" part of Theorem 16. PROOF (IF). Let (I2, U, P) be a probability space and ~0 : U ~ B a vector measure with finite variation that is absolutely continuous with respect to P. We can assume that Iq0l, the total variation measure, satisfies Iq0[(~2) = 1 so that (I-2, .T', I~0[) is also a probability space.

D.L. Burkholder

262

We can also assume that ~- is countably generated (see Johnson and Lindenstrauss [95] in this Handbook). Consequently, there is a sequence of finite partitions Hn of I2, each set in Hn being a union of sets in Hn+l, such that ~ is the smallest a-algebra containing all of the partition sets. By the classical Radon-Nikod3~m theorem, there is an integrable function g such that I~ol(C) - fc g dP for all C E ~ . We shall show below that there is a bounded B-valued ~-measurable function f ~ such that

qg(C) - fc f ~ dl~ol for all C 6 .Y.

(56)

Therefore, qg(C) - fc f ~ g dP for all C E 9t'. Consequently, B has the RNP. The existence of f ~ is proved as follows. Let H + = {A E Hn: Iqgl(A) > 0} and define fn : S-2 ---->B by qg(A) ~IA. AEH +

I~ol(a)

Note that IIin II ~ 1. Let 9t'n be the algebra generated by Hn : if B E f n , then B is the union of a family of partition sets in Hn. It is easy to see that the sequence f is a martingale relative to (~n)n)0 on the probability space (I2, 9t', I~ol), and that qg(B) = f , fn dl~ol for all B E 9t'n. So by the assumption of the "if" part of the theorem, there is a a B-valued ~ measurable function f ~ such that 2 ~> IIA - f ~ II ~ 0 almost everywhere as n increases to oo. Here define ~ ' ~ --+ B by O(C) = f ~ dl~01 and note that I~Pl ~< I~ol on 3t'. If B 6 3t'n and C E ~ , then II~0(n) - 7s(n) II <~f~ Ilfn - f ~ l l algol---> 0 and

fc

II o(c> -

II: [l o(c> -

- ~o(B)

+

II

21~ol(B A C).

But inf{l~0l(B A C): B E ~n for some n ~> 0} = 0 and (56) follows. This completes the proof of Theorem 16. [3 The martingale characterization of spaces with the RNP given in Theorem 16 should be compared with the following martingale-transform characterization of UMD spaces [30]. THEOREM 17. A Banach space B is UMD if and only if every -+-l-transform of every

L~-bounded martingale converges almost everywhere.

7. Martingale characterization of spaces with the A R N P

A complex Banach space B has the analytic Radon-Nikodym property (ARNP) if for every vector measure q9:/3[0, 2zr) --+ B of finite variation satisfying

fo ree -in~ dqg(0) -- 0 for all integers n < 0,

(57)

Martingales and singular integrals in Banach spaces

263

there is an integrable function g:[0, 2rr) ~ B such that q)(B) = fB g(O)dO for all B B[0, 27r), the o--algebra of Borel sets of [0, 27r). For example, see Bukhvalov [26], Dowling [64], and Hensgen [88]. An earlier and equivalent definition of the ARNP is contained in the work of Bukhvalov and Danilevich [27]. Let D be the open unit disk of C. The Hardy class H~ (D) is the family of analytic functions F : D ~ B satisfying

sup fo 27r IIF (re iO) ][P dO < oo, O
and H~C(D) is the family of the bounded analytic functions F. A complex Banach space B has the ARNP if and only if for each F ~ H~C(D), the limit of F(re i~ as r t 1 exists for almost all 0 6 [0, 27r). The same class of Banach spaces B is obtained if H ~ ( D ) is replaced by H~ (D) where p ~> 1. There is an important martingale characterization of complex Banach spaces with the ARNP due to Edgar [68]. Suppose that (X2, U, P) is a probability space on which is defined an independent sequence Z of Steinhaus random variables: Zn is uniformly distributed on the unit circle, the boundary 0 D of D. Let 5On be the a-algebra generated by Z0 . . . . , Zn. Then an analytic martingale (starting at x 6 B) is any sequence f of the form fn -- x + ~-~=1 vkZk where the function vk" X2 --> B is fk_l-measurable and integrable. By (1), f is a martingale. Typically, P is the Haar measure on the infinite polydisk s = (0 D) ~176 and the value of fn at a point (e i~176 , e i01 . . . . ) in s is written as fl

f , (0) = x + ~

~0k(00 . . . . . Ok-1 )e i~ 9

k=l

THEOREM 18 ([68]). A complex Banach space B has the ARNP if and only if every L 1bounded analytic martingale converges almost everywhere. This theorem and the corresponding theorem for the RNP (Theorem 16) have quite different proofs. We omit Edgar's proof. The two theorems yield an obvious corollary, which is also a consequence of (57) and the vector-valued version of the E and M. Riesz theorem due to Ryan [132], as was observed by Bukhvalov [26]. COROLLARY 19. A complex Banach space with the RNP has the ARNP. But these two properties are not equivalent. The Banach space L~ (0, 1) has the ARNP (see the next section) but it is well known that it does not have the RNP (see Diestel and Uhl [56]). Further work related to the ARNP includes Dowling and Edgar [65], Garling [77], Ghoussoub and Maurey [80], Ghoussoub, Maurey, and Schachermayer [82], Ghoussoub, Lindenstrauss, and Maurey [81 ], and Bu [24].

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8. Characterization of AUMD spaces 0 (B) the least ot 6 [1 cx~] such Here let B be a complex Banach space, 0 < p < oc, and Otp that 0 (B) IIf IIp Ilgllp ~ Ofp for all f and g where f is a B-valued analytic martingale (see Section 7) and g is a -+-l-transform of f (see Section 2). If the numbers E~ E {1 , - 1} in the definition of g are replaced by complex-valued U(~_l)v0-measurable functions v~ that have their values in the closed unit disk, then the best constant may change a little but its finiteness, or lack of it, will not be affected. There will be no change if the vk have their values in [ - 1 , 1]. The space B is an A UMD space if Ctp 0 (B) is finite for some p 6 (0 c~); equivalently, for all p ~ (0, 0o). This equivalence can be seen by using the inequality IIf*llp ~< el/pllfl]p, which holds for all p E (0, c~) and all analytic martingales [77], using the trivial inequality IIf IIp ~< IIf* IIp, and observing that the argument in the dyadic case (see Section 2 above) shows that if Ilg*llp ~< cp(B)llf*llp holds for some p E (0, cx~) and all f and g as above, then it holds for all p 6 (0, ~x~). Notice that if p E (1, oc), then Otp 0 (B) ~< tip(B) So a UMD space is an AUMD space. But the converse is not true: the space L ~ (0, 1) is AUMD but is not UMD because it is not reflexive. That L ~ (0, 1) is AUMD is an immediate consequence of the fact that C is UMD, hence is AUMD, and Theorem 9 of Garling [77]: if B is AUMD, then so is L~ (0, 1) for all p E [1, cx~). Note that Ilf* Ill ~ ellfll l, a special case of Garling's inequality for analytic martingales, implies that an AUMD space is an ARNP space" if B is AUMD and f is an L~-bounded analytic martingale, then f * is integrable so there is an integrable function foc such that fn = E (fee [ .T'n) as in (55), which implies that f converges almost everywhere. Therefore, by Edgar's theorem above, the space B is ARNE Using multipliers, Blower [ 16] proved the following characterization of AUMD. THEOREM 20. A complex Banach space B is AUMD if an only if there is a real number y such that

fo re

• n~O

mnan einO dO <, ~' fo 2rc

Z ane inO

dO

n >~O

for all sequences (an)n>~O in B with an = 0 for all large n, and all complex-valued sequences (mn)n>~O such that Imnl ~ 1 and n2[mn+l - 2mn + mn-ll ~ 1. This implies that B is AUMD if and only if for all such sequences m, the linear transformation Tm defined by Tm (e in0) = mne inO extends to a bounded linear operator on the Hardy space H 1 (0 D). Piasecki [ 120,121 ] has two quite different characterizations of AUMD. The first one can be viewed as the subharmonic analogue of the ~'-convex characterization of UMD given in Theorem 1. The other characterizes AUMD in terms of tangent analytic martingales: Piasecki proves for AUMD an analogue of McConnell's inequality [ 112] for tangent martingales in UMD spaces.

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[114] A.M. Olevskii, Fourier series and Lebesgue functions, Uspehi Mat. Nauk 22 (1967), 237-239 (Russian). [115] A.M. Olevskii, Fourier Series with Respect to General Orthogonal Systems, Springer, New York (1975). [ll6] R.E.A.C. Paley, A remarkable series of orthogonal functions I, Proc. London Math. Soc. 34 (1932), 241264. [117] A. Pdczyfiski, Structural theory of Banach spaces and its interplay with analysis and probability, Proceedings of the International Congress of Mathematicians (Warsaw, 1983), PWN, Warsaw (1984), 237-269. [118] A. Petczyfiski, Norms of classical operators in function spaces, Colloque Laurent Schwartz, Ast6risque 131 (1985), 137-162. [119] A. Petczyfiski and H. Rosenthal, Localization techniques in LP spaces, Studia Math. 52 (1975), 263-289. [120] M. Piasecki, A geometrical characterization of A UMD Banach spaces via subharmonic functions, Demonstratio Math. 30 (1997), 641-654. [121] M. Piasecki, A characterization of complex AUMD Banach spaces via tangent martingales, Demonstratio Math. 30 (1997), 715-728. [122] A. Pietsch and J. Wenzel, Orthonormal Systems and Banach Space Geometry, Cambridge University Press, Cambridge, UK (1998). [123] G. Pisier, Un exemple concernant la super-rdflexivitd, S6minaire Maurey-Schwartz, 1974-75, l~cole Polytechnique, Paris (1975). [124] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326-350. [125] A.O. Pittenger, Note on a square function inequality, Ann. Probab. 7 (1979), 907-908. [126] J. Prtiss, Evolutionary Integral Equations and Applications, Birkh~iuser, Basel (1993). [127] J. Prtiss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z. 203 (1990), 429-452. [128] M. Riesz, Sur les fonctions conjugudes, Math. Z. 27 (1927), 218-244. [129] J.L. Rubio de Francia, Fourier series and Hilbert transforms with values in UMD Banach spaces, Studia Math. 81 (1985), 95-105. [1301 J.L. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, Probability and Banach Spaces (Zaragoza, 1985), Lecture Notes in Math. 1221, Springer, Berlin (1986), 195-222. [131] J.L. Rubio de Francia and J.L. Torrea, Some Banach techniques in vector valued Fourier analysis, Colloq. Math. 54 (1987), 273-284. [132] R. Ryan, The F. and M. Riesz theorem for vector measures, Indag. Math. 25 (1963), 408-412. [133] ES. Scalora, Abstract martingale convergence theorems, Pacific J. Math. 11 (1961), 347-374. [134] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton (1970). [135] E.M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Princeton University Press, Princeton (1970). [136] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton (1993). [137] G. Wang, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab. 23 (1995), 522-551. [1381 J. Wenzel, Mean convergence of vector-valued Walsh series, Math. Nachr. 162 (1993), 117-124. [1391 J. Wenzel, Ideal norms associated with the UMD-property, Arch. Math. 69 (1997), 327-332. [1401 E Zimmermann, On vector-valued Fourier multiplier theorems, Studia Math. 93 (1989), 201-222. [1411 A. Zygmund, Trigonometric Series I, II, Cambridge University Press, Cambridge (1959).

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CHAPTER 7

Approximation Properties Peter G. Casazza Department of Mathematics, The University of Missouri, Columbia, MO 65211, USA E-mail: pete @casazza, math. missouri, edu

Contents 1. The basis property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The approximation property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The bounded approximation property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The commuting bounded approximation property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The 7r-property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The finite dimensional decomposition property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. The uniform approximation property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. The compact approximation property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Approximation properties in non-separable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H A N D B O O K OF THE G E O M E T R Y OF BANACH SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved 271

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The approximation property, which already appeared in Banach's book in 1932 [5], plays a fundamental role in the structure theory of Banach spaces. The first systematic study of the variants of the approximation property was initiated by Grothendieck in 1955 [35]. At that time the main properties were the approximation property, the bounded approximation property and the basis property. It was not until 1972 that Enflo [23] produced the first example of a Banach space which fails the approximation property. Today, there is a "chain" of approximation properties in between the standard ones. Our goal is to introduce the important variants of the approximation properties and the relationships between them. In the process we will consider the various counter-examples and how they fit into the theory without giving the details of their constructions. We will not emphasize the applications of the approximation properties, but many of them can be found throughout the handbook. As we will see, there is a rich theory surrounding the approximation property for Banach spaces and the counter-examples merely illustrate the delicate nature of the relationships between the various properties. What one wants from this theory is to be able to prove that a given space has the weakest possible form of the approximation property and then be able to conclude that it actually has the strongest possible form. For example, we will see that a Banach space has the bounded approximation property if and only if it is a complemented subspace of a Banach space with a basis. But, there are Banach spaces with the bounded approximation property with fail to have bases. However, a complemented subspace of L p for 1 < p < cx~ (which immediately has the bounded approximation property) actually has a basis. Also, Lusky [69] has shown that the Disk Algebra A has a basis without using any deep results from complex function theory, but instead relies on results from the approximation property. Our emphasis will be on building the theory necessary to pass the various approximation properties from a space to its complemented subspaces, or from the dual space to the space etc. and finding the minimal conditions for passing from one of these properties to another. We will give only some representative proofs illustrating the basic techniques in the area. The counter-examples are quoted mostly to illustrate that these properties really are distinct in the general case, to show that positive results are best possible, and to explain some of the technical difficulties encountered in the proofs. These examples also make contact with other important areas such as Banach algebras, C*-algebras et al.

1. The basis property Basis theory is a large subject, and we will not attempt to cover even a fraction of it here. We will concentrate on those results which relate basis theory to the approximation properties. We refer the reader to the Basic Concepts section of this Handbook [47] for the basic definitions and results concerning bases. Here we use the term basis to mean Schauder basis. This concept was introduced by Schauder in 1927 [90]. But Schauder assumed his bases were normalized and that the dual functionals were continuous. Later, Banach [5] removed these unnecessary assumptions. The basis problem appeared in Banach's book [5] in 1932, p. 111: "On ne sait pas tout espace du type (B) separable admet une base?" But it was not until 1972 that Enflo answered this question in the negative [23]. Later, Szarek [95] constructed a Banach space X

P.G. Casazza

274

with a finite dimensional decomposition (see Section 6) which fails to have a basis. Also, Read [86] constructed a Banach space satisfying the commuting bounded approximation property which fails the :r-property (see Section 5). Both of these spaces are Banach spaces which fail to have bases despite being isomorphic to a complemented subspace of a Banach space with a basis, namely C2 (see Definition 1.10 and Proposition 6.7). Whenever possible we will refer to the space of Szarek since Read has not submitted his manuscript for publication. Although every separable Banach space embeds into a Banach space with a basis, namely C[0, 1], there are separable Banach spaces which never embed complementably in a space with a basis. One class of such spaces are the Banach spaces which fail the approximation property discussed in Section 2. There is a simple criterion due to Petczyfiski and Wojtaszczyk [82] (later improved independently by Petczyfiski [80] and Johnson, Rosenthal and Zippin [50]) for checking when a Banach space is complemented in a space with a basis. To state this result, we need a definition.

DEFINITION 1.1. A sequence of non-zero finite rank operators (Ai) from a Banach space X to itself is called a (unconditional)finite dimensional expansion of the identity if for all xEX, x -- Z

Aix, i

(and the series converges unconditionally). It is immediate that a finite dimensional expansion of the identity is the same as the bounded approximation property (see Section 3). Requiring the operators to be non-zero in Definition 1.1 is a convenience but not a necessity. We now have

PROPOSITION 1.2. A Banach space X is isomorphic to a complemented subspace o f a Banach space with a basis if and only if X has a finite dimensional expansion o f the identity. PROOF. (=~) If Y is a Banach space with a basis (Xi,X*) and X is a complemented subspace of Y with projection P : Y -+ X, let I -- {i: Pxi 7/= 0}. Now, the operators A i x -- x i*( x ) P x i , for i 6 I, form a finite dimensional expansion of the identity for X (,=) Here we will just do the case where the operators Ai are rank 1 and in Section 6 we will do the general case. We let Y be the family of sequences (xi), xi ~ A i ( X ) for which Y~i xi converges in X. We define addition and multiplication coordinatewise in Y and let the norm be given by ]l(xi)ll- supn II ~in___lxillx. Then Y is a Banach space with a basis (Yn) consisting of any choice of Yn = (xi) with xi = 0 for all i ~ n, and Xn :7/=O. Finally, Ux = (Anx) is an isomorphic embedding of X into Y and P(xi) = (An(Y'~i xi)) is a bounded linear projection of Y onto X. [5] Similarly, one can show that a Banach space X is isomorphic to a complemented subspace of a Banach space with a unconditional basis if and only if X has a unconditional 1-dimensional expansion of the identity. It is known [50] that if X* has a basis, then so does X. In fact, in this case X has a shrinking basis and so X* has a boundedly complete basis. To see that the converse of

Approximation Properties

275

this fails, we need a basic tool for constructing examples which is a generalization due to Lindenstrauss [63] of a result of James [38]. PROPOSITION 1.3. If X is a separable Banach space, there is a separable Banach space Z with Z** having a boundedly complete basis and Z * * / Z is isomorphic to X. Moreover, Z*** ~- Z* 9 X*. PROOF. Let (xn) be a sequence of which is dense in the boundary of the unit ball of X. The space Z consists of all the sequences of scalars z - - (ai) for w h i c h Y~i aixi = 0 and Pj

j=l

Z +1aixi

2) 1/2 <~,

(1)

i=pj-i

where the supremum is taken over all choices of integers m and 0 = P0 < pl < " " < Pm. It follows that for every z = (ai) E Z, the s e r i e s ~-~i aixi converges in X. The main point of the proof is to show that Z** can be identified with the space of all sequences for which (1) holds. Then the operator T: Z** --+ X given by T (ai) = ~--~iaixi is bounded, and by the density of (xi) in the boundary of the unit ball of X, it is a quotient map. Moreover, the unit vectors of Z form a boundedly complete basis of Z. The moreover part of the theorem comes from the fact that for every Banach space Z there is a norm 1 projection from Z*** onto Z*. The projection is just the map which takes every bounded linear functional on Z** to its restriction to Z. D Now it is simple to construct a Banach space with a basis whose dual fails to have a basis (even the approximation property). PROPOSITION 1.4. There is a Banach space X with a basis so that X* is separable and fails the approximation property.

PROOF. We take a separable reflexive Banach space X failing the approximation property (see Section 2) and choose Z with Z** having a basis and Z*** ~ Z* @ X*. Then Z*** fails to have a basis - e v e n the approximation property. D Johnson and Schechtman (see [48]) have shown that there are subspaces of co with bases whose dual spaces fail to have bases (or even the approximation property). Also, there are subspaces of g l which have bases whose dual spaces fail the approximation property (see Corollary 6.12). Obviously, any subspace of a Hilbert space (being itself a Hilbert space) has a basis. This property does not characterize Hilbert spaces but spaces having this property must be very close to Hilbert spaces. Kwapiefi's theorem [59] states that any Banach space which is both type 2 and cotype 2 must be isomorphic to a Hilbert space. But, Szankowski [91] shows that if every subspace of a Banach space X has the approximation property then for every e > 0, X has type ( 2 - e) and cotype (2 + e). Johnson [44] showed there are nonHilbert spaces with the property that every subspace (even every subspace of every quotient

276

P. G. C a s a z z a

space) has a basis. The first such example was the convexified Tsirelson space T 2. In 1972, at the same time that Enflo gave the first counter-example to the approximation problem, Tsirelson [97] constructed the first Banach space which contains no subspaces isomorphic to co, or g p, for 1 ~< p < o~. Figiel and Johnson [28] gave an analytic description of the norm on the dual space of the space constructed by Tsirelson, and this soon became known as Tsirelson's space T. The p-convexification of T was also constructed by Figiel and Johnson [28]. To define these spaces we fix 1 ~< p < oo, and let (tn) be the unit vectors in the linear space of finitely non-zero sequences Coo. For two subsets E, F of the natural numbers, we will write E < F to mean max E < min F. Also, for E C N and x - - ~-]m an tn we write E x =: Y~n~E antn. For x = Y~n antn, we inductively define a sequence of norms (ll" IIm)mC~_-0 on Coo by

IIx Iio : max lan I, /7

and for all m ~> 0,

Ilxllm+l - max

IIEjxll

IlXllm, ~ max

,

(2)

j:l

where the inner max is taken over all choices of finite subsets k <~ E1 < E2 < ... E~ of N. Since (llx [Im )m= oo 1 is increasing we can let

IIxll -

lim IIxllm

m---> oo

for all x for which this limit is finite. Now, p-convexified Tsirelson space TP is the completion of Coo in this norm. A parallel theory developed for modified Tsirelson's spaces where we assume the (Ei) are just disjoint subsets of N with k <<,Ei. However, Casazza and Odell [17] showed that these two constructions produce equivalent norms. The unit vectors (tn) form a 1-unconditional basis for T p and none of these spaces contains a subspace isomorphic to co or gq, 1 ~< q < oo. Johnson [44] showed that every subspace of every quotient space of (a subspace of) T 2 has a basis. When stronger properties of these spaces were developed [18], it was discovered that the subspace constructed by Johnson was isomorphic to the whole space T 2. It follows easily from the definition that any n disjointly supported normalized blocks (yi)in=l o f (tk)Ln are 2-equivalent to the unit vector basis of ~2. Duality arguments show that this property holds also in T*, which also yields norm 2 projections onto the blocks spanning ~ in T 2. Casazza, Johnson and Tzafriri [12] showed that we can iterate this process (see also [18,85]) to exponentially increase the number of blocks one can take supported after n and still have a good copy of the unit vector basis in a Hilbert space. Now, applying Proposition 7.3, we can show that every n-dimensional subspace E of span(tk)~=n is 8-complemented and 8-isomorphic to ~ . Now, if E is any 2n-dimensional subspace of T 2, and F -- E N span(tk)~__n+l then dim F ~> n. It follows that every 2n-dimensional subspace of T 2 contains an n-dimensional subspace which is 16-isomorphic to a Hilbert space and 16-complemented in T 2. Pisier

277

Approximation Properties

[84] introduced the notion of a weak Hilbert space and gave several equivalent formulations for the definition. One of them is that X is a weak Hilbert space if there is a constant M > 0 so that every 2n-dimensional subspace of X contains an n-dimensional subspace which is M-complemented in X and M-isomorphic to s So by the discussion above, T 2 is a weak Hilbert space. Nielsen and Tomczak-Jaegermann [78] have shown that every separable weak Hilbert space which is a Banach lattice is very much like T 2. In particular, every subspace of every quotient space of these spaces has a basis. Pisier [84] showed that all weak Hilbert spaces have the approximation property (see Section 7 for a discussion of weak Hilbert spaces and the uniform approximation property). It is an open question whether every separable weak Hilbert space has a basis. But Maurey and Pisier (see [74]) have shown that every separable weak Hilbert space has a finite dimensional decomposition (see Section 6). The converse is also an open question. That is, if X is a separable Banach space and every subspace of X has a basis, must X be a weak Hilbert space? There is some strong evidence towards a positive answer to this question given by Mankiewicz and Tomczak-Jaegermann [72,73]. They showed that a Banach space X for which every subspace of X and every quotient space of X has a basis (with uniformly bounded basis constants) is a weak Hilbert space. The results in [73] are much stronger than we have stated and actually are finite dimensional in nature and give quantitative estimates of the weak type 2 and weak cotype 2 constants in terms of the uniform basis constants for all subspaces (respectively quotient spaces) of X. Although there are infinite dimensional spaces which are never complemented in spaces with bases, the finite dimensional situation is much better. This is a result of Petczyfiski [80] (proved independently without the exact constants in [50]). PROPOSITION 1.5. For every finite dimensional space E and every e > O, there is a finite dimensional space F D E, with a norm one projection P from F onto E, so that F has a basis with basis constant <~ (1 + e). n forE PROOF. Let dim E -- n and choose an Auerbach system ( X i , X i*)i=l rn + j, 0 <~ r < n, 1 <<.j <<.n, put Ui(x) -- -ff1X* j(x)xj such that yi E Ui (E). Define a norm on F by

IlYll =

max

l<~k<~n 2

For e a c h i - -

Let F be all sequences y -- (yi)/.n21

~Yi

i=1

I f w e choose ei E U i ( E ) for 1 ~
1

I IE, IIg-~ll ~< 1 and IIv II ~ 1 + n2.

E by U ( y i ) -- ~ yi, we see F]

The situation for unconditional bases is much more delicate. Although every Banach space contains a subspace with a basis, Gowers and Maurey [33] have shown that they need not contain unconditional basic sequences. There are classical Banach spaces, such as C[0, 1] and L 110, 1] which do not embed into any Banach space with a unconditional

278

P.G. Casazza

basis. Also, Lindenstrauss [60] observed that el has a subspace D with a basis given by the vectors (en - 1 (ezn -k- e2n+l)), while D fails to have a unconditional basis. It is immediate that if X has a unconditional basis and X* is just separable, then el does not embed into X and so the unconditional basis for X is shrinking and hence its dual functionals form an unconditional basis for X*. Also, any pre-dual X of el without an unconditional basis (such as C (co'~ provides an example of a Banach space which fails to have a unconditional basis but its dual space has a unconditional basis. Proposition 1.2 becomes more delicate in this setting also. PROPOSITION 1.6. I f X is complemented in a Banach space with a unconditional basis, then X has a unconditional finite dimensional expansion o f the identity. The converse fails. However, if X has a unconditional finite dimensional expansion of the identity by rank 1 operators, then X is complemented in a Banach space with a unconditional basis. PROOF. The first part of Proposition 1.5 is exactly the same as the first part of the proof of Proposition 1.2. The last statement in Proposition 1.5 is also done exactly as in the proof of Proposition 1.2 with the slight modification in the definition of the space Y where we require the series Y~i xi there to converge unconditionally. For the converse, Kalton and Peck [54] have constructed a Banach space Z which fails to have a unconditional basis but is an unconditional sum of two dimensional Banach spaces. That is, there is a sequence En of subspaces of Z with dim En = 2, for all n = 1, 2 . . . . and for every z E Z there is a unique sequence of vectors Zn E En with z -- Y~n Zn and this series converges unconditionally. It follows that the operators An z = Zn give an unconditional expansion of the identity by rank 2 operators while it is known [46] that Z is not complemented in any Banach space with a unconditional basis. [] There is a partial converse for Proposition 1.6 stating that a Banach space with a unconditional finite dimensional expansion of the identity is isomorphic to a subspace of a Banach space with a unconditional basis (see Section 6). The different results we are getting for complemented subspaces of spaces with bases and spaces with unconditional bases can be explained by some fundamental results of Gordon and Lewis [32]. We say that a Banach space X has GL-LUST (Gordon-Lewis local unconditional structure) if there is a constant K > 0 so that for every finite dimensional subspace E of X there is a Banach space F with a 1-unconditional basis and operators T : E --+ F and S : F --+ X with S T = liE and IISIIIITll ~< g . It is easily checked that a complemented subspace of a Banach space with a unconditional basis must have GLLUST. Gordon and Lewis [32] and a host of other authors explored some deep consequences of a space having GL-LUST. By Proposition 1.5, every finite dimensional space is 1-complemented in a finite dimensional space with a basis whose basis constant is at most 2. One consequence of the work on GL-LUST is that if we fix K, we cannot find for every finite dimensional space E a finite dimensional space F so that E O1 F has a K-unconditional basis. The picture for complemented subspaces of Banach spaces with unconditional bases is quite unsettled at this time. The one positive result here is due to Kalton and Wood [55].

Approximation Properties

279

THEOREM 1.7. If X is a complex Banach space with a 1-unconditional basis and Y is a 1-complemented subspace of X, then Y has a 1-unconditional basis. Rosenthal [87] and Flinn [29] have given separate proofs of this result. Benyamini, Flinn and Lewis [6] have constructed a (real) Banach space without a 1-unconditional basis which is 1-complemented in a space with a 1-unconditional basis. But, their complemented subspace does have a unconditional basis. So the general question is still open. PROBLEM 1.8. Does every complemented subspace of a Banach space with a unconditional basis have a unconditional basis? This is the atomic version of the general question: Is a complemented subspace of a Banach lattice isomorphic to a Banach lattice? More than likely this question has a negative answer. A weaker conclusion might have a chance of holding. PROBLEM 1.9. If Y is complemented in a Banach space with a unconditional basis, does Y have a complemented subspace with a unconditional basis? Special cases of these problems are also of interest. For example, if Y is a complemented subspace of Lp[O, 1], 1 < p < cx~, does Y have a unconditional basis? Complemented subspaces of Lp which are not isomorphic to a Hilbert space are called s and it is known [50] that separable s have bases. In fact [79], s have bases (xi) so that supnd(span(xi)in=j, g~) < oo. It is also known [15] that if gp r X has an unconditional basis, then X has an unconditional basis. One way to give a positive answer to the s would be to answer the corresponding question for Lp[O, 1]. If Lp 9 X has a unconditional basis, must X have an unconditional basis? This would answer the s question since it is known [4] that if X is complemented in Lp, then L p "~ L p 9 X. Although every subspace of T 2 has a basis, this space has subspaces without unconditional bases. This comes from the GL-LUST considerations of Komorowski [53]. We ask PROBLEM 1.10. If every subspace of a Banach space X has a unconditional basis (or just GL-LUST), must X be isomorphic to a Hilbert space? There is evidence to suggest that Problem 1.10 has a positive answer [53,57]. We end this section with a discussion of a class of universal spaces with bases constructed by Johnson [42]. We choose a sequence of finite dimensional spaces (En) which are dense in the Banach-Mazur distance in the family of all finite dimensional spaces. Also, let (Fn) be dense in the family of finite dimensional spaces with a 1-unconditional basis. DEFINITION 1.1 1. For 1 ~< p < cx~ let

n

~p

n

~p

and for p - 0 , the sums above are replaced by c0-sums.

P.G. Casazza

280

We note for later reference that C p has the property that for every finite dimensional space G and for every e > 0 and for every natural number m, there is an n > m and an operator T: G --+ En such that II T II II T -1 II ~< 1 + e. To see that we can assume this l property, let Cp = (Y~n @Gn)ep, where (Gn) is a listing of the (En) where each En appears ! ! infinitely many times in the sequence (Gn). Then clearly Cp is isometric to ( ~ E3Cp)ep. Note that C p is isometric to a subspace of C p' . Also, given any Ek and any m, the space (Y~in_ 1 ~)Ei)~pEDpEkis (1 -+- e)-isomorphic to one of the Ei's and we must have i > m just ! by dimension considerations. It follows that C p is isometric to a complemented subspace of Cp and hence by the Petczyfiski decomposition method Cp ~ C'p" An application of Proposition 1.5 shows that C p has a basis. The simplest way to do this is to observe that we can rearrange the En so that each Ezn G E2n+! has a basis (x~)~" l

~,~,~t'xn~kni)i:l)n~

with basis constant <~ 2. Then l__ is a basis for (a space isometric to) Cp with basis constant ~< 2. Johnson [42] showed the universality of C! and U1. PROPOSITION 1.12. If X is a separable Banach space, then X* is isometric to a norm 1

complemented subspace of C~. If X is separable and has GL-LUST then X* is isometric to a norm 1 complemented subspace of U~. PROOF. Pick a sequence G l C G2 C ... of finite dimensional subspaces of X whose union is dense in X. We can find natural numbers kl < k2 < ... and operators Tn : Gn --+ Ek, with II Zn II II T~- 1 II <~ 1 + 1. Define T" C1 --+ X by T (Xn) = Y~n Tn 1xk,. Then T is a quotient n map, and so T* is an isometry. Also, T* T-1 is a projection onto the range of T*. The other case is similar. D

2. The approximation property If a Banach space X has a basis (Xn)n~__l with basis projections Pn, then for any Banach space Y and any compact operator T E B(X, Y), l i T - PnTll ~ O. The question o f w h e t h e r the converse was true (known as the approximation problem) was an open problem for quite some time until until Enflo [23] gave a counter-example in 1972. At the end of this section we will discuss the various examples concerning the approximation property. NOTATION 2.1. For Banach spaces X, Y, we denote by U ( X , Y), the family of finite rank operators T : X --+ Y, and let U ( X ) = f ' ( X , X). DEFINITION 2.2. A Banach space X is said to have the approximation property (AP for short) if for every compact set K in X and every e > 0, there is a T 6 9t'(X) so that IIT x - x II ~< e, for every x E K. Every Banach space with a basis has the approximation property. To see this, we just observe that if K is a compact set in X, then l i m n ~ I1(I - en)lK II = 0, where (Pn) are the basis projections. In this proof, the finite rank operators approximating the identity on K are actually projections. It is an open question whether the approximation propertyis equivalent to this "projection form" of the approximation property. A simple calculation

Approximation Properties

281

shows that every complemented subspace of a space with the approximation property must have the approximation property. Grothendieck [35] initiated the study of the variants of the approximation property and the relations between them. One important tool he used was the topology of uniform convergence on compacts sets. NOTATION 2.3. If X and Y are Banach spaces, we let r denote the topology on B(X, Y) of uniform convergence on compact sets in X. The topology r is the locally convex topology on B(X, Y) generated by the seminorms of the form IITII~c - s u p { l l T x l l : x ~ K } , where K ranges over the compact subsets of X. We write (B(X, Y), r) for this space. Grothendieck [35] identified the dual space of (B(X, Y), r). PROPOSITION 2.4. The continuous linear functionals on (B(X, Y), r) consist of all func-

tionals 4) of the form oo

0(3

~ ( r ) -- Z y * ( r x i )

'

xi 9 X

'

y* 9 X* , Z

Ilxi IIIlyi* II < ec.

l

i=1

i=1

PROOF. If 4) has such a representation, assume 0 r Xi and choose Oti > 0 tending to ec so that ~-~i Oli Ilxi III Y* II - M < ec. Then K -- xi /Ilxi Iloti U {0}, is compact and

I (r i

liy*lillx ll llW(xi/llxiil e)ll

<<.

MilrgiK.

i

Conversely, if 4) is a linear functional on B(X, Y) satisfying ]4~(T)] ~< MIITIIx, with K compact, a well known construction (see, e.g., [66, Proposition 1.e.2]) shows that we may assume that K = c--o-~(xi) where ]]xi ]]--+ 0. Let S: B(X, Y) --+ (Y | Y G . . ")co = Z be given by S(T) = ( T x l , Tx2 . . . . ). Since ]4~(T)I ~< MIIS(T)I], there is a linear functional on S B ( X , Y) so that 4~(T) = ~ ( S T ) . By the Hahn-Banach theorem, we can extend this to a linear functional on Z, and hence to an element of (Y* G Y* G . . ")e~. That is, there must exist Yi 9 Y* so that Z i IlYi II < oc and ~ ( T ) -- Y~i Yi T ( x i ) . I--I The next result due to Grothendieck [35] relates the approximation property to the question of approximating compact operators. THEOREM 2.5. For a Banach space X, the following statements are equivalent:

(1) X has the approximation property. (2) There is a net (Tk))~A of finite rank operators on X converging to the identity op-

erator on X uniformly on compacta, i.e. T)~ --+ I in (B(X), r). (3) For every Banach space Y, .F(Y, X) is dense in (B(Y, X), r). (4) For every Banach space Y, U ( X , Y) is dense in (B(X, Y), r). (5) For every choice of (xi) in X and (x[) in X* such that ~ i []xi I] ]ix*l] < oc and

Y~i x*(x)xi - O, for all x 9 X, we have Y~.i x*(xi) - O. (6) For every Banach space Y, every compact operator T 9 B(X, Y) and every e > O, there is a T 9 .F(X, Y) with IIT - T1 II < 8.

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PROOF. The equivalence of (1) and (2) is immediate from the definitions. The equivalence of (1) and (5) is a consequence of Proposition 2.4. (1) means that the identity operator is in the r-closure of U(X) in (B(X, Y), r). This happens if and only if every r-continuous linear functional which vanishes on the operators of rank 1, also vanishes on the identity operator. But, in light of Proposition 2.4, this is exactly what (5) is saying. Letting X = Y in (3) and (4) shows that they both imply (1). Now assume (1). Let T 9 L(Y, X). For every compact set K in Y, T (K) is compact in X. Hence, given e > 0, by (1) there is a T1 9 9t-(X) so that IIT~Ty - Tyll <<,e, for all y 9 K. Since Tl T 9 9t'(Y, X), this proves (3). A similar construction proves that (1) implies (4). The equivalence of (1) and (6) can be found in [35,66]. [] Theorem 2.5 has several important consequences. First, if X fails the approximation property, letting Y = span(x/) in (5) of Theorem 2.5, we see that for every subspace Y C Z C X, Z fails the approximation property. That is, if X fails the approximation property, then there is a separable subspace Y C X so that for every subspace Y C Z C X, Z fails the approximation property. Another immediate consequence of the equivalence of (1) and (5) in Theorem 2.5 is PROPOSITION 2.6. If X* has the approximation property, then so does X. Hence, if X is reflexive, then X has the approximation property if and only if X* has the approximation property. Condition (6) of Theorem 2.5 raises an obvious question, which is still unsolved. Namely, is there an internal characterization of the approximation property in terms of approximating compact operators by finite rank operators on X ? Formally, we are asking. PROBLEM 2.7. Let X be a Banach space with the property that for every e > 0 and for every compact operator T 9 B(X), there is a T1 9 9t-(X) with liT - T1 II ~< E. Must X have the approximation property? Given the symmetry of (3) and (4) in Theorem 2.5, it is natural to ask what happens if we reverse the rolls of X and Y in Condition (6). The answer is another result of Grothendieck

[35]. PROPOSITION 2.8. l f X is a Banach space, then X* has the approximationproperty if and only if for every Banach space Y, every e > 0 and every compact operator T 9 B(X, Y), there is a T1 9 ~ ( X , Y) such that IIT - T111 ~< e. During his investigation of the approximation problem Grothendieck found many interesting equivalent formulations of the problem. Although we now have counter-examples to the general problem, Grothendieck's results still have interest since they show how to transfer these examples to some examples in classical analysis. PROPOSITION 2.9. The following three assertions are equivalent:

Approximation Properties (1) Every Banach space has the approximation property. (2) Every matrix A - - ( a i j ) i,~ j= l of scalars, f o r which limi

283

a i j - - O,

and }-~i maxj laij l < 0(3 and A2 __ O, also satisfies trace A - - Z i (3) Every continuous function K ( r , s ) on [0, 1)| [0, 1],forwhich

f o r all i - 1,2 . . . . . a i i - - O.

fo K ( r , s ) K ( s , t ) d s - - O ,

for every r and t, satisfies fd K (s, s) ds - 0. We now pass to a discussion of the various counter-examples to the approximation property and some of their consequences. After Enflo [23] constructed the first Banach space failing the approximation property, this was quickly followed by a number of important examples concerning the approximation property. Almost immediately, Figiel [25] and Davie [20,21 ] gave greatly simplified constructions for spaces failing the approximation property and showing that there are subspaces of g p, 2 < p, failing the approximation property. Szankowski [92] then constructed subspaces of g p, 1 ~< p < 2, which fail the approximation property. We will start, however, at the end of the story, since it puts these results into proper context. There is a general criterion for a space to fail the approximation property. This is a modified version of a criterion used by Enflo in his original solution to the approximation problem. This criterion actually identifies spaces which fail the weaker property the compact approximation property (compact AP for short) studied in Section 8. A Banach space is said to have the compact approximation property if it satisfies Definition 2.2 with the finite rank operators there replaced by compact operators. PROPOSITION 2.10. Let X be a Banach space and assume there are sequences ( x i ) in

X and (x*) in X*, a sequence (Fi) offinite subsets of X and an increasing sequence of integers (ki ), so that the following hold: (1) x i (xi ) -- 1, f o r every i -- 1, 2 . . . . . (2) sup/Ilxi II < ~ . :r (3) x i --+ 0 in the weak-star topology. (4) For every T E B ( X , Y), if we let rio(T) = 0 and f o r n ~ 1 1

kn

fin(T)-- kn ) ix;(Txi)' then f o r every n = 1,2 . . . . we have I/~n(T)-/~n-l (T)I ~< sup{llTxll: x E Fn}. (5) E n sup{ IIx I1: X E F~ } < ~ . Then X fails to have the compact approximation property (and hence the approximation property).

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P. G. Casazza

Although this criterion looks formidable, in practice it allows the construction of spaces designed specifically to satisfy this criterion to produce counter-examples to the approximation property. In an important series of constructions, Szankowski carried out this program to produce a spectacular collection of counter-examples to the approximation property. The first was the construction of a Banach lattice which fails the approximation property [91 ]. PROPOSITION 2.11. For every 1 <<, r < p < e~, there is a sublattice o f (Lr[0, 1] 9 Lr [0, 1] 0 " " ")ep which fails to have the compact approximation property. The lattice Lr[0, 1] itself cannot have any sublattices which fail the approximation property since its sublattices are all order isometric to L p ( v ) , for a suitable measure v. Szankowski's next construction [92] (see [66, Theorem 1.g.2, p. 103]) was of subspaces of p, p r 2, which fail the approximation property. The real significance of this construction is the surprisingly exact representation of the set of vectors in e p which span a subspace failing the approximation property. PROPOSITION 2.12. For every 1 <~ p :/= 2 < cx~, there is a subspace o f g,p which fails the compact approximation property. PROOF. We will work with the case 1 ~< p < 2. We let X be the space of all sequences x -- ( a i ) ~ 4 so that

Ilxll =

Z =-

lail2

BEAn

where An is a carefully chosen partition of {2n, 2 n -+- 1 . . . . . 2 n+l - 1}. Since X is an gpsum of e~'s, it follows that X "~ ep. We let (ei)~xz=4be the unit vector basis of this space and let Z be the closed subspace of X spanned by the sequence Zi =

e2i

--

e2i+l -+- e 4 i

-+-

e4i+l + e4i+2 + e4i+3.

Then Z fails to have the compact approximation property. Let us recall a result of Krivine [58] and Maurey and Pisier [76]. For a Banach space X, if p(X) _ sup{p: X is of type p},

q (x) _ inf{q" X is of cotype q },

then X contains almost isometric copies of s p ( X ) and s q ( X ) , for n - 1 , 2,

....

Because of the

structure of the proof of Proposition 2.12, we can see that any Banach space which contains subspaces uniformly isomorphic to g np . for. n .- 1. 2 .. , will have such a construction on it. This observation combined with the preceding discussion yields a major theorem in this area.

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285

THEOREM 2.13. If X is a Banach space and every subspace of X has the compact approximation property, then X is of type 2 - E and cotype 2 + e, for every e > O. For later reference, we mention that because of the structure of compact sets in sums of Banach spaces, the approximation property passes through gp-sums. PROPOSITION 2.14. If (Xn) is a sequence of Banach spaces with the approximation property, then for every 1 <~ p < cx~ the space (Y-~,I | (and (~-~n | has the approximation property. Now we consider other counter-examples concerning the approximation property. Since there are reflexive spaces (even subspaces of g p, p r 2) which fail the compact approximation property, using the construction of Section 1, following Proposition 1.3, we see that there is a Banach space Z so that Z** has a boundedly complete basis while Z*** fails to have the compact approximation property. It follows that the dual space of the universal space Cl (which has a basis - see Definition 1.10) fails to have the compact approximation property. We also have the construction of Johnson and Schechtman (see [48]) of a subspace of co with a basis whose dual space fails to have the approximation property. Johnson [44] has constructed spaces of the form X - ( ~ n ~_ ~k,, p,,)e2, for carefully chosen Pn $ 2 and kn t c~ so that every subspace of X has the approximation property, but X has subspaces without bases. Casazza, Garcfa and Johnson [10] combined Enflo's construction [23] with Johnson's construction just cited to produce Banach spaces X, Y so that every subspace of every quotient space of both X, Y have the approximation property (even a finite dimensional decomposition) but X @ Y has a subspace failing the compact approximation property. Since there are Banach spaces with bases whose duals fail the approximation property, it follows from Theorem 2.5 and Proposition 2.8 that there exists Banach spaces X, Y and a compact operator T e B ( X , Y) so that T is a r-limit of finite rank operators but T is not a norm limit of finite rank operators. Figiel (unpublished) observed that if X is reflexive and Y is either separable or reflexive, then any compact operator T E B ( X , Y) which is a r-limit of finite rank operators is also a norm limit of finite rank operators. Since there is a Banach lattice which fails the compact approximation property, it follows that U~ fails the approximation property by Proposition 1.12. In particular, there is a Banach space with a 1-unconditional basis whose dual space fails the approximation property. Szankowski [93], in yet another spectacular construction, resolved the much worked on question of the approximation property for B (H). THEOREM 2.15. For infinite dimensional Hilbert spaces H, B ( H ) fails the approximation property. This was the first naturally occurring example shown to fail the approximation property. Godefroy and Saphar [31] observed that B ( H ) / K ( H ) fails the approximation property. It is still unknown at this time if H ~ has the approximation property. In [83,84], Pisier gave a solution to a problem of Grothendieck related to the approximation property. Grothendieck raised the question: If X and Y are Banach spaces for which the injective and projective tensor products coincide (i.e., X + Y - X ~ Y), must either X

286

P G. Casazza

or Y be finite dimensional? Pisier [83,84] constructs an infinite dimensional Banach space X so that X + X -- X ~ X, both algebraically and topologically. Such a space must fail the approximation property. THEOREM 2.16. Any Banach space Y of cotype 2 can be embedded isometrically into a Banach space X of cotype 2 such that (1) X + X - X ~ X . (2) X and X* are both of cotype 2. (3) Every operator from X into X which is in the norm closure of the finite rank operators is nuclear. Moreover, if Y is separable then the same is true for X. It is known [84] that if X has the approximation property and both X and X* are of cotype 2, then X is isomorphic to a Hilbert space. Pisier then used this construction to answer a question of Lindenstrauss" Does every separable Banach space contain uniformly complemented subspaces uniformly isomorphic to g~, ~ or g ~ ?

COROLLARY 2.17. There is an infinite-dimensional Banach space X and a number 6 > 0 such that every finite rank projection P on X satisfies

II P [I >~ 6 (rank P) 1/2. We end this section with a brief discussion concerning the impact of counter-examples to the approximation problem on the study of Banach algebras. A left approximate identity in a Banach algebra A is a net (e~)~eA of elements of A such that limx Ilexx - x II - 0, for all x E A. One consequence of Theorem 2.5 (2) is that if a Banach space X has the approximation property, then f ( X ) (the closure of the finite rank operators in B ( X ) ) has a left approximate identity. The converse is an open question related to Problem 2.7. The other variants of the approximation properties we will examine in later sections also have formulations in terms of (bounded) approximate identities. We will not examine this topic in detail but refer the reader to [11,22,34,89] or books on Banach algebras for the current results on this subject. There has been little attention paid to the approximation property for Banach lattices and whether there are stronger results in this setting. We refer to Nielsen [77] for the most recent results on this subject. We ask PROBLEM 2.18. For a Banach lattice X, does the approximation property imply the positive approximation property? Does the bounded approximation property imply the positive bounded approximation property (see Section 3)?

3. The bounded approximation property The definition of the approximation property (Definition 2.2) puts no restriction on the norm of the finite rank operator which approximates the identity on a compact set. Yet, as

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287

we saw in the case of X having a basis, the basis projections are providing the required approximation and this set of projections is uniformly bounded (independent of the compact set K). Now we introduce this stronger form of the approximation property. DEFINITION 3.1. Let X be a Banach space and 1 ~< )~ < cx~. We say that X has the h - b o u n d e d approximation property (X-BAP for short) if for every e > 0 and every compact set K in X, there is a finite rank operator T (i.e. T E f ( X ) ) so that IITll ~< )~ and IIT x - x II ~< e for every x E K. We say that X has the b o u n d e d approximation property (BAP for short) if X has the )~-BAP, for some ~. Finally, a Banach space is said to have the metric approximation property (MAP for short) if it has the 1-BAP. The compact sets in Definition 3.1 can be replaced by finite sets. To see this, let K be a compact set in a Banach space X and let e > 0 be given. We can find a finite set (Xi)i__ n 1 SO that K C U i ~ l B ( x i , e/3)~). Now, if T is a finite rank operator with IlTxi - xill <<.e / 3 , for 1 ~< i <~ n, then II Z x - x II ~< e, for every x E K. It is immediate that a space with a basis has the bounded approximation property and that a space with the bounded approximation property has the approximation property. The converse of both of these implications is false. For the first implication, we observe that if X has the X-BAP and if P is a bounded linear projection from X onto Y then Y has the X II P IIBAP. Now let X be a reflexive Banach space which is complemented in a space with a basis while X itself fails to have a basis (see Proposition 6.7). Then X has the bounded approximation property but fails to have a basis. A counter-example to the converse of the second implication will be done later in this section. As we have seen in Section 2, a space with a basis has the bounded approximation property and the same argument shows that a space with a monotone basis has the metric approximation property. The converse of these results fails as we will see later in this section. All/2p-spaces have the bounded approximation property. If ( X , ) is a sequence of Banach spaces, let X = (y~ | (or a c0-sum). Then it is immediate that X has the bounded approximation property if and only if the Xn all have the bounded approximation property with a uniform bound on their BAP constants. We now show that finite rank operators approximating the identity close enough on a finite dimensional subspace can be perturbed to equal the identity on that subspace. LEMMA 3.2. Let X be a Banach space, let E be an n-dimensional subspace o f X and let T : X --+ E be onto. Let k ~ n and let F be a k-dimensional subspace o f X such that ]]T[F -- IF[] < e < 1, where (1 - e) -1 ek < 1. Then there is a rank n operator S on X such that SIF = I F , I I S - Tll < (1 - e ) - l e k ] l T l l , and S * ( X * ) = T * ( X * ) . Moreover, if T is a projection then S can be chosen to be a projection. PROOF. By our assumption, U = T IF is an invertible operator on F with 11U [I < 1 + e and II U- 1 II < (1 - e ) - 1. Hence, [IU - 1 _ I IF II < e(1 - e ) - 1. By a result of Kadec and Snobar (see Proposition 9.12 in [96]) we can choose a projection P from E = T X onto T F with IIP II ~< k = dim T F. Let V = U-1 p + l i e - P, and S = V T. Now IIV - l i E II = [[U - 1 P - PII < ke(1 - e) -I < 1. Hence V : E -+ X is one-to-one and so T* and S* have the same r a n g e - since the null space of T is a finite co-dimensional subspace of X. Clearly SIF = IIF and S is a projection if T is. The rest of the lemma is easily derived from here. D

288

P.G. Casazza

As an immediate consequence of L e m m a 3.2 we have the following THEOREM 3.3. The following properties f o r a Banach space X are equivalent: (1) X has the bounded approximation property. (2) There is a uniformly bounded net (T~) of finite rank operators on X which tends strongly to the identity on X. (3) There is a ~ ~ 1 so that for everyfinite dimensional subspace E C X there is a finite rank operator T on X such that IIT ]1 <<.)~ and T x = x, f o r all x ~ E. In the case of separable spaces, we can use induction on this procedure to prove

COROLLARY 3.4. Let X be a separable Banach space. Then X has the h-bounded approximation property if and only if there is a sequence of finite rank operators (Tn) on X converging strongly to the identity so that Tm Tn -- Tn, f o r all n < m, and lim SUPn IITn II <~ ~. We will call such a sequence (Tn) an approximating sequence (or a h-approximating sequence if the ~ is important). What actually occurs in Corollary 3.4 is slightly stronger than we state. In particular, if (Tn) is a sequence of finite rank operators on a separable Banach space X which converges strongly to the identity on X, then there is an approximating sequence (Sn) on X so that for some k l < k2 < ..- lim, ]lTk, -- S, II = 0. Since there are spaces with bases whose duals fail the approximation property, we see that the bounded approximation property does not pass to dual spaces. However, it does pass from dual spaces down to the space. This was observed by Grothendieck [35] for the metric approximation property and by Johnson [41 ] for the general case. We first introduce some notation. A Banach space X is said to have the h-duality bounded approximation property provided there is a net (T~) of finite rank operators on X uniformly bounded by )~ which converges strongly to the identity on X and so that (T*) converges strongly to the identity on X*. Equivalently, X has the )~-duality bounded approximation property provided that, for each e > 0 and each pair of finite dimensional subspaces E of X and F of X*, there is a finite rank operator T on X with IIT II ~< ~, IITx - x ll ~< e]lx II, for all x 6 E and liT*x* - x*ll ~< ellx*l] for all x* 6 F. PROPOSITION 3.5. I f X* has the h-bounded approximation property than X has the )~duality bounded approximation property. PROOF. An application of Helly's Theorem (or local reflexivity) shows that we may assume that X* has the )~-bounded approximation property given by weak-star continuous operators. So choose a net of finite rank operators (T~) on X so that lim sup~ IIT~ II ~< )~ and lima T ' x * = x* for all x* 6 X*. Then (T~) converges weakly to the identity on X and so there are disjoint finite convex combinations of the T~ (which must be chosen carefully since this is only a net) which form a net and converge strongly to the identity on X. [] We will see later in this section that there are Banach spaces with the approximation property which fail the bounded approximation property. First, we recall another important

Approximation Properties

289

result of Grothendieck [35] that for separable dual spaces, the approximation property implies the metric approximation property. We refer the reader to Lindenstrauss and Tzafriri [66] for the best proof of this result. THEOREM 3.6. A separable dual space with the approximation property has the metric approximation property.

There are several important consequences of Theorem 3.6 which we will now explore. THEOREM 3.7. Every reflexive Banach space with the approximation property also has the metric approximation property. PROOF. This is immediate from Proposition 3.5 and Theorem 3.6 in the separable case. For the non-separable case we also need Proposition 9.6. [] In light of the counter-examples to the various forms of the approximation property, we see that Theorem 3.6 is quite a powerful result in that it uses the existence of a large number of finite rank operators on a separable dual Banach space to draw the conclusion that there is an equally large number of uniformly bounded finite rank operators on the space. The proof of Theorem 3.6 has always been a little mysterious. It relies on the fact that for a separable dual Banach space X, the sets Bx and Bx, are compact metric in their respective weak-star topologies. Because of this, Grothendieck's result has never been generalized to non-separable duals. PROBLEM 3.8. If X is a (non-separable)dual space with the approximation property, must X have the metric approximation property? Now we will consider the relationships between the approximation property, the bounded approximation property and the metric approximation property. The main example here comes from a construction of Figiel and Johnson [27]. PROPOSITION 3.9. Fix 1 <~ )~ < cx~. If X is a Banach space which has the )~-bounded approximation property in every equivalent norm, then X* has 2)~(1 + 4)~)-bounded approximation property. In the case of the metric approximation property, Proposition 3.11 was strengthened by Johnson [42] to: If X has the metric approximation property in every equivalent norm, then X* has the metric approximation property. If X is a Banach space with a separable dual X* having the approximation property, then for every equivalent norm I" I on X, (X*, 1. I) still has the approximation property and hence the metric approximation property. In light of Proposition 3.5 we now have THEOREM 3.10. Let X be a Banach space with a separable dual space. The following are equivalent: (1) X* has the approximation property.

290

P.G. Casazza

(2) X* has the metric approximation property. (3) X has the metric approximation property in all equivalent norms. In particular, co has the metric approximation property in every equivalent norm. Proposition 3.9 will allow us to construct several important examples. COROLLARY 3.1 1. (1) There is a Banach space which has the bounded approximation property but fails the metric approximation property. (2) There is a separable Banach space (which even has a separable dual) with the approximation property which fails to have the bounded approximation property. PROOF. Let X be a separable Banach space with a basis whose dual is separable and fails to have the approximation property (see Proposition 1.4). By Proposition 3.11, for every natural number n, there is an equivalent norm I. I, on X so that (X, I" I,) fails the n-bounded approximation property. Now (X, l" I,) is a Banach space with the bounded approximation property which fails the metric approximation property. Also, (y~, q)(X, I" I,))e2 fails the bounded approximation property (see the discussion following Definition 3.1) but has the approximation property by Proposition 2.14. D Left open here is another important question. PROBLEM 3.12. If a Banach space X has the bounded approximation property, must X have the metric approximation property in an equivalent norm? Does g l have the metric approximation property in every equivalent norm? Does g~ have the metric approximation property in every equivalent dual norm? There is a very nice classification of spaces which have the bounded approximation property proved independently by Petczyfiski [80] and Johnson, Rosenthal and Zippin [50]. Although we cannot prove this result until Section 6 (see Proposition 6.9), we will state it now.

THEOREM 3.13. A separable Banach space has the bounded approximation property if and only if it is isomorphic to a complemented subspace of a space with a basis. Moreover, if X is reflexive and has the bounded approximation property then X is isomorphic to a complemented subspace of a reflexive space with a basis. The classifications of the approximation property due to Grothendieck and appearing in Theorem 2.5 have corresponding formulations for the bounded approximation property. The proof is the corresponding bounded versions of the proof of Theorem 2.5. THEOREM 3.14. For a Banach space X, the following statements are equivalent: (1) X has the )~-bounded approximation property. (2) For every Banach space Y, the finite rank operators from Y to X of norm <~)~ are r-dense in the unit ball of (B(X, Y), r).

Approximation Properties

291

<

(3) For every K > O, every Xn E X, x* E X* with Y~n IlXn IIIIx~ II ~ , and satisfying I ~ x ~ ( T x ~ ) l <~ KIITII for every finite rank operator T on X, must also satisfy )~g.

I~ x~,(xn)l <.

There are unconditional forms of the metric approximation property [13]. The motivation for this concept comes from the observation that if X is separable and has an approximating sequence (Tn), then letting An = Tn - Tn-1 (where To = 0 ) we have for all x 6 X, x = Y~n An (x). We say that a separable Banach space X has the unconditional metric approximation property (unconditional MAP for short) if X has an approximating sequence (Tn) for which l i m n ~ I I I - 2Tn II = 1. The justification for this terminology comes from the following result of Casazza and Kalton [ 13]. THEOREM 3.15. A separable Banach space X has the unconditional metric approximation property if and only if for every e > 0 there is an approximating sequence (Tn) so that if An = Tn - Tn-1 (and To = O) then

sup

~OiAi[I

~ 1 +s.

N,On=-+- I i=1

4. The commuting bounded approximation property As we will see, having an approximating sequence (Tn) whose elements commute will yield much stronger results. This property was first isolated by Johnson and Rosenthal

[51]. DEFINITION 4.1. A Banach space X has the )~-commuting bounded approximation property ()~-commuting BAP for short) if there is a net (T~)~eI of finite rank operators on X converging strongly to the identity such that lim sup~ IIT~ II ~< )~, and for all or, fl we have T~ T/~ = Tr T~. We say that X has the commuting bounded approximation property (commuting BAP for short) if it has the )~-commuting bounded approximation property for some X~l.

Clearly the commuting bounded approximation property implies the bounded approximation property. The converse of this fails in the non-separable case (see Corollary 9.4) and is an open question in the separable case. PROBLEM 4.2. Does every separable Banach space with the bounded approximation property have the commuting bounded approximation property? Later in this section we will see that, in the separable case, the metric approximation property implies the commuting metric approximation property. In the case of separable spaces, we can greatly simplify the commuting bounded approximation property. This comes from a result of Casazza and Kalton [ 13]. Recall the definition of a )~-approximating sequence on X given in Corollary 3.4.

P.G. Casazza

292

PROPOSITION 4.3. Let X be a separable Banach space with a ~-approximating sequence (Tn). If ~--~ m [[TnTn+l - Tn+l Tn 1[ < oo, then X has a )~-approximating sequence ( S n ) satisfying Sn Sm = Smin(n,m), for all n ~ m. As an immediate consequence we have THEOREM 4.4. For a separable Banach space X, the following are equivalent: (1) X has )~ the commuting bounded approximation property. (2) There is a s sequence (Tn) on X with Tn Tm = Tmin(n,m), for n =/=m. (3) There is a s sequence (Tn) on X with

lim llZn Zm - Zm Tn ll = 0,

for all m.

n

(4) There is a s

sequence (Tn) on X with

lim IITn Tm - Tm Tn [I = O. f/,m

Problem 3.13 asks if the bounded approximation property implies the metric approximation property in an equivalent norm. Johnson [42] showed that the answer is yes for the commuting bounded approximation property. PROPOSITION 4.5. Every separable Banach space with the commuting bounded approximation property can be renormed to have the commuting metric approximation property. PROOF. Let (Tn) be an approximating sequence satisfying Tn Tm = Tmin(n,m), for n ~ m. We define a new norm on (X, [[. [[) by: [x[ = SUPn [[Tnx[[. For each natural number n let In(n-k-l)

S~=-

1 n

~

Ti,

i=l (n-1)n+l

Then (Sn) is an approximating sequence, with SnSm = Smin(n,m) ISnl---> 1.

for n ~ m and D

Casazza and Kalton [13] gave a strong converse to Johnson's result. THEOREM 4.6. Let X be a separable Banach space with the metric approximation property. Then X has the commuting metric approximation property. PROOF. Let X be a Banach space and (Tn) be an approximating sequence (recall that this implies that Tm Tn = Tn for m > n) satisfying I]Tn II <~ 1 + an, where Y~n an = / 3 < oo. For t > 0 define the operators

Vn(t ) -e

- n t exp

t

Tk k=l

--e-n' Z j--0

~(T1 +".+

Tn) j.

Approximation Properties

293

Then

IIW~(t)II <~e-ntexp( t~[lTkll)k-O <~ e/~t"

One shows that S(t)x = limn~e~ Vn(t)x, exists for all x e X. A calculation shows that S(t) is compact for all t > 0. Also, if x E Tn (X) then S(t)x = Vn (t)x --+ x, and as t ~ 0, IIS(t)ll --+ 1. Finally, since X has the metric approximation property, there are finite rank operators Rn so that IIRn - S(1/n)ll--+ O. Since S(t) has the semigroup property (i.e., S(t + s) = S(t)S(s)), the S(1/n) satisfy the requirements of Theorem 4.4(3). D The previous two results show that the metric approximation property has an isomorphic equivalent form: the commuting bounded approximation property. There is also a reverse metric approximation property. We say that X has reverse metric approximation property if there is an approximating sequence (T,) with l i m , ~ III - Tnl[ -- 1. It can be shown [13] that the reverse metric approximation property implies the commuting bounded approximation property. We also have the following result of Casazza and Kalton [ 13] concerning renormings of a space with the commuting bounded approximation property which we will need later. THEOREM 4.7. Let X have the commuting bounded approximation property and 1 <<,ot <

2. Then X can be renormed to have a commuting approximating sequence (Tn) with lim n----~ o o

IIT~ II :

lim n----~ o G

III

-

ot

Tn II =

l,

and

lim sup] Tn- T~J ~ .

1

n---~ o o

In particular, X can be renormed to have simultaneously the metric approximation property and the reverse metric approximation property. As we have seen, it is an open question whether the bounded approximation property implies the commuting bounded approximation property. Now we turn our attention to some cases where this conclusion is valid. The following result is due to Johnson, Rosenthal and Zippin [50], although it was tacitly contained in earlier work of Johnson [41 ]. PROPOSITION 4.8. If X is separable and X* has the )~-bounded approximation property

(or if X has the )~-bounded approximation property and is isomorphic to a dual space) then X has the X commuting bounded approximation property. In particular, every separable reflexive Banach space with the approximation property has the metric commuting bounded approximation property.

294

P G. Casazza

PROOF. Assume X* has the bounded approximation property. By Proposition 3.5, there is a net (Tc~) of finite rank operators on X which converges strongly to the identity on X while its net of adjoints (T*) converges strongly to the identity on X*. By induction, we extract a sequence (Tn) from the net with the property that limm--,~ IITm Tn - Tn II -- 0 and limm~ec IIT* T* - T* I I - 0 for all n = 1, 2 . . . . . Taking adjoints through the second limit we see that limm--,oc IITnT m - Tn II - 0 , for all n = 1, 2 . . . . . and so the hypotheses of Theorem 4.5(3) are satisfied. The other case is handled similarly. For the last part we just apply Proposition 2.6 and Theorem 3.6. M Zippin [99] has shown that a Banach space has a separable dual if and only if it embeds into a space with a shrinking basis. Combining the above results, we obtain the complemented version of Zippin's theorem. THEOREM 4.9. Let X be a Banach space with a separable dual. The following statements are equivalent: (1) X* has the bounded approximation property. (2) X embeds complementably into a space with a shrinking basis. (3) X has the shrinking commuting bounded approximation property. That is, there is a commuting approximating sequence (Tn) on X with (T*) converging strongly to the identity on X*. PROOF. It is immediate that (2) =~ (3) =~ (1). The implication (1) =~ (2) follows from Proposition 6.9 (or from the results of [50]). D Our next result [ 13] is useful for our constructions in Section 6. PROPOSITION 4.10. Let X be a separable Banach space and suppose there is a separable reflexive Banach space Y so that X 9 Y has the commuting bounded approximation property. Then X has the commuting bounded approximation property.

PROOF. Let (Sn) be an approximating sequence for X ~3 Y with SnSm = Smin(n,m), for all n 7~ m, and let P be the projection of this space onto X. Since (P Sn) converges strongly to zero on the reflexive space Y we have that limn (I - P)* (P Sn)*X* = 0, weakly for every x* 6 X*. It follows that PSn (I - P) converges weakly to zero in B(X, Y). Thus, we may pass to convex combinations Rn of (Sn) (which is still an approximating sequence for X • Y) but so that ( P R n ( I - P)) converges to zero in norm. Letting Tn = PRn, a calculation shows that limn,m-+~ IITn Tm - Tm Tn II - 0. Applying Theorem 4.4(4) finishes the proof. D We end this section with the unconditional version of the commuting metric approximation property similar to the unconditional metric approximation property encountered in Section 3. We say that X has the commuting unconditional metric approximation prop-

Approximation Properties

295

erty if for every s > 0 there is a commuting approximating sequence (Tn) for which if An = Tn - Tn-1 (where To = 0 ) w e have

sup

N,Oi=-+-I

~OiAi

~
i=1

There is a good classification of the commuting unconditional metric approximation property due to Godefroy and Kalton [30]. THEOREM 4.1 1. Let X be a separable Banach space with the approximation property.

Then the following conditions are equivalent: (1) X has the unconditional metric approximation property. (2) X has the commuting unconditional metric approximation property. (3) For every s > O, X is isometric to a 1-complemented subspace of a Banach space with a (1 + s)-unconditional finite dimensional decomposition.

5. The re-property In Section 3 we observed that if X has a basis, then it has the bounded approximation property with an approximating sequence given by projection operators. Lindenstrauss [61 ] isolated this property which today is called the re-property. DEFINITION 5.1. A Banach space X is said to have the rez-property if there is a net of finite rank projections (S~) on X converging strongly to the identity on X with lim supc~ IlSc~II ~< )~. The case )~ = 1 is called the metric re-property. A space with the rezproperty for some )~ is said to have the re-property. It is clear that every space with a basis has the the re-property given by the basis projections. The converse is false due to the example of Szarek [95] (see Proposition 6.7). Clearly, the re-property implies the bounded approximation property. The converse of this fails due to an important example of Read [86] of a reflexive Banach space with the bounded approximation property (and hence the commuting bounded approximation property) which fails the re-property. Actually, Read's space gives more. PROPOSITION 5.2. There is a complemented subspace of fails the re -property.

C2 (see Definition

1.10) which

We ask PROBLEM 5.3. For a separable Banach space, does the re-property imply the commuting bounded approximation property? Separable is needed here since in Section 9 we will show that this implication fails for non-separable spaces. In the case of the metric re-property, we do have an answer to Problem 5.3 due to Johnson [39].

296

P.G. Casazza

PROPOSITION 5.4. If X is separable and has the metric re-property, then X has the commuting metric re-property (and hence the finite dimensional decomposition property). PROOF. Let (P/7) be an approximating sequence of finite rank projections on X with 17 IIPnll = 1 + en and ]-In~__l(1 + e/7) < cx~. For n > j let Sj = PjPj+l "" Pn. For each j , Sj = l i m n ~ Sj exists pointwise and gives a commuting sequence of finite rank projections on X converging strongly to the identity. D Applying L e m m a 3.2 we obtain a stronger form of the zr-property for separable spaces. PROPOSITION 5.5. For a separable Banach space X, the following are equivalent: (1) X has the rex-property. (2) There is a X-approximating sequence of finite rank projections (Sn) on X for which SmSn -- Sn, for all m ~ n. The re-property does not pass to the dual space since there are Banach spaces with bases whose dual spaces fail the approximation property (see Section 1). The next result shows that the only reason the re-property does not pass to a dual space is that the dual space may fail the bounded approximation property. This is a result of Johnson, Rosenthal and Zippin [501. THEOREM 5.6. If X* has the rex-property, then so does X. If X has the rex-property and X* has the bounded approximation property, then X has the shrinking reu-property for some #. That is, there is a net of finite rank projections (S~) converging strongly to the identity on X so that (S*) converges strongly to the identity on X*. PROOF. The severe technicalities of the proof in [50] come from trying to give the best calculation for the respective )~. If we ignore this, we can get quickly to the basic idea of how one produces projections on X from projections on X*. If X* has rex-property then by Proposition 3.5, X has the bounded approximation property. So if E is a finite dimensional subspace of X and F is a finite dimensional subspace of X*, we can choose a finite rank operator S on X with IISII ~< x + 1 and so that SIE = IE and then choose a finite rank projection Q on X* with IIQII ~< )~ + 1 and with QIF = IIF and S*(X*) C Q(X*). A simple application of local reflexivity to L e m m a 3.2 shows that we may as well assume that there is a projection P on X with P* = Q. Now, P ' S * = S* implies that S = S P. This fact yields that the operator S + P - S P is a projection on X which is the identity on E and its adjoint is the identity on F. If we consider the set of pairs (E, F) so that E is a finite dimensional subspace of X and F is a finite dimensional subspace of X*, this becomes a net when it is ordered by inclusion of the subspaces. By the above, we can construct projections PE,F on X with PE,F -- I on E and P{,F -- I on F. It follows that (PE,F) gives the shrinking re-property. Assume now that X has the rex-property and X* has the bounded approximation property. If we choose finite dimensional subspaces E C X and F C X*, we can find a finite rank operator S on X and a finite rank projection P on X so that PIE -- I[E and S*[F -- IF

Approximation Properties

297

and and the norms of the operators are functions of)v and the bounded approximation property constant alone. Now, if Q = S + P - SP, then Q is our finite rank projection on X which equals the identity on E and Q* = S* + P* - P* S* is our finite rank projection on X* which equals the identity on F. [5 The central theme in the 7r-property is whether we can construct large classes of uniformly bounded finite rank projections on a Banach space. Read's example [86] says that even if we have a large number of uniformly bounded finite rank operators on a space (even a commuting family) this is not enough to induce a large number of projections on the space. So we now turn to the question of constructing projections on a Banach space. It is not surprising that a bounded operator which is sufficiently close (in norm) to its square will induce a projection on a Banach space. The exact measure of closeness required was found by Casazza and Kalton [ 13] and is surprisingly large. PROPOSITION 5.7. Let X be a Banach space, T a bounded linear operator on X, and assume that I ] T - T2[] = 0 < 1/4. Then there is a projection P on X such that {x: T x = x } C P ( X ) C T ( X ) and 1( I+2]ITII ) IIPII <~ ~ 1 + ( 1 - 4 0 ) 1 / 2 " PROOF. Define

m,

S

mzO

and P--

21(I - (I - 2T) S)

Power series manipulation and some calculating show that this works. Now we have the result of Casazza and Kalton showing that 1/4 is the best constant. THEOREM 5.8. Let X be a separable Banach space with an approximating sequence (Tn) f o r which limsuPn__+oo ]]Tn - T2]] < 1/4. Then X has the 7c-property. Moreover, there is a Banach space X which fails the 7c-property, but f o r which there is an approximating sequence (Tn) on X satisfying 1 lim sup] Tn - T2[] -- ~. n---+ oo

PROOF. By Corollary 3.4 and the statements following it, we may assume that our approximating sequence (7",) has the additional property that Tm Tn = Tn, for all m > n. Now the result follows easily from Proposition 5.7. For the second part of the theorem, we observe that Read's example [86] of a reflexive Banach space with the commuting bounded approximation property but failing the 7r-property, has an approximating sequence (T,) with lim s u p , ~ I]Tn - T2 ]l -- 1/4 by Theorem 4.7. U

298

P G. Casazza

An important tool for constructing finite rank projections on a Banach space is due to Johnson [40]. This construction, and its variations, have had widespread application in the area. So we will now consider this technique in detail. Recall the definition of the universal spaces with a basis Cp given in Definition 1.10. THEOREM 5.9. I f X has the bounded approximation property, then X 9 Cp has the yrproperty, f o r all 1 <<,p < ~ and p = O. PROOF. Recall that Cp = (Y~i ~[~Ei)gp where the Ei are dense in all finite dimensional spaces. Fix a finite dimensional subspace F of X and let H -'- Y ~ in: I OEi for a fixed n. Let T be a finite rank operator on X so that T I F - - I and IIT[I ~< )~ + 1. Let E = (I - T ) T ( X ) and choose an m > n so that d (E, Em) < 2, and let L be an isomorphism from E onto Em with IlL II = 1 and ILL-1 II < 2. Let P be the natural norm 1 projection of C p onto H and let Pm be the natural projection of Cp onto Em. Define S : X 9 Cp --+ X @ Cp by:

S(x, y) - ( T x + L -1 PraY, L ( I - T ) T x + L ( I - T ) L -1 PmY + PY). Then S is a finite rank projection on X G C p which is the identity on both F, H and with norm a function of the norm of T alone. [] In the next section we will see some strengthenings of Theorem 5.9. In particular, if X has the bounded approximation property then X @ C ~ has a basis. If X has the commuting bounded approximation property then X @ C p has a basis for all 1 ~< p ~< oo. A consequence of Theorem 5.9 and Proposition 4.10 is that Problems 4.2 and 5.3 are equivalent in the sense that either both have positive answers or both have negative answers. There are many variations of the above construction which have been adapted for special uses. Casazza [7] used a variation of this technique to show that James' space J is primary. Casazza [8] gave another variation of this technique showing that whenever Y is a complemented subspace of a reflexive Banach space X with a (sub) symmetric basis then X r Y has a basis. Yet another variation of this method was used by Casazza [9] to show that for separable Banach spaces X with the commuting bounded approximation property, there is a subspace Y of X so that both Y and X r Y have finite dimensional decompositions.

6. The finite dimensional decomposition property A Schauder basis decomposes a Banach space into a direct sum of one-dimensional subspaces. A cruder form of this is to decompose the space into finite dimensional subspaces. DEFINITION 6.1. A sequence (En) of finite dimensional subspaces of a Banach space X is called a (unconditional)finite dimensional decomposition for X (FDD (UFDD) for short) if for every x ~ X there is a unique sequence Xn ~ En so that x -- Y~n Xn (and this series converges unconditionally). In this case we will write X = Z n ~ En and say X has a FDD (finite dimensional decomposition).

Approximation Properties

299

As observed in the Basic Concepts Chapter of this Handbook [47], all the basic properties for bases hold in this setting. The decomposition constant is sup~ [1Pn 1] < oc, where the Pn are the natural decomposition projections given by Pn (y~oc n Xi. We i=1 X i ) = ~i=1 have the notions of shrinking and boundedly complete finite dimensional decompositions and the standard results that a unconditional finite dimensional decomposition for X is shrinking if and only if el does not embed into X and that it is boundedly complete if and only if co does not embed into X. It is clear that every space with a basis has a finite-dimensional decomposition (even into 1-dimensional subspaces). The converse is false as Szarek [95] has constructed a space with a finite dimensional decomposition which fails to have a basis. Also, a complemented subspace of a space with a finite dimensional decomposition (or even a space with a basis) need not have a finite dimensional decomposition. This is an example of Read [86] (see Proposition 6.7). It is also clear that a space with a finite dimensional decomposition has the Jr-property given by the finite dimensional decomposition projections. The converse is an open question. PROBLEM 6.2. Does every separable Banach space X with the Jr-property have a finite dimensional decomposition? We see from the definitions that X has a finite dimensional decomposition if and only if there is an approximating sequence (P~) of commuting projections on X, and in this case the finite dimensional decomposition is just X = ~-,n G(Pn - P n - l ) ( X ) where P0 = 0. Casazza [9] isolated these two properties as classifying spaces with finite dimensional decompositions. THEOREM 6.3. A separable Banach space has a finite dimensional decomposition if and only if it has both the commuting bounded approximation property and the Jr-property. PROOF. Let (jrn) be an approximating sequence of finite rank projections on X and let (Tn) be a )~-commuting approximating sequence of finite rank operators on X (so Tn Tm -rmin(n,m), for all n 7~ m). Let Y be

Y -- U T~X* -- span U T~x*. The second equality follows from the fact that Tin*T~* = T~*, for all m > n. This property also yields that (T~*) is a commuting approximating sequence for Y. By Lemma 3.2, we can choose n l < n2 < ... and finite rank operators (Si) on X with (1) IIsi II ~< 2)~. (2) Si Jri = Jri .

(3) S*l (X*)= T~ (X*) c Y. Now let Qi -- Jri Si. A direct calculation shows that the Qi are projections which by (2) above converge strongly to the identity on X. Now, even though X* may fail the approximation property, we can use Lemma 3.2 to mimic the construction in the proof of Theorem 5.6 using Y in place of X*. Now, by induction we can find uniformly bounded finite

300

P.G. Casazza

rank projections Pn on X so that Pm Pn = Pn and Pm*P* - Pn for all m >~ n, and Pn*(X*) C Y. Finally, (Pn) is the required commuting sequence of finite rank projections on X converging strongly to the identity on X and (Pff) converges strongly to the identity on Y. [3 This result makes Problem 6.2 equivalent to Problems 4.2 and 5.3. In this setting, Proposition 5.4 and Proposition 3.8 can be formulated as THEOREM 6.4. A separable Banach space X with any of the following properties has a finite dimensional decomposition: (1) X has the metric Jr-property. (2) X* has the re-property. (3) X has the re-property and is isomorphic to a dual space. In particular, every separable reflexive space with the Jr-property has a finite dimensional decomposition. (4) X has the 7r-property and X* has the bounded approximation property. The importance of finite dimensional decompositions does not stem from the fact that they have properties common with spaces with bases. Their importance comes from the fact that there are results for spaces with finite dimensional decompositions whose analogues are either false or unknown for spaces with bases. For example, Johnson [44] has constructed a Banach space X with the property that every subspace of X has a finite dimensional decomposition, but not every subspace has a basis. Also, it is unknown if every separable weak Hilbert space has a basis. But Maurey and Pisier (see [74]) have shown that every separable weak Hilbert space has the 7r-property, and hence a finite dimensional decomposition. It is unknown if every separable Banach space has a subspace with a basis so that the corresponding quotient space has a basis. But it is known [51 ] that every separable Banach space X has a subspace Y so that both Y and X~ Y have a finite dimensional decomposition. Moreover, if X* is separable, both these finite dimensional decompositions can be chosen to be shrinking. The above constructions can be found in Lindenstrauss and Tzafriri [66]. Also, the above constructions use an important technique in this area called "blocking techniques" which can also be found in [66]. One use for finite dimensional decompositions is in the construction of Banach spaces with bases. PROPOSITION 6.5. Let X = Y~n GEn be a finite dimensional decomposition (with FDD constant K ) f o r a Banach space X. If each (En) has a basis (xn)~" 1 with basis constant bounded by M, then ((xn)~n 1)n~__l is a basis for X with basis constant ~ K M . This yields our earlier stated result of Petczyfiski [80] and Johnson, Rosenthal and Zippin [50]. THEOREM 6.6. For a separable Banach space X the following are equivalent: (1) X has the bounded approximation property. (2) X has a finite dimensional expansion of the identity. (3) X is isomorphic to a complemented subspace of a space with a basis.

Approximation Properties

301

PROOF. (2) =~ (1) is obvious and (3) =~ (2) follows from the proof of Proposition 1.2. So let X have (1) and we prove X is complemented in a space with a basis. By the argument of the proof of Proposition 1.2 (where we now allow the operators Ai to be just finite rank) we conclude that X is complemented in a space Y with a FDD, say Y ~- Y]~n @En. By Propositions 1.5 and 6.5, we conclude that Y • C p has a basis. M Now we will take a closer look at the examples of Szarek [95] and Read [86]. PROPOSITION 6.7. (1) There is a complemented subspace of C2 with a finite dimensional decomposition which fails to have a basis. (2) There is a complemented subspace of C2 which fails the Jr-property. PROOF. These constructions are quite difficult and cannot be covered here. But we will discuss at an intuitive level how they are carried out. We choose a sequence (En) of finite dimensional Banach spaces with the property that the basis constant of En G ~2 is >/n. To get a space with a finite dimensional decomposition which fails to have a basis, we take X -- (y-~'GEn)e2. It follows that X is a complemented subspace of C2. We now have a space with a unconditional finite dimensional decomposition and it cannot have a basis taken from the En's. The main problem is that this space might have a basis coming from Proposition 1.4. To prevent this from happening, we choose the En's carefully as subspaces of s with Pn ,~ 2. The main point then is to show that we can pick the En's so that En @ s fails to have a good basis, and choose p~ so that any subspace of s of dimension n-1

~< Y]~i=l dim Ei is a good Hilbert space. Now, in order to build a good basis for E~, the span of {E1 . . . . . En- 1} is too small to be of help, while elements chosen from the span of En+l . . . . have the property that any set of dim En of these elements are spanning a good Hilbert space and therefore cannot help either. For (2) we have the same subspaces En, only now we need to glue them together more carefully. We now need to form the En's into a Banach space in such a way that we do not introduce new projections. We do this by overlapping the En's carefully and gluing them together in such a way that there are uniformly bounded finite rank operators passing through the overlaps but no good projections. This program is carried out by carefully embedding the E~ 's (with overlap) into C2. This is a very delicate operation and the complexity of Read's paper reflects this. In both cases above, the space X we construct is isomorphic to a complemented subspace of C2 and X has the bounded approximation property. Since C2 is reflexive, X has the commuting bounded approximation property and so by Proposition 6.8, X | C2 has a basis. That is, in both cases above, our space X is complemented in a space with a basis. E3 Proposition 6.5 does not hold for unconditional finite dimensional decompositions since the Kalton-Peck space [54] has a unconditional finite dimensional decomposition into two dimensional subspaces but fails to have a unconditional basis (or even be complemented in a space with a unconditional basis). However, a space X has a unconditional finite dimensional decomposition if and only if it has a unconditional finite dimensional expansion of the identity. Also, every Banach space with a unconditional finite dimensional decomposition embeds into a Banach space with a unconditional basis (see [66, Theorem 1.g.5, p. 51 ]). We ask

302

P.G. Casaba

PROBLEM 6.8. If X has a unconditional finite dimensional decomposition and X has GLLUST, must X have a unconditional basis? Casazza and Kalton [14] have given a partial positive answer to Problem 6.8: If X has GL-LUST and a unconditional finite dimensional decomposition X = ~ O En with SUPn dim En < ~ , then X has an unconditional basis (whose basis elements may be chosen from the En's). In particular, what the Kalton-Peck space [54] is missing for it to have a unconditional basis is GL-LUST. Another general method for constructing bases uses the universal spaces C p of Definition 1.10. This is an immediate consequence of the construction of Johnson (see Theorem 5.9). PROPOSITION 6.9. If X is separable and has commuting bounded approximation property, then X G Cp has a basis, for all 1 ~ p <<,cx~. Moreover, if X G Cp has a basis for some 1 < p < cx~, then X has the commuting bounded approximation property. PROOF. By Theorem 5.9, X 9 Cp has the Jr-property. Since it clearly has the commuting bounded approximation property, we could apply Theorem 6.3 to show it has a basis. However, if we go back to the proof of Theorem 5.9, we can see directly that the projections constructed there are commuting as long as our original approximating sequence (Tn) was commuting. Either way, X @ C p has a finite dimensional decomposition ~n G En. But Cp ~- Cp @ Cp and so we just need to observe that ( ~ n GEn) G Cp has a basis. But this is a simple application of Propositions 1.5 and 6.5. The moreover part of the proposition follows from Proposition 4.10. [] An important question is whether Proposition 6.8 holds (for 1 < p < cx~) with only the assumption that X has the bounded approximation property. By Proposition 4.10, this question is equivalent to Problems 4.2 and 5.3. Lusky [68] gives a positive answer to this question for p = 0 (see also Lusky [70]). PROPOSITION 6.10. If X is separable and has the bounded approximation property, then X G Co has a basis. It was well known for some time (see [95] for a proof) that every separable space X containing co has the local basis property. That is, there is a K >~ 1 and finite dimensional subspaces E1 C E2 Q ... of X with U En dense in X and each Ei has a basis with basis constant ~< K. If X is a Banach space for which every subspace has the local basis property, then there is a constant K > 0 so that for every finite dimensional subspace E of X there is another finite dimensional subspace F of X so that span (E U F) ~2 E G F and E • F has a basis with basis constant ~< K. That is, if every subspace of X has the local basis property that every finite dimensional subspace of X is well complemented in a finite dimensional subspace of X with a good basis. One consequence of Lusky's proof of Proposition 6.10 is that every/21-space has this property, i.e., every subspace of a s has the local basis property.

Approximation Properties

303

PROPOSITION 6.1 1. There is a constant C > 0 so that f o r every 1 <<, p <, 2 and every finite dimensional subspace E o f L p[O, 1], there is a finite dimensional subspace F o f ~1 so that the basis constant o f E q) l F is <, C. PROOF. Let (E,)n~=l be a sequence of finite dimensional quotient spaces of ~ for m = 1, 2, 3 . . . . which are dense (in the Banach-Mazur distance) in the family of all finite dimensional quotients of s Let

Y =

GEn n-- 1

9 co

Then Lusky [68, Corollary 2.2] shows that the space Y has a shrinking basis (xn, xS) with basis constant, say M. In particular, Y* has a basis (x*). But

Y* '~

@E* n--1

61

is isomorphic to a subspace of el and (E*) is dense in the family of all finite dimensional subspaces of g l. Also, g p is uniformly finitely representable in ~1 with constant, say K. Hence, for any finite dimensional subspace E of L p[O, 1], E is K-isomorphic to some space E* which in turn is 1-complemented in the space Y* with a basis. This is all we need for the proposition. [] COROLLARY 6.12. There is a subspace X o f g~l with a basis so that X* fails the approximation property. PROOF. Let X = Y* from the proof of Proposition 6.11. Now, Y has a basis and we claim that Y* fails the approximation property. To see this, choose a subspace Z of el which fails the approximation property and choose a sequence of finite dimensional subspaces Fl C F2 C ... of Z whose union is dense in Z. Then there are natural numbers (kn)~ec=1 so that Fn ~ 2 Ek*' 9Hence if

w-

eF, n=l

61

then by the proof of Proposition 1.12 we have that W* fails the approximation property and clearly W is isomorphic to a complemented subspace of Y. So Y fails the approximation property. [-1 We do not know if every subspace of L p[O, 1] has the local basis property for any 1 < p 7~ 2 < ec. These results concern constructing (local) bases from inside a Banach space. In general, this is a very delicate operation. There are a few bright spots here however. Let X be reflexive and have a symmetric basis and let Y be a complemented subspace

304

P.G. Casazza

of X. Then X G Y has a basis [8]. Also, if X is separable and has the commuting bounded approximation property, then there is a subspace Y of X so that both Y and X @ Y have finite dimensional decompositions [9]. However, Mankiewicz and Nielsen [71 ] have shown that there is a superreflexive space with a finite dimensional decomposition such that for every subspace Y of X, the space X ~3 Y fails to have a basis. Finally, if Y is any reflexive subspace of a space with a unconditional finite dimensional decomposition, then Y is isomorphic to a complemented subspace of a space with a unconditional finite dimensional decomposition if and only if Y has the approximation property [24]. In general, it is very difficult to discover if a space has a unconditional finite dimensional decomposition even if we assume it has a basis. PROBLEM 6.13. If X is complemented in a space with an unconditional basis, does X have a unconditional finite dimensional decomposition? What if we assume that X has a finite dimensional decomposition? Problem 6.11 is unsolved even in concrete cases such as the Orlicz spaces or the Lorentz spaces. There is one positive but technical result due to Alspach and Carothers [3] which solves this problem in very special circumstances. We end this section with a more general concept of an infinite decomposition for a Banach space. A sequence (X/7) of closed subspaces of a Banach space X is called a Schauder decomposition for X if for every x E X there is a unique sequence x/7 ~ Xn so that x - ~ n Xn. If the Xn are finite dimensional, this becomes a finite-dimensional decomposition. Many basis properties generalize trivially to this setting. For example, we have a uniformly bounded sequence of projections Pn Y~i=l xi) Y~i=l xi Conversely, every bounded sequence of projections (Pn) on a Banach space X for which Pn Pm = Pmin(n,m) and l i m n ~ Pnx = x for all x 6 X determines a Schauder decomposition Xn = (Pn - P n - 1 ) ( X ) for X (where P0 - 0). We also have the notion of shrinking and boundedly complete and unconditional decompositions. It was an open question until just recently whether every separable Banach space must have a Schauder decomposition. A counterexample was observed by Allexandrov, Kutzarova and Plichko [2]. OO

m

/7

.

PROPOSITION 6.14. There is a separable Banach space with no Schauder decomposition. PROOF. Since the hereditarily indecomposable space of Gowers and Maurey [33] contains g~ uniformly, it must have a subspace X failing the approximation property. If (X/7) is a Schauder decomposition for X, then there is an no so that X/7o is infinite dimensional (otherwise, this would be a finite dimensional decomposition for X and so X would have the approximation property). Now, the natural projection onto X/70 decomposes X which is a contradiction. 71

7. The uniform approximation property The property which corresponds to the bounded approximation property in the local theory of Banach spaces is the uniform approximation property (UAP). The uniform approxima-

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305

tion property was introduced by Petczyfiski and Rosenthal [81] where they showed that L p(l z) has it for all 1 ~< p ~< cx~. DEFINITION 7.1. A Banach space X has the uniform approximation property (UAP for short) if there is a constant K and a function f (n) on the natural numbers so that for all n and all n-dimensional subspaces E C X, there is an operator T : X --+ X with rk T ~< f (n) such that 11T1[ ~< K and T [E = I[E. If for every 0 < e < 1, we can replace the K above by 1 + e, we say that X has the metric UAR And if the operators T can be chosen to be projections, we say that X has the uniform projection approximation property (UPAP for short). Given K > 1, let

kx(K,n)

-

sup

-

inf{rk T" T e L(X), IITII <~ K, TIE -- liE].

E C X, dim E = n

So X has the uniform approximation property if and only if there is a constant K such that k x ( K , n) is finite for all n, and in this case we say that X has the K-UAP. The uniform approximation property is much harder to check than the other approximation properties. The list of spaces known to have this property is quite small and includes only a few large classes of spaces: the L p-spaces and the reflexive Orlicz spaces [65] (both classes actually have the UPAP), and spaces obtained from these. Samuel [89] shows that ( ~ OLp)eq has the uniform approximation property. This was generalized by Heinrich [37] (see Theorem 7.5 below). A deep result of Jones [52] is that H 1 has the uniform approximation property. Szankowski [94] shows that the matrix spaces Sp fail the uniform approximation property (for p > 80). It is unknown if the disk algebra has the uniform approximation property. If true, this would be even more useful and interesting in light of the fact that it is unknown if H ~ has the approximation property. Clearly, if H is a Hilbert space then k/4 (1, n) = n. Hence if X is isomorphic to a Hilbert space, then there is a constant K such that

k x ( K , n) -- n

for all n.

The converse of this is also true by the complemented subspaces theorem of Lindenstrauss and Tzafriri [64]. From our discussion concerning the weak Hilbert space T 2 in Section 1, we have that T 2 has the uniform projection approximation property with KT2 ( 8 , n ) - - 2n (or even K T z ( K , n ) -- (1 + e)n for any fixed e and K -- K(e)). That is, for any 2ndimensional subspace E C T 2, E C span{(ti)in_=l , F}, where F is an n-dimensional subspace of span(ti)i~=n+l. But now F is 8-complemented in T 2. Johnson and Pisier [49] showed that such linear behavior of kx actually characterizes weak Hilbert spaces. THEOREM 7.2. A Banach space X is a weak Hilbert space if and only if there are constants K, C such that

k x ( K , n) <~Cn

for all n.

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P.G. Casazza

The following result of Johnson (which appeared in [64]) reduces the checking of the uniform approximation property (at least in Banach lattices) to working with disjoint elements. PROPOSITION 7.3. There is a function N(k, e) <<,(2k2/e) k so that for any order complete Banach lattice L, and every k-dimensional subspace F of L, there are, for every e > O, N -- N(k, e) disjoint elements (xi)N1 in L and a linear operator T" F --+ X =: span(xi)Ul such that

II Tx - x II ~ ex for all x E F. Johnson [43] used this to show there are reflexive Banach spaces which are not sufficiently Euclidean (i.e., reflexive Banach spaces which do not contain uniformly complemented copies of g~). The disadvantage of this result is that it immediately jumps the value of the uniformity function to f ( K , n ) ~ n 2n. The advantage is that it trivially implies that spaces like gp and L p have the uniform projection approximation property since disjointly supported sequences in these spaces span 1-complemented subspaces. Local reflexivity implies that X has the uniform approximation property if and only if X** has the uniform approximation property. It was open for a long time whether the uniform approximation property passes from a space to its dual. This was done by Heinrich [37] using ultra-power techniques. Recently Mascioni [75] gave a direct proof with the best estimates of the UAP constant for X* given the UAP constant for X. THEOREM 7.4. A Banach space X has the uniform approximation property if and only if X* has the uniform approximation property. Heinrich [37] demonstrated that ultra-product techniques represent a powerful tool for working with the approximation property. THEOREM 7.5. Let X be a Banach space.

(1) If X has the uniform approximation property and 1 ~ p ~ ~ then L p (lZ, X) has the uniform approximation property. (2) X has the (K + e)-uniform approximation property (respectively uniform projection approximation property), for all e > 0 if and only if each ultrapower (X)u has the )~-uniform approximation property (respectively uniform projection approximation property) if and only if each ultrapower (X)u has the ~-bounded approximation property (respectively Jrz-property).

We end this section by considering the relationship between the uniform approximation property and the other approximation properties. Certainly the uniform approximation property implies the bounded approximation property. There are spaces (even spaces with unconditional bases) which fail the uniform approximation property. For one, since the uniform approximation property passes to all dual spaces of the space, any space with a basis whose dual space fails the approximation property cannot have uniform approximation property. In particular the spaces C1, U1 of Definition 1.10 fail to have the uniform

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307

approximation property. It follows from Proposition 4.8 that a separable space with the uniform approximation property also has the commuting bounded approximation property. This result fails in the non-separable case since goc has the metric uniform approximation property but fails the commuting bounded approximation property by Corollary 9.4. And if X is any Banach space with the uniform approximation property and the re)~-property, then by Theorem 5.6, all the duals of X have the rex-property. There are many natural open questions here, some of which would resolve earlier problems concerning the approximation properties. PROBLEM 7.6. Does the uniform approximation property imply the metric uniform approximation property? If the answer to Problem 7.6 is yes, then we can drop altogether the notion of K-UAE PROBLEM 7.7. Does the uniform approximation property imply the metric approximation property? Note that restated, Theorem 7.5 implies that Problems 7.6 and 7.7 are equivalent. Also, if these problems have a negative answer, then we can give a negative answer to Problem 3.9. That is, if X is a Banach space with the uniform approximation property and failing the metric approximation property, then there is an ultraproduct Y of X so that Y has the uniform approximation property and fails the metric approximation property. Now, Y* has even the uniform approximation property while failing the metric approximation property. Lindenstrauss and Tzafriri [65] have given a positive answer to Problem 7.6 for uniformly convex spaces. THEOREM 7.8. A uniformly convex Banach space with the uniform approximation property also has the metric uniform approximation property. We can also ask: PROBLEM 7.9. Does the uniform approximation property and the re-property imply the uniform projection approximation property? Or does the uniform approximation property itself imply the uniform projection approximation property? Because of the ultraproduct characterization of the uniform approximation property there should be a relationship between the uniform approximation property and g~-sums. PROBLEM 7.10. If X is a Banach space and ( ~ @X)e~ has the bounded approximation property does X have the uniform approximation property? Heinrich [37] has also introduced a uniform version of the approximation property. We say that a Banach space X has the uniform Grothendieck approximation property if there are uniformity functions f(., .) and g(., .) satisfying for every e > 0 and for every/3 = (/3n) with/3n --+ 0 and for every x, ~ X with IIx~II ~< r there is a finite rank operator T on X

308 with rk T ~< f (fl, e), proves [37]

P.G. Casazza

II T II ~ g(fl, e), and IITxn - Xn II ~ e for every n = 1, 2 . . . . . Heinrich

PROPOSITION 7.1 1. Let X be a Banach space. Then the following hold: (1) X has the uniform Grothendieck approximation property if and only if X* has it. (2) X has the uniform Grothendieck approximation property implies that Lp(#, X) has it for all 1 <<,p <<,oc. (3) X has the uniform Grothendieck approximation property if and only if each ultrapower (X)u has the approximation property. It is immediate that the uniform approximation property implies the uniform Grothendieck approximation property. We ask PROBLEM 7.12. Does the uniform Grothendieck approximation property imply the uniform approximation property? The answer to Problem 7.12 is "yes" in the super-reflexive setting [37]. Again, if Problem 7.12 has a negative answer, then by Proposition 7.11 we can construct a counterexample to Problem 3.8. We just take a space X with the uniform Grothendieck approximation property and failing the uniform approximation property. Then there is an ultrapower Y of X which has the uniform Grothendieck approximation property and fails the bounded approximation property. Now, Y* has the uniform Grothendieck approximation property (and hence the approximation property) while failing the bounded approximation property.

8. The compact approximation property The compact approximation property was somewhat mysterious until recently when the first major results were done. This was in part due to the fact that it was not clear that this property was actually distinct from the approximation property. DEFINITION 8.1. A Banach space X has the compact approximation property (compact AP for short) if for every e > 0 and every compact set K in X there is a compact operator T ~ L(X) so that I[Tx - x[I ~< e for all x 6 K. We say X has the )~-bounded compact AP if these approximating compact operators can be chosen with IIT II ~< )~ (independent of the compact set K). If )~ = 1 works, we say X has the metric compact AP.

If X has the )~-bounded compact approximation property for some ~, we say X has the bounded compact approximation property. The spaces failing the approximation property we encountered in Section 2 all fail the compact approximation property. It is immediate that the approximation property implies the compact approximation property. In fact, Theorem 2.5 yields PROPOSITION 8.2. A Banach space X has the )~-bounded approximation property if and only if X has both the approximation property and the )~-bounded compact approximation property.

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309

Willis [98] has shown that the approximation property and the compact approximation property are not equivalent. PROPOSITION 8.3. There is a separable reflexive Banach space having the compact approximation property but failing the approximation property. Willis' construction is really another factorization theorem. Johnson [40] showed that if an operator T E B(X, Y) is a norm limit of finite rank operators then T factors compactly through C p (see Definition 1.10). Figiel [26] shows that every compact operator T E B(X, Y) factors compactly through a subspace of Cp. A close examination of Willis' proof yields THEOREM 8.4. Every compact operator T E B(X, Y) factors through a reflexive Banach space which has the metric compact approximation property. The basic properties of the compact approximation property are not resolved yet. PROBLEM 8.5. If X* has the compact approximation property, must X have the compact approximation property? The Lindenstrauss and Tzafriri proof [66] of Grothendieck's Theorem 3.6 [35] works for a special case of the compact approximation property. THEOREM 8.6. If X* is separable and has the compact approximation property given by weak-star continuous operators, then X* has the metric compact approximation property. But the general case here is open. PROBLEM 8.7. If X* has the compact approximation property (even just for X* separable) must X* have the metric compact approximation property? Casazza and Jarchow [ 11 ] have shown that the bounded compact approximation property does not pass from a dual space to the space. PROPOSITION 8.8. There is a Banach space which has the approximation property, does not have the bounded compact approximation property, but all of its duals are separable and have the metric compact approximation property. PROOF. Let Y be the reflexive space of Willis [98] which has compact approximation property and fails the approximation property. By Proposition 1.3, there is a Banach space Z with Z** having a basis and Z*** ~ Z* @ Y*. Now, Z*** fails the approximation property but has shrinking compact approximation property (since Z* has a shrinking basis and Y is reflexive and has the compact approximation property). By Proposition 3.11, we can

310

P G. Casazza

renorm X with I" In so that it fails the n-bounded approximation property (and hence it fails the n-compact approximation property by Proposition 8.2). Now,

x-

e ( z * * , I. In) E2

does the trick.

E]

Godefroy and Kalton [30] have extended Theorem 4.6 to show that a separable space with the metric compact approximation property has the commuting metric compact approximation property. The compact approximation property shows up naturally in many settings. For example, Dixon [22] shows that a Banach space has the )~-bounded compact approximation property if and only if K (X) has a )~-bounded left approximate identity. The compact approximation property shows up in the study of M-ideals. A subspace M of a Banach space X is called an M-ideal if there is a projection P on X* with M • as its kernel and for all x* E X* we have IIx*ll- IIPx*ll + I1(I - P)x*ll. M-ideals were first studied by Alfsen and Effros [ 1]. Harmand and Lima [36] proved that if X is a Banach space such that K (X) is an M-ideal in L(X), then X has the metric compact approximation property. Cho and Johnson [ 19] proved a strong converse to this by showing that if X is a subspace of ~p for 1 < p < cx~, with the compact approximation property, then K (X) is an M-ideal in L(X). After work by several authors, Kalton [53] has classified the separable Banach spaces X such the K (X) is an M-ideal in L(X). Gowers and Maurey [33] have exhibited classes of Banach spaces X for which every bounded operator on X is a strictly singular perturbation of the identity. It is still an open question whether there are Banach spaces X for which every bounded operator on X is a compact perturbation of the identity. Godefroy and Saphar [31 ] have asked whether there are any infinite dimensional Banach spaces X for which K(X) is reflexive? They then observed that the answer to this question is positive if there is a Banach space X which fails the compact approximation property and so that L(X) = K (X) G I.

9. Approximation properties in non-separable spaces Many of of the proofs of our earlier results do not depend upon the separability of the Banach space. We will list here which results (proofs) carry over to this setting and the few for which problems occur and why. THEOREM 9.1. Let X be a non-separable Banach space. If X* has the approximation property (respectively)~-bounded approximation property, zcz-property, the uniform approximation property, the uniform projection approximation property) then so does X. Missing from Theorem 9.1 is the compact approximation property (for which this is an open question even in the separable case) and the bounded compact approximation

Approximation Properties

311

property since this implication fails even in the separable case (see Proposition 8.8). Also missing from Theorem 9.1 is the commuting bounded approximation property. This also fails as shown by Casazza, Kalton and Wojtaszczyk [ 16]. To put this result into its proper framework, we first need a definition. DEFINITION 9.2. A non-separable Banach space X is said to have the separable complementation property if for every separable subspace Y in X, there is a separable subspace Z with Y C Z C X and Z is complemented in X. We now have [ 16] THEOREM 9.3. Let X be a non-separable Banach space with the commuting bounded approximation property. Then X has the separable complementation property. PROOF. Let X have the commuting bounded approximation property and let Y be any separable subspace of X. Choose an increasing sequence (En) of finite dimensional subspaces of Y whose union is dense in Y. By induction, we can find a sequence (Tn) of finite rank operators on X so that Tn Tm - - T m i n ( n , m ) for all n r m and Tn IEn -- IEn. Now, let Z be the separable subspace of X which is the closed linear span of the Tn (X). By switching to a pointwise convergent subnet of (Tn), we obtain a bounded linear operator P mapping X onto Z and a calculation shows that P - p2. IS] Since co is complemented in every separable Banach space containing it, we have COROLLARY 9.4. If X has the commuting bounded approximation property, then every subspace of X which is isomorphic to co is complemented in X. Hence, ~ does not embed into any Banach space with the commuting bounded approximation property

Of course, both the dual and the pre-dual of g ~ have the commuting bounded approximation property. A very useful tool for working in non-separable spaces is the "Lindenstrauss subspace" defined in [61 ]. PROPOSITION 9.5. Let X be a Banach space and let E be a finite dimensional subspace of X. Let k be an integer and let e > O. Then there is a finite dimensional subspace F of X containing E such that for every subspace Y of X containing E with dim Y / E -- k there is a linear operator T : Y --+ F with IITII ~< 1 + ~ and T x = x for every x ~ E. By refining Proposition 9.5 in the reflexive case, Lindenstrauss [61 ] shows THEOREM 9.6. Every reflexive Banach space has the separable complementation property with norm one projections. Actually, Lindenstrauss' method shows more. For a reflexive Banach space X and any separable subspaces Y C X and Z C X*, there is a projection P on X with separable range such that Y C P ( X ) , Z C P*(X*) and IIPII -- 1. We observed in Section 2 that if every

312

P.G. Casazza

separable subspace of X is contained in a separable subspace with the approximation property, then X has the approximation property. Johnson [42] proves the corresponding result for the bounded approximation property. This proof is based on ideas of Lindenstrauss [61 ] which are interesting in that it uses discontinuous, non-linear operators to construct a family of approximating bounded linear operators. THEOREM 9.7. Let X be a non-separable Banach space. Then X has the X-bounded approximation property (respectively the fez-property, the uniform approximation property, the uniform projection approximation property) if and only if every separable subspace of X is contained in a separable subspace with the ),-bounded approximation property (respectively the rc)~-property, the uniform approximation property, the uniform projection approximation p rope rty). PROOF. We first check the bounded approximation property. If X has the )~-bounded approximation property and Y is a separable subspace of X, let (En) be an increasing sequence of finite dimensional subspaces of Y whose union is dense in Y. Now we can construct by induction a sequence of finite rank operators (Tn) on X with the property that Tn+l IEnUTn(X) = I. Now letting Z be the closed linear subspace of X (which contains Y) generated by (Tn (X)) will do it. Conversely, assume that every separable subspace of X is contained in a separable subspace with the ~.-bounded approximation property. Let E be any finite dimensional subspace of X. By Proposition 9.5, there exists a separable subspace Y of X so that, given any finite dimensional subspace F of X with E C F, there exists an operator TF : F --+ Y with IITFll ~< 1 + 1/dim F and TFIE -- IE. By our hypotheses there is a separable subspace Z of X with Y C Z and Z has )~-bounded approximation property. Choose a finite rank operator L: Z --+ Z so that IILI[ ~< )~ and LIE = IE. Now consider the net SF : X --+ L Z of (non-linear and discontinuous) functions defined by

LTFx SFX --

0

x 6F, otherwise,

where the direction on the net is by inclusion on F. Since dim L Z < oc, a compactness argument yields a pointwise convergent subnet to say Lx for each x 6 X. Then, L is linear, llLll ~< )~, L has finite rank, and LIE = IE. The zrz-property is done similarly. For the uniform approximation property, the if part can be done as in the bounded approximation property case. For the only if part, choose a separable subspace Y of X and let (Yn) be dense in Sy with each Yn repeated countably many times in the sequence. Let (En) be the spans of all finite subsets of (yn). For every n, choose an operator Tn with IlTn II ~< K, rank Tn <~f ( d i m E n ) (where f ( n ) is the uniformity function for X) and so that 11(I - Tn)lEn II ~< 1/dimEn. Let Y1 = span Un Tn(X). Repeat this procedure on Y1 to get Y2 and so on. Let Z = Ui Yi. Then Y C Z, Z is separable and Z has the uniform approximation property. D Since all the approximation properties except the basis property and the finite dimensional decomposition property pass to complemented subspaces, Theorem 9.7 holds trivially for all these properties in the case that X is reflexive by Theorem 9.6. Another simple

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313

consequence of Lindenstrauss' result is [18] that every weak Hilbert space X can be decomposed as X = H @ Y, where Y is a separable weak Hilbert space and H is isomorphic to a Hilbert space.

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[48] W.B. Johnson and T. Oikhberg, Separable lifting property and extensions of local reflexivity, Preprint. [49] W.B. Johnson and G. Pisier, The proportional u.a.p, characterizes weak Hilbert spaces, J. London Math. Soc. 44 (1991), 525-536. [50] W.B. Johnson, H.E Rosenthal and M. Zippin, On bases, finite dimensional decompositions, and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488-504. [51] W.B. Johnson and H.E Rosenthal, On co*-basic sequences and their applications in the study of Banach spaces, Studia Math. 43 (1972), 77-92. [52] E Jones, BMO and the Banach space approximation problem, Amer. J. Math. 107 (4) (1985), 853-893. [53] N.J. Kalton, M-ideals of compact operators, Illinois J. Math. 37 (1993), 147-169. [54] N.J. Kalton and N.T. Peck, Twisted sums of sequences spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979), 1-30. [55] N.J. Kalton and G.V. Wood, Orthonormal systems in Banach spaces and their applications, Math. Proc. Cambridge Philos. Soc. 79 (1976), 493-510. [56] R. Komorowski, On constructing Banach spaces with no unconditional basis, Proc. Amer. Math. Soc. 120 (1994), 101-109.

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[57] R. Komorowski and N. Tomczak-Jaegermann, Banach spaces without local unconditional structure, Israel J. Math. 89 (1995), 205-226. [58] J.L. Krivine, Sous espaces de dimension fnite des espaces de Banach reticules, Ann. of Math. 104 (1976), 1-29.

[59] S. Kwapiefi, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients, Studia Math. 44 (1972), 583-595. [60] J. Lindenstrauss, On certain subspaces ofs Bull. Acad. Polon. Sci. 12 (1964), 539-542. [61 ] J. Lindenstrauss, Extensions of compact operators, Mem. Amer. Math. Soc. 48 (1964). [62] J. Lindenstrauss, On non-separable reflexive Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 967-970. [63] J. Lindenstrauss, On James' paper "separable conjugate spaces", Israel J. Math. 9 (1971), 279-284. [64] J. Lindenstrauss and L. Tzafriri, On the complemented subspaces problem, Israel J. Math. 9 (1971), 263269. [65] J. Lindenstrauss and L. Tzafriri, The uniform approximation property in Orlicz spaces, Israel J. Math. 23 (1976), 142-155. [66] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, Springer, Berlin (1977). [67] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Function Spaces, Springer, Berlin (1979). [68] W. Lusky, A note on Banach spaces containing cO or C ~ , J. Funct. Anal. 62 (1985), 1-7. [69] W. Lusky, On Banach spaces with the commuting bounded approximation property, Arch. Math. (Basel) 58 (6) (1992), 568-574. [70] W. Lusky, On Banach spaces with bases, J. Funct. Anal. 138 (2) (1996), 410-425. [71 ] E Mankiewicz and N.J. Nielsen, A superreflexive Banach space with a finite dimensional decomposition so that no large subspace has a basis, Israel J. Math. 70 (1990), 188-204. [72] P. Mankiewicz and N. Tomczak-Jaegermann, Schauder bases in quotients of subspaces of s American J. Math. 116 (1994), 1341-1363. [73] P. Mankiewicz and N. Tomczak-Jaegermann, Structural properties of weak cotype 2 spaces, Canadian J. Math. 48 (1996), 607-624. [74] V. Mascioni, Some remarks on the uniform approximation property for Banach spaces, Studia Math. 96 (1990), 243-253. [75] V. Mascioni, On the duality of the uniform approximation property in Banach spaces, Illinois J. Math. 35 (1991), 191-197. [76] B. Maurey and G. Pisier, Series de variables aleatoires vectorielles independantes et proprietes geometriques des espaces de Banach, Studia Math. 58 (1976), 45-90. [77] N.J. Nielsen, The positive approximation property of Banach lattices, Israel J. Math. 62 (1988), 99-112. [78] N.J. Nielsen and N. Tomczak-Jaegermann, Banach lattices with property (H) and weak Hilbert spaces, Illinois J. Math. 36 (1992), 345-371. [79] N.J. Nielsen and E Wojtaszczyk, A remark on Bases in 12p-spaces with an application to complementably universal 12~-spaces, Bull. de l'Acad. Polon. Sci. 23 (1973), 249-254. [80] A. Pe~czyfiski, Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basis, Studia Math. 40 (1971), 239-242. [81] A. Petczyfiski and H.E Rosenthal, Localization techniques in Lp-spaces, Studia Math. 52 (1975), 263-289. [82] A. Petczyfiski and E Wojtaszczyk, Banach spaces with finite dimensional expansions of identity and universal bases of finite dimensional subspaces, Studia Math. 40 (1971), 91-108. [83] G. Pisier, Counterexample to a conjecture ofGrothendieck, Acta Math. 151 (1983), 180-208. [84] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, C.B.M.S. Amer. Math. Soc., Vol. 60 (1985). [85] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math. 94, Cambridge University Press (1989). [86] C.J. Read, Different forms of the approximation property, Unpublished notes. [87] H.P. Rosenthal, On one complemented subspaces of Banach spaces with a one unconditional basis, according to Kalton and Wood, Israel Seminar in the Geometrical Aspects of Functional Analysis, Univ. of Tel Aviv (IX) (1983-1984), 1.1-1.16. [88] C. Samuel, Exemples d'espaces de Banach ayant la propriete de projection uniforme, Colloq. Math. 43 (1980), 117-126.

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[89] C. Samuel, Bounded approximate identities in the algebra of compact operators on a Banach space, Proc. Amer. Math. Soc. 177 (1993), 1093-1096. [90] J. Schauder, Zur Theorie stetiger Abbildungen in Funkionalraumen, Math. Zeit. 26 (1927), 47-65, 417-431. [91] A. Szankowski, A Banach lattice without the approximation property, Israel J. Math. 24 (1976), 329-337. [92] A. Szankowski, Subspaces without the approximation property, Israel J. Math. 30 (1978), 123-129. [93] A. Szankowski, B(H), does not have the approximation property, Acta Math. 146 (1981), 89-108. [94] A. Szankowski, On the uniform approximation property in Banach spaces, Israel J. Math. 49 (1984), 343359. [95] S.J. Szarek, A Banach space without a basis which has the bounded approximation property, Acta Math. 159 (1987), 81-98. [96] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Pitman Monographs, Longman Scientific and Technical, New York (1989). [97] B.S. Tsirelson, Not every Banach space contains g.p or co, Funct. Anal. Appl. 8 (1974), 138-141. [98] G.A. Willis, The compact approximation property does not imply the approximation property, Studia Math. 103 (1992), 99-108. [99] M. Zippin, The embedding of Banach spaces into spaces with structure, Illinois J. Math. 34 (3) (1990), 586-606.

CHAPTER

8

Local Operator Theory, Random Matrices and Banach Spaces

Kenneth R. Davidson Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 E-mail: krdavids @math. uwaterloo, ca

Stanislaw J. Szarek Equipe d'Analyse, University Paris VI, Case 186, 4, place Jussieu, 75252 Paris, France Department of Mathematics, Case Western Reserve University, Cleveland, 0H44106-7058, USA E-mail: sjsl 3 @po.cwru.edu

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

319

1. Local operator theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

321

1.1. A l m o s t c o m m u t i n g H e r m i t i a n matrices 1.2. Unitary orbits of normal matrices 1.3. Quasidiagonality

.................................

....................................

.............................................

1.4. Extensions of pure states and matrix paving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Hyper-reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. R a n d o m matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

321 326 330 333 338 341

2.1. T h e overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Concentration of m e a s u r e and its c o n s e q u e n c e s ............................ 2.3. N o r m of a r a n d o m matrix and the S l e p i a n - G o r d o n l e m m a . . . . . . . . . . . . . . . . . . . . . . .

342 346 350

2.4. R a n d o m matrices and free probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. R a n d o m vs. explicit constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

354 358

Acknowledgments References

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Introduction Connections between Banach space theory and classical operator theory on Hilbert space are multifold. First, one can generalize notions and results involving linear operators on a Hilbert space to the Banach space context. Second, more often than not, the study of the latter area also involves linear operators, and so one can use methods developed in one of the fields to attack questions in the other. Thirdly, spaces of operators are typically Banach spaces, and so we may study them as such using the methods of geometry of Banach spaces. Of course, this is not a complete list of connections. In this survey we shall focus on the second and the third of the links hinted above, and on those of their aspects which are related to the local theory of Banach spaces and to the asymptotic properties of finite dimensional normed spaces, as the dimension approaches infinity. More specifically, we shall address the following two groups of issues; the corresponding parts of the article (Sections 1 and 2) can be read essentially independently. 1. Problems in operator theory which are local in the sense that they reduce to questions about finite matrices, and moreover the resulting questions are similar in flavor to ones studied in the Banach space context. For example, they might involve estimates independent of the dimension of the space on which the matrices act. Or, they may require the determination of approximate isomorphic or almost isometric asymptotics for some quantity as a function of the dimension. At the same time, the estimates needed are usually not optimal, a feature that would be characteristic to a representative Banach algebra or C*-algebra problem. 2. The asymptotic behavior of large dimensional random matrices, with particular attention to their spectral properties. This is a topic which has been of major interest both in the geometry of Banach spaces and in the theory of operator algebras. It is also related to numerous other fields, and has lately become a very hot subject in view of apparent analogies between the distribution of eigenvalues of large random matrices and of zeros of the Riemann ff function. This article will not focus on these connections, but will mention them where appropriate. More generally, we shall bring up examples where the methods of one of the fields appear to have relevance to the other. This relevance may be described by a theorem, or just hinted by a vague analogy. As these points of contact between Banach space theory and operator theory are known to relatively few people on both sides, we hope that popularizing them will give a boost to both areas. Before proceeding, we shall clarify what we mean by isomorphic, almost isometric and isometric estimates through this simple example. Let Tn denote the triangular truncation map on the n x n matrices which sends a matrix A = [aij ].n~,j--1 to "-l-nA -- [bij ] where

bij

_ J aij I0

i f / ~< j , ifi > j .

Suppose that we are interested in the behavior of IlTn I1 as n tends to infinity. Several answers of varying degrees of precision (respectively isomorphic, almost isometric and isometric) are possible:

K.R. Davidson and S.J. Szarek

320

(a) lien II is of order log n, meaning that there are universal constants c, C > 0 such that for all n, c log n ~< IIT~ II ~< C log n. (b) IIT~ II is equivalent to ~1 log n [8], meaning that IIT~ II

1

lim = --, n--+~ log n Jr

(c) liT. II = f(n), where f is an explicit function. We shall also say that (a) and (b) describe, respectively, the rough and precise asymptotic order of IIT~ II. As above, and unless indicated otherwise, c, cl, C, C t etc. will always stand for universal positive constants whose numerical values may vary between occurrences. We now describe those topics which we will address in this article. In Section 1, we will treat questions of local operator theory focusing on the following areas: 9 Almost commuting versus nearly commuting problems: given a pair of (e.g., selfadjoint) matrices whose commutant is small in norm, is it close to a pair of (selfadjoint) matrices which exactly commute? 9 Spectral distance problems: given two matrices (normal, unitary, etc.) which are close in norm, are their spectra close in some appropriate sense? 9 Approximation of large matrices by direct sums of smaller ones: for example, the Herrero-Szarek result on non-reducibility of matrices and the examples of Szarek and Voiculescu of exotic quasidiagonal operators. 9 The Kadison-Singer problem (reformulated as the paving problem): does there exist a positive integer k such that, given a square matrix A with zero diagonal, it is always possible to find diagonal projections P1, P2 . . . . . Pk such that k

k

and i=1

~

Pi APi

1

~IIAII?

i=1

9 Questions about hyper-reflexivity: in particular, do Tn | Tn, the tensor products of two copies of the algebra of n • n upper triangular matrices, have a common distance estimate analogous to Arveson's distance formula for nests? We shall not elaborate on the Sz. Nagy-Halmos similarity problem recently solved by Pisier, see [138] or [64]. Even though it conforms quite well to the spirit of this article, it is, in its heart, a complete boundedness/operator space question. We do not embark on those areas as both them and the similarity problem itself are well covered in the literature, cf. [134,140,139] and (in this collection) [141] for the former and [61] for the latter. On the other hand, we shall mention other less known questions about finite matrices that fit into our framework. In each case, we shall try to sketch the connections to infinite operator theory. In particular, the above problems relate respectively to the Brown-Douglas-Fillmore theory, similarity and perturbations of operator algebras, approximation and distance estimates from spectral data, quasidiagonality, and the uniqueness of extensions of pure states. Depending on whether a problem has been solved or not, we shall either hint at the ingredients of the proof or describe the current state of the art and the consequences for the original problem.

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We point out that, in addition to their significance for the infinite dimensional theory, many of the problems stated above are also clearly important from the numerical linear algebra point of view. In Section 2, we shall signal various ways in which random matrices interact with operator theory and the geometry of Banach spaces. We start with an overview of the subject by presenting a sampler of classical (and not so classical) results on asymptotic spectral properties of large random matrices. We shall show how some of these classical results can be rather routinely obtained using standard methods of Banach space theory, e.g., by exploiting the measure concentration phenomenon and various tools of probability in Banach spaces. Although some of these arguments are folklore in the Banach space circles, they are generally not known to the wider mathematical public. Our purpose is to just illustrate this approach to the topic of random matrices, and so we shall not aim at the strongest or most general results that could be so obtained. We shall also sketch the links between the subject of random matrices and Voiculescu's free probability and state some other subtle and interesting questions (mostly of the almost isometric nature) about norms and similar parameters of random matrices. These questions are not necessarily directly relevant to Banach space theory, but conceivably they can be approached with the aid of standard tools of the area. We feel that it would be a worthwhile project to review and unify the closely related random matrix results and rather dissimilar methods pertaining to those results coming from, among others, mathematical physics, probability and Banach space theory. In addition to clarifying the picture, the benefits could include stating results in a form immediately utilizable in other fields. This is often a problem. For example, large deviation and mathematical physics formulations frequently involve rescaling the quantities in question and then investigating their asymptotics as the dimension tends to c~. On the other hand, applications to the geometry of Banach spaces, convexity and computational complexity typically require estimates that are valid for a wide range of parameters in any given dimension. Finally, in the last section, we mention some of the technologies used in operator theory to exhibit phenomena analogous to the ones obtained in the Banach space theory via random methods. These technologies typically involve representation theory, but there are also links to graph theory and arithmetic geometry, to name a few.

1. Local operator theory We will discuss this area issues through a few important problems which are in various states of solution, as mentioned in the introduction.

1.1. A l m o s t commuting Hermitian matrices There has been dramatic progress made on the following matrix problem: PROBLEM 1.1. Given e > 0, is there a 6 > 0 so that: whenever A and B are n x n Hermitian matrices of norm one, for any n >~ 1, such that ]lAB - BA]l < 6, then there are Hermitian matrices A1 and B1 which exactly commute such that IIA - A lll < e and lib - B1 II < ~?

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K.R. Davidson and S.J. Szarek

The condition IIAB - BAll < 6 colloquially says that the matrices A and B almost commute; while the conditions IIA - All] < e and IIB - B~ II < e for a commuting Hermitian pair A1 and B1 say that A and B nearly commute. So the question becomes:

Are almost commuting Hermitian matrices nearly commuting ? An interesting reformulation of the problem tums this into a question about normal matrices. The matrix T -- A + iB satisfies [T, T*] := TT* - T*T = - 2 i ( A B - BA). Thus the Hermitian pair {A, B } is almost commuting when T is almost normal, and exactly commuting when T is normal. Hence the pair is near to a commuting pair precisely when T is close to a normal matrix. Conversely, if T is an almost normal or nearly normal matrix, then the real and imaginary parts A = (T + T * ) / 2 and B = (T - T * ) / 2 i are almost commuting or nearly commuting, respectively. Thus another equivalent question is: Are almost normal matrices nearly normal? As with many of the problems we discuss in this article, the crucial point is the dimension-free character of the estimates. Indeed, the problem could be reformulated for compact Hermitian matrices. However that was not the appropriate way to go for the solution, as we shall see. Dimension dependent results were obtained early on by Pearcy and Shields [135] who establish that 6 = eZ/n will suffice. This was sharpened by the second author [158] to 6 = cel3/Z/x/-ff. We shall see however that these estimates are not nearly of the correct order. It is interesting to note that there is a negative answer to the analogue of Problem 1.1 for arbitrary operators on Hilbert space. Consider a weighted shift on ~2 with basis {ek: k ~> 1 } given by

Snek=

min{k, n}

ek+l

fork~>l.

It is easy to see that 1 [s.,s*~]--~P.

where Pn is the orthogonal projection onto span{el . . . . . en}. Thus for large n, this operator has small self-commutator; and thus its real and imaginary parts are almost commuting. However, every Sn is a compact perturbation of the unilateral shift $1. This operator is a proper isometry which is Fredholm of index - 1 and has essential spectrum tre(S1) -- 7s Therefore each Sn is also Fredholm of index - 1 with essential spectrum ere (Sn) = ql", as these properties are invariant under compact perturbations. It is also straightforward to show that any operator T with II$1 - T II < 1 has Fredholm index - 1. So again, this is also true for Sn. A normal operator N has index 0 because ker N = ker N*. Thus each Sn is at least distance 1 from a normal operator, and in fact this is exact. There are many variants of our problem, and most of them have turned out to have negative answers. Here is one example which has received a lot of attention. Fix a basis

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323

{ej" 0 <~ j <~ n} for C n+l, and consider the matrices 2j A e j -- ( 1 - --ff-)ej

for0 ~< j ~
2

Bej -- - f f - ~ / ( j 4- 1)(n -- j ) e j + |

for0 <~ j ~ < n - 1,

Ben = 0 . Easy computations show that A is Hermitian, II[B, B*]II < 4 / n and II[A, B]II < 2 / n . Voiculescu [ 173] showed that the triple {A, Re B, Im B } are almost commuting but not near to a commuting triple of Hermitian matrices. A refinement of this by the first author [55] showed that {A, B} was not close to a commuting pair {A', B'} where A' is self-adjoint but B' is not required to be normal. Finally Choi [51 ] showed that this pair is far (about 0.5) from any commuting pair with no conditions on A' or B'. Choi's obstruction involved the determinant of a polynomial in A and B. Curiously, the matrix B above is close to a normal matrix; and the pairs {A, Re B } and {A, ImB} are also near commuting. This was shown in [55] using a method introduced by Berg [28]. Berg developed a technique for perturbing direct sums of weighted shifts that are almost normal to obtain normal operators when no index obstruction prevents it, as in the example cited above. This was a special case of a deep problem in operator theory. This leads to a connection with the Brown-Douglas-Fillmore theory [47,48] of essentially normal operators. An operator T is essentially normal if [T, T*] is compact. The original operator theory question they analyzed was to find when there is a compact perturbation of T which is normal. It is evident that if T - )~I is Fredholm, then the Fredholm index must be zero. The operator T has image t = 7r(T) in the Calkin algebra s The Calkin algebra is a C*-algebra, and t is a normal element of it. Thus C* (t) is isomorphic to C(X) where X = o-(t) = ae(T). For each bounded component Ui of C \ o-(t), the index r t i = ind(t - )~i) is independent of the choice of a point )~i E Ui. A normal operator will have ni = 0 for every 'hole'. And it is possible to construct an essentially normal operator T with ae(T) = X and any prescribed indices r / i - - ind(T - )~i I ) . The BrownDouglas-Fillmore theorem establishes that these are the only obstructions. THEOREM 1.2 (Brown-Douglas-Fillmore). Suppose that T is an essentially normal operator Then there is a compact operator K such that T - K is normal if and only if ind(T - )~I) -- Of o r every )~ in C \ ae(T). Surprisingly, the proof of BDF did not follow usual operator theoretic lines. Instead they observed that for any operator T above, there is a short exact sequence 0

>/C

i

> C*(T)+/C

7/"

> C(X)

> O.

Thus T determines an extension of the compact operators by C(X). Conversely any such extension determines, up to a compact perturbation, an essentially normal operator such that 7r(T) = z, where z is the identity function on X. A third viewpoint identifies this extension with the monomorphism that identifies C(X) with the subalgebra C* (t) of the Calkin algebra. They put a natural equivalence on extensions which corresponds to identifying operators which are unitarily equivalent up to a compact perturbation. The set of

K.R. Davidson and S.J. Szarek

324

equivalence classes Ext(X) becomes a semigroup under the operation of direct sum. Moreover this construction makes sense for any compact metric space X, not just subsets of the plane. They established that, in fact, Ext(X) is a group. Moreover it is a generalized homology theory paired in a natural way with topological K-theory. This opened the door to a new era in C*-algebras in which K-theory and topological methods came to play a central role. Even the existence of a zero element is non-trivial. In the case of a single operator or planar X, this reduces to the Weyl-von Neumann-Berg theorem [27] which shows that every normal operator is a small compact perturbation of a diagonalizable operator. This is easily generalized to arbitrary metric spaces, and it shows that any representation of C(X) on Hilbert space is close in an appropriate sense to a representation by diagonal operators. BDF show by elementary methods that any essentially normal operator T is unitarily equivalent to a small compact perturbation of T 9 D, where D is a diagonal normal operator with a ( D ) = ae(D) = ae(T). This leads us somewhat afield from the original problem. However the connection is made via quasidiagonality. A set of operators T is quasidiagonal if there is an increasing sequence Pk of finite rank projections converging strongly to I such that l i m n ~ II[T, P]II = 0 for every T E T. In particular, if T is quasidiagonal and e > 0, then there is a sequence Pk such that

T-- Z(Pk-- Pk-1)T(Pk-- Pk-1)= K k>/1

is compact and IlK II < e. Indeed, one just drops to an appropriate subsequence of the original one. Thus T - K has the form ~ke> 1 Tk where T~ act on finite dimensional spaces. Such operators are called block diagonal. Salinas [149] showed that the closure of the zero element in Ext(X) in the topology of pointwise-norm convergence, where we think of an extension as a monomorphism from C(X) into/2(7-{)//(7, is the set of all quasidiagonal extensions. In particular, unlike the planar case, the trivial element need not be closed. He used this to establish homotopy invariance of the Ext functor. In the case of a single operator, this shows that if T is essentially normal with zero index data, then it has a small compact perturbation which is block diagonal, say

T-K-T'-

ZeTk. k>~l

Moreover, it follows that [T', T'*] -- y~ke> 1[Tk, Tk*]. Therefore lim

k---+ (x)

II t:Tk, :T;J II

-

0

A positive solution to our original question would yield normal matrices Nk such that limk~ IIT~ - N~ II - 0. Hence N -- Y~ke>1 Nk is a normal operator such that T - N is compact. This was the motivation for the work of the first author [55]. If T -- y~,e k~>l Tk is essentially normal with ae(T) = X, then after a small compact perturbation, T may be replaced

Local operator theory, random matrices and Banach spaces

325

with T G D where D is any normal operator with eigenvalues in X. Thus each summand Tk can be replaced by Tk 9 D~, where Dk is diagonal. An absorption theorem is proven: THEOREM 1.3 (Davidson). I f T is any n • n matrix, then there are an n x n n o r m a l matrix N a n d an 2n • 2n n o r m a l matrix M such that

IIM - Z 9 Nil ~< 75[1[T, Z*]l[ 1/2. One difficulty in applying this to the BDF problem is that any holes in the spectrum of T are obliterated by summing with a normal with eigenvalues dense in the unit disk. In [29], Berg and the first author showed that similar operator-theoretic techniques can be used to show first that every essentially normal operator with zero index data is quasidiagonal. And then refinements of the absorption principle which control the spectrum were used to prove the planar version of BDF. This had the advantage of providing quantitative estimates for 'nice' spectra. But the correct positive answer to our question came again from a more abstract approach. Huaxin Lin [113] considered T = ~ke>~l Tk as an element of the von Neumann algebra 93l -- Ilk~>l 9)tnk. A positive solution is equivalent to constructing a normal operator N = Y'~ke>1Nk asymptotic to T, that is, verifying l i m k ~

[ITk -- Nk ] l - 0. This is a per-

turbation by the ideal 3 -- ~ k ~>1 93ln~ of elements J = ~ ek/> 1 Jk where limk~ oc I[Jk 11= 0. Thus T determines an element t = q ( T ) in the quotient algebra 9Jt/~. The problem is reduced to lifting each normal element of 93I/3 to a normal element of 93t. This was accomplished by a long tortuous argument. THEOREM 1.4 (Lin). Given e > O, is there a ~ > 0 so that w h e n e v e r A a n d B are n x n H e r m i t i a n matrices o f n o r m one, f o r any n >~ 1, a n d ]lAB - BAll < 6, then there are H e r m i t i a n matrices A1 a n d B1 w h i c h exactly c o m m u t e such that ]IA - All] < e a n d

IIB - B1 II < ~. A remarkable new proof of Lin's theorem is now available due to Friis and ROrdam [78]. They take the same approach, but base their argument only on two elementary facts about 93l/3. It has stable rank one, meaning that the invertible elements are dense. And every Hermitian element can be approximated by one with finite spectra. Both of these results can be obtained by using well-known facts about matrices on finite dimensional spaces. From this, a short argument cleverly using no more than the basic functional calculus yields the desired lifting of normal operators. Thus the original question has a positive solution. (See Loring's book [ 115] for a C*-algebraic view of this problem and its solution.) In a second paper [79], they show how the planar case of the Brown-Douglas-Fillmore Theorem follows. An important matrix question remains: PROBLEM 1.5. What is the dependence of 3 on e for almost commuting Hermitian matrices?

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Optimal estimates for 3 should be 0(82) as in the absorption results. However the Lin and Friis-Rcrdam results yield only qualitative information, and do not provide information on the nature of the optimal 6 function. A related question deals with pairs of almost commuting pairs of unitary matrices. Voiculescu [173] constructed the pair Uej --ej+lmodn and Vej --e2zrij/nej for 1 ~< j ~< n. It is evident that [I[U, V]ll < 2rc/n. However he showed that this pair is not close to a commuting pair of unitaries. He speculated that both this example and the commuting triples example occur because of topological obstructions. Asymptotically, the spectrum approximates a torus (or a two-sphere for Hermitian triples), and thus there is a heuristic justification for this viewpoint. Loring [ 114] showed that this was indeed the case for this pair of unitaries by establishing a K-theoretic obstruction coming from the non-trivial homology of the two-sphere. Exel and Loring [74] obtained an elementary determinant obstruction that shows that this pair of unitaries is far from any commuting pair, unitary or not. 1.2. Unitary orbits of normal matrices Consider a normal matrix N in 9Xk. By the spectral theorem, there is an orthonormal basis which diagonalizes N. So N is unitarily equivalent to diag(~,l . . . . . )~k) where the diagonal entries form an enumeration of the spectrum of N including multiplicity. Two normal matrices are unitarily equivalent precisely when they have the same spectrum and the same multiplicity of each eigenvalue. A natural question from numerical analysis and from the approximation theory of operators asks how the spectrum can change under small perturbations. The answer one gets depends very much on what kind of perturbations are allowed. In particular, if the perturbed matrix is also normal, then one obtains a more satisfying answer than if the perturbation is arbitrary. Consider first the case of a Hermitian matrix A. Then the eigenvalues may be enumerated so that ,kl ~> ... ~> ~.k. If B is a small perturbation of A, say IIA - B 1[ < e, then the Hermitian matrix R = (B + B*)/2 also satisfies IIA - R[[ < e, and so we may assume that the perturbation B is Hermitian to begin with. In that case, write the eigenvalues as # l ~> " " ~> #k. Then a 1912 argument due to Weyl [183] gives: THEOREM 1.6 (Weyl). Let A and B be k x k Hermitian matrices with eigenvalues ordered as )~l ~ "" >1 )~k and # l ~ "" >~ #k, respectively. Then sup [#j - ,kj [ ~< [] A - B II. l~j~k

Moreover, f o r each A, this value is attained by some such B. Indeed, suppose for example that )~j ~ # j . Consider the spectral subspace for A corresponding to {)~1. . . . . ~j } and the spectral subspace of B for {#j . . . . . #k }. Since the dimensions of these spaces add up to more than k, they contain a common unit vector x. Then (Ax, x) lies in the convex hull of {~1 . . . . . ~,j }, whence is ~> )~j. Similarly (Bx, x) <~ lZj. Hence

IIA -

BI[

~ ((A -

B ) x , x ) - ~(Ax, x) - (Bx, x) ~ ~,j - # j .

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Weyl's theorem shows that a perturbation of Hermitian matrices of norm e cannot move any eigenvalue more than e. One can reformulate this to say that if A and B are Hermitian with eigenvalues as above, then IIA - B II is bounded below by the spectral distance suPl <~j<~kI#j -- ~'j I. This spectral distance is achieved when A and B are simultaneously diagonalizable with eigenvalues matched in increasing order. The unitary orbit bt(A) of a matrix A is the set of all matrices which are unitarily equivalent to it. In finite dimensions, this is a closed set because the unitary group is compact. However, for operators this is no longer the case. The unitary orbit of a normal matrix consists of all normal matrices with the same spectral data. Thus one must be able to express the distance between these two unitary orbits solely in terms of the two spectra. Weyl's result shows that in the Hermitian case, this distance is precisely the spectral distance. More recently, significant attention has been paid to the more difficult normal case. Here is it no longer trivial to define a spectral distance. However, if M and N are normal matrices which are simultaneously diagonalizable, then this diagonalization amounts to a pairing of the two sets of eigenvalues. Among all possible permutations, there is one of least distance, and this is called the spectral distance: sp.dist(~.,/z) := inf

m a x [~,j yr CSk 1~ j ~ k

~Tr(j)[,

where ~ a n d / t represent the eigenvalues repeated according to their multiplicities. (This is really the quotient metric on ~ divided by the natural action of the symmetric group on cn.) Evidently, dist(L/(M), L/(N)) ~< sp.dist(a(M), a ( N ) ) , where a (M) denotes the spectrum of M including multiplicity. Many results in the literature (cf. [36,155]) show that under special hypotheses, the spectral distance is again the exact answer. See [33 ] for a full treatment of these ideas. However a recent result of Mueller (private communication) shows that even in three dimensions, it is possible that dist(L/(M), b/(N)) < sp.dist(a(M), a ( N ) ) . His example is explicit, but it is fair to say that we do not really understand why this strict inequality occurs. So it is of great interest that a result of Bhatia, Davis and McIntosh [35] obtains bounds independent of dimension: THEOREM 1.7 (Bhatia-Davis-McIntosh). that for all normal matrices M and N,

There is a universal constant c (> 1/3) such

dist(lg(M), lg(N)) ~ c sp.dist(a(M), a ( N ) ) . This result is obtained from a study of the Rosenblum operator on 9Ytk given by rM,N(X) = N X - X M . It is well known that this map is injective if a ( M ) and o'(N) are disjoint. In the case in which M and N are normal, there is a useful integral formula for the inverse which allows norm estimates. This integral formula requires finding a function f in LI(]R 2) with Fourier transform f satisfying f ( x , y) -- 1/(x + iy) for x 2 + y2 ) 1. The constant depends on 11f Ill, which may be chosen less than 3 [34].

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Estimates of this type allow one to estimate the angle between spectral subspaces of nearby normal operators. For example, if E = EM(K) and F = EN(L) are the spectral projections of M and N respectively for sets K and L with dist(K, L) = 3, then

1 IIEFII ~ ~ I I A - nil. c~ When this is small, the ranges of E and F are almost orthogonal. These results can be extended to normal operators on infinite dimensional space. Here the unitary orbit of a normal operator is not closed. The unitary invariants are given by the spectral theorem in terms of the spectral measure and a multiplicity function. However, the Weyl-von Neumann-Berg theorem [27] shows that the closure of the unitary orbit of every normal operator contains a diagonal operator. The multiplicity and spectral measure do not play a role in this closure except for isolated eigenvalues. Indeed, two normal operators have the same unitary orbit if and only if they have the same essential spectrum, and the same multiplicity at each isolated eigenvalue. (See [60, Theorem 11.4.4].) It turns out that one can define an analogue of the spectral distance for spectra of normal operators. Once this is accomplished, the finite dimensional methods extend to show that the distance between unitary orbits is equal to the spectral distance for Hermitian operators [15] and greater than a universal constant times the spectral distance for normal operators [56]. Some of these results are valid for other norms on the set of matrices. Although we are usually interested in the operator norm because of its central role, other norms occur naturally. For example, the Schatten p-norms are defined as the gP norm of the (sequence of the) singular values. In particular, p = 2 yields the Hilbert-Schmidt norm I1" 112, sometimes also called the Frobenius norm, which is particularly tractable. More generally, there is a large class of norms which are unitarily invariant in the strong sense that II A II ~ = II U A V II for all unitaries U and V; for 9Ytn, there is a one-to-one correspondence between such norms and symmetric norms on C n (or ~n) applied to the sequence of singular values. For example, the Ky Fan norms are given by the sum of the k largest singular values. The set of (properly normalized) unitarily invariant norms is a closed convex family and the Ky Fan norms are the extreme points. One can define the spectral distance between the unitary orbits of two normal matrices with respect to any unitarily invariant norm by sp.dist~ (k, p,) " - inf Ildiag()~j - #~(j))II 7rESk

z"

(Again, this is really a quotient metric induced by the symmetric norm o n C n related to I1" II~.) A classical result for the Hilbert-Schmidt norm is due to Hoffman and Wielandt [96]. Note that there is no constant needed. THEOREM 1.8 (Hoffman-Wielandt). If M and N are normal matrices, then sp.dist2 (a (M), a ( N ) ) ~< I I M - NIl2.

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For general unitarily invariant norms, a result of Mirsky [ 125] extends Weyl's Theorem for Hermitian matrices. A more recent result of Bhatia [32] extends this to certain normal pairs. THEOREM 1.9 (Mirsky-Bhatia). Suppose that M and N are normal matrices such that M - N is also normal (e.g., M and N Hermitian). Then f o r every unitarily invariant norm, sp.distr (or(M), or(N)) <, i I M - NIIr. For unitary matrices, Bhatia, Davis and McIntosh [35] show that THEOREM 1.10. Suppose that U and V are unitary matrices. Then f o r every unitarily invariant norm, 2

--sp.distr

(or(U), o'(V)) ~< [ I U - Vlir.

Jr

More results along these lines may be found in Bhatia's books [31,33]. When matrices are not normal, there are generally no good estimates of distance in terms of their spectra. A simple example is to take an orthonormal basis el . . . . . en for C n, and set Jn to be the Jordan nilpotent Jnei = ei+l for 1 ~< i < n and Jnen 0. Then consider T -- Jn + ~(', en)el. We see that liT - J n [ I - E while cr(Jn) - {0} and or(T) consists of the nth roots of e. So the spectral distance is e 1/n. This is much more dramatic than the normal case. We refer the reader to the books [31,33 ] for a lot more material on these types of questions. Likewise when calculating the eigenvalues of a normal matrix numerically, we are dealing with its perturbation or approximation which is not necessarily normal. A small perturbation of a normal matrix may have spectrum which does not approximate the spectrum of the normal at all. Indeed, Herrero [91] shows that there is a normal matrix N in 9)tk with 1 E or(N) and a nilpotent matrix Q such that lIN - Qll < 5 k-1/2. This allowed him to establish an important infinite dimensional result: =

THEOREM 1.1 1 (Herrero). Suppose that N is a normal operator on Hilbert space with connected spectrum containing O. Then N is the norm limit o f a sequence o f nilpotent operators. One way to construct such examples is to take Q to be a weighted shift Qei --aiei+l for 1 ~< i < k and Q ek = 0. This is evidently nilpotent. However, if the weights slowly increase from 0 to 1 and back down again, it will follow that IIQ II = 1 and II Q* Q - Q Q* II is small on the order of O(k-1). We saw in the previous section that the matrix Q must be close to a normal matrix. For weighted shifts, a method known as Berg's technique [28] (cf. [91 ]) allows explicit perturbations to a normal matrix with estimates on the order of C II Q* Q - Q Q* IIl/R, which yields the desired example. Self-adjoint examples can be constructed, but the estimates are strikingly different. Hadwin [90] solved this problem by relating the problem of approximating a Hermitian matrix by a nilpotent one to the norm of triangular truncation, mentioned in the introduction.

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THEOREM 1.12 (Hadwin). For k >>.1, let 6 ~ - i n f { l l A - QII" A, Q ~ 9Ytk, A - A*, IIAI] = 1, Q~ = 0 } . Then l i m k ~

6n logn = zr/2.

However, for the numerical analysis problem of calculating eigenvalues of a matrix which is a priori known to be normal, this deterioration of estimates is inessential. Indeed, if A0 is a given small but not necessarily normal perturbation of an unknown normal matrix A, and A1 is any normal approximant of A0, then by Theorem 1.7, the spectra of A and A 1 are close. However this argument is not constructive, as it does not tell us how to find A1. This leads to a refinement of Problems 1.1 and 1.5: given a matrix Ao which is known to be nearly normal, find a specific normal approximant with an explicit error estimate. In infinite dimensions, the description of those operators which are limits of nilpotents is a central result in the approximation theory of operators due to Apostol, Foia~ and Voiculescu [10], cf. [91, Chapter 5]. THEOREM 1.13. An operator T is the limit of a sequence of nilpotent operators if and

only if (i) T has connected spectrum and essential spectrum containing O, and (ii) whenever T - )~I is semi-Fredholm, the Fredholm index is O. These conditions are easily seen to be necessary by elementary means. The special case of normal limits mentioned above follows from the finite dimensional approximations and an application of the continuous functional calculus. However, the general results require a good model theory for a dense set of Hilbert space operators. This requires interplay between infinite dimensional methods and analytic function theory. We return to some considerations of this result in the next section.

1.3. Quasidiagonality The class of quasidiagonal operators and block diagonal operators were introduced in Section 1.1. A priori, one might expect that quasidiagonal operators behave more like matrices than arbitrary operators. However, in many ways, it has proven to be a difficult class of operators to deal with. Still, they do occur in important ways in both operator theory and C*-algebras. Here we will limit our attention to questions related to questions about matrices. The class of block diagonal operators of the form T = ~j(t3>/1 Tj which most closely mimics matrices are those in which the summands Tj are of uniformly bounded dimension, say m. Such operators, after a rearrangement of basis, can be written as a direct sum of k x k matrices with diagonal operator entries for 1 ~< k ~< m. The Weyl-von Neumann-Berg Theorem shows that every normal operator is the norm limit of diagonalizable operators. More generally, the same methods show that any finite set of commuting normal operators

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can be simultaneously approximated by diagonal operators. Thus the closure of the set of operators which are k x k matrices with diagonal entries is the set of k x k matrices with commuting normal entries. These operators are called k-normal. From a more algebraic point of view, the C*-algebra generated by an m-normal operator has the property that there is a family of irreducible representations of dimension at most m which separate points in the algebra. Conversely this condition implies that the operator is the direct sum of k-normal operators for k ~< m. In this section, we will concern ourselves with a variety of questions dealing with the approximation of operators in the quasidiagonal class. There are also many connections with C*-algebras which will have to be neglected. Consider first the question of whether every quasidiagonal operator can be approximated by m-normal operators with m unlimited. The first attempt would be to try to approximate large finite matrices by a direct sum of smaller matrices. This works very well for weighted shifts using Berg's technique [28]. This method has been exploited by Herrero [93] in obtaining good estimates for the distance between unitary orbits of weighted shifts. With Berg, the first author [29] expanded this method to work for general block tri-diagonal forms in their proof of a quantitative Brown-Douglas-Fillmore Theorem [47,48]. Further evidence comes from the infinite dimensional setting where Voiculescu's celebrated generalized Weyl-von Neumann Theorem [172] implies, among other things, that every operator is the norm limit of operators with many reducing subspaces. Indeed, every operator T is an arbitrarily small compact perturbation of an operator unitarily equivalent to Tt = T G A G A | where A is the image of T under a ,-representation of C* (T) which annihilates K~(7-/) n C* (T). This is especially good since T and T t have the same closed unitary orbit (whence T ~ is also the limit of operators unitarily equivalent to T). In spite of this positive evidence, the desired decomposition of large matrices into smaller ones is not possible [95]. THEOREM 1.14 (Herrero-Szarek). There is a computable constant 6 > 0 so that f o r every n >~ 2, there is an n x n matrix An o f norm one which cannot be approximated within 6 by any matrix which decomposes as an orthogonal direct sum o f smaller matrices. The proof is probabilistic. It measures the set of reducible operators, and shows that small balls centered there cannot cover the whole ball of 9Jtn. It is based, in particular, on precise estimates for metric entropy of the unitary groups U (n) and Grassmann manifolds [ 156,157] obtained by the second author while studying problems in geometry of Banach spaces (see [163] for similar more recent results motivated by questions in operator algebras and free probability). Herrero conjectured that the optimal value of 6 is probably 1/2, or at least close to 1/2. However, the 6 found in [95] was quite small, 1.712 x 10 -7. The approach from [24] (cf. Section 2.5) will certainly yield a much better, but still not quite optimal, constant. This negative evidence is not sufficient in itself to answer the question of approximation by m-normals. Nevertheless, the union of the set of m-normals over all finite m is not norm dense in the set of quasidiagonal operators. This was established by two different methods. The second author [159] pushed the probabilistic argument harder and was able to show the existence of block diagonal operators which cannot be uniformly approximated by m-normals.

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Voiculescu [ 174,175] used more algebraic methods to find an obstruction to approximation. He showed that if T can be approximated by m-normals, then the injection of C* (T) into/3(7-/) is a nuclear map. This is equivalent to saying that C* (T) can be imbedded as a subalgebra of a nuclear C*-algebra. Due to the work of Kirchberg, this class is known to be the set of exact C*-algebras. Voiculescu then shows by non-constructive methods that many block-diagonal operators do not have this property. In the second paper, property T groups are exploited to yield a more constructive approach. These papers describe a path leading to explicit examples by using explicit representations of explicit groups with property T such as S L3 (Z). However, they stop just short of actually producing such an example. See Section 2.5 for further discussion of "constructivity" in this and other contexts. On the other hand, Dadarlat [53] has positive results which show that this exactness condition could be precisely the obstruction. He shows that if T is quasidiagonal and C* (T) contains no non-zero compact operators, then T is the limit of m-normal operators if and only if C*(T) is exact. Voiculescu's Weyl-von Neumann Theorem is an important tool in the approximation theory of Hilbert space operators. So when working with quasidiagonal operators, it is natural to want to use this result and stay within the set of quasidiagonal operators. Generally one wants to approximate T by operators unitarily equivalent to T G A or T G A ~ ) where A = p(rc(T)) and p is a faithful representation of C*(T)/(/C(7~) n C*(T)). We mention in passing that this somewhat resembles the "Petczyfiski decomposition method" used in Banach space theory. So the question arises whether p can be chosen to be quasidiagonal when T is quasidiagonal. Indeed a C*-algebra is called quasidiagonal if it has a faithful quasidiagonal representation. Clearly this property is preserved by subalgebras, but our question asks about specific quotients. It turns out to have a negative answer. Wassermann [ 180,181 ] constructs several counterexamples. He uses the fact that the reduced C*algebra of a non-amenable group cannot be quasidiagonal. His first example uses a connection of the free group to Anderson's example [7] of a C*-algebra 92 for which Ext(92) is not a group. His second paper exploits property T. On the other hand, in [62] it is shown that the quotient of a nuclear quasidiagonal C*-algebra by the compact operators remains quasidiagonal. Theorem 1.13 of Apostol, Foia~ and Voiculescu [10] characterized the closure of the nilpotent operators. An interesting corollary is that the closure of the algebraic operators, those satisfying a polynomial identity, is the set of operators satisfying only the second condition (ii) whenever T - XI is semi-Fredholm, the Fredholm index is 0. Another deep result of Apostol, Foia~ and Voiculescu [9] characterizes this class as the set of biquasitriangular operators. A triangular operator T is one which has an orthonormal basis {en: n ~> 1} in which the basis of T is upper triangular. Say that T is quasitriangular if it is the limit of triangular operators, normally with respect to different bases. An operator is biquasitriangular if both T and T* are quasitriangular. Since every matrix can be put into triangular form, it is easy to see that every block diagonal operator is triangular. Consequently, every quasidiagonal operator if biquasitriangular. The converse is far from correct. Within the class of quasidiagonal operators, one may ask for a description of the closure of all nilpotent or all algebraic quasidiagonal operators. Since quasidiagonal operators trivially satisfy condition (ii) of Theorem 1.13, the natural conjecture might well be that

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property (i) should describe the closure of quasidiagonal nilpotents while the closure of quasidiagonal algebraic operators might be all quasidiagonal operators. Unfortunately, the answer to the nilpotent question is no; and the algebraic question remains open. Herrero [94] considered the operator [ ~ / ] where D is diagonal with eigenvalues dense in [0, 1]. This operator is the direct sum of 2 • 2 matrices, and so is 2-normal. The spectrum and essential spectrum is [0, 1] which is connected and contains the origin. So this operator is the norm limit of nilpotent operators. However, there is a trace obstruction to being the limit of nilpotent quasidiagonal operators. A natural finite dimensional question could play a central role in solving this problem. PROBLEM 1.15. If T is an n x n matrix, can the distance from T to the set Nilk of nilpotent matrices of order k be estimated in terms of the quantity IITk II ~/ ~ ? On the other hand, it is easy to exhibit Herrero's operator as the limit of quasidiagonal algebraic operators- just approximate D by diagonal operators with finite spectrum. So he asked whether every quasidiagonal operator is the limit of quasidiagonal algebraic operators. In [62], a special case of this conjecture is verified. THEOREM 1.16 (Davidson-Herrero-Salinas). Suppose that T is quasidiagonal, its essential spectrum does not disconnect the plane, and C*(T)/(IC(~) A C*(T)) has a faithful quasidiagonal representation. Then T is the limit of quasidiagonal algebraic operators. We mention a problem from [62] in the spirit of this survey article. A positive solution would provide a positive answer to the algebraic approximation question. PROBLEM 1.17. Given e > 0, is there a constant C (e) independent of n such that for every matrix T with [[T [[ ~< l, there exists a diagonal operator D and an invertible operator W such that

IT -

WDW -1

and

]]W[[ [[W-' [1 ~ C(e)?

A compactness argument shows that for each fixed dimension n, there is a constant C(e, n) which works.

1.4. Extensions of pure states and matrix paving In this section, we will discuss a well-known problem of Kadison and Singer [99], which asks whether every pure state on the algebra D of diagonal operators on ~2 with respect to the standard orthonormal basis extends uniquely to a (necessarily pure) state on s163 We note that 79 is a generic discrete maximal abelian subalgebra or masa. It is proved in [99] (see also [4]) that the answer to the uniqueness question is negative for non-discrete masas. The interest in the Kadison-Singer problem lies, in particular, in the fact that a positive solution would shed new light on the structure of pure states on s163 In fact it would go a long way towards a simple characterization of such states. Indeed, pure states on 79 ~ s _~

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334

C (fiN) are just point evaluations at elements of fiN, the Stone-Cech compactification of N. The canonical extension of a state from 79 to s is obtained by composing the state with the conditional expectation E which takes each T in s to its diagonal part in 79. In other words, if L/is the ultrafilter associated to a pure state q) on 79, then the extension 7t to/~(/~2) is given by ~ ( T ) = ( p ( E ( T ) ) = lim(Ten,en). n~/g

This suggests the open question raised in [99, w whether every pure state on/~(~2) is an extension of a pure state on some discrete masa, or indeed a given masa. However, it has been shown in [6], cf. [4, w that the two questions are in fact equivalent. See also [5] for a related positive result for the Calkin algebra. On the other hand, every pure state on/~(~2) is of the form qg(T) = limncU (Txn, Xn) for some sequence (Xn) of unit vector in ~2 and some ultrafilter in N [ 186]. Moreover, if q9 is not a vector state, the weak limit w-limneU Xn is necessarily 0. Since weak-null sequences in ~2 can be refined to be asymptotically orthogonal, it follows that the difficulty in settling both this and the uniqueness question lies in the difference between limits over ultrafilters and regular limits. Thus, at the first sight, the Kadison-Singer problem seems to be a strictly infinitedimensional question. However, it was shown by Anderson [4] that it is equivalent to a finite dimensional question known as the paving problem. PROBLEM 1.18. Does there exist a positive integer k such that, for any n >~ 1 and any matrix A 6 9Jtn with zero diagonal, one can find diagonal projections P1, P2, 9.., Pk 6 9Jtn such that

(i)

EPi=I

and

PiAPi

(ii)

i=1

1

~ IIAII?

i=1

This is clearly the kind of question that fits into our framework. To clarify the connection we will sketch the argument. Observe first that Problem 1.1 8 is formally equivalent to the following: Given s > O, does there exist a positive integer k such that, for any matrix T in s one can find diagonal projections P1, P2, ..., Pk such that ~ 1 Pi - I and k

k

E( ))Pi

(ii') i=1

E

PiTPi - E(T)

~<slITII?

i=1

Indeed, one gets s in place of 1/2 in the condition (ii) via iteration, with k depending on s. Passing from finite matrices with a uniform estimate on k to infinite matrices considered as operators on ~2, one obtains the partition of N corresponding to the decomposition of the identity on ~2 into a sum of projections from finite partitions via a diagonal argument. Let q9 be an extension to s of a pure state on 79. We claim that qg(D1TD2) = q)(D1)qg(T)q)(D2)

for D1, D2 E 79, T E s163

(l)

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We note that a priori ~p is multiplicative only on 79 as a point evaluation on C (fiN) _~ 79. We postpone the proof of (1) and show first how it allows to complete the argument showing that an affirmative answer to Problem 1.1 8 implies that q9(T) -- q9(E (T)) for T E/2 (~ 2 ). Let e > 0. Fix T ~ E(g2) with IIT II <~ 1, and let P1, P2 . . . . . Pk be the projections given by the affirmative answer to infinite variant of Problem 1.1 8. We now identify 79 with ~oo. P1, P2 . . . . . Pk correspond then to a partition of unity in ~ into a sum of k indicator functions of subsets. In this identification, qglz~ is obtained as a limit with respect to certain ultrafilter b/. It follows that among the numbers q9(Pi), i = 1, 2 . . . . . k, exactly one is equal to 1 and the others are all 0, depending on whether the corresponding subset belongs to b/ or not. Accordingly, by (1),

(P

Pi TPi i--1

-- Z

(P(Pi TPi) -- Z

i=1

(p(Pi)(p(T)(p(Pi) = (p(r).

i--1

Therefore

IIk

e >~ ~ _ P i T P i - E ( T )

q)

Pi TPi

- ~o(E(T))

i=1

Since e > 0 was arbitrary, it follows that qg(T) = qg(E (T)), as required. It remains to prove (1). To this end, consider the GNS representation of (E(g2), ~p). There is a Hilbert space H, a norm one vector x 6 H and a ,-representation 7r of s on/2(7-/) such that for all T in s =

An elementary argument shows then that x must be an eigenvector for each 7r(D), D ~ 79 with an eigenvalue qg(D). Hence, for any T in E(g2),

~o(D1TD2) = (Tr(D1TD2)x, x) -- (Tr(D1)rc(T)Tr(D2)x, x) = (Tr(T)Tv(D2)x 7r(D1

x)--(Tr(T)(qo(D2)x) 9o(D1)x)

= qo(D2)qo(D1)(rc(T)x, x ) = qo(D2)qo(D1)qo(T). The converse, uniqueness implies paving [4, (3.6)], can be proved in a similar spirit. One way is to show first that paving is implied by the relative Dixmier property, which says that for T 6 E(s

79(T) := conv{U*TU: U ~ D, U unitary} M 79 ~- 0. This immediately implies that E (T) belongs to 79(T). This is done in very much the same way as the argument presented above. On the other hand, if E ( T ) is not in 79(T), then E(T) can be separated from 79(T) by a functional qg. Heuristically, in view of the balanced

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nature of 79(T), q9 may be assumed to be positive and with more work, an extension of a pure state on D. By construction, q) is different form the canonical extension obtained by composing with the conditional expectation E. There has been a lot of work on Problem 1.18 in the intervening years, mostly yielding partial and related results. In particular, it was shown by Berman, Halpern, Kaftal and Weiss [30], cf. [43,98], that the answer is positive for matrices with nonnegative entries. A major progress was achieved by Bourgain and Tzafriri. Their work was motivated principally by applications to local structure of L p-spaces, see [98] in this collection for more details. First, in [42], they showed that for Problem 1.18, there exists a diagonal projection P with IIP A P II ~< 89IIA II and rank P ~ 6n, where 6 is a universal positive constant. A closely related result was obtained earlier by Kashin [ 102]. It then clearly follows by iteration that there is a decomposition of identity P1, P2 . . . . . Pk verifying the conditions (i) and (ii)of Problem 1.18 such that k = O(logn). Then in [44], they obtained by far the strongest results to date. Problem 1.18 is solved in the affirmative when the absolute values of entries of the matrix A are relatively small, specifically O(1/(logn) l+~) for some 7/> 0 (Theorem 2.3). Their solution also applies to the cases of Hankel and Laurent matrices with certain regularity properties But the major accomplishment is a saturation result that follows. As earlier, we identify s with the algebra 79. Similarly, the power set P(N) --= {0, 1}r~ is thought of as a subset of s In particular, a C N is associated with the sequence (aj) ~ s the indicator function of a , and with the diagonal projection P,, in 79. THEOREM 1.1 9 (Bourgain-Tzafriri). Given ~ > 0 and T ~ /2(~2), one can find a positive measure v supported on the w*-compact set

K - K(r, of total mass

Ilvll ~

c N: IIP (r - E(r))P

[I <

c

CE -2 f o r which

fK aj dr(or)/> 1 f o r all j >~ 1, where C is a universal numerical constant.

For clarity, we state also the finite dimensional version of Theorem 1.19 from which the theorem easily follows by a diagonal argument. PROPOSITION 1.20. There is a universal constant c > 0 so that given ~ > O, n ~ N and a matrix A E 9J'(n with zero diagonal, there exist a finite sequence of nonnegative weights (ti) with E i ti = 1 and diagonal projections (Pi) such that

IleiAPi II ~ EIIAII f o r all i >~ 1

and

E

ti Pi ~ c~ 2 I. i

Let us point out first that the last condition in this proposition implies by trace evaluation that the rank of at least one of the Pi's is at least ce2n, thus recovering the result from [42]

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mentioned above with the best possible dependence on ~. Note also that, except for that optimal dependence, it is enough to prove Proposition 1.20 for some ~ E (0, 1). We wish to emphasize that Theorem 1.19 is very close to implying the affirmative answer to the Kadison-Singer problem. Indeed, if we somewhat carelessly apply a (pure) state q9 to both sides of the assertion of Theorem 1.19, we "obtain" (2)

~O(fKadV ) -- fKcP(a)dv ~ 1.

Hence there is a a E K such that qg(a) = qg(Po) > 0. Thus qg(Po) = 1 because q9 is multiplicative on 79 and Po is idempotent. We now proceed as in the derivation of the original Kadison-Singer problem from the paving problem. As a 6 K, one has IIP~(T - E(T))P~II <~~IITII a n d s o -

E(T))I

:

(T -

E(T )P )I

IITII

for all e > 0. Therefore (p(T) = (p(E(T)), whence qg(T) is determined by its diagonal part. The weak point of this "argument" lies in the fact that the integral fK a d r ( a ) makes only weak* sense, while in equality (2) we implicitly used weak convergence. The argument would work if the measure v was atomic, though. Still, Theorem 1.19 shows that there is an abundance of diagonal projections P~ verifying [[P~ (T - E(T))P~ 11~< ellT[[. That abundance just isn't formally strong enough to guarantee that the collection of such a ' s will intersect every ultrafilter. We show now a simple example to that effect. Let d E 1~ and let Id be the set of all words of length 2d in the alphabet {A, B } consisting of d A's and d B's. So n := #Id = (2d)!/(d!) 2. Next, for s = 1, 2 . . . . . 2d, let as be the set of those words in Id whose sth letter is A. Clearly, Z'd :-- {al, 0"2 . . . . . CrZd } provides a cover of Id. It is not a minimal cover, but d + 1 sets are required. Let r[t be the hereditary subset of the power set 79 (Id) generated by Zd, namely X~ := {a: a C as for some 1 ~< s ~< 2d}. Then every subcover, or partition, of Id consisting of elements of r ~ must have at least d + 1 -- O(logn) elements. On the other hand, it is easily seen that 1 ~-~,2~l Xo, -- 89Thus the set of projections {Po" a E X~} verifies the condition in the assertion of Proposition 1.20, but does not verify the condition of Problem 1.18 with k independent of n. An infinite example verifying the condition of Theorem 1.19, but not that of the infinite variant of Problem 1.18 is routinely obtained by identifying N with ~ d Id and setting Z'--{aCI~:

aAIdEr~foralld~l}.

Of course, this is just a combinatorial contraption. There is no a priori reason why r produced above would correspond to an actual operator T E s163 The methods of [42-44] are quite sophisticated. Without going into details, we mention that the first step in finding large subset a C { 1, 2 . . . . . n} for which IIP~ A P~ I[ is small involves a random procedure. The first approximation is a = {j: ~j = 1 }, where

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~n is a sequence of independent Bernoulli selectors. These are random variables satisfying P(~j = 1) = 1 - P(~j = 0) - - ~ for j = 1,2 . . . . . n for some properly chosen E (0, 1). The norm of P~ A P~ is then the maximum of a random process, which is analyzed using decoupling inequalities (see, e.g., [106]), majorized by a more manageable maximum of a Gaussian process, and estimated via metric entropy and Dudley's majoration (see [112, (12.2)]). This works if the entries of the matrix A are rather small, such as the O ( 1 / ( l o g n ) l+u) bound mentioned earlier. Such random subsets also yield a partition of {1,2 . . . . . n }. In the general case, one only obtains an estimate on the norm of P~ A P~ as a map from s to ~ . In the context of Proposition 1.20, one has instead a weighted s The final step uses the Little Grothendieck Theorem [ 137]. Alternatively, some of the steps may be done by using the measure concentration phenomena (cf. Section 2.2) already employed in [103] or majorizing measures, cf. [165]. See [98] in this collection for more details on some of the above arguments and related issues. We conclude this section by commenting on the type of examples that need to be analyzed in hope of further progress. To be a potential counterexample, a matrix A -- (aij) ni j__ 1 (meaning a sequence of n • n matrices that together provide a counterexample) must have the following features: (i) ]lAB] must be much smaller than ]](]aij ]) ]], or otherwise we could apply the argument that works for nonnegative entries. (ii) ]aij] do not admit a (uniform) O ( 1 / ( l o g n ) l+u) bound. (iii) On the other hand, the substantial entries of A must be sufficiently abundant, or otherwise one could avoid them by the same combinatorial argument that works for nonnegative entries. (iv) The combinatorial structure of the substantial part of A must be quite rigid to distinguish between the conditions from Proposition 1.20 and Problem 1.18. One structure that comes to mind is related to adjacency matrices of Ramanujan graphs (see, e.g., [116]). Let B = (bij) be such a matrix corresponding to a d-regular graph on n vertices. Then []B ]] = d, and it is achieved on the eigenvector (1, 1 . . . . . 1), while all the remaining eigenvalues are bounded by 2~/d - 1. So the n • n matrix ( 89(bij - k ~ n ) / ~ / d - 1) is of norm at most 1, and appears to enjoy the features (i)-(iv), some of which are admittedly vague. The question would then be to determine whether matrices obtained this way from various constructions of Ramanujan graphs can be paved. It seems at the first sight that new techniques are required for any kind of answer. ~1, ~2 . . . . .

1.5. Hyper-reflexivity If r is an operator algebra contained in s then L a t A denotes the lattice of all of its invariant subspaces. Dually, given a collection 12 of subspaces, Alg/2 denotes the algebra of all operators leaving each element of 12 invariant. The algebra ,A is reflexive if Alg Lat,4 = A. There is a quantitative version of reflexivity which has proven to be a powerful tool when it is available. Notice that if PL is the projection onto an invariant subspace L of .,4, then for any operators T E s and A 6 .A,

II

II- II

A)P li

-

All.

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339

Hence the inequality

fls

"-- sup

IIPL TPL LI

dist(T, A).

LE/Z

The algebra A is hyper-reflexive if there is a constant C such that dist(T, A) <~ Cfls

for all T E/3(7-/).

We also say that an operator A is reflexive or hyper-reflexive if the unital WOT-closed algebra W ( A ) generated by A is reflexive or hyper-reflexive respectively. In finite dimensions, the question of which operators are reflexive was solved by Deddens and Fillmore [69] in terms of the Jordan form. Evidently, every reflexive subalgebra of 9Ytn is also hyper-reflexive; but there is no a priori estimate of the constant even in two dimensions. For example, it is very easy to show that the 2 • 2 diagonal matrices 792 are hyper-reflexive with constant 1. Similarity by an operator S will preserve hyper-reflexivity, but can change the constant by as much as the condition number [IS I] IIS-1ll 9Thus it may not be too surprising that the hyper-reflexivity constant of St 792S t | increases to +e~ with t it

when St - [0 1 ]. There are two classical situations in which the existence of hyper-reflexivity has played a key role. The first is nest algebras. A nest A/" is a chain of subspaces containing {0} and 7-/ which is closed under intersections and closed spans. The nest algebra is T(A/') = AlgA/" is the algebra of all operators with an upper-triangular form with respect to this chain. In particular, the algebra T~ of n x n upper triangular matrices with respect so some basis is the prototypical finite dimensional example. An early result of Ringrose [ 145] shows that T(A/') is reflexive, as well as the fact that LatT(A/') --A/'. Arveson [13] showed that nest algebras are hyper-reflexive with constant 1, so that one obtains the distance formula: THEOREM 1.21 (Arveson). Let jV" be a nest, and let A be an arbitrary operator in 13(7-[). Then dist(A, T ( H ) ) -

sup

[IPAPNII.

NcA/"

An easy proof [144], cf. [58], can be based on the well-known matrix filling lemma of Parrott [ 130] and Davis-Kahane-Weinberger [68] valid for matrix or operator entries:

inf xcs

A

B

-max{ll[A

n]ll,

o

In particular, Lance [108] used this distance formula to establish that close nests are similar. Likewise, this fact was a key ingredient in the first author's Similarity Theorem [54] stating that two nest algebras are similar if and only if there is a dimension preserving isomorphism between the underlying nests. The search for generalizations of this will lead to an open question below.

340

K.R. Davidson and S.J. Szarek

The second context in which this concept occurs is the von Neumann algebras. Christensen [52] showed that many von Neumann algebras (specifically all whose commutant do not have certain type H1 summands) are hyper-reflexive. This is closely related to derivations. Recall that a derivation is a map 8 of an algebra .A into an .A-bimodule A4 satisfying the product rule 8(ab) -- aS(b) 4- 6(a)b. The derivation is inner if there is an element x e .A4 such that 8(a) = ax - xa. Christensen showed: THEOREM 1.22. For a v o n Neumann algebra 92 contained in s the following are equivalent: (i) Every derivation 8 o f 92~ into s is inner. (ii) There is a constant C such that dist(X, 92) ~< C l l S x l ~ , l l f o r all X ~ s The second condition actually asserts that 92 is hyper-reflexive. The invariant subspaces of any C*-algebra correspond to projections in the commutant. Every von Neumann algebra is spanned by its projections. Thus von Neumann's famous double commutant theorem, which asserts that (92~)~ = 92, shows that every von Neumann algebra is reflexive. Moreover, the convex hull of the symmetries 2P - I, for P projections in 7)(92), is the whole unit ball of the self-adjoint part of 92. Now a simple calculation shows that

l[(2P -

I)X-

X ( 2 P - l)ll -

2llpxP"

- P'xP]l

2max{llexp•

Ile xell}.

Thus since every operator is the sum of its real and imaginary parts, 118xl~'ll % 4

sup IIP'xPII"

P e7)(9~)

So (ii) is equivalent to hyper-reflexivity. For all injective von Neumann algebras, the constant is at most 4 [52]. And for abelian von Neumann algebras, the constant is no more than 2 [ 146]. Even for the 3 x 3 diagonal matrices, the constant is greater than 1. Indeed, it is exactly ~/-3/2 [63]. So unlike the nest case, most examples involve non-trivial constants, not exact formulae. Now we turn our consideration to the class of woy-closed reflexive algebras which contain a masa, known as CSL algebras. The terminology is short for commutative subspace lattice algebra because any invariant subspace is, in particular, invariant for the masa, and therefore corresponds to a projection in the masa. So the orthogonal projections onto all of these invariant subspaces commute with each other. These algebras were introduced in a seminal paper by Arveson [14]. The finite dimensional versions occur in many contexts, and are also called incidence algebras or digraph algebras in other parts of the literature. A masa in ~)T~n is unitarily equivalent to the diagonal algebra 7)n. Any algebra containing Dn is determined by the standard matrix units Eij which belong to the algebra. Moreover, since this is an algebra, the set of matrix units is transitive in the sense that if Eij and Ejk belong, then so does Eik. Thus one may associate a directed graph to the algebra with n vertices representing the

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341

standard basis, and including a directed edge from node j to node i if Eij lies in the algebra. In this way, we obtain the graph of a transitive relation. Not all CSL algebras are hyper-reflexive. Davidson and Power [67] constructed finite dimensional examples with arbitrarily large distance constants. This is more subtle than the easy example mentioned earlier. However, Larson [109] showed that this construction was part of a general mechanism for producing examples of this type. On a positive note, Davidson and Pitts [65] showed that if there is a dimension preserving lattice isomorphism between two CSL algebras, then the two lattices are approximately unitarily equivalent. This was established for nests by Andersen [3], and was a key step in obtaining similarity invariants. Pitts [ 142] was able to obtain good perturbations results for the algebras without using hyper-reflexivity in spite of the fact that hyper-reflexivity was used in an important way in the nest case. But other possible extensions of results for nests to this more general context have been hampered by the lack of hyper-reflexivity. For this reason, attention has been focussed on an important finite dimensional case (see [58,59]). The algebra ,An = T~ | Tn represents the subalgebra of ~lY~n2 consisting of n x n upper-triangular matrices with n x n upper-triangular matrix entries. The problem becomes: PROBLEM 1.23. Is there is a finite upper bound to the hyper-reflexivity constants of all of the algebras ,An ? This is a typical situation where one understands well the one dimensional case, but not the higher dimensional or multivariable case. Similar difficulty appears when studying, e.g., multi-indexed orthogonal expansions. A few other examples of hyper-reflexivity are known and are worth mentioning. The unilateral shift generates the analytic Toeplitz algebra, and the first author [57] showed that it is hyper-reflexive. Then with Pitts [66], he showed that the algebra generated by the left regular representation of the free semigroup on n letters is also hyper-reflexive. Popescu [143] generalized this to a wider family of semigroups. Recently, Bercovici [26] used predual techniques to establish hyper-reflexivity constants for the large class of algebras with property Xo,• This property, which we do not define precisely, allows approximation of any weak-, continuous functional on the algebra by a sequence of rank one functionals tending to infinity in a very strong sense. These notions arose in using predual techniques to establish reflexivity for single operators beginning with the celebrated theorem of Scott Brown [49] that every subnormal operator has invariant subspaces. Olin and Thomson [128] showed that subnormal operators are reflexive. And we mentioned above that the prototypical subnormal operator is hyper-reflexive. Many subnormal operators fit into Bercovici's criterion, but the general question of hyper-reflexivity for subnormal operators remains open.

2. Random matrices In this chapter, we shall present a selection of classical and not-so-classical results on asymptotic spectral properties of random matrices that are related one way or the other to

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the geometry of Banach spaces. As was the case with other topics, these connections come in several flavors. First, facts about random matrices are being applied to the geometry of Banach spaces. Second, the methods of the Banach spaces yield new results or offer an alternative perspective on random matrices. Finally, there are connections via vague analogies between the fields.

2.1. The overview Random matrices appeared in Banach space theory in an explicit way some time in the 70s, for example in [23,25] and [101], even though one can claim that their spirit was already present, e.g., in probabilistic proofs of Dvoretzky theorem, cf. [81] in this collection. Let us quote here a result from [23]: THEOREM 2.1. Let q E [2, oc) and f o r some m, n E N, let A = ( a i j ) be an m x n random matrix whose entries aij -- aij (co) are independent, mean-zero real random variables with [aij [ <<.1 f o r all i, j. Then

Ell A

~ ~qm

II

~

K m a x i m 1/q n 1/21

where E denotes the expectation and K -= K (q) is a numerical constant depending only onq.

This is a very typical statement as far as the asymptotic theory of finite dimensional Banach spaces is concerned. We have an estimate giving the correct rough asymptotic order (see the introduction for terminology) and involving a universal numerical constant. Usually we do not know, and often do not really care about, the optimal value of that constant, i.e., the precise asymptotic order. This could be viewed as an unsatisfactory situation from the point of view of other related fields. We shall return to this issue later on. On the other hand, in spite of the "asymptotic" qualification, we do have above an inequality valid for any m, n, a crucial detail for applicability to fields like geometry of Banach spaces, computational complexity and approximation theory. On the other hand, random matrices were of interest to statisticians at least since the 20s, and to theoretical physicists at least since the 50s, see [121]. Perhaps the most celebrated result coming from the latter area is the Wigner's Semicircle Law [184,185] which says that, under some weak regularity assumptions, the spectra of large symmetric random matrices are, in a sense, virtually deterministic and their spectral densities are, after proper normalization, asymptotically semicircular. More precisely, Wigner proved a somewhat weaker statement: THEOREM 2.2. For each n E N, let A -- A (n) (co) be an n x n random matrix whose entries a~; ) (co)" I2 --+ IR are symmetric real random variables satisfying f o r each n ~ 1 and l<.i,j~n, (i) Elaij 12 --- 1/n.

aij =

(ii) a~; ) are independent f o r 1 <. i <. j <. n. (n) (iii) a{~.) - - a j i .

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343

(iv) For each m ~ 1, there is a tim < CXDindependent of n so that E[aij Im ~ tim. Then, f o r any v E N, lim ETr(A v) -- fR xv d#(x),

(3)

n---->oo

where Tr is the normalized trace in the respective dimension and # is the measure on I[{ whose density w is given by

w(x) --

1 x/4_x 2 TY 0

i f x E [ - 2 , 2], if x ~ [ - 2 , 2].

The measure # above is often referred to as the standard semicircular distribution. Theorem 2.2 says in effect that, for large n, the random measure l // # n - - # n (CO) "-- -- Z (~)~j(A), n j=l

where 6x is the Dirac measure at x and ~.l(B) ~> ..- ~> ~.n (B) are eigenvalues of B e 9J~)]a, is approximately standard semicircular when considered as a measure on ~2 x • (which is the same as convergence in distribution in the free probability sense, as explained in Section 2.4). In fact, as was implicitly assumed by physicists and proved later (cf. [12]), a much stronger statement holds: for large n the random measures # , (co) are "close" to # with probability close to 1 (convergence in probability). In particular, #n (co), the empirical measure associated to the spectrum of A (n), is nearly deterministic. We emphasize that this is far from being a formal consequence of Theorem 2.2. It is possible in principle that the assertion of the theorem holds while, for any co E ~2, A (co) is a multiple of identity: just consider a scalar random variable ~ whose law is standard semicircular and a random matrix A :-- ~ I. However, we do have THEOREM 2.3. In the notation and under the hypotheses of Theorem 2.2 one has, f o r any interval I C ~, lim n--->oo

#{eigenvalues of A (n) contained in I }

= #(Z)

almost surely.

n

There are similar results for the asymptotic distribution of singular values of random non-selfadjoint matrices; e.g., when the hypothesis (ii) in Theorem 2.2 is replaced by (ii) a~; ) are independent f o r 1 <, i, j <, n and the hypothesis (iii) is dropped. The limiting distribution is then a quartercircle law (singular values being necessarily nonnegative), supported in the interval [0, 2] and given by the density ?1 ~ / 4 - x 2 Similarly, one may consider (see [118,179]) large rectangular matrices such that the ratio of the sides is roughly fixed, cf. Theorem 2.13, and many other "ensembles" (cf. the remarks preceding Theorem 2.7).

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K.R. Davidson and S.J. Szarek

The example preceding Theorem 2.3 notwithstanding, the implication Theorem 2.2 =~ Theorem 2.3 is in fact a formal consequence of rather standard local theory techniques, even though this was not the way how it was historically shown. We shall sketch the argument later in this chapter, after Theorem 2.7; see also Corollary 2.12 and Theorem 2.17. As a bonus, one obtains fairly strong estimates on the probabilities implicit in the almost sure part of Theorem 2.3. That proof, and many other arguments presented here, while being a folklore among Banach space theorists, are not exactly a common knowledge in the random matrix circles. As opposed to the techniques emphasized here, standard tools used in that area involved the moment method (roughly, working directly - via a heavy duty combinatorics- with the moments involved in the limit in (3), cf. the remarks following Problem 2.18), the more precise Stieltjes transform method introduced in [118] (cf. [132]) and later adopted in and developed by the free probability approach, or, in the case of classical random matrices, analyzing the explicit formulae for joint densities of eigenvalues or singular values, see (8), which leads to the so-called orthogonal polynomials method, see, e.g., [121]. In other directions, it has been determined that much weaker assumptions on regularity of the entries suffice: one needs just a little bit more than the existence of second moments, with uniform estimates (more precisely, a Lindeberg type condition), and the symmetry hypothesis may be replaced by Eaij = 0 (see, e.g., [ 131,82]). We point out that assertions of Theorems 2.1 and 2.3 are not comparable. On one hand, Theorem 2.3 does not say anything about the norms of matrices. It is consistent with its assertion that o(n) largest (in absolute value) eigenvalues are far outside the interval [ - 2 , 2], the support of w. On the other hand, as indicated earlier, Theorem 2.1, being "isomorphic" in nature, does not give the precise order of the norm (even when q -- 2 and m = n), and it does not address the question of the asymptotic distribution of the eigenvalues (or singular values in the non-selfadjoint case). Again, common strengthenings of results of the two kinds have been obtained (see, e.g., [46,80,18]), but we do not know of any really satisfactory argument that encompasses simultaneously the two aspects of the picture. In the opposite direction, precise estimates were obtained on the (very small) probability that a specific eigenvalue (or singular value) of a random Gaussian matrix is far away from its theoretical value predicted by the corresponding Semicircle Law result. A sample such large deviation result, motivated by questions in geometry of Banach spaces and numerical analysis, is Theorem 2.4 below. Before stating the theorem, a few words about the level of generality of our discussion. More often than not, we shall concentrate on the central Gaussian case, and most of the arguments will be specific to that case. In particular, throughout this chapter G = G (n) will stand for an n x n random matrix whose entries gij, 1 <~ i, j <<,n, are independent identically distributed Gaussian random variables following the N(0, 1/n) law. Similarly, A = A (n) will usually stand for a Gaussian selfadjoint matrix verifying the hypotheses of Theorem 2.2. However, both the results and the methods employed are representative of much more general setting. It would be a useful project to work out in detail the consequences of general concentration results for product measures (see [166,167] and their references) and related "probability in Banach spaces" tools to the setting of random matrices. As indicated earlier, very few papers produced in the random matrix circles employ those methods and, to our knowledge, no vigorous research in this direction was attempted. It is

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345

clear that some consequences would be just a formal repetition of the arguments sketched in this section. Chances are that, to get best results, one may need to apply those tools rather creatively. The payoff can be substantial since important part of research in random matrices turns around "universality" of the limit laws, i.e., their independence of the particular probabilistic model involved, the "symmetries" of that model being the only meaningful parameter (the uniw;rsality conjecture). THEOREM 2.4 ([161,162]). Let G = G (n) be as above and S1 ~ $2 ~ "'" ~ Sn be its singular values. Then

(

dl

"7-0 Sn-d+l > ~-n

~

exp(-cfi2d2),

(

d)

s~-d+l <~~-n

<~ (C~) a2

f o r 1 <~ d <~ n, fl >~ [3o and ot ~ O. Above, c, C and flo are universal positive constants. Apart from the precise values of the constants, the estimates are optimal except possibly f o r the f o r the first one when n - d -- o(n). Analogous results can be obtained in the complex case and, for eigenvalues, in the selfadjoint case. The optimality of the theorem means here that there are similar lower estimates with c, C replaced by other positive universal constants. However, being an isomorphic result, the Theorem doesn't detect smaller deviations from the values predicted by the Quartercircle or Semicircle Law. Much more precise results of the same nature were obtained very recently in the case when d is asymptotically a fixed proportion of n [22]. One of the concepts employed there is the free (noncommutative) entropy [ 178], arrived at by following on the ideas sketched in Section 2.4 below. It is quite likely that the methods generalize to the case d = o(n), but probably not to the edge of the of the spectrum, i.e., the case when d is nearly equal to n; cf. Theorem 2.8 below and the comments following it. It would be potentially useful to clarify the relationship between Theorem 2.4 and these results. To complete the overview, we shall mention that, besides the large deviation results mentioned above and the results in the spirit of Wigner's Semicircle Law (the so-called global regime), there is a large body of research dealing with micro-local analysis of the spectrum {)Vl(A(co)) . . . . . )vn (A(co))}, and particularly the gaps )vj -- )v j + 1 in that spectrum (the local regime). These are natural features to consider in view of the original physical motivation, the eigenvalues being interpreted there as energy levels. Further attention to this direction was brought by the largely experimental results [ 127,77] relating the properly defined distribution of such gaps and the corresponding gaps between zeros of the Riemann ~" function; cf. [126,148]. In what follows, we shall state a result in the spirit of [122], the fundamental work concerning the local regime. It deals with a complex analogue of A (n) called the Gaussian unitary ensemble or GUE (defined more precisely in item (iii) the next section), for which Theorems 2.2 and 2.3 hold with the same limiting semicircle distribution. We need to introduce first some notation: given B ~ 93tsa with eigenvalues )vl (B) ~> 9.. ~> )vn (B) and x E R, we define 6B(x) to be the length of the interval ()vj-1 (B),~,j(B)] containing x, with the convention that 6 ~ ( x ) = oo if x > ~1 (B) or X ~ )vn(B ) and 68(x) -- 0 if x is a multiple eigenvalue.

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346

THEOREM 2.5. There exists a probability density cr supported on [0, cx~) such that when1 ~ / 4 - x 2 the density of the standard semicircle distribuever x E ( - 2 , 2) and w(x) - ~-y tion, then, f o r any s ~ O,

(

s)/0

lim 79 6GUE(X) <<, n~ nw(x)

--

a(u) du.

We point out that the scaling of 6GVE(X) by the quantity n w ( x ) in Theorem 2.5 was to be expected: by Theorem 2.3, for large n and a short interval 2- containing x, the number of eigenvalues of the random matrix that belong to 2- will be, with probability close to 1, approximately n f l w ( u ) d u ~ nw(x)12-1, and so the gaps should average about 1 / ( n w ( x ) ) . Added in proof There was some very recent progress in the direction of the universality conjecture, i.e., generalizing results from classical ensembles of random matrices to those whose distributions of the entries are more generic. For example, large deviation results related to Theorem 2.4 and [22] were obtained in [87]. In the language of Theorem 2.7 below, a version of the inequality (5) for the extreme eigenvalues )~l, )~n was shown in [ 105]. Both of these papers use the measure concentration phenomenon and specifically techniques of Talagrand [ 166,167], thus addressing some elements of the program sketched in this survey. Particularly [87] is a tour de force. Some reasonably general results related to Theorem 2.8 below were obtained in [154]. Finally, some initial progress on the topics relevant to Theorem 2.5 was achieved in [97].

2.2. Concentration of measure and its consequences In this section we shall sketch some applications of the measure concentration phenomenon, well known and widely applied in local theory of Banach spaces (see the article [ 150] in this collection), to the subject of random matrices. Most of these applications have been a folklore among some of the experts in the former field, but to the best of our knowledge, they haven't been presented anywhere in a systematic fashion. One of possible starting points is the Gaussian isoperimetric inequality [39], which we shall present here in the functional form (see [ 136] for the last assertion). Recall that a function F defined on a metric space (X, d) is called Lipschitz with constant L if IF(x) - F(x')l <~ L d ( x , x ~) for all x, x ~ E X. Let y -- Vn be the standard Gaussian measure on ~ n with density (27r) -n/2 e -Ixl2/2, where I" I is the usual Euclidean norm. As usual, ~ ( t ) "-- ~'1 ((-cx~, t]) is the cumulative distribution function of the N (0, 1) Gaussian random variable. THEOREM 2.6. Let F be a function on ~n which is Lipschitz with constant L with respect to the Euclidean metric and let M = MF be the median value of F with respect to Vn. Then, f o r any t > O, 79 (F ~ M + t) ~ 1 - ~ ( t ) < e x p ( - t 2 / 2 L 2 ) .

(4)

One has the same upper estimate e x p ( - t 2 / 2 L 2) if the median M is replaced by the expected value fR, F dyn.

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347

For future reference we point out here that, for a convex function, its median with respect to a Gaussian measure does not exceed the expected value; see [71,107] or Corollary 1.7.3 in [76]. The relevance of Theorem 2.6 to random (Gaussian) matrices is based on the elementary and well-known fact, described in Section 1.2 of this article, that spectral parameters like singular values (respectively eigenvalues in the selfadjoint or unitary case) are Lipschitz functions with respect to the matrix elements; see [35,33] for related more general results. In particular, for each k c {1. . . . . n}, the k-th largest singular value sk(X) (respectively 17 the eigenvalue ,kk(X)) is Lipschitz with constant 1 if X (Xjk)j,k=| is considered as an - -

element of the Euclidean space R n2 (respectively the submanifold of R n2 corresponding to the selfadjoint matrices). If one insists on thinking of X as a matrix, this corresponds to considering the underlying Hilbert-Schmidt metric. Accordingly, in the context of applying Theorem 2.6, the Lipschitz constant of sk (G (n)) is 1/~/-n (because of the variances of the entries being l / n , which corresponds to the identification G = G (n) = 1/x/~ X). Respectively, for a Gaussian selfadjoint matrix A = A (n) verifying the hypotheses of Theorem 2.2, the Lipschitz constant of )~k(A(~)) is v/2/n. The additional 2 is a consequence of the same variable appearing twice, in the jkth and kj th position. The above comments, and hence the results below, carry essentially word for word to the following often considered variants, all Gaussian unless explicitly stated otherwise. (i) A variant of A = A (") in which variances of the diagonal entries are assumed to be 2In rather than l / n , called often the Gaussian orthogonal ensemble or GOE. This is in fact the same as v/2 times the real part of G (n) . Note for future reference that GOE can be represented as Y + A, where Y is a (diagonal) Gaussian random matrix independent of A, and so many results for GOE transfer formally to A. (ii) The complex non-selfadjoint case, all the entries being independent and of the form g + ig', where g, g' are independent real N(O, 1/2n) Gaussian random variables. (iii) The complex selfadjoint case: formally the same conditions as in Theorem 2.2 (except for the obvious modification in the symmetry condition), but the abovediagonal entries are as in (ii) while the diagonal entries are real N(0, 1/n)'s. Again, this is the real part of the matrix in (ii) times v/2, and is frequently referred to as the Gaussian unitary ensemble or GUE. (iv) Rectangular, real anti-symmetric or complex anti-selfadjoint matrices. (v) Orthogonal or unitary matrices distributed uniformly on SO(n), respectively U (n). It is also easily seen that in the first three cases above the Lipschitz constants are respectively 2 x / ~ , 1 / v / ~ and 1/v/-n. The anti-symmetric/anti-selfadjoint/rectangular cases are treated the same way, and there are equally useful results for orthogonal/unitary matrices cf. [124,86]. There appears to be no easily available exposition of the unitary case, even though all ingredients are available, cf. [176]. Still, a word of caution is needed. As noted in Section 1.1, eigenvalues are not very regular functions of general (non-normal) matrices. Combining the above remarks and Theorem 2.6 we get THEOREM 2.7. Given n E N, there exist positive scalars Sl, $2 Sn such that the singular values of the n x n real Gaussian random matrix G = G (n) satisfy .

79(Isk(G)- ski ~> t) < 2 e x p ( - n t 2 / 2 )

.

.

.

,

(5)

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K.R. Davidson and S.J. Szarek

f o r all t ~ 0 and k -- 1, 2 . . . . . n. This holds, in particular, if sk's are the medians or the expected values of sk(G). Similar results hold f o r the eigenvalues of the real symmetric Gaussian random matrix A -- A (n) with - n t 2/4 in the exponent on the right side of (5). Moreover, the corresponding deterministic sequence ~.l, )~2 . . . . . )~n can be assumed to be symmetric, i.e., )~k = --)~n-k+l f o r k = 1, 2 . . . . , n. Likewise, related results hold f o r other Gaussian random matrices, in particular those described in (i)-(iii) above.

A shortcoming of the above result is that it doesn't see the possible relationships between the deterministic sequences sl, s2 . . . . . Sn (or)~1, ~.2 . . . . . )~n) for different n's. Still, it allows to formally deduce statements in the spirit of Theorem 2.3 from the corresponding Theorem 2.2 like results. Consider, as an illustration, the ensemble A = A(n). Once we know that there is a rough estimate on, say, EIIAII (e.g., of the type of Theorem 2.1), the analogue of (5) for A implies via an elementary calculation ETr(A v) - ~ x v d#(n)(x) <~

Ci)

(6)

for v 6 N, where #(n). ~1 Z j n= I 3)~j is the deterministic measure involving the expected values or medians of the eigenvalues of A and Cv is a constant depending only on v. Combining this inequality with Theorem 2.2 we deduce that #(n) __+ #, the semicircular distribution from Theorem 2.2, weakly as n --+ ec. The estimate on the right-hand side of (5) and the Borel-Cantelli lemma imply then the assertion of Theorem 2.3 for our ensemble A (n) or for any ensemble for which a result of the Theorem 2.7 variety holds. We emphasize that this is independent from how the matrices A(n) are stochastically related for different n's. We shall present even stronger results (Corollary 2.12, Theorem 2.17) in the next two sections. Another consequence of Theorem 2.7 is that, in the normalization we use, fluctuations of singular values or eigenvalues of n x n (Gaussian) random matrices are at most O(n-1/2): if t >> n-l/2, the exponent in (5) becomes "large negative". We need to point out that, in all likelihood, this is not an optimal result, and certainly not for the full scale of parameters. For example, it is conjectured that fluctuations of eigenvalues in the bulk of the spectrum (i.e., neither k = o(n) nor n - k = o(n)) are of the order O(n-1), and the conjecture is supported by the large deviation results quoted in the preceding section (see Theorem 2.4 and the paragraph following it). There are several partial results in that direction of varying degrees of generality and strength (see, e.g., [16,17]). In the Gaussian case, an improvement to the O(n -1/2) result "nearly" follows from the approach presented and the results stated in this article. For example, if we knew that the differences between the consecutive deterministic )~j's from Theorem 2.7 were of order 1/n (which we "almost" do, cf. Theorem 2.5), we could argue that, for a )~k(A) to be 0 or more away from its central value )~k, approximately the same would have to be true for con neighboring eigenvalues and so, by Theorem 1.8, the square of the Hilbert-Schmidt deviation from the most likely spectral picture would be of order t 2 = 03n; substituting this value, and L 2 = 2In, into an estimate of type (4) we would obtain a meaningful estimate whenever O/n -2/3 was large. Another promising approach would be in exploiting the representation of G (n) (which works also _

Local operator theory, random matrices and Banach spaces

349

for A (')) found in [152]. A related fact which is worth pointing out is that there are two sources contributing to the quantity in (6): the deviation of the Aj(A)'s from their central values k j ' s , and the difference between the deterministic measures #(n) and the limit semicircular distribution #, and both of them are of interest. For the very edge of the spectrum, i.e., the largest or smallest eigenvalues and perhaps a few adjacent ones and in particular for the norm of the matrix, the fluctuations are, in some cases, known to be of order n -2/3. A sample result (see [170]) is THEOREM 2.8. There exists an increasing function (p on IR such that the largest eigenvalue A I(GOE) o f the n • n Gaussian orthogonal ensemble satisfies lim 79(A1(GOE) <~ 2 + rn -2/3) --99(r)

(7)

t/----> o o

f o r r ~ JR, the convergence being uniform on compact subsets o f ]t{. Similar result (with a different 99) holds f o r the Gaussian unitary ensemble.

The reader will notice that exactly this size of fluctuations is predicted by the semicircle law itself: if we choose s > 0 so that 1/27r f22_,, x/4 - x 2 dx -- 1/n, then s ~ (3rc/(Zn)) 2/3. On the other hand, it is easy to see that (7) does not accurately predict the order of the probabilities without passing to the limit. For example, if r ~> n 1/6, which corresponds to t ~> 1 in Theorem 2.7, (5) yields the correct asymptotic order of l o g 7 9 ( k l ( G O E ) <, 2 + t), which, because of a difference in scaling, is inconsistent with the behavior that would have been suggested by (7). In view of possible applications in local theory, it would be potentially useful to recover the asymptotics on the probabilities involved, given by the arguments of [170], for the full range of the parameters, a similar project to the one suggested in the paragraph following Theorem 2.4 in the context of large deviation results. The proof of Theorem 2.8, and similarly, the proofs of Theorems 2.5, 2.4 and the other large deviation results, uses the explicit formulae for joint densities of eigenvalues (respectively singular values), which are of the form

l~j
l<~k<~n

where /3 = 1 or 2 depending on whether the context is real or complex and c(fl, n) is the normalizing numerical coefficient. Accordingly, one can not expect that it generalizes much beyond the Gaussian and some other classical random matrices. However, there is a strong circumstantial evidence that at least Theorem 2.5 (see [133,70]) and Theorem 2.8 are much more universal. Concerning the latter, it has been shown in [153] that the moments E Tr(A v), where A = A (') are real symmetric matrices satisfying any of our variance and independence assumptions and such that the laws of the appropriately normalized matrix elements are uniformly sub-Gaussian, exhibit, as n, v --+ oo, an asymptotic behavior which is consistent with the assertion of the theorem. A concentration result for the norm in similar degree of generality but going in a somewhat different direction can be found in [45]. As suggested earlier, it would be an interesting project to figure out relevant consequences of measure concentration phenomenon so successfully exploited in local theory.

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The phenomenon signalized in Theorem 2.8 has not been exploited in local theory of Banach spaces, and so it is not known whether, conversely, any of the methods of that area are relevant here. It would be extremely interesting to find a general concentration principle which would imply the small fluctuations of the norm and/or eigenvalues in the bulk of the spectrum. Techniques and ideas that may be relevant here (and, more generally, when investigating the "local regime") include hypercontractivity, logarithmic Sobolev or Poincar6 type inequalities, transport of measures and various concepts of information theory, see [21,38,110,111,129,168].

2.3. Norm of a random matrix and the Slepian-Gordon lemma In this section we shall present a simple argument giving exact asymptotics for extreme singular values (basically norms) and eigenvalues of some symmetric Gaussian random matrices (in particular all real Gaussian matrices considered here). The argument is based on the well-known Slepian's lemma from probability and its generalization due to Gordon. For greater transparence we shall state the relevant special cases of both variants. LEMMA 2.9. Let ( X t ) t ~ T and (Yt)tcT be two finite families of jointly Gaussian mean zero random variables such that

(a)

IIX, - Xt, II2 ~ IIYt - Yt, l[2 f o r t , t' ~ T.

Then E max Xt ~< E max Yr. toT

(9)

t~T

Similarly, if T = U s e s Ts and

(b)

IIXt

(c)

IIXt - Xt, ll2 ~ IIYt - Yt' 112

-

Xt, ll2 ~ IIYt - Yt'll2

if t ~ Ts, t' ~ Ts, with s =/: s', if t, t' ~ Ts for some s,

then one has E max min Xs,t ~ E max min Ys,t. soS t~Ts

soS tCTs

REMARK 2.10. Lemma 2.9 is usually stated with an additional hypothesis IlXtl[2 = IIYtll2 for t 6 T, and yields the stronger assertion for any k E R, 79(maxt~T Xt > )~) <~ 79(maxt~T Yt > k); (9) is then an immediate consequence. Analogous comment applies to the second part of the lemma. Let us also point out that when all Ts are singletons, the second part of the lemma (the Gordon version) reduces to the first (the Slepian version), which in our formulation coincides with a result of Fernique [75].

Local operator theory, random matrices and Banach spaces

351

A typical context in which L e m m a 2.9 shall be applied is the probability space (~N, VN) with linear functionals as random variables. For u E R N, set Zu := {., u}; for future reference we point out that the map ~N ~ U ~ Zu E L2(]t~N, }IN) is an isometry. If K C R N, one has the gauge, or the Minkowski functional of the polar of K given by maxuEK Zu = I[" 11/~~ The quantity E maxu E/~ Zu may also be expressed in terms of the mean width of K. These relationships provide a link between Gaussian processes and convexity or geometry of Banach spaces. It is also clear that in this context T could b e any bounded set, as long as we replace max and min by sup and inf. In particular, if qJ" T --+ T is a surjective contraction (possibly nonlinear) between subsets of two Euclidean spaces, then Eli. liT~ ~ Eli 9I1~o. In our setting, the norm [1. lifo is the operator norm on n x n real matrices identified with N n2 For such a matrix X and u v E R n we have o

<Xu, v} - tr(X(v | u)) - <X, u | V}tr = Z , | where, as usual, v | u stands for the rank one

matrix (Ujl)k)j,k=l, that is, the matrix of

the map x --+ (x, v)u and {X, Y)tr " - tr(XY T) is the trace duality, (sometimes referred to as the Hilbert-Schmidt scalar product), which can also be thought of as the usual scalar product on R n2" note t h a t - as opposed to the remainder of the p a p e r - we use here the standard, i.e., not normalized trace. Accordingly IlXll-

max

{Xu, v } =

U, u E S n - I

max

z.|

lg, u E S n-1

The Gaussian process Xu,v " - Zu| u, v E S n-1 is now going to be compared with Yu,v := Z(u,v), where (u, v) is thought of as an element of R n x R n - R en. In view of prior remarks, to show that the Slepian's lemma applies, one only needs to verify that, for U , 1), bl t

V I E S n-1

(~o) an elementary exercise. On the other hand, for (x, y) E ]Rn x R n , Z(u,v)(x, y) = {x, u) + (y, v) whence max

u,I)ES n-I

Z(u,v)(x, y ) = Ixl +

lYl

(this is just saying that II-II(uxv)o - I1" IIuo + II" IIvo) and so the assertion of L e m m a 2.9 translates to

n'/2Ell

II 2

txl dyn<X

352

K.R. Davidson and S.J. Szarek

By comparing with the second moment of l. I, the last integral is easily seen to be < n 1/2 9 In fact, it equals v ~ F (--U-) n + l / F (~). n Note also that I I2 is distributed according to the familiar x2(n) law. The same argument, applied just to symmetric tensors u | u, allows us to analyze X I(GOE), the largest eigenvalue of the Gaussian orthogonal ensemble. The case of )~1 (A(n)) can then be deduced formally (see the comment after the definition of GOE). In combination with Theorem 2.6 and the remark following it, the above shows: 9

THEOREM 2.1 1. Given n ~ I~I consider the ensembles of n x n matrices G, A and GOE. If the random variable F equals either lIGI], X1 (A) or X1 (GOE), then MF < E F < 2 , where MF stands for the median of F. Consequently, for any t > O, 7~(F ~ 2 + at) < 1 - ~ ( t ) < e x p ( - n t 2 / 2 ) ,

(11)

where rr - 1 in the case of llGll and V ~ f o r XI(A) or XI (GOE).

The beauty of the above result lies, in particular, in the inequalities being valid for all n rather than asymptotically. However, Theorem 2.8 shows that asymptotically (11) is not optimal 9We need to mention that the gist of Theorem 2.11 and the argument given above can be extracted from the work of Gordon [83-85]; the same applies to Theorem 2.13 that follows. The relevance of the approach to the "standard" results in random matrices was noticed and publicized by the second author. Similarly as in the preceding section, it is possible to derive from (11) (via the BorelCantelli lemma) results on convergence in probability, for example lim X1 (A ~n)) -- 2

almost surely.

//---+ cx3

Indeed, (11) implies that l i m s u P n ~ )~1 (A(n)) ~< 2 almost surely. The reverse inequality for liminf is even easier: by Theorem 2.3, for any ~ > 0, l i m n ~ 79(3k Xk(A (n)) E [2 -~, 2]) -- 1 and so, for n large enough, the median X1 of X1 (A (n)) is ~> 2 - E. We now get the conclusion by appealing to (5). It is now rather routine to deduce the following more precise version of Theorem 2.7. COROLLARY 2.12. Given n ~ N, set ~,k "-- F -1 ( ~ ) , k - 1. . . . . n, where F is the cumulative distribution function of the Semicircle Law described in Theorem 2.2 (i.e., the n measure #(n) . _ _ n1 Y~k=l 6~k is the 'best' approximant of the semicircle distribution among measures with n atoms). Then lim max [Xk(A(n)) - ~-kl--0 n ~ 1~
Local operator theory, random matrices and Banach spaces

353

An argument just slightly more involved than the proof of Theorem 2.11 allows an analysis of the extreme singular numbers of rectangular Gaussian matrices with independent entries. It has been known for quite a while, cf. [118,179], that as the size of such matrices (appropriately normalized, with entries not necessarily Gaussian) increases with the ratio of the sides approaching/~ E (0, 1), then, as in Theorem 2.3, the empirical measures corresponding to the singular values converge in distribution, or almost surely, to a deterministic measure supported on the interval [1 - ,,/-fl, 1 + v/-fl] and often referred to as the M a r c h e n k o - P a s t u r distribution (or, more recently, in the fi'ee probability context, the free Poisson distribution, cf. the next section). Somewhat later it was determined [80,152, 187,20] that, under appropriate assumptions on the distribution of the entries, the extreme singular values do converge almost surely to the endpoints of the interval above. Here we present the following special but elegant fact. THEOREM 2.13. Given m, n E I~ with m <~ n, p u t 1~ -- m / n and consider the n • m random matrix F whose entries are real, independent Gaussian random variables following the N(O, I / n ) law. Let the s i n g u l a r v a l u e s be sl ( F ) >~ . . . >~ sin(F). Then (12)

1 - w/~ < E s m ( f ) < Ms,(F) < Esl ( F ) < 1 + and consequently, f o r any t > 0,

max{79(s, ( F ) >~ 1 + ~

+ t), V ( s m ( r )

,)1

<~ 1 -

< 1 - q~ (t) < e x p ( - n t 2/2). The proof of the upper estimate in (12) is essentially the same as that of the first assertion of Theorem 2.11. For the lower estimate, we consider the same families: Xu,v "-- Zu| and Y,,v " - Z(,,v) w i t h u E S m-1 , y E S n-1 , but then we set T, -- {(u, v)" y E S n-I }. The hypothesis (c) is satisfied since (10) is an equality if u -- u', and so we may use the second part of Lemma 2.9 to obtain

nl/2E

max uEsm-1

min yES n- 1

(f u, v) <~fR Ixldym(x)- fR Ixldgn(x). m

n

The quantity on the right does not exceed ~ v/-n. (This actually requires some work. What is clear is that the difference between the two expression --+ 0 as m, n --+ ~ . ) Since for any F , max

min ( f u, v) ~ - s m ( F ) ,

btCsm-1 t ~ E S n - I

the first inequality in (12) follows. It remains to appeal again to Theorem 2.6. The arguments of this section can be modified to treat other classes of real matrices, even if the outcomes may be less elegant. It would be nice and potentially very fruitful to find an approach to the complex case(s) that is based on similar ideas. As complex (or symplectic) matrices can be thought of as real matrices with a special structure, the

354

K.R. Davidson and S.J. Szarek

problem at hand is equivalent to considering simple expressions in real Gaussian matrices with matrix coefficients. For example, the complex analogue of G (n) ((ii) on our list) can be identified with a real matrix

~/~

GI

G

] ,(El0] - ~

0

1

|

E0 1] ) 1

0

|

,

(13)

where G and G f are independent copies of G (n). Accordingly, the proper context for the question appears to be that of operator spaces; more about related issues in the next section. In another direction, it is quite clear that variations of the methods of this sections can be applied to a wide range of questions including random factorizations, and particularly, estimating virtually any reasonable norm applied to Gaussian matrices. For example, one could consider for the operator norm with respect to underlying norms on ]1~n different than the Euclidean norm. One just needs to replace S n-1 by the spheres corresponding to the norms in question. This has been noticed early on, see [50,25,84]. However, more often than not, one only gets this way an approximate asymptotic order up to a constant like in Theorem 2.1, rather than the precise asymptotics obtained in this section. We mention here an important question which perhaps did not receive enough publicity in the Banach space circles. PROBLEM 2.14. Show the existence and find lim rt----> o o

Emax~c{-1,1}, (A(n)~, ~) n

The quantity in the numerator, which is close to and of the same order as the norm of A (n) considered as an operator from ~ to ~ , is related to spin glass theory [169, 123]. Even the existence of the limit is not clear. However, it is generally believed that it does exist and equals approximately 1.527. Mimicking the proof of Theorem 2.11 yields x/g-/Tr ~ 1.596 as an upper estimate for every n, and hence for the lim sup, just slightly worse than the best known rigorous upper estimate of 2 - x/e/4 ,~ 1.588 for the latter, due to B. Derrida. The best rigorous lower estimate is 4/Jr ,~ 1.273 [ 1]; we suspect that this one is most susceptible to attack due to improvements in technology in the intervening years. A related, possibly simpler, problem would be calculating the exact asymptotic order of, say, the norm of G (n) considered as an operator from ~pn to ~q. n To the best of our knowledge, this is known only for very special values of p and q. Or, perhaps more appropriately, one could consider the analogous question for the matrices F from Theorem 2.13.

2.4. Random matrices and free probability A significant recent development was the emergence, due largely to efforts of D. Voiculescu, of the area of free probability, and the realization of its connections to random matrices. Even though, a priori, free probability seems relevant only to the macroscopic features of the asymptotic spectral pictures of random matrices, it is hard to overestimate its effect

Local operator theory, random matrices and Banach spaces

355

on clarifying the subject. Several books on the topic appeared in recent years or are in the works and, accordingly, we shall only sketch here the basic ideas and mention a few results and problems that are relevant to the remainder of our discussion. A noncommutative probability space is a pair (A, qg), where A is an algebra, usually over C, with unit I and a state qg. A state is a linear functional verifying qg(1) -- 1. In the classical case, we have an algebra of (measurable) functions on a probability space and the expectation. There are also Banach algebra, C*-algebra etc. probability spaces where the algebra in question is endowed with the appropriate additional structure, and the state q9 respects that structure. Elements of A are thought of as random variables, and the distribution of a random variable x is the linear functional #x on the algebra of C[X] of complex polynomials in variable X defined by # x ( p ) := qg(p(x)). In the C*-algebra context, the distribution of a normal element is actually represented by a measure supported on the spectrum of x (by the Gelfand-Neumark theorem). In the classical context, this measure is necessarily the law of x. One defines similarly joint distributions as functionals on the algebra of polynomials in the appropriate number of noncommuting variables. The convergence in distribution is the w e a k - , convergence: #x,, (p) ~ #x (P) for all polynomials p. This can be defined even if the xn's belong to different probability spaces. A typical statement concerning convergence in distribution is Theorem 2.2. Three important examples of probability spaces are: (i) The C*-algebra 93tn of n x n matrices with the normalized trace Tr; the distribution 1 n of A 6 9Jtn is represented by the measure # a -- n ~-~j--1 6~j(a). (ii) The *-algebra A (n) of random n x n matrices A (co) on some sufficiently rich classical probability space ($2, r , P ) verifying EllA(co)ll p < ~ for all p < cx~, with q9 = ETr. (iii) An operator algebra A c L(7-/) with a vector state qg~(X) := (X~, ~), where ~ 6 7-/ is a norm one vector. The fundamental concept of the theory is that of freeness (or free independence). It is modelled after that of classical independence, which may be restated as follows: commuting subalgebras A l , A2 are independent if q g ( a l a 2 ) = 0 whenever aj E A j with qg(aj) -- 0 for j = 1, 2. By analogy, one says that a family ( A j ) j E J are free if for any reduced product a - - a l a 2 . . . am where neighboring elements come from different A j ' s , one has qg(a) = 0 whenever qg(ak) -- 0 for k = 1 . . . . . m. A family of random variables (or sets of such) is free (respectively, *-free) if the algebras (respectively, *-algebras) generated by them are free. An early result of free probability was a free central limit theorem (CLT): if (Xj)jEN is a sequence of free random variables normalized by qg(Xj) -- O, qg(X2) -- 1, j E 1~, and

satisfying some mild technical conditions such as supjcl ~ Iqg(x~)l < cx~for all k ~ N, then as N --+ cx~, xl+...+xx converges in distribution to the standard semicircular distribution. The mysterious appearance of that distribution in this context was elucidated only after Theorem 2.16 below was proved. It is not clear from the above discussion that nontrivial free random variables actually do exist. A prototype of free algebras is as follows. Let G l, G2 be discrete groups and G = G I * G 2 their free product. The group algebras ~ j = C ( G j ) , i.e., the formal linear combinations of elements of the corresponding group, considered as subalgebras of C ( G ) are free with respect to the state which assigns to an element of C ( G ) the coefficient of the

356

K.R. Davidson and S.J. Szarek

unit e. Equivalently, one may consider C(G) as a canonical subalgebra of L(g2(G)) with the vector state corresponding to ~ = ~e, the unit vector supported at e. However, a more useful model is provided by the creation/annihilation operators on the full Fock space. Given Hilbert space 7-/, the corresponding Fock space f(7-/) is defined as the orthogonal 9 7_/| of all tensor powers of 7-/. The 0th power is identified with CI2, direct sum Y~k>~0 where 12 is a fixed unit vector, which is thought of as an empty tensor product and is called the vacuum vector. The corresponding vector state ~0s2 is referred to as the vacuum state. Given h 6 7-/, one defines a shift operator g(h) by g ( h ) r / = h | r/for elementary tensors r/. We have FACT 2.15. In a probability space (L(f(7-/)), qgs2) the following hold (i) If (~"[j)jEJ is a family of orthogonal subspaces ofT-t, then the corresponding family ({t~(h): h E 7-[j })jEJ of subsets of L(gt'(7-/)) is ,-free. (ii) If h E 7-[ is a norm one vector, then s(h) := g(h) + g(h)* has the standard semicircular distribution. The crucial link between random matrices and free probability is provided by the following result of Voiculescu ([ 176], see also [ 177]): THEOREM 2.16. For each n E N, let (A j(n)~J j

EJ

be independent copies of the random n • n

matrix A (n), considered as elements of the noncommutative probability space (A (n) , E Tr) defined above. Then, as n --+ oo, (A~))jEJ converges in distribution to (Sj)jEJ, a family of free random variables, each of which has the standard semicircular distribution. The same is true for GOE, GUE and any ensembles of random matrices verifying the hypotheses of Theorem 2.2. The fact that, for any j ~ J, A j(n) converges in distribution to sj is of course equivalent to Theorem 2.2. The new ingredient is the asymptotic freeness of large random matrices. It is worthwhile emphasizing that this asymptotic freeness, in combination with the free CLT and the infinite divisibility of the Gaussian distribution, does imply that the only possible limit distribution is semicircular. As we indicated, free probability in general and Theorem 2.16 in particular open new vistas on the subject of random matrices. For example, if P is a polynomial in m noncommuting variables, it follows that, for any v E N, lim ETr(P(A(1 n) . . . . . A(mn))v)

- - (fl$2 ( P ( s 1

.....

Sm)V),

(14)

t/---+ o o

where Aj(n) and S j a r e as in Theorem 2.16. In particular, if the polynomial P is formally selfadjoint (the variables themselves are assumed to be selfadjoint here), this leads to a measure concentrated on the spectrum of the bounded selfadjoint operator P(Sl . . . . . Sm) as the asymptotic spectral distribution of the random matrix P (A (1n) . . . . . A (mn)).Asymptotic means a priori in the sense of Theorem 2.2, but proceeding as in Section 2.2, one deduces: THEOREM 2.17. In the notation of Theorem 2.16, let P be a selfadjoinf polynomial in m noncommuting variables and let n ~ N. Then there exist scalars )~1, )~2. . . . . )~n such that, for k = 1 . . . . . n and t >>.O,

357

Local operator theory, random matrices and Banach spaces

7 (l k

A ~ ) ) ) - ~'kl >~ t) < C e x p ( - c p n min{t 2, t2/d})

(~5)

where d is the degree of P, c e > 0 depends only on P and C is a universal constant. One necessarily has maxl<~k~
~

9 o

.

PROBLEM 2.18. In the notation of Theorem 2.17, does )~j converge to the maximum of the spectrum of # e ? An affirmative answer would imply the almost sure convergence of the eigenvalues )~k(P(A(I n) . . . . . A ~ ) ) to the same quantity and, more importantly, the fact that, asymptotically, (A(1n), . . . , A~ )) ar~ equivalent to (sl, . .., Sm) not only in distribution as in (14) or Theorem 2.17, but also in ~he C*-algebra sense. Problem 2.18 is of similar nature as Problem 2.14: in both cases w~: do know, for somewhat similar reasons, the order of growth of the quantity in question, bm we do not know exact asymptotics. However, the former seems to be much more accessible than the latter because of direct applicability of spectral methods. This is because, in the Euclidean case, []B[[ ~ (Tr(B~)) 1/~ if B is an n x n matrix and v is an even number such that v/log n is large, and so the problem reduces to properly

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estimating E T r ( P ( A I n) , . . . , A(mn))v) for such v, n, in principle a straightforward combinatorial question. This approach, already present in the original Wigner's paper [ 184], was exploited by a number of authors, perhaps in the most sophisticated way in [38,89]. Another possibility would be to somehow use the ideas behind Theorem 2.7 or 2.11 to show by an a priori argument that )~k's of Theorem 2.17 vary slowly with k. This could be interesting for other reasons as the distribution of quantities like sl (G (n)) - s2(G (n)) are of interest in the theory of computational complexity, cf. [ 162]. (It is also conceivable that to efficiently study the gaps in the spectrum in the spirit of Theorem 2.5, one needs to consider functions of matrices and not only matrices as such.) For the quantity mentioned here, the needed information can probably be extracted from the methods of [ 171 ]. But, to our knowledge, no careful examination of the relevant arguments was made. A nontrivial but possibly accessible test cases for any approach would be to find the exact asymptotic behavior of II(G(~))~II for v ~> 2 or IIU + V + U* + V*ll, where U, V are independent and uniformly distributed on U(n) or SO(n). The predictions given by the free probability are v/(V + 1)v+l/vv ,~ ~ for large v, and 2V/-3. Both were actually determined before the era of free probability, see [182,104]. The latter reference treat the random walk on the free group, see [ 178] a clarification of the connection. A known (asymptotic) estimate on [l(G(n))~[[ is v + 1 [19].

2.5. Random vs. explicit constructions It is a quite frequent occurrence that existence of mathematical objects possessing a certain property is shown via nonconstructive methods (the probabilistic method). Roughly speaking, one produces a random variable whose values are those objects and then proves that the property in question is satisfied with nonzero (cf. [2]), or close to 1, probability. Two fields where this principle has been successfully applied are combinatorics and analysis, particularly harmonic analysis and local theory of Banach spaces. Many developments in the latter area are described elsewhere in this collection [ 117,81,41 ], and some spillovers to local operator theory were mentioned in this survey. Here we shall introduce only the details needed to address some philosophical aspect of the issue. We start by recalling a remarkable result of Kashin [101], cf. [160,164], motivated by questions in approximation theory, which roughly asserts that the space s is an orthogonal sum (in the s sense) of two nearly Euclidean subspaces. More precisely: THEOREM 2.19. Given n -- 2m e N, there exist two orthogonal m-dimensional subspaces El, E2 C ]1~n s u c h that 1

1

IIx 112 ~ - - 7 IIx II1 ~ IIx 112 for all Xi E E i, i -- 1, 2. ~/n

(16)

Moreover, for large n, this holds for nearly all decompositions E1 • E2 with respect to the Haar measure on the Grassmann manifold Gn,m. The existence of such a decomposition was surprising because, when considered on the entire space IRn, the ratio between the ~1 and e e norms varies between 1 and v/ft.

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Because of the wealth of examples of various objects (e.g., random Banach spaces) arising from or related to the one above, it would be of interest to have, for starters, an explicit example of a Kashin decomposition. Having explicit Ek's would likely lead to more natural examples of finite dimensional Banach spaces with various extremal properties than the ones obtained by the probabilistic method. Indeed, an explicit example must have an explicit reason, and this should presumably be reflected by a presence of some additional structure. It may also conceivably lead to some useful algorithms. It may be the right place to comment here on what exactly we mean by an explicit construction. We shall not give a general definition, but, in the case at hand, admissible descriptions of E1 9 E2 would include an algorithm yielding, for a given n, a basis of E1 or the matrix of the orthogonal projection onto El. That algorithm would need to have a reasonable worst case performance, preferably polynomial in n. The worst case requirement is needed to exclude a strategy that would involve choosing E1 @ E2 at random and then somehow efficiently verifying whether it satisfies (16). We note that, at least at the first sight, even checking (16) for a given Ek appears to be an exponentially hard problem. To the best of our knowledge, the largest explicit subspace E of g'~, for which an assertion of type (16) holds is of dimension approximately ~ as opposed to m/2 in Theorem 2.19 - a long way to go. This follows, e.g., from the construction in [ 147], using finite fields and difference sets (i.e., the so called finite geometries), and giving, for an even integer p ~> 4, an exact Ap-set. This leads to another question: finding explicit exact Ap-sets for other values of p; for probabilistic results see [40,165] and cf. [41]. An example of a probabilistic construction followed by an explicit one is the work on approximation of quasidiagonal operators mentioned in Section 1.3 [159,175,181]. The explicit approach used representations of groups with property T. It has been observed very recently by Benveniste and the second author [24] that the argument of [ 175] can be refined to yield explicit matrices poorly approximable by reducible matrices (cf. [95]), also mentioned in Section 1.3. Their argument suggests specific subspaces of g~n as candidates for E from the previous paragraph (a rather "long shot", but nevertheless a good starting point). A somewhat similar example comes from another field, involving random graphs. Following the seminal work of Erd6s [73], this became a powerful technique to show existence of graphs with various extremal properties. As some questions about graphs are very practical optimization problems (like design of a network), it was important to have explicit solutions. This is particularly relevant because verifying that a given large graph has some required property is sometimes computationally not feasible. For some important questions, explicit solutions were found, see [120,116] and much earlier work [119]. The constructions were based on properties of some arithmetic groups, sophisticated tools from arithmetic geometry or, again, the property T. Some of the graphs in question (Ramanujan graphs) were already mentioned at the end of Section 1.4. The final topic we propose to analyse is suggested by the preceding section on free probability. Theorem 2.16 can be interpreted as an assertion that two large random matrices are nearly free, in the sense of the pattern of the moments of monomials in the two matrices with respect to the normalized trace being approximately the same as it would have been the case if the two matrices were free. In fact, the same is true for, roughly speaking, one fixed and one random matrix [ 177]. A natural problem is to come up with explicit matrices

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having the same property. (It cannot happen that two finite matrices are exactly free, except for the trivial cases.) Some examples to that effect were produced in [176] and [151 ]. While they are not fully satisfactory in some respects, it is nevertheless conceivable that they may also constitute a step towards an explicit Kashin decomposition (even though they do not work directly). The ideas from [37], involving representations of symmetric groups, may also be relevant here. This and the preceding two paragraphs do broadly suggest some of the areas of mathematics that might be pertinent to other explicit constructions.

Acknowledgments The first author was partially supported by a grant from the Natural Science and Engineering Research Council (Canada). The second author was partially supported by grants from the National Science Foundation (USA). The authors thank Professor L. Pastur for helpful comments on the second part of the article.

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[136] G. Pisier, Probabilistic methods in the geometry of Banach spaces, C.I.M.E. Varenna 1985, Lecture Notes in Math. 1206, Springer Verlag, Berlin (1986), 167-241. [ 137] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conference Series in Mathematics 60, Amer. Math. Soc., Providence, RI (1986). [138] G. Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc. 10 (1997), 351-369. [ 139] G. Pisier, Similarity Problems and Completely Bounded Maps, Lect. Notes in Math. 1618, Springer-Verlag, Berlin (1996). [140] G. Pisier, Operator spaces and similarity problems, Proc. Inter. Congress Math., Vol. I (Berlin, 1998), Doc. Math., Extra Vol. I (1998), 429-452. [141] G. Pisier, Operator spaces, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), to appear. [142] D.R. Pitts, Close CSL algebras are similar, Math. Ann. 300 (1994), 149-156. [143] G. Popescu, A generalization of Beurling's theorem and a class of reflexive algebras, J. Operator Theory 41 (1999), 391-420. [ 144] S.C. Power, The distance to upper triangular operators, Math. Proc. Camb. Phil. Soc. 88 (1980), 327-329. [145] J.R. Ringrose, On some algebras of operators, Proc. London Math. Soc. 15 (1965), 61-83. [146] S. Rosenoer, Distance estimates for von Neumann algebras, Proc. Amer. Math. Soc. 86 (1982), 248-252. [147] W. Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203-227. [148] Z. Rudnick and P. Sarnak, Zeros of principal L-functions and random matrix theory, A Celebration of John F. Nash, Jr., Duke Math. J. 81 (1996), 269-322. [149] N. Salinas, Homotopy invariance of Ext(A), Duke Math. J. 44 (1977), 777-794. [150] G. Schechtman, Concentration, results and applications, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), to appear. [151] D. Shlyakhtenko, Limit distributions of matrices with bosonic and fermionic entries, Free Probability Theory (Waterloo, ON, 1995), Fields Inst. Commun. 12, Amer. Math. Soc., Providence, RI (1997), 241252. [152] J.W. Silverstein, The smallest eigenvalue of a large-dimensional Wishart matrix, Ann. Probab. 13 (1985), 1364-1368. [ 153] Ya.G. Sinai and A.B. Soshnikov, A refinement of Wigner's semicircle law in a neighborhood of the spectrum edge for random symmetric matrices, Funk. Anal. i Prilozhen. 32 (1998), 56-79 (Russian). English translation: Funct. Anal. Appl. 32 (1998), 114-131. [154] A. Soshnikov, Universality at the edge of the spectrum in Wigner random matrices, Comm. Math. Phys. 207 (3) (1999), 697-733. [155] V.S. Sunder, Distance between normal operators, Proc. Amer. Math. Soc. 84 (1982), 483-484. [ 156] S.J. Szarek, The finite dimensional basis problem with an appendix on nets of Grassmann manifolds, Acta Math. 151 (1983), 153-179. [157] S.J. Szarek, Nets of Grassmann manifold and orthogonal groups, Proceedings of Banach Space Workshop, University of Iowa Press (1982), 169-185. [158] S.J. Szarek, On almost commuting Hermitian operators, Proc. Seventh GPOTS (Lawrence, KS, 1987), Rocky Mountain J. Math. 20 (1990), 581-589. [159] S.J. Szarek, An exotic quasidiagonal operator, J. Funct. Anal. 89 (1990), 274-290. [160] S.J. Szarek, On Kashin's almost Euclidean orthogonal decomposition of g~]~,Bull. Acad. Polon. Sci. S6r. Sci. Math. Astronom. Phys. 26 (1978), 691-694. [161] S.J. Szarek, Spaces with large distance to gn and random matrices, Amer. J. Math. 112 (1990), 899-942. [162] S.J. Szarek, Condition numbers of random matrices, J. Complexity 7 (1991), 131-149. [163] S.J. Szarek, Metric entropy of homogeneous spaces, Quantum Probability (Gdafisk, 1997), Banach Center Publ. 43, Polish Acad. Sci., Warsaw (1998), 395-410. [164] S.J. Szarek and N. Tomczak-Jaegermann, On nearly Euclidean decomposition for some classes of Banach spaces, Comp. Math. 40 (1980), 367-385. [165] M. Talagrand, Sections of smooth convex bodies via majorizing measures, Acta Math. 175 (1995), 273300. [166] M. Talagrand, New concentration inequalities in product spaces, Invent. Math. 126 (1996), 505-563.

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[167] M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes l~tudes Sci. Publ. Math. 81 (1995), 73-205. [168] M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal. 6 (3) (1996), 587-600. [ 169] M. Talagrand, Huge random structures and meanfield models for spin glasses, Proc. Inter. Congress Math., Vol. I (Berlin, 1998), Doc. Math., Extra Vol. I (1998), 507-536. [170] C.A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), 151-174. [171] C.A. Tracy and H. Widom, On orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177 (1996), 727-754. [172] D. Voiculescu, A non-commutative Weyl-von Neumann theorem, Rev. Roum. Math. Pures et Appl. 21 (1976), 97-113. [173] D. Voiculescu, Asymptotically commuting finite rank unitary operators without commuting approximants, Acta Sci. Math. (Szeged) 45 (1983), 429-431. [174] D. Voiculescu, A note on quasidiagonal operators, Topics in Operator Theory, Constantin Apostol Memorial Issue, Operator Theory: Advances and Applications 32, Birkhatiser-Verlag, Basel (1988), 265-274. [175] D. Voiculescu, Property T and approximation of operators, Bull. London Math. Soc. 22 (1990), 25-30. [ 176] D. Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104 (1991), 201-220. [177] D. Voiculescu, A strengthened asymptotic freeness result for random matrices with applications to free entropy, Inter. Math. Res. Notices 1 (1998), 41-63. [178] D. Voiculescu, K.J. Dykema and A. Nica, Free Random Variables. A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups, CRM Monograph Series 1, Amer. Math. Soc., Providence, RI (1992). [179] K.W. Wachter, The strong limits of random matrix spectra for sample matrices of independent elements, Ann. Probab. 6 (1978), 1-18. [180] S. Wassermann, A separable quasidiagonal C*-algebra with a nonquasidiagonal quotient by the compact operators, Math. Proc. Cambridge Phil. Soc. 110 (1991), 143-145. [181] S. Wassermann, C*-algebras associated with groups with Kazhdan's property T, Ann. Math. 134 (1991), 423-431. [182] R. Wegmann, The asymptotic eigenvalue-distribution for a certain class of random matrices, J. Math. Anal. Appl. 56 (1976), 113-132. [183] H. Weyl, Der asymptotische Verteilungs gesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann. 71 (1912), 441-479. [ 184] E.P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. Math. 62 (1955), 548-564. [185] E.P. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. Math. 67 (1958), 325327. [186] I.M. Wils, Stone-Cech compactifications and representations of operator algebras, Ph.D. thesis, Catholic Univ. of Nijemgin (1968). [187] Y.Q. Yin, Z.D. Bai and P.R. Krishnaiah, On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix, Probab. Theory Related Fields 78 (1988), 509-521.

CHAPTER

9

Applications to Mathematical Finance

Freddy Delbaen Department of Mathematics, E.T.H., Ziirich, Switzerland E-mail: delbaen @math. ethz. ch

Walter Schachermayer Department of Mathematics, Vienna University of Technology, Vienna, Austria E-mail: WSchach @statl, bwl. univie, ac. at

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Strategies and arbitrage possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The fundamental theorem of asset pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The continuous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Changes of num6raire and a related Banach space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Weighted norm inequalities and closedness of a space of stochastic integrals . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Abstract We give an introduction to the theory of Mathematical Finance with special emphasis on the applications of Banach space theory. The introductory section presents on an informal and intuitive level, some of the basic ideas of Mathematical Finance, in particular the notions of " N o Arbitrage" and "equivalent martingale measures". In section two we formalize these ideas in a mathematically rigorous way and then develop in the subsequent four sections some of the basic themes. O f course, in this short handbook-contribution we are not able to give a comprehensive overview of the whole field of Mathematical Finance; we only concentrate on those issues where Banach space theory plays an important role.

HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved 367

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1. Introduction The field of Mathematical Finance has undergone a remarkable development since the seminal papers by E Black and M. Scholes [4] and R. Merton [40], in which the famous "Black-Scholes Option Pricing Formula" was derived. In 1997 the Nobel prize in Economics was awarded to R. Merton and M. Scholes for this achievement, thus also honoring the late E Black. The idea of developing a "formula" for the price of an option actually goes back as far as 1900, when L. Bachelier [2] wrote a thesis under the supervision of H. Poincar6 with the title "Th6orie de la spdculation". His aim was precisely to obtain such a formula to be used on the stock market in Paris, which was also booming at the previous fin de si~cle. Bachelier's contribution was remarkable in several respects: firstly he had the innovative idea of using a stochastic process as a model for the price evolution of a stock. He had an almost mystic belief in the "law of probability": Si, dt l'dgard de plusieurs questions traitdes dans cette dtude, j ' a i compard les rdsultats de l'observation ~ ceux de la thdorie, ce n'dtait pas pour vdrif ier des formules dtablies p a r les mdthodes mathdmatiques, mais pour montrer seulement que le marchd, ~ son insu, obdit ~ une loi qui le domine: la loi de la probabilitd. For a stochastic process (St)0~ 0 is a fixed parameter, and, that for 0 ~< to ~< -.. <~ tn ~< T the random variables (Sti - St;_~)i~l are independent. If we also fix the real number So we have well defined a stochastic process, namely Brownian motion, which in the present context is interpreted as follows: So is today's (known) price of a stock (say a share of company XYZ to fix ideas) while for the time t > 0 the price St is a normally distributed random variable. Bachelier's use of Brownian motion in the context of finance was thus prior to the use of this process in the context of physics by A. Einstein [21] and, independently, by M. Smoluchowski [50] some five years later. Having fixed the model, Bachelier now turned to the question of pricing an option (to be precise: a European call option); such an option is the right to buy one unit of the underlying stock at a fixed time 7" (the expiration time) and at a fixed price K (the strike price). This determines the value CT of the option at time T as a function of the (unknown) value ST of the stock at time T, namely CT = (ST - K ) + .

(1)

Indeed, a moment's reflection reveals that the option is worthless (at time T) if ST <~ K, and is worth the difference between ST and K if ST > K: in the latter case the holder of the option will exercise the option to buy one unit of stock at a price K and can immediately resell it at the present market price ST to make a profit ST - K. (In practice, a cash settlement is often made.)

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Hence we know, conditionally on the random variable St, what the option will be worth at time t = T. But what we really want to know is what the option is worth today, i.e., at time t = 0. To determine this quantity Bachelier simply took the expected value, i.e., Co " - E [CT] -- E [(ST -- K)+]

(2)

which leads to a simple explicit formula for Co involving the normal distribution function. The approach of formula (2) has been used in actuarial mathematics for centuries and is based on a belief in the law of large numbers: if the stochastic model (St)o< 0 obeying the stochastic differential equation

(3)

dSt = # S t dt + a St dBt,

where (Bt)o<~t<~T is a standard Brownian motion starting at B0 --0. Using It6's formula one quickly verifies that the solution to (3) is given by the process

St - So exp

#-

-~

t + o Bt ,

(4)

which is called geometric Brownian motion. If returns are defined through the logarithm, then the price process St gives returns that are stationary and are independent when taken over disjoint time periods. It is a well known result that the only L6vy processes with this property and continuous paths are Brownian motions with drift. This immediately leads to (3) or (4). The rationale of (3) is that the drift as well as the noise term driving the process St is proportional to St; in other words, the difference between Brownian motion with drift, i.e., St = So + at + b Bt and geometric Brownian motion with drift as defined in (4) is similar to the difference between the methodology of calculating linear or compound interest. Clearly, in the short run there is little difference between the linear and the exponential

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point of view (in the deterministic as well as in the stochastic case) while, in the long run, the difference is very noticeable; it also is obvious, that the exponential point of view is economically more meaningful. The model (4) of geometric Brownian motion today became the standard reference model to describe the price evolution of a stock; although promoted by Samuelson, it now is often called the Black-Scholes model or even the Black-Scholes world. It has many very appealing properties but its match to reality is not very good in many aspects (as is very well-known to practitioners): there are several key properties observed in real financial data (e.g., heavy tails, volatility clustering etc.) which are not captured by this model. Many alternative models have been proposed (e.g., in the early work of B. Mandelbrot [38]). Hence, turning back to question (1) raised a b o v e - whether Brownian motion is the good m o d e l - it is generally agreed that geometric Brownian motion with drift is economically more reasonable than Bachelier's original choice (note, however, that in the short run there is little difference), but the question whether (geometric) Brownian motion is a "good model", cannot be answered with a simple yes or no: it depends on the context and purpose of the modeling. We now turn to question (2) pertaining to the economic soundness of the pricing argument based on the law of large numbers: it was the merit of Black and Scholes [4] and Merton [40] to have replaced this argument by a "no arbitrage" argument, which is of central importance to the entire theory. The basic principle underlying this idea is nicely explained in the subsequent little story quoted from a paper of Dupire [20]: Imagine you are offered a strange bet. A coin is tossed; you win $60 if it comes up heads and lose $40 f o r mils. Would you accept the bet ? It looks attractive: the expectation is (0.50 x 60) + (0.50 x - 4 0 ) = 10, a healthy bias. A more thorough examination may lead to second thoughts, however. Perhaps the pain o f losing $40 would overweight the pleasure o f making $60. What if it were $60 000 and $40 000? What if the coin were unbalanced, or tossed by a manipulator who can get tails three times out o f four? In other words, your decision would depend on both preferences (risk aversion) and expectations, and it may well be wiser to refuse this apparently attractive proposition. Let us now set the stage f o r a very similar situation. You are in a casino, at a simplified roulette table. There are only two possible outcomes: red or black. There is no zero and it is possible to play with the casino on a 1:1 basis. The proposed bet is: win $60 if red comes up and lose $40 f o r black. Would you accept ? At first sight, this seems equivalent to the previous bet, so you would decline the invitation. But the fact that it is possible to play with the casino makes a real difference. You can hedge the bet and build up a certain gain, regardless o f the outcome - an arbitrage, in other words. Just accept the bet and, at the same time, put a stake o f $50 on black with the casino. I f red does come up, you make $60 through the bet and lose $50 with the casino. If black wins, you lose $40 through the bet and make $50 with the casino.

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In either case, you are better off by $10: the expected, potential gain has been converted into a certain one. Most important, this is true regardless o f the real probabilities o f red versus black, which could very well be 10: 90.

Summing up: an arbitrage- which is the concept crystallized in the second part of the above story - is a riskless way of making a profit with zero net investment. This intuitive notion will be mathematically formalized in Section 2 below. An economically very reasonable assumption on a financial market consists of requiring that there are no arbitrage opportunities. (To convince yourself that this assumption is indeed reasonable, just think for a moment, what would happen in a liquid financial market offering arbitrage opportunities.) The remarkable fact is that this simple and primitive "principle of no arbitrage" allows already to determine a unique option price in the BlackScholes model: to be slightly more formal, we assume that in our financial market there are two traded assets, the stock S = (St)0<~t<~T whose price process is given by (4) above and a riskless bond B whose price process (Bt)o<~t <~T evolves deterministically with a constant interest rate r, i.e., Bt -- Boe rt . For (mainly notational) simplicity we assume r = 0. We denote by (ft)O<~t<~T the (right continuous, saturated) filtration generated by the process S (which is defined on some stochastic base (1-2, ,T, I?)). We then have that every 9t'r-measurable random variable f (satisfying an appropriate integrability condition) can uniquely be written as

f -- c +

f0 T H u d S u - - c + ( H . S ) T

(5)

for some constant c and a predictable process (Ht)o
Applications to mathematical finance

373

Ht to rewrite this as an integral on S martingale. It is trivial, using (3) and letting Ht -- ~-g, (remember that # = 0):

f--

/0 T-H, dB, --

/~, cr dS, S. -

H. dS..

(6)

Hence, admitting the above cited martingale representation theorem, we get the representation (5) for an arbitrary f E L2(X2, f , 1?) where c = E l f ] . Hence we essentially find the same methodology as used by Bachelier and by actuaries: calculate prices by taking expectations. The reasoning behind this procedure however is now completely different; the argument involving the strong law of large numbers has been replaced by the (economically much more compelling) no arbitrage argument. What happens in the case # -76-0? Here the crucial trick is to pass from the original measure 17 to an equivalent measure Q such that under Q the process S becomes a martingale. The determination of the Radon-Nikodym density of the measure Q with respect to the original measure IP is the theme of the Cameron-Martin-Girsanov theorem (see, e.g., [43]): under the above assumptions the measure Q << 17 under which S is a martingale is unique, and its density is given by dQ ( # 1#2 ) d1? - e x p ---Brcr - 2 ~ - j T .

(7)

dQ Note that i--Y > 0 1?-a.s. so that the measures Q and 1? are equivalent, i.e., have the same nullsets. Now apply the above reasoning to the process S defined on (s 5c, (ft)0~
c = EQ[f],

(8)

where E~[. ] denotes expectation taken with respect to Q. Using again the principle of no arbitrage we find that c is the only possible price for f not allowing arbitrage opportunities (in the mathematical formalization of the no arbitrage principle given below we shall see that this principle remains unchanged under an equivalent change of measure; the only thing that matters are the sets of measure zero). For a European call option f = (ST - K)+ the constant c in (8) can be explicitly calculated yielding the celebrated Black-Scholes option pricing formula. Summing up the above methodology: we have passed from the original measure 17 to a so-called risk-neutral measure Q, which in the Black-Scholes model turned out to be unique, and then have used this measure to calculate the price of a derivative f via (8); we have claimed that this is the unique arbitrage-free price for f . The above presentation was deliberately informal and should only motivate the questions which are dealt with in full mathematical rigor in the subsequent sections. For example: What happens for more general stochastic processes (St)0~
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The latter question is the theme of the so-called Fundamental Theorem of Asset Pricing w h i c h - roughly speaking- states that a process S = (St)0~
2. Strategies and arbitrage possibilities As mentioned in the introduction, agents can use investment strategies, that are time dependent. This leads to the use of stochastic integrals. Of course this requires then some regularity on the trajectories of the stochastic processes used to model prices. Also as we will see, not all investment strategies can be used. We will first give some definitions and then we will give the interpretations. As stated in the introduction the main ingredient in arbitrage theory is stochastic integration. In this section we will give a mathematical formulation of the objects referred to in the introduction. All processes are defined on a stochastic

Applications to mathematical finance

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basis (s ( f t ) t >~0, P) where f t describes an increasing family of o--algebras on I2. The probability measure l? is defined on the o--algebra f e c = V t ~>0f t . o-algebra f t can be seen as the mathematical description of the information available at time t. As common in stochastics, we will also assume that the filtration ~ satisfies the usual hypotheses, i.e., the filtration is right continuous, meaning that f t = ['-'ls>t f s , and the o--algebra f 0 contains all the null sets of foc. To avoid trivialities at time t = 0 we assume that f 0 only consists of the null sets and their complements, set, the set of natural numbers or even a finite set. By taking R+ as the time set we aim for the most general situation. It is obvious that models with finite time horizon can be imbedded in a model where the time set is R+. The time an agent wants to buy or sell a financial instrument can be a random variable, prices coincide. Of course such random times can but it may only depend on past information and hence it must be a stopping time. DEFINITION 2.1. A stopping time T is a random variable T :s -+ R+ U {+co} such that for each t ~> 0, we have that {T ~< t} E Ut. If T1 ~< T2 are two stopping times then the stochastic interval ~T1, T2~ is the set

~T1,T2~- {(t, co) It

9 R+. T~ (co) ~
(9)

Other intervals ~T1, T2~, ~T,, T2[[, ~T1, T2~ are defined in an analogous way. The smallest o--algebra on R+ x s containing the stochastic intervals of the form ~T1, T2]] ( T1 and T2 are stopping times) is called the predictable o--algebra 72. A process H :R+ • s --> R d is called predictable if it is measurable with respect to the predictable o--algebra. Mathematically a process S, modeling the price evolution of d stocks, is best described by a d-dimensional semi-martingale S : R + x I2 --+ Rd. The set R+ is the set of times where trading can take place. As in the introduction we also suppose that there is a riskless bond (Bt)0~0 is locally bounded if there is a sequence of stopping = times (Tn)n~l such that Tn ~ Tn+l, l i m n ~ Tn = +cx~ almost surely, and S is bounded on the intervals ~0, Tn~ , i.e., there are kn E R such that for all t, 0 <. t <. Tn (co): ISt (co)l ~< kn. A stochastic process S which is c~dl?ag (i.e., right continuous with left limits) is a semimartingale if we can build a stochastic integration theory with it. To give a precise definition requires some extra notations and definitions. DEFINITION 2.2. A process H : R + x ~ -~ R d is called simple predictable if there is a finite sequence of stopping times 0 = To <~ T1 <~ ... <~ Tn+l < oe, Rd-valued functions n f0 . . . . . fn such that each f j is 3rrj-measurable and H - ~-~j=l f J 31~;,T~+~1]. It is easily seen that such a simple process is indeed measurable for the predictable oralgebra. Let us denote by G ~ the space of all bounded simple predictable process. On | we define the norm IIHII~ - supt,~o IHt(co)l.

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376

If S is a d-dimensional c?adl~g process then we can define the stochastic integral in an elementary way: f

( H . S)~ = ] Hu dSu J[o,oo] d

(10)

n

- ~Zfj(

Tj+! _

_

T;) .

S i

S i

(11)

i--1 j----1

The indefinite integral defines a cadl?ag process. f

( H . S)t -- I

H, dSu

(12)

,t] _

--Z

S i fj(T;+,At

_

S TjAt)" i

(13)

i=1 j = l

The notion of stochastic integral has an immediate financial interpretation. If the trader decides to hold the position (fj) during the time period ~Tj, Tj+l]] then the value of the portfolio is easily seen to be described by the process ( H . S). The final value of the process ( H . S ) ~ is then given by ( H . S ) ~ = limt__,~(H. S)t = (H. S)Tn+~. Stochastic integration is a natural tool in financial modeling. Predictable processes serve as strategies. DEFINITION 2.3. A c?~dl?ag adapted process S ' R + • ~2 --+ R d is a semi-martingale if for every T < cx~ the mapping | --+ L0(S2, U ~ , P); H --+ ( H 9S)T is continuous for the sup-norm topology on G ~ . This means that if suPt,~o IHtn (co) l --+ 0 then ( H n 9S)T ~ 0 in probability. If S is a semi-martingale then the notion of stochastic integral ( H . S) can be extended to all bounded predictable integrals H. This is the subject of the general stochastic integration theory, see, e.g., [ 17,32,42]. Exactly as in the deterministic, one-dimensional case, the notion of integral can be extended to some unbounded integrands H. See [32,42,45] for details. If ( H . S) can be defined, then we say that H is S-integrable. Integrands that are S-integrable can be seen as general strategies. If the limit ( H 9S ) ~ - l i m t ~ ( H 9S)t exists, then we can say that ( H 9S ) ~ is the final outcome of the strategy H. As observed in [29] not all S-integrable processes H can be accepted as reasonable strategies (from the financial point of view). Indeed we should avoid so-called doubling strategies. The most classical example of such a doubling strategy can be constructed using Rademacher functions. EXAMPLE 2.4 (Doubling strategy (discrete time)). On (S-2, U ~ , P) we take (gn)n>/1 t o be a sequence of independent variables, so that P[en - 4-1] - 1 which we may think of as the gain or loss of a risky investment revealed at time n. As filtration we take (Un)n>>O where Un = a (el . . . . . en). Sn is defined as el -+-...-+- en. The reader can easily change the

Applications to mathematical finance

377

definition in such a way that (.Tt)0~
Lt -- exp(Wt - 89 is easily seen to be the unique solution of the stochastic differential equation dLt = Lt dWt, with initial condition L0 -- 1. Hence we find that

1 -

Lt

f]0,t]

( - L u ) dW, = (H . W)t.

-

-

In both the discrete and the continuous example, we get that ( H 9S) is a martingale but not a uniformly integrable martingale. To exclude the p h e n o m e n o n described by the above examples (where in both cases it was crucial that the process ( H 9S)t was not uniformly b o u n d e d from below) one m a y adopt the following concept: DEFINITION 2.6 ([30] and [11]). An S-integrable predictable process H : R + x $2 --+ R d is called a-admissible for some a ~> 0 if, (i) ( H 9S) ~> - a i.e., the process always remains above the ("budget") level - a , (ii) l i m t ~ ( H S)t = ( H . S ) ~ exists. We call H admissible if it is admissible for some a ~> 0. It is clear that the set of a-admissible integrands forms a cone but not necessarily a vector space. We define: K = { ( H . S)oc I H admissible}, c

-

(14)

h L+InL --(X-L+)nL

interpretation, it is simply the cone of all outcomes of admissible strategies (starting with an initial wealth zero). The set C has a similar interpretation. It is a philanthropic version of K in the sense that after having obtained ( H . S)oc the investor can give away a nonnegative amount of m o n e y thus making the gain uniformly bounded. Before continuing, let us point out that in the case of non-locally b o u n d e d processes it m a y happen that K is reduced to {0}. Indeed, for an easy example take St = 0 for t < 1 and St - X for t ~> q 1 where X is a standard normal variable. The filtration (.Tt)0~
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predictable integrand H we have that H1 -- constant. It follows that ( H . S ) ~ = ( H . S)l is of the form )~X where )~ ~/1~ and H is only admissible if H1 = 0 which yields ( H . S ) ~ = 0. In this example we obtain that C - - L + . We now have enough material to define the No Arbitrage properties. Roughly speaking these properties say that it is impossible to make money out of nothing (no money pump, no arbitrage, no free lunch ...). Because of doubling strategies we have to restrict to admissible integrands. DEFINITION 2.7 ([11]). The semi-martingale S satisfies the No Arbitrage (NA) property if K A L + = {0} or, equivalently, C fq L + -- {0}. The idea of this notion is that, by using admissible strategies, the investor can only make money if he/she is willing to face the possibility of also losing money. A standard and very intuitive example of (NA) is given by: PROPOSITION 2.8. If there is aprobability measure Q "- IP underwhich S is a martingale, then S satisfies (NA). PROOF. Let H be admissible and Q a probability measure, Q ~ P; then ( H . S) is a Qlocal martingale by Emery's theorem (see [25]). Since ( H . S) is bounded below it is a Q-super martingale and hence E Q [ ( H 9S)e~] ~ 0. It follows from the assumption Q "~ P that P [ ( H . S ) ~ < 0] > 0 as soon as P [ ( H - S ) ~ > 0] > 0 i.e., (NA). [] The fundamental theorem of asset pricing is the appropriate converse of Proposition 2.8. It is easily seen that in the case of a finite set #2, the cone K is a vector space and an easy separation argument yields the existence of a risk neutral probability Q such that S becomes a martingale (compare the more general arguments below). However in the general case, it is not clear that the sets K or C are closed (in an appropriate topology). Also in a general context it can be shown that it is impossible to obtain a martingale measure Q for a process S. This is of course related to the integrability properties of the process S. The generalizations we need are described in the following definition. DEFINITION 2.9. If (S-2, (~t)O~t,~) is a filtered probability space, then a process S :IR+ x #2 ~ IR (or IRd) is called a local martingale if there is a non-decreasing sequence of stopping times 0 ~< To <~ Tl ~< T2 <~ ... such that Tn --+ ~ and for each n we have that the stopped process S Tn is a uniformly integrable martingale. We recall that S T is defined as the process S f = StAT. In case the time interval is finite, e.g. [0, 1], we require that limP[Tn = 1] = 1. A standard example of a uniformly integrable (!) process S that is a local martingale and not a martingale is given by the inverse of a Bessel process in 3 dimensions (see, e.g., [43]). The process is defined as follows. We start with a 3-dimensional Brownian motion X : [ 0 , 1] x S2 --+ R 3. The filtration is the standard filtration generated by X. We suppose that X0 = (1, 0, 0). The Bessel process R is defined as R = IlSll where II II is the Euclidean norm on IR3. By the well known property of Brownian motion in 3 dimensions

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we have that R -r 0 and S -- 1/R is therefore well defined. It can be checked that S is a local martingale (roughly speaking because 1/ll R II is harmonic on IR3\{(0, 0, 0)}), that (St)o<<.t<<.1is a uniformly integrable set (in fact bounded in any L p space with p < 3) and that E [St ] < 1. The latter shows that S cannot be a martingale. If we replace the filtration by the smaller filtration generated by S, then it can be shown that IF' is the only probability that turns S into a local martingale. The reader can check that Proposition 2.8 also holds for local martingales and hence we get that S satisfies (NA) (see [43] for details on Bessel process). For non-continuous processes the concept of local martingale is not yet sufficient as is shown in [15]. We need the even more general concept of sigma-martingale. These processes were introduced by Chou and Emery (see [25]). DEFINITION 2.10 ([15]). A process S ' R + x S-2 ~ IR defined on the filtered probability space (S-2, (~t)0~
3. The fundamental theorem of asset pricing We start this section with a generalization of the concept of No Arbitrage, which can be seen as a topological version of (NA). DEFINITION 3.1. With the notation of Section 2 we say that the semi-martingale S'IR+ x S2 --+ R d satisfies the No Free Lunch with Vanishing Risk (NFLVR) property if n L + -- {0} where d is the closure of C in Loc with respect to the norm II IIoo. There are different topological versions of (NA) pertaining to weaker topologies than the sup-norm (and therefore yielding stronger assumptions). We refer the reader to [ 11 ] for a discussion of the different concepts. THEOREM 3.2. A semi-martingale S satisfies (NFLVR) if and only if S satisfies the following two properties: (a) S satisfies (NA). (b) If en > 0 is a sequence of positive numbers tending to zero, if (Hn)n>>.l is a sequence of en-admissible strategies (i.e., H n 9S ~ - e n ) then (H n 9S)oc --+ 0 in probability.

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One may give examples showing that (a) does not imply (b) and (b) does not imply (a). For the former we defer the discussion to Section 4 below. For the latter we may take the Bessel process R in three dimensions (see [ 12]). We drop the details. The reader can now see why the property is called No Free Lunch with Vanishing Risk. The No Free Lunch with Vanishing Risk property is also equivalent to a boundedness property. In fact we have: THEOREM 3.3. A semi-martingale S satisfies (NFLVR) if and only if S satisfies the following two properties: (a) S satisfies (NA). (b) The set K = {(H-S)oc I n is 1-admissible} is bounded in probability. The boundedness property also has an immediate economic interpretation. If strategies are chosen so that the losses are bounded uniformly (i.e., in Lot), then the gains are bounded in probability (i.e., in L0). This can also be seen as some continuity property. Of course, the relation between items (b) of the two preceding theorems is immediately noticeable. In some cases the analytic definition (i.e., through a stochastic differential equation) may help. Property (b) can easily be checked in the case of continuous processes. We will see this later on. The fundamental theorem of asset pricing in its most general form can now be stated (see [11] and [15]). THEOREM 3.4. For a semi-martingale S :IR+ x s --+ IRd the following two properties are equivalent: (a) S satisfies the (NFLVR) property. (b) There is an equivalent probability measure Q ~ P such that under Q the process S is a sigma-martingale. If we assume that the semi-martingale S is locally bounded (respectively bounded) the term "sigma-martingale" in (b) may be replaced by the term "local martingale" (respectively "martingale "). In case the (NFLVR) property is satisfied we denote by l~/I[e the set of all equivalent sigma-martingale measures. The set M is the set of all absolutely continuous sigmamartingale measures. In case S is locally bounded NI -- M e, where the closure is taken with respect to the norm of L1 (I?), but with general case NI ~ M e. The proof of this theorem is lengthy and we cannot repeat it here. But we will give a sketch of the different points that relate the theorem to functional analysis. The first and most delicate step consists in proving that under the assumption of (NFLVR) the set C is already weak*-closed. Yan's theorem [52] (whose proof consists of a combination of a Hahn-Banach and an exhaustion argument) (see, e.g., [48]) then yields the existence of an equivalent probability measure Q such that, for all f 6 C, E q [ f ] ~< 0. Interpreting Q as a linear functional on L ~ , it separates C from the positive cone L + . In the case where S is locally bounded this already implies that S is a local martingale for the measure Q. In the general case one has to make an additional (nontrivial) step

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that consists in showing that, for e > 0, there is a measure Q0 such that Q0 is equivalent to P, that ]IQ - QoIJ ~< e and that under Q0 the process S is a sigma-martingale. See [ 15] for details and for an example that shows that one cannot do better than a sigmamartingale. As indicated above, the central point of the proof is the fact that C is weak*-closed. This is done using the Krein-Smulian theorem, also called the Banach-Diedonn6 theorem. This theorem says that a convex set C in the dual X* of a Banach space X is ~ (X*, X) (i.e., weak*-closed) if and only if C n (nBx,) is ~(X*, X) closed for each n ~> 1. If X = L1 and X* = L ~ we can, using the characterization of relatively weakly compact sets in L1 as uniformly integrable subsets of L1, make this even more precise. A convex set C C L ~ is weak*-closed if and only if, for each sequence ( f , ) ~ > l in C that is uniformly bounded and converges in probability to a function f , we have that f E C. Since in our context the set C is a cone we have to show the following fact. CLAIM 3.5. Let (Hn)n>~j be a sequence of 1-admissible integrands, let (fn)n>~l be a sequence in L ~ ~ , ~) such that - 1 <~ fn <<.(H n" S)~, which tends to f in probability; then under the assumption of (NFLVR) there is a 1-admissible integrand H such that f <~ ( H . S)~. To prove this claim we have to replace the integrands by a better choice. Let us define the following concept: _

_

m

DEFINITION 3.6. An element f E C1 is called maximal if g E C and g >~ f a.s. implies g - f . Here C1 denotes the set C1 - {h ] h 6 Co, h ~> - 1 } and the bar refers to the closure in the space L0 (i.e., with respect to convergence in probability). m

Our assumptions on (NFLVR) show that C1 is bounded in L0 and hence C1 has sufficiently many maximal elements. In fact every g E C1 is dominated by such a maximal element. To prove the claim we reduce it to the following" CLAIM 3.7. The maximal elements o f C1 are already in C1. So let us fix a sequence ( H n ) n ~ l such that H n is 1-admissible and such that ( H n 9 S ) ~ --+ f in probability, where f is a maximal element in C1. The first trick is to replace the probability P by an equivalent probability (still denoted by I?) such that the function supn~> 1 supt I(H n 9S)t] is in L2(]P). That this is possible follows from an upcrossing type lemma and the maximality of f . After an additional technical reduction, which we skip, the D o o b - M e y e r decomposition theorem now allows to decompose S into a martingale M and a predictable process of finite variation A, i.e., S - M + A. The D o o b Meyer decompositions of ( H n 9S) are then given by ( H " 9M) and ( H ~ 9A). In order to control the jumps of M and A we use the following generalization of an inequality of Stein.

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PROPOSITION 3.8. Let (.Tn)n=0 ..... N be a discrete time filtration on the probability space (12, ~ , I?), let ( fn)n--1 ..... U be adapted to (.Tn)n=O ..... N, let gn be the predictable projection, i.e., gn = E [fn I f ' n - l ] then we have

EI(ZJgn[q)P/q]I/P~2E[(~]fnlq)p/q]1/p whenever 1 ~ p <. q <~ cx~. In other words the mapping

L p (l N)

> L p (lqN),

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(fn)n>~l t

> (E[fn I.T'n-1])n>~l

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has norm less than or equal to 2.

The next step is to show that the convex hull of ( ( H n. M)~)n>>l and of ( ( H n. A)~)n>>l are both bounded (in a good sense). Since ( ( H n . S ) ~ ) n ~ l is bounded in L2(I?) both sets are at the same time bounded or unbounded. But if they are unbounded we are faced with the fact that ( H n 9A)t increases in a "linear" way whereas ( H n . M ) t , due to the orthogonality of its increments, grows only in a way related to the square root. This leads to a contradiction. Once the boundedness is proved, it is a straightforward track to find convex combinations K n of (Hn)n>~l that converge to a predictable process K such that (K 9S ) ~ = f . The latter technique is a combination of techniques of Memin (see [39]). The separation argument in the proof of the fundamental theorem can be exploited further. It yields to the following duality result (see [24,9]). THEOREM 3.9. A s s u m e that S is a locally bounded semi-martingale such that M e is not empty. Then f o r f ~ 0 we get sup E Q [ f ] : inf{ot I 3g ~ K with ot 4- g ~ f } : : or0.

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Furthermore, if the quantities are finite, the infimum is a minimum and there is a maximal element g E K max with oto 4- g ~ f .

4. The continuous case

When the process S is continuous or, more generally, locally bounded, the assumption on S to be a semi-martingale turns out to be a necessary condition for the conclusion of the fundamental theorem of asset pricing to hold true. In fact we can prove the following result (see [ 11 ]), stated under a finite time horizon, say [0, 1]. THEOREM 4.1. I f the locally bounded process S = (St)o<~t <<1. is not a semi-martingale then there is a function f >~ 0, I?[f > 0] > 0, and a sequence o f simple strategies Hn such that (Hn . S) >>.- e n , en --+ O, (Hn . S) 1 ~

f .

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We recall that a strategy H is simple, if there is a finite sequence of stopping times

0 <, To <~T1 <.... <, Tn < cx~ as well as functions (gk)k>~0, gk being 9t'r~ measurable and H __ ~n--1 k=0 gk 2~rk,Zk+~" These strategies are most elementary and the definition of their

stochastic integral (H 9S) does not involve any calculus. The theorem is not true for processes with unbounded jumps, since in this case the boundedness condition on (H 9S) might imply that H ~ 0. Theorem 4.1 has some nice consequences for continuous processes. In particular fractional Brownian motion fails to be a semi-martingale (with the exception of Brownian motion, of course) and therefore allows, by Theorem 4.1, this kind of arbitrage. This implies that these models are useless in the context of pricing and hedging derivative securities by no-arbitrage arguments. This fact was known since a long time, but even after publication of the above theorem, people still continued to ask about fractional Brownian motion. Rogers [44] decided to settle the problem by giving a fairly explicit strategy H such that (H.S) l ~>0and~[(H.S)I>0]>0,(H.S)>/-1. The strategy is not simple (one can show that for simple strategies (NA) is satisfied), but it is "semi"-simple in the sense that restricted to each interval [0, t] with t < 1, it is simple. Fractional Brownian motion, however, has some nice long range dependencies, observed in financial data. In forthcoming work of Cheridito [5] the reader can find a way out to this dilemma. For continuous processes the concept of (NA) can be analyzed further. So let us assume that S is a continuous semi-martingale with respect to a filtered space (I2, (~t)0~
~u f olt h 2 d(M, M)u), the so-called Dol6ans-Dade or stochastic exponential of ( - h 9M),

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also written as s 9M). Of course the existence depends on the finiteness of the integral f0 h] d(M, M)u. If the integral is not almost surely finite then there are essentially two possibilities. To distinguish them we introduce the stopping time

{If0 ' h u2 d(M , M)u -- e~ }.

T--inf t

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On the set T ~ 1 we then have either f o h ] d ( M , M)u --cxz or we have, Ve > 0, fTT+ e hu2 d ( M , M)u = oc.

Technically the latter case is more difficult but it has a simple solution: there is a very special kind of arbitrage, namely there is H supported by ~T, 1~ and such that (H 9S) ~> 0 i.e., the outcome at every time is nonnegative, and iP[(H 9S)T+e > 0] > 0 for all e > 0. dt + d Wt where W is Brownian motion An example of this nature is given by dSt - -~

defined in its natural filtered space. A strategy of the form above is given by Kt -1/(~/71og(t-1)) and then we stop the process ( K . S) either when it reaches a level 1 or when it hits 0 for the first time after 0. The iterated logarithm theorem implies that, immediately after 0 the process (K 9S) is strictly positive! The next result is closely related to these arguments. It was shown in [12] and, independently and under slightly stronger hypothesis, in [37]. THEOREM 4.2. I f the continuous martingale S satisfies (NA) then there is a measure Q absolutely continuous with respect to iP and such that S is a local martingale under Q. The support o f the measure Q can be chosen to be equal to {L1 > 0} where Lt -- e x p ( hu dMu - I h2 d ( M , M)u) stopped when h2 d ( M , M)u hits oc, or what is the same: when L hits O.

fo

fo

fo

Even when S satisfies (NA) and fo h] d(M, M}u < ec a.s., the process L need not be a martingale, i.e., it can happen that E[L1] < 1. This means that the measure Q is not necessarily given by dS = L1 diP. See [47] and [ 16]. We also have the following: PROPOSITION 4.3. I f S is a continuous semi-martingale decomposed as dS = d M + h d ( M , M ) then the set KI = {(H. S ) ~ ] H is 1-admissible} is bounded in Lo(IP) if and only if f 0 h2, d ( M , M)u < cx) a.s.

The 3-dimensional Bessel process shows that this does not rule out arbitrage (compare

[13])~ 5. Changes of num~raire and a related Banach space Throughout this section we suppose that the process S modeling the price of d stocks is a d-dimensional semi-martingale that is locally bounded. We also suppose that the process

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S admits a local martingale measure. We repeat that the set of equivalent local martingale measures is denoted by M e, the notation M being reserved for the closed convex set (in fact the closure of M e) of absolutely continuous local martingale measures for S. Ka will denote the cone of outcomes of (a-)admissible strategies, K - - U a Ka. As seen before, maximal elements play an important role in the proof of the fundamental theorem. However, more can be said. If V = 1 + (H 9S) is a strictly positive process satisfying V~ > 0 a.s., then we might use V as a new money unit (say $ instead of CHF or Euro). The market 1 s ) . Economic interpretation is now described by the (d + 1)-dimensional process X - (V, leads us to conjecture that S satisfies (NA) if and only if X satisfies(NA): It does not matter whether you do the bookkeeping in $, CHF or in Euro! However since the admissible strategies may change, we have to be more careful. THEOREM 5.1. If S satisfies (NFLVR) then the following assertions are equivalent for a process V = 1 + (H . S), where H is admissible and V ~ > 0 a.s.: 1 s (1) X -- (V, V ) satisfies (NA), (2) there is an equivalent local martingale Q measure f o r the process S, such that V is a uniformly integrable Q-martingale, i.e., E q [ V ~ ] = 1 or equivalently E q [ ( H 9 S)~] =0, (3) (H . S ) ~ is maximal in K1 (and hence also in K). It is not true that in this case V is a uniformly integrable martingale for each element R in the set of equivalent local martingale measures M e, see [47] or [16] for this surprising fact. The above theorem also yields another proof of the theorems of Jacka [31 ] and AnselStricker [ 1]. THEOREM 5.2. If S satisfies (NFLVR) and f is a positive random variable, then the following conditions are equivalent: (a) f = ~ + (H . S ) ~ with (H . S ) ~ maximal in K, (b) there is Q E 1 ~ e such that E q [ f ] = sup{E R[f][R E M} < cx~. The Bishop-Phelps theorem now immediately implies THEOREM 5.3. If S satisfies (NFLVR) and continuous local martingale measures are already equivalent, then M is reduced to a singleton. PROOF. As the set of absolutely continuous local martingale measures M is a bounded, closed and convex subset of L1, the set { f I f attains its supremum on M} is a norm dense subset of L ~ . The set of all elements of the form c~ + ( H - S ) ~ , where the process ( H . S) is bounded is therefore dense in L ~ and because it is closed (this follows essentially from claim 3.5) we have that it equals L ~ . This implies that all elements of L ~ are constant on M, hence M can only be a one-point-set. [-1 Another application of Banach space theory is given by James' theorem on weakly compact sets. We state the result in its negative form, see [9] for details.

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THEOREM 5.4. Suppose S is continuous, satisfies (NFLVR) and suppose that all martingales with respect to (.Yt)o<<.t are continuous (i.e., all stopping times are predictable). Then we have (a) either M is a singleton, (b) or M is so big that it has no extreme points. It turns out that (a) occurs if and only if 1Vgis weakly compact and the proof uses James' theorem. In the case when S is only assumed to be locally bounded (and not necessarily continuous), the above theorem is false and the implications of M being weakly compact are not yet fully understood. Turning back to the theme of Theorem 5.1 above, we find that the maximal elements generate, in a natural way, a Banach space. THEOREM 5.5. The set K max of maximal elements forms a cone in L1 (~). This is not obvious. Indeed if g, f 6 K, E Q [ f ] = 0 and E R[g] = 0 for Q, R 6 M e then how can we find an element Q' ~ 1~]ie with E Q , [ f + g] = 0? The proof uses the numeraire Theorem 5.1 above. With the convex cone K max we may construct the vector space G = K max - K max. On G there is natural norm, if g = f - h where f, h ~ K max then we put Ilgll--inf{a I g - f - h, f, h E gamaX}. Surprisingly (G,

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II II) is complete and

211gll = sup{llgllL,(Q) I Q ~ l~/i[e}.

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Although the constructions of K max and of G make sense economically, the implications of this result in mathematical finance are not yet clear. Examples in [14] show that the space G can have different natures, it can be an L1 space but it can also contain a complemented L ~ space. Also the space G is different from the space of functions f so that sup{ IIf IIL I(Q) I Q ~ Me } < 00. Indeed take f 6 K1 \ K max (such elements exist) then f ~ G (this requires a proof!) and supQ~IVl[e IIflIL~(Q) ~< 1. A side result of the theory is the following THEOREM 5.6. I f f E K max then { Q I Q 619I[, E Q [ f ] - 0 } is a dense G~-set in M. Hence { Q I Q ~ Me, E Q [ f ] = 0} is a dense G~-set in M e.

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6. Weighted norm inequalities and closedness of a space of stochastic integrals In this section we investigate a topic initiated by the work of F611mer and Sondermann [27] and F611mer and Schweizer [26], to deal with the problem of hedging in incomplete markets, i.e., when there is no uniqueness of the equivalent martingale measure. To motivate the idea, first suppose for simplicity that S is an R-valued martingale under the original measure I? which we also assume to be bounded in L 2 ( ~ , .)c-, iP). We denote by G the subspace of L 2 ( ~ , f', iP) consisting of the functions of the form

f --

fo T Hu dSu

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= ( H . S)T,

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where H is a predictable process such that the stochastic integral makes sense in L 2, i.e., such that ((H- S)t)t~
II/0THu dS,

L2 (~,.)c',IP)

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= IIHIIL2(S-2•

=

Ht2 (co) d(S)t (09) diP

.

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Here 79 denotes the sigma-algebra of predictable subsets of s x [0, T] and diP d(S) denotes the finite measure on 79 induced by the quadratic variation process of S. As the space L2(s x [0, T], 79, diP d(S)) is obviously complete and can be identified via the above isometry with thesubspace G of L2(~2, .T, iP) we find that G is closed in L2(I2, .T, iP). We denote by G the space spanned by G and the constant functions which clearly again is closed in L 2. Note in passing that in the case when (St)0~
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where c is a constant, g may be written as g - ( H . S)T while h is orthogonal to G. We may associate to g and h the martingales E[g I~'t]O<~t<~T= ((H. S)t)O<~t<~rand (E[h I bet ])0~t ~
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So far we have used the simplifying assumption that the price process S - (St)0~
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We have seen in Section 4 that under the condition of (NFLVR) the IRd-valued measure dA is absolutely continuous with respect to d(M) (taking values in the non-negative definite operators on 1Rd) and we may find an IRd-valued predictable process )~ = ()vt)0~
dAt -- d(M)t)~t

1?-a.s. for 0 ~< t ~< T.

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A formal application of the Girsanov-theorem indicates that the probability measure Qmin defined via the density process

Lt := s dQ min d1?

9M)t,

and

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:= LT" = C(-)v. M)7",

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where C denotes the Dol6ans-Dade exponential of a local martingale, is a candidate for an equivalent local martingale measure for the process S (provided all the involved limiting procedures make good sense and L7" really defines a density of an equivalent measure). F611mer and Schweizer have called this measure the "minimal" martingale measure Qmin (if it exists). Another natural choice from the set l~/I[e (S) of equivalent local martingale measures for S is QOpt, the "variance-optimal" measure, which minimizes the L 2 (S2, bt-, 1?)-norm among all elements of ~/]~e(S) (again, provided it exists). In many cases, e.g., when S has independent increments, Qmin and QOpt coincide, but in general they are different. We now turn again to the theme of closedness of a (properly defined) space of stochastic integrals. To introduce the notion of predictable trading strategies appropriate in the present context, we call a predictable IRd-valued process L2-admissible if H . M as well as H . A make sense in L 2, i.e., if H . M is an L 2 (S-2, ~ , 1?)-bounded martingale and E [ IHu dAu I)2] < oo. The definition is chosen in such a way that we may define the stochastic integral H . S - H . M + H . A and that ( H . S)7~ is in L2(12, ~ , 17). Denote again by G the subspace of L 2 (12, ~ , 17) formed by the functions ( H . S)7", where H runs through the L2-admissible trading strategies.

(fo

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To formulate the theorem we recall that a measure Q .~ I? with density process Lt -E [ ...~ I fi't_], satisfies the reverse HOlder condition Rp, for 1 < p ~< oc, if there is a constant C such that E

~

I.Tt

1

~CE

~l.Tt

]'

forl~t~
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The name stems from the fact that by HOlder's inequality the reverse inequality is always satisfied with C = 1. The reverse H61der condition is dual to the Muckenhaupt condition (see [ 18]). We now can formulate the central result of [ 10]: THEOREM 6.1. Let S = (St)o<<.t<.T be a continuous semi-martingale and suppose that there exists an equivalentprobability measure Q with ~dQ E L2(I2, f ' , 17) under which S is a local martingale. The space G is closed in L 2 (I-2, .T, 1?) if and only if there is an equivalent local martingale measure Q for S which satisfies the reverse HOMer condition R2. In this case the so-called "variance-optimal" measure Qopt, which has minimal L 2 (s .T, 1?)-norm among all equivalent local martingale measures, is well-defined and satisfies the reverse HOlder condition R2. There are some variants of the above theorem in terms of the process )~. M. A necessary condition for Theorem 6.1 to hold true (under the assumptions given there) is that the process )~. M is in BMO but this condition is not sufficient. In fact )~. M is in BMO is equivalent to the completeness of G with respect to the stronger norm defined via the maximal function II(H. S)ll "--Ilsup(H. M), IlcZ(s~,~-,?)-

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t dQ min

On the other hand the condition that the minimal m e a s u r e ~min defined via o~ = s M) exists and satisfies the reverse H61der condition R2 is sufficient but not necessary for the theorem to hold true. In fact, this stronger condition is equivalent to the fact that G is closed and, in addition, that the (in general not orthonormal) projection re from L 2 ( ~ , f ' , P) onto G with ker(zr) -- M • is well defined and continuous. Here M • denotes the subspace of L 2 orthogonal to the space M generated by the constants and the stochastic integrals on the local martingale part M of the process S. If Jr is well-defined and bounded then Jr induces the F611mer-Schweizer decomposition as it splits a general contingent claim f E L2($2, f , 17) into the hedgeable part 7r(f) which may be written as Jr ( f ) - const + f o 14, dSu for an L 2 -admissible trading strategy H and the non-hedgeable part f - Jr ( f ) which induces a martingale strongly orthogonal to the local martingale part MofS. We refer to [10] for a detailed presentation of these topics for the above discussed closedness of G in LP for p = 2 and the case of Rd-valued continuous semi-martingales S. Extensions to the case 1 < p < cx~ as well as to the case of general Rd-valued semimartingales were obtained in [28,6] and [7].

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References [1] J.E Ansel and C. Stricker, Couverture des actifs contingents etprix maximum, Ann. Inst. Henri Poincar6 30 (1994), 303-315. [2] L. Bachelier, Th~orie de la speculation, Ann. Sci t~cole Norm. Sup. 17 (1900), 21-86. English translation: The Random Character of Stock Market Prices, E Cootner, ed., MIT Press, Cambridge, MA (1964), 17-78). [3] T. Bj6rk, Arbitrage Theory in Continuous Time, Oxford University Press (1998). [4] E Black and M. Scholes, The pricing of options and corporate liabilities, J. Political Econom. 81 (1973), 637-654. [5] E Cheridito, Arbitrage for fractional Brownian motion, ETH working paper (2000). [6] T. Choulli, L. Krawczyk and C. Stricker, On Fefferman and Burkholder-Davis-Gundy inequality for Emartingales, Probab. Theory Related Fields (1997). [7] T. Choulli, L. Krawczyk and C. Stricker, E-martingales and their applications in mathematical finance, Ann. Probab. 26 (1998), 853-876. [8] R.C. Dalang, A. Morton and W. Willinger, Equivalent martingale measures and no-arbitrage in stochastic securities market model, Stochastics Stochastics Rep. 29 (1990), 185-201. [9] E Delbaen, Representing martingale measures when asset prices are continuous and bounded, Math. Finance 2 (1992), 107-130. [10] E Delbaen, E Monat, W. Schachermayer, M. Schweizer and C. Stricker, Weighted norm inequalities and hedging in incomplete markets, Finance and Stochastics 1 (3) (1997), 181-227. [ 11 ] E Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Math. Ann. 300 (1994), 463-520. [ 12] E Delbaen and W. Schachermayer, The existence of absolutely continuous local martingale measures, Ann. Appl. Probab. 5 (4), (1995), 926-945. [ 13] F. Delbaen and W. Schachermayer, An inequality for the predictable projection of an adapted process, Sem. de Probabilites XXIX, Lecture Notes in Math. 1613, J. Azma, M. Emery, EA. Meyer and M. Yor, eds, Springer (1995), 17-24. [14] E Delbaen and W. Schachermayer, The Banach space of workable contingent claims in arbitrage theory, Annales de 1' I.H.P. 33 (1) (1997), 113-144. [15] E Delbaen and W. Schachermayer, The fundamental theorem of asset pricing for unbounded stochastic processes, Math. Ann. 312 (1998), 215-250. [16] E Delbaen and W. Schachermayer, A simple counterexample to several problems in the theory of asset pricing, Math. Finance 8 (1998), 1-12. [17] C. Dellacherie and P.A. Meyer, Probabilit~s et Potentiel. Th~orie des Martingales, Hermann, Paris (1980). [18] C. Dol6ans-Dade and EA. Meyer, In~galitis de normes avec poids, S6m. de Probabilit6 XIII, Lecture Notes in Math. 721, Springer (1979), 313-331. [19] D. Duffle and C.-E Huang, Multiperiod security markets with differential information; martingales and resolution times, J. Math. Econom. 15 (1986), 283-303. [20] B. Dupire, Model Art, Risk 6 (9) (1993). [21 ] A. Einstein, On the moment of small particles suspended in stationary liquid demanded by molecular-kinetic theory ofheat, Ann. Phys. 17 (1905). [22] A. Einstein, Zur Theorie der Brownschen Bewegung, Ann. Phys. 19 (1906), 371-381. [23] A. Einstein, Investigations on the Theory of the Brownian Movement, Dover, New York (1956). [24] N. E1Karoui and M.C. Quenez, Dynamic programming and pricing of contingent claims in an incomplete market, SIAM J. Control Optimiz. 33 (1995), 29-66. [25] M. Emery, Compensation de processus ~ variation finie non localement int~grables, S6m. de Probabilit6 XIV, Lecture Notes in Math. 784 (1980), 52-60. [26] H. F611mer and M. Schweizer, Hedging of contingent claims under incomplete information, Applied Stochastic Analysis, M.H.A.Davis and R.J.Elliott, eds, Gordon and Breach, London (1991), 389-414. [27] H. F611mer and D. Sondermann, Hedging of non-redundant contingent claims, Contributions to Mathematical Economics in Honor of G6rard Debreu, W. Hildenbrand and A. Mas-Colell, eds, North Holland, Amsterdam (1986), 205-223. [28] P. Grandits and L. Krawczyk, Closedness of some spaces of stochastic integrals, Seminaire de Probabilites XXXII (1998), 73-85.

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[29] J.M. Harrison and D.M. Kreps, Martingales and arbitrage in multiperiod securities markets, J. Econom. Theory 20 (1979), 381-408. [30] J.M. Harrison and S.R. Pliska, Martingales and stochastic integrals in the theory of continuous trading, Stochastic Process. Appl. 11 (1981), 215-260. [31] S.D. Jacka, A martingale representation result and an application to incomplete financial markets, Math. Finance 2 (1992), 239-250. [32] J. Jacod, Calcul Stochastique et Problbmes de Martingales, Lecture Notes in Math. 714, Springer, Berlin (1979). [33] I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer (1997). [34] A.N. Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung, Erg. Math., B.2 (1933). [35] D. Kreps, Arbitrage and equilibrium in economies with infinitely many commodities, J. Math. Econom. 8 (1981), 15-35. [36] D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, Chapman & Hall (1996). [37] S. Levental and A.S. Skorohod, A necessary and sufficient condition for absence of arbitrage with tame portfolios, Ann. Appl. Probab. 5 (1995), 906-925. [38] B.B. Mandelbrot, Forecasts of future prices, unbiased markets and "martingale" models, J. Business 39 (1966), 242-255. [39] J. Memin, Espaces de semi-martingales et changement de probabilitY, Z. Wahrscheinlichkeitsth. verw. Geb. 52 (1980), 9-39. [40] R.C. Merton, Theory of rational option pricing, Bell J. Econom. Manag. Sci. 4 (1973), 141-183. [41] M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling, Applications of Mathematics, Springer (1997). [42] E Protter, Stochastic Integration and Differential Equations, Springer, Berlin (1990),. [43] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, Berlin (1991). [44] L.C.G. Rogers, Arbitrage from fractional Brownian motion, Math. Finance 7 (1997), 95-105. [45] L.C.G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, Vol. 2: ItO Calculus, Wiley, Chichester (1987). [46] EA. Samuelson, Proof that properly anticipated prices fluctuate randomly, Industr. Manag. Review 6 (1965), 42-49. [47] W. Schachermayer, A counter-example to several problems in the theory of asset pricing, Math. Finance 3 (1993), 217-229. [48] W. Schachermayer, Martingale measures for discrete time processes with infinite horizon, Math. Finance 4 (1) (1994), 25-55. [49] A.N. Shiryaev, Essentials of Stochastic Finance. Facts, Models, Theory, World Scientific (1999). [50] M. Smoluchowski, Drei Vortriige iiber Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen, Phys. Z. 17 (1916), 557-585. [51] C. Stricker, Arbitrage et lois de martingale, Ann. Inst. H. Poincar6 Probab. Statist. 26 (1990), 451-460. [52] J.A. Yan, Caracterisation d' une classe d'ensembles convexes de L 1 ou H 1, Seminaire de Probabilites XIV, Lecture Notes in Math. 784 (1980), 220-222.

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CHAPTER

10

Perturbed Minimization Principles and Applications Robert Deville Department of Mathematics, University of Bordeaux, Bordeaux, France E-mail: deville@ceremab,u-bordeaux.fr

Nassif Ghoussoub Department of Mathematics, University of British Columbia, Vancouver, BC, Canada E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Dentability and perturbed minimization principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. A general perturbed minimization principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. First examples of admissible classes of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Martingales and admissible cones of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Perturbed minimization and compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Perturbed minimization and differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Smooth minimization principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Smooth norms and smooth bumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Generic differentiability of convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. The existence of separating polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Non-smooth calculus in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. First order sub- and super-differentials for continuous functions . . . . . . . . . . . . . . . . . . . 4.2. The sub-differential of the sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Second order differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Stability of subdifferentials and superdifferentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Mean value theorems . . . . . . ..................................... 5. Application to critical point theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Palais-Smale minimizing sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The mountain pass theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Applications to Hamilton-Jacobi equations in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . .

395 397 397 399 401 403 405 406 407 409 412 414 415 418 419 422 422 423 423 424 426

6.1. The m a x i m u m principle for stationary first order Hamilton-Jacobi equations . . . . . . . . . . . . 6.2. The m a x i m u m principle for parabolic Hamilton-Jacobi equations . . . . . . . . . . . . . . . . . .

426 428

6.3. Uniqueness in second order Hamilton-Jacobi equations . . . . . . . . . . . . . . . . . . . . . . . .

430

6.4. The existence of viscosity solutions for Hamilton-Jacobi equations . . . . . . . . . . . . . . . . . . References

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H A N D B O O K OF T H E G E O M E T R Y OF B A N A C H SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved 393

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Abstract Given a bounded below, lower semi-continuous function q~ on an infinite dimensional Banach space or a non-compact manifold X, we consider various possibilities of perturbing q~ by an element p of a reasonable class of functions A in such a way that for the new functional 4) - P, the minimization problem infx (q~ - p) is well-posed (i.e., every minimizing sequence is convergent). These perturbed minimization principles are quite powerful and have a wide array of applications in Banach space theory, potential theory, non-smooth analysis, non-linear analysis, the variational approach to differential equations as well as in the theory of viscosity solutions for Hamilton-Jacobi equations.

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1. Introduction A Perturbed Minimization Principle is a statement of the following type:

Let A be a class of functions on a complete metric space X and consider a bounded below, lower semi-continuous function ~0:X --+ R U {+co}. Keeping in mind t h a t - in g e n e r a l - such a function does not necessarily attain its minimum, is it then possible to perturb ~0 by an element p of A in such a way that the new functional q) - p attains its minimum? a unique minimum? or that the minimization problem (X, ~0- p) is well-posed (i.e., every minimizing sequence is convergent). The problems involved in looking for such a statement revolve around the appropriate classes of functions A and the Banach spaces where it is possible. These choices are of course motivated by their scope of applicability albeit to differential equations, non-linear analysis or to Banach space theory. The first perturbed minimization principle is of course the celebrated: THEOREM 1.1 (Bishop-Phelps). Let C be a nonempty, closed, convex and bounded subset o f a real Banach space X. Then, the set M c o f all elements o f X* that attain their supremum on C is norm dense in X*. The connection with our topic is more clear if we state the following reformulation due to Br6ndsted and Rockafellar. COROLLARY 1.1. Let C be a nonempty closed convex subset o f X and let q) : C --+ R be lower semi-continuous, convex function that is bounded from below. Then, f o r any e > O, there exists x* ~ X*, Ilx* 11~< e such that q) + x* attains its minimum at some point xo ~ C.

Later, Ekeland [32] established the following very general and ultimately much more applicable theorem. THEOREM 1.2 (Ekeland). Let (X, d) be a complete metric space and consider a function q):X ~ ( - c o , +ec] that is lower semi-continuous, bounded from below and not identical to +ec. Let E > 0 and )~ > 0 be given and let x ~ X be such that q)(x) <, infx q9 + e. Then there exists xe ~ X such that:

(i) qg(xe) ~ qg(x). (ii) d ( x , x e ) <~ 1/)~. (iii) For each y 7~ xe in X, q)(y) > q)(xe) - e)~d(xe, y). Note that the class of perturbations in Ekeland's theorem is the space A of all Lipschitz functions, while in the Bishop-Phelps theorem, it is the much nicer space of continuous linear functionals, provided of course we limit ourselves to the problem of minimizing only convex functionals which is quite restrictive for most purposes. There are also other important differences between the 2 results. In Ekeland's theorem, the minimization problem becomes well-posed after perturbation, and the procedure can be localized around any given approximate minimum. This is not the case in the Bishop-Phelps theorem. However, we shall exhibit in this paper a whole range of admissible classes of perturbations that are

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sometimes larger but more flexible than the class of continuous linear functionals but can also be smaller but more regular than the class of Lipschitz functions. First, let us show how Ekeland's result implies Bishop-Phelps', modulo the H a h n - B a n a c h Theorem. PROOF. Let q9 as in the above corollary. According to Ekeland's principle, there exists xo E C such that qg(x) ~> ~o(xo) - ~llx - xoll for all x 6 C. Consider in the Banach space X 9 R the subsets: C1 - {(x, t) E C @ R; t/> ~o(x)}, C2 - - { ( x , t) E X ~) R; t ~< ~p(xo) -

EIIx - xoll ].

They are both closed and convex and the interior of C2 is non empty and disjoint from C1. Using the H a h n - B a n a c h theorem, we can find G = (y*, c~) 6 X* E3 R = (X G R)* such that (y*, c~) r (0, 0), and m E R such that: (1) if x ~ C and t >~ qg(x), then y * ( x ) + at >~m; (2) if x 6 X and t <~ qg(xo) - ellx - x0ll, then y * ( x ) + at <<,m. Note that c~ = 0 is impossible. Otherwise, using condition (2), y * ( x ) <<,m for all x E X, which implies y* - - 0 and contradicts the fact that (y*, or) ~ (0, 0). The case c~ < 0 is not possible either: take x -- x0 and t large enough to get a contradiction in (1). Thus ot > 0. Replacing if necessary G by ~1 G , we can assume ot - 1. Since (x0, qg(x0)) 6 Cl n C2, m = y* (x0) + ~0(x0), (1) and (2) then imply: (3) for all x E C, y * ( x ) + qg(x) >~ y*(xo) + qg(x0); (4) for all x E X, y * ( x ) + qg(xo) - ellx - x0ll <~ y*(xo) + qg(xo). Relation (3) shows that y* + ~o attains its minimum on C at x0 and (4) shows that y* (x xo) ~< ellx - xoll for all x c X, whence Ily*[I ~< e. D To establish now the Bishop-Phelps theorem, let x* 6 X* and e > 0 be fixed. By the above, there exist y* ~ X*, Ily*ll ~< e, such that - x * + y* attains its minimum on C at some point x0 6 C, so z* = y* - x* attains its supremum on C and it is e-away from x*. We now describe an important improvement of Ekeland's theorem due to Borwein and Preiss [8]. Consider the class Q of all real-valued functions on X of the form 1 ~ q (X) -- -~ Z

, )2 # n d (x Vn. ,

n=l

where #n >~ 0, ZnC~__l # n - - 1 and where (Vn)n is some convergent sequence in X. The following result states that the Lipschitz perturbations obtained in Ekeland's theorem can be replaced by quadratic perturbations in the class Q. The relevance of this improvement becomes clear in the case when X is a Banach space admitting a differentiable norm (away from the origin), since the functions in the class Q are then differentiable everywhere. THEOREM 1.3 (Borwein-Preiss). Let qg : X --+ (-cx~, +c~] be a lower semi-continuous function that is bounded from below and not identical to +cx~ on the complete metric

Perturbed minimization principles and applications

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space (X, d). Fix e > O, X > 0 and assume that u E X satisfies qg(u) < infx q9 + e. Then there exist q E Q and v~ ~ X such that: (i) qg(v~) < infx ~o + e.

(ii) d(u, v~) < X. (iii) For all x E X, qg(x) + 2eX-Zq(x) >~qg(v~) + 2eX-Zq(v~). In view of the three perturbed minimization principles discussed above, it is natural to inquire about the classes of functions on a complete metric space X that are suitable to be perturbation spaces for a minimization principle. In the next section, we shall isolate a fairly general condition on a class A of continuous functions on X that makes it eligible to be a perturbation space. We shall then say that A is an admissible class. We will see that the perturbations used in Ekeland's principle as well as the one of Borwein-Preiss, readily satisfy our condition. The same will hold for various spaces of smooth functions defined on a suitable Banach space. However, the proofs of the facts that, on certain Banach spaces, the spaces of linear functionals or cones of plurisubharmonic functions are admissible classes, are more involved. The rest of the paper contains various applications of these principles to Banach space theory, potential theory, non-smooth analysis, non-linear analysis, the calculus of variations as well as to the theory of viscosity solutions for Hamilton-Jacobi equations.

2. Dentability and perturbed minimization principles 2.1. A general perturbed minimization principle Let (X, d) be a metric space and let (A, 3) be a metric space of real valued functions defined on X. For any subset F of X, we shall denote by ,AF the class of functions in A that are bounded above on F. For f E AF, and t > 0, we denote by S(F, f, t) the following slice of F

S(F, f, t) = {x E F" f (x) > s u p f ( F ) - t}. DEFINITION 2.1. The space (X, d) is said to be uniformly A-dentable if for every nonempty set F C X, every f E AF, and every e > 0, there exists g 6 ~AF such that 6(f, g) <<,e and a non-empty slice S(F, g, t) of F such that diam S(F, g, t) ~< e. In the sequel, we associate to the metric space (X, d), the space X - X • R equipped with the pseudo-metric

~l((x, X), (y, #)) -- d(x, y). We also associate to (A, 3) the class A of functions of the form f "-- (f, - 1 ) which act on X via (f, - 1)(x, )~) -- f ( x ) - X. We equip A with the distance ~ ( f , ~,) - 3(f, g). DEFINITION 2.2. Say that a function qg" X --+ R U {+cx~} strongly exposes y in X if:

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(i) qg(y)= inf{q)(x); x 6 X} and (ii) d(y, Yn) --+ 0 whenever q)(Yn) --+ q)(Y). We then say that y is a strong minimum for q). Note that, in particular, y is then a unique minimum for q). Denote by D(qg) the set {x ~ X; qg(x) < +c~}. We can now state the following general perturbed minimization principle. THEOREM 2.1. Let A be a class of bounded and continuous functions on a complete metric space (X, d) equipped with a metric 6. Suppose ,,4 satisfies the following conditions: (i) There is a K > 0 such that 6 ( f , g) >~ K sup{If(x) - g(x)l; x e X} f o r all f, g ~ A. (ii) (A, 6) is a completemetric space. (iii) The product space (X, d) is uniformly A-dentable. Then, f o r any lower semi-continuous function q):X --+ R U {+cx~} that is bounded below with D(~o) 7~ 0, the set {g E A; q9 - g attains a strong minimum on X}

is a dense G a subset G of,4. We shall then say that (,4, 6) is an admissible class of perturbations for the space (X, d). PROOF. We claim that the set /.gn = {g c A ; 3Xn e X with ( ~ o - g ) ( x n ) < inf{(~o - g)(x): d ( x , x n ) >~ I / n } } is an open dense subset of A. Indeed, ~n is open because of assumption (i). To see that/gn is dense, let g E A and e > 0. We need to find h E A, 6(h, g) < e, and Xn E X such that

(~o - h)(xn) < inf{(~o - h)(x); d ( x , x n ) >~ 1/n}.

(*)

To do that, note that the functional (g, - 1 ) in A is bounded above on the epigraph F of q) in ~', hence for any e' > 0, there exists a non-empty slice S - S(/t, F, t) of F with diameter less than e ' and determined by a function h = (h, - 1 ) ~ A verifying 6(h, g) < d. Take e ~ < min{e, I / n } and pick any (Xn,)~n) E S. For any x E X such that d ( x , x n ) 1/n, we have that (x, qg(x)) ~ F \ S, so that h (x) - q9(x) ~~ q)(Xn), we obtain that the function h verifies (,). Since A is a complete metric space, G - On>~l L/n is a dense G~-subset of A, by the Baire category theorem. We claim now that if g ~ G, then q) - g attains a strong minimum on X. Indeed for each n ~> 1, there exists Xn E X such that

(~o - g)(xn) < inf{(cp - g)(x); d ( x , x n ) >f 1/n}.

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We necessarily have that d (Xp, Xn) < 1/n for each p > n. Indeed, if not then by the definition ofxn, we have (qg-g)(xp) > (q)-g)(xn). On the other hand, we willhaved(xn,Xp) >I 1/n >~ 1/p which gives, by the definition of Xp, that (q9 - g)(xn) > (q9 - g)(xp). This is clearly a contradiction. Thus (Xn)n is a Cauchy sequence converging to some x ~ 6 X and we claim that x ~ is a strong m i n i m u m for 99 - g. Indeed, since 99 is lower semi-continuous,

(~p - g ) ( x ~ ) <~ liminf(tp - g)(Xn) <~liminfinf{(~o - g)(x)" d(x,xn) >~ I/n} <~ inf{ (q9 - g)(x); x e X \ { x ~ } }. Moreover, let (y~) be a sequence in X such that (q9 - g)(yn) converges to ((p - g)(x~). Let us assume that (Yn) does not converge to x ~ . Extracting, if necessary, a subsequence, we can assume that there exists e > 0 such that for all n, d (y,, x ~ ) >~ e. Thus there exists an integer p such that d (Xp, y,) ~ 1/p for all n. Consequently

(cp - g ) ( x ~ ) <~ (q) - g)(Xp) < inf{(~o - g)(x); d ( x , x p ) > l / p } <~ (~o - g)(yn) for all n, which contradicts the fact that limn ((p - g)(y,) = (q9 - g)(x~).

2.2. First examples of admissible classes of perturbations One can easily recover Ekeland's result as well as the theorem of Borwein and Preiss [8] from T h e o r e m 2.1. For that, we m a y consider the cones r (respectively .A2) consisting of those functions f on (X, d) of the form

f(x)--~)~,~d(x,xn)

(respectivelyf(x)--Z)~nd2(x,x~))

n

n

with Xn >~ 0. We assume here, without loss of generality, that d is a b o u n d e d metric on X. Let us check that they are admissible cones. Indeed, suppose F is a closed subset of ~" - X • R such that/~ -- ( h , - 1 ) 6 .A1 is b o u n d e d above on F. For small r > 0, consider a point (x0, so) 6 F such that h(xo) - so > suPF/~ -- r 2. Let k be the functional in .A1 defined by/r -- (h - r d (, x0), - 1) and let S be the slice of F given by

s - { (s, ~) e F; /,(s, ~) > ~(xo, so) - r2}. It is easy to see that the d - d i a m e t e r of S is less than r, which means that X is .A1-uniformly dentable. A similar proof works for .A2. Note that Ekeland's result would then follow from T h e o r e m 2.2 and the triangular inequality. Note also that we could have used the space .A - Lip(X) of Lipschitz functions on X as an admissible space. We shall now investigate the possibility of having other classes of functions as perturbation spaces. For simplicity, we shall only deal, in the sequel, with the case where X is a Banach space.

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A function b ' X --+ R is said to be a bump function on X if it has a bounded and nonempty support. PROPOSITION 2.1. Let ,A be a Banach space o f continuous functions on a Banach space

X satisfying the following properties: (i) For each g 6 A, IlgllA t> Ilgll~ = sup{lg(x)]; x E x}. (ii) A is translation invariant, i.e., if g E A and x ~ X, then rxg : X --+ R given by rxg(t) = g(x + t) is in A and II~xgllA = IlgllA. (iii) A is dilation invariant, i.e., if g ~ A and ot ~ R then g~ : X --+ R given by g~ (t) = g(ott) is in A. (iv) There exists a bump function b in A. Then A is an admissible space o f perturbations f o r the space X. PROOF. According to (ii) and (iv), we can find a bump function b in A such that b(0) r 0. Using (iii) and replacing b(x) by otl b(oQx) with suitable coefficients C~l, ot2 E R, we can assume that b(0) > 0, Ilbll~ < e and b(x) - 0 whenever Ilxll ~> e. Let now ~ - (g, - 1) E A be bounded above on a closed subset F of X and let (x0, so) F be such that g(xo) - so > sup F ~ - b(0) and consider the function h(x) -- b(x - xo) and k = (g + h , - 1 ) . By (ii), h 6 A and Ilhll.a - IlbllA < e, which implies that Ilg - ~:IIA < e. On the other hand, consider the following slice of F,

S--

{

]

(x, s) 6 F; k(x, s) > sup ~ . F

It is non-empty since it contains (xo, so). On the other hand, if (x, s) 6 F and IIx - xoll ~ e, then b(x - x0) = 0 and (x, s) cannot belong to S. It follows that the d-diameter of S is less than 2e. Consequently, A is an admissible family of perturbations. E] COROLLARY 2.1 (Localization). Assume A is a Banach space o f bounded continuous functions on X satisfying conditions (i)-(iv) above. Then, f o r some constant a > O, depending only on X and A, the following holds: I f 99 : X --+ R U { + ~ } is lower semi-continuous and bounded below with D(99) ~: 0 and if Yo ~ X satisfies qg(y0) < infx q9 + ae 2 f o r some e ~ (0, 1), then there exist g ~ A and xo ~ X such that (i) Ilx0 - y0ll ~< e, (ii) Ilgll,a ~< e, (iii) q9 + g attains its minimum at xo. PROOF. We can clearly assume that there exists a bump function b in A with b(O) - 1, 0 ~ Ilbll~ - 1. Let a -- 1 / 4 M and suppose that e and Y0 are given. Define the function

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We have qS(yo) < infx ~o - a~ 2 and qS(y) >~ infx ~o whenever Ily - yoll ~> e. From Proposition 2.1 and Theorem 2.1, we can find xo E X and k E A such that []kl[A ~< min{e/2, ae2/2} and q5 4- k attains its minimum at xo. The above conditions imply that ]Ix0 y0ll < e. Thus, the function g e ,A defined by g(x) -- - 2 a e Z b ( x--xo ----E- ) + k(x) satisfies claims (i), (ii) and (iii) of the corollary. E] REMARK 2.1. Again, we can recover Ekeland's minimization principle on Banach spaces from the (easily verifiable) fact that the space ~41 of all bounded Lipschitz functions on X equipped with the norm [

[If[lA, = sup{ If(x)l; x E X} + sup{ /

If(x) IIx -

f(y)l

; x=/=y}

yll

satisfies the conditions of Proposition 2.1 and hence it is an admissible space of perturbations. Note that, as an additional bonus, we get that the perturbation is also small in the uniform norm as well as in the Lipschitz norm. To derive an analogue of the Borwein-Preiss Theorem, we can consider the space .42 of all bounded Lipschitz functions f on X that also verify the following second order condition

I[fl[

_ supJ If(x + 2h) - 2 f ( x + h) + f ( x ) [ 9x, h E X } / h2

<ec.

The space .,2[2 equipped with the norm IIf ]IA2 = IIf [IA~ + Ilf II is also an admissible space of perturbations. Clearly, the above norms will correspond to the C 1 and C2-norms whenever the functions are differentiable. But since X is in general infinite dimensional, we have to deal with two types of difficulties: firstly, the appearance of various different and generally nonequivalent types of differentiability and secondly, the problem of admissibility of these spaces of differentiable functions which requires extra assumptions on the Banach spaces involved. We deal with some of these problems in Section 3.

2.3. Martingales and admissible cones of perturbations In this section, we briefly sketch the relationship between admissible cones of perturbations and the theory of balayage associated to such a cone. Let (s I7, P) be a probability space and let (17n)n be an increasing sequence of suba-fields of I7 (i.e., r n C 17n+1). Recall that a sequence (Fn)n of real-valued integrable random variables on s is said to be a martingale (respectively a submartingale) with respect to ( r n ) n if for each n E N, (i) Fn is 17n-measurable, and (ii) fA Fn dP -- fa Fn+l dP (respectively fa Fn dP <~fa Fn+l d P ) for every a E r n .

402

R. Deville and N. Ghoussoub

In other words, Fn = E[Fn+I; r n ] (respectively Fn <~ E[Fn+I; r n ] ) where the latter denotes the conditional expectation of Fn+l with respect to the a-field r n . To extend the above notions to a nonlinear setting, we proceed in the following way. Suppose that X is a complete metric space and let A be a convex cone of real valued continuous functions on X. DEFINITION 2.3. Say that a sequence (Fn) n of X-valued random variables on ~2 is an A-martingale if for any ~0 E A, the process (~p o Fn)n is a real-valued submartingale. As stated above, there is a close relation between the almost everywhere convergence of X-valued A-martingales and the admissibility of A as a cone of perturbations for a minimization principle on X. This program has been developed extensively in the upcoming monograph [48]. We shall here describe briefly a few - particular but i m p o r t a n t - examples that have been developed in Ghoussoub and Maurey [46,47]. If X is a Banach space, then one can define the Bochner integral of an X-valued random variable F as well as its conditional expectation, first by defining them for step functions and then by taking limits in the usual fashion and as in the real valued case. Therefore, one can say that a sequence (Fn)n of X-valued Bochner integrable random variables is a vector martingale if for all n E N, we have that almost surely E[Fn+I; r n ] = Fn, where both are regarded as X-valued random variables. It is easy to check that if X is a Banach space, the notion of X-valued martingales coincides with the notion of A-martingales where A is the cone of continuous convex (or linear) functionals on X. The convergence of uniformly bounded Banach space valued martingales does not always hold and it is closely related to the geometry of the space involved. Actually, one can show the following result. THEOREM 2.2 (Bourgain-Stegall). If C is a closed convex bounded subset of a Banach space X, then the following properties are equivalent: (i) Every C-valued martingale converges almost surely. (ii) The space X* of continuous linear functionals is an admissible space of perturbations on C. Such sets are then said to possess the Radon-Nikodym property. See for example [46]. Now, if X is a manifold or just a quasi-Banach space (as in the case of L p (0 < p < 1)) the notion of a vector integral is not well defined in general, and one can already see that the concept of an A-martingale is more appropriate. We first recall the relevant concepts. Let X be a vector space. A map x ~ llx II from X into R + is called a quasi-norm if (i) Ilx l] > 0 when x -r 0.

(ii) II~x II - I ~ 1 IIx II, for ~ ~ C, x ~ X. (iii) Ilxt + x211 ~< C(llxl II + IIx211) for all xl, x2 6 X. Here C/> 1. We call II II a p-norm for 0 < p ~< 1, if in addition it is p-subadditive, that is (iv) Ilxl +x211 p ~< Ilxt IIp + IIx211p f o r x l , x 2 E X. The Aoki-Rolewicz theorem asserts that every quasi-norm is equivalent to a p - n o r m for some p (0 < p ~< 1). A complete quasi-normed vector space X will be called a quasiBanach space. If the quasi-norm on X is also p-subadditive, we say that X is a p-Banach

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space. For the basics about non-locally convex vector spaces, we refer the reader to the book [57]. Let A = {z 6 C, Izl < 1 } be the open unit disc. Denote by 0A or T the unit circle {z C; Iz[ = 1} and by z~ = A U 0A. DEFINITION 2.4. Let g : X --+ [ - c o , + c o ) be an upper semi-continuous function. It is said to be plurisubharmonic on X if for every x, y 6 X we have

g(x) ~

f0 zrg(x + ei~

dO 2re

We denote by PSH(X) the class of all such functions. LIPp(X) will denote the set of functions g on X satisfying for some K > 0, Ig(x) - g(Y)l ~< Kllx - y[I p for all x, y in X. We shall write PSHp(X) for PSH(X) A LIPp(X) and we shall equip it with the following norm Ilgllp -- max{Ig(O)l, sup{Ig(x) - g(y)l/llx - YlIP; x, y 6 X, x - r y}}. Here is the Plurisubharmonic perturbed minimization principle [47]. See also [45]. THEOREM 2.3 (Ghoussoub-Maurey). Let X be a p-Banach space for some p (0 < p <, 1). The following assertions are equivalent: (i) All bounded X-valued PSHp-martingales converge almost surely. (ii) The space PSHp(X) is an admissible class of perturbations for any closed bounded subset C of X: that is, for every bounded below, lower semi-continuous function q) on C, the set

{ g E PSHp (X); 99 - g attains a strong minimum on C } is a dense G~ in PSHp(X). Examples of quasi-Banach spaces that verify the above principle are Lebesgue spaces LP, Hardy spaces HP or the Schatten spaces C p where 0 < p ~< 1. Note that when 1 < p < ~ , the above mentioned spaces are all reflexive Banach spaces and therefore one can apply the stronger linear perturbation principle (Theorem 2.2).

2.4. Perturbed minimization and compactifications Suppose now that C is a subset of a compact metric space (K, d) a n d - for simplicity- let A be a Banach space of continuous functions on K equipped with a norm II II. A natural question is to identify the topological structure of C as a subset of K that insures that A is an admissible cone of perturbations on C. It turns out that the concept of As-sets - defined b e l o w - is the appropriate notion for this description.

404

R. Deville and N. Ghoussoub A

We start by associating to any subset L of K, the function ~/~ as follows:

A

{

j

qJL (x) -- sup h (x) - sup h; h 6 A and IIh II ~< 1 . L D E F I N I T I O N 2.5. With the above notations:

(i) A closed subset L of K is said to be A-convex in K if it is of the form

L-

N { x E K; hi(x) ~ ~,i} i6I

where the functions hi belong to A and )~i E R. (ii) A subset C of K is said to be an A~-set in K if K \ C is a countable union of compact A-convex sets (Kn)n. (iii) C is a strict A~-set in K if in the above representation, we have infx~c qJKn (x) > 0 for each n 6 N. Modulo the appropriate hypothesis, one can show in quite a general setting that A is an admissible space of perturbations on any strict A~-subset C of K. An interesting aspect of this approach is that the perturbed minimization principle also holds not just for lower semi-continuous functions but also for strict A~-functions: i.e., those whose epigraph in K x R is a strict A~-set. See examples below. Conversely, if one starts with a general complete metric space C and a Banach space A of continuous functions on C that forms an admissible vector space of perturbations, then one can find a compactification K of C in which C sits as a strict A~-set. Again, this program is developed in its full generality in the upcoming monograph [48] and we shall only state here the linear c a s e - studied extensively in [46] - where C is a closed convex subset of a Banach space X and A is the dual space X*. The case where A is the space of weakly continuous harmonic functions was considered in [49]. In the linear case, C is assumed to be a subset of a dual space Y* (usually the double dual X** of the Banach space X containing C. A-convexity coincides then with the usual convexity, meaning that A-convex compact sets are exactly the convex weak* compact subsets of Y*. In this case, A~-sets are called w*-H~-sets (i.e., their complement is a countable union of convex weak* compact sets) since A-convex compact sets are - in this setting - just intersections of hyperplanes determined by linear functionals in Y. Here is the main result of the linear theory [46]. THEOREM 2.4 (Ghoussoub-Maurey). Let K be a w*-compact convex set in some dual Banach space Y* and let C be any strict w*-H~ -subset of K. For any strict w*-H~-function llt on C, we have: (1) The set {y ~ Y; lp + y attains its maximum on C} is dense in Y. (2) If K is w*-metrizable, then the set {y ~ Y; ~ + y exposes C from below } is a dense G~ in Y: i.e., (Y, el Hi) is an admissible space of perturbations for (C, w*). (3) If C is norm separable then the set {y ~ Y; ~p + y strongly exposes C from below} is a dense G~ in Y: i.e., (Y, ]1 I]) is an admissible space of perturbations for (C, I] I])-

Perturbed minimization principles and applications

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Moreover, if in (2) and (3), ~p is assumed to be concave then the minimum is (uniquely!) attained on an extreme point o f C.

Here are a few basic examples of strict w*-H~-sets and functions. PROPOSITION 2.2. Let K be a weak*-compact convex subset o f a dual Banach space Y*. Then (1) A countable intersection o f open half-spaces in K determined by functionals in Y is a strict w*-H~-set in K. (2) I f f is a functional in Y** that is in the first Baire class f o r the weak*-topology, then it is a strict w*-H~-function on K and any half-space determined by f in K is a strict w*-H~-subset o f K. (3) If 99 is convex, w*-lower semi-continuous and norm-Lipschitz on K, then lp = -cp is a strict w*-H~-function on K. (4) The supremum o f a bounded sequence o f strict w*-H~-function on K is also a strict w*-H~ -function. (5) I f C is a strict w*-H~-set in K and if K is weak*-metrizable then any relatively weak*-closed subset D o f C is a strict w*-H~-set in K and any weak*-lower semicontinuous function on C is a strict w*-H~-function. (6) I f C is a norm separable strict w*-H~-set in K, then any norm closed subset o f C is a strict w*-H~-set in K and any norm-lower semi-continuous function on C is a strict w*-H~-function. In particular, if K is norm separable, then any norm-lower semi-continuous function on K is a strict w*-H~-function.

Finally, we have the following compactification theorem [46]. THEOREM 2.3 (Ghoussoub-Maurey). Let C be a closed convex bounded subset o f a separable Banach space X. The following are then equivalent: (i) C has the R a d o n - N i k o d y m property. (ii) (X*, II II) is an admissible space o f perturbations f o r (C, II II). (iii) There exist a separable subspace Y o f X* and an isometric embedding T : X --+ Y* such that T (C) is a strict w*-H~ in Y*.

3. Perturbed minimization and differentiability Let again q9 be a function defined on a Banach space X with values in R U { + ~ }. In principle, we are only concerned here with the concepts of Frgchet and G~teaux-differentiability. However, since the arguments in both cases are similar, we shall avoid repetition, by working with the notion of differentiability associated with any bornology on X. Recall that a bornology fl is just a class of bounded subsets of X such that the topology r/~ on X* that corresponds to the uniform convergence of linear functionals in X* on the sets of/3 defines a locally convex topology on X*. Note that if/3 is the class of all bounded subsets (respectively all singletons) of X, then r/~ coincides with the norm (respectively weak*) topology on X*.

406

R. Deville and N. Ghoussoub

DEFINITION 3.1. Say that q9 is ~-differentiable at xo ~ D ( f ) with/3-derivative qg'(xo) =

p 6 X* if for any A 6 t3, lim t -1 [qg(x0 + th) - ~p(xo) - (p, th)] - 0 t--+0

uniformly for h E A. When 13 is the class of all bounded subsets (respectively all singletons) of X, the 13differentiability coincides with the classical notion of Fr~chet-differentiability (respectively GCtteaux-differentiab ility). We shall denote by cl~(x) the space of all real valued, bounded Lipschitz and /3differentiable functions g on X such that g~:(X, II II) ~ (X*, ~ ) is continuous, endowed with the norm IIg II~ = sup { Ig (x)l; x E x } + sup { IIgt (x)ll; x E x ] = IIg II~ + IIg' II~ . We shall use the notation c l ( x ) (respectively c l ( x ) ) when we are dealing with Fr6chet (respectively G~teaux) differentiability. Analogously, we can define the spaces C~ (X) (respectively C~ (X)) equipped with the C2-norm. 3.1. Smooth minimization principles Two immediate applications of Theorem 2.1 and Proposition 2.1 are the following [24]: THEOREM 3.1 (First order smooth minimization principle). Suppose X is a Banach space on which there exists a C1-Frechet smooth (respectively, G~teaux-differentiable) and Lipschitz continuous bump function. Then C~ (X) (respectively C1F(X)) is an admissible space of perturbations: i.e., for each e > 0 and f o r each lower semi continuous and bounded below function ~p: X --+ R U {+cx~ } such that q9 is not identically equal to +c~, there exists a C 1 (respectively, GCtteaux differentiable), Lipschitz continuous function p such that: (1) IIPlI~ = sup{Ip(x)l; x E X} < E, (2) IIP'll~ = sup{llP'(x)llx*; x E X} < e, and (3) q9 + p has a strong minimum at some point xo E X. THEOREM 3.2 (Second order smooth minimization principle). Suppose X is a Banach space on which there exists a C2-Frechet smooth bump function b with Lipschitz derivative. Then C2 (X) is an admissible space of perturbations: i.e., f o r each e > 0 and f o r each lower semi continuous and bounded below function qg : X --+ R U {+cx~} such that 99 is not identically equal to +cx~, there exists a C 2-function p on X with pt Lipschitz continuous such that: (1) IIPlI~ = sup{Ip(x)l; x ~ X} < e, (2) IIP'll~ = sup{llP'(x)llx*; x E X} < e, (3) IIP"II~ = sup{llP"(x)llB(x); x ~ X} < e, and (4) q9 + p has a strong minimum at some point xo E X.

Perturbed minimization principles and applications

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REMARK 3.1. It is worth comparing the above with the result of Borwein and Preiss (Theorem 1.3). Indeed, if X is a Hilbert space or more generally an LP space (1 < p < ec), then the perturbation q in Theorem 1.3 can be taken to be of the form q ( x ) = 89 - wll 2 for some w (usually not equal to re). This quadratic perturbations is more explicit than the above, however it lacks the boundedness properties of the perturbations in Theorem 3.2.

3.2. Smooth norms and smooth bumps The existence of a smooth bump function is a key hypothesis in the smooth minimization principles stated above. From now on, we shall use the following notation: (H1) There exists a C 1-smooth, Lipschitz continuous bump function b: X ~ R. (H1G) There exists a Gfiteaux-differentiable, Lipschitz continuous bump function b : X - - + R.. (H2) There exists a C2-smooth bump function b : X ~ R with Lipschitz derivative. We begin this section by pointing out that these assumptions are necessary for the validity of the smooth minimization principles. PROPOSITION 3.1. Let qg: X --+ R be a (Lipschitz continuous)function on X satisfying 99(x) > Of o r all x E X and l i m l l x l l ~ qg(x) = 0. (1) I f there exists a C 1-smooth, Lipschitz continuous function g such that 99 - g has a global minimum attained at some point xo, then X satisfies (HI). (2) I f there exists a Gfiteaux-differentiable, Lipschitz continuous function g such that 99 - g has a global minimum attained at some point xo, then X satisfies (H1G). (3) I f there exists a C2-smooth function g with Lipschitz derivative such that 99 - g has a global minimum attained at some point xo, then X satisfies (H2). Our next result, due to Leduc [59], asserts that the existence of a smooth bump function can be characterized by the existence of a smooth function looking like a norm, but not convex in general. PROPOSITION 3.2. Let X be a Banach space. The following assertions are equivalent: (i) There exists on X a Lipschitz continuous function d : X --+ R +, which is C l-smooth on X\{0}, and a constant K > O, such that [Ixll ~< d ( x ) <~ Kllxll f o r all x E X, and d(s = s f o r all x E X and all s ~ O. (ii) X satisfies (H1).

PROOF. Let b : X --+ R be a C 1-smooth, Lipschitz continuous bump function, with support in the unit ball and such that 0 ~ 0. The function d : X ~ R defined by d(0) - 0 and d ( x ) - ( f + ~ b ( t x ) dt) -1 for x ~ 0 satisfies the required properties. Conversely, let r :R --+ R be a continuously differentiable function such that r = 0 on (-cx~, 1] U [3, +cx~) and r(2) ~ 0. The function b : X --+ R defined by b(x) = r ( d ( x ) ) for x E X is a continuously Fr6chet differentiable bump on X. E] An immediate application of the above proposition (and a similar proof in the case of Gfiteaux-differentiability) is that the existence of a smooth norm implies the existence of a

R. Deville and N. Ghoussoub

408

smooth bump function. That is if X has an equivalent Fr6chet-differentiable norm (respectively, G~teaux-differentiable norm), then X satisfies (HI) (respectively, (HI G)). A consequence of the above result is that every reflexive space and every space with separable dual satisfy (HI), (because these spaces admit an equivalent Frechet-differentiable norm). Finally, we note that Haydon [52,54,53] constructed an example of a Banach space which satisfies (HI) (this space even admits a C ~ - s m o o t h Lipschitz continuous bump function), but has no equivalent Fr6chet-differentiable norm, thus showing that the converse is not true in general. On the other hand, for separable Banach spaces, a converse is available: THEOREM 3.3. Let X be a separable Banach space. The following conditions are then equivalent: (1) X satisfies (H 1). (2) There exists on X an equivalent Frdchet-differentiable norm. (3) The dual X* of X is separable. PROOF. (3) implies (2) is a classical result of Kadec and Klee. Since X* is separable, we let N* (x *) --

Ilx*II+ ~

2-n dist(x *, En ),

n

where (En)n is an increasing sequence of finite dimensional subspaces such that X* = Un En. It is easy to see that N* is weak*-locally uniformly convex: that is, a sequence (Xn*)n on the N*-unit sphere of X* norm-converges to x*, whenever it weak*-converges to x* and N* (x*) - 1. The predual norm N on X is then Frdchet differentiable on X \ {0} and its derivative is (norm to norm)-continuous from X \ {0} to X*. For more details, we refer the reader to the monograph [25]. The proof that (1) implies (3) uses the smooth minimization principle, so we shall provide a sketch. Let b be a C l-smooth Lipschitz continuous bump function on X. Denote U -- {x E X; b(x) ~ 0} and define f ' U --+ R by f ( x ) = 1/b(x). We claim that f is C 1smooth on U and that {ft (x); x E U} is norm dense in X*. Indeed, fix p E X* and e > 0. The function q)" X --+ R U {+oo} defined by qg(x) = - p ( x ) + f ( x ) if x E U and qg(x) -- cx~ otherwise, is lower semi continuous on X. By the first order smooth minimization principle, there exists a C 1-function g on X such that IIg~Iloc < e and q9 + g has a minimum at some point x0 E U. We have f ' ( x o ) - p + g'(xo), hence I[f'(x0) - Pll < e which proves the claim. On the other hand, since f t is continuous and U is separable, {f~ (x); x 6 U} is separable. Together with the claim, this implies that X* is separable. D REMARK 3.2. As noted above, if X is a Banach space which admits an equivalent G~teaux-differentiable norm, then there exists on X a Lipschitzian GSteaux differentiable bump function b such that b t is norm to weak* continuous. In particular, X satisfies (H1G). The class of Banach spaces satisfying (H1 G) is very large: Banach spaces which are weakly compactly generated (in particular, all reflexive Banach spaces and all separable Banach spaces) admit an equivalent G~teaux-differentiable norm. Therefore, they satisfy (H1G). On the other hand we shall see in Section 3.3 that goc does not satisfy (H1G).

Perturbed minimization principles and applications

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We conclude this section by showing that the existence of a smooth bump function implies the existence of other bump functions enjoying special properties. PROPOSITION 3.3. Let X be a Banach space on which there exists a C 1-smooth Lipschitz continuous bump function b, then: (1) There exists a C 1-smooth Lipschitz continuous bump function bl such that b~ ~ 0 and bl (x) = 1 = maxx bl f o r x in a neighbourhood o f O. (2) There exists a C 1-smooth Lipschitz continuous bump function b2 which attains its strong maximum at O. (3) I f moreover X is separable and infinite dimensional, then there exists a C l - s m o o t h bump function b3 such that b~3(X) -- X*. PROOF. (1) By translation and replacing if necessary b by - b , we can assume that b(0) > 0. Let ~0: R --+ R be a C l - s m o o t h Lipschitz continuous function such that ~p(t) = 0 if t <~ 0 and ~0(t) = 1 if t >~ b(0)/2. The function bl = ~o o b has the required properties. (2) Apply the smooth minimization principle to a constant function to get a function g which attains a strong minimum at some point x0. Replacing b by b(. - x0), we can assume that x0 = 0. Let a > f (0) such that IIx II < 1 whenever f (x) < a. Let ~ : R --+ R be a Cl-smooth Lipschitz continuous function such that gr(t) = 0 if t/> a and 7r' (t) < 0 if t < a. The function b2 = ~ o b has the required properties. (3) The construction of b3, which is given in [2], requires a series of functions of the form x --+ cnbl (anx - Xn).Yn, where an, Cn > O, Xn C X and Yn E X* are suitably chosen. Observe here that under the assumptions of (3), there exists on X an equivalent norm I1.11 which is Frechet-differentiable on X\{0}. If b : X --+ R is of the form b(x) = qg(llx II), where q) is a C l bump function on R with support in (0, + e c ) , then, according to James theorem, the set {a.b'(x); a > O, x ~ X} coincides with X* if and only if X is reflexive. Thus the bump function b3 cannot be of this form in general. D For higher order smoothness, the situation is more delicate. Let us mention here that for 2 ~< p < + ~ , the norm of the LP spaces is C2-smooth and its derivative is Lipschitzian on the unit sphere. Therefore these spaces satisfy (H2) and the second order minimization principle can be applied to these spaces. On the other hand, if X satisfies (H2), then X is superreflexive of type 2 (see [25]). If one does not require a Lipschitz condition on the derivative, the situation is different: for instance, there exists on spaces of continuous functions on countable compact, an equivalent C ~ - s m o o t h norm [53], although these spaces are not reflexive. Actually one can even construct an equivalent analytic norm on these spaces. Finally, in spaces not containing co, the existence of a C ~ - s m o o t h bump function is equivalent to the existence of a polynomial P on X such that P (0) = 0 and P (x) >~ 1 whenever Ilx II = 1.

3.3. Generic differentiability o f convex functions We show in this section how the smooth minimization principle is an efficient tool to establish generic differentiability of convex functions. The following is a theorem of Ekeland and Lebourg [32,33,59].

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THEOREM 3.4. A s s u m e X satisfies (HI), then it is an A s p l u n d space, i.e., every convex and continuous function ~p " X --+ R is differentiable on a dense set. PROOF. Fix xo E X and e > O. Set F ( x ) --

1 - q ) ( x ) + b2(m(x_xo) ) +cxz

if b ( M ( x - xo)) 7~ O, otherwise.

The function F is lower semi-continuous, and bounded below for M large enough. By the smooth minimization principle, there exists g" X --+ R, C 1-smooth, such that F + g attains its m i n i m u m at some point Xl where b ( M ( x l - xo)) ~ O. If M > 0 is large enough, then Ilx 1 - x0 II < e. Moreover, if we let

f (x) -- b 2 ( M ( x _ xo)) + g ( x ) ,

the function q) - f has a local m a x i m u m at x l, so q9 is superdifferentiable at the point X l and hence at each point of a dense subset D of X. Since q9 is convex and continuous, q) is subdifferentiable at every point (by the H a h n - B a n a c h theorem). Therefore, q) is differentiable at every point of D. D An immediate application of T h e o r e m 3.4 is: COROLLARY 3.1. A s s u m e that the norm o f X satisfies f o r every x ~ X,

lim sup h--,0

Ilx + h II + IIx - h II - 21Ix II

> 0.

(,)

Ilhll

Then X does not satisfy (HI) (i.e., there exists no bump b" X --+ R which is C l - s m o o t h and Lipschitz continuous).

PROOF. Indeed, a norm satisfying (,) is an example of a nowhere differentiable convex continuous function. This corollary allows to deduce that a certain number of spaces do not satisfy assumption (HI): indeed, the usual norm on the space C([0, 1]) satisfies

lim sup h~0

IIx + hll + IIx - h l l - 211xll

= 2

Ilhll

for every x 6 C ([0, 1]). Therefore C ([0, 1]) does not satisfy (HI). The same remark applies to the space g l (N) and to the space L 1([0, 1]). The fact that C ([0, 1]), g 1(N) and L 1([0, 1]) do not satisfy (H 1) also follows from Proposition 3.3, since these spaces are separable with non separable duals. D

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411

PROBLEM 1. Is the converse of Theorem 3.4 true? The answer is yes for separable spaces, since the class of separable Asplund spaces and the class of separable Banach spaces satisfying (H 1) both coincide with the class of spaces with separable dual. Thus, any counterexample should be a non separable Banach space. Possible candidates for a counterexample is C ( K ) , where K is the compact space constructed by Kunen, or the long James space J(col). Here is the "Gfiteaux" counterpart of Theorem 3.4 which can be proved in a similar way. THEOREM 3.5. A s s u m e that X satisfies (H1G). Then every convex and continuous function ~p" X ---> R is G~teaux-differentiable on a dense set. Let X -- s (N) and ~0"X ---> R be the convex continuous function defined by ~o(x) -limsup ]xn]. Since for every h 6 X, ]]hl]- 1 and for every x 6 X, we have lim sup

~o(x + th) + ~p(x - th) - 2~0(x)

t-~O

= 2,

t

the function ~0 is nowhere Gfiteaux-differentiable. Therefore there exists no Lipschitz continuous bump function b" X --+ R which is Gfiteaux-differentiable at every point of s REMARK 3.3. Let X be a Banach space and let A1 be the Banach space of all bounded Lipschitz functions on X (equipped with the norm given in Remark 2.1). Recently, Bachir [4] proved that the supremum norm on the Banach space ~41 is generically Fr6chetdifferentiable on ,AI. His proof relies on the general perturbed minimization principle and on a duality result involving a new notion of conjugacy. THEOREM 3.6. A s s u m e that there exists a bump b" X --+ R which is differentiable at every point, Lipschitz continuous and such that, f o r some p ~ (1,2], sup lim sup x6X

b ( x + h) + b ( x - h) - 2b(x)

< +ec.

Ilhll p

h-+O

Then, f o r every convex and continuous function ~p" X ~ o f X such that f o r every x ~ D, ~p is differentiable and

lim sup h~O

IIx -4- hll-4-IIx - h l l -

211xll

R, there exists a dense subset G

< +o~.

IIhlIP

A consequence of this result is that on s 1 ~< q < 2, there is no bump function which is differentiable at every point, Lipschitz continuous and such that, for some p E (q, 2], sup lim sup x~X

h--+O

b ( x + h) + b ( x - h) - 2b(x)

< +oc.

]]h]lP

The following theorem follows from Theorem 3.6 with p -- 1.

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THEOREM 3.7. Let X be a Banach space satisfying (H2) and let qg: X --+ R be a convex

continuous function. Then the set of points xo ~ X such that q9 is differentiable at xo and such that, for h small enough, Iqg(x0 + h) - qg(x0) - qg'(x0).hl ~< Cllhl] 2, is dense. In other words, a concave continuous function defined on a space satisfying (H2) has a non empty proximal subgradient on a dense set. For a study of proximal subgradients in Hilbert spaces, we refer to [ 15], and to [ 16] for a more extensive study with application to optimal control. PROBLEM 2. Does Theorem 3.7 remain true if the assumption "X is a Banach space satisfying (H2)" is replaced by the weaker hypothesis "there exists on X a C2-smooth bump function"? In particular, since there exists a C ~ - s m o o t h bump function on c0(N), is it true that a concave continuous function defined on co (N) has a non-empty proximal subgradient on a dense set? We conclude this section with an important open problem. Recall that Alexandroff's theorem states that a convex continuous function in R n has second order expansions almost everywhere, i.e., for almost every x ~ R n, there exists (p, Q) ~ R n • S(n) such that f (x + h) = f (x) + (p, h) + Q(h, h) + o(llhl12). PROBLEM 3. Does Alexandroff's theorem extend to an infinite dimensional setting? More precisely, if H is a separable Hilbert space and if f : H ~ R is a convex continuous function, does there exist (a dense set of) points x 6 H such that there exists (p, Q) 6 H x S(H) satisfying f (x + th) = f (x) + t.(p, h) + t2.Q(h, h) + o(t 2) for every h 6 H. Note that the convex continuous function f ( x ) = I[x+l[ 2 on s shows that we cannot hope to have f (x + h) = f (x) + (p, h) + Q(h, h) + o(llhll2), and the function defined by the same formula on t~2(F), with F uncountable, shows that it is necessary to assume that H is separable.

3.4. The existence of separating polynomials Let X be a real Banach space, let p >t 1 be a real number and let f be a real valued function defined on X. We say that f has a Taylor expansion of order p at the point x E X, if there is a polynomial P of degree at most n, where n = [p] is the integer part of p, verifying

[f(x + h ) -

f(x)-

P(h)[-o(IihliP).

We say that f is TP-smooth if it has a Taylor expansion of order p at every point. Note that if f is m-times Fr6chet differentiable on X, then from Taylor's theorem we have that f is T P-smooth for 1 ~~ 1 for all x in the unit sphere of X. It is known (and easy to prove) that if X admits a separating polynomial, then X admits a C ~ - s m o o t h bump function.

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Recall that the norm II. II on a Banach space X has modulus o f convexity o f p o w e r type p if there exists a constant C > 0 such that for each e E [0, 2], 6(e) "-- inf{ 1 -

x+y 2

9 x, y ~ x

Ilxll ~ 1; Ilyll ~ 1; IIx

- y ll/> e} ~

cE p

The following result is due to Deville et al. [30]. We include its proof since it uses the linear minimization principle of Bourgain-Stegall. THEOREM 3.8. Let p >~ 1 be a real number and let X be a Banach space. Assume: (1) There exists on X a T P-smooth bump function. (2) The norm o f X is uniformly convex with modulus o f convexity o f p o w e r type p. There exists then on X a separating polynomial o f degree <, [p]. In particular, there is a C~176 bump function on X. The following corollary improves a result of Bonic and Frampton [7]. COROLLARY 3.2. Let p >~ 1 be a real number which is not an even integer. Then there is no T P-smooth bump function on L p as long as it is infinite dimensional. PROOF OF COROLLARY 3.3. Assume there exists a TP-smooth bump on Lp. Since the modulus of convexity of the norm in L p is of power type p, Theorem 3.8 shows that there is a separating polynomial P on L p of degree less or equal than p. The polynomial Q defined by Q (x) = P (x) + P ( - x ) is separating, even and of degree less or equal than the degree of P. Since p is not an even integer, the degree of Q is strictly less than p. This contradicts the fact that there exists no separating polynomial of degree < p on LP. [3

For the proof of Theorem 3.8, we shall use the following elementary lemmas from Fabian et al. [37]" LEMMA 3.1. Let 6(e) be the modulus o f convexity o f the norm I1.11of X. Let x , h ~ X and f ~ X* such that f ( x ) - ]]xl[ - ]if I ] - 1, f ( h ) - - 0 a n d e <, ]]hl] ~< 2. Then"

IIx +hll~>l +6(2 ) . Consequently, if the norm has a modulus o f convexity o f p o w e r type p and if x, h E X, f E X* are such that f ( x ) - ]]xl] -J: 0, ]if I] - 1, f ( h ) - 0 and ]thai ~< 2]]xl], then:

IIx -+-hll- Ilxll/> Cllxlll-pllhll p. LEMMA 3.2. Let k ~ 2 be an integer and let F be a finite codimensional subspace o f a Banach space X. Assume that P is a polynomial on X o f degree <, k which is a separating polynomial on F. Then there is a separating polynomial o f degree <~ k on X.

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414

PROOF OF THEOREM 3.8. Let b be a bump function on X with Taylor expansion of order p at any point and such that b(0) = 0. Let qg:X ~ R U {+cx~} be the function defined by: qg(x) = 1/b(x) 2 if b(x) ~ 0 and qg(x) = +cx~ otherwise. The function q0 - I1.11is lower semicontinuous, bounded below and identically equal to + ~ outside a bounded set. On the other hand, since X is uniformly convex, it has the R a d o n - N i k o d y m property. According to the Bourgain-Stegall minimization principle, there exists g E X* (actually a dense G~ in X* of such g's) such that q9 - II. II + g attains its minimum at some point x. So for every hEX: qo(x + h) - I I x + hll-4- g(x + h) >~~o(x) - I l x l l + g(x). Consequently"

qo(x + h ) - ~o(x) + g(h) ~ IIx + h l l -

Ilxll.

Next x r 0 because qg(0) = + ~ . Let f 6 X* such that Ilfll = 1 and f ( x ) = Ilxll. Using L e m m a 3.1, we get for every h 6 K e r ( f ) n Ker(g)"

q0(x + h ) - qg(x)/> IIx + h l l - Ilxll ~ Cllxll~-Pllhll p. If we denote C(x) -

CIIxll l-p,

qg(x + h) - qg(x) ~

we thus have:

C(x)llhll p,

We now use the fact that q9 has a Taylor expansion of order p at x: there exists a polynomial P of degree ~< [p] and a function R such that qg(x + h) - qg(x) = P(h) + R(h), P(O) = 0 and limh~0 R(h)/llhll p = O. We fix e > 0 such that R(h)l ~< (C(x)/2)llhll p whenever h E X, IIh II -- e. Therefore, if h E K e r ( f ) n Ker(g) and IIh II = ~, then:

C(x) P(h) ~> 2 IlhllP" 2 The polynomial Q defined by Q(h) = C(x)cp P(eh) is a separating polynomial of degree ~< [p] on K e r ( f ) N Ker(g). By L e m m a 3.2, there is a separating polynomial of degree ~< [p] on X. [2

4. Non-smooth calculus in Banach spaces The aim of this section is to show how smooth minimization principles allow to define a subdifferential calculus for lower semicontinuous functions in smooth Banach spaces. We start by defining a generalized gradient. Note that this is well-known for convex functions qg:X --+ R U {+cx~} where the definition for any x ~ D ( f ) reads as follows: the subdifferential of q9 at x is the set:

D-q~(x) -- {p E X*; ~o - p has a local minimum at x}.

Perturbed minimization principles and applications

415

However, non-convex functions often appear in many problems from non-linear partial differential equations (as we shall illustrate in the last section) and a non-smooth calculus for semi-continuous functions is needed. Several notions of generalized gradients can be (and has been) defined in a natural way. In this paper, we define notions of subdifferentials and superdifferentials that are directly motivated by the smooth minimization principles. As shown in Section 6, these will play a key role in the resolution of Hamilton-Jacobi equations.

4.1. First order sub- and super-differentials for continuous functions Let X be a Banach space and q9 : X --~ R U { + ~ } be an arbitrary function. If x E D(qg), we define:

D - q g ( x ) - - {g'(x); g ' X --+ R i s C 1 and q9 - g has a local minimum at x}. It is not difficult to check that whenever ~p is convex, this definition coincides with the definition in the sense of convex analysis given above. We say that 99 is subdifferentiable at x if D-qg(x) ~ 0 (for x ~ D(qg), we define D-qg(x) = 0). We can define in a similar way the superdifferential of q9 at x:

D+~p(x) - {g'(x); g ' X --+ R i s C c~ a n d q g - g has a local m a x i m u m at x}. It is elementary to check that the sets D-qg(x) and D+qg(x) are convex and norm-closed, and that, if D-~o(x) and D +qg(x) are both non-empty, then q9 is Fr6chet-differentiable at x, and in this case D-~o(x) = D +qg(x) = {qg'(x) }. We now relate subdifferentials and superdifferentials with differentiability. The following result is proved in [24]. PROPOSITION 4.1 (Characterization of subdifferentials). Assume that X is a Banach space with a C l, Lipschitz continuous bump function b. Let ~p : X --+ R be an arbitrary function, x E X and p ~ X*. The following assertions are equivalent: (1) p 6 D-qg(x); (2) liminfllhll~0((qg(x + h) - qg(x) - (p, h))/[lhll) ~ O. PROOF (Sketch). (1) implies (2) is easy. Conversely, modulo replacing r by the function h -+ sup{cp(x + h) - ~ p ( x ) - (p, h ) , - 1 } we can assume, without loss of generality, that x = 0, p = 0, ~p(x) = 0 and ~0 is bounded below on X. Under these assumptions, condition (2) becomes: liminfjihll_~o(~o(h)/llhll ) >~ O. Denote by p, pl, P2 : R + --> R the functions defined by p(t) - inf{~0(h); Ilhll ~< t}, p~ (t) = f2t p(s) ds and p2(t) - f2t Pl (s) ds. The function P2 is non increasing, C l - s m o o t h and satisfies limt-+o p 2 ( t ) / t - - 0 . If we define g : X --> R by g(x) = p2(d(x)), where d is the function from Proposition 3.2, we get that g is C 1-smooth and that ~0 - g has a minimum (equal to 0) at 0. D From this, we deduce:

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PROPOSITION 4.2 (Criterium for differentiability). I f X is a Banach space which admits a C 1 and Lipschitz continuous bump function b and if qg"X --+ R is an arbitrary function. Then q9 is differentiable at a point x if and only if D-qg(x) ~ 0 and D+qg(x) ~ 0. In this case, D-qg(x) = D +~o(x) = {~p'(x) }. It follows from the above criterium that in a space satisfying (HI), a convex continuous function is differentiable at some point x if and only if D +~0(x) ~ 0.

We shall see that it is possible to define a calculus involving sub- and super-differentials. A first difficulty is that even in finite dimensions, the sets D-qg(x) and D+~o(x) can be empty at many points. In fact, the sets D - = {x e X; ~p is subdifferentiable at x} and D + -- {x E X; q9 is superdifferentiable at x } need not be residual in X as shown by the following simple example: REMARK 4.1. If ~0:X --+ R is a continuous nowhere differentiable function on X (such functions do exist even when X -- R), then D - n D + = 0. In infinite dimensions, a Lipschitz function can be nowhere subdifferentiable and nowhere superdifferentiable: take for instance the function ~0:~ 1(N) --+ R defined by

n even

n odd

However, the smooth perturbed minimization principle yields that under some geometrical assumptions on the space X, lower semi-continuous (respectively upper semicontinuous) functions on X are subdifferentiable (respectively superdifferentiable) on a dense set. A key geometrical assumption on X is of course the existence of a smooth bump function from X. We shall start with the following result, originally due to Borwein and Zhu under the assumption of the existence of an equivalent Frechet-differentiable norm. In its present form, it is due to Bachir [5]. THEOREM 4.1 (Minimization of the sum of two functions). Let X be a Banach space satisfying (HI). Let u l, u2 :X --+ R U {+cx~} be two lower semi continuous and bounded below functions such that limn~ 0 inf{ul (Xl) + u2(x2); Ilxl - x211 < n} < + ~ . Then, f o r each e > O, there exist xl, x2 E X, Pl E D - u l (Xl) and P2 e D - u 2 ( x 2 ) such that: (1) Ilxl -x211.(1 + IlPl II + liP211) < e, (2) IIel + P2 II < e, and (3) Ul(Xl) + u2(x2) < limo~o inf{ul(y) + u2(z); IlY - zll < rl} + ~. PROOF (Sketch). Let d be the Lipschitz continuous function given by Proposition 3.2. For each a > 0, define Wa :X • X --+ R by Wa(X, y) = Ul (X) + ue(y) + ad2(x - y). Wa is a lower semi continuous and bounded below function on the Banach space X x X and B(x, y) = b ( x ) b ( y ) is a Lipschitz continuous, cm-smooth bump function on X • X.

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417

A c c o r d i n g to the s m o o t h m i n i m i z a t i o n principle, there exists a Lipschitz continuous, C 1s m o o t h function g" X x X --+ R such that: (a) wa 4- g has a strong m i n i m u m at s o m e point (Xa, ya) 9 X x X , (b) Ilgll~ - - s u p { l g ( x , Y)I; (x, y) 9 X x X} < l / a , (c) sup{ll ~Og (X , y)][ ~ " (X , y ) 9 X x X} < 1 / a and 0g (X, y)[] ~ ; (x, y) 9 X x X} < 1 / a (d) sup{ II~S

[]

Og D e n o t e Xl -- Xa, X2 -- Ya, Pl = - a ( d 2 ( x - Y))' -~(Xa, Ya) and p2 - a ( d 2 ( x - y ) ) ' o__~g ay (Xa, Ya). For a large enough, properties (1), (2) and (3) are satisfied. COROLLARY 4.1. Let X be a B a n a c h space satisfying ( H I ) a n d let ~p : X --+ R be a lower s e m i - c o n t i n u o u s a n d b o u n d e d below f u n c t i o n such that D(qg) ~ 0. Fix e, )~ > O, then, f o r every xo 9 X satisfying qg(xo) < infx q9 + eL, there exist Xl 9 X a n d Pl 9 D - q g ( X l ) such that IlPl Ilg* < ~ a n d []Xl - xoll <

e.

PROOF. C h o o s e )~o 9 (0,)~) such that e)~o - (qg(xo) - infx qg) > 0. Define h ( x ) - ~.o(llx xoll - e) and let 0 < a < e)~o - (qg(xo) - infx qg). C h o o s e eo 9 (0, e) in such a way that, if h ( x ) < qg(xo) - e)~o + a - infx qg, then Ilx - xoll < eo. A p p l y i n g T h e o r e m 4.1 with U l - q9 and u2 = h, there exists Xl, x2 9 X and there exist p l 9 D - q g ( X l ) , P2 9 D - h ( x 2 ) such that: (a) Ilxl - x2 II < e - eo, (b) IIP~ + P2 II < Z - Zo, and (c) qg(Xl) + h(x2) < l i m o ~ o inf{qg(x) + h ( y ) ; IIx - yll < r/} + a. Since h is Lipschitz c o n t i n u o u s with Lipschitz constant )~o, we have liP211 ~< )~o. B y (b), we obtain that IlPl II < ~. Since q9 is lower s e m i - c o n t i n u o u s a n d h is Lipschitz continuous, we have

lim inf{qg(x) 4- h ( y ) ; 0----~0

IIx - yll < ~} -

inf {qg(x) 4- h ( x ) ; x 9 X}

<<, ~o(xo) + h(xo). Therefore, h(x2) < qg(xo) 4- h(xo) 4- a - ~O(Xl) <<,99(x0) - e)~o 4- a - infx qg. Thus, ]Ix2 xo II < eo. Together with (a), this implies that IlXl - xo 11 < e. []

COROLLARY 4.2. A s s u m e that X satisfies (HI), a n d that qg" X --+ R is lower semi contin-

uous. Then q9 is sub-differentiable on a dense set.

PROOF. Fix x0 9 X and e > 0. A p p l y i n g T h e o r e m 4.1 with U l = q9 and u2 the function defined by uz(x0) = 0 and u z ( x ) = +cx~ if x r x0, we obtain points X I , X 2 9 X such that IlXl - x21l < e and vectors p l 9 D - q g ( X l ) and p2 9 D - u z ( x 2 ) . Necessarily, x2 = xo. So q9 is subdifferentiable at a point X l such that IIx~ - xo II < e. E3

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4.2. The sub-differential o f the sum Formulae for the subdifferential of the sum of two functions defined on a Banach space have been investigated by various authors (Ioffe [55], Fabian [34,35], Deville and E1 Haddad [27]). We present here an extension, observed by Bachir [5], of a result obtained by Borwein and Zhu [9] in the context of Banach spaces admitting an equivalent Fr6chetdifferentiable norm. We need the following: DEFINITION 4.1. Let u l, u 2 : X ~ R W {+cxz} be lower semicontinuous functions. We say that the pair (u l, u2) is locally uniformly lower semicontinuous if, for every x 6 X and every uniformly continuous function qg:X • X ~ R, there exists r > 0 such that, inf ul (y) + u2(y) -+- ~o(y, y) yEB

= liminf{ul(y)+u2(z)+qg(y,z); 77--+0

I l y - z l l < rl, y , z ~ B}.

where B = B x (x , r). This condition is clearly stable if we perturb u l and U2 by uniformly continuous functions, and is always satisfied when dim(X) < +cx~, or when one of the functions is locally uniformly continuous. THEOREM 4.2. Let X be a Banach space satisfying (H1). Let ul, U2 : X ~ R U {+cx~} be lower semicontinuous functions. Assume that the pair (Ul, u2) is locally uniformly lower semicontinuous. Suppose that xo E X and p E D - (u 1 + UZ)(X0) are given. Then, f o r each e > O, there exist xl, x2 E X, Pl E D - u l (Xl) and P2 E D - u z ( x 2 ) such that: (1) Ilxl - xoll < e, Ilx2 - xoll < e and Ilxl - xzll.(llpl II -+- [Ipzll) < e, (2) lUl (Xl) - Ul (xo)l < e and lu2(x2) - u2(xo)l < e, and (3) IlPl + p2 - Pll < e. T h e o r e m 4.3 is a non trivial consequence of Theorem 4.1 on the minimization of the sum of two functions. A Rademacher-Preiss type theorem for uniformly continuous functions in spaces which admit a smooth bump function can be deduced from this formula. Let us recall that according to Rademacher theorem, every Lipschitz continuous function in R n is differentiable almost everywhere. Preiss [62] has extended this result to an infinite dimensional setting. He proved in particular that if X is an Asplund space (in particular if X satisfies (H1)), then every locally Lipschitz continuous real valued function defined on X is differentiable on a dense subset of X. The following result from [27] can be seen as a weak form of Preiss theorem for uniformly continuous functions. COROLLARY 4.3. Let X be a Banach space satisfying (HI). Let u be a uniformly continuous function defined on X. Then f o r every x E X and every e > O, there exist x l, x2 E X, there exist p - ~ D - u ( x l ) and p+ ~ D+u(x2) such that: (1) Ilxl - x II < e and [Ix2 - x II < e, (2) lU(Xl) - u(x)l < e and ]u(x2) - u(x)] < e, (3) liP- - P+II < e.

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419

In order to prove Corollary 4.3, it is enough to apply Theorem 4.2 with U l ~ - U and u2 = - u , and to observe that D - ( - u ) ( x 2 ) -- D+u(x2). Let us here stress the fact that Preiss' result is considerably harder to prove. PROBLEM 4. If the function u of Corollary 4.3 is nowhere differentiable, then the points Xl and x2 are necessarily different. It is unknown whether one can take p - -- p+ when u is an arbitrary uniformly continuous function (the answer to this question is positive if d i m X = 1). PROBLEM 5. Counterexamples show that the assumption "(u l, u 2 ) is locally uniformly lower semicontinuous" cannot be dropped in Theorem 4.1 (see for instance [29]). These counterexamples involve extended valued functions. It is an open problem to know whether a formula for the subdifferential of the sum of two lower semi continuous, real valued functions in infinite dimensions always holds true. REMARK 4.2. Theorem 4.2 will be applied in the next section in the proof of uniqueness of the viscosity solution of a particular Hamilton-Jacobi equation. The conclusion "llxl x211.(llpl II + lip211) < e" in Theorem 4.2 is usually needed in the case of Hamilton-Jacobi equations coming from optimal control or differential games. Most results can be established for weaker types of differentiability. Indeed, let u : X R tO {+cx~} be an arbitrary function. If x 6 X, we may define for example the G-viscosity subdifferential of u at x: DGU(X) -- {gt(x); g ' X --+ R is G~teaux-diff. and u - g has a local min. at x}.

The G-viscosity superdifferential of u at x is defined similarly. The following result is the analogue of Theorem 4.2. 4.3. Let X be a Banach space on which there exists a Lipschitz continuous, GCtteaux-differentiable bump function b on X such that b t is norm to weak* continuous. Let u l, u2 be two real valued functions on X such that u l is lower semi continuous and u2 is uniformly continuous. Suppose that xo E X and p E DG(Ul + uz)(x0) are given. Then, f o r PROPOSITION

each e > 0 and each weak* neighbourhood V o f p, there exist xl, x2 E X, Pl E DGUl(Xl) and P2 E DGU2(X2 ) such that: (1) [[Xl - x0 [[ < e and [[x2 -- X0 [[ < 6, (2) [ul(xl) - ul(xo)[ < e and [u2(x2) - u2(xo)[ < e, (3) P l + p 2 E V .

4.3. Second order differentials We now investigate analogous results for second order sub-differentials defined as follows: Let X be a Banach space and qg:X --+ R U { + ~ } . The Second order subdifferential and

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the Second order superdifferential of q9 at x, are the sets: De'-~p(x) = { (g'(x), g" (x)); g ' X ---> R is C 2 and ~o - g has a local minimum at x } and De'+99(x ) = { (g'(x), g ' ( x ) ) ; g ' X --> R is C 2 and ~p - g has a local maximum a t x } . The second order smooth minimization principle tells us that whenever X is a Banach space which admits a C2-smooth bump function with Lipschitz derivative, then for every s function ~0"X ~ R, D2'-~p(x) r 0 for every x in a dense subset of X. On the other hand, it is possible that both D2'-~o(x) and D2'+~0(x) are non empty, and yet q9 is not twice differentiable at x" take ~o'R --+ R defined by: ~0(x) " - x 2 s i n ( I / x ) if x ~ 0 and f ( 0 ) - 0. Although a second order minimization principle is available in infinite dimensions, not much more is known in this context and our goal in this section is to present some results that are valid in finite dimensions, hoping to motivate the problem of their extensions to an infinite dimensional setting. For example, we present a formula for the second order subdifferential of the sum of two lower semicontinuous functions that is available only in finite dimensions (see [28]). The main ingredient of the proof is the following theorem due to Jensen who established uniqueness of viscosity solutions of Hamilton-Jacobi equations of order two [56]. THEOREM 4.3 (Jensen) (Second order information in a minimization problem). Let u" R n -+ R be a function such that x --+ u ( x ) - ~llxll 2 is concave (such a f u n c t i o n u is called semi-concave). A s s u m e that u has a strong m i n i m u m at xo. Then, f o r every e > O, there exists x ~ R n with IIx - x011 < e and (p, Q) ~ R n • S(n) with Ilpll < E a n d 0 <~ Q <. 2)~I such that u ( x + h) -- u ( x ) 4- (p, h) + Q(h, h) 4- o(llhl12).

Note that Alexandroff's theorem (See Problem 3) applies to a semi-concave function so that it has order expansions almost everywhere, that is, there exist (a dense set of) points x E R n such that there exists (p, Q) 6 H • S(R n) satisfying f ( x + t h ) - f ( x ) + t . ( p , h ) + t 2 . Q(h, h) + o(t 2) for every h E R n . Jensen's theorem gives more information on the value of p and Q. The idea of the proof is the following: Fix r > 0 and consider u the greatest convex function on Br dominated by u (u is called the convex lower envelope of u on Br). Jensen proved that if u is semiconcave, then the Lebesgue measure of the set A - {x ~ Br ; u ( x ) = u (x)} is positive (see [18]). Therefore, by Alexandroff's theorem, there exists x 6 A and (p, Q) ~ R n x S(n) such that u ( x + h) - u ( x ) + (p, h) + Q(h, h) + o(llhl12). Because u ( y ) >~ u ( x ) for all y and u ( x ) = u (x), we have Q ~> 0. And because ofthe semi-concavity

of u, we also have Q ~< 2)~I. The conditions IIx - x0ll < e, Ilpll < e can be obtained by choosing r > 0 small enough.

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PROBLEM 6. Is it possible to extend Jensen result mentioned above to an infinite dimensional setting? To our knowledge, the work [ 18] gives the best information in this direction. As in the first order sub-differentials, we have the following two results which are only established in a finite dimensional setting. THEOREM 4.4. Let u l, u2 be two real valued, bounded below, lower semicontinuous functions on R n. Then, f o r every e > O, there exist x 1, x2 E R n , (Pl, Q l) E D 2'- (u 1) (Xl) and (p2, Q2) E D 2'- (uz)(x2) such that: Ilxl - x211 < ~,

Ilpl q- p2ll < ~

and

II Ol Jr- Q21I < ~.

THEOREM 4.5 (Deville-E1 Haddad). Let u l, u2 be two real valued lower semicontinuous functions on R n. Suppose that xo and (p, Q) E D Z ' - ( U l + uz)(x0) are given. Then, f o r every ~ > O, there exist Xl,X2 E R n, (Pl, Q1) E D2'-Ul(Xl) and (P2, Q2) E o Z ' - u z ( x 2 ) such that: (i) ]lxl - xo II < e and Ilx2 - xo l[ < e, (ii) lul (xl) - ul (xo)] < e and ]u2(x2) - u2(xo)l < e, (iii) IlPl + p2 - Pll < e and II Q1 + Q2 - QII < e. Unfortunately, this result does not extend to infinite dimensional Banach spaces. For instance, let X = g2(N). For x = (xn) E g2(N), denote x + = sup{x, 0} and x - = - inf{x, 0}. If u l ( x ) = [Ix+ll 2 and uz(x) = I[x-II 2, then (Ul + uz)(x) = I[x[]2. So ( 0 , 2 I ) E DZ'+(Ul + u2)(0). On the other hand, for every Xl,X2 E g2(N) and every (pl, Q1) E D2'+(Ul)(Xl), ( P 2 , Q 2 ) E D2,+(uz)(x2), w e have [IQ1 + Q 2 - 2111 >~ 2. We just give a sketch of the proof of T h e o r e m 4.5 so as to motivate the need for the extension of Alexandroff's and Jensen's theorems to infinite dimensions. For more details, we refer to [28]. As in the first order case, one first reduce to the case (p, Q) = (0, 0) and u 1 -Jr- U2 has a m i n i m u m at xo. We then minimize the function equal to Ul (x) + u2(y) -+- Mllx - yl[2 in a n e i g h b o u r h o o d of (xo, xo) and equal to +cx~ outside of this neighbourhood. This function attains its m i n i m u m at some point (x l, x2). We get then all the claims but the condition II Q1 + Q2 - Q II < ~. In order to overcome this difficulty, the first step is to regularize by using Inf convolutions: one introduces the functions Ui,Z (X) = infzERn {Ul (Z) + 89IIz - x II2 } for i = 1, 2. The functions u l,z and u2,z are difference of convex functions and are approximations (as)~ --+ cx~) of u 1 and u2. The idea is then to prove T h e o r e m 4.5 for u l,z and u2,z and then to pass to the limit. The key ingredient for proving T h e o r e m 4.5 for u 1,z and u2,z is the result of Jensen mentioned above (see [ 18] for details). The following corollary of T h e o r e m 4.5 can be seen as a weak version of Alexandrov's theorem for continuous functions. C O R O L L A R Y 4.4. Let u be a continuous function defined on R n. Then f o r every x E R n and f o r every e > 0 there exist x l , x 2 E R n, ( p - , Q - ) E D Z ' - u ( x l ) and (p+, Q+) E oZ'+u(x2) such that: (i) IlXl - x II < e and ][x2 - x [I < e, (ii) ]u(xl) - u(x)[ < e and [u(x2) - u(x)l < e, (iii) liP- - P+ II < ~ and II Q - - Q+ II < e.

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4.4. Stability of subdifferentials and superdifferentials The following stability result is from [22] where one can find a proof and variants of this result, as well as references on other stability results. We only state results for superdifferentials, since the case of subdifferentials is similar. THEOREM 4.6. Let X be a Banach space satisfying (HI). Let q) and (q)n, n ~ N) be upper semi continuous functions from X into R. Assume that (~On) converges locally uniformly to ~p. If x E D(qg) and p E D+qg(x), then there exist (Xn) C X and Pn E D+qgn(Xn) such that (Xn) converges to x, (~O(Xn)) converges to qg(x) and (Pn) converges to p. We now present a stability result of viscosity superdifferentials for the upper semicontinuous envelope of the supremum of a family of upper semi-continuous functions. This result can be used to show that, under very mild assumptions, a supremum of viscosity subsolutions is a viscosity subsolution. This last result is a key ingredient to prove the existence of a viscosity solution for Hamilton-Jacobi equations via the Perron method (see Section 6.4). THEOREM 4.7. Let X be a Banach space satisfying (HI). Let $ be a locally uniformly bounded family of upper semi-continuous functions from X into R. Let u* be the upper semi continuous envelope of the function u(x) -- inf{g(x); g E S}. If x E X and p E D+u*(x), then there exist (Xn) C X, (q)n) a sequence of functions in S and Pn E D+q)n(Xn) such that (Xn) converges to x, (q)(Xn)) converges to u*(x) and (Pn) converges to p. Under assumption (H1G) (respectively (H2)), similar stability results hold for the G~teaux superdifferential (respectively second order superdifferential).

4.5. Mean value theorems We now turn to mean value theorems. There are many! The first and the last theorems presented below are valid for nonsmooth functions defined in (HI) spaces. The first is from [22], the second is from [2], while the third can be found in E1Haddad's thesis. THEOREM 4.8. Let X be a Banach space satisfying (HI), b / a n open convex subset of X and q) : bl --+ R be lower semicontinuous. Assume that there exists a constant K > 0 such that for all x E bl and for all p E D-qg(x), IlPll <~ K. Then ~o is Lipschitz continuous, with Lipschitz constant K. The following result holds in arbitrary Banach space. Observe that below, we only refor one p E D-q)(x) (instead of all p E D-q)(x)).

quire IlPll ~< M

THEOREM 4.9 (Mean value theorem for subdifferentiable functions). Let X be a Banach space, bl an open convex subset of X and q) : bl --+ R a continuous G~teaux subdifferentiable function. Suppose that there exists M >/0 such that for every x E lg, there exists p E D-qg(x) with IlPll <<,M. Then q) is Lipschitz continuous, with Lipschitz constant K.

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The following mean value inequality is a variant of the mean value inequality of Clarke and Ledyaev (see [ 13,14]). It is obtained formally from the mean value inequality of Clarke and Ledyaev by considering, in the Banach space E = X • R, the point (to, x0) and the (unbounded) convex C = {0} x X. It is useful to prove uniqueness results for viscosity solutions of parabolic Hamilton-Jacobi equations. COROLLARY 4.5 (A mean value inequality). Let X be a Banach space satisfying (HI) and let ~p: [0, T) • X --+ R be lower semicontinuous and bounded below. Assume that f o r all x E X, ~p(O,x) ~ 0 and fix (to, xo) E (0, T) • X. Then f o r every e > O, there exist (t, x) E (0, T) • X and (a p) E D-~o(t x) such that a < ~(to,xo) + e and Ilpll < e. to '

'

5. Application to critical point theory Many problems of Mathematics and Physics amount to the search of a critical point of a differentiable map q9 on a Banach space X: that is points x E X such that qg'(x) = 0. The simplest way of finding critical points is to consider minimization problems. Our first result gives the existence and the localisation of an "almost critical point" and is an immediate application of Ekeland's theorem.

5.1. Palais-Smale minimizing sequences PROPOSITION 5.1. Let X be a Banach space and ~p : X --+ R be a differentiable mapping such that ~p is bounded below. Let F be a closed subset o f X such that infx 99 -- infF q9 = c. Then there exists a sequence (Xn) C X such that limqg(Xn) = c, lim II~0'(xn)ll- 0 and lim dist(xn, F) = 0.

The proof follows from Ekeland's theorem applied to any minimizing sequence (which can be chosen in F) and the condition qg(x) - qg(xn) ~> - 1 / n l l x - xnll which implies that II~o'(Xn)llx* <~ 1/n. In order to get a critical point, we need to insure that this sequence of "almost critical points" located around F, is convergent, hence we are led to introduce the Palais-Smale around F, at altitude c ((PS)F,c), which simply means that any sequence satisfying the claim of the above proposition, is relatively compact. Whenever F -- X, we shall write (PS)c instead of (PS)x,c. With this condition, we clearly obtain an x E F such that qg(x) = c and qg'(x) = 0. If we assume some smoothness on the domain X, then the first order smooth perturbed minimization principle allows to extend the above result to functions that are merely lower semi-continuous: PROPOSITION 5.2. Let X be a Banach space satisfying (HI) and let ~p : X --+ R be a brunded below, lower semi continuous function. Let F be a closed subset o f X such that infx q9 = infF qg. Then there exist a sequence (Xn) C X and Pn E D-~p(xn) such that: limqg(Xn) = inf~p, x

lim Ilp~ Ils* = 0

and

lim dist(xn, F) = 0.

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The next result gives second order information. The proof is similar to the above, but uses the second order smooth minimization principle. PROPOSITION 5.3. Let X be a Banach space satisfying (H2) and let r : X --+ R be a bounded below function which is twice differentiable at every point. Then f o r every e > 0 and any ye ~ X such that qg(ye) <~infq9 + e 2, there exists xe ~ X such that: (1) (2) (3) (4)

Ilx~ - Y~II ~< e, qg(xe) ~< infq9 + e, II~0'(x~)IIx, ~< E, and ~p"(x~)(h,h) ~ -ellhll2 f o r a l l h ~ S.

PROBLEM 7. In view of Proposition 5.1 which is valid in any Banach space, the following question is quite natural: does Proposition 5.3 remain true without the assumption "X satisfies (H2)"?

5.2. The mountain pass theorem The assumption "q9 is bounded below" can be replaced by an assumption about the existence of a mountain range separating two points. We then obtain the mountain pass theorem of Ambrosetti and Rabinowitz. The following result is a variant of that theorem with an additional information about the location of the sequence of "almost critical points". THEOREM 5.1 (Ghoussoub-Preiss). Let X be a Banach space and let qg : X --+ R be a differentiable mapping. For a, b ~ X, consider F to be the set o f all continuous functions y : [0, 1] --+ X such that y (0) = a and y (1) = b. Denote c -- inf max ~0(y (t)). yeF

t e [ O , 1]

Let F be a closed subset of X such that a, b ~ F, F C {q9 ~ c} and F separates a and b, i.e.,for every y ~ F, there exists t ~ ]0, 1[ such that y ( t ) ~ F. Then there exists a sequence (Xn) C X such that limqg(Xn) = c, lim llqg'(Xn) I[ = 0 and limdist(xn, F) = 0. I f moreover q9 satisfies the (PS)F,c, then there exists x ~ F such that qg(x)= c and ~o' (x) = O.

PROOF (Sketch). Fix e < min{ 1, dist(a, F), dist(b, F) }, pick a continuous path g : [0, 1] --+ X such that max{qg(y(t)); t 6 [0, 1]} < c + e Z / 4 a n d d e f i n e L = {t 6 [0, 1]; dist(g(t), F) ~> e}. L is a closed subset of [0, 1] and the set F0 = {p 6 C([0, 1], X); PIE = )tiE} is a closed subset of the Banach space C ([0, 1], X) (endowed with its usual supremum norm) that is included in F . Finally, define I :L --+ R by I (p) -- max {(r + s

t E [0, 1]\L },

where g : X --+ R is given by the formula g(x) = max{0

; 62 -

e.dist(x, F) }.

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425

The functional I is continuous, bounded below and defined on the closed subset of a Banach space. According to Ekeland's minimization principle, there exists )to 6 F0 such that: (i) I ( y o ) <<.I(y); (ii) IlYo - zll ~< e/2; (iii) I ( p ) >1 I ( y o ) - ~ l i p - zoll for all p E Fo. Denote M = {t ~ [0, 1]; (~p + g)(yo(t)) = 1(9/o)}. There exists then t E M such that: (a) c <<,~p(yo(t)) <<,c 4- 5s2/4; (b) dist(y0(t), F) ~< 3e/2; (c) II~o'(zo(t))llx, < 5e/2. Since this can be done for every e > 0, we obtain Theorem 5.1. See [50] for details. [] The localisation in the above mountain pass theorem can be used for studying the structure of the set of critical points of q9 (see [42-44,38-40] and [50]). Let us show here how it can be used to construct nice paths. COROLLARY 5.1 (Taubes). Let X, ~p, a, b E X, F a n d c be as in Theorem 5.1. A s s u m e that qg(a) < c, ~p(b) < c, a n d that q9 satisfies (PS)c. Then, f o r each s > O, there exists a continuous p a t h y :[0, 1] --+ X such that y (0) = a, y (1) = b, a n d 6 > 0 such that:

PROOF. Since q9 is uniformly continuous on compact subsets of X, it is enough to prove that for each e > 0, there exists a continuous path y :[0, 1] ~ X such that y (0) = a, y(1) = b and ~0(y(t)) = c :=~ II~o'(Z(t))llx* < e. If not, then there exists e > 0 such that the set F -- F, -- {x ~ X; Ilqg'(x)lls, ~ ~ and ~o(x) -- c} is a closed subset of X that separates a and b. By the preceding theorem, there exists x 6 F such that ~0~(x) = 0, which is a contradiction. M REMARK 5.1. There exist also non-smooth versions of the mountain pass theorem. A first result in this direction has been obtained by Chang [12]. More recently, Corvellec, de Giovanni and Marzzochi [17] and Fang [39] introduced the notion of weak slope which allows them to develop critical point theory for lower semi continuous functions.

As in minimization problems, it is possible to give second order information in the mountain pass theorem. This is done in [41,42] under some conditions of uniform continuity of the first and second order derivatives of 99. These conditions do not seem to be severe in applications. THEOREM 5.2 (Fang-Ghoussoub). Let X be a B a n a c h space a n d let qg : X --+ R be a C 2functional. Let a, b ~ X, F, c a n d F be as in Theorem 5.1. Suppose that q9t a n d r are H61der continuous on a n e i g h b o u r h o o d o f {r = c}. Then there exists a sequence (Xn) C X such that:

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426

(1) limqg(Xn) = c; (2) limllqg'(xn)]l = 0 ; (3) limdist(xn, F) = O; 1 2 for every h in a subspace En of codimension at most 1. (4) ~o"(xn)(h, h) >1-~llhll

6. Applications to Hamilton-Jacobi equations in Banach spaces General first order Hamilton-Jacobi equations are of the form

H (x, u(x), Du(x)) = 0 in the stationary case, and of the form

H (t, x, u t, x), OuSt, x)) = o in the evolution case. They arise in optimal control theory and in differential games. We refer to [ 19] and [20] for a detailed discussion of these equations, and, for instance, to [6] for an introduction to the theory of Hamilton-Jacobi equations. The problem is to deal with existence and uniqueness of a global "solution" of these equations, under suitable hypothesis on the Hamiltonian H (monotonicity of H in the variable u, continuity properties of H, and in some cases, convexity properties of H in the variable Du(x)). In this section, we shall concentrate on a couple of examples that will illustrate the power of the smooth minimization principles described previously.

6.1. The maximum principle for stationary first order Hamilton-Jacobi equations Let I2 be a bounded open subset of R n and H" S2 x R n ~ R be an arbitrary continuous function ( H is called an Hamiltonian). Consider the following Hamilton-Jacobi equation: u(x) + H(x, Du(x)) = 0 u (x) -- 0

for all x 6 S2, for all x 6 OS-2.

(H J1)

We first examine the maximum principle for classical solutions. Let u ~ C ( ~ ) N C 1 (if2) be a (classical) subsolution: i.e., u(x) + H(x, Du(x)) <~0

u(x) <<,0

for all x ~ S'2, for all x ~ 0 $2,

and let v 6 C(I2) N C1 (12) be a supersolution" i.e., v(x) + H(x, Dr(x)) ~ 0

v(x) >10

for all x 6 Y2, for all x 6 O12.

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427

Assume supse(u - v) > 0. Then there exists x0 E I2 such that u(xo) - v(xo) = sups2 (u - v) > 0. So Du(xo) = Dv(xo). Since u(xo) + H(xo, Du(xo)) <<.0 and v(xo) + H(xo, Dv(xo)) >~ O, we get by subtracting u(xo) - v(xo) <<.O. This contradiction means that for all x E S-2, u (x) ~< v (x). Assuming that u and v are bounded, a similar proof still works on unbounded domains or in infinite dimensions, but since we no longer have compactness, sups ~ (u - v) is not necessarily attained, and we therefore need to apply perturbed minimization principles. It is therefore necessary to impose more restrictive hypothesis on the Hamiltonian H (for instance H is uniformly continuous) in order to estimate H(xo, Du(xo)) - H(xo, Dv(xo)). Indeed, the following counterexample shows that some hypothesis on H is needed. The linear equation: u(x) - x(1 + IxI)u'(x) = 0 on R has 0 and x / ( 1 + Ixl) as two classical bounded solutions on R, that is the maximum principle is violated in this case. In general, classical solutions do not exist, but viscosity s o l u t i o n s - as defined b e l o w do exist under relatively general assumptions. We shall then see that, using the calculus involving sub- and superdifferentials, the above proof of the maximum principle can be adapted to viscosity subsolutions and supersolutions. Let us recall the definition of viscosity solutions for Hamilton-Jacobi equations of the form given above. DEFINITION 6.1. A function u :X --+ R is a viscosity subsolution of (HJ1) if u is upper semi continuous and, for every x 6 X and every p E D+u(x):

u(x) 4- H ( x , p) <<.O. The function u is a viscosity supersolution of (HJ1) if u is lower semi continuous and, for every x 6 X and every p E D - u (x):

u(x) § H ( x , p) ~ O. Finally, u is a viscosity solution of (HJ 1) if u is both a viscosity subsolution and a viscosity supersolution of (HJ 1). Let X be a Banach space and H :X • X* --+ R be a continuous Hamiltonian satisfying the following condition: 1 p)

-

H(x2, P)l ~ 0

as IIx~ - x211(llpll + 1) --+ 0,

IH(x, P l )

-

H ( x , P2)I--+ 0

as IlPl - P2 II ~ 0.

IH(xl,

(*)

If H is uniformly continuous, then H satisfies condition (,) which in turn implies the continuity of H. We now prove the uniqueness of a bounded continuous viscosity solution u : X -+ R of equation (HJ 1): THEOREM 6.1. Let u, v be two real valued bounded functions defined on X. Assume that the Banach space X satisfies (H1) and that the Hamiltonian H satisfies (,). If u is a 1 Thiscondition is satisfiedby the Hamilton-Jacobi equations arising in deterministic optimal control problems and in differential games.

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viscosity subsolution of (HJ1) and if v is a viscosity supersolution of (HJ1), then u <<.v on X. PROOF. Fix e > 0. The functions v a n d - u are lower semicontinuous continuous and b o u n d e d below. According to T h e o r e m 4.1 on the minimization of the sum of two functions, for each e > 0, there exist xl, x2 E X and pl E D+u(xl), P2 E D-v(x2) satisfying: (1) (1 + Ilplll + IIp211), ]lxl - xzll ~ e, (2) V(X2) -- U(Xm) < l i m o s 0 inf{v(y) -- u(z); IlY -- zll < 0} + e, and (3) liP2 - Pl II < e. Condition (2) implies that: v(x2) - U(Xl) < inf{v(x) - u(x); x 6 X} + e. Since u is a viscosity subsolution of u + H(x, Du) = 0, we have U(Xl) + H(xm, Pm) ~< 0. Since v is a viscosity supersolution of v + H(x, Dr) = 0, we have v(x2) + H (x2, P2) ~> 0. Consequently inf(v

-

u)

>

v(x2)

-

U(Xl)

-

6

x

H(xj,

Pl)

--

H(x2,

P2)

-

e

>/ H(xl, P l ) - H(x2, Pl) -+- H(x2, P l ) - H(x2, P2) - e.

(1)

Since Ilxl - x 2 l l ( l l p l II + 1) ~ e, we obtain that H(xl, Pm) - H(x2, P l ) ~ 0 as e --+ 0. Since IIp~ - p2 II ~< e, we get that H (x2, Pl) - H (x2, P2) ~ 0 as e --+ 0. Plugging this information in (1), we get: inf(v - u) ~> 0.

[]

x

REMARK 6.1. It is important to have comparison theorems for semi-continuous functions rather than for uniformly continuous solutions for several reasons: This need appears in the existence results via Perron's method where discontinuous viscosity subsolutions are used in the proof as well as in optimal control with boundary conditions where the value function can be discontinuous.

6.2. The maximum principle for parabolic Hamilton-Jacobi equations Our aim here is to prove uniqueness of a b o u n d e d uniformly continuous viscosity solution u : R + • X --+ R of the following evolution equation:

ut + H(x, Ux) -- O, u(O, x) - uo(x), where u o : X ~ tinuous.

(HJ2)

R is the initial condition which is assumed bounded and uniformly con-

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DEFINITION 6.2. Say that a function u : R + x X --+ R is a viscosity subsolution of (HJ2) if u is upper semi continuous and, for every (t, x) 6 X and every (a, p) E D + u ( t , x):

u(O, x) <<. uo(x). Likewise, we can define the viscosity supersolutions for (HJ2). THEOREM 6.2. Let u, v be two real valued uniformly continuous functions defined on R + x X. Assume that the Banach space X satisfies (H1) and that the Hamiltonian H satisfies (.). I f u is a viscosity subsolution o f ( H J 2 ) and if v is a viscosity supersolution o f (HJ2), then u <<.v on X. PROOF. Fix T E (0, §

(v-u)

inf

and, aiming for a contradiction, let us assume that <0.

[0,T)xX

Fix s > 0. The function v - u is uniformly continuous and non-negative on {0} • X, hence b o u n d e d below on [0, T) x X. Thus, there exists (to, x0) E (0, T) x X such that

(v - u)(to, xo) <

inf

(v - u) § s T

and

(v - u)(to, xo) < O.

[0,T)•

According to the mean value inequality, there exists (t, x) E (0, T) x X and (a, p) E D - (v - u) (t, x) such that a<

(v - u)(to, xo) to

+s

and

[IP[l<s.

By the formula for the subdifferential of the sum applied to U l = 1) and U 2 "-- m U , there exist ( t l , x l ) , (t2, x2) E (0, T) x X, there exist (al, P l ) 6 D - v ( t l , x l ) and (a2, P2) E D+u(t2, X2) satisfying: (1) [Ixl - x0[I < s, Ilx2 - xoll < s, Itl - to[ < s and It2 - tol < s, (2) IIpl - p2 _ pll < e and lal - a2 _ al < e, and (3) (llpl II + lall + Ilpzll + [azl).(llxl - xzl[ + Itl - tzl) < s. The function u is a viscosity subsolution of u t + H (x, Ux) : 0, so: a2 § H1 (x2, P2) ~< 0. The function v is a viscosity supersolution of vt § H (x, Vx) = 0, so:

al + H ( x l , p l ) ~ O. Consequently

inf[o,T)•

>

(v - u)(to, xo)

mS>

T > a-2s>al-az-3s/>H(xz,

(v - u)(to, xo)

m s

to pz)-H(xl,pl)-3s.

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Observe that (3) implies that (llplll 4- [Ip2ll).llxl - x211 < e. In the same way as in the stationary case, using the fact that H satisfies ( . ) , and sending e to zero, we get: inf[0,r)xX(V - u) ~> 0,

which is a contradiction.

6.3. Uniqueness in second order Hamilton-Jacobi equations We conclude by explaining how to use T h e o r e m 4.5 to prove uniqueness of viscosity solution for second order Hamilton-Jacobi equations. Our model is the following stationary equation: u +

(HJ3)

where H is a uniformly continuous Hamiltonian. DEFINITION 6.3. A function u : R n --+ R is a viscosity subsolution of (HJ3) if u is upper semi continuous and, for every x 6 R n and every [p, Q) ~ D2'+u(x):

u(x) 4- H ( x , p, Q) <. O. The function u is a viscosity supersolution of (HJ3) if u is lower semi continuous and, for every x 6 R n and every (p, Q) E DZ'-u(x):

u(x) 4- H ( x , p, Q) >~O. Finally, u is a viscosity solution of (HJ3) if u is both a viscosity subsolution and a viscosity supersolution of (HJ3). THEOREM 6.3. Let u, v be two bounded real valued functions defined on R n. If u is a viscosity subsolution o f ( H J 3 ) and v is a viscosity supersolution of(HJ3), then u <. v. PROOF. Fix e > 0, the function u - v is upper semicontinuous and b o u n d e d above, so by the smooth minimization principle, there exist xo 6 R n and (p, Q) 6 D 2'+ (u - v)(xo) such that I]P]I < e, II QII < e and (u - v)(xo) > sup(u - v) - e. Applying T h e o r e m 4.5 with ul = v and u2 = - u , there exist Xl, x2 in R n, (pl, Q1) E D2'-V(Xl) and (p2, Q2) E D2,+u(x2), such that: (1) [[x l - xo [[ < e and [Ix2 [[ < 6. (2) IV(Xl) - v(x0)ll < E and lU(X2) u(x0)ll < e. (3) liP2 - Pl - pll < e and II Q2 - - Q 1 - QII < e. The function u is a viscosity subsolution of u 4- H (x, Du, D2u) = 0, so --

XO

--

u(x2) 4- H(x2, P2, Q2) ~< 0.

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431

The function v is a viscosity supersolution of v + H (x, D r , D2v) - - 0 , so v ( x l ) + H ( x l , p l , Q1) >/0. Consequently sup(u - v) <<, (u - v)(xo) + e < u(x2) - l)(Xl) + 3e < H ( X l , Pl, Q1) - H ( x 2 , p2, Q2) + 3e. Moreover, Ilxl-x2ll ~ Ilxi-xoll-+-llxo-x21[ < 2e, I I p 2 - p l II ~ I I p 2 - p l - p l l - + - I l p l l < 2e and IIQ2 - Q1 II ~< IIQ2 - Q1 - Q II + IIQ II < 2~. Using the uniform continuity of H and sending ~ to zero, we get that sup(u - v) ~< 0. E2 REMARK 6.2. Using these techniques, the reader will now be able to deal with more general Hamiltonians and to prove uniqueness results for second order parabolic HamiltonJacobi equations. We have omitted several topics of interest as we did not mention boundary conditions and how they can be interpreted in the viscosity sense. We did not mention either the uniqueness results for viscosity solutions of Hamilton-Jacobi equations in some classes of unbounded functions. For more on these topics, see [6,18], and references therein.

6.4. The existence o f viscosity solutions f o r H a m i l t o n - J a c o b i equations A classical method to show the existence of an harmonic function w on an open subset 12 of R n satisfying a given boundary condition (say w = f on Ol2) is due to Perron: the idea is to first establish the existence of a subharmonic function u0 and of a superharmonic function v0 such that u0 ~< v0 on S-2 and u0 = v0 - f on OY2, and then to show that the supremum of all subharmonic functions w satisfying u0 ~< w ~< v0 is actually harmonic and satisfies the desired boundary condition. Our aim here is to show, following Ishii, that this method can be adapted to yield the existence of viscosity solutions for Hamilton-Jacobi equations under very general hypotheses. In order to do this, we introduce some notation. If I2 is an open subset of a Banach space X and if u is a function defined on S-2, the upper semicontinuous envelope u* of u is u* = inf{v; v is continuous on 12 and v >~ u on $2 } and the lower semicontinuous envelope u. is defined similarly. We can now state: THEOREM 6.4. Let X be a Banach space satisfying (H1) and let 12 be an open subset o f X. Let H : 1-2 • R x X* --+ R be continuous. Let uo, vo be respectively a viscosity subsolution and a viscosity supersolution o f H (x, u, D u ) -- 0 on S-2 such that uo <<.vo on 1-2. Then there exists a function u : S-2 --+ R such that uo <<.u ~ vo on S-2, u* is a viscosity subsolution o f H (x , u, D u ) = 0 on S-2 and u , is a viscosity supersolution o f H (x , u, D u ) = 0 on S-2.

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PROOF. Let S be a family of functions which are locally uniformly bounded above and which are viscosity subsolution of H ( x , u, Du) -- 0 on S2. Let u = sup{w; w 6 S}.

CLAIM. u* is a viscosity subsolution of H (x, u, Du) -- 0 on I2. In particular, when S = {ul, u2}, we obtain that the supremum of two viscosity subsolutions of H ( x , u, Du) -- 0 on S-2 is a viscosity subsolution of H (x, u, Du) -- 0 on S-2. Indeed, let x E s and p ~ D+u*(x). According to the stability of superdifferentials (Theorem 4.7), there exist (Xn) C S-2, (Un)a sequence of functions in S and Pn ~ D+un(xn) such that: (1) (Xn) converges to x, (2) (un(xn)) converges to u*(x), and (3) (Pn) converges to p. Since Un is a viscosity subsolution of H ( x , u, Du) -- 0 on I2, we have H(xn, Un(Xn), Pn) -0. By continuity of H , we obtain, as n tends to infinity, H ( x , u*(x), p) = 0. Therefore, u* is a viscosity subsolution of H (x, u, Du) = 0 on S2. Define now

S = { w ' X --+ R; u0 ~ w ~< v0 and w is a viscosity subsolution of H (x, u, Du) -- 0 on S2 } and let u - - s u p { w ; w E S}. According to the above claim, u* is a viscosity subsolution of H (x, u, Du) -- 0 on S2. Let us now prove that u, is a viscosity supersolution of H (x, u, Du) = 0 on s Indeed, otherwise, there exist x0 6 S2 and a C l-smooth function g ' X --+ R such that: (i) u,(xo) - g(xo) -- 0 and u , ( x ) - g(x) >~0 for all x 6 X'2, (ii) H(xo, u,(xo), g'(xo)) < O. Observe that necessarily g(xo) < v(xo). Indeed, if not, then g(x) <~ u , ( x ) <<. v(xo) and g(xo) = u,(xo) - vo(xo) which imply that v0 - g attains its infimum at x0. Since v0 is a viscosity supersolution of H ( x , u, Du) = 0 on S2, H(xo, vo(xo), g~(xo)) ~ O, which contradicts (ii). Denote B(xo, 3) the open ball centered x0 and with radius 6 and B(xo, 6) its closure. By continuity of H and g~, there exist 6 > 0 such that B(xo, 23) C S2, and a bump function B ' X --+ R with support in B(xo, 6), such that b(xo) > 0 and which satisfies

H (x, g(x) + b(x), g'(x) + b'(x)) < 0

for all x E B(xo, 26),

and

g(x) + b(x) <~ vo(x)

for all x E S-2.

Indeed, this is possible if 6 > 0, and if [[bl[~ and [Ib'[[~ are small enough, because g(xo) < v(xo) and g(x) <~ v(x) for all x E $2. Define now

w(x)=

max{g(x) + b(x), u(x)}

u(x)

if x E B(x0, 26), if x ~ S-2\ B (x0, 26).

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433

By the above claim, w is a viscosity subsolution of H (x, u, Du) -- 0 on I21 = B(x0, 26). On the other hand, if x 6 S'22 -'- S - 2 \ B ( x o , (~), then w ( x ) = u(x). Therefore, w is also a viscosity subsolution of H ( x , u, D u ) - 0 on S22. Since S21 and ~f'22 are open, w is a viscosity subsolution of H ( x , u, Du) = 0 on S-2 = S21 U S22. Since u0 ~< w ~< v0, we obtain that w ~~ w (x) ~> g (x) + b (x) on B(x0, 6) which implies u , ( x ) ~ g(x) + b(x) on B(xo, 6), and this contradicts the fact that u,(xo) = g(xo). [] REMARK 6.3. A slight modification of the above argument yields an existence result for second order H a m i l t o n - J a c o b i equation on a Banach space satisfying (H2). COROLLARY 6.1. Let X be a Banach space satisfying ( H I ) and let I-2 be an open subset o f X. Let H :I-2 x X* --+ R be a continuous Hamiltonian satisfying (,). Assume moreover that the function x --+ H (x, O) is bounded. Then there exists a unique bounded function u : X --+ R such that u is a viscosity solution o f u + H (x, Du) = 0 on I-2. PROOF. By hypothesis, there exists constants m, M such that, for all x E I2, m ~< H ( x , O) <~ M. Thus the constant functions equal to - m and - M are respectively viscosity supersolution and viscosity subsolution of H ( x , u, Du) = 0 on I-2. According to T h e o r e m 6.4, there exists a function u:I-2 --+ R such that - M ~< u ~< - m on s u* is a viscosity subsolution of H ( x , u, Du) - - 0 on I2 and u , is a viscosity supersolution of H ( x , u, Du) - - 0 on I2. Since u is bounded, u , and u* are bounded. According to the m a x i m u m principle (Theorem 6.1), u* ~< u , . Since obviously u , ~< u ~< u*, we get that u = u* - - u , is a continuous viscosity solution of (HJ1). Again the m a x i m u m principle shows that u is the unique b o u n d e d viscosity solution of (HJ 1). D

References [1] J.P. Aubin and H. Frankowska, Set Valued Analysis, Birkh~iuser, Basel (1990). [2] D. Azagra and R. Deville, Subdifferential Rolle's and mean value inequality theorems, Bull. Austr. Math. Soc. 56 (1997), 319-329. [3] D. Azagra and R. Deville, Starlike bodies in Banach spaces and a new characterization of separable Asplund spaces, Preprint. [4] M. Bachir, On generic differentiability and the Banach-Stone theorem, Preprint. [5] M. Bachir, Subdifferential calculus and smooth variational principles, Preprint. [6] G. Barles, Solutions de viscositd des dquations de Hamilton-Jacobi, Mathematiques & Applications (Paris), Springer-Verlag (1994). [7] R. Bonic and J. Frampton, Smooth functions on Banach manifolds, J. Math. Mech. 15 (1966), 877-898. [8] J.M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), 517-527. [9] J.M. Borwein and Q. Zhu, Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, SIAM J. Control Optim. 34 (1996), 1568-1591. [ 10] K-C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. [ 11 ] M. Choulli, R. Deville and A. Rhandi, A general mountain pass principle for non differentiable functionals, Rev. Mat. Apl. 13 (1992), 45-58. [12] F. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247-262. [13] F. Clarke and Y. Ledyaev, Mean value inequalities, Proc. Amer. Math. Soc. 122 (1994), 1075-1083.

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CHAPTER 11

Operator Ideals Joe Diestel Department of Mathematics, Kent State University, Kent, OH, USA E-mail: diestel@ mcs. kent. edu

Hans Jarchow Department of Mathematics, University of Ziirich, Ziirich, Switzerland E-mail: [email protected]

Albrecht Pietsch Department of Mathematics, Jena University, Jena, Germany E-mail: [email protected]

Contents 1. Basic concepts and closed ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Banach and quasi-Banach ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Approximation numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Ultraproducts, maximality and trace duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. S u m m i n g operators and some of their relatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. L p-factorable operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Grothendieck's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Concrete operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Rademacher type and cotype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. U M D operators, Haar type, and uniform convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Extension of classical theorems to vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . 12. Operator ideals and tensor products, the approximation property . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Operator ideals

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1. Basic concepts and closed ideals Standard references: [ 135, Part 1], [42]. 1.1. The theory of operator ideals was born in 1941 with the following observation of Calkin [23]: The ring B of bounded everywhere defined operators in Hilbert space contains nontrivial two-sided ideals. This fact, which has escaped all but oblique notice in the development of the theory of operators, is of course fundamental from the point of view of algebra and . . . . As examples of two-sided ideals in 13 we may mention here the class of all operators A such that R(A), the range of A, has a finite dimension number, the class of all operators of Hilbert-Schmidt type, and the class of all totally continuous operators. Let H be an infinite dimensional separable Hilbert space, and write ~ ( H ) instead of B to denote the ring of (bounded linear) operators from H to itself. Then the main results of Calkin's paper can be stated as follows: 9 ~ ( H ) , the set of finite rank operators, is the smallest proper ideal in ,~(H). 9 ~3(H), the closed set of completely (totally) continuous operators, is the largest proper ideal in ~ ( H ) . 9 Ideals between ~ ( H ) and ~23(H) exist in abundance; we take particular note of the Schatten-von Neumann classes ~ p (H), the most prominent of which is 6 2 ( H ) , the class of Hilbert-Schmidt operators; see [ 166,169]. Identifying H with 12, operators T E ~52(/2) are characterized by the property that the representing matrix (rhk) has finite double norm; see [66,171]:

o'2 (T) "--

IZ'hk 12

.

h--1 k=l Obviously, ~52 (/2) is a Hilbert space with this norm, which differs from the operator norm. Schur's inequality [ 173] says that

IXnl2

~< o'2(T),

where (Xn) denotes the eigenvalue sequence of T 6 ~52(H). In 1949, Weyl [190] proved a natural generalization of this inequality to operators T ~ ~ p ( H ) with 0 < p < ~ . Since their inception, Schatten-von Neumann operators have become an indispensable tool of spectral theory in Hilbert spaces; see [55,157,168] and [181]. 1.2. Schatten and von Neumann actually dealt with more general classes of operators, so-called cross-spaces of linear transformations. They phrased their results in the language of direct (now tensor) products of Banach spaces [166,167] and [169]. Historically, this work was one of the starting points of Grothendieck's fundamental investigations. His

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famous Rdsumg [60] became the source of modern Banach space theory. However, after a first attempt made by Amemiya and Shiga [2], an incubation period of more than a decade was needed to understand and appreciate Grothendieck's ideas. In the late sixties, Lindenstrauss and Petczyfiski published a pioneering paper [104], which stresses the following point of view: Though the theory of tensor products has its intrinsic beauty we feel that the results of Grothendieck's paper and their corollaries can be more clearly presented without the use of tensor products. Already the revised theory of nuclear locally convex spaces [129] and the study of absolutely summing operators had shown that there is a very handy alternative to tensor products: operator ideals. 1.3. Before we proceed, let us fix some notation: Throughout, t_ -- {X, X0, Y, Y0, Z . . . . } stands for the class of all (real or complex) Banach spaces, and IN will denote the scalar field, be it IR or C. The norm of a Banach space X is denoted by I1" II, but sometimes we use the symbol I1" I X II for better distinction. We denote the set of all (bounded linear) operators from X (domain) into Y (codomain) by t3(X, Y), and write t2 " - ~ t2(X, Y), where the union ranges over X, Y E t_. As usual, I[TI[ or I [ T ' X --+ YII stands for the operator norm of T E t3(X, Y). The identity map of a Banach space X will be denoted by Ix, or just by I. Furthermore, I will also refer to canonical embeddings like I ' l l --+ loc. Given any functional x~ in the dual space X* and any element y0 in Y, we let x 0 | y0" x ~ (x, x 0) y0. We stress that C and _ are used in the same way as < and ~<; that is, the symbol C indicates proper inclusions. 1.4. The notion of an operator ideal was invented once it became clear that restriction to the ring t2(X) of operators on a fixed Banach space X is too limiting. Rather, in contrast to the Hilbert space setting, operators between different Banach spaces must be taken into account. The basic concepts were first presented by Pietsch [ 131,133], in the preliminary version of his monograph Operator Ideals. The subject has experienced, and continues to experience, considerable development accompanied and inspired by striking applications in diverse areas of mathematical endeavour. Suppose that, in every 13(X, Y), we are given a subset 9A(X, Y). Then

~1 "-- U 9A(X, Y) X,Y

is said to be an operator ideal, or just an ideal, if the following conditions are satisfied: (Oil)

x* | y E 9,1(X, Y) for x* E X* and y E Y.

(OI2)

S + T E 9,1(X, Y) for S, T E 9A(X, Y).

(013)

B T A E P,I(X0, Y0) for T E 9A(X, Y), A E 13(X0, X) and B E t3(Y, Y0).

Operator ideals

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Since (OI3) implies that )~T E 9A(X, Y) for T E 9A(X, Y) and ~ E K , every component 9A(X, Y) is a linear subset of s Y). For brevity, we shall write 9A(X) instead of ~(X,X). 1.5. An illustrative example might convince the reader of the power inherent in ideal theory. Given any complex Banach space X, the Fredholm resolvent set p ( T ) of an operator T E ~ ( X ) consists of all )~ E C for which I - )~T has an inverse in ~ ( X ) . In the setting of integral equations the Fredholm resolvent is defined by F(~) :-- T ( I - s -1 . Then ( I - ~T) -1 = I + LF()~), and we have the advantage that F(L) E 9A(X) whenever T belongs to an ideal 9A(X). 1.6. We emphasize that the strength of ideal theory lies in its concrete examples. Many of them will be discussed in the course of this article. An operator T E ~ ( X , Y) has finite rank if rank(T) "-- dim{Tx" x E X} is finite. Such an operator admits a finite representation

T--

xk |

with x 1* . . . . . x n* E

X*

and Yl . . . . . Yn E Y.

(1.6a)

k--I

These operators form the smallest ideal, denoted by ~. The symbol | used above and the formal identification ~ ( X , Y) = X* N Y indicate the close link to the theory of tensor products, which will be presented in Section 12. 1.7. The dual ideal ~..[dual consists of all operators T such that T* belongs to the given ideal 9,1. We say that 9,1 is symmetric if 9,1 -- 9,1dual. The closed hull ~ of an ideal 9,1 has the components 9A(X, Y); the closure refers to the uniform topology on s Y). Obviously, 9,1 is also an ideal; and ideals 9A such that 9,1 -- 9,1 are said to be closed. Apart from the very last paragraph, the rest of this section is devoted to closed ideals. m

1.8. For obvious reasons, we refer to operators in ~ as approximable. Using the principle of local reflexivity, Hutton [70] proved that ~ is symmetric. That even the elementary concepts above have some power, can be illustrated by sketching how to reduce a certain infinite dimensional problem to a problem of linear algebra. Suppose we are given an operator T E ~ ( X ) . Choose some A E ~ ( X ) such that Q - T - A has norm less than 1. Then I - Q is invertible and, as was already noted by Schmidt [171, Part II], the equations x - T x - - y and x - ( I - Q ) - l A x ( I - Q ) - l y are equivalent. Since ~ ( X ) is an ideal, we have (I - Q ) - I A E ~ ( X ) . Hence the element x that we are looking for can be found by solving a linear system in finitely many variables. Considerations like these were among the starting points of functional analysis. 1.9.

In the classical work of Hilbert [66, p. 200], a scalar-valued function f on the U3

l/)

closed unit ball of 12 is called vollstetig if Xn --+ x implies f (Xn) --+ f (x). We use Xn --+ x to indicate that Xn tends to x coordinatewise. However, since we are in a bounded set, this

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is just weak convergence. Hilbert, following the tradition of Weierstrass and Frobenius, applied his definition to bilinear and quadratic forms. Soon thereafter, Riesz [155, p. 96] introduced the concept of a substitution complktement continue acting in 12. Nowadays, an operator T 6 13(X, Y) is called completely continuous if it carries weakly convergent sequences to norm convergent sequences. These operators form a non-symmetric closed ideal, denoted by ~3. Already Riesz [155, p. 113, footnote] observed that ~J(12) - ~(12). On the other hand, we know from Schur [174] that ~J(ll) = t3(11). 1.10.

In 1916, Riesz [156] extended Hilbert's spectral theory to operators acting on

C[a, b] and, in fact, to Banach spaces: Die in der Arbeit gemachte Einschr~inkung auf stetige Funktionen ist nicht von Belang. Der in den neueren Untersuchungen tiber diverse Funktionalr~iume bewanderte Leser wird die allgemeinere Verwendbarkeit der Methode sofort erkennen. As a first step, he had to find a suitable class of operators, since certain components of the ideal ~ are too large. We do not know his reasoning. Nevertheless, the crucial Riesz lemma says that a Banach space is finite dimensional if and only if all bounded subsets are relatively compact. This basic observation suggested to consider those operators that map bounded sets into relatively compact sets. For many years such operators were called alternatively vollstetig (Schauder) or opdrations totalement continues (Banach). It was Hille [67, p. 14] who proposed to use the label compact. The compact operators form a symmetric closed ideal, which will be denoted by .~. The symmetry expresses a classical result of Schauder [170] who proved the equivalence of T E N and T* 6 .~. We also note that an operator T :X --+ Y is compact if and only if every bounded sequence in X admits a subsequence whose image is norm convergent in Y. The celebrated Riesz-Schauder theory says that the determinant-free results of Fredholm's theory of integral equations carry over to operators T E N(X) and their duals. So it is extremely important to verify the compactness of concrete operators. We illustrate the facility and elegance of ideal theoretic considerations by a typical application. Let S2 be a bounded domain in IRn with a smooth boundary. Denote by W2 (I-2) the Sobolev space consisting of all functions which, together with their second weak derivao

tives, belong to Lp (I2), where 1 < p < ec. Form W2 (I2), the subspace of all functions in W2 (g2) that vanish on the boundary. Then the Laplacian A defines an isomorphism between IV 2p (I-2) and L p (I2). Moreover, the embedding map I" W2 (I2) --+ L p (S2) is compact: A-1

Lp(~2) ,

A

o

~ W2p(S-2)

I

,. Lp(S-2).

Thus the ideal properties of .~ tell us that I A -1 is a compact operator on L p(I2), and the Riesz-Schauder theory applies. Classically, this was done by representing I A -1 as an integral operator defined by the Green's function. 1.11. An operator T 6 t3(X, Y) is called weakly compact if it takes bounded subsets of X to weakly relatively compact subsets of Y. These operators form a closed ideal, which

Operator ideals

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we denote by 913, with symmetry reflecting a now classical result of Gantmacher [50]. She also knew that weak compactness is characterized by the property that T** maps X** into Y, viewed as a subspace of Y**. Thanks to the Eberlein-Smulian theorem, T" X --+ Y is weakly compact if and only if every bounded sequence in X admits a subsequence whose image is weakly convergent in Y. Weakly compact operators were introduced by Kakutani [82] and Yosida [193] when they proved a Banach space version of the mean ergodic theorem: if the norms ]lTn ]l of T ~ 9I]'(X) are uniformly bounded, then the averages x+Tx+"+vn-'x converge in norm to a limit Px for all x 6 X, and x ~ Px defines a n projection from X onto T's set of fixed points. 1.12.

The infinite summation operator ~ E ~(/1, lo~) defined by

r " (~k) k=l

is not weakly compact; indeed, it is in a sense the prototype of all operators that fail to be weakly compact. An operator To" X0 --+ Y0 not contained in an ideal 9,1 is called CgA-universal if it factors a

through all operators T ' X --+ Y in the complement of 9,1. More precisely, To'Xo X

T

Y B ~ Y0. In this sense, Z" is C~3-universal, while the embedding map I ' l l --+ lo~ turns out to be C.~-universal; see [104] and [79]. More examples will be given in 1.18 and 11.6. On the other hand we know from [52] that a C93'-universal operator does not exist. 1.13. With every operator ideal we associate the collection of all Banach spaces X 6 k such that Ix, the identity map of X, belongs to 9A. The class A so obtained has the following properties"

(s~) (s2) (s3) (s4)

A contains all finite dimensional Banach spaces. If X belongs to A, then so does any isomorphic copy. A is stable when passing to complemented subspaces. A is stable under the formation of finite direct sums.

Our general strategy will be to use Gothic bold uppercase letters to denote operator ideals, while the associated classes of spaces are denoted by the corresponding sans serifs. For example: ~ --+ F, .~ --+ K, 93 --+ V and 913 --+ W. Banach spaces in V are said to have the Schur property: norm and weak convergence of sequences coincide. In particular, 11 6 V. Moreover, W is the class of reflexive spaces. Since F - K, different ideals may well define the same class of spaces. This is in particular true for 9A and 9,1. So the language of operator ideals offers manifold opportunities to describe specific properties of Banach spaces, giving this landscape a much more colourful view. Let A be any class of spaces satisfying (S1) to ($4). Then all operators factoring through some member in A form an ideal, which is said to have thefactorization property; see [65]. In this way, 913 can be reconstructed from W: identities of reflexive Banach spaces provide the prototypes for all weakly compact operators [31 ]. Further examples are treated in 1.16, 1.18 and 11.6.

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1.14. So far, we have dealt with the oldest generation of closed ideals: ~ , .~, ~3 and ~ ' . For further consideration it will be advantageous to have a few additional abstract concepts in hand. Let 9,1 and f13 be ideals. The (X, Y)-component of the product ~ o 9,1 consists of all A

B

operators admitting a factorization T ' X >M > Y, where M is a suitable Banach space, A 6 9,1 and B 6 f13. We have ~3 o ~ = .~. An ideal 9,1 is said to be idempotent if 9,1 = 9,1 o 9,1. The ideals ~, ~ , .~ and ~ 3 enjoy this property, but ~23 does not [135, p. 60]. Every ideal with the factorization property is idempotent. But, as shown by .~, the reverse implication may fail. The (X, Y)-component of the quotient f13 -1 o 9,1 consists of all operators T 6 ~ ( X , Y) T such that, regardless of how we choose Y0 and B 6 fl3(Y, Y0), the composition X > y

8

Y0 is a member of 9,1. Similarly, the (X, Y)-component of the quotient f13 o 9,1-1 is A

T

the set of all operators T 6 s Y) such that X0 >X > Y belongs to f13 whenever A 6 9A(X0, X). It is no problem to verify that ~23 o 9.1, f13-1 o 9.1 and f13 o ~.[-- 1 are ideals. 1.15. Grothendieck [58] was the first to define a subclass of Banach spaces using operator ideals. A space X is said to have the Dunford-Pettis property if every weakly compact operator T" X ~ Y is completely continuous, for any choice of Y. The collection of these spaces will be denoted by DP. Dunford and Pettis [41 ] proved that L1 (M, #) 6 DP and, more than 10 years later, Grothendieck added the relation C ( K ) E DP. He even showed that 93(C(K), Y) -- ~ 3 ( C ( K ) , Y) for all Y. Therefore the Riesz-Schauder theory extends to completely continuous operators on C (K), since their squares are compact. A Banach space X has the Dunford-Pettis property if and only if Ix belongs to 53 ~3 "-- ~ 3 - 1 o f13. So it is natural to use the same terminology when referring to operators T" X --+ Y in this quotient. A necessary and sufficient condition is that the scalar sequence ((Txn, y~)) tends to zero whenever (Xn) is a weak null sequence in X and (Yn*) is a weak null sequence in Y*. As an easy corollary we get X)9~3dual ___53r hence DP dual c_ DP. If X - - Y --/2, then T E ~D~]3(/2) means that the bilinear form (x, y*) ~ (Tx, y*) on 12 • 12 is vollstetig in Hilbert's original sense. 1.16. An operator T : X --+ Y is said to have the Banach-Saksproperty if every bounded sequence in X admits a subsequence whose image has norm convergent arithmetic means. The class of these operators, denoted by f13~, is a non-symmetric closed ideal with the factorization property; see [3,47,7] and [65]. Closedness is proved with the help of a Ramsey type theorem. The terminology refers to a classical result of Banach and Saks [4] who showed that L p E [35 if 1 0 such that ]lJx[] >~ cIIx[[ for all x 6 X. In the case when [[Jx[] = IIx[[, we speak of a metric injection. For every (closed linear) subspace M of X, the canonical embedding from M into X is denoted by JM x. An important example of a metric injection is the canonical map K x : X ~ X**, which assigns to every element x in X the functional x* w-~ (x, x*) in X**.

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445

An operator Q from X onto Y is called a surjection. In the case when the open unit ball of X is mapped onto the open unit ball of Y, we speak of a metric surjection. For every (closed linear) subspace N of Y, the quotient map Q~v from Y onto Y / N has this property. 1.18. One may ask whether there exist larger classes of operators for which the basic results of the Riesz-Schauder theory remain true. This question led Kato [85] in 1958 to introduce the notion of a strictly singular operator. Such operators are defined by the property that T j X cannot be an injection for any infinite dimensional subspace M. In other words, if IITxll ~> cllxll for all x 6 M and some c > 0, then dim(M) < cx~. These operators form a closed ideal, which will be denoted by 6 . Petczyfiski [121 ] found a dual counterpart, the closed ideal ~s of strictly cosingular operators. By definition, T E if(X, Y) if Q~v T cannot be a surjection for any infinite dimensional quotient Y/N. Bourgain and Diestel [ 17] showed that an operator T is strictly cosingular if T* carries weak* convergent sequences to norm convergent sequences. It turns out that .~ C ~ and N C ~s The righthand ideals are much larger. For example, the summation operator 27:11 --+ 1~ is strictly singular, and Kco'co --+ c~* is strictly cosingular. Moreover, ~d,at C ~ and ~5d"at C ~. The concept of strict singularity admits various generalizations. For 1 ~< p < c~, an operator T E ~ ( X , Y) is said to be lp-singular if T J x cannot be an injection for any subspace M isomorphic to l p. This produces a 1-parameter scale of pairwise different closed ideals, denoted by ~ [ p . In the limiting case p = cx~,the space I~ should be replaced by co. Thus we get the ideal ~c0. From Rosenthal's ll-theorem [159] it follows that an operator belongs to ~11 if and only if the image of every bounded sequence contains weak Cauchy subsequences; such an operator is thus said to have the Rosenthalproperty. Since completely continuous operators map weak Cauchy sequences into norm convergent sequences, it is readily seen that ~ !l ~3"-I o .~. Any isometric embedding from ll into 1~ is CESll-universal. An operator T E s Y) is c0-singular if and only if ~ k ~ J Txk converges in norm for all sequences (xk) such that )--~--11(x~,x*)l < ~ whenever x* E X*; see [151, p. 270]. Those operators are also called unconditionally summing (or converging). The canonical embedding from co into 1~ is C~Sc0-universal. Every/1-singular operator factors through a Banach space not containing any isomorphic copy of 11. That is, ~511 has the factorization property; see [48] and [189]. On the other hand by [53], the ideals ~5 and ~ c 0 even fail to be idempotent. 1.19. In continuation of 1.7 and 1.14, we now describe a few more methods to produce new operator ideals from old. We say that an operator T E ~ ( X , Y) belongs to the regular hull ~.[reg(X, Y) if K r T E 9A(X, Y**). Here Ky denotes the canonical map from Y into Y**. The injective hull 9...[inj (S, Y) consists of all operators T E ~ ( X , Y) that become a member of 9A by extending T J the codomain: X >Y > Y0, where J is an injection into a suitable space Y0. Similarly, the surjective hull 9As"~(X, Y) consists of all operators T E ~ ( X , Y) that become a member of 9,1 by lifting the domain: X0 Q X T > Y, where Q is a surjection defined on a suitable space X0. Due to the extension and lifting properties, in the definitions above we may take Y0 - loc (I[) and X0 -- l l (]I) with an appropriate index set. It is clear that 9Areg, ~inj and 9As"~ are ideals and that ~[reg C ~inj. m

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An ideal is called regular if 9,1 = p.~lreg, injective if 9,1 - - ~[inj, and surjective if 9,1 = ~sur. Every dual ideal is regular. Moreover, if 9,1 is injective, then 9,1dual is surjective; similarly, if 9.1 is surjective, then p_~dual is injective. However, in neither case can one conclude the converse. By Enflo's [46] negative solution of the approximation problem, ~ is properly contained in .~, but it is easy to see that ~ inj __ -~sur __ ~.. Obviously, f13sur - ~ . Moreover, Weis [189] proved that ~5 s"r - ~inj_ ~ [ 1 . Injectivity and surjectivity of 9,1 imply that A, the associated class of Banach spaces, is stable w h e n passing to subspaces and quotients, respectively. In case of symmetry, X E A and X* 6 A are equivalent. 1.20.

Properties of ideals considered so far are listed in Table 1.

Table 1 Ideal

Symmetric

Regular

Injective

Surjective

Factorization property

Idempotent

yes yes no yes no no no no no no

yes yes yes yes yes yes yes no yes yes

no yes yes yes no yes yes no yes yes

no yes no yes no yes no yes yes no

no no no yes ??? yes no ??? yes no

yes yes no yes ??? yes no ??? yes no

J~ ~9" ~gt ~3~ ~; ~I1 ~5c0

1.21. Let H denote an infinite dimensional Hilbert space and (. I') its inner product. With e 6 H we associate the functional e* "x --+ (xle). Every operator T E ~ ( H ) admits a Schmidt representation oo

T--

men|

n,

(1.21a)

n=l

where (en) and (fn) are suitable orthonormal sequences and t = (rn) is a null sequence. We m a y arrange by simple manipulations that r l / > 1:2 ~> . . . >/0, and then the coefficients rn are uniquely determined. Indeed, (1.21 a) implies that Ten = rn fn and T* fn = rnen where, deviating from our usual notation, T* stands for the Hilbert space adjoint of T. r nen, and rn is the nth eigenvalue of the positive operator ~/T--;T. We Hence T* Ten 2 refer to Sn (T) := rn as the nth s-number (singular number) of T. Italics indicate that lp and co are viewed as Banach spaces in the usual sense. However, in order to stress the analogy with operator ideals, Gothic bold lowercase letters will be used to denote sequence ideals. These are permutation invariant (rearrangement invariant) -

-

Operator ideals

447

ideals a in the ring of bounded scalar sequences. That is, in addition to the ideal property we require that (rk) E a implies (rrr(k)) E a for every permutation Jr of N. Obviously, there is no other proper and closed sequence ideal in lec than co. Thus we have a counterpart of Calkin's theorem, which asserts that t3(H) contains one and only one proper and closed operator ideal: m

~(H)--.~(H)--93(H) ....

.

Consequently, proper ideals a and 9.1(H) are contained in co and . ~ ( H ) , respectively. Given any proper sequence ideal a, we consider the collection of all operators T E .~(H) having a Schmidt representation with t E a. This process yields a proper ideal in t2(H). Conversely, with every proper ideal 9.1(H) in s we may associate the set of all sequences t = (rn) such that the operator defined by (1.21a) belongs to 9.1(H). Clearly, this property does not depend on the special choice of the underlying orthonormal sequences. Thus we get a one-to-one correspondence: a +-~ 9.1(H). In particular, c0 +-~ .~(H). In this way, ideal theory in ~ ( H ) is reduced to the theory of permutation invariant ideals in [oc. This may be considered as a significant simplification, since a non-commutative situation is turned into a commutative one. Unfortunately, sequence ideals exist in abundance. For our purposes, the ideals [p and co are most important. The above procedure associates with [p the Schatten-von Neumann ideal ~ p(H). Given any Schmidt representation of r r (~p(H), put tip(T) := Iltllp II and note that this quantity is well-defined. It takes some doing to show that tip defines a norm when 1 <~ p < ec and a p - n o r m when 0 < p < 1. In both cases, ~ p (H) becomes complete with respect to the corresponding metric. Similarly, the Lorentz sequence ideals [p,q generate the operator ideals ~p,q(H). Several well-known results on [p have natural substitutes for ~Op(H). A m o n g others, a non-commutative analogue of HOlder's inequality holds [69]: if 0 < p, q, r < oc and 1/r = 1/p + 1/q, then

ST E ~Or(H)

and

tir(ST) <<.tip(S) tiq(T)

for S E ~Op(H) and T E ~Oq(H). Furthermore, every R E ~Or(H) can be written in the form R = ST, where S and r are as above and tir(R) = tip(S)tiq (T). Of particular interest are the Hilbert-Schmidt class ~ 2 ( H ) and the trace class ~ l (H). Originally, the latter was defined via I~51(H) = ~ 2 ( H ) o ~ 2 ( H ) . For 1 < p < oc and 1 / p + 1/p* = 1, the standard duality between [p and [p, also has a counterpart: trace duality. However, the presentation of this needs some preparation, and we postpone the discussion to Section 4. In the context of his Grundlagen der Quantenmechanik (1932), von Neumann looked for traces of infinite operators. His early reasoning was quite heuristic. Later, in collaboration with Schatten, he laid the theoretical background, based on previous work in the finite dimensional case [116]. Von Neumann's influence on the early period of the theory of operator ideals can hardly be overestimated; Calkin, Murray and Schatten were all von Neumann students.

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2. Banach and quasi-Banach ideals Standard reference" [135, Part 2]. 2.1. By an ideal norm defined on an ideal PA we mean a rule a that assigns to every operator T 9 P,l a non-negative number a ( T ) such that the following conditions are satisfied: (IN1)

a(x* | y) = ]]x*l] ]]yl] for x* 9 X* and y 9 Y.

(IN2)

~(S 4- T) ~< r

(IN3)

o t ( B T a ) <<,IIBI] or(T) ]]all for T 9 9A(X, Y), a 9 ~(X0, X) and B 9 s

4- or(T) for S, T 9 PA(X, Y). Y0).

Since (IN3) implies that a ( ~ T ) -- ])~]ot(T) for T 9 9A(X, Y) and )~ 9 ]K, we have indeed a norm. Moreover, ]ITI] ~< a ( T ) . For greater clarity, we shall sometimes write a ( T " X --+ Y) instead of a ( T ) . In some examples, we will only have a weak form of (IN1), namely" or(T) > 0 if T r O, which is equivalent to ot (IK) > 0. If the later quantity is different from 1, then a is said to be non-normalized. In this case, (IN1) can be ensured by passing to a(IK)-la. Occasionally, more general concepts will be needed. We call a an ideal quasi-norm if the triangle inequality is replaced by a quasi-triangle inequality: ot(S + T) ~< c[a(S) + a ( T ) ] for S, T 9 PA(X, Y). Here the constant c ~> 1 does not depend on the underlying spaces X and Y. We speak of an ideal p-norm with 0 < p ~< 1 whenever ot(S 4- T) p <~ or(S) p 4ot(T)P. In this case, the quasi-triangle inequality holds for c :-- 21/p-1. Conversely, given some ideal quasi-norm a, a classical renorming process yields an equivalent ideal p-norm

~ ( T ) "= inf

~(Ti)P i=1

where p is defined by c

Ti, T1 . . . . . Tn 9 PA(X, Y), n 9 N

9T =

,

i=1

-- 2 l/p-1.

2.2. A Banach ideal is an ideal 9,1 equipped with an ideal norm a such that all components PA(X, Y) are complete with respect to the associated convergence. Quasi-Banach and p-Banach ideals are defined analogously. As it turns out, the ideal quasi-norm of a quasiBanach ideal is unique up to equivalence. Consequently, we may refer to a quasi-Banach ideal 9.1 without specifying or. In addition, the definition of a concrete quasi-Banach ideal automatically leads to a natural ideal quasi-norm, under which it becomes complete. However, there are cases where one and the same ideal can be obtained in different ways, and then we have, indeed, 'die Qual der Wahl'; see 3.5 and 9.10. We also stress that there are ideals which in no way can be made a quasi-Banach ideal. The most prominent example is ~, the ideal of finite rank operators. Of course, every closed ideal is a Banach ideal with respect to the operator norm. Every ideal p-norm ot defined on an ideal 9A induces a corresponding ideal p-norm otdual, Otreg, Olinj and asur on 9,1dual, ~reg, ~2[inj and PAsur, respectively. The basic intention of the above and the following definitions consists in preserving completeness. Given an ideal p-norm ot on 9,1 and an ideal q-norm fl on ~3, then fl-1 o ~ ( T ) " - - i n f { ~ ( B T ) " B e ~ ( Y , Y0), fl(B) <~ 1, Yo 9 L},

Operator ideals

449

provides an ideal p-norm on the quotient ~3-1 o ~.1. Similarly, we may get an ideal q-norm on ~3 o 9A-1 . Less obvious is the case of products. Then the expression

fl ore(T)'--inf{fl(B)ot(A)" T = B A ,

A E 9A(X, M),

B E~(M,Y),

}

M EL

yields an ideal r-norm on ~3 o 9,1, where 1/r - 1 / p + 1/q. This value of r can be improved in many concrete cases. 2.3. We now treat the most prominent non-closed Banach ideal 911, which consists of the 1-nuclear, or simply nuclear, operators. The index '1' is used with an eye on later generalizations. Passing from finite to infinite representations, we look at operators T" X --+ Y that can be written in the form O(3

T --

eo

E*

Xk | Yk

such that

k=l

x *1 , x 2* , . .

.

E

X*

Z IIx~ II IlYk II < ~,

(2.3a)

k=l

and yl, y2 . . . . E Y. It is readily seen that the infinite series of the terms

x[~ | yk converges in It(X, Y). The 1-nuclear ideal norm is given by eo

IIx~ II IlYk II,

vl (T) "-- inf Z k=l

where the infimum ranges over all 1-nuclear representations (2.3a). Clearly, 9 1 1 is the least Banach ideal, and for all nuclear operators T and any ideal norm a, associated with a complete ideal, we have 0t(T) ~< vl (T). It will be helpful to have at hand an alternate description of nuclearity. By normalization, we get eo

r-

|

with IIx~ll- 1, IlY~II- 1 a n d t - ( r k )

E [1.

k=l

This yields the factorization T 9X A > leo Dt> 11 B > Y. Here A : x ~ ((x,xk)), , Dt'(~k) w-~ (rk~k) and B ' ( o k ) ~ Y~C~__10kYk. Obviously, Dt is 1-nuclear and v l ( D t ) ]]Dt ]] = ]ltlll ]]. Hence diagonal operators from leo to ll are the prototypes for all 1-nuclear operators. The concept of nuclearity was introduced in the early fifties for two reasons. Inspired by the theory of distributions, Grothendieck looked for an abstract version of Schwartz's kernel theorem; this explains the origin of the term nuclear. Secondly, Grothendieck [61 ] and Ruston [160] extended Fredholm's determinant theory to operators in Banach spaces. To this end they needed a sufficiently large class of operators for which the concept of a trace is meaningful. For a Hilbert space H, the Schatten-von Neumann class ~ l (H) turned out to be appropriate. Since 6 1 (H) = 911 (H), it was thought that 9tl would be

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450

the right choice for general Banach spaces; indeed the nuclear operators were referred to as opdrateurs gt trace or trace class operators; see [57] and [162, p. 166]. Ironically, Enflo's negative solution of the approximation problem implied that there were trace class operators without trace! We now address this anomaly. 2.4.

The trace of an (n, n)-matrix T = (rh~) is defined by

trace(T) := ~

r~k = ~

k--1

~k,

k--1

where ~,1 . . . . . )~n are the eigenvalues. With a little bit of linear algebra, the concept of a trace can be extended to finite rank operators T acting from a Banach space X to itself. It turns out that the expression H

trace(T) := Z

(x/~, x/~)

k=l

does not depend on the finite representation

T --

x k @ xk

with x 1* . . . . . x n* ~

X*

and Xl . . . . . Xn ~ X.

k----1

In this way, we get a linear functional T ~ trace(T) on every component ~ ( X ) , which is uniquely determined by the condition that trace(x* | x) - (x*, x) for x* 6 X* and x E X. Very important is the commutative law: trace(AT) = trace(TA)

for T 6 ~(X, Y) and A 6 ~(Y, X).

This formula relates traces defined on components ~ ( X ) and ~(Y). It is natural to think of extending the trace functional by continuity. Using Auerbach's lemma, we obtain the inequality Itrace(T) I ~< nllTll whenever rank(T) ~< n. Since the identity map of l~ shows that the factor n cannot be improved, the trace functional is discontinuous with respect to the operator norm in infinite dimensions. The next idea that comes to mind is to replace the operator norm by a larger ideal norm. But even the largest, namely vl, is not good enough, as we shall explain next. Given T ~(X, Y), put

Ilxk II Ily~ll,

I~1 (T) "-- inf k=l

where the infimum only ranges over finite representations (1.6a). We obviously have [trace(T)] ~< v~(T) for T 6 ~(X). Since the definition of Vl(T) permits representations that are not necessarily finite, we can only conclude that I~1 ( T ) ~ I ~ ( T ) . Even more: the

Operator ideals

451

norms vl (T) and v~ (T) may be non-equivalent on ~(X, Y) for certain spaces X and Y. In the language of tensor products, the situation can be described by stating that the completion of X* | Y -- ~(X, Y) with respect to the norm v~ is canonically mapped onto 9tl (X, Y), but this surjection may fail to be one-to-one; see 12.8. 2.5. If the underlying Banach space X has the approximation property, then the above problems disappear, and every operator in r (X) admits a well-defined trace. This is in particular the case when we work in a Hilbert space H. Then r (H) = 1~1 ( O ) , and the term trace class operator is indeed justified. But there is more. The eigenvalue sequence ()~k) of T 6 ~ l (H) is absolutely summable, and we have LidskiFs trace formula [99], [137, Section 4.2]: oo

trace(T) = Z

)vk.

k=l

2.6.

Every operator T E t~(ll) can be identified with its representing matrix (rhk) via oo

for ( ~ ) ~ l l. k=l

Then o<)

IlZll--

~lrhkl

sup

l~
(x)

and

vl(T)-- Z

sup

I~hkl.

h = l 1~
Infinite matrices for which the right-hand expression is finite were already considered in von Koch's theory of infinite determinants [87] and by Dixon [38, p. 191] when he solved linear systems in infinitely many unknowns. Since 11 has the approximation property, every T = (rhk) 6 r (ll) admits a trace, and we get oo

trace(T) -- Z

rk~,

k=l

as it should be. But even in this comfortable setting, pathologies occur. It was known to Grothendieck [59, Chapter I, p. 171] that the existence of an operator T 6 r (11) with the properties T 2 = O and trace(T) = 1 would yield a negative answer to the approximation problem. However, operators with these 'almost contradictory' properties were constructed only much later [46]. Being nilpotent, T cannot have any eigenvalue different from 0, but its trace is 1. Hence, in this case, Lidskii's trace formula fails. In conclusion, a satisfactory trace theory can only be developed using ideals that are strictly smaller than 9tl. Since 9tl is the smallest Banach ideal, we realize that there is no way to avoid quasi-Banach ideals.

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452

2.7. Grothendieck [59, Chapter II, p. 3] was the first to find non-trivial ideals with a trace. For 0 < p ~< 1, he introduced opdrateurs de puissance p.bme sommable, which are defined by the property that (X3

T -- ~

rk x ; | Yk

with IIx~ll- 1, Ilykll- 1 and t - ( r ~ )

e lp.

k--1

They form the smallest p-Banach ideal ~p; the underlying ideal p-norm being given by ~ p ( T ) = inflltllpll, where the infimum ranges over all admissible representations. The factorization T 9X A > loc Ot> 11 B > Y can be adapted from 2.3, but now we have t -(r~) 6 lp and 4~p(Dt)= IItllpll, Every operator T ~ ~ p ( X ) is compact, and its eigenvalue sequence ()~k) belongs to [r, where 1/r = 1/p - 1/2. Actually, for 0 < p < 1 one can do slightly better: ()~) 6 Ir, p; this is then best possible; [90, pp. 105, 124], [137, p. 163]. Consequently, in order to ensure that ()~) 6 11, we need 0 < p ~< 2/3. This suggests that ~2/3 might be a good candidate for a trace ideal, and indeed, it is. But still a disadvantage arises: if we work on a Hilbert space H, too many operators get lost. In fact, since ~ p ( H ) = ~Sp(H) for 0 H > Y such that A is 1nuclear and B is bounded. This ideal coincides with the product t32 o 9 t l , where 132 denotes the Banach ideal of all operators factoring through a Hilbert space H; see 6.3. Now we have t2~2 o r (H) = t ~ 1 (H).

3. Approximation numbers Standard references: [25], [135, Part 3], [137, Chapter 2]. 3.1. The non-increasing rearrangement (r*) of (r~) 6 co is defined as follows: If the sequence has infinitely many terms r~ 7~ 0, then these are ordered such that Irrr(1)l ~> Irrr(2)l > ~ ' " . Otherwise we proceed in the same way as long as possible and fill up with zeros 9

Throughout, ql" stands for the unit circle {( 6 C: Iffl = 1}. In the standard fashion, C ( 2 ) is identified with the Banach space of all 27r-periodic continuous functions f on R. Let E n ( f ) := inf{llf - PII: deg(P) < n}, where the infimum ranges over all trigonometric polynomials P of degree less than n. The numbers En ( f ) form a monotone null sequence whose rate of convergence may serve as a measure of smoothness of the function f . Motivated by these concepts from Fourier analysis and approximation theory, Pietsch [ 128] defined the nth approximation number an(T) "-- inf{ liT - All" A e ~(X, Y), rank(A) < n}

for T e 13(X, Y).

This yields a monotone sequence: IlZll -- al (T) ~> a2(T)/> . . . / > 0. Occasionally, we will write an(T" X --+ Y) instead of an(T).

Operator ideals

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3.2. Precise computation of approximation numbers is rare; however, for some diagonal operators Dt(~k) -- (r~k) this can be achieved. Let 1 <~ q < p <. ec, 1 / r -- 1 / q - 1 / p and t -- (rk) E [r such that Z'I ~ /72 ~ " ' " ~ 0. Then

an(Dt'lp ~ l q ) -

(L),r r~

k--n In the case p -- q and t - (rk) E C0, we have a n ( D t ' l p ~ lp) -- rn. This result includes an observation of Allekhverdiev [1] who showed that, for compact operators in a Hilbert space, the approximation numbers are just the s-numbers. 3.3.

For 0 < p < ec, the operators T with

Otp(T) "--

<(X3

an(T) p n=l

form the quasi-Banach ideal PAp, which is a natural extension of the Schatten-von Neumann class l~Sp(H). There are operators with a n ( T * ) < a n ( T ) . However, if T is compact, then we have equality. Hence the ideal PAp is symmetric. Since Otp satisfies a quasi-triangle inequality with c -- 21/p, there is an equivalent ideal r-norm, 1/r = 1 4- 1 / p . Unfortunately, apart from the construction described in 2.1, no explicit and aethically pleasing expression of such an r-norm is known. We see from Dt E ~q(lec, ll) 0 t E [q and Dt E PAp(lec, ll) ~ t E [r,p that ~q(Icc, ll) ~ f21p(lec,ll) whenever 0 < r < q ~< 1. Since ~q is the smallest q-Banach ideal, the aforementioned renorming result cannot be improved [ 137, pp. 84 and 110]. 3.4.

It is easy to verify that am+n-1 ( S T ) <~ a m ( S ) a n ( T )

for T E ~ ( X , Y) and S E ~(Y, Z).

Hence we get an analogue of H61der's inequality" if 0 < p, q, r < ec and 1 / r -- 1 / p -Jr-1 / q , then S T E PAr(X, Z)

and

Otr(ST) <~ 21/r 19lp(S) olq(T)

for T E ~.[q(X, Y) and S ~ f21p(Y, Z ) . In fact: PAp o P..lq - Pdr. 3.5. Since Lorentz spaces naturally appear in this context, we introduce the quasiBanach ideal 9.1p,q constituted by all operators T such that ( a n ( T ) ) E [p,q. The following important criterion is adopted from approximation and interpolation theory. An operator T E 13(X, Y) belongs to ~lp,q if and only if it admits a dyadic (p, q)-representation O(3

T -- Z k=0

Ak

with rank(Ak) ~< 2 k,

(3.5a)

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J. Diestel et al.

where the operators Ak e ~(X, Y) satisfy the condition (2k/PllAkll) ~ [q; see [136] and [ 137, p. 84]. An equivalent ideal quasi-norm is obtained by tglp,q (Z) "=

inf]l (2~/PllA~ll) l lq ll,

the infimum being taken over all possible representations (3.5a). In this case it is indeed a o matter of taste which ideal quasi-norm, Olp,q "-- II(an(. )) I [p,q II or Otp,q, should be considered as more natural; see 2.2. 3.6. Fix T E PAl (X) and choose a dyadic (1,1)-representation as described above. From ]trace(Ak)l ~< 2kllAkl] it follows that OQ

trace(T) "-- Z

trace(Ak)

k=0

yields a continuous trace on PAl. Comparing 9--[1 and ~2/3 shows that PAl (12) is larger than ~2/3(12), whereas PAl (l~, ll) is smaller than ~2/3(1~, 11). 3.7.

A remarkable ideal is given by PA0"--

F]

PAP"

0
Operators in 9,10 are characterized by the property that the sequence of their approximation numbers tends rapidly to zero. If 0 < p < cx~ and 1/r -- 1 + 1/p, then PAr C ~r C PAp. Hence 9-10 coincides with ~0
II

cn(T)"-- inf[ z J ~

II cod(M)

< n}

and the Kolmogorov numbers

dn(T)"= inf{I I

QNTII dim(N)

< n},

respectively. The asymptotic behaviour of the resulting monotone sequences provides measures of compactness of the operator T 6 g ( X , Y). Other quantities with similar properties are the Weyl numbers and the entropy numbers. For further information, the reader is referred to the Gluskin, Gordon and Pajor article [54].

Operator ideals

455

4. Ultraproducts, maximality and trace duality Standard references 9[35, Chapter 6], [106], [135, Chapter 8]. 4.1. Roughly speaking, local theory of Banach spaces deals with those properties that can be expressed in terms of finite dimensional spaces 9The most usual way is to consider quantitative relations that are assumed to hold uniformly for any choice of n elements and/or functionals. The constants involved may or may not depend on n. Local ideal theory generalizes this, transferring such techniques to operators. For a Banach space X the collection of all finite dimensional subspaces is denoted by DIM(X), while COD(X) stands for the collection of all finite codimensional (closed) subspaces. The idea is to gain access to the structure of a given operator T :X ~ Y through an understanding of its 'elementary parts'

QYTJ x

9

M gkx

7">

Y

QY> Q / N

with M E DIM(X) and N E COD(Y). Remarkably, such access is available if we use ultraproduct techniques 9 4.2. Given a family (Xi)i E~ of Banach spaces and any ultrafilter U on the index set IT, all equivalence classes (xi) u = {(x?)" U-limi IIx? - xill = 0} generated by bounded families (xi) such that xi E Xi for i E ~ form a Banach space (Xi) u under the norm II(xi)Ull := U-limi Ilxill. If the family (7~)iE~ with 7) 6 13(Xi, Yi) is bounded, then (Ti)U:(xi) u --+ (Tixi) u defines an operator from (Xi) u into (Yi) u such that ll(7~)ull = U-limi IlTi II. These new objects are referred to as ultraproducts. If Xi -- X, Yi = Y and 7) = T, then we speak of ultrapowers, denoted by X u, y u and T u, respectively. Ultraproducts were introduced by Dacunha-Castelle and Krivine [30]; they constitute a powerful tool in Banach space theory. A few years later, Pietsch [134] showed how to adapt this notion to operators; however, the most important applications to ideal theory are due to Heinrich [63,64]. 4.3. Given an ideal 9.1, we denote by ~[super the collection of operators T for which all ultrapowers T u belong to 9,1. Clearly, 9,1super is again an ideal. It turns out that properties such as closedness, injectivity and surjectivity carry over from 9,1 to 9,1super. We say that 9,1 is ultrapower-stable if 9,1 = 9Asuper. Naturally, 9Asuper is always ultrapower-stable. Let ~Sep denote the closed ideal formed by all operators with separable range 9 Then [super super ~r _..~. So .i~ is ultrapower-stable. Further examples, namely ~ ' l , ~c 0 and ~ZBsuper, will be discussed in 9.9 and 10.9. 4.4. We now choose as our index set the collection of all pairs (M, N) with M E DIM(X) and N 6 COD(Y). Fix an ultrafilter U which contains the final sections {(M, N)" M D M0 and N _c No} associated with M0 E DIM(X) and No E COD(Y). Let X M,N := M, YM,N "-- Y / N and TM,N "-- Q YNT J~l" Define

J~r "x ~ (XM,N)u

with XM,N "--

x

i f x E M,

0

ifx~M,

456

J. Diestel et al.

and QC~ . (YM,N)U w+ H- lim K r y~M N" M,N

Here (YM,N) is some bounded family in Y such that Qruy~M,N -- YM,N. Apply the canonical map K r to pass from Y to Y**. Now Alaoglu's theorem implies the existence of the righthand limit with respect to the weak*-topology. We have I[J~ l] -- 1 and IIQ~ I I - 1. The upshot is the following commutative diagram: ( TM,N ) Lt

(XM,N) u

X

> (YM, N) u

T

> Y

Ky

> Y**

Roughly speaking, we may say that every operator is the ultraproduct of its elementary parts. 4.5. The maximal hull ~2,[max of a quasi-Banach ideal 9A consists of all operators T 13(X, Y) for which

olmax(r) :=sup{ot(QrNTJfvl) 9 M E DIM(X), N ~ COD(Y)} is finite. Since olmax(T) ~ or(T) for T E 9A(X, Y), we have 9A ___9,1max. Clearly, ~max is again a quasi-Banach ideal with respect to Ogmax, and maximality means that 9A = (2[max. If 9.1 is maximal, then 9 A - (9.1dual) dual. We stress that a maximal quasi-Banach ideal is uniquely determined if we just know the values of its ideal quasi-norm for all operators between finite dimensional spaces. Lotz showed in his Lecture Notes [ 106] how the theory of maximal Banach ideals can be derived from the Rdsumd. A quasi-Banach ideal 9A is called ultraproduct-stable if 7) 6 P-d(Xi, Yi) and sup/~(T/) < c~ imply (7~)u ~ 9A((Xi)U, (Yi)U) for all ultrafilters H on arbitrary index sets ~. The central result says that a quasi-Banach ideal is maximal if and only of it is regular and ultraproductstable. Among the ideals considered so far, only t2 and PAp are maximal. The concept of ultrapower-stability makes sense for all ideals, while that of ultraproductstability requires an ideal quasi-norm. Ultraproduct-stable quasi-Banach ideals are all the more ultrapower-stable. The closed ideal ~ shows that the converse implication is not true. 4.6. The real significance of maximality lies in its close ties to trace duality, a topic we broach next. In this context it is natural to restrict the considerations to B a n a c h ideals. Duality of sequence spaces is based on the bilinear form (s, t) " - Y~__ 1 cr~r~ defined for s = (crk) and t -- (r~) in I~. In order to guarantee the convergence of the series we assume st El1. Replacing [1 by ~ 1 (H), there exists a perfectly good working counterpart. In fact, letting (S, T) := trace(ST) whenever S, T E s and S T E ~ 1 (H) yields a duality for

Operator ideals linear subsets of s

457

Note the following striking analogies:

c0* = 11 +-~ .~(H)* = ~ 1 (H) , IT = l ~ ~ ~5~ ( H ) * = ~ ( H ) ,

[p -=- [p. +-~ ~;,p(H)* -- ~;,p.(H),

1 < p <(x~.

What about a generalization to Banach spaces? We know already that 9"tl cannot be used as a substitute for ES1 (H), since there exists no trace on ~ 1 (X) unless X has the approximation property. But local ideal theory offers a possibility to get around this problem by appropriate reduction to a finite dimensional setting. First, for finite dimensional spaces E and F, we easily get the right-hand counterparts of the left-hand relations:

([~)* -- [~ ~ ~ ( E , F)* = 9t, (F, E), (l~)* -- In ~ r

(F, E)* = s

F).

Even more is so: in this case every norm ot on s s F) defined by

E) gives rise to an adjoint norm on

ot* (T " E --+ F ) : = sup{ Itrace(ST)l" ot(S " F ~ E) <<.1 }. Let now 9A be a Banach ideal and T 6 ~ ( X , Y). The construction of the maximal hull suggests that we put

o t * ( T ) ' = s u p { o t * ( Q Y x T J ~ ) 9M 6 D I M ( X ) , N 6COD(Y)}. The operators for which this quantity is finite constitute the adjoint Banach ideal 9.1". Of course, 9A* is maximal. Moreover, 9A** = p..[max. It turns out that a* (T) can be obtained as the infimum of all constants c >~ 0 such that [trace(ASoBT)l <~ c [IAllot(So)lIBll regardless of how we choose the spaces X0, Y0 and the operators B 6 ~(Y, Y0), So E 93(Y0, X0), A ~(X0, X). This characterization is quite flexible, since the finite rank operators A S o B T , So B T A, B T A So and T A So B have the same trace. If 9A and ~3 are Banach ideals, then 9A _c ~3 and fl ~< c o~ imply ~3" _c 93* and ~* ~< c #*. In particular, it follows from 9..1 _c 93** that 93* = 93"**, isometrically. Analogous relations hold within the setting of vollkommene Folgenriiume, in the sense of K~3the [9 1] and Toeplitz. 4.7.

As a basic example, we describe the ideal 9t~* -- ~"t' ml a x 9Grothendieck defined

1-integral, or simply integral, operators T ~ ~ ( X , Y) by the following property. Viewing (Tx, y*) as a bilinear form on X x Y*, he assumed the linear form induced on X | Y* to be continuous with respect to the plus petit | raisonnable [59, Chapter I, pp. 126-127], [60, pp. 9, 12]. This just means that there is a constant c >~ 0 such that [trace(ST) I ~< cllSI] for S 6 ~(Y, X). The 1-integral operators form a Banach ideal 31 with the ideal norm ll (T) := infc. Although the inequality [trace(ST)l ~< c[ISII looks much simpler than the inequality Itrace(ASoBT)[ <~ c[]Allot(So)llBll considered in the previous paragraph both

J. Diestel et al.

458

are equivalent for oe : II 9 II. Hence t3" = ~ 1 . The formula 9t~ -- 13 follows from the fact that I~1 : I ~ for all operators between finite dimensional spaces, and tilT* - - ~1 is then obvious. m-1

m-1

4.8. For every quasi-Banach ideal 9A, we have ~ m a x __ ~ o 9,1 o ~ . This formula can be used to define the maximal hull also in the case when 9.1 is any ideal. Similarly, we get a minimal kernel by letting ~min .__ ~ o ~ o ~ . It turns out that ~min C ~ C ~max i m p l i e s ~min __ ~ m i n and ~2[max -- ~ m a x . If 9,1 is a Banach ideal, t h e n ~ m i n ( x , Y) consists of all operators T E/3(X, Y) that can be written in the form CO

(3O

such that

T -- Z B k T k A k k=l

Z Ilnkll~(Zk)llAkll < k=l

and X Ak> Ek Tk> Fk 8k> y. Here El, E2 . . . . and F1, F2 . . . . are supposed to be finite dimensional. Moreover, P-dmin is a Banach ideal under the ideal norm (3O

~min(T) "-" inf ~

IIB~II~(T~)IIAklI,

k=l

where the infimum ranges over all representations described above. This criterion and its obvious generalization to p-Banach ideals implies that, in analogy to 4.5, a minimal quasiBanach ideal is uniquely determined if we just know the values of its ideal quasi-norm for all operators between finite dimensional spaces. As an example and a counterpart of ~'~Tax - - 21, we get ~min~,l- - r . Surprisingly, the ideals Pip are not only maximal, but also minimal.

5. Summing operators and some of their relatives

Standard references: [35,73,124], [135, Chapter 8]. 5.1. A classical theorem due to Dirichlet (1829) asserts that all rearrangements of a scalar series Y ~ l ~ converge (even to the same sum) if and only if Y ~ l I~k[ < ~ - I n other words, unconditional and absolute convergence of series coincide. Looking at series in a finite dimensional Banach space and replacing the absolute value I" I by the norm I1" II does not change the statement. However, troubles ensue if we pass to the infinite dimensional setting. Still, absolute convergence implies unconditional convergence, but the converse cannot be generally true: look at orthogonal series in 12. Mazur and Orlicz asked in The Scottish Book [108, Problem 122] if every infinite dimensional Banach space contains a series that converges unconditionally, but not absolutely. A positive answer to this question was given by Dvoretzky and Rogers [44]. On the other hand, Orlicz [ 118] proved already in 1929 that every unconditional convergent series ) - ~ - 1 fk in L p satisfies Y~_-i [Ifk II2 < ~ if 1 ~< p ~< 2 and ~ - - 1 IIA IIp < c~ if 2 <~ p < ~ ; see 6.1 and 9.4.

Operator ideals

459

5.2. It turned out to be extremely fruitful to extend such investigations to operators. By analogy with the notion of complete continuity, we may consider operators that improve the convergence of specific sequences or series. Let 0 < p ~< q < cx~. An operator T E ~ ( X , Y) is said to be absolutely (q, p)-summing, or just (q, p)-summing, if it takes weak lp-sequences (xk) of X to strong (absolute) lqsequences (Txk) of Y; precisely: if ((xk,x*)) ~ lp for all x* E X*, then ([[Txkl[) ~ lq. It is important to realize the local character of this concept, which is hidden in this definition. In fact, by the closed graph theorem, T is (q, p)-summing if and only if there exists a constant c ~> 0 such that, for any choice of n E 1~ and Xl . . . . . Xn E X,

1/p IlTxkll q k=l

~< c

sup ](xk,x*)lP IIx*ll~
We put ~q,p(T) : = infc, where the infimum is taken over all constants c ~> 0 with the above property. The class of (q, p)-summing operators, denoted by ~q,p, is an injective and maximal Banach ideal for 1 ~< q < cx~ and a q-Banach ideal for 0 < q < 1. Most important are the ideals ~3p :-- ~]3p,p whose members will be called p-summing. In view of the preliminary remarks, we stress that an operator is 1-summing, or just summing, if and only if unconditionally converging series are carried to absolutely converging series. If 1/p - 1/q <~ 1/r - 1/s and p ~< r, then ~q,p c ~s,r" The 1-parameter scale ~3p increases strictly on [1, cx~), but is constant on (0, 1); see [109]. Absolutely 1-summing operators were introduced by Grothendieck as applications prdintdgrales (semi-intdgrales ?~) droite [59, pp. 160-161], [60, p. 33]. He also considered operators T 6 ~ ( X , Y) that admit a factorization of the form T : X --+ C (K) --+ H --+ Y. In a communication addressed to the ICM in Moscow (1966), Pietsch identified these applications prohilbertiennes gauches, also called applications ayant la propridtd de prolongement hilbertien, as absolutely 2-summing operators [60, p. 36]. Most important:/t'2 proved to be the natural ideal norm on ~]32. At the same congress, Mityagin and Pelczyfiski [ 114] defined the general concept of a absolutely (q, p)-summing operator. 5.3. The central part of the theory of p-summing operators was developed by Pietsch [130]. In particular, he established a criterion for an operator to be p-summing, which has become a standard result in Banach space theory. Its proof can be found in many monographs: [35, p. 44], [73, p. 55], [135, p. 232], [188, p. 47], [192, p. 203] and in the Johnson and Lindenstrauss article [80]. Recall that Bx,, the closed unit ball of X*, is a compact Hausdorff space in the weak* topology. So, thanks to the Riesz representation theorem, positive linear functionals on C(Bx,) are just given by regular Borel measures. Pietsch's criterion may be formulated as follows.

J. Diestel et al.

460

An operator T 6 s Y) is p-summing if and only if there exists a constant c >/0 and a (regular) Borel probability # on B x , such that

IlZxll <~ c

for all x E X.

](x,x*)[ p d/ztx*)

(5.3a)

X*

In this case, rrp(T) is the infimum, and even the minimum, of all constants c >~ 0 for which such a measure can be found. In order to make the support of/z small, we may pass to any weak* compact subset W of B x , such that Ilx[I - SUpx,cW [(x,x*)l. For example, if X is C ( K ) , then the set of all Dirac measures 6~" f ~ f ( ~ ) with ~ E K has this property. For 1 ~Y

> loc(Br*) (5.3b)

C(Bx,)

I

> L p ( B x , , Ix)

Indeed, let A take x to (x, .), viewed as a continuous function on B x , , while J sends y to ((y, y*)), viewed as a bounded family on Br,. Moreover, consider the canonical map I " C ( B x , ) --+ L p ( B x , , lz). Then inequality (5.3a) means that []JTxl[ <~ c [llAxlLpl] for all x E X. Hence, I A x w-~ J T x defines an operator on a linear subset of L p ( B x , , lz) and, by the metric extension property of loo(Br,), we can find B such that J T -- B I A and Ilnll ~< c. From (5.3a) and (5.3b) we easily conclude that p-summing operators are completely continuous and weakly compact: ~3p c ~ and 213p ___~[i'. 5.4.

The kth Rademacher function rk takes alternately the values + 1 and - 1 on the

dyadic intervals of length 1/2 ~. That is, rk(t) :-- ( - 1 ) j-1 whenever t 6 A~j) .= [_St_, ) j - 1 ffj and j = 1 . . . . . 2 k. A particular case of Khintchine's inequality says that, for ~l . . . . . ~n E ]~, r/

I~kl 2

Z ~ k r l ~ ( t ) dt.

~< ~/2

(5.4a)

k=l

k=l

The constant ~/~ is best possible [184] and [96]. Moreover, since (r~) is an orthonormal system in L2[0, 1), we have Parseval's equality

I~kl 2 k=l

-

(So

2

n

k=l

~kr~(t)

d')

1/2

(5.4b)

Observe that p ' t w-, (r~(t)) defines a Borel measurable map from [0, 1) into the closed unit ball of l ~ - l~, equipped with the weak* topology. Look at the image of Lebesgue's measure under p to see that (5.4a) and (5.4b) are in fact special cases of (5.3a) applied

Operator ideals

461

to the embedding l : l l --+ 12. Hence this map is 1-summing, and afortiori 2-summing. We even have rrl (I : ll --+ 12) = v/2 and ~r2 (I : ll --+ 12) -- 1. These examples show that

q3p ~ J~. 5.5. Let 1 ~< p, q, r < c~ and 1 / r - - 1/p + 1/q. Then HOMer's inequality for summing operators asserts that

ST E 9i3r(X, Z)

and

rtr(ST) <<,rrp(S) rtq(T),

whenever T E q3q(X, Y) and S E ~

ST E 9tl (X, Z)

and

Z). In the case p = q = 2 we even have

vl (ST) <~/t'2(S) rt2(T).

We mention that the composition formula gtp o gtp C ~D"o ~ = ~ yields an ideal theoretic proof of the Dvoretzky-Rogers theorem discussed in 5.1: Pp -- F. 5.6. In the rest of this section, # will denote a probability measure 9 Sometimes # is d e f n e d on an abstract measurable space M; however, should the underlying set be a compact Hausdorff space K, then we a s s u m e / z to be a regular Borel probability. The canonical maps I : L ~ ( M , / z ) --~ Lp(M, #) and I : C ( K ) --~ Lp(K, #) belong to ~13p. Moreover, in both cases, the p-summing norms are equal to 1. As discrete counterparts, we consider diagonal operators D t : l ~ -+ lp induced by sequences t E [p. Then ltp(Dt) = IIDtll = Iltllpll. 5.7.

Let 1 ~ lp B ~ Y with t E [p. These operators form the Banach ideal r the ideal norm being given by vp(T) : = infllAII IIDt II IIBll, where the infimum extends over all admissible factorizations. In the limiting case p - cx~, we use co instead of lp. Operators T E 91p(X, Y) are characterized by the property that they admit a p-nuclear

representation oo

T --

rk x k | Yk, k--1

:~ . . ~ E S :~ and Yl, Y 2 , . . . :r II ~ 1 and x~,x 2, E Y satisfy the conditions sup~ Ilx~ II((yk, y*)) I lp, II ~< Ily*ll for all y* 6 Y*. In addition, (rk) E lp. The connection between p-summing and p-nuclear operators is given by trace duality: 9tp - ~3p,. To preserve this formula in the limiting case p - 1, we let g t ~ " - ,!3. In this way the correspondence [p +-~ ~13p extends to p -- cx~. It is clear that the counterpart of c0 in such a setting is ~3, the ideal of completely continuous operators. In the language of tensor products, 2-nuclear operators were first introduced by Saphar [163]. Later, Chevet [26] and Saphar [164] generalized this concept to exponents 1 ~< p < cx~.

where

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5.8. Let 1 ~< p ~< cx~. An operator T e s Y) is said to be strictly p-integral if it factors through a map I : C ( K ) ~ L p ( K , lz). These operators form a Banach ideal. As an example we mention the embedding map from l l into co, which can be factorized in the form l "/l R*> L ~ [ 0 , 1 ) I > LI[0,1) R > c0, where R ' f ~ - + (f o f ( t ) r k ( t ) d t ) . M o r e important are the p-integral operators that are supposed to admit a factorization of the following form: X

C(K)

T

Ky

>Y

I

> Y**

> L p ( K , #)

The lower arrow may be replaced by L ~ ( M , #) I> L p ( M , / ~ ) , where (M,/z) is an abstract probability space. This definition yields the Banach ideal 2p equipped with the ideal norm tp(T) := inf IIBII ]IA[[; the infimum ranges over all decompositions as above. If p = 1, then we get the 1-integral operators, which were already considered in 4.7. The formulas ~max __ ,'~p and ~min = q~j~p reflect the philosophy of passing from finite or infinite - -p --p sums to integrals. Sophisticated examples show that when p ~ 2 there exist p-integral operators that fail to be strictly p-integral [ 153]. Of course, the reader may wonder why we prefer the more p --- ~.~ p , . complicated concept. The answer is given by trace duality: ~3p - - "~ Jp, and ~J '~'* Persson and Pietsch [ 126] defined strictly p-integral operators. Later, Pietsch [ 132] observed that regularity is indispensable for a smooth theory. So, following Grothendieck's pattern for p = 1, the natural embedding Ky was incorporated in the definition of operators T ~ 3 p ( X , Y). ~inj

5.9. Diagram (5.3b) implies that ~Jp -- ~]3p, and every p-summing operator T" C ( K ) --+ Y is even strictly p-integral. The following example due to Petczyfiski [ 123] indicates the fruitful interplay between ideal theory and harmonic analysis. Given any subset A of Z, we denote by C a (~1") and L pA ('T) the subspaces of C ('I[') and L p (T), respectively, that are spanned by all characters e int with n E A. Then the p-summing embedding map I" cA (T) ~ LpA (T) is p-integral if and only if L A p ('~') is complemented in L p ('IF). Since, for p -~ 2, there are A's failing this property, ~ p is properly contained in glp. 5.10. The case p - 2 is something special. The Banach ideals of strictly 2-integral, 2-integral and 2-summing operators coincide isometrically. Moreover, Jr2 (T) = v2 (T) for 2-nuclear operators T. The formula ~]3~ = ~]32 can be rephrased by saying that ~]32 is selfadjoint. The sole defect is a missing theorem of Schauder type. Indeed, the embedding from 11 into 12 is 1-summing though neither its predual nor its dual is p-summing for any 0
Operator ideals

463

T 6 ~ 2 ( X , Y). This is an immediate consequence of the fact that the spaces L ~ ( M , #), which appear in strictly 2-integral factorizations, have the metric extension property 9 Next, we show that the ideal of 2-summing operators not only has a beautiful theory; it is also extremely useful. Garling and Gordon [51] observed that zr2(E) = ~ for every n-dimensional Banach space E. An elementary proof of this fact is due to Kwapiefi; see [135, p. 385]. Using a strictly 2-integral factorization of the identity map yields an operator T such that

IE 9E T ) 12n T - I ) E and 11T1111T - i l l ~< x/-d. In this way, we have obtained John's theorem [77], which states that the Banach-Mazur distance d (E, l~) is at most v/ft. Moreover, if E is a subspace of X, then the Hahn-Banach theorem for 2-summing operators provides us with a projection P 6 s E) such that IIPI] ~< nr2(P) ~< ~ (Kadets-Snobar theorem) [81]. K6nig [89] observed that the square ~ 2 o ~ 2 admits a trace, which can be obtained as the sum of the eigenvalues of the operators under consideration; see also [137, p. 182]. We stress that this ideal coincides isomorphically with the product ~ 2 o r which was already discussed in 2.7. 5.11. One of the basic problems is to determine the Hilbert space components of a given ideal 9A. For example, we have ~ p ( H ) = ~ 2 ( H ) if 0 ~~ 1/2, then ~ q , p ( H ) = ,~(H). The case 1/p - 1/q < 1/2 splits into two subcases: On the one hand, ~q,p(H) -- ~13r,2(H ) = ~r(H) for 1/r - 1/2 = 1/p - 1/q and 0 < p ~< 2. On the other hand, it can be shown that ~q,p(H) = ~2q/p,q(H) whenever 2 < p < cxz. Some of these results are proved by interpolation theory [138]. However, the inclusion ~q,p(H) C ~J2q/p,q(H) was obtained by considering random matrices; [9] and [10]. Use of probabilistic techniques to prove existence results has become a staple in the arsenal of functional analysis. 5.12. Ideal quasi-norms ot and fl defined on different ideals 9A and ~3 cannot be compared in general. The situation improves when the considerations are restricted to finite rank operators. Then we always find constants cn ~> 1, which may also depend on a and fl, such that

or(T) <~cn fl(T)

whenever rank(T) ~< n.

It is of interest to know best possible values of the numbers Cn, because their asymptotic behaviour may be used to measure the distance between a and ft. Here is a short list of known and useful relations from [24], which hold for operators with rank(T) ~< n:

Otp(T) <~n llp-1/q Otq(T)

i f 0 < p < q ~< cx~,

r

if0
~ n 1/p []T]]

1,

tp(T) ~ n max{1/p'l/2} []T[]

i f l ~< p ~< c~,

tp(T) ~ n jl/p-1/21 lrp(T)

i f l ~< p ~< cx~.

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It is worthwhile to point out some analogies with other fields. For example, if x -- (~k) has at most n non-zero coordinates and 0 < p < q ~< o~, then H61der's inequality implies that [[xllp[I <~ n 1/p-1/q [[xllq[I. For trigonometric polynomials P with deg(P) ~< n and 1 ~< q < p <~ c~, NikolskiY's inequality says that [[PILp[[ <<.Cp,q n l/q-lIp [[PILq [[. As in the theory of function spaces such results can be used to prove embedding theorems. Applying the inequality zr2(Ak) ~< 2k/2[[A~[[ to a dyadic (2,1)-decomposition (3.5a) gives

oo oo ~2(T) ~ Z ~2(Ak) ~< Z 2k/2llAkll" k=0 k=0 Hence 9---[2,1 ~ ~]~2" This result is the best possible. Indeed, it follows from Dt 9-12,q(12, 11) <:~ t E [1,q and Dt 6 ~]32 (/2, 11) <:~ t 6 [1 that 9---[2,1+~ ~ ~ 2 if e > 0; see [137, pp. 64 and 110]. Similarly, ~p(T) ~ n 1/p IIT[[ for rank(T) ~< n and 0 < p ~< 1 yields 9ap _c ~p. 5.13. An ideal quasi-norm a is only defined for operators contained in the underlying ideal 9,1. Trivially, we could put or(T) := (x~ whenever T ~ 9,1. But this is quite unsatisfactory. Petczyfiski [ 125] made a much better proposal, one which we entertain presently. Recall that the 2-summing norm Jr2(T) is the least constant c ~> 0 such that

1/2 ~< c k=l

sup

.(Xk,X*). 2

IIx*II~<1 k--1

for n E N and xl . . . . . Xn E X . If T r ~ 2 ( X , Y), then we can find no constant that works for all n. However, for a fixed n we can always find some c ~> 0 such that the above inequality obtains so long as we consider no more than n members of X. Let g~n)(T) denote the infimum of such constants. What results is a non-decreasing sequence (Jr~n) (T))

with ~ n ) (T) ~< ~

IITll, whose growth measures just how non-2-summing the operator T is. All identity maps of infinite dimensional spaces X are examples of the worst case: g~n) (X) -- X/rn. On the other hand, for the summation operator, ~r~n) ( r ' l l --+ 1~) behaves like log n. The most beautiful and powerful theorem along this line was proved by Tomczak-Jaegermann [ 187]: n'2(T) ~< ~/2jr~n)(T)

wheneverrank(T) ~< n.

6. Lp-factorable operators Standard references: [35, Chapter 9], [135, Chapter 19], [188, w 6.1. So far we have mainly dealt with small ideals; that is, ideals 9,1 for which Ix ~ 9,1 implies dim(X) < oo. Whenever A -- F, good spectral properties as known from the RieszSchauder theory may be expected. Now we turn to the study of large ideals. This means

Operator ideals

465

that A contains infinite dimensional spaces too. Here classification of Banach spaces is the major objective. For example, q32, l gives P2,1, the class of spaces with the Orlicz property; see 5.1. 6.2.

Let 1 <~ p < c~. An operator T : X --+ Y is strictly L p-factorable if there exists A

B

a factorization of the form T : X > Lp (M, #) > Y, where (M, #) is some measure space. Among others, we may take a set IT equipped with the counting measure. From our experience with p-integral operators we know that using Kr T instead of T is advantageous. This leads to the definition of an Lp-factorable operator: K r T :X

a

B

Lp(M, Iz) > Y**. We get the maximal Banach ideal 13p equipped with the ideal norm ~,p(T) :---- infllBll IIAII, where once more the infimum extends over all factorizations as above. If p = cx~, then we factor through a suitable space C(K), which may be replaced by L ~ ( M , #) or l~(]l). However, the ideal of strictly L~-factorable operators is smaller than that of strictly C-factorable operators. Note that 13~ -- 3 ~ . The maximality of 13p is by no means obvious. Deep representation theorems from the theory of Banach lattices, due in the main to Bohnenblust [13] and Kakutani [83,84] in tandem with ultraproduct techniques are called for; see also [95, pp. 131-139]. Whereas it is readily seen that .f~dual ~_p -- 13p,, it requires some doing to verify that 13p, -~.~dual p o g3p,. Moreover, if 1 < p < oc, then t32 ___13p. This follows from the fact that 12 is isomorphic to a complemented subspace of Lp[0, 1]. The theory of Lp-factorable operators is essentially due to Kwapiefi [93]. 6.3. Most important is the case p -- 2. Then L2 (M, #) can be replaced by an abstract Hilbert space, and there is no need of the natural embedding Kr. That is, L2-factorable operators are even strictly L2-factorable. Operators admitting a factorization T : X ~ H --+ Y were already investigated in Grothendieck's R~sum~. Lindenstrauss and Petczyfiski [ 104] referred to such operators as Hilbertian. Based on their work, Kwapiefi showed that T E 13(X, Y) is Lz-factorable if and only if there exists a constant c ~> 0 such that 1/2

~-~ trhkTXk h= l

k=l

C

IlXk II2 k-1

for all n E 1~, unitary (n, n)-matrices (O'hk) and xl . . . . . Xn E X. In this case, ~,2(T) -- infc, where the infimum is taken over all constants c ~> 0 with the above property. The ideal 132 is injective, surjective and symmetric. 6.4. The class kp consists of all Banach spaces X for which X** is isomorphic to a complemented subspace of some Lp(M, #). Lindenstrauss [103] found a space in t-1 that has no complemented copy in any L I(M, #). Bourgain and Delbaen [ 16] have constructed an infinite dimensional Banach space X E k ~ with the Schur property. Hence X admits no complemented copy in any C(K). So, for p = 1 and p -- cx~, passage from X to X** is really needed. Of course, for 1 < p < cxz, all spaces in Lp are reflexive, and we have 12 c Lp. In particular, 12 consists of those Banach spaces that are isomorphic to a Hilbert space.

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In their seminal papers, Lindenstrauss, Petczyfiski and Rosenthal [ 104,105] studied the class s which may be slightly smaller. More precisely, we have 12p U/~2 : Lp and s N /~2 : F for 1 < p < cx~ and p ~- 2, but/~1 : L 1 , / ~ 2 : L2 and s = L~. 6.5.

An application of trace duality leads to s

o r.,_ dpu a l C_

"~p

. There is also an in-

dual - 1

verse formula, s -- 2p o [~13p ] . By definition of quotient ideals, this means that T e ~ ( X , Y) belongs to ,P_,p(X, Y) if and only if TA e "3p(Xo, Y) regardless of how we choose f t inj

dual - 1

X0 and A e ~dual(Xo.,_ p , X). Passing to the injective hulls, we get ,,p -- ~13p o [gtp ] In the setting of spaces, we end up with the following characterization: a Banach space X is isomorphic to a subspace of some Lp(M, #) if and only if, for any choice of the Banach space X0, every operator T e s X) with a p-summing dual is itself p-summing. When p = 2, this yields an isomorphic characterization of inner product spaces, which was the starting point of Kwapiefi's investigations on Lp-factorization [92]. In 5.10 we have stated ~13p'S lack of symmetry. However, the above criterion is based on just this lack! 6.6.

We conclude this section by presenting another type of factorization. An operator A

B

s Y) is called lattice-factorable if Kr" T can be written in the form Kr T : X >L > Y**, where L is a suitable Banach lattice. Passing from L to L**, we may even suppose that the lattice is order complete. The class of these operators, denoted by s is a maximal Banach ideal with respect to the ideal norm Lust(T) := inf IIBII IIA II. As usual, the infimum ranges over all possible decompositions as described above. This theory goes back to Gordon and Lewis [56]. Solving a problem raised by Grothendieck [60, p. 72], they proved that 1-summing operators need not factor through L1. That its, ~31 ~ s A consequence of this result will be discussed in 12.10. On the other hand, we have gtl o s C ~1. 6.7. Banach spaces belonging to Lust are said to have local unconditional structure in the sense of Gordon and Lewis. This term is justified by the fact that, from the local point of view, those spaces behave just like Banach spaces with an unconditional basis. The most prominent examples of spaces X ~ Lust are the disk algebra [124, p. 25] and the Schattenvon Neumann classes ~5p(12) with p r 2; see [56] and [98].

7. Grothendieck's theorem Standard references: [32, Section 14], [35], [135, Chapter 8], [148, Chapter 5]. 7.1. Henceforth, Lp denotes any member of Lp. In particular, we may think of a classical space Lp(M, #). By defnition, the C(K)'s are included in t_~. Let 9,1 and ~3 be maximal quasi-Banach ideals. Then for all Lp and Lq, we have P,.l(Lp, Lq) c_ ~ ( L p , Lq) if and only if P,.[(lp, lq) c ~ ( l p , lq). This, in turn, occurs pren cisely when fl(T) <, cot(T) holds for T e ,g,(lnp, lq) and n - 1,2 . . . . . where c ~> 1 only

7.2.

depends on ot and ft.

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By far, the most striking result of the theory of operator ideals is Grothendieck's theorem, which he obtained as a corollary of his thdorkme fondamental de la thdorie mdtrique de produits tensoriels [60, p. 59]. It can be stated in various ways. The usual 7.3.

form is

t2(L1, L2) = gj31(L1, L2). We prefer, however, to work with inequalities, since this point of view yields additional information about the involved constants: (G1)

~ l ( T ' l 7 --~ l~) <~cllT'l 7 ~

z ll,

(G2) (G3)

Vl ( r "l~ --+ l ~ ) <~ c k 2 ( r "l~ ~ l ~ ) ,

(G4)

,~"a' o,~2(T'l~ -, zg) ~ ~IITZL -+ 1711.

Trace duality tells us that (G1)C~(G2) and (G3)CC,(G4). In order to prove (G3)=~(G1), just embed l~ almost isometrically into some 1u; and (G1)=~(G3) is straightforward. All inequalities hold with the same constant c > 0, and the infimum is denoted by KG; different values obtained in the real and complex case will be distinguished by the superscripts R and C. The exact Grothendieck constants are still unknown; see [32, pp. 174 and 269]: 1.27323... < K~ < 1.40492... < 1.57079... < K~ < 1.78222 . . . . Further we have the estimates

(gl) (g2)

~ ( T ' l ~ ~ l~) ~<~11T "l~ ~ ~l"

The later can be restated as the little Grothendieck theorem:

~(L~,L2)--~2(L~,L2). Obviously, (G1) implies (gj). The equivalence (gl) r rr2(S) for S E/3(1~) and the formula

~2(T)- sup{rrz(TA)" [IA'l~

(g2) follows easily from N2(S*) =

~ E[ ~ 1, n ~ 1~l,

which holds, in particular, for all operators T : E --+ F between finite dimensional spaces. c are known: KgR _ ~ / z r / 2 In this case, the best possible constants, denoted by KgR and Kg, 1.2344... and Kgc _ v/T/zr _ 1.1284 . . . . Finally, we deduce from (G4) that (H)

X2(T'I~ ~ iT) ~ c[IT.z ~ ~ zT[[,

2 ~ KG ~< K 2KH ; see [32, and the smallest constant will be denoted by KH. T h e n Kg p. 173]. It would be interesting to know whether KH is less than KG.

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7.4. Grothendieck's theorem can also be stated in the form ,~2 o ,~1 ~__ ~1~1" We even have ,~2 : ~ 1 o ~11 . In particular, s X) -- ~131(11, X) implies that X e / 2 . On the other hand, let ~5'Eb " - s o g t 1. Then X e GTh if and only if s = ~31 (X,/2). In this case X is said to satisfy Grothendieck's theorem. Kislyakov [86] and Pisier [143], [148, Chapter 6] showed that this property is shared by all quotients of L1 modulo a reflexive subspace N. If, moreover, N is infinite dimensional, then L 1/N ~ L1. Hence L1 is properly included in GTh, and we have obtained a new and interesting class of Banach spaces. Bourgain [ 15] proved one of the most spectacular results in this area: L 1 ('~) / H1 (T) E GTh. 7.5. Grothendieck's theorem has many applications in Banach space geometry. We present but two of them. A classical result asserts that every Banach space X is isometric to a subspace of some C (K) and to a quotient of some l l (]I). Interchanging the roles, Grothendieck asked: What spaces X are simultaneously quotients of a C ( K ) and subspaces of an L1, say via the surjection Q : L ~ --+ X and the injection J : X --+ L 1? Then J I x Q E ~ ( L ~ , L 1) = ~ 2 ( L ~ , L1). Injectivity and surjectivity of s imply that Ix e s So X is isomorphic to a Hilbert space. Let (x~) be an unconditional normalized basis in l l. Then every x E l l admits a unique representation x -- ~ = 1 ~x~ such that the right-hand sum converges unconditionally. From l l e P2,1 it follows that x w-> (~k) takes l l into 12. By Grothendieck's theorem, this map is 1-summing. Hence ( ~ ) e ll. Conversely, every sequence ( ~ ) E l l defines an element x -- ~ - 1 ~kx~ in 11. So all unconditional normalized bases of l l are equivalent to the standard basis.

8. Concrete operators 8.1. Determining precisely when a concrete operator of prescribed form belongs to a given ideal often provides considerable enlightenment about the operator, its domain and codomain, and the ideal. For example, every continuous kernel K defined on the unit square induces a nuclear integral operator g(t) ~ fd K (s, t)g(t) dt from C[0, 1] to itself, but may fail to be nuclear as an operator from L2[0, 1] to itself. We are going to present a small but tasty selection of concrete operators, which make the subject of ideal theory more attractive. 8.2. The simplest examples are identity maps I'lnp --+ lq. These operators belong to t/ every quasi-Banach ideal 9.1 and o t ( I ' l p/7 ~ lq) is well-defined for the underlying ideal quasi-norm ct. The asymptotic behaviour of these quantities as n --+ oo offers valuable information, in particular, when we compare different quasi-Banach ideals. Here are two typical examples:

Vl

.

(I " lp --+ l

q) = {" 1-1/p+l/q 11

rtp(I " lq, ---->lq) ~ (n logn) 1/q

i f l <<.q <~ p <<.cx:, i f l <<.p <<.q <<.c~. ifl~

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O p e r a t o r ideals

8.3. Next, consider diagonal operators Dt : (~k) w-~ (rk~k) generated by a sequence t = (rk) E c0. If 1 ~< q ~ 0. The limit order )~(9A, p, q) of an ideal 9A is defined to be the infimum of all )~ > 0 such that D ~ E 9d(lp, lq). Note that Dt E Pd(lp, lq) whenever t E lr, where 1/r > )~(gA, p, q). If 9A is a quasi-Banach ideal with respect to a , then n )~(9A, p, q) - inf{)~ > 0: ot(I'Ipn --+ lq) <. cn z for some c > 0}.

As an example, we give the limit order )~(~13r , p, q). Viewed as a function of 1 / p and 1/q, it is illustrated in the following diagrams [135, pp. 312-313]" l~
2

1/q

l/q

8 9~§

l/r

~1 - ~1 + ~ 1 1/2

I,

l/r I

p*

p*

1/p

lip

D,

l/r*

l/r*

_ )~

"

1 _jr_ ( l / r * - l l p ) ( l l q - 1 / r ) l12-11r

r

8.4. Closely related to diagonal operators are embedding maps of function spaces. As indicated in 1.10, properties of I ' W 2 (s ~ L p ($72) can be used in the theory of partial differential operators. Given a b o u n d e d domain s in R n with a smooth boundary, )~ > 0 and 1 ~< p < c~, then Wpz (S-2) denotes the Sobolev space if )~ E N and the SlobodetskiY

space if )~ r N. If p - c~ and 0 < )~ < 1, then we mean by W)(S-2) the space of all H61derLipschitz continuous functions: CZ(s The limit order )~se(PA, p, q) is d e f n e d to be the infimum of all )~ > 0 such that the e m b e d d i n g map from Wpz (S-2) into Lq (S-2), which exists whenever )~/n > m a x { 1 / p - 1/q, 0}, belongs to 9A. It was shown by K6nig [88] that, for quasi-Banach ideals, both limit orders are connected as follows:

)~se(PA, p, q) - )~(9A, p, q) + 1/p - 1/q. R o u g h l y speaking, the inclusion I E f21(W~p(X-2),Lq(S-2)) and the asymptotic estimate

ot(I "lpn __+ lq)n < c n ~- 1/p+ 1/q for n - 1, 2,. .. are 'almost' equivalent. 8.5.

Let (M, # ) and (N, v) be ~r-finite measure spaces. If 1 ~< p, q < ec, then a Hillex N --+ C such that

Tamarkin kernel is a / z x v-measurable function K ' M 1/p (fNIfM

IK(s, t)lq d v ( t ) l P / q d # ( s ) )

< (X).

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Then

TK "g(t) --+ fN K(s, t)g(t) dr(t) defines a p-summing operator from Lq.(N, v) into Lp(M, #); for p -- q -- 2, we get the classical Hilbert-Schmidt operators. 8.6.

The convolution operators

Tf "g(t) ---->f 9 g(s) -- ~

mf+

f (s - t)g(t) dt

are especially important examples of integral operators 9If f 6 C (qI'), then it follows from T f 9L l (7~) r f> C (7~) I > L1 (T) that Tf E 9tl (L1 (T)) C ~2(L1 ('IF)). For f E Cz(T) we can do better. Approximation by trigonometric polynomials gives En ( f ) ~< c n -z, which in turn implies aen(Tf" LI(T) -+ C(T)) ~ c n - A . Hence Tf E 9132 o 9.lp,c~(Ll(T)) with 1 / p - - )~. Finally, employing the theory of eigenvalue distributions [90, pp. 213-214], [137, p. 310], we arrive at a well-known result of Bernstein [ 11 ]" if f E C z (~) and 1/2 < ~. < 1, then f has an absolutely and uniformly convergent Fourier series.

8.7. Let 0" D -+ D be a analytic function on the open unit disk D. Then the composition operator

Co " f (w ) --> f o O(z) "- f (d#(z)) maps every Hardy space Hp (D) into itself. Shapiro and Taylor [ 180] were the first to find criteria that guarantee that a composition operator belongs to a given ideal. For instance, if (1 -14~1) - 1 has integrable boundary values and 1 ~< p < oo, then C 4) ~ O43p(Hp(D)). In the case 2 ~< p < oo and only in this case the condition above is also necessary. We know that Ce maps Hp (D) into H ~ (ID) if and only if ~b(II)) is contained in a disk r D with radius r less than 1. An equivalent property is the compactness of C~ : H ~ (D) --+ H ~ (D). Moreover, under this assumption, C4):Hp(D ) ~ Hp(D) is nuclear and even a member of 9.1o. Using Nevanlinna's counting function

N4)(w)'--

Z

l~

~1

for w --# 0 (0)

ZEr (tO)

Shapiro [ 178] answered the question of how much r has to compress ID into itself in order to insure that C~ compresses bounded subsets of Hp (I3) into relatively compact ones: Cr ~ ~(Hp(]D)) r

lim Nr Iwl~ 1 log

1

Iwl

=0.

Operator ideals

471

We stress that the right-hand condition is independent of 1 ~< p < ~ . In the case of H2 (D), this criterion was extended by Luecking and Zhu [107]: dA(w)

Ccb E ~r(H2(D)) ~ fD[Ncb(t~ r/2

(1 - Iwl2) 2

log ~

<ex~

if0
where dA(w) denotes the area measure. The symbol 4~ of a compact composition operator C 4, always has a unique fixed point z0 6 D such that I~P'(z0)[ < 1. If 4~'(z0) r 0, then the spectrum consists of 0 and a rapidly decreasing sequence of simple eigenvalues: 1, 4~'(z0), 4r (z0) 2 . . . . . Otherwise, we only get {0, 1}. Composition operators have been investigated in other spaces of analytic functions. In many respects, Bergman spaces Ap(D) can be used to replace Hardy spaces and, sometimes, they behave better [192, p. 23]. The most general result obtained so far is due to Domenig [39]. He constructed for every analytic function 4) :D --+ D and 1 ~< p, q < a sequence t 6 [~ such that, for every maximal Banach ideal 9,1, the statements C~ 9A(Ap(D), Aq(D)) and Dt c 9,1(lp, lq) are equivalent. For more information about composition operators, we refer to [28,29,76,177] and [ 179]. 8.8. Specific components of Banach ideals provide a useful arsenal of non-classical Banach spaces. The most prominent examples are the Schatten-von Neumann spaces ~r(/2), which fail to have local unconditional structure for r ~ 2; see 6.7. The components ~r(lp, lq) were also investigated; see [100,149] and [172].

9. Rademacher type and cotype Standard references: [35, Chapter 11], [113, Part II], [135, Chapter 21], [139, Chapter 4], [148, Chapter 3], [175], [188, w 25]. 9.1. Let (rk) be the sequence of Rademacher functions defined in 5.4, and assume that 1 ~< p ~< 2 ~< q < oo. Then Parseval's equality (5.4b) implies that

I~klq

dt

k=l

~<

k=l

I~k]p k=l

t

Just as in the theory of summing operators, we may ask what happens when the scalars ~l . . . . . ~n E K are replaced by arbitrary elements Xl . . . . . xn of a Banach space X and the absolute value l-I by the norm I[. 11. The upper and the lower part of the preceding inequality are now used separately to introduce new classes of operators. An operator T 6 ~ ( X , Y) is said to have Rademacher type p if there exists a constant c/> 0 such that, for any choice of n 6 1~ and Xl . . . . . xn E X,

• Is0

k--1

Txkrk (t)

2)j2 dt

~< c

[Ixk[Ip k=l

t

.

(9. la)

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We put p'rp(T) "= infc, where the infimum is taken over all constants c ~> 0 with the above property. The class of these operators, denoted by 9lff;p, is an injective, surjective and maximal Banach ideal. On the other hand, the inequality 2

IITxk IIq

~< c

k=l

k--1

1/2

d'1

~xkrk(t)

(9.1b)

yields operators of Rademacher cotype q. An injective and maximal Banach ideal, denoted by 9lffq, is obtained in this way; the ideal norm prq being defined in an analogous fashion. We readily see that 9q;'Ep~ C ~C~p2 if 1 ~< P2 < pl ~< 2 and ~C~ql C ~C~q2 if 2 ~< qi < q2 < oo. Clearly, 9~ff71 = .!3, and it is convenient to put 9~ff~ : - s 9.2. The Kahane inequality is obtained by extending the classical Khintchine inequality to the vector-valued case" for 0 < s < ec, there are constants As and Bs such that, regardless of the Banach space X, the number n 6 N and the elements x l . . . . . Xn ~ X, we have

As

(s0

2

xkrk(t) k=l

1/2

d't

<,

(s0 (s0

xkrk(t)

k=l

<~ Bs

s),s II ) dt

2

~_~xkrk(t) k=l

1/2

dt

Consequently, the quadratic average in the above definitions can be replaced by any Lsaverage. Sometimes, it is advantageous to use the same exponents on both sides of (9. l a) and (9. l b), respectively. Equivalent, but non-normalized ideal norms are produced in this way on 9l~gp and 9lffq. 9.3. Our definitions can be repeated with random variables other than the Rademacher functions. For instance, take a sequence of independent standard Gaussian random variables defined on some probability space, and use in (9.1a) and (9.1b) quadratic Gaussian averages. Then the preceding strategy leads to the Banach ideals ~ g p and ~Sffq. The operators so obtained are said to be of Gaussian type p and Gaussian cotype q, respectively. As for type, there is nothing new: ~5ffp = 9l~gp, with equivalent norms. On the other hand, in the case of cotype, we only get l~l~q C 9cl;l~q C ~l~q§ for e > 0. Further variants of type are obtained from any sequence of independent, identically distributed s-stable random variables with 0 < s < 2 [97, Chapter 5]. 9.4. Investigations on type and cotype originally started on the level of spaces. In this setting we have not only GTp -- a T p, but also GCq -- aCq. It is readily seen that

L p e RTp if 1 ~ p ~< 2

and

L q E RT2 if 2 ~< q < oo,

(9.4a)

L p E a c 2 if 1 ~< p ~< 2

and

Lq E RCq if 2 ~< q < oo,

(9.4b)

Operator ideals

473

and these statements are best possible. Moreover, ~,-~dual C ~ p * ; this inclusion is strict, since l l E RG2, but neither co nor l ~ have Rademacher type for any 1 < p ~< 2. From l l 6 RC2 we also conclude that ~ q fails to be surjective. It can be shown [185] that ~ l ~ q C ~ q , l " We even have RCq -- Pq,1 for 2 < q < oo, while RG2 C P2,1. Therefore (9.4b) generalizes a classical result of Orlicz [ 118]" L p E P2,! if 1 ~< p ~< 2

and

L q E P q,1 if 2 ~< q < c~.

9.5. The notions of type and cotype emerged around 1972: 9 Hoffman-JCrgensen [68] studied sums of independent vector-valued random variables. 9 Dubinsky, Petczyfiski and Rosenthal [40] showed that all operators from C(K) into a Banach space X are 2-summing if X* has subquadratic Gaussian averages. 9 Maurey [109] proved that, for 0 < p < 1, all ideals ~13p coincide, and, in particular, 9 Kwapiefi [94] characterized inner product spaces via an abstract two-sided Khintchine inequality. These are the historical roots of the pioneering work of Maurey and Pisier [112] who introduced type and cotype as we use it today. 9.6. In our language, Kwapiefi's amazing Hilbert space characterization takes the form of a formula: RT2 A RC2 = L2. In terms of operator ideals, a more general version is available: ~ I ~ 2 o ~lq~'7~"2 = ~2- In particular, Maurey showed that every operator from a Rademacher type 2 space into a Rademacher cotype 2 space factors through a Hilbert space. This implies that ~ ( L q , Lp) = ,~2(Lq, Lp) whenever 1 ~< p ~< 2 ~< q < oo. Luckily, since every such space L p embeds into some L1, the excluded case q = cx~ is covered by Grothendieck's theorem: ~ ( L ~ , L1) = ~ 2 ( L ~ , L1). In order to get a unified approach, one may ask whether ~ ( X , Y) = ~ 2 ( X , Y) if X* and Y are of Rademacher cotype 2. Under the additional assumption that X or Y has the approximation property, Pisier [ 144], [ 148, p. 41 ] gave indeed an affirmative answer. Moreover, if X is a C*-algebra, then X* 6 RG2, and the required formula even holds for all Y ~ RG2; see [ 186,75]. However, in the general case, Pisier [146], [148, p. 143] exhibited a counterexample. 9.7. If ~ ) ~ 1 ~ "-- ~ 2 o ,~x~1, then DPR is the class of all Banach spaces Y such that , ~ ( L ~ , Y) -- ~ 2 ( L ~ , Y) for every L ~ , and the Dubinsky-Petczyfiski-Rosenthal theorem says that RT~ _ DPR. Maurey [110, p. 116] improved this result by showing that RC2 ___ DPR. On the other hand, DPR _ P2,1. Since RC2 is a proper subclass of P2,1, we only know that DPR belongs somewhere in between. In the case when Y E RCq and 2 < q < oe, a weaker result is true:

~ ( L ~ , Y) -- ~ q , 2 ( L ~ , Y) = ~,~q+e(Lc~, Y) if e > 0, but not necessarily if e - 0.

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9.8. The theory of Rademacher cotype and that of p-summing operators are intimately related due mainly to discoveries of Maurey [110]. Here are the basic results underlying these relations [35, pp. 222-224]" ~ 1 (X, Y) ~---~ 2 ( X , Y)

if X 6 FIC2 and Y E L,

~.~I (X, Y) -- ~.~p(X, Y)

if X 6 FiCq and Y E L, where 1 < p < q* < 2,

~ 2 ( X , Y) -- ~ q (X, Y)

if X E I and Y E FIC2, where 2 ~< q < oo.

Consequently,

~'~1 (X, Y) = g3p(X, Y)

if X, Y E FIC2 and 1 < p < oo.

We stress that, by trace duality, the relation ~ I ( X , Y ) - ~ 2 ( X , Y ) t~ec(Y, X) = ~132(Y, X). Hence 13(L~, X) = ~ 2 ( L o o , X) if X 6 FIG2.

translates into

9.9. In 5.13 we have associated with ~r2 the sequence of the ideal norms ~r~n) , which are defined for all operators. This construction will now be repeated in the case of pr2. Let (n) p32 (T) "-- infc, where the infimum is taken over all constants c >~ 0 such that inequality (9.1a) holds for p -- 2 and fixed n. Then

pr~ n) (T) <. v/-ff IITII. The

operators T with

limn~c~ p'r 2(n) (T)/~/-ff = 0 form the injective, surjective, symmetric and closed ideal 9q2E. An operator T E s Y) does not belong to 9~t~E if and only if there exists a constant c ~> 0 with the following property [5]" For n -- 1, 2 . . . . . we can find elements xl n),..., X(nn) ~ X such that

~ k=l

~x~ ~

whenever ~1 . . . . . ~n ~ K. k=l

k=l

This implies that Mn "--span{xl n) . . . . . X(nn) } and span(Txl n) ..... Tx~ n)} are uniformly isomorphic to l~', and cllTxll >>Ilxll for all x ~ M,,. In the setting of spaces, X r FIT means that X contains the l~ 'S uniformly. Spaces in FIT are also referred to as B-convex; see also 11.5. On the other hand, we conclude from 1.18 that an operator T E s Y) does not belong to O lt if and only if there exist a constant c >t 0 and a sequence of elements x t, x2 . . . . ~ X such that oo

k=l

OQ

~< ~-~ I~kl ~
~k Txk

The criteria above and the formula 9~t~E -and global concepts.

whenever (~k) E l i .

~[~uper illustrate

Already Pisier [140] observed that the sequence

P~2(mn) (ST) <<.plr~m) (S)p'c~n) (T)

(pr~n))

the connection between local

is submultiplicative:

for T E ~ ( X , Y) and S 6 ~(Y, Z),

Operator ideals

475

and he employed this to prove U1 0 such that, for any choice of n 9 1~ and x l . . . . . Xn 9 X,

dt

~< c sup (xk,x*) 2 IIx*ll<~l k-1

We put 7rp (T) := infc, where the infimum is taken over all constants c ~> 0 with the above property. The class of p-summing operators, denoted by ~3p, is an injective and maximal Banach ideal. We have ~13p ___~]3p if 1 ~< p < cx~, and ~132(X, Y) = ~3p(X, Y) if X 6 L and Y e RC2. Replacing the Rademacher functions by Gaussian random variables yields ~]3• the Banach ideal of y-summing operators. In fact, we obtain nothing new: !:13• = ~13p. The underlying ideal norms ~r• and rrp are different, but equivalent. For every finite dimensional Banach space E and e > 0, the Dvoretzky dimension D(E, e) is defined to be the largest n such that we can find an n-dimensional subspace En of E whose Banach-Mazur distance to l~ is less than 1 + e. The main result, implicitly proved by Pisier [147, Chapter 1] says that

a(e)r~y(E) <~v / D ( E , e) <~b(e)rry(E), where the constants a(e) > 0 and b(e) > 0 do not depend on E; see also [113]. The following asymptotic relations are due to Linde and Pietsch [ 102], while the corresponding results for D(l N, e) were found by Figiel, Lindenstrauss and Milman [49]:

rr• N) ~ N 1/2 if l ~

and

rr•

N)~N

1/q if 2 ~
Moreover, Jr• (l~) N • x/log N. This is the worst case, since rr• (EN) ~ C x/log N for all N-dimensional Banach spaces E N and a universal constant c > 0. As an immediate consequence we obtain the celebrated Dvoretzky theorem [43]: every infinite dimensional Banach space contains almost euclidian subspaces of finite dimension as large as we please. 9.11. The S~minaire Laurent Schwartz 1969/70 was devoted to applications radonifiantes. This subject is based on the fact that cylindrical measures on Banach spaces may behave very badly. The most prominent example is the standard Gauss measure Y on L2 [0, 1], which fails to be ~r-additive. So the question arose whether the quality of a cylindrical measure improves under certain transformations. It can be shown that the integration

operator S" f (t) --+

f0 Sf (t) dt

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is y-summing as a map from L2[0, 1] into CZ[0, 1] whenever 0 < ~ < 1/2 and, as a consequence, it turns out that the Wiener measure, which is the image of ~, under S, extends to a Radon measure on the Borel a-algebra of CX[0, 1].

10. U M D operators, Haar type, and uniform convexity Standard references: [33, IV], [139, Chapter 7, Section 8.5]. For k -

10.1. by

X~j) (t)

1, 2 . . . . and j - 1, 2 . . . . . 2 k - l , the Haar functions are defined on [0, 1)

:=

2j-2

2j-1

-+-2(k-l)/2

if

--2 (~-~)/2

if 2j -- 1 < ~ t < - -2 j 2~ 2 k' otherwise.

0

2k

~< t <

2~

In order to get an orthonormal basis of L2[0, 1), the function X0~l) (t) :-- 1 must be added. An X-valued Haar polynomial (of length n), n

2 ~-1

+ZZ

,

k=l j=l is a function f , " [0, 1) --+ X that takes constant values on the dyadic intervals A~j) := [

,

) with j = 1 . . . . . 2n. In other terms, f , must be Dn-measurable, where Dn denotes

the algebra generated by

A (1)

, ..

A(2") . . , ,._.~n

Writing

m

fm " = ~ d k

with do "-- x 0( 1 ) and dk

"--

--2k-I Z j = I x~J)x~J)

k=0

yields a Walsh-Paley (dyadic) martingale (fo .... , fn). Indeed, if m < n, then fm is the conditional expectation of fn with respect to 79m. The functions do . . . . . d, so obtained are said to be martingale differences. 10.2. Let 1 0 such that

LTd,(t) k=0

I1I (SoL dt

e~d~(t)

<<.c

1/p dt

(10.2a)

k=0

for n E N, any choice of X-valued martingale differences do . . . . . dn and arbitrary signs eo,..., en = -+-1. The class of these operators, denoted by )AgJIX), does not depend

Operator ideals

477

on p and is an injective, surjective, symmetric and maximal Banach ideal; the ideal norms II.p(T) "-- infc being equivalent, but non-normalized for p 7~ 2. Burkholder [21] showed that I X p ( I K ) - m a x { p - 1, p * - 1}, which is the unconditional constant of the Haar basis in Lp[O, 1); see also [192, p. 63]. The notation UMD stems from the fact that inequality (10.2a) expresses an unconditionality property f o r martingale differences. UMD spaces were considered for the first time by Maurey [ 111, Expos6 2]. He already observed that L p E UMD if 1 < p < cx~. However, the real break-through only came a few years later. In a series of papers Burkholder [18,19,22] found a characterization by a geometric property, called ~"-convexity, and he discovered the connection with the vectorvalued Hilbert transform to be discussed in 11.4. 10.3. Let 1 ~ 0 such that /7

y ~ Tdk (t)

<. c

(s01

Ildk(t)ll p dt

(10.3a)

k=0

k=O

for n E N and any choice of X-valued martingale differences do . . . . . dn. The class of these operators, denoted by 9A'Ep, is an injective, surjective and maximal Banach ideal. Its ideal norm OtTp(T) "-- infc is non-normalized for p 7~ 2. Clearly, ~1'~1 - - ,t~. Setting 2k-1

do

.

0.

and dk . .

2-(k-1)/2 Z

-

XkXk

"(j)

--

Xkrk

with Xl . . . . . x/7 E X,

j=l

inequality (10.3a) is equivalent to (9.1a). Hence 9.1~gp __ 9Vgp. The summation operator I7 :ll --+ l ~ shows that this inclusion is proper. Indeed the difference is quite large, since there exist spaces in FIT2 that have no Haar type p > 1; see [72] and [ 150]. Put a~/7) (T) "-- infc, where the infimum is taken over all constants such that inequality (10.3a) holds for p -- 2 and fixed n. Then a~/7) (T) ~< ~/n + 1 lITII. The operators T with limn--,~ ot~/7) ( T ) / ~ / B -- 0 form the injective, surjective, symmetric and closed ideal 9.1ff7. If 1 < P2 < Pl ~< 2, then 9A'Epl C ~'['~p2 C ~.1'~. As in the context of Rademacher functions, a Banach ideal ~9.11~qformed by operators of Haar cotype q can be introduced for 2 ~< q < ec, and a closed ideal 9.1ff is obtained in analogy with 9~ff. But we now encounter a situation of immaculate duality: p~l . . .r'Cf . dual p - 9 A f f p . and p_l . . .ord . qual --9.1~gq,. In the limiting case, we even have 9~l"~dual _ f21 ~ - - ~ 1 ~ . - - 9.1 l~. dual .

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10.4. The order relations between the ideals treated in Sections 9 and 10 are illustrated in the following diagram, where the arrows indicate strict inclusions:

~'~2

~'~P

)-~J~

Pdl~q

~'7~"2

~lq~ 2

~,[l~2

10.5. Let 1 ~~ 0 such that l l y + Txll p + IlY- Txll p 2 Let 2 ~< q < cx~. An operator T 6 s constant c ~> 0 such that

_ ilyllP)1/p

cllx II for all x 6 X and y 6 Y.

Y) is called uniformly q-convex if there exists a

x++x_

T x + - T x - II <~c ( llx+ llq -+2

q)l/q

for all x+ 6 X.

10.6. We are also interested in the limiting cases of p-smoothness as p--+ 1 and of q-convexity as q --+ ~ . An operator T e s Y) is called uniformly smooth if for every e > 0 there exists 6 > 0 such that IlY+ Txll + I l Y - Txll

- 1 ~ EIIxll

whenever IlYll- 1 and Ilxll ~ ~.

An operator T 6 ~ ( X , Y) is called uniformly convex if for every e > 0 there exists 6 > 0 such that

Tx+2

ii~<e

whenever

Ilxi II -

1 and

2 x++x ii> ~ 1 - 3 .

10.7. The operators just defined form one-sided ideals that are related to the Haar type and cotype ideals in the following way: 9 T ~ 9A~Ep(X, Y) if and only if Y admits a renorming such that T becomes uniformly p-smooth.

Operator ideals

479

9 T E PA~q(X, Y) if and only if X admits a renorming such that T becomes uniformly q-convex. These criteria remain true in the limiting cases as well. 10.8.

The following examples should be compared with (9.4a) and (9.4b).

Lp E ATp if 1 ~< p ~< 2

and

Lq EAT2 if2~
(10.8a)

L p E A02 if 1 < p ~< 2

and

Lq E AC,q if 2 <~ q < cx~.

(lO.8b)

The easy parts follow from Clarkson's inequalities I l f + g l L p l l p-+- IIf - gltpll p ~ 2(llflLpll p + llglLpll p)

ifl~p~2

and IIf + gltq IIq -+- IIf - gltq IIq ~ 2(llfltq IIq + Ilgltq IIq)

if 2 ~< q < cx~,

whereas the rest is more involved. 10.9. It turns out that PA~ = PAX coincides with ~ s u p e r , the ideal of super weakly compact operators. The local nature of this concept is stressed by the following criterion: An operator T E ~ ( X , Y) fails to be super weakly compact if and only if it factors n An> T Bn> the finite summation operators uniformly. That is, Z'n "l 1 X ~Y In such that SUPn IIB~ II IIAn II < ~ , where

~:. : ( ~ ) k=l

l <~m<~n

The global counterpart of this property was already discussed in 1.12 when we observed that the infinite summation operator Z :ll --+ l ~ is C~-universal. 10.10. The historical starting point of all renorming results stated above is James's theory of super reflexivity [71]. Enflo [45] showed that every super reflexive space can be given an equivalent uniformly convex norm, and Pisier [142] developed a quantitative approach in terms of Walsh-Paley martingales. Moreover, Beauzamy [6] introduced the ideal of uniformly convexifiable operators. The inclusion 3.193ED C 9A~Eextends a result of Pisier [ 141, Annexe 2] which says that every UMD space is super reflexive. 10.11. As already mentioned in 1.15, the (C(K), Y)-components of ~3 and ~/3' coincide. Diestel and Seifert [36] and Pe~czyfiski [120] observed that the same is true for the ideals ~3~5 and ~5c0 defined in 1.16 and 1.18; see also [37, p. 160]. This list can be enriched by adding the corresponding super ideals.

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Let 9,1 be a quasi-Banach ideal, then an operator T 9 s Y) belongs t o ~inj, the injective hull of 9,1, if and only if there exist an operator To 9 s Y0) and a function N:IR+ --+ IR+ such that IITxII ~ N(e)llToxll-4- ellxll --inj

see [74, pp. 473-476]. Obviously, ~1

for x 9 X and e > O; ---~inj

__C"'2 ~ ~

-----~]'' However, using Niculescu's

characterization [117] of weakly compact operators on C(K), we get -'~[nJ(c(K), Y) -~ 3 ( C ( K ) , Y) for all Banach spaces Y. Hence the (C(K), Y)-components of the ideals ~. inj -~ inj ~J1 , ""2 , ~

and ~ 3 coincide. This observation connects, among others, the non-

ultrapower-stable ideal ~

with the ultrapower-stable ideal

~. inj D 1

.

So

"" T 9 ~Z~3(C(K), Y) ==~ T 9 ~;nJ(C(K), Y) ==~T u 9 ~J1. i n J ( c ( K ) t ~ y U ) :=~ TU 9 ~ ( C ( K ) U , yU) :=~ T 9 ~:Ijsuper(c(K), Y),

and all weakly compact operators on spaces C(K) are even super weakly compact. Moreover, using the ideal 9 l ~ defined in 9.9, Beauzamy [7] showed that ~]1~super C ~]~ 0 ~J~"~ C f13~ C ~ . Hence ~ j ~ s u p e r _ _ ~ s u p e r . Let X be a reflexive quotient of C(K), and fix a surjection Q : C ( K ) --+ X. Then Q e ~ 3 S"per, which in turn implies that X is super reflexive. However, super reflexive spaces have finite Rademacher cotype. Therefore Q is q-summing and hence strictly q-integral for some 2 ~< q < oo. In conclusion: we find a probability measure # such that X becomes a quotient of Lq (K, #). This is an ideal theoretic proof of the dual version of a theorem due to Rosenthal [158].

11. Extension of classical theorems to vector-valued functions

Standard references: [37,135]. 11.1. A major goal of functional analysis consists in investigating how much of scalar variable theory might be extended to functions taking their values in Banach spaces. Some examples have been discussed earlier: the relation between absolute and unconditional convergence of series, the theory of Rademacher type and cotype as well as its counterpart in the context of Haar functions. Ensuring the validity of classical results in the vectorvalued setting often requires to impose quite strong restrictions on the underlying Banach spaces. In this section, we will address such problems with respect to the Fourier transform, the Hilbert transform, the Riesz projection, the Rademacher projection and the RadonNikod3~m theorem. 11.2. As one of the first steps Bochner [12] developed his Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind. For 1 ~< p < oe, we denote by [Lp(M, #), X] the Banach space of all Bochner p-integrable X-valued functions defined on a a-finite measure space (M, #); see also [37, Chapter II]. As usual, functions are

Operator ideals

481

identified when they coincide almost everywhere. In a natural fashion, the algebraic tensor product L p ( M , #) @ X can be viewed as a (dense!) subset of [Lp(M, #), X]. Some of the deepest questions in Banach space theory have their roots in the following simple situation: if S : L p ( M , #) --+ Lq(N, v) and T :X --+ Y, then

Z

fk | x~ w-~

k--1

Sfk @ Txl( k=l

defines a map

S | T : L p ( M , #) @ X --+ Lq(N, v) @ Y; when does there exist a continuous extension

[S, T] : [Lp(M, #), X] --+ [Lq(N, v), Y]? 11.3.

The Hausdorff-Young theorem asserts that the Fourier transform

f'" f (t) ~ fA(s) "-- F

~ f (t) e ist dt OG

induces an operator from L p (R) into L p. (]R) if 1 ~ p ~< 2. Accordingly, T 6 E(X, Y) is said to have Fourier type p if the extension [9r , T] exists. These operators form the injective, surjective, symmetric and maximal Banach ideal ~57p. Its ideal norm t#'rp(T) := 11[f', T]:[Lp(IR), X] ~ [Lp, (IR), Y]]], is non-normalized if 1 < p ~< 2. Babenko and Beckner [8] have computed the exact value of ~'rp(/K) = I1~-: Lp(]R) ~ Lp*(]R)II. The following characterization is more flexible and displays, in particular, the local nature of the concept under consideration. An operator T 6 ~ ( X , Y) has Fourier type p if and only if there exists a constant c ~> 0 such that, for any choice of n 6 1~ and x l . . . . , Xn ~ X,

(1

fi

Txk e ikt

dt

~< c

k-1

IIx~ IIp k--1

Putting 4}1:p(T) "-- infc yields an equivalent ideal norm. Banach spaces of Fourier type were first considered by Peetre [ 119] who used tools from interpolation techniques to show that L p and Lp. are contained in FTp. The fundamental Kwapiefi theorem [94] asserts that / 2 -- FT2. However, it seems likely that ~2 is a proper subclass of ~ff72. 11.4.

The periodic Hilbert transform is defined by

_

7-/" f (t) ~-+ f (s) "-- ~

f (t) cot 7/"

(s_,) 2

dt.

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If f e L p (~) and 1 < p < c~, then the right-hand integral exists almost everywhere in the sense of Cauchy's principal value, and the Riesz theorem tells us that 7-/yields an operator from Lp (qF) into itself. An operator T 6 ~ ( X , Y) is said to be compatible with the Hilbert transform if the extension [7-/, T] exists. In this way we get another injective, surjective, symmetric and maximal Banach ideal ~ E , which does not depend on the choice of p. All ideal norms Xlrp(T) : = 1117-/,T]: [Lp(7~),X] ~ [ L p ( ~ ) , Y]ll are equivalent, but non-normalized for p -~ 2. Pichorides [127] has computed the exact value of X rp(IK) = 117-t: Lp(T) Lp(qF)ll. Local characterizations are available as well. For example, T e ~ ( X , Y) is in ~ E if and only if there exists a constant c ~> 0 such that

( io"

11 Z Txk c o s kt k=l

' )"' ( i0 • dt

~< c

rr

1 --

2

1/2

xk sin kt

k=l

for n e N and Xl . . . . . Xn e X. Again the smallest constant provides an equivalent ideal norm. The same ideal can be obtained by working, instead of the periodic Hilbert transform, with the Hilbert transform on the real line, some discrete versions thereof, or the related Riesz projections. If 1 < p < ex~, then L p 6 HT, but l l, l ~ ~ H'I'. The remarkable Burkholder-Bourgain theorem [20,14] asserts that UMD = HT. We do 9

not know whether this result extends to operators: JI93tX) -- ,9'E. 11.5.

Khintchine's inequality implies that the Rademacherprojection

~" f (t) w-> Z

(fo'

f (t)rk(t) d t ) rk(s)

k=l

maps L p[O, 1) into itself whenever 1 < p < ex~. An operator T e ~ ( X , Y) is compatible with the Rademacherprojection if there exists a continuous extension [7~, T]. The injective, surjective, symmetric and maximal Banach ideal 9~t~13 so obtained does not depend on the choice of p. All ideal norms prtp(T) := [117~, T]:[Lp[O, 1), X] --+ [Lp[0, 1), Y][[ are equivalent, but non-normalized for p # 2. A local characterization reads as follows: an operator T e s Y) belongs to 9~r if and only if there exists a constant c ~> 0 such that

(s0 s

k=l

)

T fn(t)rk(t) dt rk(s)

as

~< c

IIf~ (t) llp dt

for n 6 N and all X-valued Haar polynomials fn; see 10.1. Then the smallest admissible constant coincides with the ideal norm prrp defined above.

Operator ideals

483

Let us return to the relation r_ __p ual C 91ffp, stated in 9.4. Though we do not have equality, there is something like a reverse: 91~dual o 9 1 ~ C 91(~'.p. Moreover, 91gt C 91~. In the setting of spaces, we even have FIP = FIT; a formula which is usually phrased by saying that K-convexity and B-convexity are the same; see [ 145] and 9.9. 11.6. Next, we look for a vector-valued version of the Riesz representation theorem. Given a compact Hausdorff space K and a Banach space X, we want to represent operators A : C ( K ) --+ X by integration against X-valued measures. There are a few problems in the general case, but weak compactness of A is characterized by the existence of a regular X-valued Borel measure m on K such that

A f -- s

f (~)dm(~)

for all f 9 C(K).

Moreover,

A 9 gt 1(C(K), X) ~ m has a finite variation Iml, A 9 011 (C(K), X) ~ m has a derivative with respect to [ml. The latter condition means that there is a function a 9 [L 1(K, #), X] such that

m(B) = fB a(~) dlml(~)

for all Borel sets B in K.

An operator T 9 t2(X, Y) is said to have the Radon-Nikodym property if A 9 ~ 1 (C[0, 1], X) implies TA 9 011 (C[0, 1], Y). The same holds then with any C(K) instead of C[0, 1]. A large variety of equivalent properties is known; see [37, pp. 217-219]. For example, T f T carries absolutely continuous functions f :[0, 1] --+ X to functions g : [ 0 , 1] >X >Y that are differentiable almost everywhere. In this case, we have b

g(b) - g(a) --

fa

g' (t) dt

for0 ~< a
The operators with the Radon-Nikod))m property form an injective and closed ideal, 910t. From l l E FIN and co, loc ~ FIN it follows that 9101 and 9101 dual are incomparable. We have ~ C 9101 C ~ r and ~ C ~r C l~[1. Since all reflexive spaces are in FIN, a deep theorem of Grothendieck [59, p. 132] follows: ~ o ~1 -- 9 t l , isometrically. If ~ e p denotes the closed ideal of all operators with separable range, then ~5r dual C ~r and Stegall [182] proved that 919l duat- ~ r o l~r -1 . In particular, FIN dual coincides with the class of Asplund spaces, which are defined by the property that their separable subspaces have separable duals. The ideal 919t dual behaves better than 919t. It is not only injective, but also surjective; fJlgt dual has the factorization property, whereas 919t even fails to be idempotent; see [ 182, 53]. Moreover, there are C9101 dual -universal operators. For example, we may take the map

484

J.

Diestel et al.

from ll into L ~ [ 0 , 1) given by (~k) ~ Y~'~--1~kXk. Here (Xk) is any enumeration of the Haar functions, normalized with respect to the L~-norm. The difference between 9~9t and ~r disappears when passing to the associated super ideals; in both cases we get ~super. Beginning in 1936 with a pioneering paper of Clarkson [27], the study of the RadonNikod3~m property was mainly carried through on the level of spaces. There is a geometric part based on the concept of dentability [154]; see [37, Chapter VII, pp. 208-219] for a historical survey. Our presentation has emphasized the analytic point of view, which is closely related to vector measures. Operators with the Radon-Nikod3~m property were first considered by Reinov [152] and Linde [101].

12. Operator ideals and tensor products, the approximation property Standard references: [32,34], [37, Chapter VIII], [ 135, Chapter 10], [ 106,162,167]. 12.1. In this final section, we describe how ideal theory grew out of the metric theory of tensor products. Nowadays it seems that the novice outstripped the master. Let us compare some pros and cons: 9 The language of operator ideals is more appropriate for applications. 9 Operators can be composed, which is not so easy for tensors and bilinear forms. 9 Tensor product techniques yield minimal and maximal ideals, but there are plenty in between. This is in particular true when we consider closed ideals. 9 9tl is the least ideal that can be obtained from a tensor norm. However, smaller ones are needed, for example, in trace theory. 9 Tensor products provide a deeper insight into trace duality. 9 Tensor products proved to be an indispensable tool for the understanding of all phenomena related to the various approximation properties. Since mathematics is not a one-way street, it may well happen that somewhere in the future this evaluation will be refuted. 12.2. In 1936, Murray and von Neumann [115, Chapter II] invented direct products of Hilbert spaces to be used as a tool in their fundamental studies of rings of operators; their notation: H1 | | Hn. Two years later, the name tensor product was introduced by Whitney [ 191 ] when he investigated abelian groups: If the linear spaces are topological, a topology may be introduced into the tensor product . . . . In the case of Hilbert spaces, there is a natural definition of the topology in the product. In the intermediate case of Banach spaces, probably the norm I~1 may be defined as the lower bound of numbers ~ Igillhil for expressions y~ gi 9hi of Or. 1 1 This definition was suggested to me by H.E. Robbins.

Around 1950, Schatten [165,167] and Ruston [160,161] developed the basic theory. The most significant contributions, however, are due to Grothendieck. His main achievements in this field include

Operator ideals

485

9 introducing localization techniques in the theory of tensor products and in the theory of Banach spaces as a whole; 12.6, 9 enriching the theory through his 14 natural tensor norms; 12.10, 9 stressing the significance of approximation properties and giving various equivalent reformulations; 12.8, 9 pointing out the use of Banach spaces in the theory of (nuclear) locally convex spaces. 12.3. The theory of tensor products was developed on three levels. The simplest way is to work on a fixed algebraic tensor product. On the other hand, we can consider all X | Y's with X 6 t_ and Y 6 !_ simultaneously. More elementary, it is possible to start with tensor norms that are only given on all finite dimensional domains: E | F. We begin by describing the most general approach. By a tensor norm (| we mean a rule r that assigns to every element u = ~ = l xk | Yk in the algebraic tensor product X | Y of any pair of Banach spaces a nonnegative number o~(u) such that the following conditions are satisfied: (TN1)

a(x @ y) --Ilxll IlYll for x 6 X and y E Y.

(TN2)

ot(u + v) ~< or(u) + or(v) for u, v ~ X | Y.

(n

ot ~ A x k (TN3)

@ Byk

)

~ IIAII Ilnll~

(n)

k--1

~xk

Q Yk

k=l

17

for ~ x k

| y~ ~ X N Y, A ~ t2(X, Xo) and B E t3(Y, Yo).

k=l

Since (TN3) implies that a()~u) = I~l~(u) for u 6 X | Y and )~ 6 K , we have indeed a norm. For greater clarity, we shall sometimes write ot(ulX | Y) instead of ~(u). The objective of the theory is to investigate the Banach spaces X | Y that are obtained by completing X @ Y with respect to any tensor norm a. The duals (X | Y)*, formed by bilinear functionals, are considered as well. 12.4. If (TN3) is supposed to hold only for functionals instead of arbitrary operators A and B, we get

Z ( x * , x k ) ( Y k , Y*) <~ IIx*ll IlY*ll~

xk | Yk

/7

for Z x k | Yk 6 X @ Y, x* 6 X* and y*

y*.

k=l

In the terminology of Schatten, a norm on a single algebraic tensor product is called a crossnorm if it satisfies (TN1). He also used condition (TN~). Following Grothendieck, we refer to a norm with both properties as reasonable. Schatten defined the greatest crossnorm, now the projective tensor norm, /7

rr(u) "-- i n f ~ k=l

Ilxkll IlYkll,

J. Diestel et al.

486

where the infimum ranges over all representations of u, and the least reasonable crossnorm, now the injective tensor norm,

e(u) " - s u p {

~-~(x*, xk) (Yk,Y*) 9 IIx*ll ~ 1, Ily*ll ~ 1 }. k=l

That is, for any reasonable norm t~, we have e(u) <<.or(u) <<.rt(u). We know that L 1(M, #) | X = [L 1(M,/z), X] for every measure space (M,/z). On the other hand, C ( K ) | X coincides with [C(K), X], the Banach space of all X-valued continuous functions on a compact Hausdorff space K. More generally, if 1 ~< p < c~, then Bochner's norm

IlflLpll " -

IIf(~)ll p d/z(~)

restricted to L p ( M , lZ) | X is reasonable. We even have

~-~A | Tx~ILp

~< IITII

k--1

•

fk | Xk [ Lp

k=l

for T ~ 13(X, Y), f l . . . . , fn E L p ( M , lZ) and xl . . . . . Xn ~ X. On the other hand, if 1 < p, q < c~, then there exist operators S" L p ( M , lZ) --+ L q ( N , v) and Banach spaces X such that F/

H

~
(12.4a) k=l

does not hold for any constant c > 0, independent of fl . . . . . fn ~ L p ( M , Iz) and x 1. . . . . Xn ~ X. Hence Bochner's norm can be derived from a tensor norm only for p -- 1. It was just this deficiency that called forth the introduction of the ideals 9l~gp, 9,1~gp, ~ g p , tlgJED, etc. For example, if S" lp --+ L2[0, 1) is given by (~k) w-~ Y~'~--1 ~krk, then (12.4a) defines spaces of Rademacher type p; see 9.1 and 11.2. 12.5. Already Schattten observed that it is advantageous to define crossnorms not only on a single X | Y, but on all algebraic tensor products. In this case, he spoke of a general crossnorm. However, without any additional property, crossnorms defined on different domains X N Y were unrelated and could be chosen at random. So we need some connecting link, and this is just condition (TN3) in 12.3. Schatten referred to crossnorms satisfying this inequality as uniform [167, p. 29], [60, p. 9]. Gradually, it turned out that (TN3) and (IN3) are the decisive keys of both, the theory of tensor products and the theory of Banach operator ideals.

Operator ideals

487

12.6. We stress that Grothendieck used the term tensor norm (| only in the finite dimensional setting. However, in a subsequent step he constructed extensions of such ot's. This process will be described next. L e t u -- ~ = 1 x~:Nyk E X | Y. If M 6 DIM(X) and N 6 DIM(Y) such that Xl . . . . . Xn U_ M and Yl . . . . . Yn E N , then u can also be considered as an element of M | N. We put A

ot ( u l X @ Y) "-- i n f a ( u l M @ N), where the infimum ranges over all M and N described above. In this way we get the greatest extension of or, which is defined on all tensor products X @ Y. The least extension is given by

v

/( n

ot ( u l X N Y) "-- sup t~

i

~Q~xk|

X/M N Y/N

)

" N 6 COD(Y)

"

k=l

12.7. Let F "-- {E, F . . . . } denote the set of finite dimensional Banach spaces. Identifying E* | F and 13(E, F) shows that, in this case, the theories of tensor norms and ideal norms can be translated into each other. In particular, the tensor norms ~r and e correspond to the ideal norms t l - P l - - P7 and II" INrespectively. In the infinite dimensional setting, this relationship becomes much more complicated. For example, there exist operators T such that tl (T) < P l ( T ) and v~ (T) < v ~ ( T ) , respectively. Hence ideal norms over F admit different extensions over L; and the same phenomena occur for tensor norms. This situation will now be discussed in detail. Our starting point is some bisexual 0t as indicated in the middle of the following picture: V

I

Jot

~.[max

ol

m,

least extension

I

maximal ideal

tensor

I ideal norm

finite dimensional setting

A

Qot

~min

greatest extension

minimal ideal V

A

Then, by 12.6, 4.5 and 4.8, the extensions a and a as well as the Banach ideals ~[max and f21min are uniquely determined. Every tensor u -- Z ~ = I Xk* | Yk ~ X* @ Y defines an operator S" X -+ Y by 17

S ' x ~--~ X,Xk)Yk. k--1

J. Diestel et al.

488

It turns out that the map u w-~ S extends continuously to a metric injection

Jot" X* ~ ~ Y ---+ f2[max ( x , Y);

(12.7a)

see [32, p. 205]. Since ~max (X, Y) may contain non-approximable operators, the range of Jot is in general a proper subset. In this way, we can identify X* ~ e Y and ~ ( X , Y). The situation changes significantly when t~ is replaced by &. Then the continuous extension of u w-~ S becomes a metric surjection

Qot" x* ~

Y --+ }2[min(x, Y);

(12.7b)

see [32, p. 290]. This explains why Grothendieck refers to operators in ~2[min as a-nuclear. In view of the relationship between OIl and 31, it is justified to say that operators in ~max are a-integral. In the R~sum~, however, we find the term application de type or. Furthermore, with every functional u* 6 (X* | Y)* we associate an operator T ' Y --+ X** via (x*, Ty) -- (x* Q y, u*)

forx*6X*andyEY.

The correspondence u* --+ T is an isometry between (X* ,~A y)* and 9A*(Y, X**). ~ot If Qot in (12.7b) is one-to-one, then the predual X* | Y can be identified with

~,[min(x, Y). Unfortunately, this does not always happen. However, if Y has the approximation property, then everything becomes fine. In this case, the equation (2[min(x, Y)* "-9,1" (Y, X**) means the following. Since S ~ (2[min(x, Y) is approximable, an operator S ~176 can be defined by viewing S** as a map from X** into Y. Then S~176~ OIl(Y) for T E 9,1"(Y, X**), and we may put (S, T) "-- trace(S~176 This formula is the historical source of trace duality; see [176] and [32, p. 293]. 12.8. A Banach space X is said to have the approximation property if the following (equivalent) conditions are satisfied: 9 For every compact subset K and every e > 0 there exists T 6 ~ ( X ) such that mix T x II ~< e for x E K. 9 The natural map from X* ~ X into X* ~ e X or s is one-to-one. 9 The natural map from Y* ~ X into Y* ~ e X or s X) is one-to-one for all Y E k. 9 vl (T) -- v~ (T) for all T ~ ~ ( X ) . 9 Vl (T) -- v~ (T) for all T E ~(Y, X) and all Y 6 L. 9 The functional T ~-~ trace(T), defined for finite rank operators, admits a continuous extension to 9 t 1(X). 9 ~(Y, X) - .~(Y, X) for all Y ~ L. It seems to be open whether we may add: ~ ( X ) - ~ ( X ) . A Banach space X has the metric approximation property if, in the first condition of the above list, T 6 ~ ( X ) can be chosen such that IJTJl ~< 1. Now we are able to describe precisely the relationship between t l, vl and v 1o .

489

Operator ideals

t l (T) = P l (T) for all T 9 ~(Y, X) and all Y 9 k if X is reflexive. 9 Vl (T) - v~ (T) for all T 9 ~(Y, X) and all Y 9 k if and only if X has the approximation property. 9 t l (T) -- v~ (T) for all T 9 ~(Y, X) and all Y 9 k if and only if X has the metric approximation property. Approximation properties were introduced by Grothendieck in 1955. He also discovered various reformulations. For a long time many experts believed that all Banach spaces would have this property. So it occurred as a big surprise when Enflo [46] constructed a counterexample. Later, Szankowski [183] showed that even ~(12) fails this property. 9

12.9. In general, the natural map from X | Y to X | Y is neither one-to-one nor onto. Grothendieck conjectured that isomorphy only occurs when at least one of the spaces is finite dimensional [59, Chapter II, p. 136], [60, p. 74]. But this is not the case! Pisier [146] constructed an infinite dimensional space X such that X | X - X | X, algebraically and topologically. In addition, he arranged X and X* to have Rademacher cotype 2. Translating this result in the language of ideals yields ~ ( X * , X) -- 9tl (X*, X), and John [78] observed that .~(X, X * ) - 9tl (X, X*). Moreover, ~ ( X ) - 9tl (X). Loosely speaking, X admits only very few operators. It is still open if .~(X) -- 9tl (X) can be true for some infinite dimensional Banach space X. 12.10. Taking Grothendieck's point of view, the following considerations are carried out for operators acting between finite dimensional spaces. Given ideal norms ot and /3, the symbol ot -< 13 means that r ~< c f l ( T ) for all T 9 t2(E, F), where the constant c >~ 1 does not depend on E 9 F and F 9 F. If ot -
Ndual X ~1 o ~2

and

gdual o 1r2 x ~1 o ~oc.

In 2.2 and 4.6 we have defined the rules a w-> Otdual Ol ~ Olinj, Ol ~ Olsur and c~ ~ or* which assign to every given ideal norm ot a new ideal norm Olnew. The following list contains two more examples, namely ot ~ ot ext and a ~ ot lift, where the superscripts refer to ' e x t e n s i o n ' and 'lifting' otdual(T) "-- or(T*), otadJ(T) "-- or* ( T ) -- sup{ Itrace(ST)l" a ( S " F --+ E) ~< 1 }, otinJ(T) "-- sup{ot(BT)"

liBf--> l~[ I <<.1,

a'"r(T) - - sup{u(TA)" [IA " l'~ --+ oteXt(T)

El

1, n -- 1, 2 . . . . },

9-- inf{ot(To)llAIl" T -- ToA, E

wlift(T) " - i n f { l l B l l c ~ ( T o ) "

T--BTo,

n - 1,2 . . . . },

E

A >1~

To> F, n~>rank(T)},

To> 11n

B

> F, n ~> rank(T) }.

Taking the operator norm as a starting point, Grothendieck observed that there exists a minimal set of 14 n a t u r a l ideal n o r m s , which is stable with respect to ot ~ ot dual . . . . .

490 ot ~

J. Diestel et al. ot lift provided that we identify uniformly equivalent members. Thanks to the relations

(oldual)SUr__ (olinj)dual,

oldual) dual -- Ol,

(oladj)ext-- (olinj) adj,

(oladj) adj = ~,

(olSUr)sUr--otSUr '

(olinj) inj = otinj,

(~dual)lift

(Olext)dual,

(oladj)lift--- (olsur) adj, (Olext)ext-.-Olext '

(~lift)lift--ollift '

it suffices to check the invariance under a ~-+ Otdual, Oll----> Otadj and a ~ ot inj. In the following diagram the arrows point from the smaller ideal norms to the larger ones, in the sense of the preodering -<. There are no further relations as indicated; for example, we know from 6.6 that ~.l s ~1.

lift1 t dual

7[ 1

7[

dual

o 7[2

t 7~dual

7[2

lgdual

192

~adj

t axis of symmetry

~.1

i2

O~

Otdual

II II axis of symmetry

~2

t ~ S(X)U F

II. II We stress that the above considerations were originally formulated in the language of tensor norms; [60, p. 37], [32, p. 362]. This shows why the R d s u m d was the germ-cell of ideal theory.

References [ 1] D.E. Allekhverdiev, On the rate ofapproximation of completely continuous operators byfinite dimensional operators, Azerbajzhan. Goz. Univ. Uchen. Zap. (Baku) 2 (1957), 27-37 (Russian). [2] I. Amemiya and K. Shiga, On tensor products of Banach spaces, Kodai Math. Sem. Report 9 (1957), 161-178. [3] A. Baernstein, On reflexivity and summability, Studia Math. 42 (1972), 91-94. [4] S. Banach and S. Saks, Sur la convergence forte dans le champ LP, Studia Math. 2 (1930), 51-57. [5] B. Beauzamy, Opdrateurs de type Rademacher entre espaces de Banach, S6m. Maurey-Schwartz, Expos6s 6 et 7, l~cole Polyt6chnique, Paris (1975/76). [6] B. Beauzamy, Opdrateurs uniform~ment convexifiant, Studia Math. 57 (1976), 103-139.

Operator ideals [7] [81 [91 [101 [11] [121 [131 [141 [15] [161 [171 [18] [191 [20]

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491

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CHAPTER

12

Special Banach Lattices and their Applications

S.J. Dilworth Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA E-mail: dilworth @math. sc. edu

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Lorentz L p,q spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . n into Lp,q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 9 Embedding ~q and ~p 4. Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Containment of gp and the Nikishin factorization theorem . . . . . . . . . . . . . . . . . . . . . . . . . 6. Some probabilistic applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Embedding L w, q into Lq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H A N D B O O K OF T H E G E O M E T R Y OF B A N A C H SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved 497

499 499 505 511 515 519 524 527

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1. Introduction

This article discusses certain Banach lattices of importance in analysis, particularly the Lorentz and Orlicz spaces. Special Banach lattices arise naturally in probability theory and in many areas of analysis: in interpolation theory, in Fourier analysis, and in functional analysis in the theory of absolutely summing operators, for example. They furnish a rich supply of interesting Banach spaces with nontrivial properties. Consequently, their Banach space structure has attracted considerable attention, and over time a large and diverse body of results has accumulated. The first section after this Introduction is a sketch of the basic theory of the Lorentz Lp,q spaces, covering duality, convexity and concavity, isomorphic classification of weak Lp spaces, and the connection with real interpolation. Section 3 discusses the subspace structure of the Lp,q spaces, particularly the possibility of isomorphically embedding g p and ~q into Lp,q. Section 4 is a sketch of the basic theory of Orlicz spaces, covering the Luxemburg and Orlicz norms, the duality theory of N-functions, isomorphic containment of co and g~, and the Matuszewska-Orlicz indices and their role in interpolation theory. The next section discusses the structure theory for subspaces of Orlicz sequence spaces. Here we also present the Nikishin factorization theorem, which is an important application of the space weak L p to functional analysis. This result is then used to extend to Orlicz spaces a famous result of Aldous stating that every subspace of L1 contains some g p subspace. Section 6 presents two applications of Orlicz and Lorentz spaces to probability theory. The first is a theorem of Rodin and Semenov identifying the closed linear span of a Bernoulli sequence in exponential Orlicz spaces. The second is a version of Rosenthal's inequality for moments of sums of independent mean zero random variables, a result which has had important applications to Banach space theory. The final section discusses the existence of isometric or isomorphic embeddings of Lorentz spaces into L p spaces. This article is intended to be a self-contained introduction to the area. Proofs are given of the key points of many of the results presented, although very technical issues have usually been side-stepped. These considerations as well as personal taste have inevitably influenced the selection of topics. Many results which could not be presented in detail are discussed briefly in the endnotes which conclude each section. The interested reader will find there a partial guide to the extensive literature in this area. Finally, a few words of thanks: to William B. Johnson and Joram Lindenstrauss for the invitation to write this article, and to my collaborators. Special thanks to Neal Carothers for help with the bibliography and for sending me his seminar notes which shaped the sections on the Lp,q spaces.

2. Lorentz

Lp,q spaces

For a measurable function f on (0, o<~) its distribution function df is given by df(t) = m(lf(x)l > t) (0 < t < ~ ) , where m denotes Lebesgue measure. The decreasing rearrangement of f is given by f*(x) = inf{t ~> 0: df(t) <<,x}. Note that f * is a non-negative, right-continuous, decreasing function on (0, cx~) with the same distribution function as f .

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500

For 0 < p < c~ and 1 ~< q < c~, the Lorentz space Lp,q(1), where I is a subinterval of (0, c~), is the collection of all measurable functions f on I such that

Ilfllp,q -

(fo

f*(t) q d(tq/p)

)l/q=p(fo q

c~f*(t)q t q-1 dt

Integration by parts yields a useful alternative expression for

Ilfllp,q =

(foCX~(df(t)) q/p

d(tq)

)l/q

< cx~.

(1)

IIf IIp , q "

)l/q

(2)

For 1 0

(3)

t>0

After identifying functions which are equal almost everywhere, the L p,q spaces become quasi-Banach spaces when equipped with the quasi-norms (1) and (3). For 0 < q < p, observe that the weight function w(t) - - t ( q / p ) - I is decreasing, and so in the range 0 < q < p

Ilfllp,q = sup

(f f(~r(t)) q d(t q/p) )'/q

(4)

where the supremum is taken over all measure-preserving transformations of (0, ~ ) . (For the range p < q < cx~, the weight is increasing, and so the supremum in (4) should be replaced by an infimum.) The sequence space (q < OQ) consists of all sequences (an)n~=l equipped with the norm

~p,q (cx~ )l/q II(an)l[p,q--~akq(kq/P-(k-1)q/p)

(s)

k=l

The sequence space ~ p,~ (weak e p) is equipped with the quasi-norm

II(an)llp,~ - sup nl/p a n.*

(6)

Applying Minkowski's inequality in the Lebesgue space Lq(d(tq/P)), we see from (4) that II. IIp,q satisfies the triangle inequality when 1 ~< q < p. For 1 < p < q < cx~, however, the triangle inequality is not satisfied and (1) defines only a quasi-norm. Nevertheless, this quasi-norm is equivalent to a norm, a consequence of 'Hardy's inequality'.

Special Banach lattices and their applications

501

THEOREM 1. For q ~ 1 and ~ > O, and for every non-negative measurable function g on

(0, cx~), we have g(u) du

t -~-1 dt

<~ q

[tg(t)] q t -~-1 dt

Ol

.

(7)

PROOF. Jensen's inequality applied to the probability measure d ( ( u / t ) ~/q) on [0, t] gives

g(u)du (fo')~ -,(~ -- fo'g(u)ul--4"d((u/t) ~/q) )~ Ol

< ~ ( q_Ol) q t ~ fot g(u)quq-~ d((u/t) ~/q) --

t ~ ( 1 - ~/)

g(u)qu q-~+q-1 du.

Thus,

foe~ (fot g(u) du)qt-~-I dt

~(q)~'f0 ~ t -l -4"(f0'~ug(u )1)'-+"du qu

-

Ol

1--or+ ~

=

(q)q-' ji~[ )]

=

(~)q fo~ (ug(u))qu-l-~du"

ug(u

q U-

q

(f~

-4

t

_1 _ ~

dt

)

-4 dt du

For t > 0, let f** (t) = (1 / t) fo f * (u) du, and define

[]fll(p,q) --

(fo ~ f**(t) d(tq/p) )'~q

(8)

It is readily verified that ( f + g)** ~< f** § g**, and so by Minkowski's inequality (8) defines a norm for all 1 < p < cx~ and 1 ~< q < e~. Moreover, an application of (7) with ot = q(1 - ( I / p ) ) yields

Ilfllp,q ~ [[fll(p,q) ~

P

p-1

Ilfllp,q.

(9)

For the ranges p ~< 1 and q < 1, however, L p,q is a non-locally convex quasi-Banach space. Under the natural pairing

( f ' g) -- fI f (t)g(t) dt,

S.J. Dilworth

502

the Banach space dual of Lp,q c a n be identified with Lp,,q,, where pl and q' are the H61der conjugate indices of p and q, and the dual norm is equivalent to II 9IIff,q'. More precisely, for 1 < p, q < cx~ there exists a positive constant c (depending on p and q) such that

c IIg IIp',q' ~ IIg IIL~,q ~ IIg IIp',q'.

(10)

The constant c arises from an application of Hardy's inequality which occurs in the proof of (10). In particular, the dual norm is not isometrically a n L p,q norm when q > 1. For Lp,1, on the other hand, the dual space is isometrically isomorphic to weak Lp, equipped with the norm: IIf II(p', cx3) - - sup t 1 / p' f** (t). t>O

( 11 )

The L p,q spaces arise naturally in the real interpolation method which we briefly review. An interpolation couple is an ordered pair (X0, X1) of Banach spaces of which both members are continuously embedded into some common larger topological vector space. For each f belonging to the sum space Xo 4- X1, and for each t > 0, the K-functional is defined thus:

K ( f , t; Xo, X1) = inf{ Ilgllxo + tllhllx," f - g 4- h}.

(12)

For 0 < 0 < 1 and 0 < q < cx~, the interpolation space (X0, X1)O,q consists of all f E X0 4- X1 such that

]lfllo,q --

(fo

[t - 0

K(f, t; Xo, X l ) ] q

< cx~.

(13)

For 0 < 0 < 1, the interpolation space (X0, X1)o,~ consists of all f 6 X0 + X1 such that [[fll0,e~ - supt -~ K ( f , t) < c~. t>0

(14)

For fixed t > 0, the K-functional defines a norm on X0 + X1. Thus, by Minkowski's inequality, (13) and (14) define norms on the interpolation spaces (Xo, X1)o,q and (Xo, X1)o,~. Completeness of 11. [[O,q can be shown to follow from the completeness of the norms of X0 and X1. Suppose that (Y0, Y1) is another interpolation couple and that T is a linear operator which simultaneously maps X0 into Y0 and X1 into Y1 continuously, with operator norms M0 and M1, respectively. The Marcinkiewicz interpolation theorem asserts that T is a continuous linear operator from (X0, X1)O,q into (Y0, Y1)O,q, and that the operator norm of T on (X0, X l)O,q is bounded by M 1-~ m ~ The couple (L 1, L ~ ) is especially important for the Lp,q spaces. It is easily seen that

K(f,t; L1,L~)--

fo t f * ( u ) d u - t f * * ( t ) .

Special Banach lattices and their applications

503

Hence for the couple (L1, L ~ ) , we have

IlfllO,q

-

(fo

[tl_ Of**(t)] q -idt

)

1/q -

(q)l/q Ilfll(p,q)

for p - 1/(1 - 0). The following more general result will be used in Section 3. THEOREM 2. Let 0 < Pl < P2 <~ cx~, 0 < 0 < 1, and 0 < q <~ ~ . Then, up to an equivalent norm, [Lpl (I), Lp2 (I)]o,q -- Lp,q(I), where 1/p - (1 - O)/pl + O/p2. For all q, the L p,q norm behaves like the L p norm under dilation: i.e., for )~ > 0, For each fixed p E (1, cx~), we obtain a scale of Banach spaces indexed by q E (1, ~ ) . To compare the spaces in this scale, let us compute the norm of the step function y~'km__1 akZn,k, where zn,k is the indicator function of the interval [(k - 1)/n, k/n]"

[]f('/)Ol[p,q --)l/pllf(.)llp,q.

m

m

Z ai Zn,i k=l P,q

-f -- f

lak Iq Zn,i d(t q/p) k=l

k=l

ak,q -- ak+l) *q

Zn,i ] d ( t q/p) i=1 m

*q - ak+l)kq/P -- y ~ a k*q (k q/p

-

k=l

(k - 1)q/P).

k=l

The last expression can be shown to be a decreasing function of q by arguing by induction on m. Thus, for 1 ~< ql ~< q2 ~< ~ , we obtain the following monotonicity property of the L p,q norms: Ilfllp,~ ~ Ilfllp,q2 ~ Ilfllp,q~ ~ Ilfllp,1.

(15)

A similar summation by parts argument shows that if X is a rearrangement-invariant function space on I such that IIZE IIx = m ( E ) 1/p for every indicator function XE, then

Ilfll(p,~) ~ Ilfllx ~ Ilfllp,1.

(16)

Thus, Lp,e~(1) (equipped with the norm I1" II(p,~)) and Lp,1 (I) are the largest and smallest rearrangement-invariant Banach function spaces on I whose norms coincide with the L p norm on indicator functions. Observe at this juncture that it is often convenient to use (1), as we may, to define Lp,q ($2, •, #) for an arbitrary a-finite measure space. Many of the results discussed above extend straightforwardly to this more general setting. Let us now mention a typical application of the Marcinkiewicz interpolation theorem. Recall that the Fourier transform f ~-+ f maps L1 (Nn) continuously into L ~ ( R n ) , and

S.J. Dilworth

504

that by Plancherel's theorem it is an isometry on L2(Rn). Observe that from (15) it follows that Lp,,p(I) C Lp,(l) for 1 < p < 2, where 1/p + 1/p' = 1. Applying Theorem 2 we obtain the following refinement of the classical HausdorffYoung inequality (replacing L p, by the smaller space L p,, p). COROLLARY 1. Let 1 < p < 2. The Fourier transform maps Lp(~ n) continuously into L p,, p (]Rn). Next we consider the Banach lattice properties of convexity and concavity for L p,q. THEOREM 3. If 1 <. q ~ p < eC, then Lp,q is q-convex, i.e., for all (~ )i=l n in Lp,q, we have

IJSI q

)'Jq ("ll llqq )

l/q

<.

i=1

(17)

i=1

P,q

If the f i 's are disjointly supported then there is also a lower p-estimate:

/>

i=1

(18)

IIf/ IlpP,q

i=1

P,q

For p < q, Lp,q is q-concave and satisfies an upper p-estimate for disjoint functions. PROOF. First we use the triangle inequality in Lp/q,1 to prove q-convexity:

)'Jq q

If/I q

i=1

k

~ lfil q i=1

P,q

/7 IIIj~lqllp/q,~

p/q,1

i=1

=

Z ]If/" 11qp,q" i--1

For the lower p-estimate, we use (2) and the reverse triangle inequality in Lq/p(d(tq)) (note that q / p < 1)" tl

~f/ i=1

,

p,q

--

(f0

(d~n

fi (t)) q/p d(tq)

>>.~l[df,(t)llLq/~ i--1

)

p/q

~d~(t) i--1

Lq/p(d(tq))

~ ]]f/llp,q. i=1

We conclude this section with a discussion of the isomorphic classification of Lorentz spaces. It is well-known that Lp(0, 1) and Lp(O, cx~) are linearly isometric Banach spaces. For Lp,q spaces, however, it was shown in [22,23] that (for p ~ q and q < cx~) Lp,q(O, 1) and L p,q(O, CX)) are not even isomorphic as Banach spaces. Nevertheless, using the Petczyfiski decomposition method and some intricate combinatorial arguments, Leung [90]

Special Banach lattices and their applications

505

proved the surprising result that the three classical weak L p spaces are all isomorphic to each other. THEOREM 4. Let 1 < p < cx~. The spaces gp,~, Lp,~(0, 1) and Lp,~(O, Cx3) are isomorphic Banach spaces. Notes. Hardy [58] proved the first version of Theorem i (for sequences). The spaces Lp,1 and weak L p were first introduced by Lorentz [ 100] with somewhat different notation as the spaces A(ot) (c~ = 1/p) and M(o~) (or = 1 - 1/p), respectively. The Lp,q spaces for I ~< q ~< p were introduced in [101] as members ofthe larger class of Lw,p spaces (see Section 7). The Lp,q notation was introduced by Calder6n (see [117, p. 129]). See [69] and its references for linear topological and interpolation-theoretic properties of the Lp,q spaces. The injective isometries of Lp,1 (0, 1) were determined by Carothers and Turett [31 ]. For the real method of interpolation and the Marcinkiewicz interpolation theorem (as well as the Lp,q spaces) see the texts [10] and [11]. Corollary 1 is taken from [135]. Theorem 3 was proved by Creekmore [38] who used it to determine the type and cotype of the Lp,q spaces. An example of Pisier [99] shows that Lp,q (1 ~< q < p) is not p-concave. In particular, L2,q is not of cotype 2. An interpolation argument is given in [53] to show (for 1 < p < 2 and 1/p + 1/pl = 1) that Lp,p, has Fourier type p. The isomorphic classification of the weak Lp spaces Lp,~(S-2, z~, lZ), where ($2, ZT, #) is an arbitrary measure space, was carried out by Leung in a series of papers [88,90-92]. If (S2, Z', #) is countably generated and o--finite then the corresponding weak L p space (when infinite-dimensional) is isomorphic to either s or ep,~ [91]. For an uncountable purely atomic measure space there are five distinct classes of isomorphism types [91]. The weak L p spaces corresponding to purely non-atomic measure spaces which are not countably generated are classified in [92]. In [89] it is proved that s (1 < p < cx~) contains a complemented Hilbertian subspace. Kalton [77] proved that (for 1 ~< p < 2) L p, ~ (0, 1) contains a complemented subspace isometrically isomorphic to L p (0, 1). It was proved in [39] that weak L1 has a nontrivial dual space. Lotz [ 100] proved that weak L p (1 < p < oo) is a Grothendieck space, i.e., that weak and weak-star convergence coincide for sequences in the dual of weak L p. The seminar notes [21] influenced this discussion of the Lp,q spaces.

3. Embedding s and s t i into L p,q The results of this section will be proved for Lp,q(O, ~ ) (denoted simply Lp,q) as this case requires rather careful reasoning and is more general than both L p,q (0, 1) and ~p,q. First we prove the existence of subspaces isomorphic to s The following result gives sufficient conditions for the existence of a sequence equivalent to the the unit vector basis Of s PROPOSITION 1. Suppose that 0 < p, q < c~. Let (fn) be a sequence of unit vectors in Lp,q such that f * ~ 0 pointwise. Then some subsequence of (fn) is equivalent to the unit vector basis of (q.

S.J. Dilworth

506

PROOF (Sketch). The hypothesis ensures that we can find a subsequence of (fn) which is arbitrarily close in the L p,q n o r m to a disjoint sequence of compactly supported functions. Thus, without loss of generality, we assume that the fn's are disjoint and compactly supported. Let (en) be a decreasing null sequence of positive numbers, and let Un = {Ifnl < en}, gn = fnXu,, and hn = f n - gn. Clearly, either limsup Ilgnllp,q > 0 or limsup Ilhn IIp,q > 0. If the former is the case, then (passing to a subsequence) we may assume that inf Ilgn IIp,q - - ~ > 0. We shall show that (gn) has a subsequence equivalent to the unit vector basis of s proving this first for the range p < q. By Theorem 3, there is a lower q-estimate for disjoint functions (in fact, L p , q is q-concave), and so

lanl q

•

~<

an gn

n=l

n=l

P,q

Obtaining an upper estimate will be the heart of the proof: to this end, choose (for each n ~ 1) Sn > 0 such that fSn+m(Un)

Ig*(t - Sn)] q d

(t q/p

q

(19)

) -- 2llgnllp,q.

J Sn

Clearly, Sn ~ 00, since Ign I < 6n ~ O. So (passing to a subsequence) we may assume that Sn + m(Un) < Sn+l. Now (19) implies the existence of a measure-preserving transformation cr of (0, ec) such that cr([Sn, Sn + m(Un)]) -- Un and f Sn+l

Ign(cr(t))l q d

(tq/p

q

) - 2llgnllp,q.

,1sn

Hence, by (4) for p < q, we have

• n--1

oo

an gn

<~ P,q

f0

oo

~

la~lqlgn(Cr(t))l q d(t q/p)

n=l

0(3

f Sn+1

= ~lanl q

Ig,(~(t))l q d(t q/p) a Sn

n=l o(3

oo

= 2 y ~ lan Iq Ilgn IIq,q ~< 2 Z n--1

lan Iq.

n=l

This completes the proof of the upper q-estimate. An analogous argument shows that if lim sup [Ihn IIp,q > 0, then (hn) has a subsequence equivalent to the unit vector basis of s Finally, the case q < p can be handled in a similar fashion: here the upper estimate follows from Theorem 3, and the lower estimate is obtained as above by considering a suitable measure-preserving transformation. [3 The following theorem is the main result on the containment of s subspaces: the closed linear span of every disjointly supported sequence in Lp,q contains a subspace isomorphic to

s

Special Banach lattices and their applications

507

THEOREM 5. Suppose that 0 < p, q < c~. Let (fn) be a disjointly supported sequence of

unit vectors in Lp,q. Then [fn] contains a subspace isomorphic to g,q. PROOF. After passing to a subsequence, we may assume by Helly's selection theorem [111, p. 221] that ( f * ) converges pointwise (except for a countable set of points) to some right-continuous decreasing function g, i.e., that fn -- gn -+-hn, where gn* -- g and hn ~ 0 in measure. If g = 0, then the result follows at once from Proposition 1. So we shall assume that m(lgl > 23) > 26 for some 6 > 0. Since hn --+ 0 in measure, we may also assume that m(lhnl > en) < en, where (e,) is some sequence of positive numbers satisfying ~nC~=l En < 6. This yields m(lfnl > 3) > ~ for all n. We shall prove the following claim: given e > 0 and k ~> 1 there exist n ~> k and ~--,n+N N ~> 1 such that F - z--,i=n+l fi satisfies m ( l F I / > 2ellFllp,q) ~< 2e. An obvious induc,~-~n k+ l tion will then yield a sequence Fk - ',z---,i=nk+l (V'~k+' J~)/ll ---,i=~ +1 f/II P,q such that Fk --+ 0 pointwise. Then Proposition 1 will produce an ~q subsequence of (Fk), which proves the theorem. To establish the claim, fix e > 0 and k ~> 1. Since the f~'s are disjointly supported, and since m(Ifnl > 6) > 6 for each n, we get

ellFllp,q >>,~3llx[o,N~)llp,q - eB(N6) 1/p -- (31+(1/P)g)N 1/p.

(20)

On the other hand, since IIgx[o,e/Nlllp,q --+ 0 as N --+ c~, we have for all sufficiently large N (depending on e and 6) ( 3 1 + ( 1 / P ) ~ ) N 1/p ~

-N- ) E

/p

IIgx[o,e/N) llp,q (21)

Ilgx[o,e/N)llp,~ >- g(e/N).

Combining (20) and (21) we obtain e][F]]p,q > / g ( e / N ) for all sufficiently large N. Now choose n ~> k such that Y~i=n+l Ei < E. Then for all sufficiently large N, we have n+N

m(IFl>ZellFIIp,q)

--

~

m(IJ~l>ZellFIIp,q)

i=n+l

n+N

<" Z

n+N

m(]gi] > e]]F]]p,q) + Z

i--n+l

n+N

n+N

<~ y ~ m(]gi] > g ( e / N ) ) + ~ i=n+l

m([hi] > e[]F]]p,q)

i--n+l

~i

i=n+l

(since e II F II p,q >/ gi)

<~ N ( e / N ) + e -- 2e. This completes the proof of the claim.

E]

S.J. Dilworth

508

Embedding s into L p , q is quite a different story: we shall show that there is no embedding of s into L p,q if p 5~ 2, q < cx~, and p ~ q. But first we give an answer to the local version of this question by providing a quantitative estimate of the dimension of s k subspaces of/~np,q" n ~or 1 < p
L

--

akek

k--1

a ; q (k q/p -- (k - 1) q / p )

p,q

)"q

k--1

By the Mean Value Theorem

L

a;q(k(q/P )-11

akek

k=l

P,q

(22)

k=l

where the constants of equivalence depend only on p and q. Let 13 > 0 and set m = [nr The theorem is equivalent to the statement (which we shall prove) that the spaces s contain uniformly complemented subspaces that Set v = ~ i =mnl i - 1 / P e i and, for 1 ~< j <~ n, set vj = ~im=l i - 1 / P e i + ( j _ l ) n . O b s e r v e that the vj's are disjointly supported and that they all have the same distribution. Thus, since L p,q satisfies a lower p-estimate (Theorem 3), we have are uniformly isomorphic to s n

[[Vl ]]p,q j=l

]aj] p j=l

P,q

n To complete the proof t h a t (l)j)j= 1 is uniformly equivalent to the unit vector basis of s /7 it suffices to prove the upper estimate

•

ajoj

j=l

~< CIIvl Ilp,q p,q

lajl p j--1

with C depending only on p and q. By (22)

i-q/Piq/p-I

I[vllqp,q ~ i=1

) /7

and so it is enough to prove that [[ ~ j = l

,~ logmn ~< (1 +/3) logn -~ Ilvl Ilq,q,

ajvj[]p,q

To this end, suppose that aj ) 0 and that ~ = l

n

~ [IVl]p,q whenever ~-'~j=l [ajl p -- 1.

ajP -- 1. Fix 1 <<.i <<.mn, and, for each

509

Special Banach lattices and their applications

1 ~<j ~
Kj -- max{k: ajk -1/p > i -1/p}. /7 Then Zj=I

K j represents the total number of coordinates of the

vector Y ~/7j = I p

a j v j which

n

exceed i -1/p (-- v(i), the ith coordinate of v). Thus, i -- ~-~i%1 iaj > Y~j=I K j , and so

ajvj

(i) <. i -1/p -- v(i),

j=l g/

for each i, which yields 11Zj=l a j v j [[p,q ~ [[IJllp,q as desired. The proof of uniform complementation is straightforward. Let vj be a Hahn-Banach norming functional for vj with the same support as vj, and set P/7 -- Y~=I (', v~)vj/llvj [[p,q. Then P/7 is a projection onto [vj] nj = l ' and

]]P,(x)] pP,q ~ Ll<x j-1

*>1p ~ Ilxllp,q p ,

, Vj

since Lp,q satisfies a lower p-estimate. Finally, the case p < q follows by duality.

V1

We end this section by proving the important fact that there is no embedding of g p into L p,q (except when p -- 2, p = q, or q -- oc). The proof requires the concept of a disjoint sum of functions. Given finitely many real-valued functions fl . . . . . fn defined on [0, 1] (or, more generally, defined on some probability space (s 27, P)), we define the disjoint sum, denoted }-~in 1 O f , , to be any function f on (0, oc) for which d f - ~i~=1 dfi; for example, take f (t) -- ~-~in=l fi (t - i + 1 ) X [ i _ l , i ) ( t ). The disjoint sum concept is helpful for formulating certain norm inequalities and will also be used in Section 6. PROPOSITION 2. Let 0 0 (depending only on p and q) such that

<~C i=1

P,q

Lej~ i=1

(23) P,q

for all n ~ 1 and for all fl . . . . . fn in Lp,q(O, 1). PROOF. Consider the linear mapping Tn on Lp(O, oc)defined by T n ( f ) - ~--~i%l fi, where ~ ( t ) - f ( t + i - 1)X[0,11(t). Then, for 0 < p < 1, since II" lipp satisfies the triangle inequality, we have

I T n ( f ) Ipp <~

LIj~I i=1

p

/7

-
Lej~ i=1

-II/11 pp. P

(24)

510

S.J. Dilworth

By the Marcinkiewicz interpolation theorem (Theorem 2), Tn is bounded on each L p,q space, for 0 < p < 1 and 0 < q ~< e~, with norm C 2 depending only on p and q. Thus, for 0 < p < 2, we can apply (24) in the space Lp/2,q/2 to obtain

/2)'J2ii

1/2 _.

1/2

~
s

p,q

p/2, q/2

i=1

= c s

o fi p/2, q/2

i=1

.

p,q

i=1

THEOREM 7. Suppose that 0 < p, q < cx~, that p =/=2, and that p ~ q. Then gp is not

isomorphic to a subspace of Lp,q (0, ~ ) . PROOF. We begin with the range 0 < p < 2 and p < q, which is the most interesting and delicate case of the theorem. Arguing for a contradiction, suppose that (j~) is equivalent to the unit vector basis of p. By approximation we may assume that each fi is a step function supported in a finite interval. Let (gi) be a disjointly supported sequence with gi having the same distribution as f/ for each i. Since (fi) is an unconditional basic sequence, the Maurey-Khintchine inequalities yield 1/2

lail p

s

i--1

(25)

ai fi

i-1

P,q

i=1

P,q

For each fixed n we may suppose, by applying a dilation (which is an isometry in Lp,q), that each j~ is supported on [0, 1], and hence that Z i n= I aigi has the same distribution as Y~in=1 EDai ft'. Thus, by Proposition 2 for the left-hand inequality, and the upper p-estimate for disjoint functions (Theorem 3) for the right-hand inequality, we have

in)

112 2 2

ai fi i=1

~ C ( ~_~ lai lp

aigi

<<.c

p,q

i=1

p,q

lip (26)

i=1

Combining (25) and (26) we see that (gi) is a disjoint sequence in Lp,q that is equivalent to the unit vector basis of g p, which contradicts Theorem 5 since g p does not contain a subspace isomorphic to gq. Let us now consider the case q < p < 2. Using Theorem 5 to dispose of the case of a disjointly supported sequence, a routine 'Kadets-Petczyfiski' argument [73] (cf. the proof of Theorem 14 below) shows that it suffices to prove that there is no strongly-embedded subspace of Lp,q(O, 1) (one for which the L0-topology and the Lp,q-topology coincide) isomorphic to g p. But by (15) every such strongly-embedded subspace would embed isomorphically into L p,q,(O, 1) for all q~ > p > q, which we have shown to be impossible.

Special Banach lattices and their applications

511

For the remaining case p > 2, it suffices to observe that every strongly-embedded subspace of L p,q (0, 1) is isomorphic to a Hilbert space because of the continuous inclusion Lp,q(O, 1) c L2(0, 1). F1

Notes. Proposition 1 is from [24]. Its analogue for Lorentz sequence spaces was proved in [95] (see also [6]) and for Lp,q(O, 1) in [51]. Levy [93] proved that the interpolation space [X0, X1]o,q contains ~q whenever X0 A X1 is not closed in X0 + X1. Theorem 6 is due to Carothers and Flinn [30] who also gave the upper bound of C(p, q)n/(logn) I1/p- 1/ql for the dimension of uniformly complemented s k subspaces n . Theorems 5 and 7 are taken from [24] (see also [22]). The fact that for p < 1 o f ~p,q and p < q there is no embedding of s into Lp,q(O, 1) was proved by Kalton [76]. Carothers [19,20] proved that, for p > 2 and 1 ~< q <~ p, the only rearrangementinvariant spaces on [0, 1] which are isomorphic to subspaces of Lp,q (0, 1) are Lp,q (0, 1) and L2(0, 1) (the case p - - q > 2 was proved by Johnson, Maurey, Schechtman and Tzafriri [70]). He also proved (using results of O'Neil [ 118]) that this result does not extend to the range p < q (or p < 2).

4. Orlicz spaces

Let p : [ 0 , ~ ) --+ [0, cx~) be a non-decreasing right-continuous function satisfying p(0) = limt~0+ p(t) = 0 and p(cx~) = l i m t ~ p(t) = ~ . Then t

49(t) --

L

p(u) du

is a continuous convex function satisfying limt~0+ ck(t)/t = 0 and l i m t ~ ck(t)lt = cx~. We say that 4~ is an N-function. The Orlicz class Lg (0, e~) consists of all measurable functions f on (0, cx~) for which

M
oo

L

cb(If ( x ) l ) d x < ~ .

The Orlicz space L~(0, cx~) (denoted simply L~) is the linear subspace generated by L~ (0, cxz) in the space of measurable functions. It consists of all f such that M4~(kf) < cxz for some k > 0. The set {f E L~: M4~(f) <<.1} is a convex set (from the convexity of 4~) which absorbs L~. Its gauge functional Ilfllq~ - inf{k > 0" M4~(f/k) <<.1}

(27)

defines the Luxemburg norm on L~, which is a complete norm enjoying the 'Fatou property', i.e., if fn increases pointwise to f as n --+ cx~, then [Ifn [l~ --+ I[f ]l~. The collection of functions f such that k f E L4~ for all k > 0 is a closed subspace of L~ denoted L~. The simple integrable functions form a dense subspace of L~, and L~ consists precisely of the functions f E L~ for which the norm is absolutely continuous, i.e., if Ifnl <<.If] and fn ~ 0 pointwise, then Ilfn II~ --+ 0.

S.J. Dilworth

512

More generally, given any cr-finite measure space (X, Z, #), one can define the corresponding Orlicz class Lg (X) and Orlicz spaces L~ (X) and L~ (X). Of special interest and importance are X = [0, 1] with Lebesgue measure and X = N with counting measure. The latter gives rise to the Orlicz sequence space g4~ with norm given by

II(an)ll~ - inf

~(lanl/k) < oo .

>

(28)

n=l

Let q(t) = sup{u: p(u) <~t} be the right-continuous inverse of p. Since q satisfies the same assumptions as p, t

~(t) --

L

q(u) du

is an N-function, called the complementary function to q~ (denoted also ~b*). Observe that 4~ is also the complementary function to 7t, so that 4~** = q~. The key result relating q~ and 7t is Young's inequality: for all s, t > 0, we have

st ~ ~(s) + 7t(t)

(29)

with equality if and only if either t = p(s) or s = q(t). (When p(t) is continuous and strictly increasing, one can easily check this inequality by sketching a graph of p(t) and interpreting (29) in terms of areas under graphs.) Young's inequality yields a useful alternative expression for the complementary function 7t: 7t(t) = supst - 4~(s).

(30)

s>0

The associate space (L4~)I is the collection of measurable functions g for which the Orlicz norm

IIg II~ = sup

If0

Ifgl dx: IIf I1~ ~ 1

/

(31)

is finite.

THEOREM 8. L V, = (L4)' and

Ilgllr ~ Ilgll~ ~ 211gll~ for all g ~ L~v. PROOF. To prove the right-hand inequality, suppose that Ilfll~ ~ 1 and Ilgll~ ~ 1 (so that M ~ ( l f l ) ~< 1 and Mo(lgl) ~< 1). Applying Young's inequality, with s = If(x)l and

Special Banach lattices and their applications

513

t = Ig(x)l, and then integrating yields

f

oo Ifgl dx <<.M ~ ( I f l ) + MT, (Igl) ~< 2,

and so [Igll# ~< 2. For the left-hand inequality, the Fatou property allows us to assume that g is a simple integrable function with I[gllV~- 1. Let f ( x ) --q(lg(x)l), and note that IIf [l~ < ex). From the case of equality in (29), we have

I f (x)g(x)l = q(Ig(x)l)lg(x)l

-

4>(IS(x)l)+ 7x(Ig(x)l).

(32)

Observe that from (27) and from the convexity of 4b, the Luxemburg norm of any function h satisfies Ilhll~ ~< max(l, M
1 + Mr

>~ Ilfllq~ ~

L

Ifgl dx = M r

+ Mo(g).

So M~l(g) <~ 1, i.e., Ilgll~ ~ 1. Fatou's Lemma now gives (Lo)' = L#. Observe that ]IXAI[o -- 1/cb-l(1/m(A)) --+ 0 as m(A) --+ O. Since the simple functions are dense in L~, the Radon-Nikodym Theorem yields the following corollary of Theorem 8. COROLLARY 2. (L~,

I1" I1~)* is isometrically isomorphic to (LTx, I1" I1~).

The N-function 4~ is said to satisfy the A2~ condition (respectively A~~ condition) if

limx~oCb(2x)/cb(x) < oo (respectively limx_+oodp(2x)/dp(x) < ex)). It is easily seen that L~ (0, ex)) = L~ (0, ex)) if and only if 4~ satisfies b o t h / t o and/t~o. Suppose, for example that 4b fails the '42oo-condition. Then there is a sequence Mn "Poo with y~'n~176~(Mn)/4J(2Mn) < 1. Let (An) be a sequence of disjoint measurable sets with m ( a n ) - 1/4b(2Mn). Then

Mq$ ~ M n X A n ~=l

--

~<1, n=l 4~(2M~)

II ~-~n--1-+-(MnXAn)II~ ~ 1. On the other hand, Mcb(2MnXAn)- 1, and so IlMn XZn II - 1/2. Setting fn = Mn XAn it follows that

and so

1

~ll(an)ll~

oo

Z

-+-anfn

II(an)ll

n=l

for all (an) E g-oo, so that (fn) is equivalent to the unit vector basis of ~7oo (respectively c0) in L~ (respectively L~). These considerations in combination with Theorem 8 help to prove the following theorem.

514

S.J. Dilworth

THEOREM 9. Let cb be an N-function. The following conditions are equivalent: (a) 4) satisfies the A 0 and A ~ conditions; (b) L4,(O, o c ) - L~ (0, oo); (c) L4,(O, o0) is separable; (d) L4,(O, oo) does not contain a subspace isomorphic to f ~ ; (e) L~ (0, co) does not contain a subspace isomorphic to co. COROLLARY 3. L4, (0, oo) is reflexive if and only if both d? and 4/satisfy the A 0 and A ~

conditions. Analogous results hold for L4, (0, 1) (respectively s for N-functions satisfying the A ~ (respectively A2~ condition. The following Matuszewska-Orlicz indices provide useful information about the Banach lattice structure and interpolation properties of Orlicz spaces (for simplicity we consider only the case of L4, (0, 1)):

ot~ = sup{p>O: ~

--

sup ~()Otp < cxz} ; ~,t~>l ~b()~t)

i n f { p > O : i n f qb()~)tp > 0 } . k,t/> 1 q~()~t)

(33) (34)

Note that ot~ ~
oty ~- ~

1

1

1;

1

/~y ~- ~c%

1.

(35)

It can be shown (see [99]) that ot~ is the supremum of the set of p's for which L4, (0, 1) satisfies an upper p-estimate for disjoint functions, and that fl~ is the infimum of the set of q's for which L4,(0, 1) satisfies a lower q-estimate. From the general theory of Banach lattices described in [99] it then follows that L4, (0, 1) is super-reflexive if and only if 1 < ot~ ~< fl~ < oo. By (35) the latter condition holds if and only if both O~and !k satisfy the A ~ condition, and thus (by Corollary 3) L4, is reflexive if and only if it is super-reflexive. The indices ot~ and fl~ can also be shown to coincide with the reciprocals of the Boyd indices [13] of L4,(0, 1), which gives rise to the following useful interpolation theorem (a variation of the Orlicz interpolation theorem). THEOREM 10. Suppose that 0 < p < ~ <. ~ < q <<.o~. Then every linear operator T which is boundedsimultaneously on Lp(0, 1)and on Lq(0, 1) is also bounded on L4,(0, 1). Finally, let us discuss what happens when the convexity assumption on 4~ is relaxed. A function 4~" [0, c~) --+ [0, ~ ) is said to be a ~-function if 4~ is continuous, increasing, and satisfies ~b(0) - 0. An F-norm [. [4, is defined on L4, (0, cx~) as follows: If [4, = inf{ k > 0: M4, ( f / k ) <~k ].

Special Banach lattices and their applications

515

Lr (0, ec) is locally bounded if and only if there exists k > 0 such that ~(kt) liminf-> 1 t~0 4~(t)

and

~(kt) liminf~ > 1. ,~oc ~b(t)

(36)

If 4~ satisfies (36) then L4 becomes a quasi-Banach space with quasi-norm II" 114 defined by (27). Notes. For the theory and history of Orlicz spaces we refer the reader to the texts [104] and [110]. For interpolation theorems for Orlicz spaces and a discussion of the Boyd indices see [10,97,99], and [104]. There is an extensive literature on the Banach space geometry of Orlicz spaces: for example, see [2,3] for weak compactness, [78] and [81] for uniform convexity, [37] and [56] for smoothness, [50] and [ 107] for the Kadets-Klee property, and [36] and [86] for normal structure. For isometries of Orlicz spaces, the reader may consult [ 120] and its references. 5. Containment of g p and the Nikishin factorization theorem

A fundamental theorem of Aldous [ 1] states that every closed subspace of L 1(0, 1) contains a subspace isomorphic to g p for some 1 ~< p ~< 2. Using the results of Section 3 and Aldous' theorem one obtains the corresponding result for closed subspaces of Lp,q. THEOREM 1 1. Suppose that 1 ~ 2 then X contains an isomorphic copy of g.r for some r E {2, q}. (b) If p < 2 then X contains an isomorphic copy of ~-r for some r E {q } U (p, 2]. Conversely, g-r embeds isomorphically into g p,q (0, 1)for the given ranges of r. The analogue of Aldous' theorem for subspaces of Orlicz sequence spaces had been obtained earlier by Lindenstrauss and Tzafriri [95,96], with the Matuszewska-Orlicz indices oe~ and/~2 (companions to the indices ~ c and/~c defined above) playing a crucial role: ot~ = s u p { p > 0 :

sup ~b()~t) < ec ] 9 0<)~,t~
(37)

/~ -- inflP > 0:

inf 4~0 ~ t ) 0 } . 0<~., t~
(38)

THEOREM 12 ([96]). (a) The Orlicz sequence space g a contains a subspace isomorphic to gp (when p < cx~) or co (when p = co) if and only if t~~ <<.p <<.~ . (b) Every closed subspace X of ga~ contains a subspace isomorphic to co or to g.p.

The theory of stable Banach spaces initiated by Krivine and Maurey [83] provides the machinery for extending Aldous' theorem to a large class of Banach and quasi-Banach

516

S.J. Dilworth

spaces, including the Orlicz spaces Lr (0, 1) when 4~ satisfies the A ~ condition [55]. However, in order to highlight the theory offactorization of linear operators, an important area of functional analysis in which the L p,q spaces arise naturally, we shall take a different approach to one such extension of Aldous' theorem. Let A0 be the collection of all measurable functions on (0, cx~) for which df(t) < cx~ for all t > 0. It is easily checked that A0 is a separable complete metric linear space with metric d ( f , g) =

sup m(E)=l

I f - gl dx. 1 + I f -- gl

The topology induced by this metric is the topology of convergence in measure on (0, cx~). Let L0 denote the closed subspace of A0 consisting of all measurable functions supported on [0, 1]. Recall that a quasi-norm II 9II on a vector space x is a p - n o r m (0 < p ~< 1) if it satisfies ]Ix -Jr-yIIP <~ ]]xl]p -Jr-]]yllP for all x, y 6 X. Now we can prove the fundamental factorization theorem of Nikishin [ 113]. THEOREM 13. Let T be a continuous linear operator from a p-Banach space (X, ]l" II)

into Lo. There exists a non-vanishing g E Lo such that the mapping x w+ ( T x ) / g is continuous from X into Lp,~. In particular, T can be factorized through Lp,~. PROOF. We have to prove the existence of g 6 L0 such that

m ( . T x , > g.,x,. ) tl/p ~t for all x 6 X and for all t > 0. By an exhaustion argument it suffices to prove that for every e > 0 there exists C(e) > 0 and Ae C [0, 1] with m([0, 1] \ Ae) < e such that

( C(e)l,x,,) m s ~ Ae" I(Tx)(s)l > tl/p <~t.

(39)

The fact that T is a continuous operator into L0 simply means that for every t > 0 there exists c(t) > 0 such that m{ITxl > tllxll} < c(t) for all x ~ X, where c(t) --+ 0 as t --+ oo. Observe that if (ei) are independent Bernoulli random variables (defined on a probability space (I2, r , P)) taking the values 4-1 with equal probability, then by symmetry and convexity P (I ~-~in= 1 ai eil >~max lai I) ~> 1/2 for all real scalars (ai). Let (xi) be a sequence in X. Then

m maxlTxil > t

I]xill p i=1

oo

E

siTxi

> t

i=1

2E(m(E

Ilxi

IIp

i=1

oo

i=1

siTxi

>t

6iXi i--1

517

Special Banach lattices and their applications

Given a > 0, select to such that 2c(t0) < e. Then

m m a x l r x i l > to

Ilxill p

(40)

< e.

i=1

Now choose a disjoint sequence (An) of sets of positive measure and a corresponding sequence of unit vectors (Xn) which are maximal with respect to the conditions An {ITxnl >/to/m(An) l/p} for each n >/1. Then maxn>~l IT(m(An)l/Pxn)[ >/to on Un~>l An and 0<3

OO

~_~[Im(A~)'/Px~II p <~~ m ( A n ) n--1

<<.1.

n=l

Thus m ( U An) < e by (40). Let Ae -- [0, 1] \ Un~>l An. From the maximality ofthe family (An), it follows that for each x 9 X and for each t > 0, we have t011x II

m ( s 9 As" I(Tx)(s)l > tl/P

)

which proves (39) with C(e) -- to.

[:3

REMARK 1. Since the inclusion operator Lp,oo(O, 1) r Lq(O, 1) is continuous for 0 < q < p < oo, it follows from the Nikishin Factorization Theorem that every continuous operator from a p-Banach space X (0 < p ~< 1) into L0 factorizes through Lq (0, 1) for each 0 < q < p. Assuming the validity of Aldous' theorem for subspaces of L p (0, 1), when 0 < p ~< 1, we shall extend the result to subspaces of A0. But first let us observe that A0 has a large collection of Banach and quasi-Banach subspaces. PROPOSITION 3. Let r be a continuous strictly increasing C-function. Then L4~ a is isomorphic to a closed subspace of Ao. PROOF. Clearly, A0 is (0, oo) x (0, cx~) with a is a decreasing function g(t) ~ F(s)g(t). Then, 4)

isomorphic to A0, the space of all measurable functions on finite distribution function. Setting F - l ( s ) r (so that F on (0, oo)), consider the mapping T'La~ (0, e~) --+ AO given by for all ot > 0, we have dt =

f0 ( ) f0 ( F- 1

ot Ig(t)l

dt -

m s" F (s) >

dt Ig(t)l

-- (m x m)(F(s)lg(t)[ > o~) - - ( m • rn)(lTgl > ol). This calculation shows that a sequence (fn) in L~ satisfies Ifn Ir --+ 0 if and only if Tfn ---> 0 in A0. Thus, T is an isomorphic embedding.

[]

THEOREM 14. Let X be a quasi-Banach (i.e., locally bounded) subspace of Ao. Then X

contains a subspace isomorphic to co or to g,p for some p 9 (0, e~).

518

S.J. Dilworth

PROOF (Sketch). We may suppose that the topology of X is given by a p-norm II" I1 for some 0 0 such that the composition f w-~ (T f)X[0,M] is also an isomorphic embedding. In this case X will be isomorphic (by dilation) to a subspace of L0, and thus, by Remark 1, to a subspace of Lq (0, 1) for all 0 < q < p, whence X will contain a subspace isomorphic to ~r for some r ) p. On the other hand, if the mapping f w-~ (Tf)x[O,M] fails to be an isomorphic embedding for every M > 0, then one can find an increasing sequence (Mn) of positive numbers, and corresponding unit vectors Xn ~ X, such that (after setting f~ = (Txn)X[Mn_~,Mn]) d(Txn, fn) --+ 0 rapidly enough to ensure that (Xn) and (fn) are equivalent basic sequences (see [74]). Moreover, by Helly's selection theorem we may assume that there exists a nonzero decreasing function F(t) on (0, 1] such that fn* --+ F pointwise. Passing to a subsequence to ensure that d (fn* X[0,11, F) --+ 0 rapidly enough, it will follow that (fn) is equivalent to a disjoint sequence (Fn), where each Fn has the same distribution as F. If F is bounded then (Fn) is equivalent to the unit vector basis of co. If F is unbounded, set q~(t) =dF(1/t), for t > 0, and observe that

m

(5)

anFn ) t =

n=l

5

n=l

dF(t/lan)--

5

4)(lanl/t)

n=l

for all t > 0. It will follow that the Orlicz sequence space ( ~ , II 9I10~)is a p-convex quasiBanach space isomorphic to [Xn], and then from the non-locally convex version of Theorem 12 (see [75]) that [Xn] contains a subspace isomorphic to ~r for some r ~> p. D

Notes.

For a proof of Theorem 12 see the texts [97] and [98] (as well as [96]). The set of p's such that g p embeds isomorphically into the Orlicz function space Lq~(0, 1) is determined in [63]. The corresponding problem for the function space L~(0, ec) is solved in [112] and [63]. A characterization of the Orlicz sequence spaces g0 which embed isomorphically into g~ is given in [95], where the problem of complemented g~p'S is also considered. An example is given there of a reflexive Orlicz sequence space which does not contain any gp as a complemented subspace (see also [97] and [98]). See [61-63] (and [60] for examples of Orlicz function spaces L~(0, 1) without any complemented gp) for results on complemented s in Orlicz function spaces. Kalton [75] extended Theorem 12 to the non-locally convex and the non-locally bounded cases and showed that the theory for complemented subspaces is significantly different in the non-locally convex setting. The proof given here of the Nikishin Factorization Theorem is modelled on the proofs in in [76] and [136]. The assumption that X be a p-Banach space can be relaxed (see [76]) to Rademacher type p (0 < p ~< 2). The Lorentz spaces Lp, l(#) and Lp,cc(#) also arise in the factorization theory of absolutely summing operators: Pisier [ 119] (see also the text [40]) proved that a (p, 1)-summing operator from a C(K) space into a Banach space admits a factorization (analogous to the Pietsch Factorization Theorem) through an Lp,1 (/z) space. See [44] for a recent application of this result, where it is used to show that cotype characterizes the H61der-continuity properties of the indefinite Pettis integral.

Special Banach lattices and their applications

519

See the text [57] for the theory of stable Banach spaces and for the proof of the following refinement of Aldous' theorem: a closed subspace X of L1 contains almost isometric copies of gp, where p is the infimum of the type interval for X. Garling [55] proved that L 4,(0, 1) is a stable Banach space if 4~ satisfies A ~ . His arguments readily extend to cover the quasi-Banach case of closed subspaces of L p, for 0 1) are stable. It follows that every closed subspace of these spaces contains almost isometric copies of ~r for some r. See [55] for a representation theorem for the space of 'types' on L~, and [9] for types on Orlicz and Lorentz sequence spaces. See [8] for results on the stability of quotient spaces and of interpolation spaces. Proposition 3 and Theorem 14 are adapted from [49].

6. Some probabilistic applications One area of analysis in which Orlicz and Lorentz norms arise frequently is probability theory. Indeed, a useful way of studying the integrability of a random series is to determine its closed linear span in an appropriate function space. In this section we present two important results of this type. Khintchine's inequality states that a convergent Bernoulli series has finite moments of all orders, and that the closed linear span in Lp (0 < p < oe) of a Bernoulli sequence is isomorphic to g2. The Orlicz spaces L~q, where ~q(t) - - e ta -- 1, may be used to study the exponential integrability properties of a Bernoulli series. The following result of Rodin and Semenov [ 130] identifies the closed linear span of a Bernoulli sequence in L~q (q > 2) with the Lorentz sequence space g~p,~, where 1/p + 1/q -- 1. THEOREM 15. Let (en) be a sequence of independent Bernoulli random variables defined on a probability space (S-2, r , P). Then, for each q > 2, there exist constants C1, C2 such that

1 - - II(a,)llp,~ C1

C21l(an)llp,~

~-~ an En

(41)

~q

n=l

for all scalars (an), where 1 / p + 1/q -- 1. PROOF. First we prove a fundamental estimate, known as Hoeffding's inequality [67], for the distribution function of a convergent Bernoulli series: L

P

anen > t

~<2exp

n=l

2 ~n~__l a 2

.

(42)

To prove (42), we may assume by homogeneity that Y~n= ~ l an2 -- 1. Observe that by independence of the en'S, we have Eexp t

anen n--1

-

Eexptanen<<, n----I

exp ~ n=l

-exp

,

520

S.J. Dilworth

and so by Markov's inequality exp(t 2)P

<~ exp(t 2/2),

anen > t n=l

from which (and from symmetry) (42) follows. Suppose now that II(an)I] p,~ = 1. An interpolation argument will be given to show that

P

(s

anen > t

<~ 2exp

,

(43)

n=l

for all t > 0, where Cq depends only on q. First, we may assume that (lan I) is a decreasing sequence. Secondly, it suffices to prove (43) for all t > 2q. Let m be the largest integer such that t > 2qm 1/q + 1. Then

•

m

(3o

m

(3O

an 6n

Z

n--1

n=l

n=m+l

n=l

an 8n

n=m+l

o(3

<~ (qm 1/q + l)-t-

Z

anen ,

n=m+l

and so by (42) (since we are assuming that t > 2qm 1/q + 1) P

anen > t

<~ P

n=l

anen > qml/q n=m+l

~< 2exp

2}~n~

(44)

"

But o<3

oo

2 an<~ Z

Z n=m+l

P___~ml-2/p = ~ qm n-2/P<~2_ p q--2

1-2/p

n--m+l

and it follows from (44) that

n=l

which yields (43) since m ~ (1/2q)qtq. Integration by parts and (43) give

E l[/q Ol

anen

--

=

P

oe ~_~ anen > t

d(exp(t q) - 1)

n=l

~<

f0 (,q) 2 exp Cqotq

d(exp(tq ) - 1) <~ 1

Special Banach lattices and their applications

521

for all ot < Otq. This gives the right-hand inequality of (41). For the left-hand inequality, observe that since P (Y~'~n=l m en -- m) -- 2-m there exists Cq > 0 such that

n=l

Oq

n=l

p,oc

where the second inequality follows by direct calculation. Since II(a~)ll- II ~ - - 1 anellOq defines a symmetric sequence space norm, it follows from the sequence space version of (16) that

•

CqII(an)ll p, ~ ,

an 6n

~q

nzl

which completes the proof of (41). REMARK 2. Note that it follows from (42) that for q ~< 2 the linear span of a Bernoulli sequence in L~q is isomorphic to ~2. Our second application is a result of Johnson and Schechtman [72] which extends (to the range 0 2) [131]. First let us introduce the Orlicz spaces Yp =- L49p (0, cx3) (0 < p < ~ ) , where

dpp(t)

- / t2

I tp

for 0 <~ t ~< 1, for 1 < t < cx~.

Observe that Yp = Lp(O, cx~) -+- L 2 ( 0 , ~x)) ( r e s p e c t i v e l y Lp(O, ~ ) 0 L 2 ( 0 , ~ ) ) for 0 < p < 2 (respectively 2 < p < ~ ) . Note also the duality Yp = Yq for 1 / p + 1/q = 1. The following theorem makes use of the disjoint sum notation introduced in Section 3. THEOREM 16 [72]. Let 0 < p < c~. Then for every sequence of independent random variables (fn)n~=l defined on a probability space (1-2, S , P), we have

G fn n=l

p

n=l

,

(45)

~bp

with constants of equivalence depending only on p. PROOF (Sketch). For the range p > 2 there is a short elementary proof (essentially the proof given by Rosenthal [131 ]), which we present. Making the substitutions gn = f 2 and q = p / 2 (so that q > 1), (45) will follow from

~gi i=1

~< max 2 q Z i--1

Ilgi II1,2

I[gi IIq i=1

9

(46)

522

S.J. Dilworth

(The reverse inequality (with 2 replaced by 1) does not require independence but just follows from the positivity of the gi's and an application of H61der's inequality.) To prove (46) observe that ~< 2 q-1 ( f i gq -+- f i ( j ~ / i--1 i=1

( f i ) q gi i=1

gj )q-1 gi ) 9

So by independence of the gi's, we have

~< 2 q-1 ( f i Ilgi IIq + f i ( f i e gi i=1 q i=1 i--1 ~ 2q-1

("

~ I[gi IIq + i=l

Ilgilll j~/ gj -' 9 " qq-1 Ilgi Ill

i=1

t)

1 i:, I11_,t

(by H61der's inequality and the positivity of the gi's) ~< 2 q max

Ilgi IIq,

i=1

Ilgi II1

figi i=1

i=1

q-l), q

which yields (46). The proof in the range 0 < p < 2 requires more elaborate arguments for which the reader should consult [72]. [~ Using the fact (due to Marcinkiewicz and Zygmund [105]) that the norm in L p of a sum of independent mean zero random variables is equivalent to the norm of the square function, we can reformulate the previous theorem. COROLLARY 4. Let (Xn) be a sequence of independent mean zero random variables. Then, f o r 1 <<.p < oo, we have "

E

Xn

"~

n=l

~Xn

n=l

9

4~p

As a last pretty application of Theorem 16, let us see how the closed linear span in L p of a sequence of independent identically distributed random variables may be identified with a certain Orlicz sequence space. COROLLARY 5. Let (Xn) be a sequence of independent identically distributed mean zero random variables with common distribution function F(t). Then, f o r 1 <~ p < oo, we have cx~ E ~

p anXn

p ~ ]](an)J]C~p,F ,

n=l where dPp,F(t) -- f R dpp(tS) dF(s).

Special Banach latticesand theirapplications

523

PROOF.

p) 1/p

oo

E

~

n=l

(~ an Xn

anXn n=l

~bp

--inf{k>O: ~Ec/)p(anXn) k <,1 --inflk>O: ~RfdPP(~-)dF(s)<<]ln'=l = [(an) I4)p,F. Notes.

[]

The proof of Theorem 15 is adapted from the proofs given in [106] and also [87]. See the text [ 14] for a characterization of the i.i.d, sequences for which the estimates in (41) hold. See [25] and [15] for some inequalities for i.i.d, sequences in the Lp,q spaces: it is proved in [25] that for p > 2 the closed linear span of a mean zero i.i.d, sequence is always a complemented Hilbertian subspace, and that in L2,q a mean zero i.i.d, sequence satisfies an upper g2,q-estimate if and only if 0 < q ~< 2. See [108] for probabilistic inequalities involving weak L p norms in the local theory of Banach spaces. Theorem 15 was extended to the case of vector-valued Bernoulli series, with coefficients from an arbitrary Banach space, by Montgomery-Smith and the author [48]. See [48] also for a fundamental two-sided estimate (improving (42)) for the distribution of a vectorvalued Bernoulli series, involving the K-functional for a certain interpolation couple. Theorem 16 can be formulated much more generally for sums of independent random variables in rearrangement-invariant quasi-Banach function spaces [72] (cf. [23] for the special case of Lp,q). See [72] for an important application of Theorem 16 to the uniform approximation property in L p spaces. Another way to obtain Theorem 16 in the range 0 < p < 2 is by a duality argument, using a version of the Burkholder-Davis-Gundy martingale inequalities [18]. We refer the reader to [43] for a full exposition of this approach. Rosenthal's inequality for Lp (p > 2) (first proved to exhibit new complemented subspaces of L p) in some sense characterizes L p. See [14] for a precise formulation of this fact and for further results on the linear span of independent random variables in rearrangement-invariant function spaces. See [71] and the text [114] for martingale inequalities in rearrangement-invariant spaces. The Orlicz space Yp is isomorphic to L p in the range 1 < p < cx~ [70] (and so L p has two representations as a rearrangement-invariant function space on (0, ec)). The 'other' scale of sums and intersections of Lp spaces, Mp = Lp(O, cx~)71L2(0, oo) (respectively Lp(O, oo) -k- L2(0, ~ ) ) for 0 < p < 2 (respectively 2 < p < ec), is closely related to (though isomorphically distinct from) the L p scale. Several Banach space properties of these spaces and of the L2(s spaces are obtained in [41,42], e.g., that there is no isomorphic embedding of L2(~p) into g2(gp) for p r 2. For the structure (e.g., containment of gr(gs) subspaces) of the 'mixed norm' spaces Lp(Lq) see the papers of Raynaud [ 122,123,127].

S.J. Dilworth

524

7. Embedding

Lw,q into Lq

In his seminal work [7] Banach discussed the question of the 'linear dimension' of the classical Banach spaces, that is, the question of the existence (or non-existence) of an isometric or an isomorphic embedding from one space into another. A famous question left open in [7] for many decades concerned the existence of an isomorphic embedding from ~q into Lp for 1 ~< p < q < 2. Surprisingly, the solution to this problem turned out to be probabilistic in nature: using stable processes Bretagnolle, Dacunha-Castelle and Krivine [16] proved that it is even possible to embed Lq isometrically into L p in the range 0 < p < q ~< 2 (cf. [82] for a self-contained proof). In this section we shall consider the question of embedding Lorentz spaces into Lq. It is appropriate to study this question not just for the L p,q spaces but for the wider class of Lw,q spaces, which will now be introduced. Let I denote the interval (0, 1) or the interval (0, cx~), and let w(t) be a positive decreasing weight function defined on I for which fd w(t) d t = 1 and f o w(t) dt = cx~ (the latter condition only for I = (0, cxz)). For 0 < q < c~, the Lorentz function space Lw,q(I) is the space of equivalence classes of real-valued measurable functions f on I for which the following norm (or q-norm if q < 1)

Ilf llw,q -

(f,

f * (t)q w(t) dt

)"q q

1

is finite. For 0 < q ~ 1, the condition that w should be decreasing guarantees (and, in fact, is equivalent to) the triangle inequality in Lw,q [ 101 ]. Given the existence of the stable embeddings of L p into Lq, a natural question to consider is the possibility of an isometric embedding of Lw,q (I) into Lq. It turns out that such embeddings exist for only a very restrictive class of weights [46]. THEOREM 17. Let 0 < q < oo and let I be either (0, 1) or (0, c~). There exists an isometric embedding of Lw,q (I) into Lq if and only if one of the following conditions is satisfied: (a) w(t) -- 1, in which case Lw,q (I) -- Lq (I) with equality of norms; (b) I = (0, 1), 0 < q ~< 1, and w(t) is a decreasing linear weight; that is, there exists ot E [0, 2] such that

w(t) =_ w~(t) = 1 + - -t~t. 2 The case q > 1 is easy: the space Lq is smooth, but Lw,q is not, unless w(t) ---- 1. For 0 < q < 1, the proof of Theorem 17 is quite technical as it uses the theory of Fourier transforms of distributions. For q = 1, however, a short geometrical argument can be given for the main step of the proof, which concerns finite-dimensional Lorentz sequence spaces. For a l >~ ... >~an >~0 (not all zero) and q > 0, the expression

II

.....

II -- (a' Cx )" +""

+

'/"

Special Banach lattices and their applications

525

is the norm (or q-norm if q < 1) of an n-dimensional Lorentz space denoted ~n,q. For an infinite decreasing sequence w -- (an) of positive weights, for which ~-~n~=l an -- co, the Lorentz sequence space g w,q is defined similarly. PROPOSITION 4. The s p a c e g.n,1 (n ~ 2) is i s o m e t r i c to a s u b s p a c e o f L1 i f a n d only i f al - a2 -- a2 - a3 . . . . .

an-1 - an.

(47)

PROOF. First we prove by induction that ~n,1 is isometric to a subspace of L1 only if the weights ak satisfy (47). Let E be a Lorentz space of dimension n + 1 with weights a 1. . . . . an+ 1. If E is isometric to a subspace of L 1 then (47) holds by hypothesis, and so it suffices to prove that an-1 - an = an - an+l to complete the induction. In particular, we may assume that a n - l , an and an+l are not all equal. Let B* denote the unit ball of E*. Since the unit ball of E is a polytope, and since E is isometric to a subspace of L1, it follows (see, e.g., [12]) that B* is a z o n o t o p e (that is, a Minkowski sum of line segments). By [12, Theorem 3.3], all of the two-dimensional faces of B* are centrally symmetric. It is easily seen that the extreme points of B* are all the sign-changed permutations of the vector a = (al, a2 . . . . . an+l). In particular, one of the two-dimensional faces of B* has as its vertices all of the vectors obtained by permuting the last three coordinates of a. If an-t = an or if an = an+l, then this face is triangular, which contradicts the central symmetry requirement. If an-1 > an > an+l, then the face is h e x a g o n a l , and the symmetry condition forces an-1 - an = an - an+l as required. Thus (47) is a necessary condition. To show that (47) is also sufficient, one can check that if (47) is satisfied then B* has four kinds of two-dimensional faces: two classes of quadrilateral faces, one class of octagonal faces, and one class of hexagonal faces like the face described above. The first three kinds of faces are automatically centrally symmetric without any condition on the weights, while (47) guarantees that the hexagonal faces are also centrally symmetric. So, by [12, Theorem 3.3] once again, if (47) is satisfied, then B* is a zonotope, and hence E is isometric to a subspace of L1. D The question of embedding Lw,q isomorphically into Lq is answered by the following theorem of Schtitt [132]. THEOREM 18. L e t 1 <<,q < 2. T h e n the f o l l o w i n g are equivalent: (a) Lw,q (0, 1) is i s o m o r p h i c to a s u b s p a c e o f Lq ; (b) Lw,q (0, 1) is a 2 - c o n c a v e B a n a c h lattice; (c) There exists C > 0 such that f o r all x E [0, 1], w e h a v e

fo x w ( t ) t - p / 2 dt <~ Cx -p/2 fo x w ( t ) dt. An interesting corollary of Theorem 18 is the fact that Lp,q embeds into Lq in the range 0 < q ~< p < 2. Let us use Rosenthal's inequality (Theorem 16) to sketch a short proof of this important special case of Theorem 18, following [26].

S.J. Dilworth

526

By Theorem 2, Lp,q is the interpolation space [Lr, Ls]O,q (with equivalence of norms), where 1/ p -- (1 - O) / r + 0/s. Thus,

Ilfllp,q ~

(~ ?1~--OQ

)l/q

2 -nOq g ( 2 n, f ; gr, gs) q

9

(48)

Let us denote by Lr + t 9Ls the Orlicz space Lr -Jr-Ls equipped with the equivalent norm K(t, f ; Lr, Ls). Then it follows from (48) that Lp,q is isomorphic to a closed subspace of (Zn---cx~CX~O ( t r -Jr-2 n 9Ls))q. For )~ > 0, the dilation property of the norm in L p (namely, IIf (Zx)ll p = z - l / p IIf (x)ll p), yields

g ( t , f ; Lr, Ls) = inf{~/rllg(~x)llr + t~l/Sllh(~x)lls" f - g + h}. Setting ~. = t rs/(s-r) (so that t - ~(s-r)/rs)), we have

g ( t , f ; Lr, Ls) -- ~/rinf{llg(~x)llr + IIh(~x)ll~" f - g + h }

=

Lr, Ls).

Thus the mapping f (x) w+ ts/(s-r) f (trs/(s-r)x) defines a linear isometry from Lr + t . Ls onto Lr + Ls. Thus, (Y~n~=_~ ~)(Lr + 2n" Ls))q is isometrically isomorphic to ~.q(Lr + Ls). In particular, if 0 < q < p < 2, then, setting r -- q and s -- 2, we see that Lp,q embeds isomorphically into s + L2). But by (45) of Theorem 16, the mapping oo

n=l

where fn -- f X(n- 1,n) and (Sn) is a sequence of independent random variables each having the uniform distribution on [0, 1], yields an isomorphic embedding of Yq = Lq + L2 into Lq(s Since s embeds into Lq, it follows that Lq(/~2) embeds into Lq(Lq), which in turn is isometrically isomorphic to Lq. For the other ranges of p and q (p r q) it turns out that it is not possible to embed L p,q into Lr (r < cx~). In particular, since L2,q (q ~ 2) does not have cotype 2 (or type 2) there is no embedding of L2,q into Lq. It can also be shown that there is no embedding of Lp,q into L1 when p < q (see [132] or [26]).

Notes. The Lw,q spaces were introduced by Lorentz [ 101] with different notation as the spaces A(4~, q), where ok(t) - fo w(s)ds. There is an extensive literature on the Banach space geometry of these spaces. The following property of the weight is especially important: w is said to be regularif there exists k > 1 such that 4~(2x) > k ~ ( x ) for all x > 0. For q > 1, Halperin [58] proved that Lw,q (I) is uniformly convex if and only if w is regular. The sequence space version of this result together with estimates for the modulus of convexity were obtained by Altshuler [1 ]. On the other hand, the uniform Kadets-Klee property in Lw, 1(0, cx~) was shown to be equivalent to 'uniform concavity' of 4~ [45,47], which is a stronger condition than regularity of w. It is proved in [27] (cf. [28]) that Lw,1 (0, e~) has the Fixed Point Property for non-expansive mappings on weakly compact convex sets when w is strictly decreasing. See [32] for the isometries of Lw,q (0, cx~).

Special Banach lattices and their applications

527

Reisner [ 129] proved that Lw,q(I) is p-concave for some p < cx~ if and only if w is regular. He also gave necessary and sufficient conditions on w for Lw,q to be p-concave and proved that the interval of p's for which Lw,q is p-concave is an open interval (cf. [ 115] for these results and [ 126] for more general results in which the condition that w be decreasing is relaxed). The Boyd indices for Lw,q are computed in [115] and [109]. See [126] for the representation of Lw,q as an interpolation space (Lr L?)o,q (generalizing Theorem 2). It is proved in [116] that a closed subspace of Lw,q (0, 1) either contains almost isometric copies of gq or or is strongly-embedded in L j(0, 1). Garling [54] identified the dual space and proved reflexivity of the sequence space gw,p (p > 1). Allen [4] obtained a simple description of the dual space for regular weight sequences. See [34] for proofs of the Kadets-Klee property and local uniform convexity of gw,p. The symmetric basic sequences in gw,p are characterized in [6]. Necessary and sufficient conditions are given for g w,p to have exactly two nonequivalent symmetric bases and an example is given of a subspace of a Lorentz sequence space with a symmetric basis which is not itself isomorphic to a Lorentz sequence space. Symmetric basic sequences in the dual of gw,p are considered in [33]. Bretagnolle and Dacunha-Castelle [ 17] proved that Lr is isomorphic to a subspace of L 1 if and only if 4) is equivalent to a 2-concave convex 4)-function (recall that 7t is 2-concave if g r ( ~ ) is concave), which is the Orlicz space analogue of Theorem 18. See [133] for an explicit embedding of a given 2-concave finite-dimensional Orlicz sequence space into L 1. The proof utilizes some combinatorial and probabilistic inequalities [84,85] which have also been used to compute projection constants for finite-dimensional Orlicz and Lorentz spaces [84]. Hermindez and Ruiz [65] proved that, for 0 < p < q < cx~, every Orlicz space Lr (0, cx)) whose Boyd index reciprocals lie strictly between p and q is lattice-isomorphic to a sublattice of Lp -+-Lq (see also [52]). Raynaud [123] proved that every p-convex and q-concave Orlicz space L~(0, oc) is lattice-isomorphic to a sublattice of Lp(Lq). See [128] for results on embedding symmetric spaces into L1 (gq), and [64] and [124,125] for further embedding results. A special case of a theorem of Kalton [76] asserts that the Banach space gp(gq) embeds into L0 if and only if 1 ~< p ~< q ~< 2. Briefly, let us mention a way in which Orlicz and Lorentz spaces have been integrated into a single class of spaces. Let 4) be a convex 4)-function and let w be a decreasing weight function. The Orlicz-Lorentz space A~,w(I) is the Banach space of measurable functions f on I for which

IIf II = inf, )~ > O:

f0

$(f*(t)/~.)w(t) dt

J

< e~.

See [81] for uniform convexity, [80] for extreme point structure (see also [29] for the uniqueness of the Choquet integral representation in Lw,1 (0, oc)), [79] for containment of co and g ~ , and [68] for local uniform convexity and the Kadets-Klee property (see also [ 134] for Lw, 1) of Orlicz-Lorentz spaces. See [ 109] and [ 123] for concavity properties and Boyd indices for Orlicz-Lorentz spaces.

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[95] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, II, Israel J. Math. 11 (1972), 355-379. [96] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, III, Israel J. Math. 14 (1973), 368-389. [97] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Lecture Notes in Math. 338, Springer-Verlag, Heidelberg (1973). [98] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, I. Sequence Spaces, Springer-Verlag, Berlin (1977). [99] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, II. Function Spaces, Springer-Verlag, Berlin (1979). [100] G.G. Lorentz, Some new functional spaces, Ann. Math. 51 (1950), 37-55. [101] G.G. Lorentz, On the theory ofspaces A, Pacific J. Math. 1 (1951), 411-429. [102] G.G. Lorentz, Relations between function spaces, Proc. Amer. Math. Soc. 12 (1961), 127-132. [103] H.E Lotz, Weak, convergence in the dual of weak LP, Preprint. [104] L. Maligranda, Orlicz spaces and interpolation, Seminfirios de Matem~itica 5, Universidade Estadual de Campinas (1989). [105] J. Marcinkiewicz and A. Zygmund, Quelques th~oremes sur les fonctions ind~pendantes, Studia Math. 7 (1938), 104-120. [106] M.B. Marcus and G. Pisier, Characterisations of almost surely continuous p-stable random Fourier series and strongly stationary processes, Acta Math. 152 (1984), 245-301. [107] A. Medzhitov and E Sukochev, The property (H) in Orlicz spaces, Bull. Polish Acad. Sci. Math. 40 (1992), 5-11. [ 108] V.D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer-Verlag, Berlin (1986). [ 109] S.J. Montgomery-Smith, Boyd indices of Orlicz-Lorentz spaces, Interaction Between Functional Analysis, Harmonic Analysis, and Probability (Columbia, MO, 1994), Lecture Notes in Pure and Appl. Math. 175, Marcel Dekker, New York (1996), 321-334. [110] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer-Verlag, Berlin (1983). [111] I.P. Natanson, Theory of Functions of a Real Variable, Vol. 1, Ungar, New York (1961). [112] N.J. Nielsen, On the Orliczfunction spaces L M (0, e~), Israel J. Math. 20 (1975), 237-259. [113] E.M. Nikishin, Resonance theorems and superlinear operators, Uspehi Mat. Nauk 25 (1970), 129-191; Engl. transl.: Russian Math. Surv. 25 (1970), 125-187. [ 114] I. Novikov and E. Semenov, Haar Series and Linear Operators, Kluwer Acad. Publ., Dordrecht (1997). [115] S.Ya. Novikov, Cotype and type of Lorentzfunction spaces, Math. Notes 32 (1982), 586-590. [116] S.Ya. Novikov, E.M. Semenov and E.V. Tokarev, On the structure of subspaces of the spaces A p(#), Amer. Math. Soc. Transl. 136 (1987), 121-127. [117] R. O'Neil, Convolution operators and L ( p , q ) spaces, Duke Math. J. 30 (1963), 129-142. [118] R. O'Neil, Integral transforms and tensor products in Orlicz spaces and L(p, q) spaces, J. Analyse Math. 21 (1968), 1-176. [119] G. Pisier, Factorisation of operators through L p, ~ or L p, 1 and non-commutative generalisations, Math. Ann. 276 (1986), 105-136. [120] B. Randrianantoanina, Injective isometries in Orlicz spaces, Function Spaces (Edwardsville, IL, 1998), Contemp. Math., K. Jarosz, ed., Amer. Math. Soc., Providence, RI (1999), 269-287. [121] Y. Raynaud, Deux nouveaux examples d'espaces de Banach stables, C. R. Acad. Sci. Paris S6r. I Math. 292 (1981), 715-717. [122] Y. Raynaud, Sur les sous-espaces de LP (Lq), S6minaire d'Analyse Fonctionelle 1984/1985, Publ. Math. Univ. Paris VII, 26, Univ. Paris VII, Paris (1986), 49-71. [123] Y. Raynaud, Sous-espaces s et ggomdtrie des espaces LP(Lq) et L 4~, C. R. Acad. Sci. Paris S6r. I Math. 301 (1987), 299-302. [124] Y. Raynaud, Finie reprdsentabilitg de s dans les espaces d'Orlicz, C. R. Acad. Sci. Paris S6r. I Math. 304 (1987), 331-334. [125] Y. Raynaud, Almost isometric methods in some isomorphic embedding problems, Banach Space Theory (Iowa City, IA, 1987), Contemp. Math. 85, Amer. Math. Soc., Providence, RI (1989), 427-445. [126] Y. Raynaud, On Lorentz-Sharpley spaces, Interpolation Spaces and Related Topics (Haifa, 1990), Israel Math. Conf. Proc. 5, Bar-Ilan Univ., Ramat Gan (1992), 207-228.

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[127] Y. Raynaud, A note on symmetric basic sequences in L p ( L q ) , Math. Proc. Cambridge Philos. Soc. 112 (1992), 183-194. [128] Y. Raynaud and C. Schtitt, Some results on symmetric subspaces of L 1, Studia Math. 89 (1988), 27-35. [129] S. Reisner, A factorization theorem in Banach lattices and its application to Lorentz spaces, Ann. Inst. Fourier (Grenoble) 31 (1981), 239-255. [130] V.A. Rodin and E.M. Semenov, Rademacher series in symmetric spaces, Anal. Math. 1 (1975), 207-222. [ 131 ] H.E Rosenthal, On the subspaces of L p (p > 2) spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273-303. [132] C. Schtitt, Lorentz spaces that are isomorphic to subspaces of L | , Trans. Amer. Math. Soc. 314 (1989), 583-595. [133] C. Schiitt, On the embedding of 2-concave Orlicz spaces into L l, Studia Math. 113 (1995), 73-80. [134] A.A. Sedaev, The H-property in symmetric spaces, Teor. Funkcii Funkcional. Anal. i Prilozen. 11 (1970), 67-80 (Russian). [135] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton ( 1971). [ 136] E Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press, Cambridge (1991).

CHAPTER

13

Some Aspects of the Invariant Subspace Problem

R Enflo and V. Lomonosov Department of Mathematics, Kent State University, Kent, OH 44242, USA E-mail: [email protected]

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Invariant subspaces of algebras containing compact operators . . . . . . . . . . . . . . . . . . . . . . . . 2. Generalizations of Burnside's theorem in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Generalizations of Burnside's theorem in Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .

535 535 538 543

Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Extremal vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Definitions and some basic properties of extremal vectors . . . . . . . . . . . . . . . . . . . . . . .

543 545 546

4.2. Invariant subspaces and a two sequences theorem

...........................

547

5. Operators without invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.0. Introduction and history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Outline of the original considerations and construction . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Remarks on some other constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. More examples of transitive operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

549 549 550 555 556

5.4. A transitive operator on el . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction In this survey we present some circles of ideas and results which we feel constitute progress on the theory of invariant subspaces. We are not trying to give a complete survey of the theory as it stands today. There are many important techniques and directions - especially for operators on Hilbert s p a c e - that will not be discussed here. For some of these we refer the reader to [1,6] where also more references can be found. Behind the techniques of the first four sections in this survey is a common general idea: to find on the boundary of a convex set, points which are hard to move into the set by operators from a given s e t - and the connection between such points and invariant subspaces. In Section 1, this idea is carried out in the context of locally convex spaces. There we give results on invariant subspaces of algebras containing a compact operator. In Section 2 we consider first general Banach spaces. We present a generalization of the classical Burnside theorem for operator algebras. We also give connections to invariant subspaces of the so called two sequences theorem. So far this two sequences theorem has been proved only for Hilbert space. It should be an interesting problem to decide for which Banach spaces it is true. We discuss this further in Section 2. In Section 3 we present strengthenings of these results to operator algebras on Hilbert space and an invariant subspace theorem for essentially self-adjoint operators on real Hilbert space. The techniques in Section 3, developed by A. Simoni6 [37] - use the same underlying ideas as those in the first two chapters. However Simonic develops them in an impressive way that, for now, seems to work only in Hilbert space. In Section 4 we present techniques of extremal vectors and their connection to invariant subspaces. This techniques have, so far, been developed only in Hilbert space. However, they are, in essence, Banach space techniques, and also here, it should be interesting to find out for which Banach spaces they work. The techniques present a more constructive way to find the points which are hard to move. This leads, in particular to a strengthening of the two sequences theorem in Section 2 for some special cases. In Section 5 we present counterexamples to the invariant subspace problem. We also present results where counterexamples are used to support or disprove other conjectures.

1. Invariant subspaces of algebras containing compact operators Let E be a complex locally convex space, L(E) the algebra of continuous linear operators on E and R a sub-algebra of L(E). R is said to be transitive if E does not contain nontrivial closed subspaces that are invariant under each operator in R. A subspace M in E is said to be hyper-invariant for a subset S in L(E) if it is invariant under each operator which commutes with each operator from S. In 1935 von Neumann proved that each compact operator acting on a Hilbert space is non-transitive. In 1954 N. Aronszajn and K.T. Smith [4] generalized this statement to Banach spaces. In 1966 A.R. Bernstein and A. Robinson [10], using non-standard analysis, proved that a polynomially compact operator on a Hilbert space is non-transitive. E Halmos [21] converted their proof into one that uses only classical concepts. In 1968 W. Arveson and J. Feldman [5] proved that an algebra of operators on a Hilbert space with

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one quasi-nilpotent generator is non-transitive if it contains a compact operator. In 1973 C. Pearcy and N. Salinas [27] proved Corollary 2.2 of this paper for the case of an algebra of operators with one generator. In 1973 the second author gave a new approach to invariant subspace theorems [23]. The most general result proved by this method was the following THEOREM 1.1 ([22]). Let R be a sub-algebra of L(E). The algebra R is dense in L ( E ) with respect to the weak operator topology if and only if it is transitive and its closure in this topology contains a nonzero compact operator. PROOF OF THEOREM 1.1. LEMMA 1.2. The algebra R is transitive if and only if for each nonzero vector x E E its orbit Rx = UAER A x is dense in E and if and only if for each nonzero functional f E E* its orbit UAER A* f is dense in E* in the w*-topology. Proof of the lemma is obvious. LEMMA 1.3. If a transitive algebra R E L ( E ) contains a nonzero compact operator K, then this algebra contains a compact operator with a nonzero fixed vector. PROOF. From the definition of compact operator it follows that there exists an open convex neighborhood of zero U in E such that the closure K (U) of the set K (U) is compact. Let us pick a vector y E E such that K y ~ O. Put V -- U + y. It is clear that the vector y can be chosen in such a way that the origin is not contained in the set K ( V ) , that is ( - K y ) q~ K ( U ) . From Lemma 1.2 it follows that for any element x ~ K ( V ) there exists an operator A 6 R such that A x ~ V. Therefore the pre-images A - I ( v ) of the open set V generate a covering of the compact K (V). We can choose from that covering a finite sub-covering {Ui }, i = 1 . . . . . n. By construction there exist operators A 1. . . . . An in R such that Ai (Ui) Q V. Since a compact topological space is normal, there exists a partition of the identity corresponding to the covering {Ui }, therefore there exist continuous functions f l . . . . . fn in K ( V ) such that 3~ ~> 0, Y~in_=l j~ ~ 1, f/(x) - - 0 if x q~ Ui. Consider a map 4~" K ( V ) --+ E defined by q~(x) -- y~in 1 fi(x)Aix. The right hand side of this equality is a convex combination of elements from the convex set V, therefore 4~ (x) 6 V if x E K (V). The composition K 4~ of the maps K and 4~ is a continuous map of the convex compact K (V) into itself. By the Schauder-Tikhonov fixed point theorem the map K4~ has a fixed point, therefore there exists an element xo ~ K ( V ) such that K ~ ( x o ) = xo and Y~in=l K 3 ~ ( x o ) A i x o - xo. The operator K1 -- ~Qinl ~ ( x o ) K A i is compact, is contained in R and has the fixed vector x0. Recall that x0 7~ 0 because K (V) does not contain the zero vector. [3 C O R O L L A R Y 1.4. Let A be a linear continuous operator on a complex locally convex space E which is different from a scalar operator and commutes with a nonzero compact operator in E. Then the operator A has a nontrivial closed hyper-invariant subspace, that is a subspace invariant for all operators commuting with A.

Some aspects of the invariant subspace problem

537

This seems to be the first invariant subspace result for general locally convex spaces. PROOF. Let R be the set of all operators from L ( E ) commuting with A. Assume that R is transitive. Then according to Lemma 1.3 there exists a compact operator K E R and an element x0 6 E such that K xo = xo ~: O. Let E1 be the linear subspace of E consisting of all fixed vectors of the operator K. It is clear that E1 is a finite dimensional subspace invariant for the operator A. Therefore A has an eigenvector in E with an eigenvalue )~. Let E2 be the subspace consisting of all eigenvectors of the operator A corresponding to the eigenvalue )~. This subspace is not empty and does not coincide with E because the operator A is not scalar. It is clear that this subspace is invariant for R, so R is not transitive, ff] COROLLARY 1.5 (Schur's lemma). If a transitive algebra of linear continuous operators in a complex locally convex space contains a nonzero compact operator, then the commurant of this algebra consists of scalar operators. LEMMA 1.6. If a transitive algebra R C L ( E ) contains a nonzero finite dimensional operator K, then this algebra contains a nonzero one dimensional operator F(e ~, e) such that F(e ~, e)x -- (e',x)e, where e ~ ~ E*, e, x E E. PROOF. Put E1 -- K (E). From Lemma 1.1 it follows that vectors of form K A x , where A 6 R, generate a dense set in E1 so that operators of form KA restricted to E1 generate a transitive algebra of operators in El. According to the classical Burnside theorem this algebra coincides with L (El). Therefore there exists an operator A 6 R such that K Ax = (e', x)e for any x c El, where e 6 E l , e' 6 E*. [] LEMMA 1.7. If a transitive algebra R C L ( E ) contains a nonzero finite dimensional operator, then the closure of the algebra R in the weak operator topology coincides with L(E). PROOF. Let F(et, e) be an operator which exists by Lemma 1.6. Since AF(et, e ) = F(e', Ae) and F(e ~, e)A = F(A*e ~, e), where A* is an operator dual to A, from Lemma 1.2 it follows that the closure of the algebra in the weak operator topology contains any one dimensional operator and therefore any finite dimensional operator. Finite dimensional operators are weakly dense in L ( E ) , so the lemma is proved. [-q LEMMA 1.8. Let K be a compact operator in E, p ( K ) its spectral radius. If p ( K ) < 1, then the sequence {K n } of powers of the operator K converges to zero in the strong operator topology. PROOF. Let U be an absolutely convex neighborhood of zero in E which the operator K maps into a relatively compact set and let p be the Minkowsky functional corresponding to this neighborhood. Let E u be the quotient space E~ ker(p). The norm in E u is generated by the functional p. Let the Banach space E u be the completion of the space E u . The canonical map 1 7 : E --+ E u is continuous. Since the kernel of the operator K contains

538

P. Enflo and V. Lomonosov

the kernel of the map H in the space E u , the operator K 1 is well defined by the formula Kl x = H K H - 1x. The operator K1 is extended continuously to an operator K2 acting in the space Eu. It is easy to check that the operators K1 and K2 are compact and K, KI and K2 have the same nonzero eigenvalues. Therefore the spectral radii of these operators coincide. From Gelfand's spectral radius formula it follows that A

lim p ( K ~ / T x ) - -

n - + oc~

lim p ( K ~ H x ) =

n-+oo

lim

n - + cx3

p(Knx)=O

(1)

for each element x E E. For this reason Knx ~ U, Kn+lx ~ K ( U ) for sufficiently large n and the set {Knx} is relatively compact. Let S be the set of the condensation points for the sequence {Knx}. It is clear that K ( S ) -- S and that the set S is contained in the kernel of /7 since l i m n ~ p ( K n x ) = 0. This implies that S E ker(K) and consequently S = {0}. 7q LEMMA 1.9. If a transitive algebra R contains a nonzero compact operator, then it contains a sequence which converges in the strong operator topology to a nonzero finite dimensional operator PROOF. By Lemma 1.3 the algebra R contains a compact operator K with an eigenvalue

)~l -- 1. Let )~l . . . . . )~n be all eigenvalues of K which are no less then 1 by modulus, let/Zl . . . . . /Zn be the maximal dimensions of Jordan cells in the corresponding spectral invariant subspaces and let M be the linear span of those subspaces. By the Leray-Riesz spectral decomposition Theorem [15] the space E is the direct sum of M and another subspace N, which is also invariant under K, and the spectral radius of the restriction KIN is strictly less than 1. Let Q be the projection operator onto M parallel to N. Put P()~) = (~. - 1) # 1 - 1 1 - - I i n _ 2 ( ) ~ )~i) lzi. It is clear that the operator P ( K ) has a nonzero restriction P(K)]M and P ( K ) K n x -- K n P ( K ) X = P ( K ) x for each x 6 M and for all nonnegative integers n. For each x E E put x = x l + x2 for x l 6 M, x2 E N. Then -

-

P ( K ) K n x : P ( K ) K n x l -+- P ( K ) K n x 2 = P ( K ) x l + P ( K ) K n x 2 . The second term in this formula converges to zero by Lemma 1.8. So the sequence P (K) K n converges in the strong operator topology to the finite dimensional nonzero operator P ( K ) Q . It is clear that P ( K ) K n E R, which finishes the proof. D Combining Lemmas 1.7 and 1.9 completes the proof of Theorem 1.1.

2. Generalizations of Burnside's theorem in Banach spaces In 1991 Theorem 1.1 was generalized for operator algebras in the following way [25]. In Theorem 2.1 below, let B be a Banach space, L ( B ) the bounded linear operators on B and K (B) the compact operators on B. Let IIIA III denote the essential norm of A, that is the distance from A to the space of compact operators.

Some aspects of the invariant subspace problem

539

THEOREM 2.1. Let R be a weakly closed sub-algebra of L(B), R ~ L(B). Then there exists x E B** and y ~ B*, x ~ 0 and y ~ O, such that for every A E R ](x, A'y)[ ~< IIIAIII.

(1)

For a given subset S C L ( B ) and a vector x 6 B put Sx UA6S Ax and S' = U A*. We say that S is transitive if it does not have a nontrivial invariant subspace and S is essentially transitive if the conclusion of Theorem 2.1 is false for S. We say that a subalgebra R C L (B) has the Pearcy-Salinas (PS) property if there exists net {Ac~} C R and a nonzero operator A 6 L (B) such that: =

lim(x , A ~*y ) - - ( x , A * y)

(2)

Ol

for every vector x E B** and every functional y 6 B* and lim IIIAc~ III - 0.

(3)

ot

Of course, every bounded operator is a weak limit of finite dimensional ones, so it is the assumption {A~ } C R that makes the condition nontrivial. COROLLARY 2.2. Let R be a weakly closed proper sub-algebra of L ( B ) with the (PS) property. Then the algebra R ~ is non-transitive. PROOF. Let x and y be as in Theorem 2.1. Then for every pair of operators T1, T2 in R we have

I(x,

T~A*TOy)I -

liml(x,

T2A~TOy)I <~limlIIT2A~TIlI[

~< lim IIT2 II IIZ~ IIIIIAc~ IIIo/

It is easy to see that one of the three subspaces

A ker(T*), TcR

R'y,

Span

(T~A*T;y)

Tj , T 2 c R

is a nontrivial invariant subspace for the algebra R f. From Corollary 2.2 we get the following strengthening of the Pearcy-Salinas theorem [27].

COROLLARY 2.3. Let T be a bounded operator on a Banach space and let R be its commutant. Assume that there is a nonzero operator A and net {Ac~} of operators that commute with T such that (2) and (3) hold. Then the algebra R I is non-transitive.

P. Enflo and V. Lomonosov

540

The commutant of T is a proper sub-algebra of L(B), so Corollary 2.3 is an immediate consequence of Corollary 2.2. In Hilbert space Theorem 2.1 has the following equivalent formulation. THEOREM 2.4. Let R be as in Theorem 2.1 and let B be a Hilbert space. Then there exist 11)

113

two bounded nets (x~) in B and (y~) in B such that x~ --+ x # O, y~ --+ y ~ O, and for every A ~ R (Axe, y~) --+ O. PROOF OF THEOREMS 2.1 A N D 2.4. Let B be a Banach space, Q a compact Hausdorff space, R a sub-algebra of L ( B ) , WQ the Banach space of weak*-continuous functions Q ~ B* with sup-norm, SQ the subspace of WQ consisting of the strongly continuous functions and NR the algebra of norm-continuous functions Q --+ R t. Let C = C (Q) be the algebra of all complex-valued continuous functions on Q. Every function f 6 C (Q) is a linear operator on the space WQ. We let f * denote its adjoint. DEFINITION. Let M be a subspace of WQ, invariant under the algebra C. We say that 0 6 M* is a point functional if there exists a point q 6 Q such that

f*O = f (q)O

(4)

for every function f E C. If E is a subspace in a Banach space let UE denote the unit ball of E and E • denote the annihilator of E in the dual space. We say that a functional 0 6 M* is a 6 (q, x) functional or simply 6-functional if there exists a point q E Q and an element x 6 B** such that if h 6 M then

O(h)-(x,h(q)).

(5)

LEMMA 2.5. Let 0 ~ M* be a point functional and assume M D SQ. Then the restriction, X = OIsQ is a 8-functional. Let H be a Hilbert space with orthogonal basis {en}, LIM(an) a Banach limit in the space l ~ , Q the compactification of the integers by {cx~}. Then O(h) = LIM(h(n), en) is an example of a point functional that is not 8-functional. Let Q x U8 be the topological product of the compact Q and the ball U8 with norm topology. The functions s ~ WQ and f 6 C (Q) define complex valued functions L (s) and Ll ( f ) on Q x U8 bythe formulas L(s)(q, x) = (x, s(q)), Ll ( f ) ( q , x) = f (q). LEMMA 2.6. L(s), L1 ( f ) are continuous functions on the set Q • UB. LEMMA 2.7. Let r ~ L(M)*, f ~ C(Q) and (L1 f)*cp = gcP. Then f*(L*qo) = g(L*qg).

541

Some aspects of the invariant subspace problem

PROOF. F o r a given function h E WQ we have (h, f * (L* go)) -- ( f h, L'go) -- ( L ( f h), go) -(L1 ( f ) L ( h ) , go) -- (L(h), L1 ( f ) * go) -- (L(h), Fgo) -- (h, FL*go). [] LEMMA 2.8. Let M, N be subspaces of WQ, invariant for C(Q). Let M ~ N and let 0 E M* be an extreme point of UN• Then 0 is a point-functional. The proof of this lemma is similar to de Branges famous proof of the Stone-Weierstrass Theorem [ 13]. LEMMA 2.9. Let T E K ( B ) and h E WQ. Then T*h ESQ. Let R be an algebra in L ( B ) and let h E WQ, IIh II - 1. By applying the algebra N R to h we get subspaces

N-

NR(h),

M-

Span(N, SQ).

then M D N and M and N are invariant for the algebra C (Q). Let 99 be a point functional, q9 E M*. Then, by L e m m a 2.5, golso is a 6(q, x) functional. LEMMA 2.10. If the functional go is in N • then for every operator A E R, (6)

I(x, A*h(q))l ~ 21gollllAIIIIhl.

Let OfF(T) be the subset of the spectrum of an operator T E L ( B ) consisting of the isolated points of finite multiplicity. LEMMA 2.1 1. Let the subspace L C B have finite co-dimension. Assume ~ > 0 and

Ilrxll < or. Ilxll

sup-

~L

(7)

Assume y E o-(T) and lyl > o~. Then y E CrF(T). LEMMA 2.1 2. If R is an essentially transitive sub-algebra, then R contains an operator T with thefollowingproperties" 1 E OfF(T) and ifF E cr(T)\CrF(T) then lyl < 1/2. LEMMA 2.13. Let R be a uniformly closed essentially transitive sub-algebra of L(B). Then R contains a nonzero finite dimensional projection. PROOF. Let T be an operator as in L e m m a 2.12. Since {9/" (T - F I ) - I E R} is a component of the resolvent set of T it follows that if (T - )/I)-1 exists and [VI ~> 1/2 then ( T - y i ) - I E R. By L e m m a 2.12 there exists a circle cr C C such that 1 is the only point of or(T) inside c~ and for every point y E o~, (T - y I) -1 E R. By the Riesz theorem P - -27ri 1 for (T -

V 1) -1 dy is a non-zero finite dimensional projection and P E R ~. If P -- P~ then P1 is finite dimensional projection in R. []

P. Enflo and V. Lomonosov

542

Now to finish the proof of Theorem 2.1 we need the following well-known fact [29]. LEMMA 2.14. I f a transitive algebra R contains a non-zero finite dimensional operator, then R is weakly dense in L ( B ). Essential transitivity implies transitivity, so by combining Lemmas 2.13 and 2.14 we finish the proof of Theorem 2.1. Of course Lemma 2.13 gives the possibility of obtaining different results on density. For example one of them is THEOREM 2.15. I f H is a Hilbert space and R is a uniformly closed essentially transitive sub-algebra of L ( H ) then R contains all compact operators in L ( H ) . Now we will prove the equivalence of Theorem 2.1 and Theorem 2.4 in Hilbert space. We first prove that Theorem 2.4 implies Theorem 1 if B is Hilbert space. Assume that Theorem 2.4 holds. Let C = lirl~ Ix~ I" lY~ I. If A 6 R then 0 = lim(Ax,~, y~) -- lim(A(x,~ - x), (y~ - y)) - (Ax, y) + lim(Ax, yot) 13l

lY

+ lim(Ax~, y), (y

where lim(Ax, y~) = lim(Ax~, y) = (Ax, y). ly

o/

Thus (Ax, y) = lim(A(x - x~), (y~ - y)). If K is a compact operator, then

l(Ax,

Y)I ~< li-~ I((A- g)(x -x=),

(y~ -

Y))I + l~ml(g(x -x~),

(y~ -

Y))I

I I A - KII4C. Now let z = y / ( 4 C ) . Then I(Ax, z)] ~< i~cf IIA - gll = IIIAIII, so Theorem 2.1 follows from Theorem 2.4. To prove that Theorem 2.1 implies Theorem 2.4 we use the following result of Glimm [20]. B is assumed to be Hilbert space. LEMMA 2.16. Let qJ ~ L(B)* and q/ ~ K (B) • Then there is a pair ofbounded nets (x~) and (y~) such that

Some aspects of the invariant subspace problem

543

(1) w-limaXa = 0 , w-lima Ya = 0 . (2) For every A ~ L ( B ) , q-/(A) = lirna(Axa, Ya). Now let q-'l (A) -- ( A x , y) where x, y are as in Theorem 2.1 and A 6 R. Theorem 2.1 gives that qq can be extended to Span(R, K ( B ) ) by putting q-tl ( T ) - - 0 for T E K ( B ) . Now let q-' be a Hahn-Banach extension of the qq to all of L (B). We now use L e m m a 2.5 and we get Theorem 2.4 with xa = x 4- Xa and Ya = Y - Ya. 7] Finally we will mention that in the case of a non-reflexive Banach space Theorem 2.1 gives invariant subspace corollaries only in the dual space B*. Since the first author in 1976 [16,17] showed that there are counter examples to the invariant subspace problem in general Banach spaces, this may be a sign that the following result is true: CONJECTURE 2.17 ([25]). I f A is a bounded linear operator in a Banach space then A* has a non-trivial invariant subspace. The known counterexamples do not contradict this conjecture. By Corollary 2.2, this would be true if the following is true: If A is a bounded linear operator in a Banach space then there exists an algebra with the (PS) property which contains the operator A. S.W. Brown [11] independently obtained similar results in the case of a commutative algebra.

3. Generalizations of Burnside's theorem in Hilbert spaces Given an operator A on a Hilbert space we define Re A - - A + A2 * and I m A A-A* With 2 " the normalizing conditions Ixl = lYl = 1 inequality (1) from Section 2 has the form -

I(x, A *y)[ <~ c IIIA III.

-

( 1)

In 1996 A. Simonic found an exact form of inequality (1) for operator algebras in a complex Hilbert space by proving the following THEOREM 3.1. Let R be as in Theorem 2.1 and let H be a Hilbert space. Then there exist two nonzero vectors f , g E H such that f o r every A ~ R IRe(Af, g)l <~ 111Re Alil(f, g).

(2)

Sketch o f the p r o o f

Let H be a real or complex Hilbert space. Fix a unit vector f0 6 H and choose a positive r 6 (0, 1). The set S is defined as follows: S-

{ f E H" ]if0 - fll ~< r}.

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P. Enflo and V. Lomonosov

LEMMA 3.2. Let W be a subset of S and let A be a bounded linear operator on H. Suppose that Re(A f, f0 - f ) ~> 3 ~> 0,

for all f ~ W. Then there exists a positive number lZ >~O, such that for any e ~ (0, #):

lifo - ( z + cA)f I[ < lifo -

f[[,

for all f ~ W. DEFINITION. For a fixed ball S = {f 6 H: lifo - fl[ ~< r} around the unit vector fo 6 H, the polar hyper-plane Ps, of S with respect to the origin, is defined by the following set:

P s - - f E H: (f, fo)-----1--r 2. LEMMA 3.3. The function Ao : S w-> S, defined by 1

A o ( f ) = r2 + (f, fo) f ' maps the set S weak-continuously into itself Furthermore, the set of all fixed points for Ao is equal to Ps (3 S. DEFINITION. For every operator A in L ( H ) define a real function A A " S ~ ~ as follows: 1

A A ( f ) ---- r2(1 _ r2 ) Re(Af, fo - f ) . LEMMA 3.4. Suppose f is a vector in the polar hyper-plane Rs C3S, satisfying the following strict inequality for some A E L ( H ) : AA ( f ) > IIIRe

AIIIAI(f).

Then there exists a positive number ~ > O, together with a weak neighborhood W of f , such that for every h ~ W U S: AA(h) > IIIReAIIIAI(h)I +S. PROPOSITION 3.5. Let A ~ F~(TY) be a convex subset of the bounded linear operators acting on a real or complex Hilbert space H. Fix a unit vector fo E H and choose a positive number r 6 (0, 1). Suppose that for every vector g _J_fo and I[gl[ <<,1 there exists an operator A ~ A satisfying the following strict inequality:

ReA

f0+

/ 1- r z g

,fo-

r

g > > ][[Re A [[[(1 - [[g [[2).

Some aspects of the invariant subspace problem

545

Then A contains an operator Ao with an eigenvector in the set S={f6Hlllfo-fll~ I11Re AIII. PROPOSITION 3.6. Suppose H is a real or complex Hilbert space, and )~ ~ C is a point in the spectrum of the operator A ~ L ( H ) such that I ReZl > Ill Re AIII. Then the norm closure of the algebra generated by A contains a nonzero finite-rank operator Similar techniques can be used to prove PROPOSITION 3.7. Suppose H is an infinite-dimensional complex Hilbert space and let R be a convex set of commuting essentially selfadjoint operators. Then the set of vectors x with non-dense orbits Rx is dense in H. COROLLARY 3.8 ([26]). Every essentially selfadjoint operator on infinite-dimensional complex Hilbert space has a nontrivial closed real invariant subspace. From 3.7 one can derive the remarkable THEOREM 3.9. Every essentially selfadjoint operator on infinite-dimensional real Hilbert space has a nontrivial closed invariant subspace.

4. Extremal vectors Extremal vectors were introduced in Ansari and Enflo [2] and Enflo [19]. For results on special operators see also Spalsbury [38]. Extremal vectors provide a tool to find hyperinvariant subspaces for compact and normal operators on Hilbert space in a unified, more constructive way. They also provide a tool to strengthen Theorem 2.4 in some special cases. For most of the proofs we refer the reader to [2]. Since Theorem 4.8 is presented here for the first time, we give a complete proof. We feel that the extremal vector technique for obtaining invariant subspace results is still in its beginning. The origin of this is to study best approximate solutions of the equations Tny = xo for the case when T has dense range and x0 is not in the range of T. More precisely, one considers, for e > 0 and for each n, the Yn of minimal norm such that

II

- xo 11

It turns out that T n Yn in many cases converges to a vector which is hyper-noncyclic for T, that is non-cyclic for all operators commuting with T.

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P. Enflo and V. Lomonosov

4.1. Definitions and some basic properties o f extremal vectors H will denote a Hilbert space over the field of complex numbers, R ( T ) will represent the range of T. Backward minimal vectors. Let T ' H ~ H be a bounded operator with dense range. Let x0 6 H and e > 0 with e < [Ix0ll. There is a unique vector Yn,xo such that T Y n , x o - x0l] ~< e and

]lYn,.0II- inf{llyl[-

[[Zny

-

x0]] ~ e}.

The points yen,xo are called backward minimal vectors. Forward minimal vectors. Let T" H --+ H be an injective bounded operator. Let xo 6 H and e > 0 with e < Ilxoll. There is a unique vector V.,x E o such that IIV n~, x o -- XOII ~< ~ and

I[Tn Un,xo ~ II =

inf{ I1Tn v 11" 11v - x0 l[ ~<e } 9

The points T n Un,xo e a r e called forward minimal vectors. W h e n there is no ambiguity, we will drop some of the superscripts and subscripts from Yne,xo and Vne,x0. Sometimes the vectors T n Yn and Vn will also be referred to as backward, respectively, forward minimal vectors. It is clear that [[T n Yn,xo e - xo 11= e and [IU n - - XO 11 -e. The minimality gives easily Orthogonality relations.

If rn I Yn then Tnrn I T n Yn - x o . And we get THEOREM 4.1 (Orthogonality equations). There exist 6~ < 0 such that ye = ~ T* (Tye - xo) a n d (possibly another) 6~ < 0 such that (rE - xo) -- 6~ T * T v e .

REMARK 1. From the orthogonality equations we obtain Tnyn = - S n ( I - 8 n T n T * n ) - l T n T * n x o

and

Vn : (I - 6 n T * n T n ) - l x o .

Let An - - 6 n ( I - 6 n T n T * n ) - l T n T *n and Bn - - ( I 6nT*nTn) -1. Then An and Bn are positive operators. ][An][ < 1 and [[Bn[[ = 1 for all n. Let f n ( z ) - 1-8,.~ -~,.z . Then fn (z)

Some aspects of the invariant subspace problem

547

is analytic in a neighborhood of a ( T n T *n) and a ( A n ) = f n ( a ( T n T * n ) ) . The maximum -~n'llrnll2 . It follows that point of f ( a ( T n T * n ) ) is i---~.:ii~l~

[IAnll =

- a n . IIT n I12 1 -

~n"

< 1.

IITn II2

1 Then gn is analytic in a neighborhood of a ( T n T *n) and Now let g n ( z ) 1-~n.z a ( B n ) = g n ( a ( T n T * n ) ) . Since 0 E a ( T n T *n) it can be verified that the maximum point of a (T n T *n) is 1. Hence l]Bn l] = 1. The proofs of the following results can be found in [2]. .

THEOREM 4.2. Let xo E H and xo =/: O. The functions e w+ ye and e w-~ ve are analytic on (0, Ilxoll). THEOREM 4.3. For any b o u n d e d operator T with dense range, if xo q~ R(T), then Tyen-xo

Tyen

\

IIT y~, - xoll' IIT y~, II f o r some sequence 6n ~

/ --~0

O.

4.2. Invariant subspaces and a two sequences theorem LEMMA 4.4. I f T is quasi-nilpotent then IlYnk II/llYnk+l II ~ 0 as k --+ cx~f o r some subsequence nk. This follows easily from the spectral radius formula. In the following theorem, let K be a quasi-nilpotent, compact operator without eigenvector and let (nk) be as in L e m m a 4.4. We have THEOREM 4.5. For every weak accumulation point z o f K nk Ynk, K z is hyper-noncyclic f o r K and { T z [ K T = K T } is a hyper-invariant subspace f o r K. PROOF. By passing to a subsequence, we can assume K nk+l Ynk + 1 -'-+ YO weakly. We have T K n k + 1Ynk --+ T K z

in norm.

Put Tynk -- OtkYnk+l + rk rk 3_ Ynk+l. Then, obviously, Olk --+ 0 as k --+ cx~. We get T K nk + 1Ynk --- OtkK nk + 1Ynk + l -+- Knk + l rk 9

548

P. Enflo and V. Lomonosov

Scalar multiplying both sides by K nk + 1Ynk+ 1

--

XO

gives

(TKnk+lyn k, Knk+lyn~+l -- xo) = Otk(Knk+lynk+l, Knk+l ynk+l -- XO) + (Knk+l , Ynk+l Knk+lynk+l -- XO).

The first term on the fight-hand side --+ 0 since otk --+ 0. The second term is 0 because of the orthogonality relations 9 The left-hand side --+ ( T K z , Yo - xo) as k --+ cx~. Since the yf angle between Knyn and x o - Knyn is/> ~, Ilyo][2 ~< liminflIKnynll 2 ~ I[xoll2 - e 2. Thus Yo - xo # 0 and the theorem is proved. O We also have THEOREM 4.6. Assume that T is normal with dense range R ( T ) and that T has a cyclic vector. Assume xo ~ R ( T ) 9 Then f o r each e > 0 Tny rel , X 0 converges in norm to a hypernoncyclic vector f o r T. To prove the following two sequences theorem we use the following THEOREM 4.7. Let xo ~ H, xo q~ R ( T ) . I f f o r every 61 > 0 and all 62 with 0 < 62 < 61, there exists r > 0 with (T n Yne, Tn Yne -- xo) <~ --r f o r all n >~ 1 and all e, $2 ~ 6 ~ SI, then xo is hyper-noncyclic f o r T.

From this we derive the following THEOREM 4.8. Let R be a commutative algebra of operators on H which contains a O9 O) quasi-nilpotent operator. Then there exist two sequences Xn --+ x # 0 and Yn ~ Y # 0 such that f o r every bounded sequence An of operators in R, (Anxn, Yn) --+ O. PROOF. Let T E R be quasi-nilpotent. Assume xo ~ R ( T ) . Then by Theorem 4.7 either x0 is hyper-noncyclic or, for some e > 0 and some subsequence (n~) of the integers

(1)

(T n~ Yn~ , Tn~ Yn~ - xo) --+ O,

where (Yn~) are the backward minimal vectors with respect to x0 and e. We write Tn~yn~ = ( 1 - e2)x0 + y E V / 1 - e2s0 + V / 1 - y 2 E v / 1 - e2sv,

where Ilsoll = I l s ~ l l - 1 and sv -~ O. (1) then gives ((1 - e2)xo + yew/1 - e2so + V/1 - y2eV/1 - e2s~, e x o - yv/1 - e2xo

-,/1-

o.

549

Some aspects of the invariant subspace problem Now, let Av be a bounded sequence of operators in R, let

xv = ( 1 - e2)xo + y e ~ / 1 - e2so + V / 1 - v2ev/1 - e2sv and let

yv -- exo - Y v / 1 (2)

Then xv --+ (1

-

g2)x0

-+-

e2so - ~ / 1 - y 2 v / 1 -

ge~/1

e2sv. O)

-

g2s0

--

X and yv --+ exo - y~/1

-

62s0

=

y. Moreover

I(A~x~, yv)l -- [(A~T~Y.~, T"~Y.~ xo)l- [(T"~A~y.~, T"~Y.~ -

-

I(z~(~y~

-

xo)l

+ rn~), T n ~ y n ~ - xo)]

<~ ]Oln~[I(T n~ yn~ , Tn~ yn~

_

xo>l+ [(r" v rn~ ,

Tnv

Yn~ - xo)] .

(2)

Since Oln~ is bounded, by (1) the first term on the right-hand side of (2) --+ 0 as v ~ oc. The second term on the right-hand side of (2) is 0, because of the orthogonality relations. This proves the theorem. []

5. Operators without invariant subspaces 5.0. Introduction and history In this chapter we will present examples of operators on Banach spaces with only trivial closed invariant subspaces. We will refer to them as transitive operators. In this section we will give a short history. The first construction of a transitive operator was given by the first author, who completed and circulated in preprint form a first manuscript in 1975. An outline of the construction was presented in Seminaire Maurey-Schwarz in the Spring of 1976 [16]. (See, also the survey article from 1982 by Radjavi and Rosenthal [30].) An improved manuscript of the same construction appeared in the Institute MittagLeffler report series in 1980 and was submitted to Acta Math. in 1981. This version with some improvements in the notation and presentation- appeared in Acta Math. in 1987. In 1984, B. Beauzamy [7] and C. Read [31] presented constructions of transitive operators on some Banach spaces using a similar approach and basic considerations but with the technical aspects worked out in different ways - a discussion and comparison will be given below. Beauzamy's construction gave the first example of an operator T on a Banach space B where all non-zero vectors are super-cyclic: for every y 6 B, y 5/=0,

{t~TKy l t~ 6 R, K 6 N } -- B. A slight modification of Read's first construction gave a transitive operator on s

550

P. Enflo and V. Lomonosov

Simplifications of the construction of a transitive operator on s were given by A.M. Davie (see Beauzamy [8]) and by C. Read [33], whose example we will present below. This example has recently been used by V. Troitsky to test several conjectures on invariant subspaces. Later, C. Read [34-36] constructed stronger examples as well as transitive operators on new spaces and we present some of his results below. But, as we have seen above, there is no example of a transitive operator which is the adjoint of another operator, a n d - in p a r t i c u l a r - no example of a transitive operator on a reflexive space. In Section 5.1, we outline the construction in [16-18]; in Section 5.2, we compare it to the constructions in [7] and [31]; in Section 5.3, we give the stronger examples of Read, and in Section 5.4, we discuss the recent work of Troitsky.

5.1. Outline of the original considerations and construction In this section we will outline the original considerations and construction of an operator with only trivial invariant subspaces on a Banach space. The Banach space will be constructed at the same time as the operator and will be non-reflexive. There are very serious difficulties in carrying out a similar construction in a reflexive Banach space. So we feel that the construction gives some support to the conjecture that every operator on a Hilbert space has a non-trivial invariant subspace. We now turn to the basic considerations behind this approach. It is clear that every operator with a cyclic vector on a Banach space can be represented as multiplication by x on the set of polynomials under some norm. So what we will do is to construct a norm on the space of polynomials and prove that multiplication by x under this norm has only trivial invariant subspaces. Our next basic consideration comes from the fact that one can have an operator with a dense set of cyclic vectors without having all vectors cyclic. In order to be able to make some limit procedure work, we will construct the operator so that it has the following uniformity property: Let 1 be a cyclic vector of norm 1 in B. Let (p j) be a sequence which is dense on the unit sphere of B. For every m there is a positive number Cj,m such that for every Pn with [[Pj -- Phil < 1/2 m+4 there is a polynomial s in T with [[s <~ Cj,m such that [[s 111 ~< 1/2 m. It is easily verified that such a T has only trivial invariant subspaces (1.1). It follows easily from the fact that the spectrum of an operator is non-empty that there is no operator for which Cj,m depends only on j or only on m. If we have the operator T represented as multiplication by x, then s will just be multiplication by the polynomial s From now on we shall identify B with the completion of the vector space of all polynomials with real (or complex) coefficients under a suitable norm l] 1]. This leads us to the next basic consideration. Assume that we have a norm II l[ on the space of polynomials. Assume that p is a polynomial of norm 1 and assume that [[s - 111 ~< e and 11s[lop <<,K. This gives that for every polynomial h we have the inequality I l h l l - KIIhpll ~ IIh~p- h. 111 ~ EIIhllop.

Some aspects of the invariant subspace problem

551

This implies that if

1

Ilhllop ~ ~

Ilhll,

then

Ilhll

Ilhpll ~ ~ .

(1.2)

In order that the operator also satisfies (1.1), it is of course necessary that the inequality Ilhpll ~> Ilhll/2K holds uniformly in p in every ball of size e / 1 6 on the unit sphere, at least if we put e = 1/2 m. This uniformity will lead us to the concept of "concentration at low degrees" for polynomials, defined below. The concept is used also in later constructions of transitive operators. In many norms a neighborhood of a polynomial of degree n consists of polynomials with some concentration up to degree n. There is a sense in which the inequality (1.2) is sufficient for p to be moved close to 1 by a polynomial with small operator norm. This is given by our Theorem 5.1.2 below. In order to describe this theorem, we have to tell something about the way that we construct the final norm. It should be pointed out that this sufficiency of (1.2) depends on the fact that the norm constructed is non-reflexive. We do not know whether anything similar can be done in a reflexive space. Consider all pairs (q, e) where q is an arbitrary polynomial whose coefficients have real and imaginary parts rational, and e is of form 2 -k. We enumerate all such pairs and call the sequence (qn, en). We also insist that for a fixed q, if n l, n2 . . . . are all the integers such that qn = q, then e n l > En2 > en3 . . . . . Also we assume deg qn <~ n. Our construction will be completely determined by a sequence of polynomials ~n and constants C~ > 2. el . . . . . ~k and C1 . . . . . Ck will determine a number ak+l inductively as explained below and we define a sequence of norms as in the following definitions. DEFINITION. For any polynomial p, consider all representations p -- ~ a i , ~ x i g.~l I . . . g,~nn and put [Plopn -- inf Z where cients.

lai,3l 2i (Clel l1 )~' 9. . (Cnl~ n l1 )~n

I1~ denotes the usual el norm equal to the sum of the absolute values of the coeffi-

REMARK. In the final norm the operator x will have norm ~< 2, and multiplication by s norm ~< Ck I~11. DEFINITION. For any p, consider all representations

p -- r + ~

Sk (g~kakqk -- 1). 1

Put Ipl ~ = inflr]l -+- Z~lSklopnek. Put Ipl ~ = IPll, and let a~ be determined inductively by the condition la~qk I~- 1 = 1. REMARK. [g~kakqk -- 1[ n < ek and clearly the operator norm of multiplication by g is [g[opn. We see that [[n is the maximal norm satisfying the following four properties:

P. Enflo and V. Lomonosov

552

(A1) II n

~lll, (A2) le~a~q~ - 11n ~ Ek, k = 1,2 . . . . . n, (A3) [gklop <. Ckls (A4) Ix lop <. 2. Observe that II n and I lopn are decreasing sequences of norms and hence converge to some pseudo-norms. We write II II = l i m l l nTHEOREM 5.1.1. Assume that (Cn) and (gin) are sequences which define norms l] m as above and assume that there are sequences o f positive numbers Dn /7 cx~ and Ln f cx~ so that the following holds: (I) [plm is c o n s t a n t f o r m >~ ( d e g p ) - 1. Inparticular [qn[m is c o n s t a n t f o r m ~ n - 1. (II) For any n, consider all k <. n such that 6 n - - 6 k , and ]akqk - anqn[ (n-l) < en/16. Let K be the least such k. Then [s [l = L K , Cn = DK. Then the resulting limit norm defines a space B f o r which multiplication by x has no invariant subspace. PROOF. Let q [Iq l] = 1. Let e ank qnk --+ q in Hence for k

be an element of B, which we recall is the closure of all polynomials and be a fixed negative power of 2. Choose increasing nk such that enk = e and B. We can even insist that [lankqnk - q l[ < e / 6 4 . > 1, E

lankqnk -- an,qn, I(nk-1) --]lankqnk SO by (II), Cnk <, maxm~
Dm

-

II < ~ ,

anlqnl

and I~n~l~ ~ maxn~
II~n~q - 111 ~ Ilenkan~qn~ -- 111 +

Ilen~(q

Lm

so

that [gnk Iopnk is bounded ~< A.

--ankqnk)[[

<, e 4- [g'nk[opnk Ilq - ank qnk [I ~< e + A IIq -- ankqnk II. Letting k tend to infinity, we see that 1 is within distance e of the space generated by q and hence, letting e ~ 0, we see that 1 is in that space and hence the space coincides with B.

D

We will now drop ak in our notation so when it is clear from the context we will denote akqk by qk and assume Iq~l ~-l = 1. DEFINITION. ord p = degree of lowest order term of the polynomial p. DEFINITION. Let f be a positive real valued function defined on (0, cx~). We say that ~. - ~ j > ~ o a j x n j is more lacunary than f if ord ~ = no > / f (0)

Some aspects of the invariant subspace problem

553

and nj ~> f (n j_ 1)

for every j.

Our next theorem shows that, under the assumption of an inequality similar to (1.2) we can satisfy condition (I) as soon as the polynomials en are lacunary enough. THEOREM 5.1.2. Let el . . . . . eN, C1 . . . . . CN be given with Ck > 2. Assume f o r a l l h and some B that IhlopN Ihl ~

~.

1 6N+l

==~ IhqN+l IN

Ihl ~ B

Then, given K > 4B/eN+l, there exists a lacunarityfunction f such that if (1) leu+lll ---- K, (2) the lacunarity Of eN+l ~ f , (3) CN+I > 2, then with this choice of eu+l and CN+I we have [glN+l

_

iglN

for all g with degg ~< N.

We now assume that we have two sequences D n / 7 ec and Ln / ec. Assume that ]in--1 is defined. We will then define [in according to the following rule: consider all k ~< n such that ek = en and [q~ - qn in-1 < en/16. Let K be the least such k. Then [en l1 = LK, Cn = DK. If this rule is fulfilled for all n ~< N, we say that [iN is defined in a compatible way from the sequences D~ and Ln. If for every N, [ [U is defined in a compatible way from the sequences Dn and Ln, then obviously condition (II) of Theorem 5.1.1 is fulfilled. Our next theorem combined with Theorem 5.1.2 will now enable us to get also the condition (I) of Theorem 5.1.1 fulfilled. We first make A growth function F is a function that for every n and every 3n-tuple D1 . . . . . Dn, L1 . . . . . Ln, el . . . . . en gives a positive number F(D1 . . . . . Dn, L1 . . . . . Ln, e l . . . . . e n ) , and every (3n + 2)-tuple D1 . . . . . D n + l , L1 . . . . . L n + l , el . . . . . en gives a lacunarity function f and a positive number 6. We say that the sequence {Dn, Ln, en, Cn } grows faster than F if (1) e~ and Ck are defined in a compatible way from the sequences Dn and Ln for every k. (2) For every n, Dn+l and Ln+l are > F(D1 . . . . , Dn, L1 . . . . . Ln, e l , . . . , en). (3) For every n the lacunarity of en+l /> f and the moduli of the coefficients of en+l are ~< 3 where f and 3 are given by the growth function applied to Dj, L1 . . . . , Dn+l, L1 . . . . . Ln+l, el . . . . . en. DEFINITION.

We will, by slight abuse of language, say that a number depends only on []m thus meaning that it is determined by D1 . . . . . Dm, L1 . . . . . Lm, el . . . . . em, C1 . . . . . Cm. (Obviously different such sequences could give the same ]]m.) We now have

554

P Enflo and V. Lomonosov

THEOREM 5.1.3. There is a growth function F such that if {Dn, Ln, gn, Cn} grows faster than f , then f o r every n there exists Bn depending only on II n-l, such that f o r all N ~ n ~n

Iq - qnl N <

and

16

IhlopN

1

<

Ihl ~

imply

en

Ihq] N >1 Ihl~ Bn

We now combine this theorem with Theorem 5.1.2 to give also (I) of Theorem 5.1.1. The main difficulty in the construction is to prove Theorem 5.1.3. Completion of the construction assuming Theorem 5.1.3.

For every N ~> 0, choose L N + I and Dn+l > max{F(D1 . . . . . Dn,L1 . . . . . Ln, g.1 . . . . . g.n),4Bu+l/eU+l}.

(A)

Now we assume that for every r ~< N we have defined L1 . . . . . Lr, D1 . . . . . Dr, el . . . . . ~r, C1 . . . . . Cr according to (A) and the growth function F.

(A')

Assuming this we will choose ~N+I by the following considerations (B)-(E)" Take the smallest n ~< N 4- 1 such that IqN+l -- qnl N < en/16 and en -- eN+I. Then by Theorem 5.1.3 IhlopN

<

1

EN+I

Ihl ~

==> IhqN+ll N ~ Ihl~ Bn

By the compatibility assumption and (A) we now choose Is247 l1 -- Ln > 4Bn/en,

(B)

Dn > 2.

(C)

CN+I

--

By Theorem 5.1.2 by choosing e N+I lacunary enough we then get [pl N _ ]plN+l

for degree p ~< N.

(D)

We choose ~N+I more lacunary and with smaller moduli of the coefficients than what is given by F(D1, . . ., DN+I, L1 . . . . . LN+I, e l . . . . . ~ N ) .

(E)

By choosing the sequence {Dn, Ln, s Cn } according to (A)-(E), we thus get the following: {Dn, Ln, s Cn } is compatible by (B) and (C) and it grows faster than F by (A), (A I) and (E) so it satisfies the assumptions of Theorem 5.1.3. By (B) and (C) it also satisfies (II) of Theorem 5.1.1 and by (D) it satisfies (I) of Theorem 5.1.1. Thus for the limit norm multiplication by x has only trivial invariant subspaces. So in fact Theorems 5.1.1, 5.1.2 and 5.1.3 give

Some aspects of the invariant subspace problem

555

THEOREM 5.1.4. There is a growth function F such that if {Dn, Ln, en, Cn } grows faster than F, then limll n is a norm for which multiplication by x has only trivial invariant subspaces. The proof of Theorem 5.1.3 uses Theorem 5.1.5, also proved in [ 17,18]. Let A and B be polynomials in many variables. Let Pm A consist of those terms in A of total degree ~< m and Pn B consist of those terms in B of total degree ~< n. We say that a polynomial C has concentration d up to degree m if lPm C ll ~> d lCll where Ill denotes the el-norm, that is the sum of the moduli of the coefficients. We have the following THEOREM 5.1.5. Let A and B be polynomials in many variables with concentrations dl and d2 up to degrees m and n respectively. There exist a constant K = K ( d l , d2, m, n) depending on dl, d2, m and n but not on the number of variables such that IABI~ >1 KIAI~IBI~.

After this result, there are quite a few papers dealing with concentration at low degrees and products of polynomials in many variables. For more results and more references see [3,9,41 ].

5.2. Remarks on some other constructions

In Beauzamy [7] and Read [31 ] one also constructs a norm on the polynomials, such that multiplication by x on the completion of this space is a transitive operator. As part of the definition of the norm one declares certain vectors small and certain operators small. The purpose of having certain operators small is, of course, to pass from a dense set of cyclic vectors to having all vectors cyclic. In [7] (A1)-(A4) above are fulfilled. However, in [7] the e n ' S a r e monomials, e, - - O l n xr(n) where r(n) is rapidly increasing. This makes all non-zero vectors supercyclic. In particular, OlX i

--

1 is small for certain i ls and otis.

(2.1)

In the definition of the norm, Read [31] prescribes (A4). (A2) and (A3) are replaced by "hybrid" bounds, that is bounds on some projections in the space and combinations of multiplication by x and these projections. From that one derives that (2.1) holds also in the example in [31 ], and one constructs bounded en'S with arguments like Lemma 5.2.1 below where the idea above of concentration at low degrees is important. In Section 5.1, [III is defined to be the maximal norm satisfying (A1)-(A4). However, any norm satisfying (A1)-(A4) would work: maximality is not essential. Similarly in [31] minimality is used in the definition of the norm but is in [32] replaced by another condition to get a transitive operator on el. When, in all these examples, certain elements and operators are declared small, it is important that the norm does not collapse to 0. So to make few elements and operators small one uses both above, in [7] and [31 ], and in later examples, rapidly increasing sequences.

556

P. Enflo and V. Lomonosov

In Davie's simplification of Read [32] the following concentration at low degrees result is used. LEMMA 5.2.1. Let r >~ O, 6, 6, M > O. There exists K > O, such that f o r every m, 0 <~ m <. n, every polynomial g ofdegree <. n with IPmg[1 ~ ~ and Igll <~ M there exists q o f degree <<.n such that IPn(gq)

-

and

Zmll < 6

Iqll ~< K.

5.3. More examples o f transitive operators The first examples of transitive operators were strengthened in different ways by Read [3436] and in this section we will give some of the main results of these papers. We say that a vector x is hypercyclic for an operator T on a Banach space B if x has a dense orbit, that is {Tnx I n >~ 0} -- B. The following theorem was proved in [34]. THEOREM 5 . 3 . 1 . If B is s 1 E3 W where W is a separable Banach space, then there is an operator on B f o r which every non-zero vector is hypercyclic. This result strengthens earlier results in two ways: the class of spaces is very big and the operator has a property even stronger than having all non-zero vectors supercyclic. In [35] Read gave the first examples of transitive operators on spaces not containing s To describe this class we first make some definitions. DEFINITION. Let J be the James space, the Banach space of all sequences 4~ - (ai)~=l in Co such that

l[4~ll =

sup Pl
]api+l _ a p i

[2

< 00.

i=1

Put J ~ - - s ~]~iC~=l Ji where the infinite direct sum is the s and each Ji -- J. Now, let C be the class of all Banach spaces B of the form B = Y @ W where Y is Co or J ~ and W is any separable Banach space. Then we have

THEOREM 5.3.2. I f B is in C then there is a transitive operator on B. Recently Read [36] has modified his earlier construction to get a transitive quasinilpotent operator on s

5.4. A transitive operator on s 1 In this section we will describe the transitive operator T on s given in Read [33]. We will also present the recent result by Troitsky [40], showing that there are three operators

Some aspects o f the invariant subspace problem

557

S1, 82 and K (not multiples of the identity) such that T commutes with S1, S1 commutes with $2, $2 commutes with K which is compact. This shows that L o m o n o s o v ' s invariant subspace result on operators, which commute with an operator which commutes with a compact operator, cannot be extended to longer chains. We first define the operator. Let d -- (al, bl, a2, b2 . . . . ) be a strictly increasing sequence of positive integers. Also let ao -- 1, vo -- 0, and Vn = n(an + bn) for n ~> 1. Then there is a unique sequence (ei)~= 0 C F with the following properties: (0) fo -- eo; (A) if integers r, n, and i satisfy 0 < r ~< n, i 6 [0, 13n-4] _qt_ran, we have fi -- a n - r ( e i ei-ran);

(B) if integers r, n, and i satisfy 1 <<,r < n, i ~ (ran + Vn-r, (r + 1)an), (respectively, 1 <~ n, i, ~ ( v n - 1 , an)), then fi -- 2(h-i)/x/-6-~ei, where h -- (r + 1)an (respectively, h - lan); (C) if integers r, n, and i satisfy 1 <~ r <<,n, i ~ Jr(an + bn), nan + r bn], then j~ = ei -- b n e i - b n ;

(D) if integers r, n, and i satisfy 0 <<,r < n, i ~ (nan + rbn, (r + 1)(an + bn)), then fi -- 2 ( h - i ) / ~ e i , where h = (r + 1)bn. i

Indeed, since f/ -- ~ j = 0 )~ijej for each i ~> 0 and ~,ii is always non-zero, this linear relation is invertible. Further, lin{ei" i -- 1 . . . . . n} -- lin{j~" i -- 1 . . . . . n}

for every n ~> 0.

(*)

In particular all ei are linearly independent and also span F. Define T" F --+ F to be the unique linear map such that T e i -- ei+l. Provided d increases sufficiently rapidly, that is an ~ G ( n , ao, bo, a, b . . . . . a n - l , bn-1), bn ~ H ( n , ao, bo, al, bl . . . . . a n - l , bn-1, an),

where G and H are some real-valued functions, this is a continuous transitive operator on s We can assume that all ai and bi, i ~> 1, are even integers. Now we put S1 - - T 2 s o , of course, T and $1 commute. We define S2ei--

ei

i f i is even,

0

otherwise.

Then, by checking the different possible cases we get Szj~ -

{j~ 0

i f i is even, otherwise.

This gives S 1 S z e i -- T 2 S2ei --

e i+2

if i is even,

0

otherwise.

So $2 commutes with $1. Finally define K via K f o -- fo and K f / - - 0 for i ~> 1. Then K is a rank one operator and obviously K $2 - $2 K.

558

P. Enflo and V. Lomonosov

Troitsky also proved (private communication) that T is not the adjoint of another operator, so it does not disprove Conjecture 2.17 above. He also proved [39] that if in the matrix representation of T on (fi) the coefficients are replaced by their moduli, then the new operator ITI has invariant subspaces.

References [1] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, The invariant subspace problem: Some recent advances, Rend. Inst. Mat. Univ. Trieste XXIX (Supplemento) (1998), 3-79. [2] S. Ansari and E Enflo, Extremal vectors and invariant subspaces, Trans. Amer. Math. Soc. 350 (1998), 539-558. [3] R. Aron, B. Beauzamy and E Enflo, Polynomials on many variables: Real vs. complex norms, J. Approx. Theory 74 (2) (1993), 181-198. [4] N. Aronszajn and K.T. Smith, Invariant subspaces of completely continuous operators, Ann. of Math. (2) 60 (1954), 345-350. [5] W.B. Arveson and J. Feldman, A note on invariant subspaces, Michigan Math. J. 15 (1968), 61-64. [6] W.B. Arveson, Ten Lectures on Operator Algebras, CBMS Regional Conference Series in Mathematics 55, Amer. Math. Soc., Providence, RI (1984). [7] B. Beauzamy, Un operateur sans sans-espace invariant non-trivial: simplification de l'example de P. Enflo, Integral Equations and Operator Theory 8 (1985), 314-384. [8] B. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North-Holland, Amsterdam (1988). [9] B. Beauzamy, E. Bombieri, P. Enflo and H. Montgomery, Products of polynomials in many variables, J. Number Theory 36 (1990), 219-245. [10] A.R. Bernstein and A. Robinson, Solution of an invariant subspace problem of K.T. Smith and P.R. Halmos, Pacific J. Math. 16 (1966), 421-431. [ 11] S.W. Brown, Lomonosov's theorem and essentially normal operators, New Zealand J. Math. 23 (1994), 11-18. [12] S.R. Caradus, W.E. Pfaffenberger and B. Yood, Calkin Algebras and Algebras of Operators on Banach Spaces, Marcel Dekker, New York (1974). [13] L. de Branges, The Stone-Weierstrass theorem, Proc. Amer. Math. Soc. 10 (1959), 822-824. [14] J. Diestel and J.J. Uhl, Vector Measures, Amer. Math. Soc. Math. Surveys 15 (1977). [ 15] R.E. Edwards, Functional Analysis, Holt, Rinehart and Winstone (1965). [ 16] P. Enflo, On the invariant subspace problem for Banach spaces, Seminaire Maurey-Schwarz (1975-1976). [17] P. Enflo, On the invariant subspace problem for Banach spaces, Institute Mittag-Leffler, Report no 9 (1980). [18] P. Enflo, On the invariant subspace problem for Banach spaces, Acta Math. 158 (1987), 213-313. [19] P. Enflo, Extremal vectors for a class of linear operators, Functional Analysis and Economic Theory, Springer-Verlag (1998), 61-64. [20] J. Glimm, A Stone-Weierstrass theorem for C*-algebras, Ann. Math. 72 (1960), 216-244. [21] P.R. Halmos, Invariant subspaces ofpolynomially compact operators, Pacific J. Math. 16 (1966), 433-437. [22] G.L. Litvinov and V.I. Lomonosov, Density theorems in locally convex spaces and applications, Selecta Math. Soviet. 8 (4) (1989), 323-339. [23] VT Lomonosov, Invariant subspaces of the family of operators that commute with a completely continuous operator, Functional. Anal. i Prilozhen. 7 (3) (1973), 55-56. [24] V.I. Lomonosov, Some problems in the theory of invariant subspaces, Kharkov University Dissertation (1974). [25] V.I. Lomonosov, An extension of Burnside's theorem to infinite dimensional spaces, Israel J. Math. 75 (1991), 329-339. [26] V.I. Lomonosov, On real invariant subspaces of bounded operators with compact imaginary part, Proc. Amer. Math. Soc. 115 (3) (1992). [27] C. Pearcy and N. Salinas, An invariant subspace theorem, Michigan Math. J. 20 (1973), 21-31.

Some aspects o f the invariant subspace problem

559

[28] C. Pearcy and A. Shields, A survey of the Lomonosov technique in the theory of invariant subspaces, Amer. Math. Soc. Math. Surveys 13 (1974), 219-230. [291 H. Radjavi and H.P. Rosenthal, Invariant Subspaces, Springer-Verlag (1973). [301 H. Radjavi and H.P. Rosenthal, The invariant subspace problem, Math. Intell. 4 (1) (1982), 33-37. [31] C. Read, A solution to the invariant subspace problem, Bull. London Math. Soc. 16 (1984), 337401. [321 C. Read, A solution to the invariant subspace problem on the space ll, Bull. London Math. Soc. 17 (1985), 305-317. [331 C. Read, A short proof concerning the invariant subspace problem, J. London Math. Soc. 34 (1986), 335348. [341 C. Read, The invariant subspace problem for a class of Banach spaces, 2: Hypercyclic operators, Israel J. Math. 63 (1) (1988), 1-40. 135] C. Read, The invariant subspace problem on some Banach spaces with separable dual, Proc. London Math. Soc. 58 (1989), 583-607. [361 C. Read, Quasi-nilpotent operators and the invariant subspace problem, J. London Math. Soc., to appear. [371 A. Simoni~, An extension of Lomonosov's techniques to non-compact operators, Trans. Amer. Math. Soc. 348 (3) (1996), 975-995. [381 A. Spalsbury, Vectors of minimal norm, Proc. Amer. Math. Soc. 126 (1998), 2735-2745. [391 V. Troitsky, On the modulus of C.J. Read's operator, Positivity 2 (3)(1998), 257-264. [401 V. Troitsky, Lomonosov's Theorem cannot be extended to chains offour operators, Proc. Amer. Math. Soc., to appear. [41] R. Varga, Scientific Computation on Mathematical Problems and Conjectures, CBMS-NSF Regional Conference Series in Applied Math. 60, Soc. for Industrial and Applied Math., Philadelphia (1990). [42] B. Yood, Properties of linear transformations preserved under addition of a completely continuous transformation, Duke Math. J. 18 (1951), 599-612.

This Page Intentionally Left Blank

CHAPTER

14

Special Bases in Function Spaces

T.

Figiel

Institute of Mathematics, Polish Academy of Sciences, ul. Abrahama 18, 81-825 Sopot, Poland E-mail: t.figiel @impan, gda.pl

E

Wojtaszczyk

Institute of Mathematics, Warsaw University, Warsaw, Poland E-mail: przemekw @mimuw, edu.pl

Contents 1. Systems of functions

..............................................

1.1. Systems on [0, 1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Wavelet bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Operations on orthogonal systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Positive results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

563 563 565 568 569

2.1. Unconditionality in L p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Haar system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Franklin system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

569 573 574

2.4. Walsh system

575

...............................................

2.5. Meyer's wavelets 2.6. Polynomial bases

............................................. .............................................

2.7. Rational bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

576 577 579

3. Negative results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Non-explicit existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

580 581

5. Function spaces on compact smooth manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Preliminary definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

583 583 585

5.3. Spaces on subsets of the manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

585

5.4. Spaces with boundary conditions 5.5. Decomposition of the manifold

.................................... .....................................

5.6. Decomposition of function spaces on the manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Bases with vector coefficients in spaces of vector-valued functions . . . . . . . . . . . . . . . . . . . . .

586 586 587 588

6.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

588

6.2. Spaces of functions with values in a U M D space . . . . . . . . . . . . . . . . . . . . . . . . . . . .

590

H A N D B O O K OF THE G E O M E T R Y OF B A N A C H SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved 561

562

T. Figiel and P. Wojtaszczyk

6.3. Equivalent X-bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Abstract We present several well known orthogonal systems of functions and discuss their properties as bases in selected classical function spaces. We show that in some function spaces there are no bases with additional special properties. We discuss non-explicit methods of constructing Schauder bases. On each compact, smooth manifold we construct a system of smooth functions which is a good basis in a wide range of spaces of Sobolev and Besov type. Finally we discuss systems of scalar valued functions which are bases (with vector coefficients) in some spaces of vector valued functions.

594 595

Special bases in function spaces

563

In this chapter the basis problem is treated from a perspective which is somewhat unusual in the Banach space theory. Given a set S, a linear space ,~' of scalar-valued functions on S and a sequence (fn)n~=l C X , the problem is to describe some (presumably interesting) classes of Banach spaces X, which arise as a completion of &" with respect to a norm I] 9il, in which (fn) forms a Schauder basis or an unconditional Schauder basis. For instance, let I = [0, 1] be the unit interval equipped with the Lebesgue measure. Then the classical Haar system (hn) on I forms a Schauder basis in every Lp(1) for 1 ~ 1 } in the II 9II~. On the other hand, the same Haar system fails miserably if we insist that the sequence should in addition form a Schauder basis in C (I). However, the classical Franklin system satisfies all those properties and in fact it turns out to form a basis in some other interesting function spaces as well. Knowing this, however, does not automatically provide the solution to the analogous question with a different choice of the set S. For instance, if d > 1 and S - Q - [0, 1]d is the unit d-cube with the Lebesgue measure, then to produce a sequence (fn) of functions on S that forms a Schauder basis in C (S) and in L p (Q) for 1 ~< p < which is unconditional for 1 < p < c~ requires some further work even though the Banach spaces Lp(Q) and Lp(I) are isometric for each p and the spaces C(Q) and C(I) are isomorphic. Thus our problem is not concerned with properties of any individual Banach space. It is rather a question pertaining to a family of Banach spaces which have a common dense linear subspace and may perhaps admit sort of simultaneous Schauder bases with certain additional interesting properties. This should give some idea why certain systems of functions have been introduced by various mathematicians and below we shall describe some of the relevant examples.

1. Systems of functions In this section we present the most important systems of functions. Some of them have been defined in other chapters of this Handbook and in particular in [32] but we repeat all the definitions here in order to present them in a unified and systematic fashion.

1.1. Systems on [0, 1] Such systems fall naturally into two categories. Bounded systems are such that the ratio between Ilfn 112 and Ilfn I1~ stays bounded. If this ratio is unbounded we say that the system is unbounded. 1.1.1. Bounded systems. Here we want to describe the most important uniformly bounded orthonormal systems on [0, 1].

Rademacher system.

It is defined as rn(t) -- signsin2nzrt for n = 1, 2 . . . . . It is an orthonormal system, but incomplete. In probabilistic terms rn (t) are independent, identically distributed functions.

T. Figiel and P. Wojtaszczyk

564

Walsh system. For an integer n >~ 1 we write n -- ~ - - 0 O(k) 2k where 69(k) = 0 or 1 are its dyadic digits. It is easy to see and well known that this representation is actually finite and unique. We define the Walsh system on [0, 1], Wn (t), n = 0, 1, 2 . . . . by wo(t) = 1 and for n ~> 1 we put Wn(t) -- 1-I~__0rk+l (t) (O(k). To make it well defined in every point we put 0 ~ = 1. The Walsh system is an orthonormal basis in L2[0, 1]. There is another way to look at the Walsh system. Let us consider the group { - 1 , 1 }r~ with the natural product measure, which is also a Haar measure. Then the system (g0n), where ~On(El, 82 . . . . ) = en, is equimeasurable with the Rademacher system, and the system 7z0 = 1 and for n ~> 1 7rn - I-I~_0 q9~ is equimeasurable with the Walsh system. Note that the system (Trn)n~=0 is the set of characters of the group { - 1 , 1 }rq.

Trigonometric system. The name actually denotes either of two closely related systems of functions. In the complex form we m e a n (eZrrint)n6Z or in the real form we mean the system 1 . . . . . x / 2 c o s 2 z r n t , f f 2 s i n 2 z r n t . . . . with n -- 1, 2 . . . . . The complex form of the trigonometric system can be identified with the set of all characters of the circle group qF. Either form of the trigonometric system is an orthonormal basis in L2 [0, 1]. 1.1.2. Unbounded systems. Those systems generally have a dyadic structure, i.e., they admit a natural ordering isomorphic to a dyadic tree. The dyadic rationals in [0, 1] are linearly ordered as follows. We let to -- 0, tl = 1 and for n > 1 we write n = 2 u + v with # >/0 and 1 ~< v ~< 2~ and we put tn = 2 - ~ - 1 (2v - 1).

(1)

Faber-Schauder system.

We define functions Sn (t) for n = 0, 1, 2 . . . . as follows: so(t) = 1, sl (t) = t and for n > 1 we define Sn (t) as a unique continuous, piecewise linear function on [0, 1] with nodes at points to, tl . . . . . tn such that Sn (tn) = 1 and Sn (tj) -- 0 for j < n. Note that for n > 1 with n -- 2 u + v we have supp Sn = [ t n - 2 - ~ - I , tn + 2-1z-l]. It is easily seen that span{sj}y_ 0 is exactly the space of all continuous, piecewise linear functions on [0, 1] with nodes at points {tj}N=0 . This implies that the Faber-Schauder system is linearly dense in C[0, 1]. It is also clear that if for f ~ C[0, 1] we define Pn ( f ) to be the piecewise-linear, continuous function on [0, 1] with nodes at to . . . . . tn interpolating f at those nodes, i.e., P n ( f ) ( t j ) = f ( t j ) for j - - 0 , 1 . . . . ,n, then P n ( s j ) = sj if j = 0 , 1 . . . . . n and Pn (s j) = 0 if j > n. Thus Pn is a partial sum projection of f with respect to the F a b e r - S c h a u d e r system. Since l[ Pn [l~ = 1 we infer that the Faber-Schauder system is a monotone basis in C[0, 1].

Franklin system.

The Franklin system (fn)n~__o is the G r a m - S c h m i d t orthonormalization of the Faber-Schauder system in the above order. Every continuous piecewise linear func?/ n tion f with nodes at points to, tl . . . . . tn can be written as f - ~ j - 0 a j s j - )--~j=0 bj f j . From this we see that the Franklin system is linearly dense in C[0, 1], so it is an orthonormal basis in L z[0, 1].

Haar system. In our context the easiest way to define the Haar system is to put hn (t) = IlSn~ll2 1Sn~(t) for n = 1, 2 . . . . . Since s n exists except at three points this definition holds

Special bases in function spaces

565

a.e. Clearly, for n = 2, 3 . . . . we have supphn = suppSn = [tn -- 2 - u - 1 , tn + 2 - u - l ] and hn(t) equals 2 u/2 on (tn - 2 - u - l , tn) and equals - 2 u/2 on (tn, tn + 2 - U - l ) . It follows easily from the above that two different Haar functions either have disjoint supports or the support of one (that with the higher number) is contained in an interval where the other is constant. Since f l h n ( t ) d t - 0 for n ~> 2, we infer that the Haar system is orthonormal. It can be seen from our discussion of the F a b e r - S c h a u d e r system (or directly from the definition of the Haar system) that span{hj }~=l equals the space of all step functions on [0, 1] whose discontinuities are only at points t2 . . . . . tn. This immediately shows that the Haar system is linearly dense in L p [0, 1] for 1 ~< p < cxz and that for the orthogonal (partial sum) projection we have the formula /7

P/7(f)--Z(f,

/7

hj)hj-~lljl-l

j=l

j=l

~

f(t)dtllj

(2)

J

in which Ij = [ o / j - l , o/j], where o/0, o/1 . . . . . o//7 is the increasing rearrangement of the set {to, tl . . . . . t/7}. Formula (2) immediately gives THEOREM 1. The Haar system is a monotone basic sequence in every rearrangement invariant function space X on [0, 1]. It is a monotone basis in X if and only if X is separable. 1 Note the following relation between the Walsh and Haar system: for each n -- 1 ' 2,

" " "

span{wj ~2n-| -- span{hj - - -J-"n }~_ 1" J j=0

(3)

Each of the above spaces equals the space of functions constant on all intervals (k2 -n, (k + 1)2 -n) with k = 0, 1 . . . . . 2 n - 1. This observation is the key to the proof that both those systems are linearly dense in L p[O, 1], 1 <<,p < cx~. REMARK. There are generalisations of some systems defined above where the exact dyadic structure is replaced by an "approximate" one (see, e.g., [27]).

1.2. Wavelet bases Recently a new class of orthonormal systems on R and on R d was discovered. We say that a function q/ e L2(R) is a wavelet if the system q J j k ( x ) " - - 2 J / Z t I / ( 2 J x - k) with j, k e Z is an orthonormal basis in L2(•). The system (q/jk)jkcZ is called a wavelet basis. 1 All monotone bases in Lp(#) with 1 ~

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Banach Spaces, Volume Volume 1 (North-Holland Mathematical Library)

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Banach Spaces of Analytic Functions

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Introduction to Banach spaces and their geometry

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Geometry and Nonlinear Analysis in Banach Spaces

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Geometry and Probability in Banach Spaces

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Introduction to Banach Spaces and Their Geometry

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Geometry of Phase Spaces

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Geometry of Phase Spaces

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The Isometric Theory of Classical Banach Spaces

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The - Volume of convex bodies and Banach space geometry

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The Volume of Convex Bodies and Banach Space Geometry

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The Geometry of Moduli Spaces of Sheaves

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The Geometry of Moduli Spaces of Sheaves

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Ordered Banach spaces

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Isometrics on Banach Spaces

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Classical Banach Spaces

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Handbook of Algebra : Volume 1

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Handbook of Econometrics Volume 1

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Banach spaces for analysts

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Probability and Banach Spaces

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Banach spaces for analysts

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