Analysis of Toeplitz Operators

Springer Monographs in Mathematics

Albrecht Böttcher · Bernd Silbermann

Analysis of Toeplitz Operators Second Edition Prepared jointly with Alexei Karlovich

With 20 Figures

123

Albrecht Böttcher Bernd Silbermann Technische Universität Chemnitz Fakultät für Mathematik 09107 Chemnitz, Germany [email protected] [email protected]

Library of Congress Control Number: 2006922574

Mathematics Subject Classification (2000): 47-02, 15A15, 30D55, 42A50, 45E10, 46E25, 47B35, 47L15, 65F99 ISSN 1439-7382 ISBN-10 3-540-32434-8 Springer-Verlag Berlin Heidelberg New York ISBN-13 978-3-540-32434-8 Springer-Verlag Berlin Heidelberg New York ISBN 3-540-52147-X 1st ed. Springer-Verlag Berlin Heidelberg New York (co-publication) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 © Akademie-Verlag Berlin 1990 First Edition Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the authors using a Springer TEX macro package Production and data conversion: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper

41/3100YL - 5 4 3 2 1 0

Preface to the Second Edition

Since the late 1980s, Toeplitz operators and matrices have remained a field of extensive research and the development during the last nearly twenty years is impressive. One encounters Toeplitz matrices in plenty of applications on the one hand, and Toeplitz operators confirmed their role as the basic elementary building blocks of more complicated operators on the other. Several monographs on Toeplitz and Hankel operators were written during the last decade. These include Peller’s grandiose book on Hankel operators and their applications and Nikolski’s beautiful easy reading on operators, functions, and systems, with emphasis on topics connected with the names of Hardy, Hankel, and Toeplitz. They also include books by the authors together with Hagen, Roch, Yu. Karlovich, Spitkovsky, Grudsky, and Rabinovich. Thus, results, techniques, and developments in the field of Toeplitz operators are now well presented in the monographic literature. Despite these competitive works, we felt that large parts of the first edition of the present monograph which is meanwhile out of stock - have not lost their fascination and relevance. Moreover, the first edition has received a warm reception by many colleagues and became a standard reference. This encouraged us to venture on thinking about a second edition, and we are grateful to the Springer Publishing House for showing an interest in this. The present book is a genuine second edition, which means that it differs from the original version but that it is not a completely new book. We left everything of the first edition, even in the cases where we felt that the material nowadays is not of the same brisance as two decades ago. However, we tried to incorporate as much as possible of the body of knowledge that has grown since about 1990. It is clearly impossible to take into account even only a moderate piece of all the developments in Toeplitz operators during the last ten or fifteen years. We therefore focussed our attention upon topics which are intimately related to those considered in the previous edition or on which we ourselves have worked some time. Many recent achievements were included together with a short comment into the list of references, some others, such as pseudospectra, phenomena caused by SAP symbols, or recent work on

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Toeplitz and Wiener-Hopf determinants received own new sections in the text, partly with full proofs. We hope the present second edition will be welcomed by those who had no chance of purchasing a copy of the original edition and also by all those who are looking for a first or refreshed orientation about the present state of the art in the field.

Braga and Chemnitz, December 2005

Albrecht B¨ ottcher Alexei Karlovich Bernd Silbermann

Preface to the First Edition

This book was originally intended as an extended version of our book “Invertibility and Asymptotics of Toeplitz Matrices”, which appeared in 1983. We planned to discuss several topics in more detail, but our main concern was to incorporate a whole series of new results obtained during the last few years. However, we soon realized that the program we had in mind required new thoughts from both the methodological and substantial points of view, and so we decided to attempt writing a completely new book on the analysis of Toeplitz operators. There are at least two reasons for the continuous and increasing interest in Toeplitz operators. On the one hand, Toeplitz operators are of importance in connection with a variety of problems in physics, probability theory, information and control theory, and several other fields. Although we shall not embark on these problems, the selection of the material of the book is to a certain extent determined by such applications. On the other hand, besides the differential operators, Toeplitz operators constitute one of the most important classes of non-selfadjoint operators and they are a fascinating example of the fruitful interplay between such topics as operator theory, function theory, and the theory of Banach algebras. One main purpose of this book is to elucidate some of the ideas and methods illustrating just the latter aspect. The theory of Toeplitz operators is a very wide area and even a huge monograph can deal with only some selected topics. Our emphasis is on Toeplitz operators over the circle and over the torus (or, what is the same, discrete Wiener-Hopf operators over the half-axis and over the quarter-plane) viewed as concrete operators on concrete Banach spaces, and a central problem is to establish a relation between the functional-analytic properties of Toeplitz operators and the geometric properties of their symbols. The selection of the special topics has been determined by our own interests and competence. However, having chosen a topic, we try to present it in such a way that it may be taken as a systematic, exhaustive, and modern introduction to the well-known and by now classical results as well as a readable account of some recent developments. A glimpse of the table of contents provides an overall

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view of the material covered by the book. We merely want to add the following remarks. Chapter 1 contains a series of notations and definitions. The reader need not study this chapter very carefully; it suffices to glance through it, pick up some notations, and backtrack whenever the necessity arises. In Chapter 2 we begin by stating elementary properties of Toeplitz operators and finish by proving some of their rather deep-lying properties. This chapter mainly incorporates those results whose proof needs almost no “theory.” Moreover, it may be viewed as the trunk of a tree, the boughs and twigs of which are the concern of the forthcoming chapters. In Chapter 3 we start preparing the more delicate theory of Toeplitz operators. It is devoted to matrix functions which are locally sectorial in a very sensitive sense. This chapter also includes the study of the phenomenon of the asymptotic multiplicativity of approximate identities and Sarason’s theory of piecewise quasicontinuous functions. In this chapter we also devise and give a first application of some sort of machinery (or “philosophy”) that will be employed repeatedly in the remaining chapters: algebraization, essentialization, localization, determination of local spectra. Chapter 4 is concerned with the Hilbert space theory of block Toeplitz operators. There we prove Fredholmness and compute the index of Toeplitz operators whose symbol is locally sectorial over QC, describe Axler’s transfinite localization approach to maximal antisymmetric sets for C +H ∞ , present the theory of local Toeplitz operators due to Douglas, Clancey, Gosselin, study symbols with a specified local range (in particular, symbols with two or three essential cluster points), and develop a new approach to algebras generated by Toeplitz operators and related objects. Chapters 5 and 6 deal with block Toeplitz operators on weighted H p and p
spaces, respectively. Ours is, to a great extent, a novel presentation of these topics. We provide new proofs of the classical results of the well-known monographs by Gohberg, Feldman and Gohberg, Krupnik, and we incorporate numerous results which are only known from mathematical journals, primarily Soviet ones. Chapter 7 is a self-contained and up to date theory of finite section method (reduction method) for Toeplitz operators. We prove that the finite section method is applicable to block Toeplitz operators on H 2 with symbols that are locally sectorial over QC, we develop a sufficiently simple theory for operators with piecewise continuous symbols on H p and
p , we study symbols with singularities of Fisher-Hartwig type, and we conclude by proving the very recent and noteworthy result of Treil, according to which there are invertible Toeplitz operators on H 2 to which the finite section method is not applicable. Chapter 8 is a comprehensive treatment of Toeplitz operators over the quarter-plane and is, at least to a certain degree, a novelty in the monographic literature. We emphasize that we study both the Fredholm theory and the theory of finite section methods of quarter-plane operators with discontinuous symbols.

Preface to the First Edition

IX

Chapter 9 looks at Wiener-Hopf integral operators. There we point out the common features between Wiener-Hopf integral operators and their discrete analogues (the Toeplitz operators) but also dwell on the significant differences between these two classes of operators. In this chapter we also consider operators with almost periodic, semi-almost periodic, and other kinds of oscillating symbols. Chapter 10 is a systematic and self-contained theory of Toeplitz determinants. The material presented ranges from the classical Szeg˝o-Widom limit theorems to a proof of the conjecture of Fisher-Hartwig for some important special cases. We shall demonstrate that the very attractive field of Toeplitz determinants requires results from all foregoing chapters and may thus serve as a beautiful application of the functional analysis of Toeplitz operators. In particular, we shall show that some important problems on Toeplitz determinants can be solved by working with Toeplitz operators on the spaces H 2 () and
pµ , so that passage from the Hilbert space theory to the Banach space theory does not turn out to be a purely academic matter. Let us also point out three peculiarities of the present monograph. First, Banach algebra techniques combined with local principles are our main tool for tackling Toeplitz operators. That such methods can be successfully applied to the study of the Fredholm theory of Toeplitz operators is wellknown from Douglas’ book. However, this approach has only recently proved to be a powerful technique of studying projection methods, harmonic approximation, or stable convergence (and thus index computation) for Toeplitz operators. Moreover, our consistent use of local Banach algebra technique will not only provide a unified technique of solving various problems related to Toeplitz operators, but will allow us to reformulate many classical results in pretty nice language. For instance, the well-known result that a Toeplitz operator with piecewise continuous symbol is Fredholm on H p or
p if and only if the curve obtained from the essential range of its symbol by filling in certain circular arcs does not contain the origin reads in this language as follows: the local spectrum of the operator is either a point or a certain circular arc. Secondly, we shall consider Toeplitz operators on the (Hilbert) space H2 ∼ =
2 and on the weighted (Banach) spaces H p () and
pµ . Each of these three situations has its peculiarities and requires its own techniques. While there are excellent and comprehensive discussions of the Hilbert space theory in the well-known monographs by Gohberg and Feldman, Douglas, and Nikolski, the same cannot be said about Banach space theory. We hope that the present book will fill this gap and, moreover, will also make a series of new contributions to the Hilbert space case. Thirdly, it should be explicitly noticed that our emphasis is on matrixvalued symbols. Many problems on Toeplitz operators have fairly fast solution in the scalar case, whereas substantial difficulties arise in the matrix case. Finally, to see what this book is all about, it should also be mentioned that the following topics are not touched upon: Toeplitz operators on domains, balls, or manifolds; pseudodifferential operators; Breuer-Fredholm results and

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generalized index theory; operator-valued symbols; invariant subspaces; linear algebra and computational mathematics of finite Toeplitz matrices. This list is naturally incomplete. Let it also be understood that this is a book on Toeplitz operators and not on Hankel or singular integral operators, although we pay due attention to these two classes of operators. We ventured on writing a book for both the beginner and the specialist. In the beginner’s interest we gave a rather full description of those topics which form the background to the theory of Toeplitz operators (e.g., some students of ours acknowledged the rather lengthy explanation of what a “fiber” is). We also provided the bulk of results with detailed proofs. We did this in order to teach the reader not only the results but also (or mainly) the techniques for proving them. This is, of course, not always in the beginner’s interest, but we hope the specialist will relish some details of these proofs. Some of the results and techniques are new and published here for the first time and are thus primarily addressed to the specialist. Many results are taken from the periodicals and are first cited with detailed proofs. We included a series of problems which we declared to be “open”. Some of them are well-known as open problems, some others are merely open in the sense that we have not found a solution within a few hours, days or weeks. In either case we followed the policy that we would better confess our own inability than hide something. We made the attempt of supplying all results with a source; however, the evolution of many theorems involves too many contributors, and so it may occur that our reference is not the right one. We hope that the reader will excuse our faulty referencing and we accept any criticism in this direction. Finally, we have labelled a lemma or theorem only when a name seems to have been attached to it by common usage. We wish to express our sincere appreciation to our colleagues Roland Hagen and Steffen Roch, who both read the bulk of the manuscript very carefully and eradicated not only a large number of mistakes and (sometimes serious) errors but helped with their criticism to essentially improve the book. We would also like to thank Mrs. Marianne Graupner and Mrs. Isolde Scholz for all the trouble they took in typing the entire manuscript. Finally, we are pleased to express our gratitude to the Akademie-Verlag Publishing House, especially to the Editor, Dr. Reinhard H¨ oppner, for inviting us to write this monograph and for the careful performance of the book. Special Acknowledgement. In May and June 1987, Naum Krupnik visited the Chemnitz University of Technology. During these two months he read the whole manuscript with great enthusiasm and made a large number of valuable remarks, a major part of which could still be incorporated into the text. We are extremely grateful to him for improving the book by his uncommon expertise.

Chemnitz, 1987

Albrecht B¨ ottcher, Bernd Silbermann

Contents

1

Auxiliary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Operator Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Operator Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Operator Matrices and Their Determinants . . . . . . . . . . . . . . . . 1.5 Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 C ∗ -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Local Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Lp and H p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 BM O and V M O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Smoothness Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 9 11 12 17 21 27 36 38 41

2

Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1 Multiplication Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2 Toeplitz Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3 Hankel Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.4 Invertibility of Toeplitz Operators on H 2 . . . . . . . . . . . . . . . . . . 57 2.5 Spectral Inclusion Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.6 The Connection Between Fredholmness and Invertibility . . . . . 71 2.7 Compactness of Hankel Operators and C + H ∞ Symbols . . . . 77 2.8 Local Methods for Scalar Toeplitz Operators . . . . . . . . . . . . . . . 85 2.9 Matrix Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.10 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3

Symbol Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.1 Local Sectoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.2 Asymptotic Multiplicativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.3 Piecewise Quasicontinuous Functions . . . . . . . . . . . . . . . . . . . . . . 128 3.4 Harmonic Approximation: Algebraization . . . . . . . . . . . . . . . . . . 140 3.5 Harmonic Approximation: Essentialization . . . . . . . . . . . . . . . . . 149

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3.6 3.7 3.8 3.9

Harmonic Approximation: Localization . . . . . . . . . . . . . . . . . . . . 152 Harmonic Approximation: Local Spectra . . . . . . . . . . . . . . . . . . . 157 Local Sectoriality Continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

4

Toeplitz Operators on H 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.1 Fredholmness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.2 Stable Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.3 Index Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.4 Transfinite Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 4.5 Local Toeplitz Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 4.6 Symbols with Specific Local Range . . . . . . . . . . . . . . . . . . . . . . . . 217 4.7 Toeplitz Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 4.8 The Role of the Harmonic Extension . . . . . . . . . . . . . . . . . . . . . . 240 4.9 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

5

Toeplitz Operators on H p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.1 General Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.2 Khvedelidze Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 5.3 Locally p, -Sectorial Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 5.4 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 5.5 P C Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 5.6 P2 C Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 5.7 Fisher-Hartwig Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 5.8 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

6

Toeplitz Operators on
p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 6.1 Multipliers on Weighted
p Spaces . . . . . . . . . . . . . . . . . . . . . . . . 287 6.2 Continuous Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 6.3 Piecewise Continuous Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 6.4 Analytic Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 6.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

7

Finite Section Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 7.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 7.2 C + H ∞ Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 7.3 Locally Sectorial Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 7.4 P C Symbols:
p Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 7.5 P C Symbols: H p Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 7.6 Operators from algL(H 2 ) T (P C) . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 7.7 Fisher-Hartwig Symbols: H 2 () Theory . . . . . . . . . . . . . . . . . . . . 374 7.8 Fisher-Hartwig Symbols:
pµ Theory . . . . . . . . . . . . . . . . . . . . . . . 378 7.9 Invertibility Versus Finite Section Method . . . . . . . . . . . . . . . . . 389 7.10 Pseudospectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 7.11 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

Contents

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8

Toeplitz Operators over the Quarter-Plane . . . . . . . . . . . . . . . . . 409 8.1 Function Classes on the Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 8.2 Elementary Properties of Quarter-Plane Operators . . . . . . . . . . 417 8.3 Continuous Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 8.4 The Invertibility Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 8.5 Bilocal Fredholm Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 8.6 P QC ⊗ P QC Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 8.7 Finite Section Method: Kozak’s Theory . . . . . . . . . . . . . . . . . . . . 454 8.8 Finite Section Method: Bilocal Theory . . . . . . . . . . . . . . . . . . . . . 461 8.9 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 8.10 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

9

Wiener-Hopf Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 9.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 9.2 Continuous Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 9.3 Piecewise Continuous Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 9.4 AP and SAP Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 9.5 Some Phenomena Caused by SAP Symbols . . . . . . . . . . . . . . . . 497 9.6 Other Oscillating Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 9.7 Finite Section Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 9.8 Operators over the Quarter Plane . . . . . . . . . . . . . . . . . . . . . . . . . 513 9.9 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

10 Toeplitz Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 10.1 The First Szeg˝o Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 10.2 Krein Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 10.3 Wiener-Hopf Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 10.4 The Strong Szeg˝o Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 537 10.5 Higher Order Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 10.6 Semirational Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 10.7 Nonvanishing Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 10.8 Selfadjoint Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 10.9 The Pure Fisher-Hartwig Singularity . . . . . . . . . . . . . . . . . . . . . . 570 10.10 Separation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 10.11 Fisher-Hartwig Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 10.12 More on Unbounded Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 10.13 Wiener-Hopf Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 10.14 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 10.15 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

1 Auxiliary Material

1.1 Operator Ideals 1.1. Bounded and compact operators. Let X and Y be Banach spaces. We denote by L(X, Y ) the linear space of all (bounded and linear) operators from X to Y . We let C∞ (X, Y ) ⊂ L(X, Y ) denote the collection of all compact operators from X into Y , and C0 (X, Y ) refers to the set of all finite-rank operators from X into Y , i.e., F is in C0 (X, Y ) if and only if F ∈ L(X, Y ) and dim F (X) < ∞. In the case X = Y we shall write L(X) = L(X, X), C∞ (X) = C∞ (X, X), C0 (X) = C0 (X, X). A sequence {An } of operators An ∈ L(X, Y ) is said to converge to an operator A ∈ L(X, Y ) (a) weakly, if f (An x) − f (Ax) → 0 for each x ∈ X and each functional f ∈ Y ∗; (b) strongly, if An x − AxY → 0 for each x ∈ X; (c) uniformly, if An − A → 0, where, for B ∈ L(X, Y ), B := sup Bx.

(1.1)

x≤1

Let {An } be a sequence of operators An ∈ L(X, Y ). Then one has the following. (d) If A ∈ L(X, Y ), if An x − AxY → 0 as n → ∞ for each x in a dense subset of X, and if sup An  < ∞, then An converges strongly to A. n

(e) (Banach/Steinhaus). If {An x} is a convergent sequence in Y for each x ∈ X, then sup An  < ∞, the operator A defined by Ax = lim An x n→∞

n

belongs to L(X, Y ), and A ≤ lim inf An . n→∞

2

1 Auxiliary Material

(f) If A ∈ L(X, Y ), if An → A weakly, and K ∈ C∞ (Y, Z), then KAn → KA strongly. Equipped with the operator norm (1.1) the linear set L(X, Y ) becomes a Banach space and L(X) a Banach algebra. Then C∞ (X, Y ) is a closed subspace of L(X, Y ). In general, the subspace C0 (X, Y ) ⊂ L(X, Y ) is not closed, but its closure is contained in C∞ (X, Y ). Notice the following implications. A ∈ L(X, Y ), K ∈ C∞ (Y, Z), B ∈ L(Z, V ) =⇒ BKA ∈ C∞ (X, V ), A ∈ L(X, Y ), F ∈ C0 (Y, Z), B ∈ L(Z, V ) =⇒ BF A ∈ C0 (X, V ). In particular, C∞ (X) is a closed two-sided ideal of L(X) and C0 (X) is a twosided (but in general not closed) ideal of L(X). If X and Y are separable infinite-dimensional Hilbert spaces, we shall write X = H1 , Y = H2 , and in case X = Y simply X = Y = H. We then have the following. (g) The closure of C0 (H1 , H2 ) with respect to the operator norm (1.1) coincides with C∞ (H1 , H2 ). (h) L(H) is a C ∗ -algebra and C∞ (H) is a closed two-sided star-ideal of L(H). 1.2. The s-numbers. Given an operator A ∈ L(H1 , H2 ), we define for n in Z+ := {0, 1, 2, . . .}
 sn (A) := inf A − F  : F ∈ C0 (H1 , H2 ), dim F (H1 ) ≤ n . The sequence {sn (A)}∞ n=0 is referred to as the sequence of the s-numbers of the operator A. The well known Horn lemma says that if K, L ∈ C∞ (H), n ∈ Z+ , and 1 ≤ p < ∞, then n n   spj (KL) ≤ spj (K)spj (L). j=0

j=0

1.3. The Schatten-von Neumann classes. For 1 ≤ p < ∞, the collection of all operators K ∈ L(H1 , H2 ) satisfying  1/p Kp := spn (K) <∞ (1.2) n≥0

is denoted by Cp (H1 , H2 ) and referred to as a Schatten-von Neumann class. Note that   C∞ (H1 , H2 ) = K ∈ L(H1 , H2 ) : lim sn (K) = 0 . n→∞

Moreover, K∞

 := sup sn (K) = s0 (K) is equal to the operator norm K n≥0

defined by (1.1). Here are some properties of the classes Cp (H1 , H2 ).

1.1 Operator Ideals

3

(a) For 1 ≤ p < ∞, Cp (H1 , H2 ) is a Banach space under the norm (1.2). For 1 ≤ r < s ≤ ∞, the space Cr (H1 , H2 ) is continuously and densely embedded in Cs (H1 , H2 ). The set C0 (H1 , H2 ) is dense in Cp (H1 , H2 ) for every 1 ≤ p ≤ ∞. (b) If 1 ≤ p ≤ ∞, then A ∈ L(H1 , H2 ), K ∈ Cp (H2 , H3 ), B ∈ L(H3 , H4 ) =⇒ BKA ∈ Cp (H1 , H4 ). Cp (H) is a two-sided (but, for p < ∞, in general not closed) star-ideal of L(H). (c) Suppose K ∈ Cp (H1 , H2 ), L ∈ Cq (H2 , H3 ), and 1 ≤ p, q, r ≤ ∞. If 1/r = 1/p + 1/q, then LK ∈ Cr (H1 , H3 ) and LKr ≤ Lp Kq . (d) Suppose Bn ∈ L(H3 , H4 ) converges strongly to B ∈ L(H3 , H4 ) and Cn∗ ∈ L(H2∗ , H1∗ ) converges strongly to C ∗ ∈ L(H2∗ , H1∗ ). If 1 ≤ p ≤ ∞ and K ∈ Cp (H2 , H3 ), then Bn KCn − BKCp → 0 as

n → ∞.

If K ∈ L(H) is a positive selfadjoint

operator and if {en }n≥0 is any orthonormal basis in H, then the sum n≥0 (Ken , en ) can be shown to be independent of the particular choice of {en }n≥0 . This sum is denoted by tr K and called the trace of K. (e) Let 1 ≤ p < ∞. Then K ∈ Cp (H) if and only if tr (|K|p ) < ∞, where |K| := (K ∗ K)1/2 . In that case  spn (K) = tr (|K|p ). Kpp = n≥0

For K ∈ C∞ (H), let {λn (K)}∞ n=0 denote the sequence of the eigenvalues of K, counted up to algebraic multiplicity. Recall that the algebraic multiplicity of an eigenvalue λ of K is defined as the dimension of the linear space 
x ∈ H : (K − λI)n x = 0 for some n ∈ Z+ . ∞ (f) If K ∈ C∞ (H), then the sequences {sn (K)}∞ n=0 and {λn (|K|)}n=0 coincide.

(g) (Weyl’s inequality). If K ∈ Cp (H) (1 ≤ p < ∞), then  |λn (K)|p ≤ Kpp . n≥0

1.4. Trace class operators. One can show that

for every K ∈ C1 (H) and for any orthonormal basis {en }n≥0 in H the series n≥0 (Ken , en ) converges absolutely and that its sum does not depend on the particular choice of {en }n≥0 . This sum is denoted by tr K and referred to as the trace of K. The operators

4

1 Auxiliary Material

belonging to C1 (H) are called trace class operators. Basic properties of the trace are as follows. (a) The mapping tr : C1 (H) → C is a star-linear contraction. In particular, |tr K| ≤ K1 for every K ∈ C1 (H). (b) If either A ∈ Cp (H), B ∈ Cq (H), 1 < p < ∞, 1/p + 1/q = 1, or A ∈ C1 (H), B ∈ L(H), then tr (AB) = tr (BA). (c) If K ∈ C1 (H) and if A ∈ L(H) is invertible, then tr (AKA−1 ) = tr A. 1.5. Hilbert-Schmidt operators. The operators in C2 (H1 , H2 ) are called Hilbert-Schmidt operators. They admit the following simple characterization. Let {en }n≥0 and {fn }n≥0 be any orthonormal bases in H1 and H2 , respectively. Then   K ∈ C2 (H1 , H2 ) ⇐⇒ Ken 2 < ∞ ⇐⇒ |(Ken , fn )|2 < ∞. n≥0

n≥0

1.2 Operator Determinants 1.6. Definition. Let H be a finite-dimensional Hilbert space, dim H = n, and let A ∈ L(H). Write A in the form A = I + K and denote the eigenvalues of K by λ1 (K), . . . , λn (K). So {1 + λj (K)}nj=1 is the sequence of the eigenvalues of A and thus, n  (1 + λj (K)). det A = det(I + K) = j=1

This should serve as a motivation for the following definition. Suppose now H is a separable Hilbert space and let A ∈ L(H) be an operator of the form I + K with K ∈ C1 (H). If {λj (K)}j≥0 denotes the sequence of the nonzero eigenvalues of K (counted up to algebraic multiplicity), then  |λj (K)| < ∞, j≥0

by 1.3(g). Therefore the (possibly infinite) product  (1 + λj (K)) j≥0

is absolutely convergent. We define det A = det(I + K) :=



(1 + λj (K)).

j≥0

In case the spectrum of K consists only of 0 we put det(I + K) = 1. The number det A is called the determinant of A.

1.2 Operator Determinants

5

Note that besides the just given definition there are at least two other possibilities of defining det(I + K) for abstract K ∈ C1 (H), which, however, are in a certain sense equivalent to the above definition (see Dunford, Schwartz [177] and Simon [488]). The technical details of any systematical treatment of operator determinants, i.e., the derivation of such of their properties which are analogous to the finite-dimensional situation, essentially depend on the definition of det(I + K) adopted in each case. Here we prefer the above definition, due to Gohberg, Krein [228], for its simplicity, but we remark that the perhaps most elegant approach to operator determinants is based on another definition (see Simon [488]). 1.7. Basic properties of operator determinants. Here we list some properties of operator determinants which will be needed in Chapter 10. (a) If K ∈ C1 (H), then | det(I + K)| ≤ exp(K1 ),

| det(I + K)| ≤



(1 + sj (K)).

j≥0

The mapping C1 (H) → C,

K → det(I + K)

is continuous, i.e., Kn − K1 → 0 implies that | det(I + Kn ) − det(I + K)| → 0. Moreover, for K, L ∈ C1 (H), one has | det(I + K) − det(I + L)| ≤ K − L1 exp(K1 + L1 + 1). (b) If Ω ⊂ C is open and if K : Ω → C1 (H) is an analytic C1 (H)-valued function, then det(I + K) : Ω → C is also analytic. (c) If K, L ∈ C1 (H), then det(I + K) det(I + L) = det(I + K + L + KL). If K ∈ C∞ (H), A ∈ L(H), AK ∈ C1 (H), KA ∈ C1 (H), then det(I + AK) = det(I + KA). In particular, for every K ∈ C1 (H) and every invertible operator C ∈ L(H) one has det(I + CKC −1 ) = det(I + K). (d) If P is a finite rank projection, then det P (I + K)P = det(I + P KP ), where the det on the left refers to the ordinary finite-dimensional determinant for operators acting on Im P , the image of P .

6

1 Auxiliary Material

(e) If K ∈ C1 (H), then det eK = etr K (note that eK − I = K + K 2 /2! + . . . is in C1 (H)). (f) Let K ∈ C1 (H). Then det(I + K) = 0 if and only if I + K is invertible in L(H). 1.8. Regularized determinants. Let K ∈ Cp (H), where p > 1 is an integer. A simple computation shows that then ⎞⎤ ⎡ ⎛ p−1 j  (−K) ⎠⎦ − I ∈ C1 (H). Rp (K) := ⎣(I + K) exp ⎝ j j=1 Thus, it is justified to define detp (I + K) := det(I + Rp (K)). One calls detp (I + K) the p-regularized determinant of I + K. (a) Let {λj (K)}j≥0 denote the sequence of the eigenvalues of K ∈ Cp (H), counted up to algebraic multiplicity. Then p−1     (−λj (K))l detp (I + K) = (1 + λj (K)) exp . l j≥0

l=1

(b) If K ∈ Cp (H), then   tr (K p ) detp+1 (I + K) = detp (I + K) exp (−1)p . p In particular, for K ∈ C1 (H), ⎞ p−1 j  tr (K ) ⎠ . detp (I + K) = det(I + K) exp ⎝ (−1)j j j=1 ⎛

(c) There exist constants Γp such that |detp (I + K)| ≤ exp(Γp Kpp ) for all K ∈ Cp (H). One may take Γ1 = 1, Γ2 = 1/2, and it is known that 1/p ≤ Γp ≤ e(2 + log p). The mapping Cp (H) → C, K → detp (I + K) is continuous. Moreover, for K, L ∈ Cp (H),   |detp (I + K) − detp (I + L)| ≤ K − Lp exp Γp (Kp + Lp + 1)p . (d) If K, L ∈ C2 (H), then det2 (I + K)det2 (I + L) = det2 (I + K + L + KL) exp(tr KL).

1.2 Operator Determinants

7

(e) Let K ∈ Cp (H). Then detp (I + K) = 0 if and only if I + K is invertible in L(H). (f) (Plemelj-Smithies’ formula). Let K ∈ Cp (H), where p ≥ 1 is an integer. Then detp (I + zK) is an entire function (of z) whose power series expansion in the plain is given by ⎛ (p) ⎞ σ1 n − 1 0 . . . 0 ⎜ (p) (p) ⎟ ∞  ⎜ σ2 σ1 n − 2 . . . 0 ⎟ zn ⎜ det ⎜ . detp (I + zK) = 1 + .. .. .. ⎟ ⎟ n! ⎝ .. . . . ⎠ n=1

(p)

σn (p)

(p)

(p)

(p)

σn−1 σn−2 . . . σ1

(p)

where σj := 0 for j ≤ p − 1 and σj := tr (K j ) for j ≥ p. Note that this formula expresses det(I + K) in terms of tr (K j ) (j = 1, 2, . . .), and when tr (K), tr (K 2 ), . . . , tr (K p−1 ) are set equal to zero in this formula, we just get detp (I + K). 1.9. The Ehrhardt formula. If A, B ∈ L(H) and AB − BA ∈ C1 (H) then 1

det eA eB e−A−B = etr 2 (AB−BA) .

(1.3)

Proof. We first show that the function F (λ) := eλA eλB e−λ(A+B) −I is an entire C1 (H)-valued function. Put M = AB − BA. Clearly, it suffices to show that G(λ) := eλA eλB − eλ(A+B) is an entire C1 (H)-valued function. By definition, G(λ) =

 λk+m  λn Ak B m − (A + B)n . k! m! n! n≥0

k,m≥0

There exist exactly (k + m)!/(k! m!) possibilities to form a non-commutative product which contains exactly k factors A and exactly m factors B. Let Ik,m be an index set enumerating all these possibilities and denote the corresponding product by Pk,m (j) with j ∈ Ik,m . Then   Pk,m (j) (A + B)n = k, m ≥ 0 k+m = n

j∈Ik,m

and hence G(λ) =





k,m≥0 j∈Ik,m

λk+m  k m A B − Pk,m (j) . (k + m)!

The product Pk,m (j) can be transformed into Ak B m by at most km permutations that interchange only two neighboring elements. Consequently,  Ak B m − Pk,m (j) = Xs(k,m,j) M Ys(k,m,j) s

8

1 Auxiliary Material (k,m,j)

where the sum is over at most km terms, and Xs propriate products of the A’s and B’s. In particular,

(k,m,j)

and Ys

are ap-

m−1 Xs(k,m,j) ∞ Ys(k,m,j) ∞ ≤ Ak−1 ∞ B∞ .

We claim that G(λ) =





k,m≥0 j∈Ik,m

λk+m  (k,m,j) Xs M Ys(k,m,j) (k + m)! s

is an absolutely convergent series of operators in C1 (H) that converges uniformly with respect to λ on compact subsets of C. Indeed, this follows from the estimate   |λk+m |  Xs(k,m,j) M Ys(k,m,j) 1 (k + m)! s k,m≥0 j∈Ik,m

≤

 |λ|k+m km m−1 Ak−1 ∞ B∞ M 1 (k + m)!



k,m≥0 j∈Ik,m

= M 1

 Ak−1 Bm−1 ∞ ∞ |λ|k+m (k − 1)! (m − 1)!

k,m≥1

= |λ| M 1 eλa∞ eλB∞ . 2

Hence G(λ) takes values in C1 (H) and is an entire function. From what has just been proved we infer that f (λ) := det eλA eλB e−λ(A+B) is entire. Obviously, f (λ) = 0 for λ ∈ C. We take the logarithmic derivative of f and use the well known formula  (det H(λ)) = tr H  (λ)H −1 (λ) det H(λ) with H(λ) := eλA eλB e−λ(A+B) . What results is that  f  (λ) = tr A + eλA Be−λA − eλA eλB (A + B)e−λB e−λA f (λ)  = tr A − eλA eλB Ae−λB e−λA  = tr A − eλB Ae−λB . Differentiating again we obtain that     f (λ) = tr eλB (−BA + AB)e−λB = tr (AB − BA). f (λ) Since f (0) = 0 and f  (0)/f (0) = 0, we see that  2  λ f (λ) = exp tr (AB − BA) .   2

1.3 Fredholm Operators

9

1.10. The Pincus-Helton-Howe formula. If A, B ∈ L(H) and AB − BA is in C1 (H) then det eA eB e−A e−B = etr (AB−BA) . Proof. Clearly, (1.3) implies that 1

det eA+B e−A e−B = etr 2 (AB−BA)

(1.4)

and multiplication of (1.3) and (1.4) yields the desired formula.  

1.3 Fredholm Operators 1.11. Definitions. Let X and Y be Banach spaces and let A ∈ L(X, Y ). We put
 Ker A = x ∈ X : Ax = 0 , Im A = A(X). The operator A is said to be normally solvable if Im A is closed in Y . A normally solvable operator A ∈ L(X, Y ) is called a Φ+ -operator (Φ− -operator ) if dim Ker A < ∞ (dim Coker A := dim(Y /Im A) < ∞). If A is both a Φ+ - and Φ− -operator, then it is called a Φ-operator (or Fredholm operator or Noetherian operator ) and the integer Ind A := dim Ker A − dim Coker A is referred to as the index of A. The collection of all Φ+ -operators from X to Y will be denoted by Φ+ (X, Y ), and Φ+ (X, X) will be abbreviated to Φ+ (X). A similar definition is made for Φ− (X, Y ), Φ− (X), Φ(X, Y ), Φ(X). 1.12. Basic properties of Fredholm operators. (a) For A ∈ L(X, Y ) the following are equivalent. (i) A ∈ Φ(X, Y ). (ii) There exist operators R, L ∈ L(X, Y ) such that AR − IY ∈ C∞ (Y ),

LA − IX ∈ C∞ (X).

(iii) There exists an operator B ∈ L(X, Y ) such that AB − IY ∈ C0 (Y ),

BA − IX ∈ C0 (X).

Here IX and IY are the identity operators on X and Y , respectively. Any operator B ∈ L(X, Y ) for which AB − IY ∈ C∞ (Y ), is called a regularizer of A.

BA − IX ∈ C∞ (X)

10

1 Auxiliary Material

(b) (Fedosov’s formula). Let A ∈ Φ(X, Y ) and let B be any operator satisfying (iii) of (a). Then Ind A = tr (IX − BA) − tr (IY − AB). In particular, for X = Y we have Ind A = tr (AB − BA). (c) (Atkinson’s theorem). Let X, Y, Z be Banach spaces and let A ∈ Φ(X, Y ), B ∈ Φ(Y, Z). Then BA ∈ Φ(X, Z) and Ind BA = Ind B + Ind A. (d) Φ(X, Y ) is an open subset of L(X, Y ) and the mapping Ind : Φ(X, Y ) → Z is constant on the connected components of Φ(X, Y ). If A ∈ Φ(X, Y ) and K ∈ C∞ (X, Y ), then A + K ∈ Φ(X, Y ) and Ind (A + K) = Ind A. (e) Let X, Y, Z be Banach spaces and let A ∈ L(X, Y ), B ∈ L(Y, Z). Then the following implications are true. (i) A ∈ Φ± (X, Y ), B ∈ Φ± (Y, Z) =⇒ BA ∈ Φ± (X, Z). (ii) BA ∈ Φ+ (X, Z) =⇒ A ∈ Φ+ (X, Y ). (iii) BA ∈ Φ− (X, Z) =⇒ B ∈ Φ− (Y, Z). (iv) A ∈ Φ(X, Y ), BA ∈ Φ(X, Z) =⇒ B ∈ Φ(Y, Z). (v) B ∈ Φ(Y, Z), BA ∈ Φ(X, Z) =⇒ A ∈ Φ(X, Y ). (f) Property (d) remains valid if Φ is replaced by Φ+ or Φ− and the index is allowed to assume the values ±∞. Moreover, if A ∈ Φ± (X, Y ), then dim Ker (A + C) ≤ dim Ker A,

dim Coker (A + C) ≤ dim Coker A

whenever C has sufficiently small norm. (g) Let A ∈ Φ+ (X) and let K ∈ C0 (X) be any projection of X onto Ker A. Then there is a δ > 0 such that Ax + Kx ≥ δx ∀ x ∈ X. Conversely, if for an operator A ∈ L(X) there exist operators Kj ∈ C∞ (X) (j = 1, . . . , n) and a δ > 0 such that Ax +

n  j=1

Kj x ≥ δx ∀ x ∈ X,

1.4 Operator Matrices and Their Determinants

11

then A ∈ Φ+ (X). (h) We have A ∈ Φ± (X, Y ) ⇐⇒ A∗ ∈ Φ∓ (Y ∗ , X ∗ ). Moreover, if A is in Φ(X, Y ) then dim Ker A∗ = dim Coker A,

dim Coker A∗ = dim Ker A,

whence Ind A∗ = −Ind A.

1.4 Operator Matrices and Their Determinants 1.13. Definitions. Given a linear space X, denote by XN the linear space of column-vectors of length N with components from X and let XN ×N denote the linear space of N × N matrices with entries from X. If X is a Banach space, XN can be made to a Banach space on defining a norm in XN by (x1 , . . . , xN ) XN := x1 X + . . . + xN X

(1.5)

or by choosing any norm in XN equivalent to that one. Every operator A ∈ L(XN ) may then be written as an operator matrix A = (Aij )N i,j=1 , where Aij ∈ L(X), that is, L(XN ) may be identified with (L(X))N ×N . It is easily seen that K ∈ Cp (XN ) if and only if all entries Kij of the matrix K = (Kij )N i,j=1 are in Cp (X). Let A = (Aij )N i,j=1 ∈ L(XN ). The determinant det A of A is the operator in L(X) which is defined by  det A = (−1)p(σ) AN,σ(N ) . . . A1,σ(1) , (1.6) σ

where σ ranges over all permutations of {1, . . . , N } and p(σ) is the signature of the permutation σ. If the operators Aij do not commute pairwise, then the order in which the factors in each term of the sum (1.6) are arranged is of significance. The arrangement chosen in (1.6) corresponds to the expansion of determinants with respect to the last row, i.e., one has   A11 A12 det = −A21 A12 + A22 A11 , A21 A22 ⎞ ⎛ A11 A12 A13 det ⎝ A21 A22 A23 ⎠ A31 A32 A33       A12 A13 A11 A13 A11 A12 = A31 det − A32 det + A33 det A22 A23 A21 A23 A21 A22 and so on. However, if we are only interested in whether det A is a Φ+ - or Φ− operator and if the entries of A pairwise commute up to a compact operator, then the answer will not depend on the arrangement of the factors in the terms of the sum (1.6). In the theorems below, [A, B] denotes the commutator AB − BA.

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1.14. Theorem. Let X be a Banach space and let A = (Aij )N i,j=1 ∈ L(XN ). (a) If [Aij , Alm ] ∈ C∞ (X) for j = m, then det A ∈ Φ− (X) =⇒ A ∈ Φ− (XN ), A ∈ Φ+ (XN ) =⇒ det A ∈ Φ+ (X). (b) If [Aij , Alm ] ∈ C∞ (X) for i = l, then det A ∈ Φ+ (X) =⇒ A ∈ Φ+ (XN ), A ∈ Φ− (XN ) =⇒ det A ∈ Φ− (X). (c) If the entries of A pairwise commute modulo compact operators, then A ∈ Φ(XN ) (resp. Φ+ (XN ), Φ− (XN )) if and only if det A ∈ Φ(X) (resp. Φ+ (X), Φ− (X)). This theorem says nothing about the connection between the index of A and that of det A. Under the hypothesis of part (c) both equality and inequality of these indices are possible. The following theorem states sufficient conditions for equality. 1.15. Theorem (Markus/Feldman). (a) Let H be a Hilbert space, let A be in Φ(HN ), and suppose the entries of A pairwise commute modulo C1 (H). Then det A ∈ Φ(H) and Ind A = Ind det A. (b) Let X be a Banach space, let A ∈ Φ(XN ), and suppose the entries of A pairwise commute modulo C0 (X). Then det A ∈ Φ(X) and Ind A = Ind det A.

1.5 Banach Algebras 1.16. Invertibility and spectrum. Let A be a Banach algebra with identity e. Throughout the book we assume that the scalar field is C, that ab ≤ a b for all a, b ∈ A and that e = 1. An element a ∈ A is said to be left (right, resp. two-sided) invertible in A if there exists an element b ∈ A such that ba = e (ab = e, resp. ba = ab = e). Two-sided invertible elements will be simply called invertible. The collection of all invertible elements of A will be denoted by GA. Note that GA is open and

forms a group with respect to the multiplication in A. For a ∈ A, let exp a = n≥0 an /n!. (a) If A is commutative, then an invertible element a belongs to the connected component of GA containing the identity e if and only if a has a logarithm in A, i.e., if there exists an element b ∈ A such that a = exp b. The spectrum sp a (or sp(a)) of an element a ∈ A is the set of all λ ∈ C such that a − λe is not (two-sided) invertible in A. In order to emphasize that

1.5 Banach Algebras

13

invertibility in A is meant, we shall sometimes write spA a instead of sp a. For several applications the following result will often be useful. (b) Let B be a closed subalgebra of A and suppose e ∈ B. If a ∈ B, then spB a is the union of spA a and a (possible empty) collection of bounded connected components of the complement of spA a. 1.17. Ideals. Let A be a Banach algebra. A closed subalgebra J of A is called a closed left (resp. right) ideal of A if aj ∈ J (resp ja ∈ J) for all a ∈ A and j ∈ J. The closed two-sided ideals of A are the closed ideals which are both left and right. An ideal J of A is said to be proper if J = A. A proper closed left (right, resp. two-sided) ideal J of A is called a maximal left (right, resp. two-sided) ideal if it is not properly contained in any other proper left (right, resp. two-sided) ideal of A. (a) In a Banach algebra with identity every proper left (right, resp. twosided) ideal is contained in some maximal left (right, resp. two-sided) ideal. Maximal two-sided ideals will be simply referred to as maximal ideals. If J is a proper closed two-sided ideal of A, then the quotient algebra A/J is a Banach algebra under the norm a + J := inf a + j. j∈J

(b) If A is commutative and has an identity and if J is a maximal ideal of A then the quotient algebra A/J is a field, i.e., every nonzero element of A/J has an inverse. (c) The radical R(A) of A is the intersection of all maximal left ideals of A. If A has an identity, then R(A) is a closed two-sided ideal of A and R(A) coincides with the intersection of all maximal right ideals of A. A Banach algebra with identity whose radical consists only of the zero element will be called semisimple. 1.18. The maximal ideal space. Let A be a commutative Banach algebra with identity e. A multiplicative linear functional on A is a continuous linear mapping m : A → C which preserves multiplication (m(ab) = m(a)m(b) for all a, b ∈ A) and takes the value 1 at e (m(e) = 1). The kernel of m is the set of all a ∈ A for which m(a) = 0. There is a one-to-one correspondence between the multiplicative linear functionals and the maximal ideals of A: the kernel of every multiplicative linear functional is a maximal ideal and every maximal ideal is the kernel of some (uniquely determined) multiplicative linear functional. Therefore no distinction between multiplicative linear functionals and maximal ideals will be made. We denote the set of all multiplicative linear functionals on A by M (A). The formula  a(m) = m(a) (m ∈ M (A)) assigns to each a ∈ A a function  a : M (A) → C. This function is called the Gelfand  be the set of all functions  transform of a. Let A a, for a ∈ A. The Gelfand topology on M (A) is the coarsest (weakest) topology on M (A)  continuous. It is the topology induced on M (A) that makes all functions  a∈A thought of as a subset of the dual space A∗ , the latter space provided with the

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weak-star topology. Thus, an open neighborhood base of a point m0 ∈ M (A) is formed by the sets
 Ua1 ,...,an ;ε (m0 ) = m ∈ M (A) : |ai (m) − ai (m0 )| < ε for i = 1, . . . , n 
= m ∈ M (A) : |m(ai ) − m0 (ai )| < ε for i = 1, . . . , n , where a1 , . . . , an ∈ A and ε > 0. The set M (A) equipped with the Gelfand topology is called the maximal ideal space of A. Note that M (A) is a com is a (not necessarily closed) subalgebra of pact Hausdorff space and that A C(M (A)). The mapping Γ : A → C(M (A)), a →  a will be referred to as the Gelfand map. Notice that in general Γ is neither one-to-one nor onto. The kernel of Γ coincides with the radical of A. Thus, if A is semisimple, then the Gelfand map  ⊂ C(M (A)). is one-to-one and hence an (algebraic) isomorphism of A onto A  Therefore we shall then simply write a and A instead of  a and A. Finally, recall that if A is a (not necessarily semisimple) commutative Banach algebra with identity and if a ∈ A, then the range of  a coincides with a∞ ≤ a, where  a∞ := max | a(m)|. the spectrum spA a and  m∈M (A)

1.19. Singly generated algebras. A commutative Banach algebra A with identity e is said to be singly generated if there is an a ∈ A such that the linear hull of the set {e, a, a2 , . . .} is dense in A: A = clos lin{e, a, a2 , . . .}. In that case the maximal ideal space of A is naturally homeomorphic to spA a, where spA a has the topology induced from the inclusion spA a ⊂ C and C is regarded as furnished with the usual topology. 1.20. The Shilov boundary. Let A be a commutative Banach algebra with identity e. A closed subset F ⊂ M (A) is called a boundary if max | a(m)| = max | a(m)| ∀ a ∈ A.

m∈M (A)

m∈F

The intersection of all boundaries is also a boundary. It is called the Shilov boundary of A and denoted by ∂S M (A). (a) A point m0 ∈ M (A) is in ∂S M (A) if and only if for each open neigh such that a∈A borhood U ⊂ M (A) of m0 there exists an  sup m∈M (A)\U

| a(m)| < sup | a(m)|. m∈U

(b) Let B be a closed subalgebra of A and let e ∈ B. Then each maximal ideal m ∈ ∂S M (B) is contained in some maximal ideal belonging to M (A). In other words, each multiplicative linear functional on B belonging to ∂S M (B) admits an extension to a multiplicative linear functional on A.

1.5 Banach Algebras

15

A subset N of A is said to consist of joint topological divisors of zero if inf

n 

 zai  : z ∈ A, z = 1 = 0

i=1

for each finite subset {a1 , . . . , an } ⊂ N . (c) Every maximal ideal belonging to the Shilov boundary of A consists of joint topological divisors of zero. 1.21. Maximal antisymmetric sets. Let Y be a compact Hausdorff space and let B be a closed subalgebra of C(Y ) containing the constants. A subset S of Y is said to be an antisymmetric set for B in case every function in B which is real-valued on S is constant on S. It is clear that every subset of Y consisting of a single point is an antisymmetric set. It is easy to see that of antisymmetric sets having a nonempty intersection, if {Sα } is any family  then the union α Sα is also an antisymmetric set, and that the closure of an antisymmetric set is again such a set. Thus the union of all antisymmetric sets containing a given point y ∈ Y is a nonempty closed antisymmetric set. Antisymmetric sets of this form are called maximal antisymmetric sets for B. Any two maximal antisymmetric sets are obviously disjoint and Y is the union of all maximal antisymmetric sets. It is also easily seen that the maximal antisymmetric sets are the equivalence classes of the equivalence relation on Y defined by y1 ∼ y2 if there exists an antisymmetric set containing both y1 and y2 . If A and B are closed subalgebras of C(Y ) containing the constants and if B ⊂ A, then obviously every maximal antisymmetric set for A is contained in some maximal antisymmetric set for B. For F a closed subset of Y and a ∈ C(Y ), we define dist(a, B) := inf max |a(y) − b(y)|, b∈B y∈Y

distF (a, B) := inf max |a(y) − b(y)|. b∈B y∈F

1.22. Theorem (Glicksberg). Let Y be a compact Hausdorff space and B be a closed subalgebra of C(Y ) containing the constants. Let S be the family of maximal antisymmetric sets for B. Then, for every a ∈ C(Y ), dist(a, B) = max distS (a, B) S∈S

and the maximum is attained at some S0 ∈ S. Proof. It suffices to prove that there is an S0 ∈ S such that dist(a, B) = distS0 (a, B). Let F denote the collection of all nonempty closed subsets F of Y for which dist(a, B) = distF (a, B). Obviously Y ∈ F. If C is a subfamily of F also belongs to F . Indeed, F totally ordered by inclusion, then E := F∈ C

if b ∈ B, then the set {y ∈ F : |a(y) − b(y)| ≥ dist(a, B)} is compact and nonempty (since distF (a, B) = dist(a, B)), and hence the intersection of the

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chain of all such sets, namely {y ∈ E : |a(y) − b(y)| ≥ dist(a, B)}, is also nonempty; as this holds for each b ∈ B, it follows that distE (a, B) = dist(a, B), i.e., that E ∈ F. Zorn’s lemma therefore guarantees the existence of a minimal element S0 of F , and to complete the proof of the theorem it remains to show that S0 is an antisymmetric set for B. Suppose this is false. Then there is a b ∈ B such that b is both real-valued and non-constant on S0 . Replacing b by a suitable real-linear combination of b and 1, we may assume that min b(y) = 0 and max b(y) = 1. Put y∈S0

 2 U = y ∈ S0 : 0 ≤ b(y) ≤ , 3

y∈S0

  1 ≤ b(y) ≤ 1 . V = y ∈ S0 : 3

Since U and V are nonempty proper closed subsets of S0 , the minimality of S0 implies that there exist c, d ∈ B with max |a(y) − c(y)| < dist(a, B), y∈U

max |a(y) − d(y)| < dist(a, B). y∈V

(1.7)

n

For n ≥ 1, set bn = (1 − b)2 and gn = bn c + (1 − bn )d. Clearly bn , gn ∈ B and 0 ≤ bn ≤ 1 on S0 . Since at each point of S0 the value of gn is a convex combination of c and d, it follows from (1.7) that max |a(y) − gn (y)| < dist(a, B).

(1.8)

y∈U ∩V

Now on U \ V , where 0 ≤ b < 1/3, we have n

bn = (1 − bn )2 ≥ 1 − 2n bn ≥ 1 −

 2 n 3

,

while on V \ U , where 2/3 < b ≤ 1, we have n

bn = (1 − bn )2 ≤

 3 n 1 1 ≤ . n ≤ n 2 n n (1 + b ) 2 b 4

Thus gn converges uniformly to c on U \ V and uniformly to d on V \ U . Combining this with (1.7) and (1.8), we see that if n is taken large enough then max |a(y) − gn (y)| < dist(a, B), contradicting the fact that S0 belongs y∈S0

to the family F .   1.23. Corollary (Shilov/Bishop). Under the hypothesis of the preceding theorem, if a ∈ C(Y ) then a ∈ B ⇐⇒ a|S ∈ B|S

∀ S ∈ S.

Proof. Immediate from Theorem 1.22.   Comment. Here a|S ∈ B|S means that there is a b ∈ B (depending on S) such that a(y) = b(y) for all y ∈ S.

1.6 C ∗ -Algebras

17

1.24. Fibers. Let A be a commutative Banach algebra with identity e and suppose B is a closed subalgebra of A containing e. Consider the mapping τ : M (A) → M (B),

α → α|B,

which assigns to each functional in M (A) its restriction to B. A little thought shows that τ is continuous. For β ∈ M (B), put
 Mβ (A) = α ∈ M (A) : α|B = β . The set Mβ (A) is referred to as the fiber of M (A) over β. Since τ is continuous and Mβ (A) = τ −1 ({β}), Mβ (A) is always a compact subset of M (A). Of course, it can happen that Mβ (A) = ∅. There are two situations of particular importance. (a) Suppose that for each β ∈ M (B) the fiber Mβ (A) is either empty or a singleton. Then τ is one-to-one and because M (A) is compact, it follows that τ is a homeomorphism of M (A) onto the compact subset τ (M (A)) of M (B). By identifying M (A) with τ (M (A)), we may therefore think of M (A) as a subset of M (B). (b) Suppose A = C(Y ), where Y is a compact Hausdorff space, and let B be a closed subalgebra of A which contains the constant functions and separates the points of Y . We then have M (A) = ∂S M (A) = Y . For β in M (B), the fiber
 Mβ (A) = y ∈ Y : b(y) = β(b) ∀ b ∈ B is either empty or a singleton, since B separates the point of Y . Therefore we may regard Y as a compact subset of M (B). Since b = max |b(y)| for y∈Y

all b ∈ B, the Shilov boundary of B is a closed subset of Y . Thus, we have ∂S M (B) ⊂ Y ⊂ M (B).

1.6 C ∗-Algebras 1.25. Definitions. A mapping a → a∗ of a Banach algebra A into itself is called an involution on A if a∗∗ = a,

(a + b)∗ = a∗ + b∗ ,

(ab)∗ = b∗ a∗ ,

(λa)∗ = λa∗

for all a, b ∈ A and λ ∈ C. A Banach algebra with an involution a → a∗ that satisfies aa∗  = a2 for all a ∈ A is called a C ∗ -algebra. If Y is a compact Hausdorff space and H is a Hilbert space, then C(Y ) and L(H) are C ∗ -algebras with their natural involutions.

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1.26. Basic properties of C ∗ -algebras. (a) (Gelfand/Naimark). If A is a commutative C ∗ -algebra with identity, then the Gelfand map is an isometric star-isomorphism of A onto C(M (A)). (b) (Gelfand/Naimark). If A is any C ∗ -algebra, then there exists a Hilbert space H such that A is isometrically star-isomorphic to some C ∗ subalgebra of L(H). (c) If A is a commutative C ∗ -algebra with identity, then ∂S M (A) = M (A). (d) If A is a C ∗ -algebra with identity and B is a C ∗ -subalgebra of A containing the identity, then an element a ∈ B is left (right, resp. two-sided) invertible in A if and only if it is so in B. (e) Let A and B be C ∗ -algebras and let ϕ : A → B be an (algebraic) star-homomorphism. Then ϕ(a) ≤ a for all a ∈ A and the range of ϕ is closed in B. If, in addition, ϕ is one-to-one, then ϕ(a) = a for all a ∈ A. (f) Let A be a C ∗ -algebra and let J be a closed two-sided ideal of A. Then J is selfadjoint (that is, J ∗ = J) and A/J provided with the involution (a + J)∗ = a∗ + J and the usual quotient norm is a C ∗ -algebra. (g) Let A be a C ∗ -algebra, let B be a C ∗ -subalgebra of A, and let J be a closed two-sided ideal of A. Then B + J is a C ∗ -subalgebra of A and the C ∗ -algebras (B + J)/J and B/(B ∩ J) are isometrically star-isomorphic. 1.27. Fibers over C ∗ -algebras. Let A be a commutative Banach algebra with identity and B a closed subalgebra of A containing the identity. (a) By 1.20(b), for β in the Shilov boundary ∂S M (B) the fiber Mβ (A) is not empty. (b) In particular, if B is a C ∗ -subalgebra of A, then ∂S M (B) = M (B) and so Mβ (A) = ∅ for each β ∈ M (B). Via α1 ∼ α2 ⇐⇒ α1 |B = α2 |B

(α1 , α2 ∈ M (A))

an equivalence relation is given on M (A). The corresponding partition of M (A) into equivalence classes is nothing else than the partition of M (A) into fibers over M (B). This set of equivalence classes equipped with a natural topology is homeomorphic to M (B). (c) Now suppose Y is a compact Hausdorff space and B is a C ∗ -subalgebra of C(Y ) containing the constant functions. Note that M (C(Y )) = Y . For β ∈ M (B), denote by Yβ ⊂ Y the fiber Mβ (C(Y )). Then the maximal antisymmetric sets for B are the fibers Yβ , β ∈ M (B). This can be proved as follows. We first show that each antisymmetric set S for B is contained in some fiber Yβ . Assume the contrary, i.e., assume there are y1 , y2 ∈ S and two distinct points β1 , β2 ∈ M (B) such that y1 ∈ Yβ1 and y2 ∈ Yβ2 . Then, for every

1.6 C ∗ -Algebras

19

b ∈ B, b(y1 ) = b(β1 ) and b(y2 ) = b(β2 ). Since B is isometrically isomorphic to C(M (B)), there is a b ∈ B such that b(β1 ) = b(β2 ). Since B is a C ∗ algebra, both Re b and Im b belong to B, and we have (Re b)(β1 ) = (Re b)(β2 ) or (Im b)(β1 ) = (Im b)(β2 ). Thus, there is a real-valued function in B taking two distinct values on S, which is impossible if S is an antisymmetric set for B. So it remains to show that each fiber Yβ is an antisymmetric set for B. But this is obvious, since B|Yβ ∼ = C and so B|Yβ does not contain non-constant functions at all. In particular, Theorem 1.22 and Corollary 1.23 are valid if S is replaced by the collection of all fibers Yβ , β ∈ M (B). (d) Thus, if Y is a compact Hausdorff space and B and A are closed subalgebras of C(Y ) containing the constants C such that C ⊂ B ⊂ A ⊂ C(Y ), then each maximal antisymmetric set for A is contained in some maximal antisymmetric set for B. Finally, note that both C and C(Y ) are C ∗ -algebras whose maximal antisymmetric sets are the whole space Y and the singletons {y}, y ∈ Y , respectively. 1.28. Restriction algebras. Let A be a commutative C ∗ -algebra with identity. By the Gelfand-Naimark theorem 1.26(a), A is isometrically starisomorphic to C(Y ), where Y = M (A). If F is a closed subset of Y , then IF = {a ∈ A : a|F = 0} is obviously a closed (two-sided) ideal of A. Moreover, it can be shown that every closed ideal of A is of this form (see, e.g., Naimark [362, pp. 247–248]). Let A|F denote the algebra whose elements are the restrictions a|F (or, more precisely,  a|F ), for a ∈ A. When endowed with the norm a|F  = max |a(y)|, y∈F

A|F is a C ∗ -algebra which is isometrically star-isomorphic to C(F ) in a natural fashion. It is called the restriction of A to F . It is not difficult to show that the mapping ϕ : A/IF → A|F,

a + IF → a|F

is an isometric star-isomorphism of the C ∗ -algebra A/IF , equipped with the natural norm a + IF  := inf a + g, g|F =0

onto the C ∗ -algebra A|F . Now let B be a closed subalgebra of A containing the identity. Via the Gelfand map, B is isometrically isomorphic to a closed subalgebra of C(Y )

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and we shall identify B with that subalgebra. It is, in general, a delicate problem to decide whether or not B|F is a closed subalgebra of A|F ∼ = C(F ). A closed subset F of Y is called a peak set for B if there is a b ∈ B such that b(y) = 1 for all y ∈ F and |b(y)| < 1 for all y ∈ Y \F . It is easily seen that a finite or countable intersection of peak sets for B is again a peak set for B (if

∞ ∞ bn “peaks” on Fn then n=0 bn /2n “peaks” on n=0 Fn ). A closed subset of Y is called a weak peak set for B, if it is the (possibly uncountable) intersection of peak sets for B. One can show that every maximal antisymmetric set for B is a weak peak set for B. The role played by weak peak sets is illuminated by the following result: If F is a weak peak set for B, then B|F is a closed subalgebra of A|F ∼ = C(F ); moreover, B|F is isometrically isomorphic to the quotient algebra B/IFB , where IFB = {b ∈ B : b|F = 0}. 1.29. Matrix norms. Let A be a normed space. A norm in AN ×N , the linear space of all N × N matrices with entries from A, is said to be admissible if there are constants m, M > 0 such that m max aij  ≤ aAN ×N ≤ M max aij  i,j

i,j

for all a = (aij )N i,j=1 ∈ AN ×N . If A is a Banach space and if AN ×N is endowed with an admissible norm, then AN ×N is also a Banach space. If A is an algebra, then AN ×N is an algebra under the usual matrix operations. If A is a Banach algebra, then a norm in AN ×N will be called a Banach algebra norm if it is admissible and if ab ≤ a b for all a, b ∈ AN ×N . Now let A be a C ∗ -algebra. Then the mapping (aij ) → (a∗ji ) is an involution on AN ×N . A norm in AN ×N will be called a C ∗ -norm if it is a Banach algebra norm and if aa∗  = a2 for all a ∈ AN ×N . (a) If A is a C ∗ -algebra with identity, then there exists exactly one C ∗ norm in AN ×N . Proof. To see this, notice first that by 
/ GAN ×N a2 := r(a∗ a) := max |λ| : λ ∈ C, a∗ a − λe ∈ a C ∗ -norm is given in AN ×N . Now let  · 1 and  · 2 be two C ∗ -norms in AN ×N and denote by A1N ×N and A2N ×N the algebra AN ×N endowed with the norm  · 1 and  · 2 , respectively. The identity mapping is clearly an algebraic star-isomorphism of A1N ×N onto A2N ×N . So 1.26(e) implies that it  is an isometry, that is, a1 = a2 for all a ∈ AN ×N .  Let the norm in CN be given by  (zk )N k=1 

:=

N 

k=1

1/2 |zk |

2

.

(1.9)

1.7 Local Principles

21

The norm in (L(CN ))∗ , the dual space of L(CN ), is 
ϕ := sup |ϕ(z)| : z ∈ CN , z = 1 . We denote by Iij (i, j ∈ {1, . . . , N }) the operator in L(CN ) which is defined by Iij : (zn , . . . , zN ) → (0, . . . , 0, zj , 0, . . . , 0) where the zj occupies the i-th place. (b) Let A be a commutative C ∗ -algebra. Then there exists exactly one C ∗ norm in AN ×N . This norm can be given by aAN ×N

!"  # N " " " ∗ = sup " aij ϕ(Iij )" : ϕ ∈ (L(CN )) , ϕ = 1 , i,j=1

A

where a = (aij )N i,j=1 . Proof. For a proof we refer to Sakai [446, 1.22.5].   Note that the linear space AN ×N can be naturally identified with the algebraic tensor product A ⊗ L(CN ). The above reference implies that there is precisely one C ∗ -norm in A ⊗ L(CN ), namely, the norm which generates the injective tensor product.

1.7 Local Principles 1.30. Definitions. Let A be a Banach algebra with identity e. A subset M ⊂ A is called a localizing class if (i) 0 ∈ / M, (ii) for any f1 , f2 ∈ M there exists a third element f ∈ M such that fj f = f fj = f (j = 1, 2). Two elements a, b ∈ A are said to be M -equivalent from the left (resp. from the right) if   resp. inf f (a − b) = 0 . inf (a − b)f  = 0 f ∈M

f ∈M

An element a ∈ A is called M -invertible from the left (resp. from the right) if there are a b ∈ A and an f ∈ M such that baf = f (resp. f ab = f ). A system {Mτ }τ ∈T of localizing classes is said to be covering if from each choice {fτ }τ ∈T (fτ ∈ Mτ ) there can be selected a finite number of elements fτ1 , . . . , fτm whose sum is invertible in A. Now suppose T is a topological space. Then a system {Mτ }τ ∈T of localizing classes will be said to be overlapping if

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(iii) each Mτ is a bounded subset of A; (iv) f ∈ Mτ0 (τ0 ∈ T ) implies that f ∈ Mτ for all τ in some open neighborhood of τ0 ;  (v) the elements of F := Mτ commute pairwise. τ ∈T

Let {Mτ }τ ∈T be an overlapping system of localizing classes. The commutant of F is the set
 Com F := a ∈ A : af = f a ∀ f ∈ F . It is clear that Com F is a closed subalgebra of A. For τ ∈ T , let Zτ denote the set of all elements in Com F which are Mτ -equivalent to zero both from the left and from the right. Note that Zτ is a closed (by virtue of (iii)) two-sided ideal of Com F which does not contain the identity (if e ∈ Zτ , then there are fn ∈ Mτ such that fn  → 0 as n → ∞, and since there exist gn = 0 in Mτ such that fn gn = gn , it follows that fn  ≥ 1, which is a contradiction). For each a ∈ Com F let aτ denote the coset a + Zτ of the quotient algebra Com F/Zτ . Finally, recall that a function f : Y → R given on a topological space Y is called upper semi-continuous at y0 ∈ Y if for each ε > 0 there is a neighborhood Uε ⊂ Y of y0 such that f (y) < f (y0 ) + ε whenever y ∈ Uε . The function f is said to be upper semi-continuous on Y if it is upper semicontinuous at each y ∈ Y . Equivalently, f is upper semi-continuous on Y if and only if {y ∈ Y : f (y) < α} is an open subset of Y for every α ∈ R. Notice that if Y is a compact Hausdorff space and f : Y → R is a bounded upper semicontinuous function on Y , then there is a y0 ∈ Y such that f (y0 ) = sup f (y). y∈Y

1.31. Lemma. (a) Let M be a localizing class, let a, a0 ∈ A, and suppose a and a0 are M -equivalent from the left (resp. from the right). Then a is M -invertible from the left (resp. from the right) if and only if so is a0 . (b) Let {Mτ }τ ∈T be a system of localizing classes having property 1.30(iii), let τ ∈ T and a ∈ Com F . Then a is Mτ -invertible in Com F from both the left and the right if and only if aτ ∈ G(Com F/Zτ ). Proof. (a) Let a be M -invertible from the left. Then there are b ∈ A and f ∈ M such that baf = f . Since a and a0 are M -equivalent from the left, there is a g ∈ M such that (a − a0 )g < 1/b. Choose h ∈ M so that f h = gh = h. Then ba0 h = bah − b(a − a0 )h = baf h − b(a − a0 )gh = h − uh = (e − u)h, where u := b(a−a0 )g, and because u < 1, we deduce that e−u ∈ GA. Thus, if we let v = (e − u)−1 b, then va0 h = h, which shows that a0 is M -invertible from the left.

1.7 Local Principles

23

(b) Let aτ ∈ G(Com F/Zτ ). Then there is a b ∈ Com F such that ba − e is in Zτ . This implies that ba is Mτ -equivalent from the left to e, and from part (a) we deduce that ba and thus a is Mτ -invertible from the left. It can be shown similarly that a is Mτ -invertible from the right. Conversely, if there are b ∈ Com F and f ∈ Mτ such that baf = f , then  (ba − e)f = 0, hence ba − e ∈ Zτ , and thus bτ aτ = eτ .  1.32. Theorem (Gohberg/Krupnik). Let A be a Banach algebra with identity, let {Mτ }τ ∈T be a covering system of localizing classes, and let a ∈ Com F . (a) Suppose that, for each τ ∈ T , a is Mτ -equivalent from the left (resp. right) to aτ ∈ A. Then a is left-invertible (resp. right-invertible) in A if and only if aτ is Mτ -invertible from the left (resp. right) for all τ ∈ T . (b) If the system {Mτ }τ ∈T has property 1.30(iii), then a ∈ GA if and only if aτ is invertible in Com F/Zτ for all τ ∈ T . (c) Let the system {Mτ }τ ∈T be overlapping. Then the mapping T → R+ ,

τ → aτ 

is upper semi-continuous. If aτ0 ∈ G(Com F/Zτ0 ) for some τ0 ∈ T , then aτ ∈ G(Com F/Zτ ) for all τ in some open neighborhood of τ . Proof. (a) If a is left-invertible, then a is Mτ -invertible from the left for all τ ∈ T and hence, by Lemma 1.31(a), aτ is Mτ -invertible from the left for all τ ∈ T. Conversely, suppose aτ is Mτ -invertible from the left for all τ ∈ T . It follows again from Lemma 1.31(a) that a is Mτ -invertible from the left for all τ ∈ T . Thus, there are bτ ∈ A and fτ ∈ Mτ such that bτ afτ = fτ . Since m {Mτ }τ ∈T is covering, we can choose fτ1 , . . . fτm so that i=1 fτi is in GA. m Put s = i=1 bτi fτi . Then    sa = bτi fτi a = bτi afτi = fτi i

i

i

−1 and it results that ( fτi ) is a left-inverse of a. (b) If aτ ∈ G(Com F/Zτ ) for all τ ∈ T , then a ∈ G(Com F ) and thus a ∈ GA by virtue of Lemma 1.31(b) and part (a) of the present theorem. On the other hand, if a ∈ GA, then clearly a−1 ∈ Com F , hence a ∈ G(Com F ) and thus aτ ∈ G(Com F/Zτ ) for all τ ∈ T . (c) Let τ0 ∈ T and ε > 0. Choose a z ∈ Zτ0 so that a + z < aτ0  + ε/2. Since z is Mτ0 -equivalent to zero from the left, there is an f ∈ Mτ0 such that zf  < ε/2. From 1.30(iv) we deduce that f ∈ Mτ for all τ in some open neighborhood U (τ0 ) of τ0 . Put y = z − zf . If τ ∈ U (τ0 ), then there exists a g ∈ Mτ such that f g = g (1.30(ii)). Consequently, yg = zg−zf g = zg−zg = 0,

24

1 Auxiliary Material

and since y ∈ Com F (1.30(v)), it follows that y ∈ Zτ for all τ ∈ U (τ0 ). Hence, aτ  ≤ a + y for τ ∈ U (τ0 ). Thus, if τ ∈ U (τ0 ), then aτ  − aτ0  < a + y − a + z +

ε ε ε ≤ y − z + = zf  + < ε, 2 2 2

which proves the upper semi-continuity of τ → aτ  at τ0 . Now suppose aτ0 ∈ G(Com F/Zτ0 ). Then there is a b ∈ Com F such that (ba − e)τ0  = ba − e + Zτ0  = 0,

(ab − e)τ0  = ab − e + Zτ0  = 0.

By what has just been proved, the mappings τ → (ba − e)τ  and τ → (ab − e)τ  are upper semi-continuous. Hence ba − e + Zτ  = (ba − e)τ  <

1 , 2

ab − e + Zτ  = (ab − e)τ  <

1 2

for all τ in some open neighborhood U (τ0 ) of τ0 . This implies that aτ is  invertible for all τ ∈ U (τ0 ).  1.33. Definitions. Let A be a Banach algebra with identity e. The center Cen A of A is the set of all elements z ∈ A with the property that za = az for all a ∈ A. Clearly, Cen A is a closed commutative subalgebra of A. Let B be a closed subalgebra of Cen A containing e. Thus, B is also commutative. If N ⊂ B is a maximal ideal of B, then JN will denote the smallest closed two-sided ideal of A containing N , that is, JN = closA

! m

# xk ak : m ∈ Z+ , xk ∈ N, ak ∈ A .

k=1

Of course, it may happen that JN = A. In case JN = A we denote by AN the quotient algebra A/JN and, for a ∈ A, by aN the coset a + JN . If JN = A, then AN will refer to the algebra {Θ} whose only element Θ is simultaneously the zero and the identity; we then let aN = Θ for every a ∈ A and make the convention that aN ∈ GAN and aN  = 0 for every a ∈ A. Note that in either case aN  = dist(a, JN ). 1.34. Lemma. Let L be a maximal left, right, or two-sided ideal of A. Then L ∩ B is a maximal ideal of B. Proof. For the sake of definiteness, suppose L is a maximal left ideal. It is clear that L ∩ B is a proper (e ∈ / L) closed two-sided ideal of B and so we are left with the proof of its maximality. Let z ∈ B \ L. Then Iz := {l + az : l ∈ L, a ∈ A} is a left ideal of A containing L properly (z ∈ / L). The maximality of L implies that Iz = A, hence e ∈ Iz , and it follows that z has an inverse modulo L (note that z ∈ Cen A). / Kz ) left ideal of A Furthermore, Kz := {a ∈ A : az ∈ L} is a proper (e ∈ containing L. Since L is maximal, we have Kz = L. In particular, if y1 , y2 are

1.7 Local Principles

25

both inverses modulo L of z, then y1 − y2 ∈ L. Thus, the inverses modulo L of z determine a unique element of the quotient space A/L. Now suppose z − λe ∈ / L for all λ ∈ C. Let y π (λ) denote the (uniquely determined) coset of A/L containing the inverses of z − λe modulo L. We claim that y π : C → A/L is an entire function. To see this, let λ0 ∈ C and let y0 ∈ y π (λ0 ) be any inverse modulo L of z − λ0 e. Then, for |λ − λ0 | < 1/y0 , the element e − (λ − λ0 )y0 is invertible in A and it is readily verified that y0 [e−(λ−λ0 )y0 ]−1 is an inverse modulo L of z−λe. Thus, for |λ−λ0 | < 1/y0 , y π (λ) = y0 [e − (λ − λ)0 y0 ]−1 + L, which implies the asserted analyticity of y π . If |λ| > z, then z−λe is actually invertible in A and, as |λ| → ∞, " n" z " 1 " " " = o(1). y π (λ) ≤ (z − λe)−1  = " |λ| λn " n≥0

Therefore, by Liouville’s theorem, y π (λ) = 0 for all λ ∈ C, contrary to the assumption that L is a proper ideal of A (for y π (0) = 0 would imply that there is a y0 ∈ L with y0 z − e ∈ L, whence e ∈ L). Hence there is some λ ∈ C such that z − λe ∈ L and, since z ∈ / L, we have λ = 0. It follows that e = λ−1 z + l for some l ∈ L ∩ B. Now assume there is a two-sided ideal I of B such that L ∩ B ⊂ I and L ∩ B = I. Then there is a z ∈ I \ (L ∩ B) ⊂ B \ L and, by what has been proved above, there exist λ ∈ C \ {0} and l ∈ L ∩ B with e = λ−1 z + l. But this implies that e ∈ I and, hence, I = B, which proves the maximality of L ∩ B.   1.35. Theorem (Allan/Douglas). Let the situation be as in 1.33. (a) If a ∈ A, then a is left (right, resp. two-sided) invertible in A if and only if aN is left (right, resp. two-sided) invertible in AN for all N ∈ M (B). (b) The mapping M (B) → R+ ,

N → aN 

is upper semi-continuous. If a ∈ A and aN0 ∈ GAN0 , then aN ∈ GAN for all N in some open neighborhood of N0 . (c) If A is semisimple, then $

JN = {0}.

N ∈M (B)

(d) If A is a C ∗ -algebra, then, for a ∈ A, a =

max aN .

N ∈M (B)

26

1 Auxiliary Material

Proof. (a) We prove the assertion for left invertibility. The proof for the right invertibility is analogous. It is clear that aN is left invertible if so is a. To prove the reverse implication assume the contrary, i.e., assume aN is left invertible in AN for all N ∈ M (B) but let a have no left inverse in A. Denote by L the maximal left ideal of A containing I := {xa : x ∈ A} (note that e ∈ / I and recall 1.17(a)). Put N := L ∩ B. By Lemma 1.34, N ∈ M (B). We claim that JN ⊂ L. Indeed, if ak xk (because x= xk ak , where xk ∈ N = L ∩ B and ak ∈ A, then x = B ⊂ Cen A), and hence x ∈ L (because L is a left ideal). Thus, JN ⊂ L. By our assumption, aN is left invertible in AN , that is, there exists a b ∈ A with ba − e ∈ JN , and since JN ⊂ L, we have ba − e ∈ L. On the other hand, ba ∈ I ⊂ L. This implies that e ∈ L, which contradicts the maximality of L. (b) Let N0 ∈ M (B) and let ε > 0. Choose a1 , . . . , an ∈ A and z1 , . . . , zn ∈ N0 such that " " n  " " ε " "a + (1.10) a z j j " < aN0  + . " 2 j=1

Define the open neighborhood Uε ⊂ M (B) of N0 as ! # n %   zj (N )| < ε 2 aj  + 1 for j = 1, . . . , n Uε = N ∈ M (B) : | j=1

and put yj = zj − zj (N )e. Then yj ∈ N and so " " n  " " " a + aN  ≤ " a y j j ". "

(1.11)

j=1

Now if N ∈ Uε , then (1.10) and (1.11) give " " " " n n   " " " ε " " " " a + − a + a y a y aN  − aN0  ≤ " j j" j j" + " " 2 j=1 j=1 " " " " n " ε " n " ε " " " " ≤" aj (yj − zj )" + = " zj (N )aj " " + 2 < ε. 2 j=1 j=1 This proves the upper semi-continuity of aN  at N0 . The second part of the assertion can be proved as the corresponding assertion of Theorem 1.32(c). (c) Let L be any maximal left ideal of A. From

Lemma 1.34 we know that N := L ∩ B is a maximal ideal of B. If x = zk ak (zk ∈ N, ak ∈ A), then x= ak zk ∈ L (because L is a left ideal), and it follows that JN ⊂ L. Let πN denote the canonical homomorphism of A onto AN = A/JN . Clearly, πN (L) is a left ideal of AN . If πN (L) would not be proper, then there would exist a y ∈ L such that πN (y) = eN , whence y − e ∈ JN ⊂ L, and thus e ∈ L, which contradicts the maximality of L. Consequently, πN (L) is a proper left

1.8 Lp and H p

27

ideal of AN . By 1.17(a), πN (L) is contained in some maximal left ideal LN −1 −1 of AN . We claim that πN (LN ) = L. It is clear that L ⊂ πN (LN ) and that −1 −1 πN (LN ) is a left ideal of A. If πN (LN ) would contain the identity e, then eN = πN (e) would belong to LN , which is impossible, since LN is maximal. −1 (LN ) is a proper left ideal of A containing the maximal left ideal Thus, πN −1 (LN ). L, and this implies that L = πN JN and let L be any maximal left ideal of A. By what Now let x ∈ N ∈M (B)

has just been proved, there is an N ∈ M (B) and a maximal left ideal LN of −1 (LN ). Because πN (x) = 0, we have πN (x) ∈ LN and AN such that L = πN hence, x ∈ L. Thus, x belongs to each maximal left ideal of A and since A was supposed to be semisimple, it follows that x = 0. & (d) If A is a C ∗ -algebra, then so is the direct sum AN . The canonN ∈M (B)

ical homomorphisms πN : A → AN produce a star-homomorphism ' AN , a → (aN )N ∈M (B) . π:A→ N ∈M (B)

Since a C ∗ -algebra is always semisimple, the above proved part (c) implies that π is one-to-one, and so 1.26(e) shows that π is an isomorphism, i.e., a = sup aN  for all a ∈ A. Finally, in view of the upper semi-continuity N ∈M (B)

of aN  (part (b)), the sup may be replaced by max.   Remark. If A and B are C ∗ -algebras, then JN = A for every N ∈ M (B). Indeed, if N0 ∈ M (B), then there is a b ∈ B such that b(N0 ) = 1 and 0 < b(N ) < 1 for N = N0 , and part (d) of the above theorem gives 1 = b =

max bN  =

N ∈M (B)

max b(N )eN  = eN0 ,

N ∈M (B)

hence eN0 = 0, and thus JN0 = A. If A = C (the continuous functions on the unit circle T) and B = CA (the disk algebra), then M (B) = clos D (the closed unit disk) and it is easily seen that, for z ∈ clos D, Jz = C if and only if z ∈ T. This is a special case of a more general result: if A is a commutative Banach algebra with identity element, then JN = A for all N belonging to the Shilov boundary of M (A).

1.8 Lp and H p 1.36. The spaces Lp . Let T be the complex unit circle. We denote by Lp (1 ≤ p < ∞) the Banach space of (classes of) complex-valued measurable functions on T summable (with respect to Lebesgue measure) in the p-th power. We let C (resp. L∞ ) denote the C ∗ -algebra of all continuous (resp.

28

1 Auxiliary Material

[classes of] measurable and essentially bounded) functions on T. Clearly, C ⊂ L∞ ⊂ Lr ⊂ Ls ⊂ L1 for 1 ≤ s ≤ r ≤ ∞. The norm in Lp (1 ≤ p ≤ ∞) will be denoted by  · p . Given f ∈ L1 we define its Fourier coefficients fn (n ∈ Z) by ( 2π 1 f (eiθ )e−inθ dθ. fn = 2π 0



Notice that f ∈ L2 if and only if n∈Z |fn |2 < ∞ and that f 22 = n∈Z |fn |2 . Moreover, L2 is a Hilbert space and if we define the functions χn (n ∈ Z) by χn (t) = tn (t ∈ T), then {χn }n∈Z is an orthogonal basis in L2 . 1.37. The harmonic extension. Let D be the open unit disk in the complex plane. The harmonic extension of a function f ∈ L1 is the function f defined in D by  f(reiθ ) = fn r|n| einθ (0 ≤ r < 1, 0 ≤ θ < 2π). n∈Z

Note that f can also be defined via the Poisson integral, ( 2π 1 iθ  f (re ) = kr (θ − t)f (eit ) dt, 2π 0 where

1 − r2 1 − 2r cos θ + r2 is the Poisson kernel. Sometimes it will be convenient to denote the harmonic extension f by hf . For fixed r ∈ [0, 1), the function f(reiθ ) may be viewed as a function given on the unit circle T = {eiθ : 0 ≤ θ < 2π}; we denote this function by hr f of fr , that is, kr (θ) =

fr (eiθ ) = (hr f )(eiθ ) = f(reiθ ) = (hf )(reiθ ). We shall occasionally think of functions in Lp as extended harmonically into D, that is, as given on the closed unit disk clos D. 1.38. Basic properties of the harmonic extension. (a) The harmonic extension f of a function f ∈ L1 is harmonic in D, i.e., f ∈ C ∞ (D) and ∆f = 0, where ∆ is the Laplace operator. (b) Let f ∈ Lp . If 1 ≤ p ≤ ∞, then sup hr f p ≤ f p r∈[0,1)

and if 1 ≤ p < ∞, then hr f − f p → 0 as

r → 1 − 0.

1.8 Lp and H p

If f ∈ C, then

hr f − f ∞ → 0 as

29

r → 1 − 0,

and if f ∈ L∞ , then hr f converges to f in the weak-star topology of L∞ = (L1 )∗ , that is, ( 2π ( 2π iθ iθ (hr f )(e )g(e ) dθ → f (eiθ )g(eiθ ) dθ (r → 1 − 0) 0

0

for every g ∈ L1 . (c) (Fatou’s theorem). Let F be a harmonic function in D, put Fr (eiθ ) = F (reiθ ), and suppose sup Fr p < ∞, where 1 ≤ p ≤ ∞. Then the non-tangential r∈[0,1)

limit f (eiθ ) := lim F (z) z→eiθ

exists and is finite almost everywhere on T, and the “boundary function” f belongs to Lp . If 1 < p ≤ ∞, then the harmonic extension f of f coincides with F , but if p = 1, then all one can say is that there is a complex measure dσ on T singular with respect to Lebesgue measure such that ( 2π 1 kr (θ − t) dσ(t), F (reiθ ) = f(reiθ ) + 2π 0 where kr is the Poisson kernel. 1.39. The spaces H p . For 1 ≤ p ≤ ∞ put H p = {f ∈ Lp : fn = 0 for all n < 0}, CA = {f ∈ C : fn = 0 for all n < 0}. Thus, H p (the Hardy spaces) and CA (the disk algebra) are closed subspaces of Lp and C, respectively. Clearly, CA ⊂ H ∞ ⊂ H r ⊂ H s ⊂ H 1

for

1 ≤ s ≤ r ≤ ∞.

Moreover, H ∞ and CA are Banach algebras. H 2 is a Hilbert space and we

2 have f ∈ H if and only if n∈Z+ |fn |2 < ∞; the set {χn }n∈Z+ forms an orthogonal basis in H 2 . 1.40. The analytic extension. If f ∈ H 1 , then the harmonic extension f is an analytic function in D and it is therefore also referred to as the analytic extension. Thus, for f ∈ H 1 one has  fn z n (z ∈ D). f(z) = n≥0

30

1 Auxiliary Material

Some properties of the analytic extension of H p functions immediately result from 1.38(b). Fatou’s theorem can be made more precise as follows. (a) Let F be an analytic function in D, put Fr (eiθ ) = F (reiθ ), and suppose that sup Fr p < ∞, where 1 ≤ p ≤ ∞. Then the non-tangential limit r∈[0,1)

f (eiθ ) := lim F (z) z→eiθ

exists and is finite almost everywhere on T, and the “boundary function” f is in H p . If 1 ≤ p ≤ ∞, then the analytic extension f of f coincides with F . That, unlike the L1 situation, the singular measure dσ does not appear in the H 1 case is, at long last, due to the following fact, which is of great importance in connection with many other problems, too. (b) (The F. and M. Riesz theorem). A function in H 1 vanishes either almost everywhere or almost nowhere on T. In what follows we shall frequently identify functions in H p with their analytic extension in D. 1.41. Inner-outer factorization. A function ϕ ∈ H ∞ is called an inner function if |ϕ(t)| = 1 for almost all t ∈ T. A function g ∈ H 1 is said to be an outer function if its analytic extension can be represented in the form   ( 2π iθ 1 e +z iθ log ψ(e g(z) = c exp ) dθ , (1.12) 2π 0 eiθ − z where c ∈ T, ψ ∈ L1 , ψ ≥ 0 a.e. on T, log ψ ∈ L1 . The inner-outer factorization theorem says the following. (a) Every function f ∈ H p (1 ≤ p ≤ ∞) which is not identically zero has a factorization of the form f = ϕg where ϕ ∈ H ∞ is an inner function and g ∈ H p is an outer function. This factorization is unique up to a multiplicative constant. A remarkable property of H 1 functions is that log |f | ∈ L1 whenever f does not vanish identically. The outer function g occurring in the inner-outer factorization theorem can be obtained through (1.12) with ψ = |f |. Examples of inner functions are χn (t) = tn (n ≥ 0),   t+1 α−t (α ∈ D). Sa (t) = exp a (a > 0), bα (t) = t−1 1 − αt Functions of the form bα1 . . . bαn (αj ∈ D) are also inner and are called finite be a sequence of complex numbers such that Blaschke products. Let {αn }∞ n=1

∞ 0 < |α1 | ≤ |α2 | ≤ . . . < 1 and n=1 (1 − |αn |) < ∞. Then the infinite product b(z) =

∞  |αn | αn − z αn 1 − αn z n=1

(1.13)

1.8 Lp and H p

31

converges uniformly in each disk |z| ≤ R < 1, one has |b(z)| < 1 for |z| < 1, and the non-tangential limits of b have modulus 1 a.e. on T. Thus b is an inner function. Each αn is a zero of b, with multiplicity equal to the number of times it occurs in the sequence, and b(z) has no other zeros in D. A function of the form  |αn | αn − z b(z) = z m αn 1 − αn z n

is called a Blaschke product. Here m ∈ Z+ , αn ∈ D, and (1 − |αn |) < ∞. The set {αn } may be finite or even empty. In the latter case b(z) := z m . In this connection notice the following: if f ∈ H 1 , f ≡ 0, and if α1 , α2 , . . . are the zeros of f in D, repeated according to multiplicity, then  (1 − |αn |) < ∞. n

A function of the form  ( S(z) = exp −

π −π

 eiθ + z dµ(θ) eiθ − z

(z ∈ D),

where µ is a positive measure on [−π, π] singular with respect to Lebesgue measure, can also be shown to be inner. Such functions are called singular inner functions. If µ is the unit mass at 0, we obtain   z+1 S1 (z) = exp . z−1 A singular inner function has no zeros in D, but the radial limits lim S(z) are z→t

zero for each t belonging to the (closed) support of the measure µ. The canonical factorization theorem states the following. (b) Every inner function ϕ ≡ 0 has a unique factorization ϕ = cBS where c ∈ T, B is a Blaschke product, and S is a singular inner function. Here are some important properties of outer functions. (c) If g ∈ H 1 and h ∈ H 1 are outer and if |g| = |h| a.e. on T, then g = λh a.e. on T with some constant |λ| = 1. (d) If f ∈ Lp (1 ≤ p ≤ ∞), f ≥ 0 a.e. on T, and if log f ∈ L1 , then there exists an outer function g ∈ H p such that f = |g| a.e. on T. In particular, if f is a real-valued function in L∞ and ess inf f (t) > 0, then there is an outer t∈T

function g ∈ H ∞ such that f = |g| a.e. on T. (e) If f ∈ H 1 is outer then f(z) = 0 for z ∈ D. If S is a set, f S denotes the set {f s : s ∈ S}. Functions of the form

32

1 Auxiliary Material N 

f n tn

(t ∈ T)

n=0

are referred to as analytic polynomials and the (linear) set of all analytic polynomials will be denoted by PA . (f) Let f ∈ H ∞ . Then the following are equivalent: (i) f is outer; (ii) f PA is dense in H 2 ; (iii) f H 2 is dense in H 2 . We finally mention some characterizations of the outer functions in H ∞ that are invertible in L∞ . (g) For f ∈ H ∞ the following are equivalent: (i) f ∈ GL∞ and f is outer; (ii) inf |f(z)| > 0; z∈D

(iii) f ∈ GH ∞ ; (iv) f H 2 = H 2 . 1.42. The Riesz projection. A function of the form N 

f n tn

(t ∈ T)

n=−N

is called a Laurent polynomial and the (linear) set of all Laurent polynomials will be denoted by P. The Riesz projection is the operator P which is defined on P by N N   P : fn tn → f n tn . n=0

n=−N

A famous theorem of M. Riesz states that cp :=

P f p f ∈P,f =0 f p sup

is finite if 1 < p < ∞ and infinite if p = 1 or p = ∞. Thus, if 1 < p < ∞ then P extends from the dense subset P of Lp to a bounded projection of the whole Lp onto H p . In 2000, Hollenbeck and Verbitsky [286] proved that cp = 1/ sin(π/ max{p, q}) where 1/p + 1/q = 1. ◦

p In the case p = 2, P is the orthogonal projection of L2 onto H 2 . Let H− p denote the kernel of P and put Q = I − P . Then L decomposes into the · ◦

p and direct sum H p +H−

H p = Im P = Ker Q,

◦

p H− = Im Q = Ker P.

1.8 Lp and H p

33

◦

◦

p p Furthermore, let H p := {f ∈ H p : f0 = 0} and H− :=H− +C. Clearly, ◦

·

p p Lp =H p + H− . It is obvious that H− = {h : h ∈ H p } (1 ≤ p ≤ ∞), where p h(t) := h(t) (t ∈ T). Therefore H− will occasionally also be denoted by H p . ◦

q (1/p + 1/q = 1) in the The dual space of H p (1 < p < ∞) is Lq / H− following sense: the general form of a functional G ∈ (H p )∗ is given by

G(f ) =

1 2π

(

2π

f (eiθ )g(eiθ ) dθ,

(1.14)

0

where g ∈ H q ; g is uniquely determined by G, and ◦ 
q . G(H p )∗ = inf g + hLp : h ∈H−

It can be easily checked that, for 1 < p < ∞, 1 gH q ≤ G(H p )∗ ≤ gH q , cq

(1.15)

where cq = P L(Lq ) . Note that c2 = 1, and consequently, for p = q = 2 we have gH 2 = G(H 2 )∗ . We also remark that every G ∈ (H 1 )∗ can be represented in the form (1.14) with some g ∈ L∞ . Finally, note that P is given in terms of Fourier coefficients by (P f )n = fn for n ≥ 0,

(P f )n = 0 for n < 0.

Also notice that the operator P − Q = 2P − I is nothing else than the Cauchy singular integral operator S over T, ( ϕ(τ ) 1 dτ (t ∈ T), (1.16) (Sf )(t) = πi T τ − t the integral understood in the Cauchy principal-value sense. 1.43. The conjugate function. If F is a real-valued harmonic function in D, then there is a real-valued harmonic function G in D such that F + iG is analytic in D. Any two such functions G differ only by a real constant and the function G for which G(0) = 0 is called the conjugate function of F and denoted by F) . If F is a complex-valued harmonic function in D, then F can be written as F = U + iV , where U and V are real-valued, and F) is defined ) + iV) . as U Now let f ∈ L1 . One can show that the conjugate function of the harmonic extension hf possesses non-tangential limits almost everywhere on T. This “boundary function” is then referred to as the conjugate function of f and denoted by f). Thus, f)(eiθ ) = lim (hf ))(z). z→eiθ

34

1 Auxiliary Material

If f ∈ L1 and (hf )(reiθ ) =



fn r|n| einθ

(0 ≤ r < 1,

0 ≤ θ < 2π),

n∈Z

then (hf ))(reiθ ) = −i



(sign n)fn r|n| einθ ,

n∈Z

where sign 0 := 0. Also notice the formula ( 2π 1 qr (θ − ϕ)f (eiϕ ) dϕ, (hf ))(reiθ ) = 2π 0 where qr (θ) =

2r sin θ . 1 − 2r cos θ + r2

If f ∈ P, then f) is also in P. Using the Riesz projection one can write f) = −i(P f − Qf − f0 ) = −i(2P f − f − f0 ).

(1.17)

We therefore deduce from the M. Riesz theorem in 1.42 that the supremum of f)p /f p over f ∈ P \ {0} is finite for 1 < p < ∞ and infinite for p = 1 and p = ∞. Thus, the operator of conjugation, f → f), is bounded on Lp for 1 < p < ∞. In particular, (1.17) makes a correct sense for arbitrary f ∈ Lp (1 < p < ∞). The conjugation operator f → f) is frequently also referred to as the Hilbert transform. It can be given by the formula   ( 2π θ−t 1 cot (1.18) f)(eiθ ) = f (eit ) dt, 2π 0 2 which is correct if the integral is interpreted in the Cauchy principal-value sense. Finally, we use (1.17) to define the Riesz projection on L1 , that is, for f ∈ L1 we define the measurable function P f as 12 (f + if)) + 12 f0 . 1.44. Lp and H p with weight. It will be convenient to denote the Lebesgue measure on T by dm. By a weight we understand a Lebesgue measurable function w : T → [0, ∞] such that the pre-image w−1 ({0, ∞}) has Lebesgue measure zero. Let w be a weight belonging to Lp (1 < p < ∞). The weighted Lp space Lp (w) is the Banach space of all functions f on T for which f w ∈ Lp , i.e., ( 1/p p p f p,w := |f | w dm < ∞. T

p

p

The weighted H space H (w) is defined as the closure in Lp (w) of the linear hull of the set {χ0 , χ1 , χ2 , . . .}. Sometimes H p (w) will also be denoted by

1.8 Lp and H p

35

◦

p Lp+ (w). H− (w) will refer to the Lp (w)-closure of the linear hull of the set ◦

◦

p (w) will occasionally be written as Lp− (w). {χ−1 , χ−2 , . . .} and H− It can be shown that P is dense in Lp (w) (1 < p < ∞). The following statements are equivalent:

(i) There is a constant cp,w depending only on p and w such that P f p,w ≤ cp,w f p,w

∀ f ∈ P.

(ii) The Riesz projection P generates a bounded operator on Lp (w). ◦

p (w) and H p (w). (iii) Lp (w) decomposes into the direct sum of H−

Of course, here the Riesz projection can be replaced by the singular integral operator (Hilbert transform). 1.45. The Helson-Szeg˝ o theorem. This theorem establishes an interesting necessary and sufficient conditions for the boundedness of the Riesz projection P on L2 (w). Suppose w is a weight in L2 . Then P is bounded on L2 (w) if and only if w can be represented in the form w = eu+)v , where u and v are real-valued functions in L∞ and v∞ < π/4. 1.46. The Hunt-Muckenhoupt-Wheeden condition. This is a necessary and sufficient condition for the boundedness of the Riesz projection P on Lp (w) whose nature is completely different from the condition occurring in the Helson-Szeg˝o theorem. Let 1 < p < ∞ and let w be a weight in Lp . Then for P to be bounded on L (w) it is necessary and sufficient that w−1 ∈ Lq (1/p + 1/q = 1) and that 1/p  ( 1/q  ( 1 1 p −q sup w dm w dm < ∞, (1.19) |I| I |I| I I p

where the supremum is over all subarcs of the circle T and |I| is the arc length of I. For concrete applications the following simple observation is useful: to show that the supremum in (1.19) is finite it suffices to show that the supremum over all subarcs I satisfying |I| ≤ δ, where δ is any positive number, is finite. If one asks for a condition on the weight w that ensures that P be bounded o type on both Lp (w) and Lq (w) (1/p + 1/q = 1), then again a Helson-Szeg˝ criterion can be given. The sufficiency portion of the following result was established by Simonenko [492], its necessary part is due to Krupnik [328]. Its proof is given in Gohberg, Krupnik [232, Chap. 12, Theorem 5.2]. Let 2 ≤ p < ∞ and 1/p + 1/q = 1. Suppose w is a weight in Lp . Then for P to be in L(Lp (w)) ∩ L(Lq (w)) it is necessary and sufficient that w be of the form w = eu+)v , where u and v are real-valued functions in L∞ and v∞ < π/(2p).

36

1 Auxiliary Material

1.9 BM O and V M O 1.47. Definitions. The Lebesgue measure of a measurable subset E of T will * be denoted by |E|, that is |E| = E dm.* For f ∈ L1 and I a subarc of T, define the mean value fI by fI := (1/|I|) I f dm. For 0 < δ ≤ 2π let ( 1 Mδ (f ) := sup |f (ζ) − fI | dm(ζ), |I|≤δ |I| I where I ranges over subarcs of T only, and put M0 (f ) := lim Mδ (f ), δ→0

f ∗ := M2π (f ).

A function f ∈ L1 is said to have bounded mean oscillation, or to be in BM O, if f ∗ < ∞, and is said to have vanishing mean oscillation, or to belong to V M O, if f ∗ < ∞ and M0 (f ) = 0. 1.48. Basic properties of BM O and V M O. (a) If f ∈ L1 and if for each subarc I of T with |I| ≤ δ there is a constant αI such that ( 1 |f − αI | dm ≤ M, |I| I then Mδ (f ) ≤ 2M . (b) If f ∈ L1 and Mδ (f ) < ∞, where 0 < δ < 2π, then f ∈ BM O and M2π (f ) ≤ (2π/δ)6 Mδ (f ) (the estimate is very generous). (c) L∞ ⊂ BM O and f ∗ ≤ f ∞ for f ∈ L∞ . (d) If f ∈ L1 and f ∗ = 0, then f =const. (e) L∞ ⊂ BM O ⊂

Lp ,

C ⊂ V M O ⊂ BM O.

1≤p<∞

(f) BM O is* a Banach space under the norm f BM O := f ∗ +|f0 |, where f0 = (1/(2π)) T f dm. V M O is a closed subspace of BM O and coincides with the closure of P in the BM O norm. There are absolute constants A1 , A2 > 0 such that for all a ∈ BM O, A1 M0 (a) ≤ distBM O (a, V M O) ≤ A2 M0 (a). Some much deeper results on BM O and V M O can be stated as follows. (g) (Charles Fefferman). BM O = {u + v) : u, v ∈ L∞ } and there is an absolute constant B such that every f ∈ BM O can be written as f = u + v) with u∞ ≤ Bf ∗ , v∞ ≤ Bf ∗ . (h) (Sarason). V M O = {u + v) : u, v ∈ C} and there is an absolute constant B such that every f ∈ V M O can be written as f = u + v) with u, v ∈ C and u∞ ≤ Bf ∗ , v∞ ≤ Bf ∗ .

1.9 BM O and V M O

37

(k) The conjugation operator (and thus also the Cauchy singular integral operator and the Riesz projection) is bounded on the following pairs of spaces: L∞ → BM O,

BM O → BM O,

C → V M O,

V M O → V M O.

(l) We have BM O = {u + P v : u, v ∈ L∞ }, and

V M O = {u + P v : u, v ∈ C}

f BM O = inf{u∞ + v∞ : u, v ∈ L∞ , u + P v = f }

is an equivalent norm in BM O. BM O(R) is defined as the set of all functions F ∈ L1loc (R) for which ( 1 F ∗ := sup |F (x) − FI | dx < ∞, I |I| I * where the sup is over all bounded intervals I ⊂ R, FI := (1/|I|) I F (x) dx, and |I| is the Lebesgue measure of I. For f ∈ L1 , define F ∈ L1loc (R) by   i−x F (x) = f (x ∈ R). i+x We are now in a position to state two further important properties of BM O. (m) f ∈ BM O ⇐⇒ F ∈ BM O(R). (n) (John/Nirenberg). If F ∈ BM O(R), then, for every interval I ⊂ R and every λ > 0,   + −cλ 1 ++ {x ∈ I : |F (x) − FI | > λ}+ ≤ C exp |I| F ∗ with absolute constants C and c. Finally, V M O(R) is the collection of all functions F ∈ BM O(R) for which ( 1 M0 (F ) := lim sup |F (x) − FI | dx = 0. δ→0 |I|≤δ |I| I If we let U C(R) denote the uniformly continuous functions on R, then (p) V M O(R) coincides with the set of all functions F ∈ BM O(R) for which
 inf F − G∗ : G ∈ U C(R) ∩ BM O(R) = 0. For F ∈ L1loc (R) and x ∈ R define Fx ∈ L1loc (R) by Fx (t) = F (t − x). (q) If F ∈ BM O(R), then F ∈ V M O(R) ⇐⇒ F − Fx ∗ → 0 as x → 0.

38

1 Auxiliary Material

We conclude the present register with a technical result. (r) Let F ∈ BM O(R) and let I and J be finite intervals of R. (i) If I ⊂ J and |J| ≤ 2|I|, then |FI − FJ | ≤ 2F ∗ . (ii) If I ⊂ J and |J| > 2I, then  |FI − FJ | ≤ c log

|J| |I|

 F ∗

with some absolute constant c. (iii) If |I| = |J|, then   dist(I, J) |FI − FJ | ≤ c log 2 + F ∗ |I| with some absolute constant c.

1.10 Smoothness Classes 1.49. Spaces of
p type. Let α, β be any real numbers and let p, r be real numbers satisfying 1 ≤ p, r < ∞. We denote by
r,p β,α the Banach space of all sequences ϕ = {ϕn }n∈Z of complex numbers for which ϕr,p;β,α :=

 ∞

1/r |ϕ−n | (n + 1) r

rβ

+

n=1

 ∞

1/p |ϕn | (n + 1) p

pα

< ∞.

n=0

p p
p,p α,α will be also written as
α (Z); an equivalent norm in
α (Z) is

ϕp,α :=



1/p |ϕn |p (|n| + 1)pα

.

n∈Z p p
p,p 0,0 =
0 (Z) will be abbreviated to
(Z) and  · p will always refer to the norm  1/p p ϕp := |ϕn | . n∈Z

The collection of all functions in L1 whose sequence of Fourier coefficients r,p belongs to
r,p α,β (α ≥ 0, β ≥ 0) will be denoted by F
α,β . The norm of a funcr,p r,p tion in F
α,β is defined as the
α,β -norm of its sequence of Fourier coefficients. p,p p p 1 F
p,p α,α and F
0,0 will be abbreviated to F
α and F
, respectively. F
and 1,1 α,β F
α,β will be denoted by W and W , respectively (the W is for Wiener). Notice that F
2 is L2 .

1.10 Smoothness Classes

39

Let
pα denote the closed subspace of
pα (Z) defined by

pα = ϕ ∈
pα (Z) : ϕn = 0 for n < 0 .
p0 will be abbreviated to
p . We shall frequently think of
pα as a space of one-sided sequences, that is, we let
pα

!   # 1/p p pα = ϕ = {ϕn }n∈Z+ : ϕp,α := |ϕn | (n + 1) <∞ . n∈Z+

Sometimes, in order to distinguish
pα from
pα (Z) more explicitly, we shall write
pα (Z+ ) and
p (Z+ ) in place of
pα and
p , respectively. We remark that a function in L2 belongs to H 2 if and only if its sequence of Fourier coefficients is in
2 =
2 (Z+ ). Finally, we let
0 (Z) (resp.
0 ) denote the sequence in
p (Z) (resp.
p ) that have a finite support. Then F
0 (Z) = P and F
0 = PA . For n ∈ Z, let en denote the element of
p (Z) which is given by {δjn }j∈Z , where δ is the Kronecker delta. If n ≥ 0, en can also be interpreted as an element of
p (Z+ ). Obviously, {en }n∈Z and {en }n∈Z+ are bases in
r,p β,α and
pα , respectively. The discrete Riesz projection is given by P : {ϕn }n∈Z → {. . . , 0, 0, ϕ0 , ϕ1 , ϕ2 , . . .}

(1.20)

where ϕ0 occupies the 0-th place. It is trivial that P is always bounded on
r,p β,α . A generalization of
pα (Z) is the Banach space
pw (Z)

!  # 1/p p p = ϕ = {ϕn }n∈Z : ϕp,w := |ϕn | wn <∞ , n∈Z

where {wn }n∈Z is any sequence of positive numbers. Given ϕ ∈
0 (Z) define ϕ ) ∈
0 (Z) by  ϕj ϕ )n = . n−j j =n

The discrete Hunt-Muckenhoupt-Wheeden theorem says the following. If 1 < p < ∞ and wn > 0, then there is a constant cp,w depending only on p and ) p,w ≤ cp,w ϕp,w for all ϕ ∈
0 (Z) if and only if w = {wn }n∈Z such that ϕ 1 sup m,n∈Z,m≤n n − m + 1 where 1/p + 1/q = 1.



n 

k=m

1/p  wkp

n 

k=m

1/q wk−q

< ∞,

40

1 Auxiliary Material

1.50. H¨ older and Besov classes. For α > 0 and 1 ≤ p ≤ ∞ we define the Besov class Bpα as ! # ( π |t|−1−αp ∆nt f pLp dt < ∞ (p < ∞), Bpα = f ∈ Lp : −π ,   α ∞ −α n B∞ = f ∈ L : sup |t| ∆t f L∞ < ∞ (p = ∞), t =0

where n is any integer such that n > α and where ∆nt := ∆t ∆n−1 , t (∆t f )(eiθ ) := f (ei(θ+t) ) − f (eiθ ), for θ, t ∈ R. Note that this definition does not depend on the choice of n, n > α. α are nothing else than the H¨ older-Zygmund classes, that The classes B∞ is, for 0 < α < 1 we have
 α B∞ = f ∈ C : |f (t1 ) − f (t2 )| ≤ Mf |t1 − t2 |α ∀ t1 , t2 ∈ T , for α = 1 we have
1 B∞ = f ∈ C : |f (ei(θ+t) ) + f (ei(θ−t) ) − 2f (eiθ )| ≤ Mf |t|

 ∀ θ, t ∈ R ,

and on denoting by C n (n ∈ Z+ ) the class of n times continuously differentiable functions on T, we can finally write 
α α−n (n < α ≤ n + 1). B∞ = f ∈ C n : f (n) ∈ B∞ α For this reason, if α is not an integer, we shell henceforth denote B∞ by C α .

Later we shall need the following two facts. (a) If 1 < p < ∞ and α > 1/p, then Bpα ⊂ C. (b) If 1 < p < ∞ and α > 0, then P is dense in Bpα . α will be denoted by cα . The closure of P in C α = B∞ α If α ≤ 0, we think of Bp as a space of sequences of complex numbers. Namely, we define Bpα as the linear space of all sequences f = {fn }n∈Z such that {(|n| + 1)−s fn }n∈Z is the sequence of the Fourier coefficients of some s−|α| function belonging to Bp , s − |α| > 0. This definition does not depend on the choice of the number s > |α|. If we identify the functions in Bpα (α > 0) with their Fourier coefficients sequence, we have the following: the mapping Is which sends a sequence {ϕn }n∈Z of complex numbers into the sequence {(|n| + 1)−s ϕn }n∈Z maps Bpα one-to-one onto Bpα+s for every 1 ≤ p ≤ ∞, α ∈ R, s ∈ R. Thus, on defining a norm in Bpα for α ≤ 0 by f Bpα = Is f Bpα+s we make Bpα become a Banach space and this definition is correct, since it is independent of the choice s > |α|. If 1 ≤ p < ∞ and s ∈ R, then the dual space (Bpα )∗ of Bpα coincides with Bq−α (1/p + 1/q = 1) in the following sense: the general form of a functional f ∈ (Bpα )∗ is given by

1.11 Notes and Comments

f (ϕ) =



ϕn fn

41

(ϕ = {ϕn }n∈Z ∈ Bpα ),

n∈Z

where {fn }n∈Z ∈ Bq−α . The space B2α admits a very simple description:  (|n| + 1)2α |fn |2 < ∞. f ∈ B2α ⇐⇒ n∈Z

Also notice that f ∈ Bpα ⇐⇒

∞ 

nαp−1 distLp (f, Pn ) < ∞,

n=1

where Pn denotes the set of all Laurent polynomials of degree at most n. Finally, let
 (Bpα )A := f ∈ Bpα : fn = 0 for n < 0 . Thus (Bpα )A is a closed subspace of Bpα . For α > 0 this space can be characterized as the class of all functions f ∈ Lp such that ( 1 (1 − r)(n−α)p−1 (hr f )(n) pLp dr < ∞ (p < ∞), 0

sup (1 − r)n−α (hr f )(n) L∞ < ∞ (p = ∞),

0
where n is any integer satisfying n > α and (hr f )(n) (eiθ ) := (d/dθ)n (hr f )(eiθ ).

1.11 Notes and Comments 1.1–1.10. Proofs of the facts presented here can be found in or can be without undue effort derived from Gohberg, Krein [228], Reed, Simon [416, Vol. IV], Dunford, Schwartz [177], Gohberg, Goldberg, Krupnik [222]. The reader is also recommended to consult the excellent article Simon [488] and the book Pietsch [392]. For 1.3(d) see Widom [569]. The formula of 1.10 was independently discovered by Pincus [395] and Helton, Howe [274] in the early 1970s, who had a quite complicated proof based on the Baker-Campbell-Hausdorff formula. The stronger formula of 1.9 and the remarkably simple proof cited in the text were found by Torsten Ehrhardt [190] in 2000. This proof makes use of a differentiation trick due to Basor [25]. Ehrhardt [190] even established the following more general result. If A1 , . . . , AR (R ≥ 3) are operators in L(H) such that A1 + . . . + AR = 0 and Aj Ak − Ak Aj ∈ C1 (H) for all j, k, then eA1 . . . eAR − I is in C1 (H) and

42

1 Auxiliary Material

 det e

A1

...e

AR

= exp

1 2



 tr (Aj Ak − Ak Aj ) .

1≤j
It is easy to check that for R = 3 this is just another formulation of formula (1.3). 1.11–1.12. These things as well as the monographs and textbooks which pay due attention to the theory of Fredholm operators are well known. In German and Russian Fredholm operators are (more correctly) called Noether operators, since Friedrich Noether [370] was the first to discover that a singular integral operator with nonvanishing continuous symbol is normally solvable and has finite kernel and cokernel dimensions; he also computed the index of such operators. The term Fredholm operator is in German and Russian used for operators with index zero. 1.13–1.15. Results of the type of Theorem 1.14 were first obtained by N. Ya. Krupnik (see Kesler and Krupnik [311]). Theorem 1.14 itself was first established by K¨ ohler and Silbermann [314], [315] and independently also by Markus and Feldman [347]. Theorem 1.15 is due to Markus and Feldman [346]. The proofs of these two theorems are in the original papers and also in Chapter 1 of Krupnik’s book [329]. (We also recommend to have a glance at Math. Reviews 87a:15006). 1.16–1.29. This material is well known and a major part of it is in almost each ˙ [587], [588]. book on Banach or C ∗ -algebras. The result 1.20(c) is Zelazko’s Theorem 1.22 was established by Glicksberg [216] and is a quantitative version of Corollary 1.23, which had already been known from earlier work of Shilov and Bishop. The proof given in the text is Ransford’s [415]. Later we shall also need vector-valued analogs of Glicksberg’s theorem (Theorem 3.6). For this topic see Burckel [127], Machado [342], Szyma´ nski [512], Ransford [415]. For 1.28 we refer to Glicksberg [216] and Gamelin [209]. All other results are explicitly proved, e.g., in Gelfand, Raikov, Shilov [212], Naimark [362], ˙ Douglas [162], Zelazko [588], Dixmier [156]. 1.30–1.35. The simplest local principle is the Gelfand theory of commutative Banach algebras with identity: if A is such an algebra and a ∈ A, then a ∈ GA ⇐⇒ a ∈ /N

∀ N ∈ M (A) ⇐⇒  a(N ) = 0 ∀ N ∈ M (A).

Thus, in a sense, invertibility in commutative Banach algebras is local in nature. Theorems 1.32 and 1.35 are what we need from the arsenal of “local techniques” for non-commutative algebras. We want not to go into the history of this topic, but only mention that Simonenko [493], [494], [495] was indisputably the first who both realized the local nature of Fredholmness of convolution and related operators and at the same time created a powerful machinery (his local principle) for tackling successfully a whole series of problems. This local principle is presented in

1.11 Notes and Comments

43

Simonenko, Chin Ngok Min [497]. Simonenko’s local principle was generalized by Kozak [318], [319], [320] to arbitrary Banach algebras. A reader who wishes to become acquainted with Simonenko’s principle is therefore recommended to consult also Kozak’s works [318], [319], [320]. Also see Chapter XV of Mikhlin, Pr¨ ossdorf [361] and B¨ ottcher, Krupnik, Silbermann [98]. Lemma 1.31 and Theorem 1.32 are due to Gohberg and Krupnik [232] (except for 1.32(c), which is due to the authors). This local principle is a modification of Simonenko’s principle, and it is distinguished for its simplicity on the one hand and for its wide area of applications on the other hand. An application of 1.32(c) is in the proof of Theorem 5.57. It is a delicate problem to say by whom and where Lemma 1.34 and Theorem 1.35 were established for the first time. As far as we know, the earliest references where these two results appeared explicitly are Allan [5], [6]. Independently, Douglas [162] stated Theorem 1.35 for the case of C ∗ -algebras and he was the first to realize its importance for the investigation of Toeplitz operators. The example in the remark in 1.35 is taken from Krupnik [329]. The comparison of several known local principles (theories) should be the subject of further investigations. Steps in this direction have been made by Clancey, Gosselin [139], Clancey, Gohberg [137], and B¨ ottcher, Krupnik, Silbermann [98]; also see B¨ottcher, Roch, Silbermann [100]. An extension of the Allan-Douglas local principle which can be applied to the study of C ∗ -algebras of operators of “non-local type” was developed by Karlovich [304] (full proofs of the results of that note are contained in Karlovich [305]). Concrete applications of this so-called local-trajectory method to C ∗ -algebras of convolution type operators with shifts and/or oscillating symbols are in B¨ ottcher, Karlovich, Silbermann [95], B¨ ottcher, Karlovich, Spitkovsky [97], Bastos, Fernandes, Karlovich [46]. Another extension of the local principle by Allan and Douglas, again for the investigation of operators of “non-local type,” but now for operators acting on Banach spaces, is in Karlovich, Silbermann [306]. The latter local method is illustrated by an application in Karlovich, Silbermann [307]. 1.36–1.46. See, e.g., Hoffman [284], Duren [178], Koosis [316], Garnett [211], Rudin [443], [444], Douglas [162], Sarason [457], Nikolski [368], [369], Rosenblum and Rovnyak [442]. 1.47–1.48. Excellent presentations of the BM O and V M O theory are in Sarason [457], Garnett [211], and Koosis [316]. See also the nicely written original works Sarason [454] and Stegenga [507]. 1.49. The discrete Hunt-Muckenhoupt-Wheeden condition is in these authors’ work [289, Theorem 10]. 1.50. These facts are taken from Peller and Khrushchev [390] and Peller [389, Appendix 2.3]. For periodic Besov spaces see Schmeisser and Triebel [462]. There one can also explicitly find 1.50(a) and (b) in Remark 3 and the Corollary of 3.5.5 and in Theorem 1 of 3.5.1, respectively.

2 Basic Theory

2.1 Multiplication Operators 2.1. Definition. If a ∈ L∞ and 1 < p < ∞, then the operator M (a) : Lp → Lp ,

f → af

(2.1)

is obviously bounded and M (a)L(Lp ) ≤ a∞ . It is called the multiplication p q operator on Lp generated by * the function a. For f ∈ L and g ∈ L (1/p + 1 f g dm. It is clear that (M (a)χ , χ ) is equal to 1/q = 1), write (f, g) := 2π j k T the (k − j)-th Fourier coefficient of a. The following proposition shows that every bounded operator with such a property is a multiplication operator. 2.2. Proposition. Let A ∈ L(Lp ) (1 < p < ∞) and suppose there is a sequence {an }n∈Z of complex numbers such that (Aχj , χk ) = ak−j . Then there is an a ∈ L∞ such that A = M (a) and {an } is the Fourier coefficient sequence of a. Moreover, M (a)L(Lp ) = a∞ . Proof. Put a = Aχ0 . Then a ∈ Lp and the n-th Fourier coefficient of a is (a, χn ) = (Aχ0 , χn ) = an . If f ∈ L∞ , then both Af and af are in Lp . We claim that (2.2) Af = af ∀ f ∈ L∞ . coefficient sequence of f . Then the j-th Let {fn }n∈Z denote the Fourier

a Fourier coefficient of af is k∈Z j−k fk . On the other hand, since the se

f χ converges to f in the Lp -norm, we deduce that the series ries k∈Z k k

j ) for each j ∈ Z. This shows that the k∈Z fk (Aχk , χj ) converges to (Af, χ

j-th Fourier coefficient of Af equals k∈Z aj−k fk , too. Thus, Af = af . We now prove that a ∈ L∞ . Let E be a measurable subset of T with positive measure on which |a| > A and let χE denote the characteristic function of E. Then, by (2.2), ( AχE p = aχE p = |a|p dm = Ap χE p . E

46

2 Basic Theory

But this is impossible and so |a| ≤ A a.e. on T. Hence a ∈ L∞ and a∞ ≤ A. Thus, since in view of (2.2) the operators A and M (a) coincide on a dense subset of Lp and both operators are bounded, it follows that A = M (a). The norm equalities are now obvious.   2.3. Definitions. Let a ∈ L1 have Fourier coefficients sequence {an }n∈Z . Given ϕ = {ϕj }j∈Z ∈
0 (Z) define the sequence a ∗ ϕ by  (a ∗ ϕ)j := aj−k ϕk (j ∈ Z). k∈Z

For 1 ≤ p < ∞, let M p denote the collection of all a ∈ L1 for which a ∗ ϕ belongs to
p (Z) whenever ϕ ∈
0 (Z) and sup

 a ∗ ϕ

p

ϕp

 : ϕ ∈
0 (Z), ϕ = 0 < ∞.

If a ∈ M p then the operator
0 (Z) →
p (Z), ϕ → a ∗ ϕ extends to a bounded operator  a ∗ ϕ, (2.3) M (a) :
p (Z) →
p (Z), ϕ → which is referred to as the multiplication operator on
p (Z) generated by the function a. The M (a) we have just defined on
p (Z) is sometimes also called the p q Laurent operator

generated by a. For ϕ ∈
(Z) and ψ ∈
(Z) (1/p+1/q = 1) put (ϕ, ψ) := n∈Z ϕn ψn . It is easy to verify that (M (a)ej , ek ) = ak−j for every a ∈ M p . Here is a converse of this. 2.4. Proposition. Let A ∈ L(
p (Z)) (1 ≤ p < ∞) and suppose (Aej , ek ) = ak−j for some sequence {an }n∈Z of complex numbers. Then there is an a ∈ M p such that A = M (a) and {an } is the Fourier coefficient sequence of a. Proof. First let 1 ≤ p ≤ 2. Put α := Ae0 ∈
p (Z). Then α = {an }n∈Z . Since
p (Z) ⊂
2 (Z), there is an a ∈ L2 whose Fourier coefficients sequence is {an }. Clearly, Aϕ = a ∗ ϕ for all ϕ ∈
0 (Z). Therefore, A = sup = sup

 Aϕ

p

ϕp  a ∗ ϕ

 : ϕ ∈
0 (Z), ϕ = 0

p

ϕp

 : ϕ ∈
0 (Z), ϕ = 0

and it follows that a ∈ M p and A = M (a). Now let p ≥ 2. The adjoint A∗ ∈ L(
q (Z)) (1/p + 1/q = 1) of A satisfies (A∗ ej , ek ) = (ej , Aek ) = aj−k . By what has been proved above, we have A∗ = M (b) for some b ∈ M q whose n-th Fourier coefficient bn equals a−n . Define a ∈ L1 by a(t) = b(t) (t ∈ T). Then the n-th Fourier coefficient of a is an and therefore (a ∗ ϕ, ψ) = (ϕ, b ∗ ψ) for all ϕ, ψ ∈
0 (Z). It follows that a ∗ ϕ ∈ (
q (Z))∗ =
p (Z) for all ϕ ∈
0 (Z) and that

2.1 Multiplication Operators

47




 sup a ∗ ϕp : ϕ ∈
0 (Z), ϕp ≤ 1 
= sup |(a ∗ ϕ, ψ)| : ϕ, ψ ∈
0 (Z), ϕp ≤ 1, ψq ≤ 1 = M (b)L(q (Z)) < ∞, whence a ∈ M p , which completes the proof.

 

2.5. Basic properties of M p . Incidentally, in the preceding proof we established that M p ⊂ L2 . Thus, if a and b are in M p , then ab ∈ L1 and now it is easily seen that actually ab ∈ M p . The conclusion is that M p is an algebra (under pointwise operations). For p = 2, the multiplication operators defined on L2 and
2 (Z) by (2.1) and (2.3), respectively, are unitarily equivalent through the isomorphism  L2 →
2 (Z), ϕn χn → {ϕn }n∈Z . n∈Z

This shows that M 2 = L∞ . It is also easy to see that    M 1 = W := a ∈ L1 : aW := |an | < ∞ . n∈Z

Indeed, if a ∈ M 1 , then {an } = M (a)e0 ∈
1 (Z), whence M 1 ⊂ W , and that 1 W is contained in M

follows from the fact that, for a ∈ W , M (a) can be written as M (a) = n∈Z an M (χn ) with   |an | M (χn )L(1 (Z)) = |an | < ∞. n∈Z

n∈Z

Moreover, the preceding argument also shows that M (a)L(1 (Z)) = aW . Here are some more properties of the algebras M p . In what follows let 1 < p < ∞ and 1/p + 1/q = 1. Put [p, q] := [min{p, q}, max{p, q}], let  · p denote the norm in L(
p (Z)) and for a ∈ L1 define a ∈ L1 by a(t) = a(t) (t ∈ T). (a) If a ∈ M p , then a ∈ M q and the adjoint M ∗ (a) ∈ L(
q (Z)) equals M (a). Proof. If a ∈ M p , then M ∗ (a) ∈ L(
q (Z)), and from the equality (ei , M (a)ej ) = (M (a)ei , ej ) we deduce that M ∗ (a) = M (a).   (b) M p = M q . If a ∈ M p , then a ∈ M p and M (a)p = M (a)p = M (a)q = M (a)q .

48

2 Basic Theory

Proof. It is easy to see that M (a) = J  M (a)J  , where J  is the isometry on
p (Z) given by J  : {ϕi }i∈Z → {ϕ−i }i∈Z . Thus, if a ∈ M p then a ∈ M p . From (a) we then deduce that a ∈ M q . The norm equalities follow from (a) and the fact that J  is an isometry.   (c) If a ∈ M p , then a ∈ M r for all r ∈ [p, q]. Proof. This results from (b) and the Riesz-Thorin interpolation theorem.   (d) If 1 ≤ p ≤ r ≤ 2, then W = M 1 ⊂ M p ⊂ M r ⊂ M 2 = L∞ and, moreover, a∞ ≤ M (a)r ≤ M (a)p ≤ aW .

Proof. The representation M (a) = an M (χn ) shows that M 1 ⊂ M r and that M (a)r ≤ M (a)1 . If p ≤ r ≤ 2, then, by the Riesz-Thorin interpolation theorem, M (a)r ≤ M (a)tp M (a)1−t q , where 1/r = t/p + (1 − t)/q and 0 ≤ t ≤ 1, and in view of (b) the right-hand side equals M (a)tp M (a)1−t = M (a)p .   p (e) If a ∈ M p and r ∈ [p, q], then γ M (a)r ≤ a1−γ ∞ M (a)p

with

γ=

p|r − 2| . r|p − 2|

Proof. Again apply the Riesz-Thorin interpolation theorem.   (f) If a ∈ L∞ is of bounded variation, then a ∈ M p and  M (a)p ≤ sp a∞ + V1 (a) , where sp is a constant independent of a and V1 (a) denotes the total variation of a. Proof. By virtue of (b) there is no loss of generality in assuming that 2 < p < ∞. For α ∈ (0, 2π) let χα (eiθ ) be 1 on (α, 2π) and 0 on (0, α). The n-th Fourier coefficient of χα is (i/(2πn))(1 − e−inα ) for n = 0. Hence, if ϕ ∈
0 (Z) then (M (χα )ϕ)j − (χα )0 ϕj equals i −ijα  eikα ϕk i  ϕk − e . 2π j − k 2π j−k k =j

k =j

From the discrete Hunt-Muckenhoupt-Wheeden theorem in 1.49 we deduce that χα ∈ M p and that there is a constant sp such that M (χα )p ≤ sp for all α ∈ (0, 2π). Here and in the following sp denotes a constant depending only on p but not necessarily the same at each occurrence. If a is a non-decreasing simple function with a(eiθ ) = αk for θ ∈ (θk−1 , θk ) (0 = θ0 < θ1 < . . . < θn = 2π), then a can be written in the form

2.2 Toeplitz Operators

a=

49

n−1 

(αk+1 − αk )χθk + α1 ,

k=1

and from what has just been proved it follows that a ∈ M p and  M (a)p ≤ sp V1 (a) + a∞ .

(2.4)

To obtain the assertion for general a it suffices to assume that a is a realvalued monotonically non-decreasing function. Then there exists a sequence {a(m) }m∈Z+ of non-decreasing simple functions converging to a uniformly on [0, 2π) as m → ∞. Since p > 2, we have "  " " M (a) − M a(m) ϕ" ≤ M (a − a(m) )ϕ2 ≤ a − a(m) ∞ ϕ2 = o(1) p as m → ∞ for every ϕ ∈
0 (Z). Because of (2.4),   M (a(m) )p ≤ sp V1 (a(m) ) + a(m) ∞ ≤ sp V1 (a) + a∞ . Thus, M (a) ∈ L(
p (Z)) and (2.4) holds for general a.   (g) M p is a Banach algebra with respect to the norm ap := M (a)L(p (Z)) . Proof. It remains to show that M p is complete. Let {a(m) }m∈Z+ be a Cauchy sequence in M p . By virtue of (d), {a(m) }m∈Z+ is a Cauchy sequence in L∞ and, consequently, there is an a ∈ L∞ such that a(m) −a∞ → 0 as m → ∞. Since L(
p (Z)) is complete, there is an A ∈ L(
p (Z)) such that M (a(m) ) − Ap → 0 as m → ∞. Because M (a)ϕ = Aϕ for ϕ ∈
0 (Z), we finally conclude that A = M (a).  

2.2 Toeplitz Operators 2.6. Definition. The operator T (a) defined for a ∈ L∞ and 1 < p < ∞ by T (a) : H p → H p ,

f → P (af )

(2.5)

is obviously bounded and T (a)L(H p ) ≤ cp a∞ , where cp is the norm of the Riesz projection on Lp . This operator is called the Toeplitz operator on H p generated by the function a. If a ∈ M p , then the operator T (a) given on
p =
p (Z+ ) (1 ≤ p < ∞) as T (a) :
p →
p ,

ϕ → P (a ∗ ϕ)

(2.6)

is clearly bounded and T (a)L(p ) ≤ M (a)L(p (Z)) . Here P denotes the discrete Riesz projection. The operator defined by (2.6) is called the Toeplitz operator on
p generated by the function a.

50

2 Basic Theory

The function a generating the Toeplitz operators (2.5) and (2.6) is usually referred to as the symbol of the corresponding operator. Toeplitz operators on
p are sometimes also called discrete Wiener-Hopf operators. For p = 2, the Toeplitz operators defined on H 2 and
2 by (2.5) and (2.6), respectively, are unitarily equivalent through the isomorphism  H 2 →
2 , ϕn χn → {ϕn }n∈Z+ . (2.7) n∈Z+

Therefore we shall frequently identify these operators without mentioning this explicitly. For f ∈ H p , g ∈ H q , ϕ ∈
p , ψ ∈
q (1/p + 1/q = 1) let (  1 (f, g) := f g dm, (ϕ, ψ) := ϕn ψn . 2π T n∈Z+

It is clear that the operators (2.5) and (2.6) satisfy (T (a)χj , χk ) = ak−j ,

(T (a)ej , ek ) = ak−j

∀ j, k ∈ Z+ .

The following theorem states that every bounded operator on H p resp.
p with this property is a Toeplitz operator and, moreover, relates the norm of a Toeplitz operator with the norm of the multiplication operator generated by the same function. 2.7. Theorem (Brown/Halmos). (a) Let A ∈ L(H p ) (1 < p < ∞) and suppose there is a sequence {an }n∈Z of complex numbers such that (Aχj , χk ) = ak−j for all k, j ∈ Z+ . Then there exists an a ∈ L∞ such that A = T (a) and {an } is the Fourier coefficient sequence of a. Moreover, a∞ ≤ T (a)L(H p ) ≤ cp a∞ ,

(2.8)

where cp is the norm of P on Lp . (b) Let a ∈ L(
p ) (1 ≤ p < ∞) and suppose there is a sequence {an }n∈Z of complex numbers such that (Aej , ek ) = ak−j for all k, j ∈ Z+ . Then there exists an a ∈ M p such that A = T (a) and {an } is the Fourier coefficient sequence of a. Moreover, T (a)L(p (Z+ )) = M (a)L(p (Z)) .

(2.9)

Proof. (a) For n ≥ 0, define bn ∈ Lp as bn := χ−n (Aχn ). Then bn p ≤ A. Since Lp = (Lq )∗ , the Banach-Alaoglu theorem implies that there is a b ∈ Lp such that bp ≤ A and some subsequence {bnk } of {bn } converges to b in the weak topology on Lp . In particular, (bnk , χj ) → (b, χj ) for all j ∈ Z, and because (bnk , χj ) = (Aχnk , χnk +j ) = aj whenever nk + j ≥ 0, it follows that (b, χj ) = aj

∀ j ∈ Z.

(2.10)

2.2 Toeplitz Operators

51

Now define the mapping B by B : P → Lp ,

f → bf.

(2.11)

If f, g ∈ P, then, by virtue of (2.10), (Bf, g) is equal to (M (χ−n )AM (χn )f, g) whenever n is chosen large enough. Hence |(Bf, g)| ≤ lim sup |(M (χ−n )AM (χn )f, g)| ≤ A f p gq n→∞

and thus
 Bf p = sup |(Bf, g)| : g ∈ P, gq ≤ 1 ≤ A f p for all f ∈ P. This shows that the linear mapping (2.11) extends to an operator B ∈ L(Lp ) with B ≤ A. Again from (2.10) we deduce that (Bχj , χk ) = (b, χk−j ) = ak−j for all j, k ∈ Z. Now Proposition 2.2 gives the existence of an a ∈ L∞ such that B = M (a) and {an } is the Fourier coefficient sequence of a. Since both (T (a)χj , χk ) and (Aχj , χk ) equal ak−j , it follows that A = T (a). Finally, because M (a) = B ≤ A = T (a), the norm equality in Proposition 2.2 gives the first “≤” in (2.8). The second “≤” in (2.8) is trivial. (b) Since Aen ∈
p for all n ∈ Z+ , it is obvious that the sequence {an }n∈Z belong to
p (Z). After defining B as ! # aj−k ϕk B :
0 (Z) →
p (Z), {ϕj }j∈Z → j∈Z

k∈Z

the rest of the proof is completely analogous to (a).

 

2.8. Corollary. (a) If a ∈ L∞ , then T (a)L(H 2 ) = T (a)L(2 ) = a∞ . (b) If a ∈ M p , then T (a)L(p ) = aM p . (c) If 1 < p < ∞, 1/p + 1/q = 1, and if a ∈ M p , then T (a) ∈ L(
q ) and T (a)L(q ) = T (a)L(p ) . The adjoint T ∗ (a) ∈ L(
q ) is equal to T (a). (d) If 1 < p < ∞, 1/p + 1/q = 1, and if a ∈ M p , then T (a) ∈ L(
r ) for all r ∈ [p, q] and γ a∞ ≤ T (a)L(r ) ≤ a1−γ ∞ aM p ≤ aW ,

where γ = p|r − 2|/(r|p − 2|). Proof. Combine 2.4, 2.5, and 2.7.  

52

2 Basic Theory

2.9. Shift operators. Recall that χn is given by χn (t) = tn (t ∈ T). The operators M (χ1 ) and T (χ1 ) are usually referred to as the bilateral and unilateral shift, respectively, because they act on
p (Z) resp.
p (Z+ ) (1 ≤ p < ∞) by the rules M (χ1 ) : {ϕi }i∈Z → {ϕi−1 }i∈Z , T (χ1 ) : {ϕ0 , ϕ1 , . . .} → {0, ϕ0 , ϕ1 , . . .}. Also as usual, we put U := M (χ1 ) and V := T (χ1 ), and, for n ∈ Z+ , we let U n = M (χn ),

U −n = M (χ−n ),

V n = T (χn ),

V (−n) = T (χ−n ).

Then obviously U ±n = (U ±1 )n , V n = (V 1 )n , V (−n) = (V (−1) )n , and U −n U n = U n U −n = I,

U ∗ = U −1 ,

V (−n) V n = I,

V ∗ = V (−1) .

Note that V n V (−n) is not the identity operator. It is clear that U ±n are isometries on Lp and
p (Z) and that the operators V n are isometries on H p (1 < p < ∞) and
p (Z+ ) (1 ≤ p < ∞). In particular, the range of V n is always closed. It is easy to see that V (−n) is onto, dim Ker V (−n) = n, dim Coker V n = n, V n is one-to-one. Thus, Ind T (χk ) = −k for all k ∈ Z.

2.3 Hankel Operators 2.10. Definitions. The definition of Hankel operators is slightly complicated by the circumstance that there is neither a definition in general use nor a unique notation for them in the literature and that there is in fact no compelling reason for adopting such a unique definition or notation. Besides the projections P and Q = I − P , we now need a third operator J, the so-called flip operator. This is the (obviously isometric) operator acting on Lp (1 < p < ∞) by the formula    1 1 (Jf )(t) = f fn t−n−1 (t ∈ T) = t t n∈Z

and, accordingly, acting on
p (Z) (1 ≤ p < ∞) by the rule (Jϕ)n = ϕ−n−1

(n ∈ Z).

For a ∈ L∞ , we define the Hankel operator H(a) on H p (1 < p < ∞) by H(a) : H p → H p ,

f → P M (a)QJf.

We let H() a) denote the operator given by

2.3 Hankel Operators

H() a) : H p → H p ,

53

f → JQM (a)P f

and refer to H() a) also as a Hankel operator (see 2.15 below). It is clear that both H(a) and H() a) are bounded whenever a ∈ L∞ and 1 < p < ∞. p Given a ∈ M (1 ≤ p < ∞) with Fourier coefficients sequence {an }n∈Z we analogously define H(a) :
p →
p , H() a) :
p →
p ,

ϕ → P M (a)QJϕ, ϕ → JQM (a)P ϕ

and call H(a) and H() a) the Hankel operators on
p generated by the function a. It is easily seen that H(a) and H() a) are bounded on
p . Moreover, it is p readily verified that their action on
can be given by the following formulas: !  # H(a) :
p →
p , {ϕj }j∈Z+ → aj+k+1 ϕk , j∈Z+

k∈Z+

H() a) :
p →
p ,

{ϕj }j∈Z+ →

! 

# a−j−k−1 ϕk

k∈Z+

. j∈Z+

If p = 2, the Hankel operators defined on H 2 and
2 are again unitarily equivalent through the isomorphism (2.7). The operators resulting from Hankel operators by omitting the flip operator, that is, the four operators ◦

p → H p, P M (a)Q : H− ◦

p QM (a)P : H p → H− ,

P M (a)Q : Q
p (Z) → P
p (Z), QM (a)P : P
p (Z) → Q
p (Z)

will occasionally also be referred to as Hankel operators. If a ∈ L∞ , then (H(a)χj , χk ) = aj+k+1 for all j, k ∈ Z+ . The following theorem describes the bounded operators on H p with this property and provides an important norm estimate for Hankel operators on H p . 2.11. Theorem (Nehari). Let A ∈ L(H p ) (1 < p < ∞) and suppose there is a sequence {an }n∈N of complex numbers such that (Aχj , χk ) = aj+k+1 for all j, k ≥ 0. Then there is a function b ∈ L∞ such that A = H(b) and the n-th Fourier coefficient bn of b is equal to an for all n ≥ 1. Moreover, distL∞ (b, H ∞ ) ≤ H(b)L(H p ) ≤ cp distL∞ (b, H ∞ ), where cp = P L(Lp ) . In particular, for p = 2 we have H(b)L(H 2 ) = distL∞ (b, H ∞ ).

(2.12)

54

2 Basic Theory

Proof. Note that AL(H p ) equals sup{|G(Af )| : f H p ≤ 1, G(H p )∗ ≤ 1}. So we deduce from (1.15) that 1 AL(H p ) ≤ Φ(A) ≤ AL(H p ) , cp

(2.13)

where Φ(A) := sup{|(Af, g)| : f H p ≤ 1, gH q ≤ 1} and 1/p + 1/q = 1. The hypothesis implies that (Af, g) = (Aχ0 , f c g)

∀ f, g ∈ PA ,

where f c (t) = f (1/t) (t ∈ T). It is obvious that f c H p = f H p . Thus, 
Φ(A) = sup |(Aχ0 , f g)| : f H p ≤ 1, gH q ≤ 1 . We claim that

 f g : f H p ≤ 1, gH q ≤ 1 = h ∈ H 1 : hH 1 ≤ 1 .

(2.14)

H¨ older’s inequality immediately gives the inclusion “⊂”. To get the reverse inclusion note first that, by 1.41(a), every h ∈ H 1 factors as h = ϕhe , where 1/p ϕ ∈ H ∞ is inner and he ∈ H 1 is outer. From 1.41(e) we see that he ∈ H p 1/q 1/p 1/q and he ∈ H q . Thus, if we let f = ϕhe and g = he , then h = f g and f H p = gH q = hH 1 . This completes the proof of (2.14). Taking into account (2.14) we obtain Φ(A) = sup{|(Aχ0 , h)| : hH 1 ≤ 1}. Hence, the mapping ( 1 1 (Aχ0 )h dm C : H → C, h → 2π T  ∗ is a linear functional belonging to H 1 and we have C 1 ∗ = Φ(A). Again from 1.42 we conclude that there is a c ∈ L∞ such that ( ( (Aχ0 )h dm = ch dm ∀ h ∈ H 1 . T

H

T

Letting h = χn (n ≥ 0), we get (Aχ0 , χn ) = an+1 = cn , and thus the function b = χ1 c ∈ L∞ has the desired property: bn = an for n ≥ 1. We are left with the norm estimate (2.12). Any extension of C to a bounded ) on L1 is given by linear functional C ( ) : L1 → C, f → C ϕf dm T

with some ϕ ∈ L∞ and we have C

H1

∗ ≤ C ) (L1 )∗ = ϕ∞ . Due to

)0 ∈ (L1 )∗ of C such that the Hahn-Banach theorem there is an extension C )0 . Thus, C = C

2.3 Hankel Operators

Φ(A) = C

H1

55

∗ = inf C ) (L1 )∗ = inf ϕ∞

) is an extension of C (where, in fact, the inf can be replaced by min). But C if and only if ( (ϕ − Aχ0 )h dm = 0 ∀ h ∈ H1 T

⇐⇒

(ϕ − Aχ0 )n = 0

⇐⇒ ⇐⇒

(χ1 ϕ)n − an = 0 χ1 ϕ − b ∈ H ∞ ,

∀n≥0 ∀n≥1

therefore, 
Φ(A) = inf ϕ∞ : χ1 ϕ − b ∈ H ∞
 = inf χ1 ϕ∞ : χ1 ϕ − b ∈ H ∞ = distL∞ (b, H ∞ ) and now (2.13) gives (2.12).   Remark. Let the hypothesis of the preceding theorem be satisfied. Then Aχ0 ∈ H p and so there exists a function a ∈ H p whose n-th Fourier coefficient is an (n ∈ N). One can now formulate the following criterion for the boundedness of A: A ∈ L(H p ) if and only if a ∈ BM O. Indeed, if A ∈ L(H p ), then, by the above theorem, a = P b for some b ∈ L∞ and thus a ∈ BM O by 1.48(k); on the other hand, if a ∈ BM O, then, by virtue of 1.48(l), there are u, v ∈ L∞ such that a = u + P v, whence u = P u, so a = P b with b = u + v ∈ L∞ , and the above theorem implies the boundedness of A. 2.12. Open problem. Establish a Nehari theorem for
p . The conjecture is that for this case in the preceding theorem “b ∈ L∞ ” must be replaced by “b ∈ M p ” and the norm H(b)L(p ) is equivalent (or, maybe, even equal) to distM p (b, H ∞ ∩ M p ). The following two sections are intended to give a first idea of the connection between multiplication, Toeplitz, and Hankel operators. Moreover, the formulas stated in 2.14, thought being very simple, will be of extreme importance for all what follows. 2.13. Decomposition of the multiplication operator. L2 decomposes ◦

into the orthogonal sum of H 2− and H 2 . Accordingly, every operator in L(L2 ) can be represented as a 2 × 2 operator matrix,  ◦     ◦  2 2 AB H − → H− . : 2 CD H H2 This applies, in particular, to the multiplication operator M (a) ∈ L(L2 ). In the corresponding matrix representation we meet operators closely related to Toeplitz and Hankel operators:

56

2 Basic Theory

 M (a) =

QM (a)Q P M (a)Q

⎛ ◦ ⎞  ⎛ ◦ ⎞ 2 H H2 : ⎝ −⎠ → ⎝ −⎠. P M (a)P H2 H2

QM (a)P

(2.15)

Clearly, the P M (a)P in the right lower corner is nothing else than the Toeplitz operator T (a). The operators QM (a)P and P M (a)Q differ from H() a) and H(a), respectively, only by the flip operator. Finally, if we define T () a) on H 2 as T () a) = JQM (a)QJ, then the QM (a)Q in the left upper corner is equal to JT () a)J. In terms of Fourier coefficients we have !  # aj−i ϕj . T () a) :
2 →
2 , {ϕi }i∈Z+ → i∈Z+

j∈Z+

Thus, the operator matrix in (2.15) can be written as ⎛ ◦ ⎞  ⎛ ◦ ⎞  2 JT () a)J JH() a) H H2 : ⎝ −⎠ → ⎝ −⎠. M (a) = H(a)J T (a) H2 H2 In more detail, if we express things via Fourier coefficients, the 6 × 6 matrix in the center of M (a) equals ⎛ ⎞ a0 a−1 a−2 a−3 a−4 a−5 ⎜ a1 a0 a−1 a−2 a−3 a−4 ⎟ ⎜ ⎟ ⎜ a2 a1 a0 a−1 a−2 a−3 ⎟ ⎜ ⎟ ⎜a a a a a a ⎟. 1 0 −1 −2 ⎟ ⎜ 3 2 ⎝ a4 a3 a2 a1 a0 a−1 ⎠ a5 a4 a3 a2 a1 a0 ◦

·

·

p Since Lp =H− + H p (1 < p < ∞) and
p (Z) = Q
p (Z) + P
p (Z) (1 ≤ p < ∞), all what has been said above applies to the case p = 2 as well. In particular, a) defined by if a ∈ L∞ or a ∈ M p , then the operators T ()

T () a) : H p → H p ,

f → JQM (a)QJf

and T () a) :
→
, p

p

{ϕi }i∈Z+ →

!  j∈Z+

# aj−i ϕj

(2.16) i∈Z+

will be bounded on H p and
p , respectively. Finally, notice the obvious identity M (a) = P M (a)P + P M (a)Q + QM (a)P + QM (a)Q.

(2.17)

This is merely a translation of the representation (2.15) into another language. The four operators on the right of (2.17) are the building stones of the operators

2.4 Invertibility of Toeplitz Operators on H 2

M (a)P + M (b)Q,

57

P M (a) + QM (b),

which are usually called singular integral operators when considered as acting on Lp and paired convolution operators when considered as acting on
p (Z). 2.14. Proposition. Let a, b ∈ L∞ resp. a, b ∈ M p . Then T (ab) = T (a)T (b) + H(a)H()b), H(ab) = T (a)H(b) + H(a)T ()b).

(2.18) (2.19)

In particular, if the positive Fourier coefficients of a = a− and the negative Fourier coefficients of b = b+ vanish, then for every c ∈ L∞ resp. c ∈ M p , T (a− cb+ ) = T (a− )T (c)T (b+ ).

(2.20)

Proof. We have T (ab) = P M (ab)P = P M (a)M (b)P = P M (a)P M (b)P + P M (a)QM (b)P = P M (a)P · P M (b)P + P M (a)QJ · JQM (b)P and this is (2.18). Similarly, H(ab) = P M (ab)QJ = P M (a)M (b)QJ = P M (a)P · P M (b)QJ + P M (a)QJ · JQM (b)QJ, which is (2.19). To complete the proof, note that the conditions imposed upon a− and b+ imply that H(a− ) = 0 and H(b.  + ) = 0.  2.15. Important remark. The ) a used in 2.10, 2.13 and 2.14 has nothing to do with the conjugate function of a (in the sense of 1.43). Given a measurable function a on T we now define ) a by ) a(t) = a(1/t) (t ∈ T). Moreover, this point of view eliminates any confusion that might arise in connection with 2.10 and definitions in 2.13. For instance, H() a) may be thought of as both JQM (a)P and as P M () a)QJ, and T () a) may be interpreted as both JQM (a)QJ and P M () a)P . In either case, both is the same. The notation ) a is in general use for the conjugate function of a as well as for the function given by ) a(t) = a(1/t) (t ∈ T). As a rule, henceforth ) a will always refer to the function ) a(t) = a(1/t) unless it is explicitly indicated that ) a means the conjugate function.

2.4 Invertibility of Toeplitz Operators on H 2 We first show how the expressions for the norms T (a)L(H 2 ) and H(a)L(H 2 ) obtained in Theorems 2.7 and 2.11 can be used to derive results on the invertibility of Toeplitz operators on H 2 .

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2 Basic Theory

2.16. Definition. Let a ∈ L∞ . If there exist a real number ε > 0 and a complex number c of modulus 1 such that Re (ca) ≥ ε a.e. on T, then a is called sectorial. It is obvious that a sectorial function is necessarily invertible in L∞ . It is also easy to see that a is sectorial if and only if so is a/|a|. Thus, sectoriality is a matter of the argument. Also notice that a ∈ GL∞ is sectorial if and only if a/|a| can be written as ei(v+c) where c ∈ R and v ∈ L∞ is a real-valued function with v∞ < π/2. Finally, it is easily verified that a function a ∈ GL∞ is sectorial if and only if distL∞ (a/|a|, C) < 1, where distL∞ (g, C) := inf{g − c∞ : c ∈ C}. 2.17. Theorem (Brown/Halmos). If a ∈ L∞ is sectorial then T (a) is invertible on H 2 . Proof. It is readily seen that there are a sufficiently small real number δ > 0 and a complex number c of modulus 1 such that 1 − δca∞ < 1. From Theorem 2.7 (or, more explicitly, from its Corollary 2.8(a)) we get I − δcT (a)L(H 2 ) < 1. This implies the invertibility of δcT (a) and thus also that of T (a).   In order to state the consequence of Nehari’s theorem we have promised above, we need two propositions which are interesting on their own and will often be applied in the following. 2.18. Proposition (Wintner). Let h ∈ H ∞ . Then T (h) ∈ GL(H 2 ) ⇐⇒ h ∈ GH ∞ . Proof. If h−1 ∈ H ∞ , then, by Proposition 2.14, T (h−1 ) ∈ L(H 2 ) is the inverse of T (h). Conversely, if T (h) ∈ GL(H 2 ), then the equation T (h)f = P (hf ) = hf = g must have a solution f ∈ H 2 for every g ∈ H 2 . Thus hH 2 = H 2 and 1.41(g) completes the proof.   2.19. Proposition. Let a ∈ GL∞ . Then T (a) ∈ GL(H 2 ) ⇐⇒ T (a/|a|) ∈ GL(H 2 ). Proof. Since a−1 ∈ L∞ , we deduce from 1.41(d) that there is an outer function h ∈ GH ∞ such that |a−1 |1/2 = |h|. So a/|a| = hah and, by Proposition 2.14, T (a/|a|) = T (h)T (a)T (h). Due to the preceding proposition T (h) and T (h) =  T ∗ (h) are invertible on H 2 and this gives the assertion at once.  Remark. This proposition reduces the invertibility problem for Toeplitz operators on H 2 to the case of unimodular symbols. In other words, the invertibility of a Toeplitz operator on H 2 is exclusively dictated by the behavior of the argument of its symbol.

2.4 Invertibility of Toeplitz Operators on H 2

59

2.20. Theorem (Widom/Devinatz). Let ϕ ∈ L∞ be a unimodular function, i.e., |ϕ| = 1 a.e. on T. Then (a) T (ϕ) is left-invertible on H 2 ⇐⇒ distL∞ (ϕ, H ∞ ) < 1, (b) T (ϕ) is right-invertible on H 2 ⇐⇒ distL∞ (ϕ, H ∞ ) < 1, (c) T (ϕ) is invertible on H 2 ⇐⇒ distL∞ (ϕ, GH ∞ ) < 1. Proof. (a) We have M (ϕ)P = P M (ϕ)P + QM (ϕ)P and since |ϕ| = 1, it follows that ) 2 f 2 = ϕf 2 = P (ϕf )2 + Q(ϕf )2 = T (ϕ)f 2 + H(ϕ)f for every f ∈ H 2 . The operator T (ϕ) is left-invertible on H 2 if and only if there exists an ε > 0 such that εf 2 ≤ T (ϕ)f 2 for all f ∈ H 2 (recall 1.12(g)). ) < 1, which, Consequently, T (ϕ) is left-invertible on H 2 if and only if H(ϕ) by Nehari’s theorem 2.11 for p = 2, is the same as " 
" ) − h"∞ : h ∈ H ∞ 1 > dist(ϕ, ) H ∞ ) = inf "ϕ " 
" )−) h"∞ : h ∈ H ∞ = inf "ϕ 
= inf ϕ − h∞ : h ∈ H ∞ = dist(ϕ, H ∞ ). (b) Since T ∗ (ϕ) = T (ϕ), this is immediate from (a). (c) Suppose T (ϕ) ∈ GL(H 2 ). Then, by (a), there is an h ∈ H ∞ such that ϕ − h∞ < 1 and it remains to show that h ∈ GH ∞ . We have (Corollary 2.8(a)) (2.21) I − T (ϕh) = 1 − ϕh∞ = ϕ − h∞ < 1, this implies the invertibility of T (ϕh) = T (ϕ)T (h) = T ∗ (ϕ)T (h),

(2.22)

and because T ∗ (ϕ) is invertible, so also is T (h). From Proposition 2.18 we deduce that h ∈ GH ∞ . Now suppose h ∈ GH ∞ and ϕ−h∞ < 1. Then (2.21) holds and therefore the operator (2.22) is invertible. By Proposition 2.18, T (h) ∈ GL(H 2 ), hence  T ∗ (ϕ) ∈ GL(H 2 ) and thus T (ϕ) ∈ GL(H 2 ).  2.21. Lemma. Suppose B is a subset of L∞ with the property that cb ∈ B whenever c ∈ C \ {0} and b ∈ B. Let ϕ ∈ L∞ be a unimodular function. Then distL∞ (ϕ, B) < 1 if and only if there are a function b ∈ B and a sectorial function s ∈ GL∞ such that ϕ = bs. Proof. If distL∞ (ϕ, B) < 1, then 1 − ϕ−1 b∞ = ϕ − b∞ < 1 for some b ∈ B. Hence ϕ−1 b is equal to a function s whose (essential) range is contained in some disk with center 1 and radius less than 1. Thus, ϕ−1 b = s with sectorial s, so ϕ = bs−1 and it remains to observe s−1 is sectorial whenever s is so.

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2 Basic Theory

Let ϕ = bs, where b ∈ B and s is sectorial. There is a c ∈ C \ {0} such that the (essential) range of cs−1 is contained in some disk with center 1 and radius less than 1. This implies that ϕ − cb∞ = 1 − ϕ−1 cb∞ = 1 − cs−1 ∞ < 1, whence distL∞ (ϕ, B) < 1.

 

2.22. Corollary. Let a ∈ GL∞ . Then the operator T (a) is left (right, resp. two-sided ) invertible on H 2 if and only if a/|a| = hs, where h ∈ H ∞ ∩ GL∞ (h ∈ H ∞ ∩ GL∞ , resp. h ∈ GH ∞ ) and s ∈ GL∞ is sectorial. Proof. Combine Theorem 2.20, Proposition 2.19, and Lemma 2.21.   For several applications the following restatement of Theorem 2.20(c) is useful. 2.23. Theorem (Widom/Devinatz). Let a ∈ GL∞ . Then T (a) is invertible on H 2 if and only if a/|a| = ei()u+v+c)

a.e. on

T,

(2.23)

∞

where c ∈ R, u and v are real-valued functions in L , and v∞ < π/2. Here u ) refers to the conjugate function of u. Proof. Put ϕ = a/|a|. By virtue of Proposition 2.19, T (a) is invertible if and only if T (ϕ) is invertible. Let T (ϕ) be invertible. Due to Corollary 2.22 there is an h ∈ GH ∞ such that ϕh is sectorial. Thus, ϕh = |h|e−iv with some real-valued function v ∈ L∞ for which v∞ < π/2. Hence ϕ = (|h|/h)eiv = (h/|h|)eiv .

(2.24)

Since h is outer, there is an analytic logarithm log h in D (see 1.41(g), (ii)). The real part of log h is u(z) := log |h(z)| (z ∈ D) and log h can be written as log h(z) = u(z) + i) u(z) + ic

(z ∈ D)

with some c ∈ R. Consequently, h(z) = eu(z) ei()u(z)+c)

(z ∈ D).

∞

Because h ∈ H , the non-tangential limit of eu(z) = |h(z)| exists a.e. on T and equals |h(t)|. Therefore the non-tangential limit of ei()u(z)+c) also exists a.e. on T and is equal to h(t)/|h(t)|. In other words, h/|h| = ei()u+c) , and (2.24) gives (2.23) with u = log |h|, which is clearly in L∞ . Now let ϕ = a/|a| be of the form (2.23). Put ψ = ei(v+c) ,

h = ei(u+i)u)/2 .

It is obvious that h ∈ GH ∞ and that ϕ = (1/h)ψh, whence T (ϕ) = T (1/h)T (ψ)T (h). But the operators T (1/h) and T (h) are invertible by Proposition 2.18, while T (ψ) is invertible due to Theorem 2.17. Thus, T (ϕ) is invertible, too.  

2.4 Invertibility of Toeplitz Operators on H 2

61

2.24. Remark. The Widom-Devinatz theorems solve the invertibility problem in H 2 for Toeplitz operators (with symbols in GL∞ ) completely. However, given an a ∈ GL∞ it is, in general, by no means easy to decide whether there is an outer function h ∈ GL∞ such that a − h∞ < 1 or to check whether a can be represented in the form (2.23). This is the reason for a great part of all further investigations devoted to the invertibility of Toeplitz operators. The main goal of these investigations is to obtain invertibility criteria, or, equivalently, descriptions of the spectrum, in terms of geometric data of the symbol. The Widom-Devinatz theorems answer the question in an analytic language. Nevertheless, there are situations in which Theorem 2.23 can be almost directly applied to decide whether a given Toeplitz operator is invertible or not. It can also be used to produce interesting examples of invertible Toeplitz operators. We shall demonstrate this in Proposition 2.26 below. 2.25. The class C(T◦ ). Let T◦ denote the punctured circle T \ {−1} and let C(T◦ ) denote the collection of all functions on T which are continuous at every point t ∈ T◦ . We denote by CU (T◦ ) the unimodular and by CR(T◦ ) the real-valued functions in C(T◦ ). Every a ∈ CU (T◦ ) can be written as a = eib with b ∈ CR(T◦ ) and b is uniquely determined by a up to an additive constant of the form 2kπ, k ∈ Z. Given a ∈ CU (T◦ ) choose any b ∈ CR(T◦ ) for which a = eib and define the real-valued function a# ∈ C(R) as   i−x a# (x) = b (x ∈ R). i+x The behavior of a# (x) as x → ±∞ provides a good picture of the argument of a near the possible discontinuity of a at −1. We write a# (+∞) = −∞ if lim a# (x) = −∞; the notations a# (+∞) = +∞, a# (−∞) = +∞, x→+∞

and a# (−∞) = −∞ are defined analogously. The function a# is said to be bounded from above (below) at −∞ if there exist both M ∈ R and x0 ∈ R such that a# (x) ≤ M

∀ x < x0

(a# (x) ≥ M

∀ x < x0 ).

It is clear that these definitions express properties of the function a that do not depend on the particular choice of the function b ∈ CR(T◦ ) which defines a# . 2.26. Proposition. (a) There are a ∈ CU (T◦ ) such that a# (+∞) = +∞, a# (−∞) = +∞ and T (a) is invertible on H 2 . (b) There exist functions a ∈ CU (T◦ ) such that a# (+∞) = +∞, a# is not bounded neither from above nor from below at −∞ and T (a) is invertible on H 2 . (c) If a ∈ CU (T◦ ) and if a# (+∞) = +∞ and a# is bounded from above at −∞, then T (a) is not invertible on H 2 .

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2 Basic Theory

(d) Let a ∈ CU (T◦ ) and suppose a# = ψ + δ, where δ ∈ L∞ (R), ψ is monotonous on (−∞, 0) and (0, ∞), and ψ(±∞) = +∞. Then if T (a) is invertible, we have a# (x) = O(log |x|)

as

|x| → +∞.

Proof. (a) Let w be a conformal mapping of D onto the region  π π . Ω1 = z = x + iy ∈ C : y > | tan x|, − < x < 2 2 There is a point t0 ∈ T such that |w(z)| → ∞ as z → t0 , z ∈ D. Without loss of generality assume t0 = −1. Define  (2.25) w(t) := lim w(z) t ∈ T◦ z→t,z∈D

and, for t ∈ T◦ , put a(t) := eiIm w(t) . Then a ∈ CU (T◦ ), a# (±∞) = +∞, and, of course, a ∈ GL∞ . Since a is of the form (2.23) with u = Re w ∈ L∞ (so u ) = Im w + const) and v = 0, we deduce that T (a) is invertible on H 2 . (b) Now let Ω2 be the region
 Ω2 = z = x + iy ∈ C : y > − cot x, 0 < x < 2π and let S be the countable union of vertical half-lines given by S=

∞   / 1 z = x + iy ∈ C : x = , y ≤ n . n n=1

Then Ω3 := Ω2 \ S is a simply connected region. Let w denote a conformal mapping of D onto Ω3 and without loss of generality suppose |w(z)| → ∞ as z → −1, z ∈ D. Define w on T◦ as in (2.25) and put a := eiIm w on T◦ . Then a has all the required properties and T (a) is invertible by Theorem 2.23. (c) Assume the contrary, that is, assume T (a) ∈ GL(H 2 ). Then a can be written in the form (2.23) and it follows from the Fefferman theorem 1.48(g) that the argument of a is in BM O. From 1.48(m) we deduce that a# is in BM O(R). But a function a# with the properties required in the hypothesis cannot be in BM O(R). This can be seen, for instance, as follows.  Assume a# ∈ BM O(R). Then the function g(ξ) := (1/2) a# (ξ)−a# (−ξ) , ξ ∈ R, is also in BM O(R). Since g is odd, gI must be zero for every I of the form I = (−x, x), and therefore we have ( 1 x |g(ξ)| dξ =: N < ∞. (2.26) sup x>0 x 0 *x Put G(x) := 0 |g(ξ)| dξ. Obviously, G(0) = 0, G(x) ≥ 0 for x ≥ 0, and (2.26) says that

2.4 Invertibility of Toeplitz Operators on H 2

G(x) ≤ N x

for

x > 0.

63

(2.27)

It is precisely the conditions that a# (+∞) = +∞ and that a# be bounded from above at −∞ which imply that g(ξ) → +∞ as ξ → +∞. This in turn ensures the existence of a (sufficiently large) x0 > 0 such that ( 2x0 G(2x0 ) − G(x0 ) = |g(ξ)| dξ ≥ (2N + 1)x0 x0

(apply the mean-value theorem). Hence, G(2x0 ) G(x0 ) + (2N + 1)x0 (2N + 1)x0 1 = ≥ =N+ , 2x0 2x0 2x0 2 which contradicts (2.27) and completes the proof. (d) As in the proof of part (c) we deduce that a# ∈ BM O(R), whence ψ = a# − δ ∈ BM O(R). Put g(x) := ψ(1/x) (x = 0). It is not difficult to see from 1.48(m) that g is also in BM O(R). The assertion can now be derived from the John-Nirenberg theorem 1.48(n) as follows. There is an x0 > 0 such that g(x) > 0 for x ∈ (−x0 , x0 ). Define ( x0 1 g(x) dx. g0 = 2x0 −x0 We now conclude from 1.48(n) that, for λ > 0, +
+ + x ∈ (−x0 , x0 ) : |g(x) − g0 | > λ + ≤ Ce−c0 λ with some constants C and c0 independent of λ. Hence, if we define x1 (λ) ∈ (0, x0 ) and x2 (λ) ∈ (0, x0 ) by g(−x1 (λ)) = g(x2 (λ)) = g0 + λ (note that g is monotonous on (−x0 , 0) and (0, x0 )), then x1 (λ) + x2 (λ) ≤ Ce−c0 λ (again use monotonicity). So xi (λ) ≤ Ce−c0 λ , whence log xi (λ) ≤ log C − c0 λ, and therefore g(±xi (λ)) = g0 + λ ≤ g0 +

log C log xi (λ) − c0 c0

(i = 1, 2).

On replacing xi (λ) by x we get g(x) ≤ A log(1/|x|) for all x ∈ (−x3 , x3 ) with some x3 > 0 (once more take into account monotonicity). Thus, ψ(x) = O(log |x|) as |x| → ∞, and consequently, a# (x) = ψ(x) + δ(x) = O(log |x|)

as |x| → ∞.  

Remark. This proposition, though being a simple consequence of the WidomDevinatz theorem 2.23 obtained by invoking some deep BM O results in a luxurious way, is already concerned with geometric data of the symbol. It says, roughly speaking, that if a ∈ CU (T◦ ) has a discontinuity of oscillating type at −1 then

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2 Basic Theory

(a), (b) T (a) may be invertible if a has the possibility of changing the orientation of the oscillation (=rotation ) into the opposite direction when passing through −1; (c) T (a) cannot be invertible if the orientation of the oscillation is preserved when passing through −1; (d) T (a) cannot be invertible if the oscillation is allowed to alter its orientation into the opposite direction when passing through −1 but is, in addition, required to be “monotonous” and “sufficiently fast”. Let us still dwell a bit on symbols a ∈ CU (T◦ ) for which a# is an even function. We saw that if a# (x) tends monotonically to infinity as x → ±∞, then T (a) is not invertible unless a# increases sufficiently slowly. We shall soon be in a position to decide whether T (a) is invertible if the limits a# (±∞) exist and are finite (this corresponds to the situation in which a ∈ CU (T◦ ) is continuous or has a jump discontinuity at −1). Much more difficulties arise for the “intermediate case,” e.g., for the cases where a# approaches +∞ sufficiently slowly or where the limits a# (±∞) do not exist at all. For instance, if a# (x) = cos x,

a# (x) = log log |x|,

or a# (x) = log log |x| + cos x

(x ∈ R, x large) we have situations of that kind. The spectral inclusion theorems we are now going to derive can be viewed as a first step towards the description of invertibility of Toeplitz operators in geometrical language.

2.5 Spectral Inclusion Theorems 2.27. Definitions. The essential range R(a) of a function a ∈ L∞ is the spectrum of a considered as an element of the C ∗ -algebra L∞ . Equivalently, R(a) is the set of all λ ∈ C such that {t ∈ T : |a(t) − λ| < ε} has positive (Lebesgue) measure for every ε > 0. Let X be a Banach space and let π denote the canonical homomorphism of L(X) onto the Calkin algebra L(X)/C∞ (X). For A ∈ L(X), the spectrum sp A of A is defined by
 sp A := spL(X) A = λ ∈ C : A − λI ∈ GL(X) and the essential spectrum spess A of A is defined as
 spess A := spL(X)/C∞ (X) (πA) = λ ∈ C : A − λI ∈ Φ(X) . The essential norm of A is given by
 Aess := πAL(X)/C∞ (X) = inf A + K : K ∈ C∞ (X) .

2.5 Spectral Inclusion Theorems

65

In order to avoid confusion, we shall sometimes write spΦ(X) A and AΦ(X) for spess A and Aess , respectively. Note that obviously spess A ⊂ sp A and Aess ≤ A for every A ∈ L(X). 2.28. Proposition. (a) If a ∈ L∞ and 1 < p < ∞, then M (a) ∈ GL(Lp ) if and only if a ∈ GL∞ . In other words, spL(Lp ) M (a) = R(a). (b) If a ∈ M p and 1 ≤ p < ∞, then M (a) ∈ GL(
p (Z)) if and only if a ∈ GM p . Consequently, spL(p (Z)) M (a) = spM p a ⊃ R(a). Proof. (a) If a ∈ GL∞ and b ∈ L∞ is the inverse of a, then M (b) ∈ L(Lp ) is the inverse of M (a). Conversely, suppose M (a) ∈ GL(Lp ). Then the equation M (a)b = 1 has a solution b ∈ Lp and we have ab = 1. Let B ∈ L(Lp ) denote the inverse of M (a). So a·Bf = f for all f ∈ P, whence Bf = bf for f ∈ P, and this implies that (Bχj , χk ) equals the (k − j)-th Fourier coefficient of b. The assertion now follows from Proposition 2.2. (b) If a ∈ GM p and b ∈ M p is the inverse of a, then M (b) ∈ L(
p (Z)) is the inverse of M (a). Now suppose M (a) ∈ GL(
p (Z)). By virtue of 2.5(a) it suffices to consider the case 1 ≤ p ≤ 2. The invertibility of M (a) implies that the equation = {ϕn } ∈
p (Z). Since
p (Z) ⊂
2 (Z), we M (a)ϕ = e0 has a solution ϕ

conclude that the function b = n∈Z ϕn χn belongs to L2 and that ab = 1. Thus, for a sequence ψ = {ψi } ∈
0 (Z) the inverse B of M (a) is given by  bi−j ψj (i ∈ Z). (Bψ)i = j∈Z

Due to the boundedness of B we !"!  # " " sup " bi−j ψj j∈Z

have " # " " : ψ ∈
0 (Z), ψp ≤ 1 < ∞, "

i∈Z p

 which implies that b ∈ M p by the definition of M p .  2.29. Proposition. (a) Suppose a ∈ L∞ and 1 < p < ∞. If M (a) ∈ Φ+ (Lp ) or M (a) ∈ Φ− (Lp ), then M (a) ∈ GL(Lp ). (b) Suppose a ∈ W . If M (a) ∈ Φ+ (
1 (Z)) or M (a) ∈ Φ− (
1 (Z)), then M (a) ∈ GL(
1 (Z)). (c) If a ∈ M p , 2 ≤ p < ∞, and M (a) ∈ Φ+ (
p (Z)), then M (a) belongs to GL(
p (Z)). (d) If a ∈ M p , 1 ≤ p < ∞, and M (a) ∈ Φ(
p (Z)), then M (a) ∈ GL(
p (Z)). Open problem. We are embarrassed to report that we have not been able to prove (c) for 1 < p < 2, although there seems to be no reason that (c) be false in that case.

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2 Basic Theory

Proof. (a) Suppose M (a) ∈ Φ+ (Lp ); otherwise pass to the adjoint operator and take into account 1.12(h). We first show that Ker M (a) = {0}. Let af = 0 for some f ∈ Lp , f = 0. Then f χn ∈ Ker M (a) for all n ∈ Z and it is easily seen that the system {f χn }n∈Z is linearly independent in Lp (if f p = 0 for some p ∈ P, then f = 0 a.e. on T). It would follow that dim Ker M (a) = ∞, which is a contradiction. Thus Ker M (a) = {0}. Consequently, Im M ∗ (a) = Im M (a) = Lq (1/p + 1/q = 1) and there is a b ∈ Lq such that ab = 1 a.e. on T, and it follows that Ker M (a) = {0}. Thus M ∗ (a) ∈ GL(Lq ), whence M (a) ∈ GL(Lp ). (b) Suppose M (a) ∈ Φ+ (
1 (Z)). As in the proof of part (a), one can see that then necessarily Ker M (a) = {0}. Thus, by 1.12(g), there is a δ > 0 / GL(
1 (Z)). such that abW ≥ δbW for all b ∈ W . Now assume M (a) ∈ −1 Then a ∈ / GW , since otherwise M (a ) would be an inverse of M (a). But the maximal ideal space, T, of the Banach algebra W coincides with its Shilov boundary. Therefore, a is a topological divisor of zero, that is, there exists a sequence {bn }∞ n=1 of functions bn ∈ W such that bn W = 1 and abn W → 0 as n → ∞. We arrived at a contradiction. Now let M (a) ∈ Φ− (
1 (Z)). Since
 c0 (Z) := ϕ = {ϕn }n∈Z : ϕn → 0 as |n| → ∞ is a predual of
1 (Z), we conclude from 1.12(h) that M (a) ∈ Φ+ (c0 (Z)). Assume M (a)ψ = 0, ψ ∈ c0 (Z), ψ = 0. Then M (a)(ψ ∗ en ) = 0 for all n ∈ Z, where (ψ ∗ en )i := ψn−i (i ∈ Z). We claim that the system {ψ ∗ en }n∈Z is linearly independent in c0 (Z). To see this, let π ∈
0 (Z) and assume ψ ∗ π = 0. Let p ∈ PA denote the polynomial whose Fourier coefficients sequence is π, assume p(t) = q(t)(t − α) (t ∈ T) with q ∈ PA and α ∈ C, and let  ∈
0 (Z) be the Fourier coefficients sequence of q. Then ξ := ψ ∗  ∈ c0 (Z) and we have ξ ∗ (e1 − αe0 ) = 0, i.e., ξn−1 = αξn (n ∈ Z). If α = 0, then ξ = 0, and in case α = 0 we have ξ−n = αn ξ0 and ξn = (1/α)n ξ0 (n ∈ Z+ ), which also implies that ξ = 0. On repeating this argument with  in place of π etc., we finally see that πn = 0 for n = 0. This proves the linear independence of the system {ψ ∗ en }n∈Z . Thus, what results is that Ker M (a) = {0} in c0 (Z). Consequently, Im M (a) =
1 (Z), hence there is a b ∈ W such that ab = 1, whence M (a) ∈ GL(
1 (Z)). (c) As in the proof of the Φ− -part of (b) we conclude that M (a) has a trivial kernel in
p (Z) whenever M (a) ∈ Φ+ (
p (Z)). Therefore M (a) is onto on
q (Z) (1/p + 1/q = 1). In particular, there is a ψ ∈
q (Z) such that M (a)ψ = e0 . Since
q (Z) ⊂
2 (Z), the function whose Fourier coefficients sequence is ψ belongs to L2 and we have af = 1. It follows that a = 0 a.e. on T. Thus, if M (a)ϕ = 0 for some ϕ ∈
q (Z) ⊂
2 (Z), then, again by passing into L2 , we have ϕ = 0. So Ker M (a) = {0} in
q (Z), hence M (a) ∈ GL(
q (Z)), and thus M (a) ∈ GL(
p (Z)).

2.5 Spectral Inclusion Theorems

67

(d) For p = 1 and 2 ≤ p < ∞ this is immediate from (b) and (c), respectively. If 1 < p < 2 and M (a) ∈ Φ(
p (Z)), then M (a) ∈ Φ− (
p (Z)), whence  M (a) ∈ Φ+ (
q (Z)) (2 < q < ∞) and (c) applies again.  2.30. Theorem (Hartman/Wintner). (a) Let a ∈ L∞ and 1 < p < ∞. If T (a) ∈ Φ+ (H p ) or T (a) ∈ Φ− (H p ), then a ∈ GL∞ . Consequently, R(a) ⊂ spΦ(H p ) T (a). (b) If a ∈ M p , 1 ≤ p < ∞, and T (a) ∈ Φ(
p ), then a ∈ GM p . Consequently, R(a) ⊂ spM p a ⊂ spΦ(p ) T (a). Proof. (a) Let T (a) ∈ Φ+ (H p ) and denote by K any (finite-rank) projection of H p onto Ker T (a). By 1.12(g), there is a δ > 0 such that T (a)f p + Kf p ≥ δf p

∀ f ∈ H p.

This implies that P M (a)P gp + P KP gp + δQgp ≥ δgp

∀ g ∈ Lp .

Hence, if we let U denote the bilateral shift, then P M (a)P U n gp + P KP U n gp + δQU n gp ≥ δU n gp

∀ g ∈ Lp ,

and since U ±n are isometries, we obtain for all g ∈ Lp , U −n P M (a)P U n gp + P KP U n p + δU −n QU n gp ≥ δgp .

(2.28)

The operators U −n P U n are uniformly bounded on Lp , and because, obviously, U −n P U n f converges in Lp to f for every f ∈ P, we deduce from 1.1(d) that U −n P U n converges strongly to the identity operator. Thus, U −n QU n → 0 strongly on Lp , U −n P M (a)P U n = U −n P U n M (a)U −n P U n → M (a) strongly on Lp . Because U n converges weakly to zero on Lp , we get, by 1.1(f), P KP U n → 0

strongly on Lp .

Thus, (2.28) gives that M (a)gp ≥ δgp for all g ∈ Lp . So M (a) ∈ Φ+ (Lp ) and Propositions 2.29(a) and 2.28(a) imply that a ∈ GL∞ . For T (a) ∈ Φ− (H p ) passage to the adjoint yields the desired result. (b) The proof is the same as that of part (a).  

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Remark 1. Propositions 2.29 and 2.28 also imply that the following implications hold: a ∈ W, T (a) ∈ Φ+ (
1 ) ∪ Φ− (
1 ) =⇒ a ∈ GW ; a ∈ M p , 2 ≤ p < ∞, T (a) ∈ Φ+ (
p ) =⇒ a ∈ GM p ; a ∈ M p , 1 ≤ p ≤ 2, T (a) ∈ Φ− (
p ) =⇒ a ∈ GM p . To prove that T (a) ∈ Φ− (
1 ) =⇒ a ∈ GW pass first to the predual c0 of
1 and notice that U −n QU n → 0 strongly on c0 . Remark 2. The Hartman-Wintner theorem shows that in Proposition 2.19 and Theorem 2.23 the hypothesis that a be invertible is redundant. Both results can be stated in the form “Let a ∈ L∞ . Then T (a) is invertible on H 2 if and only if a ∈ GL∞ and ....” Remark 3. Suppose a, b ∈ L∞ and 1 < p < ∞. If M (a)P + M (b)Q ∈ Φ+ (Lp ) or M (a)P + M (b)Q ∈ Φ− (Lp ), then a ∈ GL∞ and b ∈ GL∞ . Proof. Indeed, if, for instance, M (a)P + M (b)Q ∈ Φ+ (Lp ), then there are a K ∈ C0 (Lp ) and a δ > 0 such that " −n  " "U M (a)P + M (b)Q U n g "p + KU n gp ≥ δgp , " " n "U M (a)P + M (b)Q U −n g " + KU −n gp ≥ δgp p for all n ≥ 0 and g ∈ Lp , and because U ±n → 0 weakly on Lp as n → ∞ and U −n P U n → I,

U −n QU n → 0,

U n P U −n → 0,

U n QU −n → I

strongly on Lp as n → ∞, we have M (a)gp ≥ δgp ,

M (b)gp ≥ δgp

for all g ∈ Lp and the assertion follows as above.

 

Thus, when investigating Fredholmness or invertibility of singular integral operators (over the unit circle) we may a priori assume that the coefficients are in GL∞ . Moreover, we then have  M (a)P + M (b)Q = M (b) M (b−1 a)P + Q   = M (b) P M (b−1 a)P + Q QM (b−1 a)P + I (2.29) and since QM (b−1 a)P +I is always invertible (the inverse is I −QM (b−1 a)P ), we arrive at the following conclusion. Let a, b ∈ L∞ . Then M (a)P + M (b)Q is in Φ(Lp ) (resp. GL(Lp )) if and only if a, b ∈ GL∞ and T (b−1 a) is in Φ(H p ) (resp. GL(H p )). In the case of Fredholmness, Ind M (a)P + M (b)Q = Ind T (b−1 a).

2.5 Spectral Inclusion Theorems

69

This shows in what a sense the study of Fredholmness and invertibility for singular integral operators over the unit circle (and thus over smooth curves) is equivalent to the study of the corresponding problems for Toeplitz operators. We now extend Proposition 2.18 to the case p = 2. Note that passage to adjoints yields results for anti-analytic symbols, that is, for h ∈ H ∞ . 2.31. Proposition. (a) If 1 < p < ∞ and h ∈ H ∞ , then T (h) ∈ GL(H p ) ⇐⇒ h ∈ GH ∞ . (b) If 1 ≤ p < ∞ and h ∈ M p ∩ H ∞ , then T (h) ∈ GL(
p ) ⇐⇒ h ∈ GM p and h is outer. Proof. The implications “⇐=” follow as in the proof of Proposition 2.18. So we are left with the reverse implications. Theorem 2.30 gives that h ∈ GL∞ resp. h ∈ GM p . Thus, by 1.41(g) it remains to show that h−1 ∈ H ∞ . The identity (2.19) implies that  −1  −1  −1  H ) h +H ) h T (h). H ) h ) h =T ) h  −1   −1 But H ) h ) T (h) = 0, and h = H(1) = 0 and H ) h) = 0, whence H ) h  −1 ) = 0. Because ) h−1 = (h−1 )), since T (h) is invertible, it results that H h −1 ∞  we conclude that h ∈ H .  2.32. Proposition. Let a ∈ L∞ and 1 < p < ∞. Then T (a) is in GL(H p ) (Φ± (H p ) resp. Φ(H p )) if and only if a ∈ GL∞ and T (a/|a|) is in GL(H p ) (Φ± (H p ) resp. Φ(H p )). Moreover, if a ∈ GL∞ , then dim Ker T (a) = dim Ker T (a/|a|),

dim Coker T (a) = dim Coker T (a/|a|).

Proof. It follows from Theorem 2.30(a) that a may be assumed to belong to GL∞ . As in the proof of Proposition 2.19 we see that a/|a| = hah for some h ∈ GH ∞ . Since T (a/|a|) = T (h)T (a)T (h), the preceding proposition implies all assertions.   2.33. Theorem (Brown/Halmos). If a ∈ L∞ , then spL(H 2 ) T (a) ⊂ conv R(a),

(2.30)

where conv R(a) is the closed convex hull of R(a). Proof. Immediate from Theorem 2.17.

 

Remark. We shall see later that if E is any subarc of T and χE is the characteristic function of E, neither spL(H p ) T (χE ) nor spL(p ) T (χE ) is contained in conv R(χE ) = [0, 1] for 1 < p < ∞ and p = 2.

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2.34. Real-valued continuous functions. For p = 2, Theorems 2.30 and 2.33 together give that R(a) ⊂ spess T (a) ⊂ sp T (a) ⊂ conv R(a).

(2.31)

This is all what is needed to derive the following: if a is a real-valued continuous function, then T (a) ∈ GL(H 2 ) ⇐⇒ T (a) ∈ Φ(H 2 ) ⇐⇒ a(t) = 0 ∀ t ∈ T and

  spΦ(H 2 ) T (a) = spL(H 2 ) T (a) = min a(t), max a(t) . t∈T

t∈T

Note that both the spectrum and the essential spectrum are completely described via geometric data of the symbol. 2.35. Connectedness of the spectrum. A powerful tool for obtaining information about the spectra of Toeplitz operators are the following results. (a) (Widom). If a ∈ L∞ then spL(H p ) T (a) is connected. (b) (Douglas). If a ∈ L∞ then spΦ(H 2 ) T (a) is connected. Corollary 2.40 below implies that the boundary of spL(H p ) T (a) is contained in spL(H p ) T (a). Using this it is easy to derive the connectedness of the spectrum of a Toeplitz operator from the connectedness of the essential spectrum. Open problems. Is spΦ(H p ) T (a) always connected? We conjecture that the answer is yes and that a check of the proof in Widom [563] and Douglas [162] will indicate the modification needed to obtain the desired result. The following problem seems to us to lie essentially deeper: what can be said about the connectedness of the spectra of a Toeplitz operator on
p ? We do not know any symbol in M p generating a Toeplitz operator whose spectra are disconnected. 2.36. Real-valued symbols. If a ∈ L∞ is real-valued, then   spΦ(H 2 ) T (a) = spL(H 2 ) = ess inf a(t), ess sup a(t) . t∈T

t∈T

This result is due to Hartman and Wintner, too. Proof. Combine (2.31) and 2.35(b). There is a simple direct proof, which goes as follows. Let λ ∈ R and put b = a − λ. We must show that sign b = const whenever T (b) ∈ Φ(H 2 ). If Ind T (b) = κ, then Ind T (b) = Ind T (b) = Ind T ∗ (b) = −κ, whence κ = 0. Coburn’s theorem, which will be proved below (Corollary 2.40 for p = 2), therefore shows that we may assume that T (b) is invertible. Then the equation

2.6 The Connection Between Fredholmness and Invertibility

71

◦

2 T (b)f = 1 has a solution f ∈ H 2 . So bf = 1 + g with g ∈H− , and we obtain, for n ≥ 1, ( ( ( b|f |2 χn dm = bf f χn dm = (1 + g)f χn dm = 0. T

T

*

T

Since b|f |2 is real-valued, it follows that T b|f |2 χn dm = 0 for all n ∈ Z \ {0},  so b|f |2 = const, that is, sign b = const.  2.37. The boundary of conv R(a). For a ∈ L∞ , denote by  a the harmonic extension of a into D. The following result of Wolff is sometimes very useful to get further information about the spectrum of a Toeplitz operator. Let a ∈ L∞ and let λ belong to the boundary of conv R(a). Then λ ∈ spL(H 2 ) T (a) ⇐⇒ λ ∈ clos  a(D). An application of this result will be given in 4.75 and 4.78.

2.6 The Connection Between Fredholmness and Invertibility 2.38. Theorem (Coburn). A nonzero bounded Toeplitz operator has a trivial kernel or a dense range. The precise statement is as follows. (a) If a ∈ L∞ and if a does not vanish identically, then the kernel of T (a) in H p (1 < p < ∞) or the kernel of T (a) in H q (1/p + 1/q = 1) is trivial. (b) If a ∈ M p and if a does not vanish identically, then the kernel of T (a) in
p (1 ≤ p < ∞) or the kernel of T (a) in
q (1/p + 1/q = 1) is trivial. Proof. (a) Assume there are f+ ∈ H p , g+ ∈ H q , f+ = 0, g+ = 0 such that T (a)f+ = 0, T (a)g+ = 0. The F. and M. Riesz theorem 1.40(b) implies that ◦

f+ = 0 and g+ = 0 a.e. on T. Put f− := af+ and g− := ag+ . Then f− ∈H p , ◦

◦

◦

1 g− ∈H q , and so g− f+ ∈H 1 , g+ f− ∈H− . But g− f+ = ag+ f− , whence g+ f− = g− f+ = 0. Since f+ = 0 a.e. on T, we conclude that g− = 0 a.e. on T, and since g− = ag+ and g+ = 0 a.e. on T, it follows that a = 0 a.e. on T, which contradicts the hypothesis of the theorem.

(b) Let first 1 < p < ∞. Since the assertion is symmetric in p and q (recall 2.5(a), (b)), we may assume that 1 < p ≤ 2. Let T (a)ϕ+ = 0, T (a)ψ+ = 0, where ϕ+ ∈
p , ψ ∈
q , ϕ+ = 0, ψ+ = 0. Put ϕ− := M (a)ϕ+ and ψ− := M (a)ψ+ . Then ϕ− ∈
p (Z), ψ− ∈
q (Z), (ϕ− )n = (ψ− )n = 0 for all n ≥ 0. s (Z) (1 ≤ r ≤ ∞, 1/r + 1/s = 1) the convolution ϕ ∗ ψ For ϕ ∈
r (Z) and ψ ∈


defined by (ϕ ∗ ψ)i := j∈Z ϕi−j ψj belongs to
∞ (Z). For ϕ ∈
r (Z) define ϕ ∈
r (Z) by (ϕ)n = ϕn . Thus, we have

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2 Basic Theory

(ψ− ∗ ϕ+ )n = 0 ∀ n ≤ 0,

(ψ+ ∗ ϕ− )n = 0 ∀ n ≤ 0

and because ψ− ∗ ϕ+ = (M (a)ψ+ ) ∗ ϕ+ = ψ+ ∗ (M (a)ϕ+ ) = ψ+ ∗ ϕ− , it follows that (ψ− ∗ ϕ+ )n = (ψ+ ∗ ϕ− )n = 0 for n ∈ Z. Since ψ+ = 0, we have (ϕ− )n = 0 for all n ∈ Z. Thus, M (a)ϕ+ = 0, and since ϕ+ ∈
p (Z) ⊂
2 (Z), we deduce that af+ = 0 a.e. on T, where f+ ∈ H 2 is the function whose Fourier coefficients sequence is ϕ+ . The function f+ has a non-vanishing Fourier coefficient and therefore, by the F. and M. Riesz theorem 1.40(b), f+ = 0 a.e on T. This gives a = 0 a.e. on T and we arrived at a contradiction. Since M 1 is the Wiener algebra W , minor modifications of the proof also yield the result for p = 1. Notice that Toeplitz operators with symbols in W  are obviously bounded on
∞ .  Recall that, for a ∈ L1 , the function a is defined by a(t) = a(t) (t ∈ T). 2.39. Lemma. Let a ∈ L∞ , 1 < p < ∞, 1/p + 1/q = 1. Then T (a) is Fredholm (invertible ) on H p if and only if T (a) is Fredholm (invertible ) on H q . In the case of Fredholmness one has dim Ker T (a) = dim Coker T (a),

dim Ker T (a) = dim Coker T (a). ◦

Remark. Some care is in order, since the dual of H p is Lq / H q− and not H q . Nevertheless, all is easy. Proof. The hypothesis that a be in L∞ ensures that all operators occurring ◦

·

◦

·

p q + H p and Lq =H− + H q , we have are bounded. Since Lp =H−

T (a) ∈ Φ(H p ) ⇐⇒ P M (a)P + Q ∈ Φ(Lp ), T (a) ∈ Φ(H q ) ⇐⇒ P M (a)P + Q ∈ Φ(Lq ) and this is true with Φ replaced by GL. But (Lp )∗ is Lq and (P M (a)P + Q)∗ is easily seen to be P M (a)P + Q. This implies all assertions of the lemma.   2.40. Corollary. A Toeplitz operator is invertible if and only if it is Fredholm and has index zero. More explicitly: if a ∈ L∞ and 1 < p < ∞, then T (a) ∈ GL(H p ) ⇐⇒ T (a) ∈ Φ(H p ) and Ind T (a) = 0; if a ∈ M p and 1 ≤ p < ∞, then T (a) ∈ GL(
p ) ⇐⇒ T (a) ∈ Φ(
p ) and Ind T (a) = 0.

2.6 The Connection Between Fredholmness and Invertibility

73

Proof. The previous lemma and Theorem 2.38 imply that Ind T (a) = dim Ker T (a) − dim Ker T (a) = 0 if and only if dim Ker T (a) = dim Ker T (a) = 0.   2.41. The index of a continuous function. Let a ∈ C and suppose a has no zeros on T. Then there is a b ∈ CR(T◦ ) (recall 2.25) such that a = |a|e2πib . The increment of b as the result of a circuit around T counter-clockwise is an integer and depends only on a, i.e., it does not depend on the particular choice of b. This integer is referred to as the index (or winding number ) of a and is denoted by ind a. If a ∈ C has no zeros on T, then a/|a| is a continuous function belonging to CU (T◦ ). Therefore the limits lim (a/|a|)# (x) =: (a/|a|)# (±∞) exist, are x→±∞

finite, and its difference is an integral multiple of 2π. It is easily seen that ind a is nothing else than 1 1 0 (a/|a|)# (+∞) − (a/|a|)# (−∞) . 2π Note that ind χn = n, where χn (t) = tn (t ∈ T). Here are two important properties of the index. (a) If a, b ∈ C and a(t)b(t) = 0 for all t ∈ T, then ind (ab) = ind a + ind b. (b) If a, d ∈ C, a(t) = 0 for all t ∈ T, and d/a∞ < 1, then a(t)+d(t) = 0 for all t ∈ T and ind (a + d) = ind a. If a is continuously differentiable and does not vanish on T, then (  ( 2π  iθ 1 a (t) a (e ) iθ 1 dt = e dθ. ind a = 2πi T a(t) 2π 0 a(eiθ ) Thus, if a ∈ C has no zeros on T, then by the above property (b) and by 1.38(b), ( 1 (hr a) (t) ind a = lim ind hr a = lim dt. r→1−0 r→1−0 2πi T (hr a)(t) Also notice the following. (c) If a is a rational function without poles and zeros on the unit circle T, then ind a = z − p, where z and p are the numbers of zeros and poles (counted up to multiplicity) of a in D, respectively. In the language of Banach algebras we have the following. (d) If a ∈ GC, then ind a = 0 if and only if a belongs to the connected component of GC containing the identity. Given a Banach algebra A of continuous functions on T that contains the constants we shall say that the maximal ideal space of A is T if

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2 Basic Theory

(i) the general form of a multiplicative linear functional on A is given by ϕ : A → C, ϕ(a) = a(τ ), where τ ranges over T; (ii) the Gelfand topology on T coincides with the usual topology on T. The notion of the index allows us to specialize a result of Shilov as follows. (e) If A is a Banach algebra of continuous functions on T that contains the constants and whose maximal ideal is T, then every a ∈ GA of index zero has a logarithm log a ∈ A and, consequently, by 1.16(a), belongs to the connected component of GA containing the identity. We are now in a position to establish criteria for Fredholmness and invertibility of Toeplitz operators with continuous symbols on H p and
p . 2.42. Theorem. Let a ∈ C and 1 < p < ∞. Then (a) H(a) ∈ C∞ (H p ); (b) T (a) ∈ Φ(H p ) if and only if a(t) = 0 for all t ∈ T; if T (a) is Fredholm on H p , then T (a−1 ) is a regularizer of T (a) and Ind T (a) = −ind a; (c) T (a) ∈ GL(H p ) if and only if a(t) = 0 for all t ∈ T and ind a = 0. Proof. (a) There are an ∈ P (for example, the Fej´er means of a) such that a − an ∞ → 0 as n → ∞. Then H(an ) has finite rank and since H(a) − H(an )p = P M (a − an )QJp ≤ c2p a − an ∞ , H(a) is compact on H p . (b) The implication “=⇒” follows from Theorem 2.30. So suppose a = 0 on T. By Proposition 2.14,  −1 T (a−1 )T (a) = I − H(a−1 )H() (2.32) a), T (a)T (a−1 ) = I − H(a)H ) a and since, by (a), all Hankel operators occurring are compact, it follows that T (a) ∈ Φ(H p ) and that T (a−1 ) is a regularizer of T (a). So we are left with the index formula. Let T (a) ∈ Φ(H p ) and ind a = n. Then, by 2.41(a), ind (χ−n a) = 0. Hence, by 2.41(d), χ−n a belongs to the connected component of GC containing the identity. As T (f )L(H p ) ≤ cp f ∞ , the mapping T : GC → Φ(H p ), f → T (f ) is continuous. Consequently, the operator T (χ−n a) must be in the connected component of Φ(H p ) containing I and 1.12(d) gives Ind T (χ−n a) = 0. Because T (χ−n a) equals T (χ−n )T (a) or T (a)T (χ−n ), we deduce from Atkinson’s theorem 1.12(c) and from 2.9 that 0 = Ind T (χ−n a) = Ind T (χ−n ) + Ind T (a) = n + Ind T (a). (c) Immediate from (b) and Corollary 2.40.   Remark. Theorem 2.30 even implies that a(t) = 0 for all t ∈ T provided T (a) ∈ Φ+ (H p ) or T (a) ∈ Φ− (H p ).

2.6 The Connection Between Fredholmness and Invertibility

75

2.43. The classes Cp and M p . For 1 ≤ p < ∞, let Cp denote the closure in M p of the Laurent polynomials: Cp = closM p P. Clearly, C1 = W and C2 = C. Note that Cp is a closed subalgebra of M p . For p ∈ (1, 2) ∪ (2, ∞), let M p denote the collection of all functions a ∈ L∞ which belong to M p) for all p) in some neighborhood of p, i.e., $ (M p+ε ∩ M p−ε ). M p := ε>0

Finally, let M 1 = M 1 = W and M 2 = M 2 = L∞ . The following Proposition 2.45 intends to give an alternative description of Cp and to provide a better understanding of which functions belong to Cp . However, neither the definition of M p nor that proposition are needed to prove Propositions 2.46 and Theorem 2.47. 2.44. Lemma. Let a ∈ M p and let σn a denote the n-th Fej´er mean of a,  n   |j| (σn a)(t) = 1− a j tj n + 1 j=−n

(t ∈ T).

Then σn aM p ≤ aM p for all n ≥ 0. Proof. For θ ∈ (−π, π], let  sin2 (n + 1)θ/2 1 Kn (θ) = 2π(n + 1) sin2 (θ/2) denote the n-th Fej´er kernel and define the function ax by ax (eiθ ) := a(ei(θ−x) ). Thus, ( π

(σn a)(eiθ ) =

−π

ax (eiθ )Kn (x) dx.

(2.33)

It is easy to see that M (ax ) = D−x M (a)Dx , where Dx is the isometry Dx :
p →
p ,

{ϕj }j∈Z+ → {eijx ϕj }j∈Z+ .

Therefore, if ϕ ∈
0 (Z), then the function (−π, π] →
p ,

x → Kn (x)M (ax )ϕ

is continuous. This and (2.33) enable us to write M (σn a)ϕ as a Bochner integral: ( π M (ax )ϕKn (x) dx. M (σn a)ϕ = −π

Hence,

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2 Basic Theory

( M (σn a)ϕp ≤

π −π

M (ax )ϕp Kn (x) dx

≤ D−x p M (a)p Dx p ϕp

(

π −π

Kn (x) dx

= M (a)p ϕp ,  for all ϕ ∈ P, and consequently, M (σn a)p ≤ M (a)p .  2.45. Proposition. If 1 ≤ p < ∞, then Cp = closM p (C ∩ M p ). Proof. There is nothing to prove for p = 1 or p = 2. Thus let p ∈ (1, 2)∪(2, ∞). We first show that C ∩ M p ⊂ Cp . By virtue of 2.5(b), we may without loss of generality assume that 2 < p < ∞. Then a ∈ C ∩ M p+ε for some ε > 0. Let σn a denote the n-th Fej´er mean of a. From 2.5(e) we get γ M (a − σn a)p ≤ a − σn a1−γ ∞ M (a − σn a)p+ε ,

where 1/p = γ/(p + ε) + (1 − γ)/2. Since a ∈ C, we know that a − σn a∞ → 0 as n → ∞, and Lemma 2.44 applied to a ∈ M p+ε shows that M (a−σn a)p+ε remains bounded as n → ∞. Thus, the inclusion C ∩ M p ⊂ Cp is proved. Now it is easy to see that the asserted equality holds:  Cp = closM p P ⊂ closM p (C ∩ M p ) ⊂ closM p Cp = Cp .  2.46. Proposition. Let 1 ≤ p < ∞. (a) The maximal ideal space M (Cp ) of Cp is T. Thus, if a ∈ Cp , then a ∈ GCp ⇐⇒ a(t) = 0 ∀ t ∈ T.

(2.34)

(b) The connected component of GCp containing the identity coincides with the functions in GCp of index zero. Proof. (a) By virtue of 2.5(d), we have |a(τ )| ≤ a∞ aM p for all τ ∈ T and hence m : Cp → C, a → a(τ ) defines a multiplicative linear functional on Cp . Conversely, let m : Cp → C be a multiplicative linear functional. Put τ := m(χ1 ). It is easy to see that spM p (χ1 ) = T, whence τ ∈ T. This implies that m(f ) = f (τ ) for every f ∈ P and thus m(a) = a(τ ) for every a ∈ Cp , the closure of P. That the Gelfand topology on T coincides with the usual one on T can be checked in a standard way. (b) This follows from Shilov’s theorem 2.41(e). A proof which does not invoke that theorem is as follows. If a ∈ GCp has index zero, then, by 2.41(b), c ∈ GCp and ind c = 0 whenever c ∈ Cp and a − cM p is sufficiently small. Since P is dense in Cp , among these c’s there is a c ∈ P. Using 2.41(a) and 2.41(c) it is not difficult to see that c factors into a product of functions of the form χ1 − α and 1 − βχ−1 , where |α| > 1 and |β| < 1. But such functions are in the connected component  of GCp containing the identity, hence so is c and therefore a, too. 

2.7 Compactness of Hankel Operators and C + H ∞ Symbols

77

2.47. Theorem. Let a ∈ Cp and 1 ≤ p < ∞. Then (a) H(a) ∈ C∞ (
p ); (b) T (a) ∈ GL(
p ) if and only if a(t) = 0 for all t ∈ T; if T (a) is Fredholm on
p , then T (a−1 ) is a regularizer of T (a) and Ind T (a) = −ind a; (c) T (a) ∈ GL(
p ) if and only if a(t) = 0 for all t ∈ T and ind a = 0. Proof. (a) By the definition of Cp , there are an ∈ P such that H(a) − H(an )p ≤ a − an M p = o(1)

as n → ∞,

which gives the compactness of H(a) on
p . (b) The implication “=⇒” follows from Theorem 2.30. Taking into account (2.34), one can obtain the implication “⇐=” as in the proof of Theorem 2.42. Proposition 2.46(b) shows that the argument applied in the proof of Theorem 2.42 can be used to verify the index formula in the case at hand, too. (c) This is immediate from (b) and Corollary 2.40.   Remark 1. The index formula can also be proved with the help of the argument that will be used in the proof of Theorem 2.66 below. Remark 2. One can show, e.g., as in the proof of 2.29(b) or by using a perturbation argument (see the proof of Theorem 2.74 below), that a ∈ GCp whenever a ∈ Cp and T (a) ∈ Φ+ (
p ) or T (a) ∈ Φ− (
p ).

2.7 Compactness of Hankel Operators and C + H ∞ Symbols 2.48. Definition. For 1 < p < ∞, let
 Ap := a ∈ L∞ : H(a) ∈ C∞ (H p ) ,


 Bp := a ∈ M p : H(a) ∈ C∞ (
p ) .

2.49. Lemma. Ap and Bp are closed subalgebras of L∞ and M p , respectively. Proof. It is clear that Ap and Bp are linear spaces. From Proposition 2.14 we have H(ab) = T (a)H(b) + H(a)T ()b), which shows that ab ∈ Ap resp. ab ∈ Bp whenever a, b ∈ Ap resp. a, b ∈ Bp . Thus Ap and Bp are algebras. If an ∈ Bp , b ∈ M p , an − bp → 0 as n → ∞, then H(b) − H(an )L(p ) ≤ b − an p = o(1) as n → ∞, whence H(b) ∈ C∞ (
p ). This shows that Bp is closed. The closedness of Ap can be proved analogously.   2.50. Theorem. (a) If a ∈ Ap , then T (a) ∈ Φ(H p ) ⇐⇒ a ∈ GAp . (b) If a ∈ Bp , then T (a) ∈ Φ(
p ) ⇐⇒ a ∈ GBp .

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Proof. (a) Let a ∈ Ap . Then a, a−1 ∈ Ap and, by (2.18), T (a−1 )T (a) − I = −H(a−1 )H() a) ∈ C∞ (H p ),  T (a)T (a−1 ) − I = −H(a)H ) a−1 ) ∈ C∞ (H p ), whence T (a) ∈ Φ(H p ). Conversely, suppose T (a) ∈ Φ(H p ). From Theorem 2.30 we know that then a ∈ GL∞ and it remains to show that a−1 ∈ Ap . Let RT (a) = I + K, where R ∈ L(H p ) is a regularizer of T (a) and K ∈ C∞ (H p ). Formula (2.19) gives  −1 0 = H(aa−1 ) = T (a)H(a−1 ) + H(a)T ) , a hence, by acting with R from the left,  −1 0 = H(a−1 ) + KH(a−1 ) + RH(a)T ) , a and it follows that H(a−1 ) ∈ C∞ (H p ), as desired. (b) The proof is the same.   2.51. Definition. For 1 < p < ∞, define Cp as in 2.43 and let Hp∞ := H ∞ ∩M p . It is obvious that Hp∞ is a closed subalgebra of M p . Let alg (Cp , Hp∞ ) denote the smallest closed subalgebra of M p containing Cp and Hp∞ . It is clear that alg (Cp , Hp∞ ) coincides with alg (χ1 , Hp∞ ), the smallest closed subalgebra of M p containing the set Hp∞ and the function χ1 . The discontinuous function (1 − χ1 )iβ (β ∈ R \ {0}) can be shown to belong to Hp∞ for all 1 < p < ∞ (Theorem 6.45). This shows that alg (Cp , Hp∞ ) is strictly larger than Cp . Of course, alg (Cp , Hp∞ ) contains 
Cp + Hp∞ = f + g : f ∈ Cp , g ∈ Hp∞ . It is an absolutely unexpected fact, the discovery of which goes back to Sarason, that alg (Cp , Hp∞ ) is actually equal to Cp + Hp∞ . Our next goal is to prove this. The proof will be based on the following interesting lemma. 2.52. Lemma (Zalcman/Rudin). Let X be a Banach space and let E and F be closed subspaces of X. Suppose {Sn }n∈Z+ is a sequence of operators Sn ∈ L(X) with the following properties: (i) Sn L(X) ≤ M for all n ∈ Z+ ; (ii) Sn (X) ⊂ E for all n ∈ Z+ ; (iii) Sn (F ) ⊂ F for all n ∈ Z+ ; (iv) Sn u − u → 0 as n → ∞ for all u ∈ E. Then E + F is a closed subspace of X.

2.7 Compactness of Hankel Operators and C + H ∞ Symbols

79

Proof. Let x ∈ closX (E + F ). Then

there are uk ∈ E, vk ∈ F such that ∞ uk + vk  ≤ 1/2k for k ≥ 2 and x = k=1 (uk + vk ). For each k ≥ 2 choose k nk so that uk − Snk uk  ≤ 1/2 . If we let xk := uk + vk , then obviously xk = (uk − Snk uk + Snk xk ) + (vk − Snk vk ). We have u )k := uk − Snk uk + Snk xk ∈ E and ) uk  ≤

1 1+M + Snk xk  ≤ k 2 2k

(note that xk  ≤ 1/2k ). Furthermore, v)k := vk − Snk vk ∈ F and ) vk  ≤ ) uk  + xk  ≤

2+M . 2k

∞

∞ )k and k=1 v)k are absolutely convergent. Let Consequently, the series k=1 u u ∈ E and v ∈ F denote their sums. What results is that x=

∞  k=1

xk =

∞ 

() uk + v)k ) = u + v ∈ E + F.

 

k=1

2.53. Theorem. Cp + Hp∞ is a closed subalgebra of M p and Cp + Hp∞ = alg (Cp , Hp∞ ) = alg (χ1 , Hp∞ ).

(2.35)

Proof. We apply the preceding lemma with X = M p , E = Cp , F = Hp∞ , and Sn ∈ L(M p ) given by Sn a = σn a, where σn a denotes the n-th Fej´er mean of a. It is clear that (ii) and (iii) are satisfied. Lemma 2.44 shows that (i) is fulfilled. Finally, given a ∈ Cp and ε > 0 choose f ∈ P so that a − f M p < ε/3. Then σn a − aM p ≤ σn (a − f )M p + σn f − f M p + f − aM p ≤ a − f M p + σn f − f W + f − aM p 2ε + σn f − f W < 3 (Lemma 2.44 and 2.5(d)) and σn f − f W < ε/3 whenever n is large enough. Thus, the requirement (iv) is also met and it follows that Cp + Hp∞ is closed. Now let a, b ∈ Cp + Hp∞ . There are an , bn ∈ P + Hp∞ such that a − an M p → 0,

b − bn M p → 0 as

n → ∞.

It is obvious that an bn ∈ P +Hp∞ , and since an bn −abM p → 0 as n → ∞ and Cp + Hp∞ is closed, we conclude that ab ∈ Cp + Hp∞ . Consequently, Cp + Hp∞ is an algebra. Once we know that Cp + Hp∞ is a closed subalgebra of M p , the equalities (2.35) are obvious.  

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2 Basic Theory

2.54. Theorem (Hartman/Adamyan/Arov/Krein). Let a ∈ L∞ and 1 < p < ∞. Then distL∞ (a, C + H ∞ ) ≤ H(a)Φ(H p ) ≤ cp distL∞ (a, C + H ∞ ), where cp = P L(Lp ) . In particular, Ap = C + H ∞ , that is, H(a) ∈ C∞ (H p ) ⇐⇒ a ∈ C + H ∞ . Proof. We have 
distL∞ (a, C + H ∞ ) = inf a − f − h∞ : f ∈ C, h ∈ H ∞ = inf

inf a − f − h∞ = inf distL∞ (a − f, H ∞ )

f ∈C h∈H ∞

c∈C

≥

1 inf H(a − f )L(H p ) cp c∈C

=

1 inf H(a) − H(f )L(H p ) cp f ∈C

=

1 H(a)Φ(H p ) cp

(Theorem 2.11)

(Theorem 2.42(a)).

Now let V = T (χ1 ). Since (V n )∗ = T (χ−n ) converges strongly to zero on H p as n → ∞, we conclude that KV n  → 0 as n → ∞ for every K ∈ C∞ (H p ) (see 1.3(d)). Thus, if K ∈ C∞ (H p ) then H(a) − K ≥ (H(a) − K)V n  ≥ H(a)V n  − KV n  = H(χ−n a) − KV n  ≥ distL∞ (χ−n a, H ∞ ) − KV n  (Theorem 2.11) = distL∞ (a, χn H ∞ ) − KV n  = distL∞ (a, C + H ∞ ) − KV n , whence H(a)Φ(H p ) ≥ distL∞ (a, C + H ∞ ). Since C + H ∞ is closed, we have distL∞ (a, C + H ∞ ) = 0 if and only if  a ∈ C + H ∞.  Remark. The above theorem implies the following compactness criterion for Hankel operators: if a ∈ L∞ and 1 < p < ∞, then H(a) ∈ C∞ (H p ) if and only if P a ∈ V M O. Indeed, if a ∈ C + H ∞ , then P a ∈ P (C) ⊂ V M O by 1.48(k), and if P a ∈ V M O, then P a = u + P v with u, v ∈ C by virtue of 1.48(l), which shows that a = u + v + Q(a − v) is in C + H ∞ . 2.55. Corollary. If a ∈ C + H ∞ , then T (a) ∈ Φ(H p ) ⇐⇒ a ∈ G(C + H ∞ ). If T (a) ∈ Φ(H p ), then T (a−1 ) is a regularizer of T (a).

2.7 Compactness of Hankel Operators and C + H ∞ Symbols

81

Proof. Immediate from Theorems 2.50 and 2.54 and formula (2.18).   2.56. Open problem. Establish the analogue of Theorem 2.54 for
p . In this connection recall 2.12. It is clear that Cp + Hp∞ ⊂ Bp , but we have not been able to prove that Bp ⊂ Cp + Hp∞ . Nevertheless we shall show that the
p version of Corollary 2.55 holds (see Theorem 2.60 below). 2.57. Definition. Put 
R = p/q : p ∈ PA , q ∈ PA , q(t) = 0 ∀t ∈ T . Note that R is the restriction to the unit circle T of the set of all rational functions defined on the whole plane C and having no poles on T. 2.58. Theorem (Kronecker). (a) Let 1 < p < ∞ and a ∈ L∞ . Then H(a) ∈ C0 (H p ) ⇐⇒ a ∈ R + H ∞ . (b) Let 1 < p < ∞ and a ∈ M p . Then H(a) ∈ C0 (
p ) ⇐⇒ a ∈ R + Hp∞ . Proof. We first prove that the operator H(a) has finite rank for a ∈ R (and thus for a ∈ R+H ∞ resp. a ∈ R+Hp∞ ). This is obvious if a is a polynomial. If a = 1/(χ1 −λ) with some λ ∈ C, then H(a) = 0 for |λ| < 1 and rank H(a) = 1 for |λ| > 1. So application of the formula H(bc) = T (b)H(c) + H(b)T () c) shows that H(a) has finite rank if a is of the form polynomial /(χ1 − λ)n (n ∈ Z+ , λ ∈ C), and decomposition into partial fractions gives the assertion for all a ∈ R. Now suppose rank H(a) = r < ∞. This implies that the first r + 1 columns of the matrix (aj+k+1 )∞ j,k=0 are linearly dependent, where an denotes the n-th Fourier coefficient of a. Hence, if we let b :=

∞ 

ar+k+1 χk

(∈ H 2 ),

k=0

then there exist complex numbers λ0 , λ1 , . . . , λr such that at least one of them is non-zero and λ0 (a1 + a2 χ1 + . . . + ar χr−1 + χr b) +λ1 (a2 + a3 χ1 + . . . + ar χr−2 + χr−1 b) + . . . + λr b = 0. It follows that (λ0 χr + λ1 χr−1 + . . . + λr )b is a polynomial and therefore b must be a rational function. Since b ∈ H 2 , b cannot have poles on T, whence b ∈ R. If a ∈ L∞ resp. a ∈ M p , then χ−r−1 a − b belongs to H ∞ resp.  M p ∩ H ∞ = Hp∞ , which shows that a ∈ R + H ∞ resp. a ∈ R + Hp∞ . 

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2 Basic Theory

2.59. Corollary. (a) R + Hp∞ is an algebra. (b) If a ∈ R + H ∞ , then T (a) ∈ Φ(H p ) ⇐⇒ a ∈ GL∞ and a−1 ∈ R + H ∞ . (c) If a ∈ R + Hp∞ , then T (a) ∈ Φ(
p ) ⇐⇒ a ∈ GL∞ and a−1 ∈ R + Hp∞ . Proof. (a) This follows from Theorem 2.58 together with the formula (2.19). (b), (c) Combine the reasoning in the proof of Theorem 2.50 with Theorem 2.58 (also take into account 1.12(a), (iii)).   2.60. Theorem. Let 1 < p < ∞ and a ∈ Cp + Hp∞ . Then T (a) ∈ Φ(
p ) ⇐⇒ a ∈ G(Cp + Hp∞ ). If T (a) ∈ Φ(
p ), then T (a−1 ) is a regularizer of T (a). Proof. If a and a−1 are in Cp + Hp∞ , then H(a) and H(a−1 ) are compact, and so the argument used in the proof of Theorem 2.50 gives the implication “⇐=” and shows that T (a−1 ) is a regularizer of T (a). Now suppose T (a) ∈ Φ(
p ). Then, by Theorem 2.30(b), a ∈ GM p and it remains to prove that a−1 ∈ Cp + Hp∞ . By the definition of Cp , there are functions bn ∈ R+Hp∞ such that a−bn M p → 0 as n → ∞. If n is sufficiently large, then T (bn ) ∈ Φ(
p ) (by 1.12(d)) and so Corollary 2.59 implies that p p ∞ b−1 n ∈ R+Hp . But if bn converges to an element a ∈ GM in the norm of M , −1 −1 p −1 then bn converges to a in the norm of M . Hence a ∈ closM p (R + Hp∞ ) and since Cp + Hp∞ is closed, it follows that a−1 ∈ Cp + Hp∞ .   Our next concern is an invertibility criterion for C + H ∞ . We first need a property of the harmonic extension. Recall that, for f ∈ L∞ and 0 < r < 1, the function fr ∈ C is defined by fr (eiθ ) := f(reiθ ). 2.61. Lemma. If c ∈ C and a ∈ L∞ , then (ca)r − cr ar ∞ → 0 as r → 1 − 0. Proof. Since cr − c∞ → 0 it suffices to show that (ca)r − car ∞ → 0. Let kr denote the Poisson kernel (see 1.37). Then ( 2π 1 (ca)r (eiθ ) − c(eiθ )ar (eiθ ) = [c(eit ) − c(eiθ )]a(eit )kr (θ − t) dt. 2π 0 Given ε > 0 there is a δ > 0 such that |c(eit ) − c(eiθ )| < ε whenever |t − θ| < δ and so + +( ( 2π + + + + it iθ it ≤ ε [c(e ) − c(e )]a(e )k (θ − t) dt |a(eit )|kr (θ − t) dt + + r + + |t−θ|<δ 0 ≤ 2πεa∞ .

2.7 Compactness of Hankel Operators and C + H ∞ Symbols

83

If r is sufficiently close to 1, then +( + ( + + + + it iθ it [c(e ) − c(e )]a(e )kr (θ − t) dt+ ≤ 2c∞ a∞ kr (τ ) dτ + + |t−θ|>δ + |τ |>δ ≤ 2c∞ a∞ ε. This completes the proof.

 

2.62. Theorem (Douglas). (a) If a, b ∈ C + H ∞ then (ab)r − ar br ∞ → 0 as

r → 1 − 0.

a is bounded (b) Let a ∈ C + H ∞ . Then a ∈ G(C + H ∞ ) if and only if  away from zero in some annulus near T, i.e., if and only if there exist δ > 0 and ε > 0 such that | a(z)| > ε for 1 − δ < |z| < 1. Proof. (a) Let a = c + h, b = d + g, where c, d ∈ C and h, g ∈ H ∞ . It is clear that (hg)r = hr gr . From the preceding lemma we deduce that lim (cd)r −cr dr ∞ = lim (cg)r −cr gr ∞ = lim (hd)r −hr dr ∞ = 0,

r→1−0

r→1−0

r→1−0

and this gives the assertion at once. (b) Let a ∈ G(C + H ∞ ) and b = a−1 . Then, by (a), ar br − 1∞ → 0 as r → 1 − 0, and since br ∞ is bounded from above (by b∞ ), it follows that |ar | must be bounded away from zero if r is close enough to 1. a(z)| > ε for 1 − δ < |z| < 1. Then Now let a ∈ C + H ∞ and assume | |a| ≥ ε a.e. on T and so a ∈ GL∞ . Because a ∈ C + H ∞ , there are hn ∈ H ∞ such that χ−n hn → a as n → ∞ in the norm of L∞ . Part (a) and the fact 2n is bounded that  a is bounded away from zero near T imply that each h away from zero in some annulus 1 − δn < |z| < 1. So 1/hn is bounded and 2n can there be written as analytic in 1 − δn < |z| < 1. Consequently, 1/h the sum of a function which extends to be bounded and analytic in D and a function which extends to be bounded and analytic in |z| > 1 − δn . This decomposition yields a representation of 1/hn as the sum of a function in H ∞ ∞ ∞ and a function in C. Thus h−1 and therefore χn h−1 n ∈ C +H n ∈ C +H . ∞ ∞ −1 −1 But if χ−n hn → a ∈ GL in the norm of L , then χn hn → a in the  L∞ -norm. Since C + H ∞ is closed, it follows that a−1 ∈ C + H ∞ .  We finally compute the index of Toeplitz operators with C + H ∞ and Cp + Hp∞ symbols. By virtue of Corollary 2.40 this solves the invertibility problem for these operators. 2.63. Definition. Let a ∈ L∞ and suppose  a is bounded away from zero in some annulus near T. Then there are ε > 0 and δ > 0 such that |ar (eiθ )| ≥ ε for all r ∈ (1 − δ, 1) and θ ∈ [0, 2π). By 2.41(b), the mapping (1 − δ, 1) → Z, r → ind ar is continuous, and because (1 − δ, 1) is connected, ind ar must be constant for r ∈ (1 − δ, 1). That constant value of ind ar will be denoted by ind {ar }.

84

2 Basic Theory

2.64. Theorem. Let h ∈ H ∞ and 1 < p < ∞. Then T (h) ∈ Φ(H p ) if and only if h = bg where b is a finite Blaschke product and g ∈ GH ∞ . If T (h) is Fredholm, then h is bounded away from zero in some annulus near T and Ind T (h) = −ind {hr }. Proof. If h = bg with b ∈ GC and g ∈ GH ∞ , then T (h) = T (b)T (g) ∈ Φ(H p ) by Theorem 2.42 and Proposition 2.31. Conversely, suppose T (h) ∈ Φ(H p ). Then h ∈ G(C + H ∞ ) due to Corollary 2.55. In view of 1.41(a), 1.41(b), we have h = bSg, where b is a Blaschke product, S is a singular inner function, and g ∈ GH ∞ . Thus bS ∈ G(C + H ∞ ) and by virtue of Theorem 2.62(b), b(z)S(z)  must be bounded away from zero in some annulus near T. Since  the radial limit of S(z) vanishes at the points in the support of the singular measure defining S, it follows that S = 1, and b(z) is bounded away from zero in an annulus near T only if b is a finite Blaschke product. The index formula can be derived as follows: Ind T (h) = Ind T (b) + Ind T (g) (Atkinson) = Ind T (b) (Proposition 2.31) = −ind b (Theorem 2.42) = −ind {br } (1.38(b) and 2.41(b)) = −ind {br } − ind {gr } (1.41(g)) = −ind {(bg)r } (Lemma 2.61 and 2.41(b)).

 

2.65. Theorem (Douglas). Let a ∈ C + H ∞ and 1 < p < ∞. Then T (a) is a is bounded away from zero in some annulus near T, in Φ(H p ) if and only if  and in that case Ind T (a) = −ind {ar }. Proof. The Fredholm criterion follows by combining Corollary 2.55 and Theorem 2.62(b). So it remains to prove the index formula. There is a number ε > 0 with the following property: if b − a∞ < ε, then T (b) ∈ Φ(H p ), Ind T (b) = Ind T (a), b is bounded away from zero in some annulus near T, and ind {br } = ind {ar } (recall 1.12(d), 1.38(b), 2.41(b)). Among these b’s we can find a b ∈ C + H ∞ of the form b = χ−n h, n ∈ Z+ , h ∈ H ∞ . Because T (b) = T (χ−n )T (h) ∈ Φ(H p ), it follows that T (h) ∈ Φ(H p ) and the preceding theorem gives (2.36) Ind T (h) = −ind {hr }. The desired index formula can now be verified as follows: Ind T (a) = Ind T (χ−n h) = Ind T (χ−n ) + Ind T (h) (Atkinson) = n + Ind T (h) (2.9) = n − ind {hr } (equality (2.36)) = −ind {(χ−n )r } − ind {hr }

2.8 Local Methods for Scalar Toeplitz Operators

85

= −ind {(χ−n )r hr } (2.41(a)) = −ind {(χ−n h)r } (Lemma 2.61 and 2.41(b)) = −ind {ar }.   a is 2.66. Theorem. Let a ∈ Cp + Hp∞ and 1 < p < ∞. If T (a) ∈ Φ(
p ), then  bounded away from zero in some annulus near T and Ind T (a) = −ind {ar }. Proof. Suppose 1 < p < 2 and let 1/p + 1/q = 1. From Theorem 2.60 we know that a ∈ G(Cp + Hp∞ ). Hence, due to 2.5(c), (d), a ∈ G(Cs + Hs∞ ) for all s ∈ [p, q] and thus, again by Theorem 2.60, T (a) ∈ Φ(
s ) for all s ∈ [p, q]. Given an operator A ∈ Φ(
s ) denote by αs (A) the dimension of the kernel of A in
s and by Inds A the index of A considered as operator on
s . Since
p ⊂
2 ⊂
q , we have αp (T (a)) ≤ α2 (T (a)),

α2 (T (a)) ≤ αq (T (a)),

hence Indp T (a) = αp (T (a)) − αq (T (a)) ≤ α2 (T (a)) − α2 (T (a)) = Ind2 T (a) = −ind {ar }. On repeating this argument with a−1 in place of a we arrive at the inequality Indp T (a−1 ) ≤ −ind {(a−1 )r } = ind {ar } (recall Theorem 2.62(a)). But T (a−1 ) is a regularizer of T (a). So Indp T (a) = −Indp T (a−1 ) ≥ −ind {ar }, and we finally get Indp T (a) = −ind {ar }. If 2 < p < ∞, then 1 < q < 2 and from the equality Indp T (a) = −Indq T (a) = ind {ar } = −ind {ar } we get the desired formula.

 

2.8 Local Methods for Scalar Toeplitz Operators 2.67. The local distance at a point. Given a ∈ L∞ and an open subarc U of T denote by a|U the restriction of a to U regarded as an element of L∞ (U ). For τ ∈ T, let Uτ denote the family of all open subarcs of T containing the point τ . The local distance of a, b ∈ L∞ at τ ∈ T is defined as distτ (a, b) = inf a|U − b|U L∞ (U ) . U ⊂Uτ

Clearly, if a|U = b|U for some neighborhood U of τ , then distτ (a, b) = 0. If the finite limits a(τ ±0) exist, then distτ (a, b) = 0 if and only if a(τ −0) = b(τ −0)

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and a(τ + 0) = b(τ + 0). In particular, if a and b are continuous at τ , then distτ (a, b) = 0 if and only if a(τ ) = b(τ ). Let Rτ denote the collection of all functions f ∈ C such that 0 ≤ f ≤ 1 on T and f is identically 1 in some neighborhood of τ (depending on f ). It is not difficult to verify that distτ (a, b) = inf (a − b)c∞ . c∈Rτ

(2.37)

2.68. Theorem. Let 1 < p < ∞ and a ∈ L∞ . Assume that for each τ ∈ T there is an aτ ∈ L∞ such that distτ (a, aτ ) = 0 and T (aτ ) ∈ Φ(H p ). Then T (a) ∈ Φ(H p ). Proof. This theorem provides a good occasion of giving an application of Theorem 1.32. Put A = L(H p )/C∞ (H p ) and, for a ∈ L∞ , let T π (a) denote the coset in A containing T (a). If f ∈ C and b ∈ L∞ , then, by Proposition 2.14, T (f )T (b) = T (f b) − H(f )H()b) = T (b)T (f ) + H(b)H(f)) − H(f )H()b) and since H(f ) and H(f)) are compact on H p , we get T π (f )T π (b) = T π (f b) = T π (b)T π (f ).

(2.38)

For τ ∈ T, define Rτ as in 2.67 and put 
Mπτ := T π (f ) ∈ A : f ∈ Rτ . Using (2.38) it is easy to see that Mπτ is a localizing class in A. Given a of T we can choose a family {T π (fτ )}τ ∈T (fτ ∈ Rτ ), due to the compactness

finite subfamily {T π (fτj )}nj=1 such that g := j fτj ≥ 1 and Theorem 2.42(b) shows that T π (g) is invertible in A. Hence, {Mπτ }τ ∈T is a covering system of localizing classes in A. Also from (2.38) we deduce that T π (a) commutes with every T π (f ) in the union of all Mπτ . We finally have, again making use of (2.38), " " "  " inf " T π (a) − T π (aτ ) T π (f )"A = inf "T π (a − aτ )f "A f ∈Rτ f ∈Rτ "  " ≤ inf "T (a − aτ )f "L(H p ) ≤ cp inf (a − aτ )f ∞ = 0 f ∈Rτ

and analogously

f ∈Rτ

"  " inf "T π (f ) T π (a) − T π (aτ ) "A = 0.

f ∈Rτ

In other words, T π (a) and T π (aτ ) are Mπτ -equivalent from the left and from the right at each τ ∈ T. Since T π (aτ ) is invertible in A, it is of course Mπτ invertible from the left and from the right. Thus, we have collected together all the things allowing us to apply Theorem 1.32. The conclusion is that T π (a) is invertible in A and this yields the assertion.  

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87

2.69. Theorem. Let 1 ≤ p < ∞ and a ∈ M p . Suppose for each τ ∈ T there exists an aτ ∈ M p such that distτ (a, aτ ) = 0 and T (aτ ) ∈ Φ(
p ). Then T (a) ∈ Φ(
p ). Proof. Since M 1 = M 1 = W , the case p = 1 is covered by Theorem 2.47. The case p = 2 is contained in the preceding theorem. Thus let 1 < p < 2; for 2 < p < ∞ the assertion can be proved analogously or can be obtained immediately from the case 1 < p < 2 by taking adjoints. Put A = L(
p )/C∞ (
p ) and for a ∈ M p let T π (a) := T (a) + C∞ (
p ). It is clear that (2.38) holds for every b ∈ M p and every f ∈ Cp . Now, for τ = eiθ0 ∈ T, let
Rτ := f ∈ C ∞ : 0 ≤ f ≤ 1, there is an ε > 0 (depending on f ) such that f (eiθ ) = 1 for |θ − θ0 | < ε, f (eiθ ) = 0 for |θ − θ0 | > 2ε, f is monotonically increasing for θ0 − 2ε < θ < θ0 − ε,  f is monotonically decreasing for θ0 + ε < θ < θ0 + 2ε , 
Mπτ := T π (f ) ∈ A : f ∈ Rτ . As in the preceding proof it is readily seen that {Mπτ }τ ∈T is a covering system of localizing classes in A (note that C ∞ ∈ M p ) and that T π (a)T π (f ) = T π (f )T π (a) for every f ∈ Rτ . Since 1 < p < 2 and a, aτ ∈ M p , there is an r = rτ ∈ (1, p) such that a, aτ ∈ M r . Hence, " " "  " inf " T π (a) − T π (aτ ) T π (f )" = inf "T π (a − aτ )f " A

f ∈Rτ

≤ inf (a − aτ )f M p ≤ inf (a − aτ )f 1−γ ∞ (a − f ∈Rτ

f ∈Rτ

A

f ∈Rτ

aτ )f γM r ,

where γ = r|p−2|/(p|r−2|) and the last estimate results from 2.5(e). To prove that T π (a) is Mπτ -equivalent from the left to T π (aτ ) it therefore remains to show that (a − aτ )f M r is bounded by a constant, K, as f varies over Rτ . But from 2.5(f) we obtain (a − aτ )f M r ≤ a − aτ M r f M r   ≤ aM r + aτ M r sp f ∞ + V1 (f )  ≤ aM r + aτ M r sp · 3 =: K, since f ∞ = 1 and V1 (f ) = 2 for every f ∈ Rτ . Theorem 1.32 now completes the proof.   2.70. Remark. Let X be a Banach space and A ∈ L(X). Then A is said to be left-Fredholm (resp right-Fredholm) if A + C∞ (X) is left-invertible (resp. right-invertible) in L(X)/C∞ (X). A look at Theorem 1.32 shows that in the preceding two theorems the requirement that T (aτ ) be Fredholm for each τ ∈ T can be replaced by the requirement that T (aτ ) be left-Fredholm (resp. right-Fredholm) in order to deduce that T (a) be left-Fredholm (resp. right-Fredholm).

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The relation between left- and right-Fredholm operators and Φ± -operators is clarified by the following result of Yood [584]. (a) A ∈ L(X) is left-Fredholm if and only if A ∈ Φ+ (X) and if Im A is a complemented subspace of X. (b) A ∈ L(X) is right-Fredholm if and only if A ∈ Φ− (X) and if Ker A is a complemented subspace of X. Thus, in general, a Φ+ (Φ− )-operator need not be left-Fredholm (rightFredholm). However, for a bounded Hilbert space operator to be a Φ+ (Φ− )operator is equivalent to being left-Fredholm (right-Fredholm). 2.71. Definition. A function a ∈ L∞ is called sectorial on a subarc U of T if there is an ε > 0 and a c ∈ C of modulus 1 such that Re (ca) ≥ ε a.e. on U . A function a ∈ L∞ is said to be locally arcwise sectorial if for each τ ∈ T there is a subarc Uτ ∈ Uτ such that a is sectorial on Uτ . Since T is compact, a function a ∈ L∞ is locally arcwise sectorial if and only if T can be covered by a finite number of open subarcs Ui such that a is sectorial on each Ui . 2.72. Theorem. If a ∈ L∞ is locally arcwise sectorial then T (a) is Fredholm on H 2 . Proof. Immediate from Theorems 2.68 and 2.17.   Remark. The index of a Toeplitz operator generated by a locally arcwise sectorial symbol is, loosely speaking, minus the winding number of the “sectorial cloud” associated with the symbol. We do not make precise what this means, but shall later provide another way of computing the index, namely, via the harmonic extension of the symbol. 2.73. The algebra P C. Let P C0 denote the collection of all piecewise continuous functions on T which have at most finitely many jumps. The closure of P C0 in L∞ is denoted by P C. A function a ∈ P C possesses finite limits a(t ± 0) everywhere on T and there are at most countably many t ∈ T such that a(t − 0) = a(t + 0). Note that P C is a C ∗ -algebra of L∞ . Given a ∈ P C0 define a function a2 : T × [0, 1] → C by the formula a2 (t, µ) = (1 − µ)a(t − 0) + µa(t + 0)

(t ∈ T,

µ ∈ [0, 1]).

(2.39)

The range of a2 is a continuous closed curve with a natural orientation; it is obtained from the (essential) range of a by filling in the straight line segment [a(t − 0), a(t + 0)] for each t ∈ T at which a has a jump. If this curve does not pass through the origin, we let ind a2 denote its winding number with respect to the origin. A more precise definition of ind a2 is as follows. With each finite subset S of T we associate a function ωS : T → T × [0, 1]. To construct this function let S = {eiθ1 , . . . , eiθR }, 0 ≤ θ1 < . . . < θR < 2π, put θR+1 := θR + 2π, and for j = 1, . . . , R let

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89

θj + θj+1 θj + ϕj ϕj + θj+1 , ϕj = , ϕj = . 2 2 2 Then define, for j = 1, . . . , R and 0 ≤ λ ≤ 1,   1 i(θj +λ(ϕj −θj )) iθj 1 )= e , + λ , ωS (e 2 2 ϕj =







ωS (ei(ϕj +λ(ϕj −ϕj )) ) = (ei(θj +λ(θj+1 −θj )) , 1 − λ),   i(ϕ +λ(θj+1 −ϕ )) iθj+1 1 j j ωS (e )= e , λ . 2 Given a ∈ P C0 denote by S(a) the finite subset of T formed by the points at which a has a jump. It is easily seen that a2 (t, µ) = 0 for all (t, µ) ∈ T×[0, 1] if and only if the continuous function a2 ◦ωS(a) : T → C does not vanish on T. In that case ind a2 is defined as ind (a2 ◦ ωS(a) ), where the latter “ind ” refers to the index as it was defined in 2.41. For a ∈ P C define a2 : T × [0, 1] → C again by (2.39). It is not difficult to see that the origin belongs to the range of a2 if and only if there is a sequence of functions an ∈ P C0 such that a−an ∞ → 0 and dist(0, range(an )2 ) → 0 as n → ∞. If a2 (t, µ) = 0 for all (t, µ) ∈ T × [0, 1], we choose any sequence of functions an ∈ P C0 with a − an ∞ → 0 as n → ∞ and define ind a2 as lim ind (an )2 . It can be easily seen that this limit always n→∞ exists and that it does not depend on the particular choice of the sequence {an }. 2.74. Theorem. Let a ∈ P C. Then T (a) ∈ Φ(H 2 ) ⇐⇒ a2 (t, µ) = 0

∀ (t, µ) ∈ T × [0, 1].

If T (a) is Fredholm, then Ind T (a) = −ind a2 . Proof. If a2 (t, µ) = 0 for all (t, µ) ∈ T×[0, 1], then a is locally arcwise sectorial and therefore T (a) ∈ Φ(H 2 ) due to Theorem 2.72. Our next objective is to prove the index formula. Thus, let a ∈ P C and a2 (t, µ) = 0 for all (t, µ) ∈ T × [0, 1]. If b ∈ P C0 is sufficiently close to a in the L∞ norm, then Ind T (a) = Ind T (b), b2 (t, µ) = 0 for all (t, µ) ∈ T × [0, 1] and ind a2 = ind b2 (the latter fact per definitionem!). So it remains to show that Ind T (b) = −ind b2 . Let t1 , . . . , tn denote the points on T at which b has jumps. Choose sufficiently small neighborhoods U1 , . . . , Un ⊂ T of the points t1 , . . . , tn and put U = U1 ∪ . . . ∪ Un . Then define c ∈ C as follows: let c = b on T \ U and on Ui let c be any continuous function such that c(Ui ) = b(Ui ) ∪ [b(ti − 0), b(ti + 0)]. The function d = b/c equals 1 on T \ U and it is easy to see that d is sectorial on U if only the neighborhoods U1 , . . . , Un have been chosen sufficiently small. Thus, by (2.18), ) T (b) = T (cd) = T (c)T (d) + H(c)H(d) with H(c) compact (Theorem 2.42(a)) and T (d) invertible (Theorem 2.17). It follows that Ind T (b) = Ind T (c), and since ind c = ind b2 , Theorem 2.42(b) completes the proof of the index formula.

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We now prove the implication “=⇒”. Let T (a) ∈ Φ(H 2 ) but assume there is a (t0 , µ0 ) ∈ T × [0, 1] such that a2 (t0 , µ0 ) = 0. Then T (b) ∈ Φ(H 2 ) whenever a − b∞ is sufficiently small and among these b’s there is a b ∈ P C0 such that b2 (t, µ) = 0. If b − c∞ and b − d∞ are small enough, then T (c) and T (d) are Fredholm and Ind T (b) = Ind T (c) = Ind T (d),

(2.40)

but it is easily seen that one can find such functions c and d in P C0 which satisfy c2 (t, µ) = 0, d2 (t, µ) = 0 ∀ (t, µ) ∈ T × [0, 1] and ind c2 − ind d2 = 1. From the index formula proved above we get Ind T (d) = Ind T (c) + 1 which contradicts (2.40).   Remark. The perturbation argument used in this proof also applies to show that a2 (t, µ) = 0 for all (t, µ) ∈ T × [0, 1] if T (a) is a Φ+ - or Φ− -operator on the space H 2 . The following theorem is the “essentialization” of Theorem 2.20 and forms the basis for another local approach. 2.75. Theorem (Douglas/Sarason). Let ϕ ∈ L∞ be a unimodular function. Then (a) T (ϕ) ∈ Φ+ (H 2 ) ⇐⇒ distL∞ (ϕ, C + H ∞ ) < 1; (b) T (ϕ) ∈ Φ− (H 2 ) ⇐⇒ distL∞ (ϕ, C + H ∞ ) < 1; (c) T (ϕ) ∈ Φ(H 2 ) ⇐⇒ distL∞ (ϕ, G(C + H ∞ )) < 1. Proof. We abbreviate distL∞ to dist. (a) Let T (ϕ) ∈ Φ+ (H 2 ). If dim Ker T (ϕ) = 0, then T (ϕ) is left-invertible and so Theorem 2.20(a) gives dist(ϕ, H ∞ ) < 1. If dim Ker T (ϕ) = n > 0, then dim Ker T ∗ (ϕ) = 0 by Theorem 2.38. Thus, T (ϕ) is Fredholm of index n > 0. It follows that T (ϕχn ) is invertible, whence dist(ϕχn , H ∞ ) < 1 by Theorem 2.20(a), and thus dist(ϕ, χ−n H ∞ ) < 1. The proof of the implication “=⇒” is complete. Now suppose dist(ϕ, C + H ∞ ) < 1. Then dist(ϕχn , H ∞ ) < 1 for some n ≥ 0, and Theorem 2.20(a) yields the left-invertibility of T (ϕ)T (χn ). Since T (χn ) is Fredholm, it results that T (ϕ) is left-Fredholm. This proves the implication “⇐=”. (b) Take adjoints and apply (a). (c) If T (ϕ) is Fredholm, then T (χn ϕ) is invertible for some n ∈ Z (Corollary 2.40), hence dist(χn ϕ, GL∞ ) < 1 (Theorem 2.20(c)), and thus dist(ϕ, G(C + H ∞ )) < 1. On the other hand, if dist(ϕ, G(C +H ∞ )) < 1, then there are an n ≥ 0 and an h ∈ H ∞ ∩ G(C + H ∞ ) such that ϕ − χ−n h∞ < 1. Since T (h) ∈ Φ(H 2 )

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91

(Corollary 2.55), we have h = bg, where b is a finite Blaschke product and g is in GH ∞ (Theorem 2.64). Consequently, dist(b−1 ϕχn , GH ∞ ) < 1, and so Theorem 2.20(c) implies that T (b−1 )T (ϕ)T (χn ) is invertible. Because T (b−1 ) and T (χn ) are Fredholm, it follows that T (ϕ) must also be Fredholm.   2.76. Corollary. Let a ∈ GL∞ . Then T (a) is in Φ+ (H 2 ) (Φ− (H 2 ) resp. Φ(H 2 )) if and only if a = bs, where b ∈ C + H ∞ (b ∈ C + H ∞ resp. b ∈ G(C + H ∞ )) and s ∈ GL∞ is sectorial. Proof. Combine Theorem 2.75, Proposition 2.32, and Lemma 2.21.   Theorem 2.75 and its Corollary 2.76 do not answer the question on the Fredholmness of Toeplitz operators in terms of the geometric data of the symbol. The purpose of what follows in the next sections is to combine Theorem 2.75 with Glicksberg’s theorem 1.22 in order to make the things little bit more geometrical. Before doing this we need a few pieces of information about the maximal ideal space of L∞ and its decompositions. 2.77. M (L∞ ). The maximal ideal space M (L∞ ) of the Banach algebra L∞ will be denoted by X. Since L∞ is a C ∗ -algebra with respect to the involution a → a, where a(t) = a(t) (t ∈ T), L∞ is star-isometrically isomorphic to C(X). The Gelfand transform of a function a ∈ L∞ will also be denoted by a. Thus, if a ∈ L∞ and x ∈ X, then a(x) = x(a). The topological space X is totally disconnected in the following sense: the closure of every open set is again open. 2.78. L∞ fibers over M (C). The maximal ideal space of C is T: the general form of a functional in M (C) is given by vτ : C → C, where τ ∈ T. Let

f → f (τ ),


Xτ := Mτ (L∞ ) := x ∈ X : x|C = vτ .

It is easy  to see that the fibers Xτ are homeomorphic to each other. Because Xτ , it follows that Xτ = ∅. This is also a consequence of 1.20(b) X = τ ∈T

(recall also 1.27(b)). Given a ∈ L∞ and an open subarc U of T denote by R(a|U ) the spectrum of the restriction of a to U regarded as an element of L∞ (U ). Equivalently, R(a|U ) is the set of all µ ∈ C such that {t ∈ U : |a(t) − µ| < ε} has positive (Lebesgue) measure for each ε > 0. Finally, recall that according to 2.67 distτ (a, b) = inf a|U − b|U L∞ (U ) U ∈Uτ

while in accordance with 1.21 distXτ (a, b) = max |a(x) − b(x)|. x∈Xτ

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2.79. Proposition. (a) If a ∈ L∞ and τ ∈ T, then $ R(a|U ). a(Xτ ) = U ∈Uτ ∞

(b) If a, b ∈ L

and τ ∈ T, then distτ (a, b) = distXτ (a, b). In particular, distτ (a, b) = 0 ⇐⇒ a|Xτ = b|Xτ .

Proof. (a) A little thought shows that µ ∈ /

U ∈Uτ

R(a|U ) if and only if

∃ b, c ∈ L∞ : (a − µ)b + (χ1 − τ )c = 1.

(2.41)

If (2.41) holds, then (a(x) − µ)b(x) = 1 for all x ∈ Xτ , whence µ ∈ / a(Xτ ). On the other hand, if µ ∈ / a(Xτ ) then there is no x ∈ X such that a(x) = µ and χ1 (x) = τ . Thus, the closed ideal 
(a − µ)b + (χ1 − τ )c : b, c ∈ L∞ is not contained in any maximal ideal of L∞ , which gives (2.41). (b) Since distτ (a, b) = distτ (a − b, 0), it suffices to prove that max |f (x)| = distτ (f, 0)

x∈Xτ

for every f ∈ L∞ . By virtue of part (a), c(Xτ ) = {1} for every c ∈ Rτ (see 2.67). So max |f (x)| = max |f (x)c(x)| x∈Xτ

x∈Xτ

for every c ∈ Rτ , whence, by (2.37), max |f (x)| ≤ inf max |f (x)c(x)| = distτ (f, 0).

x∈Xτ

c∈Rτ x∈X

To establish the reverse inequality we need the following well known fact: if K1 ⊃ K2 ⊃ K3 ⊃ ... are compact nonempty subsets of a Hausdorff space, if ∞

n=1

Kn ⊂ Ω, and if Ω is open, then there is an n0 such that Kn0 ⊂ Ω.

Now put M = max |f (x)|. Given any ε > 0 we have, due to part (a), x∈Xτ ∞ $


 R(f |Un ) ⊂ z ∈ C : |z| < M + ε ,

n=1

where Un = {t ∈ T : |t − τ | < 1/n}, and since each set R(f |Un ) is compact and nonempty (as the spectrum of f |Un ∈ L∞ (Un )), it follows that there is an U0 ∈ Uτ such that R(f |U0 ) ⊂ {z ∈ C : |z| < M + ε}. Now it is clear that there exists a c0 ∈ Rτ such that f c0 ∞ < M + ε, and therefore distτ (f, 0) = inf max |f (x)c0 (x)| = max |f (x)c0 (x)| = f c0 ∞ < M + ε. c∈Rτ x∈X

x∈X

Since ε > 0 can be chosen arbitrarily, we get distτ (f, 0) ≤ M .

 

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93

Remark. Thus, a function a ∈ L∞ is continuous at a point τ ∈ T if and only if a(Xτ ) is a singleton. If a has a jump discontinuity at τ , then a(Xτ ) is a doubleton, but if a(Xτ ) is known to be a doubleton, then all one can say is that a has two essential cluster points at τ , which does, in general, not imply that a has a jump at τ . 2.80. QC. The largest C ∗ -subalgebra of C + H ∞ is denoted by QC and is referred to as the algebra of quasicontinuous functions. Thus QC = (C + H ∞ ) ∩ (C + H ∞ ). Although H ∞ ∩ H ∞ is the set of constant functions, QC is strictly larger than C. Indeed, let  1 Ω = z = x + iy ∈ C : 0 < x < 1, −2 < y < sin x and let ω map D conformally onto Ω. Then ω ∈ H ∞ and Re ω ∈ C, whence Im ω = iRe ω − iω ∈ C + H ∞ . Since Im ω is a real-valued function, Im ω ∈ C + H ∞ . But Im ω is obviously discontinuous and therefore Im ω ∈ QC \ C. Since QC is a C ∗ -algebra, we have, for c ∈ QC, c ∈ GQC ⇐⇒ c ∈ GL∞ . 2.81. L∞ fibers over M (QC). If ξ ∈ M (QC), then by virtue of 1.20(b) (or 1.27(b)) the fiber Xξ = Mξ (L∞ ) is not empty. To every ξ ∈ M (QC) there corresponds a τ ∈ M (C) = T such that ξ ∈ Mτ (QC), and it is clear that Mξ (L∞ ) ⊂ Mτ (L∞ ). We have / Mτ (L∞ ) = Mξ (L∞ ). ξ∈Mτ (QC)

Since QC = C, the partition

/

M (L∞ ) =

Mξ (L∞ )

ξ∈M (QC)

is a proper refinement of the partition / Mτ (L∞ ). M (L∞ ) = τ ∈T

Because the restriction of a function in C to a fiber Xτ (τ ∈ T) is constant, we have C + H ∞ |Xτ = H ∞ |Xτ

and

C + H ∞ |Xξ = H ∞ |Xξ .

(2.42)

We know from 1.27(c), (d) (in the setting Y = X, A = C +H ∞ , B = QC) that each maximal antisymmetric set for C + H ∞ is contained in some fiber Xξ , where ξ ∈ M (QC). Consequently, Corollary 1.23 implies that, for a ∈ L∞ ,

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2 Basic Theory

a ∈ C + H ∞ ⇐⇒ a|Xξ ∈ H ∞ |Xξ

∀ ξ ∈ M (QC).

This in turn gives that, for a ∈ L∞ , a ∈ QC ⇐⇒ a|Xξ = const ∀ ξ ∈ M (QC).

(2.43)

Note that the implications “=⇒” are trivial. Let B be a C ∗ -subalgebra of L∞ containing the constant functions. Then for each β0 ∈ M (B) the fiber Xβ0 = Mβ0 (L∞ ) is a peak set for B (recall 1.28). Indeed, because B is isometrically isomorphic to C(M (B)), there is an f ∈ B with f (β0 ) = 1 and 0 < f (β) < 1 for β = β0 , whence f |Xβ0 = 1 and 0 < f < 1 on X \ Xβ0 . In particular, for τ ∈ T, Xτ is a peak set for C and therefore for C + H ∞ , and for ξ ∈ M (QC), Xξ is a peak set for QC and thus also for C + H ∞ . So we deduce from 1.28 that C + H ∞ |Xτ and C + H ∞ |Xξ are closed subalgebras of L∞ |Xτ and L∞ |Xξ , respectively, for every τ ∈ T and ξ ∈ M (QC). Taking into account (2.42) we arrive at the conclusion that the algebras H ∞ |Xτ and H ∞ |Xξ are closed. It also follows that the algebras QC|Xτ and QC|Xξ are closed. Clearly, QC|Xξ is the complex field C, while 1.27(b) shows that M (QC|Xτ ) can be identified with Mτ (QC). We finally mention that both M (H ∞ |Xτ ) (which can be identified with the fiber Mτ (H ∞ ) of H ∞ over τ ∈ T = M (CA ) \ D) and M (H ∞ |Xξ ) are connected (τ ∈ T, ξ ∈ M (QC)). The connectedness of the first space is shown in Hoffman’s book [284] and that the second space is connected was proved by Gorkin [239, Corollary 2.9]. 2.82. Definition. Let a ∈ L∞ and let F be a closed subset of X = M (L∞ ). The Toeplitz operator T (a) will be said to be F -restricted invertible (left resp. right-invertible) if there is a b ∈ L∞ such that a|F = b|F and T (b) is invertible (left resp. right-invertible) on H 2 . If F is contained in some fiber Xτ (τ ∈ T), then T (a) is F -restricted invertible (left resp. right-invertible) if and only if there is a b ∈ L∞ such that a|F = b|F and T (b) is Fredholm (left resp. right-Fredholm) on H 2 (recall Remark 2.70). This follows from Corollary 2.40 together with the fact that continuous functions restricted to Xτ are constants. Proposition 2.79 shows that Theorem 2.68 for p = 2 may also be stated as follows: if T (a) is Xτ -restricted invertible for each τ ∈ T, then T (a) is Fredholm on H 2 . In this form the theorem was established by Douglas and Sarason [164] using a method which actually applies to prove the following much “more local” result. 2.83. Theorem (Axler). Let a ∈ L∞ and let B ⊂ C + H ∞ be a closed subalgebra of L∞ containing the constants. If T (a) is S-restricted invertible (left resp. right-invertible) for each maximal antisymmetric set S for B, then T (a) is Fredholm (left resp. right-Fredholm) on H 2 . Proof. Let T (a) be S-restricted left-invertible for some S ⊂ X. Then there is a b ∈ L∞ such that a|S = b|S and T (b) is left-invertible. By Theorem 2.30(a),

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b ∈ GL∞ and hence a(x) = 0 for all x ∈ S. Proposition 2.32 implies that T (b/|b|) is left-invertible and so Theorem 2.75(a) gives distX (b/|b|, C + H ∞ ) = distL∞ (b/|b|, C + H ∞ ) < 1. Because b/|b| equals a/|a| on S, we have distS (a/|a|, C + H ∞ ) < 1.

(2.44)

If (2.44) holds for each maximal antisymmetric set S for B, then it also holds for each maximal antisymmetric set S for C + H ∞ , since the latter ones are contained in the former ones (see 1.27(d)). Thus, Theorem 1.22 gives that distL∞ (a/|a|, C + H ∞ ) = distX (a/|a|, C + H ∞ ) < 1, from Theorem 2.75(a) we deduce that T (a/|a) ∈ Φ+ (H 2 ) and once more applying Proposition 2.32 we see that T (a) ∈ Φ+ (H 2 ). The proof for the right-Fredholmness is analogous. Finally, if T (a) is Srestricted invertible for each maximal antisymmetric set S for B, then, by what has already been proved, T (a) is in both Φ+ (H 2 ) and Φ− (H 2 ), and  hence in Φ(H 2 ).  2.84. Definitions. Let F be a closed subset of X = M (L∞ ). A function a ∈ L∞ is said to be sectorial on F if there are an ε > 0 and a c ∈ C of modulus 1 such that Re (ca(x)) ≥ ε for all x ∈ F . If a is sectorial on F , then a|F is obviously invertible in L∞ |F , and it is easy to see that a is sectorial on F if and only if so is a/|a|. Moreover, for a ∈ L∞ to be sectorial on F it is necessary and sufficient that a(x) = 0 for all x ∈ F and distF (a/|a|, C) < 1. Now let B be a closed subalgebra of C + H ∞ containing the constants. A function a ∈ L∞ will be called locally sectorial over B if it is sectorial on each maximal antisymmetric set for B. The most important special cases are B = C + H ∞ , B = QC, B = C, and B = C. So, by virtue of 1.27(c), a ∈ L∞ is locally sectorial over QC (resp. C) if and only if it is sectorial on each fiber Xξ , ξ ∈ M (QC) (resp. Xτ , τ ∈ T). The functions that are sectorial in the sense of Definition 2.16 are just the functions which are sectorial on X = M (L∞ ) or, equivalently, locally sectorial over C. Finally, from 1.27(d) (with Y = X) we deduce that if B ⊂ A, then a locally sectorial over B =⇒ a locally sectorial over A. 2.85. Theorem. If a ∈ L∞ is locally sectorial over a closed subalgebra B of C + H ∞ containing the constants then T (a) is Fredholm on H 2 . Proof. The hypothesis implies that a ∈ GL∞ and that a is locally sectorial over C + H ∞ . Hence

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2 Basic Theory

distS (a/|a|, C + H ∞ ) ≤ distS (a/|a|, C) < 1 for each maximal antisymmetric set S for C + H ∞ . Since the maximal antisymmetric sets for C + H ∞ are the same as those for C + H ∞ , Theorem 1.22 and Theorem 2.75(a), (b) can be combined to obtain that T (a/|a|) ∈ Φ+ (H 2 ) ∩ Φ− (H 2 ) = Φ(H 2 ) and Proposition 2.32 completes the proof.

 

The following proposition provides an idea of what the different notions of local sectoriality involve. 2.86. Proposition. (a) If a ∈ L∞ is locally sectorial over C + H ∞ then a can be written as f s with f ∈ G(C + H ∞ ) and s ∈ GL∞ sectorial (on T). (b) Let B be a C ∗ -subalgebra of L∞ between C and QC. Then a is locally sectorial over B if and only if a can be represented as a = bs with b ∈ GB and s ∈ GL∞ sectorial (on T). (c) For a ∈ L∞ the following are equivalent: (i) a is locally sectorial over C; (ii) a = cs with c ∈ GC and s is sectorial; (iii) a is locally arcwise sectorial. Proof. (a) Theorem 1.22 gives distL∞ (a/|a|, C + H ∞ ) < 1,

distL∞ (a/|a|, C + H ∞ ) < 1,

so Theorem 2.75 shows that distL∞ (a/|a|, G(C + H ∞ )) < 1 and Lemma 2.21 ends the proof. (b) One half of the assertion can be proved as in (a). On the other hand, if a = bs with b ∈ GB and s sectorial, then a|S = (b|S)(s|S) for each maximal antisymmetric set S for B, and since these sets are just the fibers Xβ , β ∈ M (B) (1.27(c)), b|S is a nonzero constant and hence a is sectorial on S. (c) The implication (i) =⇒ (ii) follows from part (b) and the implication (iii) =⇒ (i) results from Proposition 2.79(a). Finally, if (ii) holds, then Re (γs(t)) ≥ ε for some ε > 0, some γ ∈ C, and for almost all t ∈ T. Hence, if τ ∈ T,  Re γ/c(τ ) c(τ )s(t) ≥ ε > 0 for almost all t ∈ T and since c is continuous, Re



γ/c(τ ) c(t)s(t) ≥ ε > 0

for almost all t in some neighborhood of τ , which gives (iii).

 

2.8 Local Methods for Scalar Toeplitz Operators

97

Remark. Thus, Theorem 2.85 can also be proved as follows: if a ∈ L∞ is locally sectorial over C+H ∞ , then a = f s with f ∈ G(C+H ∞ ) and s sectorial, so T (a) = T (s)T (f )+ compact operator ((2.18) and Theorem 2.54), and since T (s) is invertible (Theorem 2.17) and T (f ) is Fredholm (Corollary 2.55), we conclude that T (a) is Fredholm. Our next concern is the index computation (and thus the solution of the invertibility problem) for Toeplitz operators whose symbol is locally sectorial over QC (or over any closed subalgebra of QC containing the constants). The key observations are Propositions 2.86(b) and the following generalization of Lemma 2.61. 2.87. Lemma (Sarason). If b ∈ QC and a ∈ L∞ , then (ba)r − br ar ∞ → 0 as

r → 1 − 0.

Proof. Let kr denote the Poisson kernel. Then + + ( 2π + 1 + [b(eit ) − br (eiθ )]a(eit )kr (θ − t) dt++ |(ba)r (eiθ ) − br (eiθ )ar (eiθ )| = ++ 2π 0 ( a∞ 2π ≤ |b(eit ) − br (eiθ )|kr (θ − t) dt 2π 0 1/2 ( 2π a∞ it iθ 2 ≤ |b(e ) − br (e )| kr (θ − t) dt 2π 0 1/2 a∞  (bb)r (eiθ ) − br (eiθ )br (eiθ ) = . 2π But if b ∈ QC, then b ∈ C + H ∞ and b ∈ C + H ∞ , whence, by virtue of  Theorem 2.62(a), (bb)r − br br ∞ → 0 as r → 1 − 0.  2.88. Theorem. If a ∈ L∞ is locally sectorial over QC, then T (a) ∈ Φ(H 2 ), the harmonic extension  a is bounded away from zero in some annulus near T, and Ind T (a) = −ind {ar }. Proof. Due to Proposition 2.86(b) we have a = bs, where b ∈ GQC and s ∈ GL∞ is sectorial (on T). So T (a) = T (b)T (s)+ compact operator ((2.18) and Theorem 2.54), and since T (b−1 ) is a regularizer of T (b) ((2.18) and Corollary 2.55) and T (s) is invertible (Theorem 2.17), it follows that T (a) ∈ Φ(H 2 ). Of course, the same conclusion might be also drawn from Theorem 2.85. Lemma 2.87 shows that ar − br sr ∞ → 0 as r → 1 − 0. If Re s ≥ ε > 0 a.e. on T, then Re s ≥ ε > 0 in D, because the Poisson kernel is positive. Hence, if s is sectorial then s is bounded away from zero in D and ind {sr } = 0.

(2.45)

If b ∈ GQC, then b is bounded away from zero in some annulus near T by Theorem 2.62. Thus, under our hypothesis,  a is bounded away from zero in some annulus near T.

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2 Basic Theory

The index formula can now be verified as follows: Ind T (a) = Ind T (b) + Ind T (s) = Ind T (b) = −ind {br } (Theorem 2.65) = −ind {br } − ind {sr } (by (2.45)) = −ind {(bs)r } (2.41(a), Lemma 2.87, 2.41(b)).

 

We finally show that for a relatively large class of symbols local sectoriality is not only sufficient but also necessary for the Fredholmness of the corresponding Toeplitz operator. 2.89. Definition. Let B be a C ∗ -subalgebra of QC containing the constants. We denote by P2 B the collection of all functions a ∈ L∞ which take at most two values on each fiber Xξ , ξ ∈ M (B). For instance, P2 C contains P C, and if E any measurable subset of T, then the functions aχE + b (χE being the characteristic function of E, a and b being in C) belong to P2 C. Further, P2 QC contains all functions of the form a=

n 

pi qi ,

pi ∈ P2 C,

qi ∈ QC.

i=1

We shall see later that P QC, the closed subalgebra of L∞ generated by P C and QC, is also a subset of P2 QC (see Remark 1 of 3.36). 2.90. Lemma. Let B = C or B = QC and let ξ ∈ M (B). If ϕ ∈ L∞ is unimodular and ϕ(Xξ ) is a pair of antipodal points, then distXξ (ϕ, H ∞ ) = 1. Proof. Without loss of generality suppose ϕ(Xξ ) is the doubleton {−1, 1}. Assume there is an h ∈ H ∞ such that max |ϕ(x) − h(x)| ≤ 1 − δ < 1. Put E± := {x ∈ Xξ : ϕ(x) = ±1}. So |1 − h(x)| ≤ 1 − δ

∀ x ∈ E+ ,

x∈Xξ

|1 + h(x)| ≤ 1 − δ

∀ x ∈ E− .

(2.46)

Let B denote the restriction algebra L∞ |Xξ , which is closed by 1.28. From (2.46) we see that the spectrum of h|Xξ in B is contained in the union of two disks with center at +1 and −1 and radius 1 − δ, and, moreover, that each of these two disks contains a point of that spectrum, i.e., that there are z1 , z2 ∈ C such that Re z1 < 0, Re z2 > 0, z1 ∈ spB (h|Xξ ), z2 ∈ spB (h|Xξ ). Now put A := H ∞ |Xξ . From 2.81 we know that A is closed and M (A) is connected (Hoffman for B = C and Gorkin for B = QC). Consequently, spA (h|Xξ ) = h(M (A)) is a connected subset of C. By virtue of 1.16(b), spA (h|Xξ ) is the union of spB (h|Xξ ) and a (possibly empty) collection of bounded connected

2.9 Matrix Symbols

99

components of spB (h|Xξ ). However, the set {z ∈ C : |Re z| < δ/2} is contained in the unbounded complement of spB (h|Xξ ), hence {z ∈ C : |Re z| < δ/2} ∩ spA (h|Xξ ) = ∅. But this is a contradiction, since together with z1 and z2 some points of the  stripe {|Re z| < δ/2} must belong to the (connected!) set spA (h|Xξ ).  2.91. Theorem. Let B = C or B = QC and let a ∈ P2 B. Then T (a) ∈ Φ(H 2 ) ⇐⇒ a is locally sectorial over B. Proof. The implication “⇐=” is immediate from Theorem 2.85 (or can be established as in the proof of Theorem 2.88). So we are left with the reverse implication. Let T (a) ∈ Φ(H 2 ). Then T (a/|a|) ∈ Φ(H 2 ) by Proposition 2.32, and hence distL∞ (a/|a|, C + H ∞ ) < 1 by Theorem 2.75(a). It follows that distXξ (a/|a|, H ∞ ) < 1 for each ξ ∈ M (B) (recall (2.42)). The preceding lemma shows that the singleton or doubleton (a/|a|)(Xξ ) cannot be a doubleton consisting of two antipodal points and this is equivalent to saying that  a/|a| (and thus a itself) is sectorial on Xξ . 

2.9 Matrix Symbols We conclude this chapter by stating some facts on Toeplitz operators with matrix symbols. We here confine ourselves to settling a few problems the solution of which merely requires minor modifications of the methods developed above for scalar Toeplitz operators. The more delicate questions on block Toeplitz operators will be deferred to the forthcoming chapters. ∞ 2.92. Definitions. Given a matrix function a = (ajk )N j,k=1 ∈ LN ×N the mulp tiplication operator M (a) is defined on LN (1 < p < ∞) by

M (a) :

LpN

→

LpN ,

(fk )N k=1

→

 N

N M (akj )fj

j=1

k=1

p (recall the notations introduced in 1.13). For a ∈ MN ×N the operator M (a) p p is defined on
N (Z) := (
(Z))N analogously. Here LpN and
pN (Z) can be regarded as being equipped with the norms

f LpN :=

N 

fj Lp ,

f = (fj )N j=1 ,

j=1

ϕpN (Z) :=

N 

ϕj p (Z) ,

j=1

or with any norms equivalent to those ones.

ϕ = (ϕj )N j=1 ,

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2 Basic Theory

Similarly, if a ∈ L∞ N ×N , the Toeplitz operator T (a) and the Hankel operap (1 < p < ∞) by tor H(a) are given on HN p p T (a) : HN → HN ,

(fk )N k=1 →

 N

N T (akj )fj

p p H(a) : HN → HN ,

(fk )N k=1 →

 N

, k=1

j=1

N H(akj )fj

, k=1

j=1

p p and for a ∈ MN ×N an analogous definition is made for T (a) and H(a) on
N (1 ≤ p < ∞). p and
pN can be viewed as subspaces of LpN and
pN (Z), The spaces HN respectively. Thus, whenever a norm in the latter two spaces is specified it will always be clear what the norm in the first two spaces is. Both the Riesz projection P : Lp → H p and the canonical projection P :
p (Z) →
p extend in a natural way to LpN and
pN (Z). We denote these projections again by P . Thus, P = diag (P, . . . , P ). If the norm on
pN (Z) is given by  1/p N N  p ϕj p (Z) or ϕj p (Z) , j=1

j=1

then obviously P L(pN (Z)) = 1. In the same fashion the projection Q and the flip operator J are defined on LpN and
pN (Z). We then have T (a) = P M (a)P |Im P,

H(a) = P M (a)QJ|Im P,

etc. In particular, formulas (2.18)–(2.20) remain true for the matrix case without any changes. The matrix function a will always be referred to as the symbol of the corresponding operator. With every ϕ ∈
pN we may associate a CN -valued sequence ψ ∈
p (Z+ , CN ) as follows: p j ∞ p if ϕ = (ϕk )N k=1 ∈
N where ϕk = {ϕk }j=0 ∈
, j N p then ψ = {ψj }∞ j=0 ∈
(Z+ , CN ) where ψj = (ϕk )k=1 ∈ CN .

So the Toeplitz operator on
pN can also be thought of as acting on
p (Z+ , CN ) by the rule ! #∞ ∞ →  a ψ , T (a) : {ψj }∞ j−k k j=0 k=0

j=0

where an (n ∈ Z) denotes the N × N matrix ((ajk )n )N j,k=1 formed by the p N Fourier coefficients of a = (ajk )j,k=1 ∈ MN ×N . Therefore Toeplitz operators on
pN are sometimes also called block Toeplitz operators. It is clear that M (a)

2.9 Matrix Symbols

101

and H(a) can be viewed as acting on
p (Z, CN ) and
p (Z+ , CN ), respectively, in a similar manner. p We define norms on L∞ N ×N and MN ×N by := M (a)L(L2N ) , aL∞ N ×N

aMNp ×N := M (a)L(pN ) .

p Provided with these norms L∞ N ×N and MN ×N (1 ≤ p < ∞) are (noncommutative) Banach algebras with identity I. Clearly, a ∈ GL∞ N ×N resp. p p ∞ a ∈ GMN ×N if and only if there is a b ∈ LN ×N resp. b ∈ MN ×N such that ∞ and ab = ba = I. It is also obvious that a ∈ GL∞ N ×N ⇐⇒ det a ∈ GL p p a ∈ GMN ×N ⇐⇒ det a ∈ GM . By virtue of 1.29(a),

= ess sup a(t)L(CN ) = max a(x)L(CN ) . aL∞ N ×N t∈T

x∈X

2.93. Theorem. (a) If 1 < p < ∞ and a ∈ L∞ N ×N , then M (a) ∈ Φ± (LpN ) ⇐⇒ M (a) ∈ GL(LpN ) ⇐⇒ a ∈ GL∞ N ×N , p T (a) ∈ Φ(HN ) =⇒ a ∈ GL∞ . N ×N p (b) If 1 ≤ p < ∞ and a ∈ MN ×N , then p M (a) ∈ Φ(
pN (Z)) ⇐⇒ M (a) ∈ GL(
pN (Z)) ⇐⇒ a ∈ GMN ×N , p p ∞ T (a) ∈ Φ(
N ) =⇒ a ∈ GMN ×N =⇒ a ∈ GLN ×N .

Proof. The assertions about the multiplication operators can be proved by the same argument as in the scalar case (2.28, 2.29). After defining the bilateral shift U on LpN or
pN (Z) as U = M (χ1 I) = diag (U, . . . , U ), the proof of Theorem 2.30 also works in the matrix case. 1 Note that the implication T (a) ∈ Φ(
1N ) =⇒ a ∈ GMN ×N can also be verified by invoking Theorem 1.14(c). Indeed, we have, by (2.18), T (f )T (g) − T (g)T (f ) = −H(f )H() g ) + H(g)H(f))

(2.47)

for all f, g ∈ M 1 , and since M 1 = W , Theorem 2.47(a) shows that the occurring Hankel operators are compact.   Important remark. A decisive distinction between the scalar case (N = 1) and the matrix case (N > 1) is that a Fredholm block Toeplitz operator of index zero is not necessarily invertible (compare Corollary 2.40). For instance, if a = diag (χ1 , χ−1 ), then obviously T (a) ∈ Φ(H22 ) although both dim Ker T (a) end dim Coker T (a) equal 1. 2.94. Theorem. (a) Let a ∈ (C + H ∞ )N ×N and 1 < p < ∞. Then H(a) is p ) and in C∞ (HN p ) ⇐⇒ det a ∈ G(C + H ∞ ); T (a) ∈ Φ(HN

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2 Basic Theory

p if T (a) is Fredholm on HN , then T (a−1 ) is a regularizer of T (a) and

Ind T (a) = Ind T (det a) = −ind {(det a)r }. (b) Let a ∈ (Cp + Hp∞ )N ×N and 1 < p < ∞. Then H(a) ∈ C∞ (
pN ) and T (a) ∈ Φ(
pN ) ⇐⇒ det a ∈ G(Cp + Hp∞ ); if T (a) is Fredholm on
pN , then T (a−1 ) is a regularizer of T (a) and Ind T (a) = Ind T (det a) = −ind {(det a)r }. Proof. The compactness of the Hankel operators can be shown as in the scalar case. This and identity (2.47) allow us the application of Theorem 1.14(c). p resp.
pN if and only if T (det a) What results is that T (a) is Fredholm on HN is so on H p resp.
p , which, by Corollary 2.55 and Theorem 2.60, is equivalent to the invertibility of det a in C + H ∞ resp. Cp + Hp∞ . That T (a−1 ) is a regularizer of T (a) follows from (2.18). In view of Theorems 2.65 and 2.66 it remains to show that Ind T (a) = Ind T (det a). To this end approximate a sufficiently close in the norm of L∞ N ×N p ∞) ∞) resp. MN by b ∈ (R+H resp. b ∈ (R+H . Then, by 1.12(d), N ×N N ×N p ×N Ind T (a) = Ind T (b),

Ind T (det a) = Ind T (det b).

Taking into account identity (2.47) and Theorem 2.58 we see that the entries of T (b) commute modulo finite-rank operators. So Theorem 1.15(b) gives Ind T (b) = Ind T (det b).   Remark. Obvious modifications of the proof show that part (b) is true for p = 1 if only Cp + Hp∞ is replaced by W = M 1 . p

2.95. Theorem. Let a ∈ MN ×N and 1 ≤ p < ∞. Assume for each τ ∈ T p

there exists an aτ ∈ MN ×N such that aτ |Xτ = a|Xτ and T (aτ ) ∈ Φ(
pN ). Then T (a) ∈ Φ(
pN ). Proof. This follows from applying Theorem 1.32. Put A = L(
pN )/C∞ (
pN ) and, p for a ∈ MN ×N , denote the coset of A containing T (a) by T π (a). For τ ∈ T, define Rτ as in the proof of Theorem 2.69 and put Fτ = {diag (f, . . . , f ) : f ∈ Rτ },

Mπτ = {T π (ϕ) : ϕ ∈ Fτ }.

It is readily seen that {Mπτ }τ ∈T is a covering system of localizing classes in A and that (2.48) T π (a)T π (ϕ) = T π (aϕ) = T π (ϕa) = T π (ϕ)T π (a)  for every ϕ ∈ Fτ . As in the proof of Theorem 2.69 one can show that T π (a) τ

and T π (aτ ) are Mπτ -equivalent from the left and from the right for each τ ∈ T. Theorem 1.32 then gives the assertion.  

2.10 Notes and Comments

103

∗ 2.96. Theorem. Let a ∈ L∞ N ×N and 1 < p < ∞, and let B be a C -subalgebra of QC containing the constants. Suppose for each ξ ∈ M (B) there is an aξ in p p L∞ N ×N such that aξ |Xξ = a|Xξ and T (aξ ) ∈ Φ(HN ). Then T (a) ∈ Φ(HN ). p p Proof. We shall derive this from Theorem 1.32. Put A = L(HN )/C∞ (HN ) p ∞ π and, for a ∈ LN ×N , let T (a) := T (a) + C∞ (HN ). For ξ ∈ M (B), define Rξ as the collection of all f ∈ B such that 0 ≤ f ≤ 1 and f is identically 1 in some neighborhood Uξ ⊂ M (B) of ξ. Then let

Fξ = {diag (f, . . . , f ) : f ∈ Rξ }, Mπξ = {T π (ϕ) : ϕ ∈ Fξ }.  It is clear that (2.48) holds for ϕ ∈ Fξ and it is easy to see that {Mπξ }ξ∈M (B) ξ

is a covering system of localizing classes in A. We now show that T π (a) and T π (aξ ) are Mπξ -equivalent from the left. Choose ε > 0, set b = a − aξ , and let 
U = η ∈ M (B) : b(x)L(CN ) < ε ∀ x ∈ Xη . Assume U is not an open subset of M (B). Then there is an η ∈ U and a net ηi in M (B) such that ηi → η and such that for each i, there exists xi ∈ Xηi with b(xi ) ≥ ε. Taking a subnet, we can suppose that there is an x ∈ X such that xi → x. Since the mapping (y ∈ X) → (y|B ∈ M (B)) is continuous, it follows that x ∈ Xη . But b(x) ≥ ε, which is impossible for x ∈ Xη and η ∈ U . This contradiction shows that U is open. It is clear that ξ ∈ U , and hence there is a closed subset S of M (B) such that ξ ∈ S ⊂ U . So there exists an f ∈ B satisfying 0 ≤ f ≤ 1, f |S = 1, f |(M (B) \ U ) = 0. If we let ϕ = diag (f, . . . , f ), then ϕ ∈ Fξ and " " π " T (a) − T π (aξ ) T π (ϕ)" = T π (bϕ) ≤ cp bϕL∞ , N ×N where cp = P L(LpN ) . Since b(x)ϕ(x) < ε for all x ∈ X, and as ε > 0 can be chosen arbitrarily, it results that T π (a) and T π (aξ ) are Mπξ -equivalent from the left, as desired. It can be shown analogously that they are Mπξ -equivalent from the right. Now Theorem 1.32 completes the proof.  

2.10 Notes and Comments 2.2–2.4. These facts are well known. The proof of Theorem 2.2 is patterned after Halmos [265, Problems 50 and 193]. 2.5. These results form only a little part of what is known about multipliers on
p and similar spaces, and significant contributions to this topic have been made by many people. For more about this see, e.g., H¨ ormander [288], Hirschman [280], Zygmund [591], Nikolski [365], Gohberg, Krupnik [232], Duduchava [169], [173], Verbitsky [538]. Inequality 2.5(f) goes back to S. B. Stechkin.

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2 Basic Theory

2.6. Commemorative articles on life and work of Otto Toeplitz may be found in Gohberg et al. [235]. 2.7. See Brown, Halmos [125] or Halmos [265] for H 2 and Duduchava [169] for
p . Let a ∈ C. Then neither the spectral radius of T (a) ∈ L(H p ) nor the spectral radius of T (a) + C∞ (H p ) ∈ L(H p )/C∞ (H p ) depend on p (see Theorem 2.42). However, one can show that T (a)L(H p ) depends on p. In particular, T (χ−1 )L(H p ) > 1 for all p = 2. Does the essential norm T (a)Φ(H p ) depend on p (see 2.27)? This problem is equivalent to the following question: Is T (χ−1 )Φ(H p ) equal 1 for all p ∈ (1, ∞)? For more about these things see B¨ottcher, Krupnik, Silbermann [98]. 2.9. We recorded these trivialities mainly to fix some notation and to have a reference. It is well known that shift operators have many remarkable and nontrivial properties. We only mention the following. Given any Hilbert space E let
2 (E) refer to the Hilbert space of all E-valued sequences {xn }n∈Z+ such

(−1) that xn 2E < ∞ and define VE as (−1)

VE

:
2 (E) →
2 (E),

{x0 , x1 , x2 , . . .} → {x1 , x2 , x3 , . . .}.

Then if A is any bounded linear operator on a Hilbert space H such that A ≤ 1 and An → 0 strongly as n → ∞, there exists a Hilbert space E (−1) with dim E = dim(I − A∗ A)H, an invariant subspace K ⊂
2 (E) of VE , (−1) and a unitary operator W : H → K such that A = W −1 (VE |K)W . Thus, shifts turn out to be “universal operators.” If I − A∗ A has rank one, we may take E = C and identify
2 (E) with H 2 . The above result is then completed by Beurling’s theorem, which states that every (closed) invariant subspace K ⊂ H 2 of V (−1) , other than H 2 , is of the form K = H 2  θH 2 with some inner function θ. The beginner should consult Rudin [443] and Halmos [265] for these things; excellent presentations of this topic and of related questions are Nikolski [368] and Rosenblum, Rovnyak [442]. 2.10. Good discussions of the main facts about Hankel operators are Nikolski [369], Partington [375], Peetre [384] (this reference also contains an outline of the life of Hermann Hankel), Peller [389], Peller, Khrushchev [390], Power [403], [404]. 2.11. For p = 2, this theorem was established by Nehari [363]. The proof given in the text is due to Sarason [457]. It is this proof which makes the extension of Nehari’s theorem to the spaces H p to a relatively simple matter (this has also been observed by Peetre [384]; also see Peller [388] and Tolokonnikov [519]). Sarason’s proof was adopted in A. Karlovich [299] to extend the Nehari theorem to the case of Hardy type subspaces or rearrangement-invariant spaces. The proof of an analogue of (2.14) in A. Karlovich [299] is based on the inner-outer factorization 1.41 and the so-called Lozanovsky factorization of L1 functions.

2.10 Notes and Comments

105

There are at least three other proofs of Theorem 2.11: Nehari’s original one (which is quite complicated), the proof using the commutant lifting theorem (see Page [374], Clancey, Gohberg [138, Chap. VIII, Theorem 5.2], and Peller [389, Appendix 1.5]), and Parrott’s proof [376]. The latter two proofs also work in the matrix case; Parrott’s proof will be given in 4.32 and 4.33. Also see Bonsall [55]. Finally notice the following characterization of bounded positive Hankel operators due to Widom [564]: if µ is a positive Borel measure on [−1, 1] and if we let ( 1 xn dµ(x) µn = −1

and H[µ] = (µi+j )∞ i,j=0 , then H[µ] ∈ L(
2 ) ⇐⇒ µn = O(1/n) (n → ∞) ⇐⇒ µ((−1, −x) ∪ (x, 1)) = O(1 − x) (x → 1). 2.14. Although similar (end equivalent) formulas had been used for a long time, identities (2.18), (2.19) appeared in Widom [569] for the first time. In connection with (2.20) we mention the following result of Brown, Halmos [125]: T (a)T (b) is a Toeplitz operator if and only if a or b is analytic. 2.17. Brown, Halmos [125] and Devinatz [151]. Corollary 4.2 generalizes this result to the matrix case. 2.18. Wintner [579]. Extensions to H p and
p are in 2.31. It is clear from the proof in 2.31 that these results extend to the matrix case. 2.19–2.23. Widom [557], Devinatz [151]. There are difficulties in the matrix case, but see 4.35–4.38 and Devinatz, Shinbrot [155] or Speck [499]. For extensions to the spaces H p see 2.32, 5.3, 5.20, 5.22. We do not know a WidomDevinatz criterion for the spaces
p . It should be noted here that the general invertibility problem for Toeplitz operators can be reduced to the special case that the symbol is of the form ω1 ω2 , where ω1 and ω2 are inner functions. Using certain factorization theorems of S. Axler, Th. H. Wolff, and D. Sarason one can show that for every function a ∈ GL∞ there exist inner functions ω1 and ω2 , an outer function h ∈ H ∞ , and a continuous function c such that a = ω1 ω2 hc and that T (a) ∈ Φ(H 2 ) (resp. GL(H 2 )) if and only if T (ω1 ω2 ) ∈ Φ(H 2 ) (resp. GL(H 2 )). However, it is by no means easy to decide whether T (ω1 ω2 ) is invertible or Fredholm. For a nice discussion of this topic see Nikolski [366], [367]. Let us also mention the following result of Lee and Sarason [335]. For an inner function ω let supp ω := {τ ∈ T : 0 ∈ ClH (ω, τ )} (see 3.72). If ω1 and ω2 are inner and supp ω1 = supp ω2 , then sp T (ω1 ω2 ) = clos D. 2.25–2.26. This material is taken from B¨ottcher [68]. See also 4.72, 4.73, 9.18. Parts (a) and (b) of Proposition 2.26 are well known. It may be that parts (c) and (d) of this proposition are known to specialists, but we have not found any reference. It should also be noted that Kats [310] considered the Riemann

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boundary value problem f + = af − +g on T with a coefficient a ∈ C(T◦ ) in the class of all functions holomorphic in |z| < 1 resp. |z| > 1, bounded in |z| ≤ 1 resp. |z| ≥ 1, and continuous in {|z| ≤ 1, z = −1} resp. {|z| ≥ 1, z = −1}. He obtained necessary and sufficient conditions for the homogeneous problem to have a finite number of linearly independent solutions, computed this number, and studied the solvability of the inhomogeneous problem. 2.28–2.29. These results are well known, but apart from the case p = 2 (Halmos [265, Problem 52], Douglas [162, 4.24]) we do not know any reference. 2.30. Hartman and Wintner [267] showed that R(a) is contained in the spectrum spL(H 2 ) T (a), and Simonenko [496] proved that a ∈ GL∞ if T (a) is a Φ+ - or Φ− -operator on H p . The proof given here is based on arguments of Widom [556]. Also see 2.93. Finally, note that the normal solvability of T (a) on H p (1 < p < ∞) implies that either a ∈ GL∞ or a ≡ 0. This was proved by Leiterer [336] for p = 2 and by Heunemann [275] for general p. 2.31–2.33. See the notes to 2.17–2.23. 2.35. In 1963, Halmos posed the following as a test question for any theory of invertibility of Toeplitz operators: Is the spectrum of a Toeplitz operator necessarily connected? Widom showed that the answer is yes (in [562] for p = 2 using the Helson-Szeg˝o theorem 1.45 and in [563] for general p without using this theorem). The connectedness of spΦ(H 2 ) T (a) was first proved by Douglas [162, 7.45]. See also 4.68. 2.36. Hartman, Wintner [267]. It has been open for a long time whether an analogous result holds for quarter-plane Toeplitz operators (see Chapter 8); in B¨ottcher [71], we observed that an argument used by McDonald and Sundberg [354] in the context of Toeplitz operators on the disk also applies to half-plane Toeplitz operators and so, by 8.13 and 8.14, proves the connectedness of both the spectrum and the essential spectrum of quarter-plane Toeplitz operators with real-valued symbols. Open problems: Is the (essential) spectrum of T 2 (a) on H 2 (T2 ) connected for all a ∈ L∞ (T2 )? As we know that this is so for real-valued symbols, are there interesting applications of Theorem 4.100 to quarter-plane Toeplitz operators? Selfadjoint operators live in another world than the generically nonselfadjoint Toeplitz operators. However, if a Toeplitz operator is selfadjoint, one is faced with the same specific questions as for general selfadjoint operators (such as diagonalization or spectral projections). These questions are treated in the pioneering papers by Rosenblum [439], [440], [441] and Ismagilov [292] and in the book Rosenblum, Rovnyak [442]. A recent remarkable contribution to the matter was made by Vreugdenhil [552]. He showed, for example, that if a(t) = |t − 1|2 (t ∈ T), then the positive square root of T (a) is  3 4∞ 4 1 1 − . π 4(j + k + 2)2 + 1 4(j − k)2 + 1 j,k=0 See also Chapter 1 of B¨ ottcher, Grudsky [86].

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2.37. Wolff [581]. See also 4.76. 2.38–2.40. Coburn [141] for p = 2, Simonenko [496] for H p , Duduchava [169] for
p . For generalizations of Theorem 2.38 and for still another proof of 2.38(b) see Volberg, Tolokonnikov [551]. Vukoti´c [553] proved a more explicit form of 2.38(a): If T (a) ∈ L(H 2 ) is a nonzero Toeplitz operator which is not one-to-one, then T (a)(lin{P Ker T (a)}) = P, where lin S denotes the linear span of the set S and P Ker T (a) is defined as {pf : p ∈ P, f ∈ Ker T (a)}. 2.41. For (e) see Gelfand, Raikov, Shilov [212]. 2.42. This theorem is the culmination of several authors including Noether [370], Mikhlin [360], Gohberg [217], Simonenko [491], Krein [322], Calder´ on, Spitzer, Widom [129], Devinatz [151]. Note that Simonenko’s 1960 work [491] actually contains Theorems 2.68 and 2.72! Stegenga [507] proved that a Toeplitz operator T (a) is bounded on H 1 if and only if a ∈ L∞ ∩ BM O1/| log t| . Recently Virtanen [546] obtained an analogue of Theorem 2.42(b) for Toeplitz operators T (a) on H 1 provided a ∈ C ∩ V M O1/| log t| . Here BM O1/| log t| and V M O1/| log t| are certain generalizations of BM O and V M O. 2.43–2.47. Theorem 2.47 was established by Krein [322] (for symbols in W ) and by Gohberg, Feldman [220] (for symbols in Cp ). Lemma 2.44 (and its proof given here) as well as Proposition 2.46 are due to Nikolski [365]. 2.50. This is an extract of arguments due to Krein [323] and B¨ ottcher, Silbermann [106, Chap. IV]. 2.51–2.53. Sarason [451] was the first to observe that C + H ∞ is closed and thus an algebra. This (at the first glance unpretentious and rather curious) discovery is certainly one of the most significant achievements in mathematical analysis during the last decades, and it has stimulated and determined subsequent developments in various fields (in particular in the theory of Toeplitz operators) essentially. See also Sarason [452], [457], Douglas [162], Koosis [316], Garnett [211]. For the Zalcman-Rudin lemma see Koosis [316, Chap. VII, 3◦ ]. The observation that Cp + Hp∞ is a closed subalgebra of M p (p = 2) is due to the authors (B¨ ottcher, Silbermann [113]). 2.54. Hartman [266] showed that the Hankel operator H(a) is compact on H 2 if and only if a ∈ C + H ∞ , and Adamyan, Arov, Krein [2] established the equality H(a)Φ(H 2 ) = distL∞ (a, C + H ∞ ). The proof given here follows Axler, Berg, Jewell, Shields [14]. It makes use of the fact that C + H ∞ is closed. There are proofs of Hartman’s result (e.g. Hartman’s original one) which do not use the closedness of C + H ∞ . Thus the fact that C + H ∞ is closed is also a consequence of Hartman’s result (see Sarason [457, p. 102]). For more about compactness of Hankel operators (s-numbers, trace class criteria, vector-valued versions of Hartman’s theorem etc.) see Page [374], Peller [385] – [388], Peller, Khrushchev [390], Nikolski [366] – [369], Rochberg

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[434], Power [403], [404], Peetre [384], Treil [524], Havin, Khrushchev, Nikolski [269]. Widom [564] showed that for positive Hankel operators the following are equivalent (recall the notes to 2.11): (i) H[µ] ∈ C∞ (
2 ), (ii) µn = o(1/n) (n → ∞), (iii) µ((−1, −x) ∪ (x, 1)) = o(1 − x) (x → 1). If a ∈ P C, then H(a)Φ(H 2 ) = distL∞ (a, C + H ∞ ) = distL∞ (a, C), which in particular implies that P C + H ∞ is a closed subset of L∞ . This was shown by Bonsall and Gillespie [56]. They also pointed out that P C + H ∞ is not an algebra. 2.55. This result (for p = 2) was obtained by Douglas [159], [162]. The proof presented here is essentially simpler than that of Douglas. 2.58. The theorem is Kronecker’s [325] and the proof in the text is after Gantmacher [210, Chap. XV, Section 10]. An alternative proof of this result (due to Axler) is contained in Sarason [457] and Peller [389, Chap. 1, Section 3]. See also the references listed in the notes to 2.54. Notice that the rank of H(a) is equal to the number of poles of the rational function P a (counted up to multiplicity). 2.59–2.60. These results are due to the authors. Corollary 2.59 appeared first in B¨ ottcher, Silbermann [106, 4.6] and Theorem 2.60 was first published in B¨ottcher, Silbermann [113]. 2.61–2.66. The results of 2.61–2.65 were established by Douglas [159], [162, 7.36] (for p = 2). Theorem 2.66 is new and due to the authors. Our presentation also relies on Sarason [452]. 2.67–2.69. Simonenko [491], [496] and Gohberg, Krupnik [232]. In the case p = 2 these results were also obtained by Douglas and Sarason [164], who used Glicksberg’s theorem 1.22 (see 2.83). 2.71–2.72. The notion of local sectoriality was introduced by Simonenko [496] and Douglas, Widom [166]. Theorem 2.72 is in Simonenko [491], Devinatz [151], Douglas, Widom [166]. 2.74. Widom [556], Simonenko [491], Devinatz [151], Gohberg [218]. 2.75. Douglas, Sarason [164]. 2.77–2.81. See Hoffman [284], Gamelin [209], Garnett [211], Gelfand, Raikov, Shilov [212]. The algebra QC was introduced by Douglas. 2.82. This definition is from Clancey, Gosselin [139]. 2.83. This theorem was established by Axler [12] using transfinite localization (Axler’s method will be described in Chapter 4). For B = C, the result goes over into Theorem 2.67 (p = 2). The proof presented here first appeared in B¨ottcher [69] and is new in the following sense: on the one hand it is not terribly new, since it mimics the argument used by Douglas and Sarason [164] to prove this theorem for B = C, and on the other hand it is strange that

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109

Axler (who wrote his dissertation under the supervision of Sarason) did not at the very least mention this possibility of proving the theorem in [12]. 2.84–2.85. The term “locally sectorial over QC” had already been used by Douglas [161], a systematic study of symbols which are locally sectorial over C ∗ -algebras between C and QC has begun in Silbermann [483]. There Theorem 2.85 was established for B = QC. For B = C + H ∞ , Theorem 2.85 is due to B¨ottcher [69]. Note that Theorem 2.85 is neither an immediate consequence of Theorem 2.83 nor of Theorems 4.63 and 4.64. 2.86. Part (c) goes back to Simonenko [491] and Douglas, Widom [166], parts (a) and (b) were probably first proved in B¨ ottcher [69]. The fact that every function which is locally sectorial over QC can be written as a product of a function in GQC and a sectorial function is nontrivial and is a key result, which simplifies the theory of Toeplitz operators with symbols that are locally sectorial over QC substantially. We remark that it was the search for a proof of this result which led us to Glicksberg’s theorem 1.22 and, subsequently, to the proofs of Theorems 2.83 and 2.85 given here. For the matrix case see 3.7, 3.8, 4.31, and the remark after 4.49. 2.87. Sarason [456]. 2.88. The result was established in Silbermann [483], the proof presented here is from B¨ ottcher [69]. 2.90. For B = C, both the result and the proof are taken from Clancey [135] (see also Clancey, Morrel [140]). Once Gorkin [239] had shown that M (H ∞ |Xξ ) (ξ ∈ M (QC)) is connected, Silbermann [483] stated this lemma for B = QC. 2.91. See Silbermann [483] for B = QC. The present proof is taken from B¨ottcher [69]. 2.93. Simonenko [492]. Also see Devinatz, Shinbrot [155]. 2.94. Douglas [160] for p = 2 and Spitkovsky [502] for H p . 2.95–2.96. Both theorems are well known. Theorem 2.96 for B = C goes back to Simonenko [492]. It is clear that 1.32 is the appropriate tool to prove 2.95. That 2.96 (in the case B = QC) can be proved with the help of 1.32 is less obvious and requires an argument which was also used by Axler [12, pp. 39–40].

3 Symbol Analysis

3.1 Local Sectoriality 3.1. Definitions. Let F be a closed subset of X = M (L∞ ) and let a be in L∞ N ×N . The matrix function a is called analytically sectorial on F if there exist a real number ε > 0 and two invertible matrices b, c ∈ CN ×N such that Re (ba(x)c) ≥ ε for all x ∈ F , that is, Re (ba(x)cz, z) ≥ εz2

∀x∈F

∀ z ∈ CN ,

and is said to be geometrically sectorial on F if conv a(F ) ⊂ GCN ×N , that is, if each matrix in the closed convex hull of a(F ) is invertible. It is easy to see that a scalar-valued function (N = 1) is analytically sectorial on F if and only if it is geometrically sectorial on F . In this case the function is simply called sectorial on F , which is in accordance with 2.84. Functions which are (analytically or geometrically) sectorial on the whole maximal ideal space X will be called (analytically or geometrically) sectorial. In the scalar case this agrees with 2.16. Let B be a closed subalgebra of C + H ∞ containing the constants. A function a ∈ L∞ N ×N will be called (analytically or geometrically) locally sectorial over B if it is (analytically or geometrically) sectorial on each maximal antisymmetric set for B. In case B is a C ∗ -subalgebra (of QC), the fibers Xβ , β ∈ M (B), occupy the place of the maximal antisymmetric sets (see 1.27(c)). 3.2. Proposition. If a ∈ L∞ N ×N is analytically sectorial on a closed subset F of X, then a is geometrically sectorial on F . 

)c) ≥ ε, then Re b λ a(x )c ≥ ε whenever λi ≥ 0 and Proof. If Re (ba(x i i i i

λ = 1. But if Re d ≥ ε > 0 for a matrix d ∈ C N ×N , then d ∈ GCN ×N . So, i i

since b, c ∈ GCN ×N , we conclude that i λi a(xi ) is invertible for all λi ≥ 0  such that i λi = 1. 

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Remark. Azoff and Clancey [16] constructed an example of a matrix function a ∈ L∞ 2×2 which is geometrically sectorial but not analytically sectorial. The following lemma is needed to prove Theorem 3.4, which represents an important special case in which geometrical sectoriality implies analytic sectoriality. 3.3. Lemma. (a) Let Jλ be the m × m ⎛ λ 1 ⎜ 0 λ ⎜ Jλ = ⎜ ⎜ 0 0 ⎝... ... 0 0

Jordan cell ⎞ 0 ... 0 1 ... 0 ⎟ ⎟ λ ... 0 ⎟ ⎟ ... ... ...⎠ 0 ... λ

and suppose the origin does not belong to the line segment [λ, 1]. Then there are B, C ∈ GCm×m and δ > 0 such that Re (BC) ≥ δ > 0,

Re (BJλ C) ≥ δ > 0.

(b) Let E, F ∈ Cm×m and suppose  det µE + (1 − µ)F = 0

∀µ ∈ [0, 1].

Then there are B, C ∈ GCm×m and δ > 0 such that Re (BEC) ≥ δ > 0, Proof. (a) Put ⎛ 0 ⎜ 0 ⎜ N =⎜ ⎜... ⎝ 0 0

1 0 ... 0 0

0 1 ... 0 0

... ... ... ... ...

⎞ 0 0 ⎟ ⎟ ...⎟ ⎟, 1 ⎠ 0

Re (BF C) ≥ δ > 0. ⎛

β m−1 β m−2 ⎜ 0 β m−2 ⎜ 0 Vβ = ⎜ ⎜ 0 ⎝ ... ... 0 0

β m−3 β m−3 β m−3 ... 0

... ... ... ... ...

β β β ... 0

⎞ 1 1 ⎟ ⎟ 1 ⎟ ⎟. ...⎠ 1

/ [λ, 1], there If β = 0, then Vβ N = βN Vβ , whence Vβ N Vβ−1 = βN . Since 0 ∈ are ν ∈ T and α > 0 such that Re ν ≥ α > 0,

Re (νλ) ≥ α > 0.

Consequently, if we let U = diag (ν, . . . , ν), then Re (U I) ≥ α > 0,

Re (U λI) ≥ α > 0.

Thus, Re (U Vβ IVβ−1 ) = Re (U I) ≥ α > 0, Re (U Vβ Jλ Vβ−1 ) = Re (U Vβ (λI + N )Vβ−1 ) = Re (U λI) + Re (βU N ) ≥

α >0 2

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113

if only β is sufficiently small. It follows that then B = U Vβ , C = Vβ−1 , δ = α/2 have the desired properties. (b) The hypothesis implies that E and F are invertible. There is a D in GCm×m such that D−1 E −1 F D = J is in the Jordan canonical form. We have det(µI + (1 − µ)J) = det(µD−1 E −1 ED + (1 − µ)D−1 E −1 F D) = det(D−1 E −1 ) det(µE + (1 − µ)F ) det D = 0 for all µ ∈ [0, 1]. The matrix J is block diagonal, J = block diag (Jλk ) with each Jλk of the form as in part (a). Because  det(µI + (1 − µ)Jλk ) ∀ µ ∈ [0, 1], 0 = det(µI − (1 − µ)J) = k

we conclude that 0 ∈ / [λk , 1] for each k. So part (a) ensures the existence of matrices Bk , Ck and of numbers δk > 0 such that Re (Bk Ck ) ≥ δk > 0,

Re (Bk Jλk Ck ) ≥ δk > 0.

If we let B  = block diag (Bk ), C  = block diag (Ck ), then Re (B  C  ) ≥ δ > 0,

Re (B  JC  ) ≥ δ > 0

with some δ > 0. Now it is easily seen that the matrices B = B  D−1 E −1 and  C = DC  have the desired properties.  3.4. Theorem (Clancey). Let F be a closed subset of X and let a ∈ L∞ N ×N be geometrically sectorial on F . If conv a(F ) is a line segment (i.e., a set of the form
 [z, w] = (1 − λ)z + λw : λ ∈ [0, 1] , where z, w ∈ CN ×N ), then a is analytically sectorial on F . Proof. Since F is compact and a is continuous on X, there are x1 , x2 ∈ F such that conv a(F ) = [a(x1 ), a(x2 )]. Because a is geometrically sectorial on F , the line segment [a(x1 ), a(x2 )] consists of invertible matrices only. So Lemma 3.3(b) can be applied to see that there are b, c ∈ GCN ×N and ε > 0 such that Re (ba(x1 )c) ≥ ε > 0, Re (ba(x2 )c) ≥ ε > 0. This implies that   Re b λa(x1 ) + (1 − λ)a(x2 ) c ≥ ε > 0 ∀ λ ∈ [0, 1], whence Re (ba(x)c) ≥ ε > 0 for all x ∈ F .  

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Important convention. In what follows we shall mainly deal with matrix functions that are analytically sectorial on closed subsets of X. Therefore a matrix function which is analytically sectorial on a set F or over an algebra B will henceforth be simply called sectorial on F or over B, i.e., in the following “sectorial” always means “analytically sectorial.” The representations stated in Proposition 2.86 for scalar-valued locally sectorial functions played a crucial role in the local theory of scalar Toeplitz operators and it is therefore desirable to have analogous representations for locally sectorial matrix functions. We first show how the arguments used to prove Propositions 2.86(a), (b) can be extended to the matrix case. 3.5. Theorem (Machado/Szyma´ nski). Let Y be a compact Hausdorff space and B a closed subalgebra of C(Y ) containing the constants. Let S denote the family of maximal antisymmetric sets for B. Then, for every a ∈ [C(Y )]N ×N , dist(a, BN ×N ) = max distS (a, BN ×N ), S∈S

where for a closed subset F of Y , distF (a, BN ×N ) :=

inf

max a(y) − b(y)L(CN ) ,

b∈BN ×N y∈F

dist(a, BN ×N ) refers to distY (a, BN ×N ), and the norm on CN is the one given by (1.9). Proof. For a proof see Machado [342], Szyma´ nski [512], Burckel [127], Ransford [415], or P´erez Carreras and Bonet [391].   3.6. Lemma. Let X = M (L∞ ) and let F be a closed subset of X. (a) A matrix function a ∈ L∞ N ×N is sectorial on F if and only if there are a d ∈ GCN ×N and an ε > 0 such that Re (a(x)d) ≥ ε for all x ∈ F . (b) A matrix function a ∈ L∞ N ×N is sectorial on F if and only if there is a d ∈ GCN ×N such that I − a(x)dL(CN ) < 1 for all x ∈ F . −1 (c) Let u ∈ GL∞ (x) = u∗ (x) for all N ×N be unitary-valued on F , i.e., u x ∈ F . Then u is sectorial on F if and only if distF (u, CN ×N ) < 1. ∞ (d) Let B be any subset of L∞ N ×N , let u ∈ GLN ×N be unitary-valued on −1 ∗ X (u (x) = u (x) for all x ∈ X), and suppose distX (u, B) < 1. Then there are a b ∈ B and a sectorial matrix function s ∈ GL∞ N ×N such that u = sb.

Proof. (a) If Re (ba(x)cz, z) ≥ δz2 for all x ∈ F with some b, c ∈ GCN ×N and δ > 0, then Re (a(x)c(b∗ )−1 b∗ z, b∗ z) = Re (a(x)cz, b∗ z) = Re (ba(x)cz, z) ≥ δz2 ≥ δb∗ −2 b∗ z2 ∀ z ∈ CN

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115

and so d = c(b∗ )−1 and ε = δb∗ −2 have the desired property. (b) The “if” portion follows from the observation that Re (a(x)d) = I − Re (I − a(x)d) ≥ I − I − a(x)d. On the other hand, if a is sectorial on F , then, by part (a), there are c ∈ GCN ×N and δ > 0 such that Re (a(x)c) ≥ δ for all x ∈ F . Put α := max a(x)c (> 0) and ε := δ/α2 . Then, for zCN = 1, x∈F

(I − a(x)c)z2 = 1 − 2εRe (a(x)cz, z) + ε2 a(x)cz2 δ2 ≤ 1 − 2εδ + ε2 α2 = 1 − 2 < 1, α which gives the assertion with d = εc. (c) From part (b) we deduce that there is a d ∈ CN ×N such that I − u(x)d ≤ 1 for all x ∈ F . Consequently, u(x) − d∗  = u∗ (x) − d = u(x)(u∗ (x) − d) = I − u(x)d < 1 for all x ∈ F , i.e., distF (u, CN ×N ) < 1. Conversely, if there is a c ∈ CN ×N with u(x) − c < 1 − δ < 1 for all x ∈ F , then I − u(x)c∗  = u(x)(u∗ (x) − c∗ ) = u∗ (x) − c∗  < 1 − δ for x ∈ F , which implies that, for zCN = 1, 2Re (u(x)c∗ z, z) = 1 + u(x)c∗ z2 − (I − u(x)c∗ )z2 ≥ 1 − I − u(x)c∗ 2 ≥ δ > 0 ∀ x ∈ F. (d) If there is a b ∈ B with u − b < 1 − δ < 1, then I − bu−1  = (u − b)u∗  < 1 − δ, hence, for zCN = 1, 2Re (bu−1 z, z) = 1 + bu−1 z2 − (I − bu−1 )z2 ≥ 1 − I − bu−1 2 > δ > 0 and thus bu−1 = s with s satisfying Re s(x) ≥ δ/2 for all x ∈ X. It follows that u = s−1 b, and because Re (s−1 z, z) = Re (s−1 sy, sy) = Re (y, sy) ≥ =

δ −1 2 δ s z ≥ s−2 z2 2 2

 s−1 must be sectorial (on X). 

δ y2 2

∀ z ∈ CN ,

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3 Symbol Analysis

3.7. Proposition. Let B be a C ∗ -subalgebra of L∞ containing the constants −1 (x) = u∗ (x) for all x ∈ X. and let u ∈ GL∞ N ×N be unitary-valued, that is, u Then for u to be locally sectorial over B it is necessary and sufficient that u be of the form u = sb with b ∈ GBN ×N and s ∈ GL∞ N ×N being sectorial. Proof. If u is locally sectorial over B, then, by Lemma 3.6(c), distXξ (u, BN ×N ) = distXξ (u, CN ×N ) < 1 ∀ ξ ∈ M (B), and Theorem 3.5 in conjunction with 1.27(c) implies that dist(u, BN ×N ) < 1. Now Lemma 3.6(d) gives that there are a b ∈ BN ×N and a sectorial ∞ s ∈ GL∞ N ×N such that u = sb. Since obviously b ∈ GLN ×N , we deduce from 1.26(d) that actually b ∈ GBN ×N . This proves the necessity portion. The sufficiency part is trivial.   We now remove the restriction to unitary-valued matrix-functions by using other (even more elementary) techniques. 3.8. Theorem. Let B be a C ∗ -algebra between C and L∞ and let a ∈ L∞ N ×N . Then a is locally sectorial over B if and only if a is of the form a = sb where s ∈ GL∞ N ×N is sectorial and b ∈ GBN ×N . Proof. It suffices to prove the “only if” portion. So suppose a is locally sectorial over B. For ξ ∈ M (B), define 
Dξ := d ∈ CN ×N : I − a(x)dL(CN ) < 1 ∀ x ∈ Xξ . Lemma 3.6(b) implies that Dξ is nonempty. It is clear that Dξ is an open convex subset of CN ×N . If d ∈ Dξ , then d ∈ Dη for all η in some open neighborhood U  (ξ) ⊂ M (B) of ξ; this follows from the upper semi-continuity of the mapping M (B) → R+ ,

ξ → max I − a(x)d, x∈Xξ

which, in turn, can be derived from Theorem 1.35(b) in the setting A = L∞ N ×N and B = {ϕIN ×N : ϕ ∈ B}. Associate with each ξ ∈ M (B) a matrix dξ ∈ Dξ and a neighborhood U  (ξ) of ξ such that dξ ∈ Dη for all η ∈ U  (ξ). Because M (B) is a compact Hausdorff space, it is a normal space and hence there are open neighborhoods U (ξ) such that ξ ∈ U (ξ) ⊂ clos U (ξ) ⊂ U  (ξ). By the compactness of M (B), there are ξ1 , . . . , ξn in M (B) such that n  U (ξi ). Consider the constant functions M (B) = i=1

fi : clos U (ξi ) → CN ×N ,

ξ → dξi .

3.1 Local Sectoriality

117

Accept for a moment the validity of the following claim: If U and V are open subsets of M (B) such that U \ clos V = ∅, if g is a continuous function on clos U with g(ξ) ∈ Dξ for all ξ ∈ U , and if d is some matrix belonging to Dξ for all ξ in some open neighborhood W (clos V ) of clos V , then there is a continuous function h on clos (U ∪ V ) such that h(ξ) ∈ Dξ for all ξ ∈ U ∪ V . Hence, letting U = U (ξ1 ), V = U (ξ2 ), g = f1 , d = dξ2 , we get a continuous function h1 on clos [U (ξ1 ) ∪ U (ξ2 )] such that h1 (ξ) ∈ Dξ for all ξ ∈ U (ξ1 ) ∪ U (ξ2 ). Then the claim for U = U (ξ1 ) ∪ U (ξ2 ), V = U (ξ3 ), g = h1 , d = dξ3 gives a continuous function h2 on clos [U (ξ1 ) ∪ U (ξ2 ) ∪ U (ξ3 )] with h2 (ξ) ∈ Dξ for all ξ ∈ U (ξ1 ) ∪ U (ξ2 ) ∪ U (ξ3 ). Continuing, we finally arrive at a continuous function h on M (B) with h(ξ) ∈ Dξ for all ξ ∈ M (B), that is, we have an h ∈ BN ×N with I − a(x)h(x)L(CN ) < 1 ∀ x ∈ X.

(3.1)

From (3.1) we see that h ∈ GL∞ N ×N , whence h ∈ GBN ×N . Also by (3.1), the matrix function s = ah is sectorial. Hence, if we let b = h−1 , then a = sb is the desired factorization. It remains to prove the above claim. Since M (B) is a normal space, there is an open neighborhood W  = W  (clos V ) of clos V such that V ⊂ clos V ⊂ W  (clos V ) ⊂ W (clos V ),

U  := clos U \ W  (clos V ) = ∅.

Put V  = clos (V \ U ) and notice that U ∪ V ⊂ U  ∪ V  ∪ (W  ∩ U ). The sets U  and V  are closed and U  ∩ V  = ∅. Hence, by Uryson’s extension theorem (see, e.g., Naimark [362, p. 43]), there exists a continuous function ϕ on clos (U ∪ V ) such that 0 ≤ ϕ ≤ 1, ϕ|U  = 1, ϕ|V  = 0. Extend g arbitrarily to a function on clos (U ∪ V ) and put h = gϕ + d(1 − ϕ). Because gϕ is continuous on clos U and vanishes identically on V  = clos (V \ U ), it follows that h is continuous on clos (U ∪ V ). If ξ ∈ U  , then h(ξ) = g(ξ) ∈ Dξ and if ξ ∈ V  , then h(ξ) = d ∈ Dξ . Finally, if ξ ∈ W  ∩ U , then h(ξ) is a convex linear combination of g(ξ) ∈ Dξ and d ∈ Dξ , which, by the convexity of Dξ ,  implies that h(ξ) ∈ Dξ . This completes the proof of our claim.  We finally state a matrix analogue of 2.86(c). 3.9. Theorem. Let a ∈ L∞ N ×N . Then the following are equivalent: (i) a is locally sectorial over C; (ii) for each τ ∈ T there are open neighborhoods Uτ ⊂ T of τ , matrices bτ , cτ ∈ GCN ×N , and an ετ > 0 such that Re (bτ a(t)cτ ) ≥ ετ

for almost all

t ∈ Uτ ;

(iii) there are a finite number of open subarcs U1 , . . . , Un of T, matrices c1 , . . . , cn ∈ GCN ×N and an ε > 0 such that Re (a(t)ck ) ≥ ε

for almost all

t ∈ Uk ;

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3 Symbol Analysis

(iv) a = sψ with ψ ∈ GCN ×N and s ∈ GL∞ N ×N sectorial ; (v) a = ϕsψ with ϕ, ψ ∈ GCN ×N and s ∈ GL∞ N ×N sectorial. Proof. The implications (iv) =⇒ (v) =⇒ (i) are trivial and the implication (i) =⇒ (iv) results from the preceding theorem. The implication (iii) =⇒ (ii) is also trivial, while the implication (ii) =⇒ (iii) is a consequence of the compactness of T and of Lemma 3.6(a). Finally, the implication (iv) =⇒ (ii) can be verified as in the proof of Proposition 2.86(c) and the implication (ii) =⇒ (i) follows from Proposition 2.79(a).   3.10. Pn C. Let a ∈ L∞ N ×N and τ ∈ T. Each point of CN ×N which belongs to the (compact) set a(Xτ ) is called an essential cluster point of a at τ . (Pn C)N,N is defined as the set of all functions a ∈ L∞ N ×N which have at most n essential cluster points at each point of T. The set (Pn C)1,1 will be abbreviated to Pn C. Example. Let τ ∈ T and let E1 , . . . , En (n ≥ 2) be pairwise disjoint measurn  Ek and U ∩ Ek has positive measure for able subsets of T such that T = k=1

each k ∈ {1, . . . , n} and each neighborhood U ⊂ T of τ . Then

n let α1 , . . . , αn be any pairwise distinct complex numbers and put a = k=1 αk χEk . Due to Proposition 2.79(a) we have a(Xτ ) = {α1 , . . . , αn } and a belongs to Pn C \ Pn−1 C. Let Ak denote the set of all x ∈ Xτ for which a(x) = ατ . It is easy to see that χEk (x) = 1 for x ∈ Ak and χEk (x) = 0 for x ∈ Xτ \ Ak . Note that, for N > 1 and n > 1, (Pn C)N,N is properly contained in (Pn C)N ×N , the collection of all N × N matrix functions whose entries are in Pn C. Indeed, if we let E1 , E2 , E3 , E4 be as in the above example and if α, β are distinct complex numbers, then the entries of the diagonal matrix function diag (αχE1 + αχE2 + βχE3 + βχE4 , αχE1 + βχE2 + αχE3 + βχE4 ) belong to P2 C, while the matrix function itself takes the four values diag (α, α),

diag (α, β),

diag (β, α),

diag (β, β)

on A1 , A2 , A3 , A4 , respectively. It is clear that P CN ×N (see 2.73) is contained in (P2 C)N ×N and that there are functions in (P2 C)N,N that do not belong to P CN ×N . Finally, it is obvious that (P1 C)N ×N = CN ×N . The following proposition characterizes the matrix functions in (P2 C)N,N that are locally sectorial over C. 3.11. Proposition. Let a ∈ (P2 C)N,N , and for τ ∈ T denote by a1τ and a2τ the essential cluster points of a at τ (it may be that a1τ = a2τ ). Then the following are equivalent: (i) a is locally sectorial over C;

3.2 Asymptotic Multiplicativity

119

(ii) for each τ ∈ T, the (possibly degenerate) line segment [a1τ , a2τ ] consists of invertible matrices only; (iii) det[(1 − µ)a1τ + µa2τ ] = 0

∀ (τ, µ) ∈ T × [0, 1].

Proof. The equivalence (ii) ⇐⇒ (iii) is trivial. To establish the equivalence (i) ⇐⇒ (ii), notice first that conv a(Xτ ) = [a1τ , a2τ ]. Therefore, by Proposition 3.2, (i) implies (ii). On the other hand, Theorem 3.4 (or Lemma 3.3(b)) shows that (i) is a consequence of (ii).  

3.2 Asymptotic Multiplicativity 3.12. The Poisson kernels. For ϕ ∈ L1 , we defined hr ϕ (0 < r < 1), the Abel-Poisson means of the Fourier series (= harmonic extension), as  r|l| ϕl eilx , x ∈ [0, 2π). (hr ϕ)(eix ) = l∈Z

Using the Poisson kernel, we have ( 2π kr (x − t)ϕ(eit ) dt, (hr ϕ)(eix ) =

x ∈ [0, 2π),

0

where kr (x) =

1 − r2 1 , 2π 1 − 2r cos x + r2

x ∈ R.

The slight change in notation (recall how kr was defined in 1.37) should not cause confusion. If we extend ϕ to a function Φ ∈ L1loc (R) periodically, i.e., if we set Φ(x) = ϕ(eix ), x ∈ R, then hr ϕ can be written as ( ∞ (hr ϕ)(eix ) = λK(λ(x − t))Φ(t) dt, x ∈ R, −∞

where K(x) = 1/(π(1 + x2 )) and λ = −1/ log r ∈ (0, ∞) (see Ahiezer [3, Section 62]). 3.13. The Fej´ er kernels. The Fej´er (or Fej´er-Cesaro) means σn ϕ, n ∈ N, of a function ϕ ∈ L1 are defined in terms of Fourier coefficients by n   (σn ϕ)(e ) = 1− ix

l=−n

|l| n+1

 ϕl eilx ,

x ∈ [0, 2π).

We also have (

2π

kn (x − t)ϕ(eit ) dt,

(σn ϕ)(eix ) = 0

x ∈ [0, 2π),

120

3 Symbol Analysis

where the (Fej´er) kernel kn is given by kn (x) =

sin2 ((n + 1)x/2) 1 , 2π(n + 1) sin2 (x/2)

x ∈ R.

Let Φ ∈ L1loc (R) denote the periodic extension of ϕ, Φ(x) = ϕ(eix ) for x ∈ R. Then ( ∞  (n + 1)K (n + 1)(x − t) Φ(t) dt, x ∈ R, (σn ϕ)(eix ) = −∞

with

2 sin2 (x/2) , π x2 (again see Ahiezer [3, Section 62]). K(x) =

x∈R

3.14. Approximate identities. The Poisson and Fej´er kernels are typical examples of what is usually called an approximate identity. Let K be a function in L1 (R) which has the following properties: ( ∞ K(x) ≥ 0, K(x) = K(−x), K(x) dx = 1, (3.2) −∞

0 < ess inf K(x) ≤ ess sup K(x) < ∞, x∈(−π,π)

(3.3)

x∈(−π,π)

there is a constant M > 0 such that K(x) ≤

M for |x| ≥ 1. x2

(3.4)

The generalized sequence {Kλ }λ∈Λ , where Λ = {l0 , l0 + 1, l0 + 2, . . .} (l0 ∈ N)

or

Λ = (r0 , ∞)

(r0 ∈ R+ ),

and Kλ (x) = λK(λx),

x ∈ R,

will be called the approximate identity generated by K. Given ϕ ∈ L1 let Φ ∈ L1loc (R) denote the periodic extension of ϕ. Then define ( ∞ (kλ Φ)(x) :=

−∞

λK(λ(x − t))Φ(t) dt,

x ∈ R,

and put (kλ ϕ)(eix ) := (kλ Φ)(x),

x ∈ [0, 2π).

Thus, for λ ∈ Λ, kλ Φ is a 2π-periodic function in L1loc (R) and kλ ϕ is a function in L1 = L1 (T). Finally we shall sometimes write kλ,t ϕ := (kλ ϕ)(t) (t ∈ T),

kλ,x Φ := (kλ Φ)(x) (x ∈ R).

The following facts are well known and can be verified without substantial difficulty.

3.2 Asymptotic Multiplicativity

121

(a) If a ∈ L∞ , then kλ a ∈ C for all λ ∈ Λ. (b) If a ∈ L∞ , then sup kλ a∞ ≤ a∞ . λ∈Λ

(c) If a ∈ L , then kλ a − aL2 → 0 as λ → ∞.

Let ϕ(x) = l ϕl eilx be a Laurent polynomial. Then ( ∞  ix λK(λt) ϕl eil(x−t) dt (kλ ϕ)(e ) = 2

−∞

=



ϕl eilx

l

=

 l

where  K(y) =

(

∞

 ϕl K

l

(

∞

λK(λt)e−ilt dt

−∞

l eilx λ

K(x)e−ixy dx =

−∞

(

(x ∈ R),

(3.5)

∞

K(x) cos(yx) dx.

(3.6)

−∞

 ∈ C(R), K(±∞)    So K = 0, K(0) = 1 and K(y) < 1 for y = 0. This can be used to derive the following fact, which will be needed later. (d) If (kλn χ1 )(eiθn ) → χ1 (τ ) = τ as n → ∞ (τ ∈ T), then eiθn → τ and λn → ∞ as n → ∞. If K is the Poisson kernel, then (d) says that eiθn → τ and rn = e−1/λn → 1 whenever rn eiθn → τ ; thus, in that case (d) is trivial. Finally, note that  K(y) = e−|y| if K is the Poisson kernel, ! 1 − |y| (|y| ≤ 1)  K(y) = if K is the Fej´er kernel. 0 (|y| ≥ 1) 3.15. Asymptotic multiplicativity. Let A and B be subsets of L∞ and let {Kλ }λ∈Λ be an approximate identity. We say that {Kλ }λ∈Λ is asymptotically multiplicative on the pair (A, B) if kλ (ab) − (kλ a)(kλ b)∞ → 0 as

λ → ∞ ∀ a ∈ A,

∀ b ∈ B.

Thus, Lemma 2.61 says that the Poisson kernels are asymptotically multiplicative on the pair (C, L∞ ), Lemma 2.87 states that the Poisson kernels are even asymptotically multiplicative on the pair (QC, L∞ ), and the statement of Theorem 2.62(c) is the asymptotic multiplicativity of the Poisson kernels on the pair (C + H ∞ , C + H ∞ ). Minor and obvious modifications of the proof of Lemma 2.61 imply that every approximate identity is asymptotically multiplicative on the pair (C, L∞ ). The purpose of what follows is to show that this remains true for the pair (QC, L∞ ).

122

3 Symbol Analysis

If an approximate identity is asymptotically multiplicative on the pair (H ∞ , H ∞ ), then it is so on the pair (C + H ∞ , C + H ∞ ). Thus, in that case the argument of the proof of Lemma 2.87 gives its asymptotic multiplicativity on the pair (QC, L∞ ). In this connection and as motivation for our further investigations notice the following. 3.16. Remark. In 1988, Wolf and Havin [580] showed that shifts and contractions of the Poisson kernel, i.e. kernels of the form K(x) =

ε 1 , 2 π ε + (h − x)2

are the only approximate identities which are asymptotically multiplicative on the pair (H ∞ , H ∞ ). Our starting point is the following characterization of QC. 3.17. Theorem (Sarason). QC = L∞ ∩ V M O. Proof. Let a ∈ L∞ ∩V M O. By 1.48(h), there are u, v ∈ C such that a = u+ v). v ∈ H ∞ . Therefore, Hence v) = a − u ∈ L∞ , and it follows that v + i) a = −i(v + i) v ) + (u + iv) ∈ H ∞ + C. If a ∈ L∞ ∩ V M O, then obviously a ∈ L∞ ∩ V M O, and the same reasoning gives a ∈ H ∞ + C. Thus a ∈ QC. Conversely, let a ∈ QC. Then a = b + ic, where b and c are real-valued u) + (v + iw) with functions in QC. Since b ∈ H ∞ + C, we have b = (u + i) ) = −w, whence u + i) u ∈ H ∞ and v + iw ∈ C. Because b is real-valued, u ) u = −u ) = w, ) and it results that b = u + v = w ) + v, where w ∈ C and v ∈ C. Again by 1.48(h), b ∈ V M O. The same argument gives that c ∈ V M O. Thus  a ∈ V M O ∩ L∞ .  3.18. Lemma. Let ϕ ∈ L1 and Φ(x) := ϕ(eix ), x ∈ R. Then (a) ϕ ∈ BM O ⇐⇒ Φ ∈ BM O(R); (b) cΦ∗ ≤ ϕ∗ ≤ Φ∗ with some absolute constant c; (c) ϕ ∈ V M O ⇐⇒ Φ ∈ V M O(R). Proof. (a), (b) It is clear that ϕ ∈ BM O if Φ ∈ BM O(R) and that then ϕ∗ ≤ Φ∗ . So let ϕ ∈ BM O and let J be a finite interval of R. Without loss of generality assume J = [−δ, 2πn], where 0 < δ < 2π and n ∈ N. Put I = [−δ, 0]. Then  ( ( ( 2π 1 1 |Φ − Φ[0,2π] | dt = |Φ − Φ[0,2π] | dt + n |Φ − Φ[0,2π] | dt |J| J δ + 2πn I 0 and we have

3.2 Asymptotic Multiplicativity

n δ + 2πn

(

2π

|Φ − Φ[0,2π] | dt ≤ 0

1 2π

(

123

2π

|Φ − Φ[0,2π] | dt ≤ ϕ∗ . 0

Further, 1 δ + 2πn

( |Φ − Φ[0,2π] | dt ≤ I

1 δ

( |Φ − ΦI | dt + I

1 δ + 2πn

( |Φ[0,2π] − ΦI | dt, I

and the first term is clearly not greater than ϕ∗ . The second term equals   2π δ δ δ ≤ |Φ[−2π,0] − ΦI | = const · log ϕ∗ |Φ[−2π,0] − ΦI | δ + 2πn 2π δ 2π (1.48(r), (i), (ii)), which is not greater than const·ϕ∗ . Now 1.48(a) completes the proof. (c) This is immediate from (a), (b) together with 1.48(f), (p).   3.19. The moving average. This is the approximate identity generated by K(x) =

1 2π

for

x ∈ (−π, π),

K(x) = 0 for x ∈ / (−π, π).

Note that ( kλ,x Φ = (kλ Φ)(x) = =

∞ −∞

λ 2π

(

 Φ(t)λK λ(x − t) dt x+π/λ

Φ(t) dt = Φ[x−π/λ,x+π/λ] . x−π/λ

Moreover, the “norm” Φ∗ on BM O(R) is nothing else than sup sup kλ,x (|Φ − kλ,x Φ|) λ>0 x∈R

and if Φ ∈ BM O(R), then, by definition, Φ ∈ V M O(R) ⇐⇒ lim sup kλ,x (|Φ − kλ,x Φ|) = 0. λ→∞ x∈R

(3.7)

The proof of the following proposition shows how (3.7) can be used to study asymptotic multiplicativity. 3.20. Proposition. The moving average is asymptotically multiplicative on the pair (QC, L∞ ). Proof. Let ϕ ∈ QC and a ∈ L∞ . Let Φ ∈ L∞ (R) and A ∈ L∞ (R) denote the periodic extensions of ϕ and a, respectively. Then with K as in 3.19,

124

3 Symbol Analysis

|(kλ ΦA)(x) − (kλ Φ)(x)(kλ A)(x)| + +( ∞ + + + [Φ(t) − (kλ Φ)(x)]A(t)Kλ (x − t) dt++ =+ −∞ ( ∞ ≤ A∞ |Φ(t) − (kλ Φ)(x)|Kλ (x − t) dt −∞

= A∞ kλ,x (|Φ − kλ,x Φ|). Theorem 3.17 and Lemma 3.18 imply that Φ ∈ L∞ (R) ∩ V M O(R) and so the assertion follows from (3.7).   3.21. Theorem. Let {Kλ }λ∈Λ be an approximate identity with generating kernel K. For ϕ ∈ L1 , define ϕK := sup sup kλ,t (|ϕ − kλ,t ϕ|). λ∈Λ t∈T

Then, if ϕ ∈ L1 ,

ϕ ∈ BM O ⇐⇒ ϕK < ∞.

Moreover, there are constants c1 and c2 depending only on K (and Λ) such that c1 ϕK ≤ ϕ∗ ≤ c2 ϕK . Proof. Put Φ(x) = ϕ(eix ), x ∈ R, and let ΦK := sup sup kλ,x (|Φ − kλ,x Φ|). λ∈Λ x∈R

Since in this definition sup may be replaced by x∈R

sup , we have ΦK = ϕK . x∈[0,2π)

Recall that Λ is either {λ0 , λ0 + 1, λ0 + 2, . . .}, where λ0 ≥ 1 is an integer, or the interval (λ0 , ∞), where λ0 > 0. First suppose ΦK < ∞. Let I be any interval of R whose length is less than δ0 := min{π, π/λ0 }: I = (x − δ, x + δ), 0 < δ < δ0 /2. Due to (3.3) there is a constant m > 0 such that K(τ ) ≥ m for (almost all) τ ∈ (−π, π). Choose a λ1 ∈ Λ so that π/(4δ) < λ1 < π/δ (this is always possible). Then, if t ∈ I ⊂ (x − π/λ1 , x + π/λ1 ), we have Kλ1 (x − t) = λ1 K(λ1 (x − t)) ≥ λ1 m > hence Kλ1 (x − t) ≥

πm 1 χ(−δ,δ) (x − t) 2 |I|

πm , 4δ

∀ t ∈ R,

and it follows that ( ( ∞ 1 1 |Φ(t) − kλ1 ,x Φ| dt ≤ |Φ(t) − kλ1 ,x Φ|χ(−δ,δ) (x − t) dt |I| I |I| −∞ ( ∞ 2 2 ΦK . ≤ |Φ(t) − kλ1 ,x Φ|Kλ1 (x − t) dt ≤ πm −∞ πm

3.2 Asymptotic Multiplicativity

125

Thus, 1.48(a) implies that Mδ0 (ϕ) = const · ΦK and 1.48(b) shows that ϕ ∈ BM O and ϕ∗ ≤ c2 ΦK . Now suppose ϕ ∈ BM O. By virtue of Lemma 3.18, Φ ∈ BM O(R). Let x ∈ R, λ ∈ Λ, and put Ij = [x − 2j /λ, x + 2j /λ] for j ≥ 0. Note that |Ij | = 2j+1 /λ. We have ( ∞ ( |Φ(t) − ΦI0 |Kλ (x − t) dt ≤ |Φ(t) − ΦI0 |Kλ (x − t) dt −∞

+

+

I0

∞ ( 

|Φ(t) − ΦIj |Kλ (x − t) dt

j=1 Ij \Ij−1 ∞ (  j=1

|ΦIj − ΦI0 |Kλ (x − t) dt.

Ij \Ij−1

(3.8)

For the first term in (3.8) we get ( ( |Φ(t) − ΦI0 | Kλ (x − t) dt ≤ K∞ λ |Φ(t) − ΦI0 | dt I0 I0 ( 2 |Φ(t) − ΦI0 | dt ≤ const · Φ∗ . = K∞ |I0 | I0 In view of (3.4) we have max

2j−1 ≤λ|x−t|≤2j

 K λ(x − t) ≤

M (2j−1 )2

=

4M . 22j

(3.9)

Therefore, the second term in (3.8) is not greater than ( ∞  4M λ j=1

22j

|Φ(t) − ΦIj | dt = Ij

( ∞  4M λ 2j+1 1 |Φ(t) − ΦIj | dt 22j λ |Ij | Ij j=1

≤ 8M Φ∗

∞  1 = 8M Φ∗ . 2j j=1

Finally, again using (3.9) we see that the third term in (3.8) is not greater than ( ∞ ∞   4M λ 8M |Φ − Φ | dt = |ΦIj − ΦI0 | I I j 0 2j 2 2j Ij j=1 j=1 ≤

∞  8M j=1

2j

const · jΦ∗

= const · Φ∗ . Thus, what we have shown is that

(by 1.48(r), (ii))

126

3 Symbol Analysis

(

∞

sup sup λ∈Λ x∈R

−∞

|Φ(t) − ΦI0 |Kλ (x − t) dt ≤ const · Φ∗ ,

where I0 = [x − 1/λ, x + 1/λ]. But ( ∞ ( |Φ(t) − kλ,x Φ|Kλ (x − t) dt ≤ −∞

∞

(3.10)

|Φ(t) − ΦI0 |Kλ (x − t) dt

−∞ ( ∞

+ −∞

|ΦI0 − kλ,x Φ|Kλ (x − t) dt

and the first integral herein admits estimate (3.10), while +( ∞ + + +  + |ΦI0 − kλ,x Φ| = + ΦI0 − Φ(t) Kλ (x − t) dt++ −∞ ( ∞ ≤ |ΦI0 − Φ(t)|Kλ (x − t) dt ≤ const · Φ∗ , −∞

the last “≤” again by (3.10), whence ( ∞ ( |ΦI0 − kλ,x Φ|Kλ (x − t) dt ≤ const · Φ∗ −∞

Thus,

(

∞

sup sup λ∈Λ x∈R

−∞

∞ −∞

Kλ (x − t) dt = const · Φ∗ .

|Φ(t) − kλ,x Φ|Kλ (x − t) dt ≤ cΦ∗

with some constant c depending only on K and Λ. It follows that ϕK =  ΦK < ∞ and Lemma 3.18(b) gives that c1 ϕK ≤ ϕ∗ .  3.22. Lemma. Let {Kλ }λ∈Λ be an approximate identity and let ϕ ∈ V M O. Then kλ ϕ ∈ V M O for all λ ∈ Λ and ϕ − kλ ϕ∗ → 0 as λ → ∞. Proof. Put Φ(x) := ϕ(eix ) for x ∈ R. Then Φ ∈ V M O(R) by Lemma 3.18. For y ∈ R, let Φy (x) := Φ(x − y). Since ( ∞ ( ∞ (kλ Φ)(x) = Φ(t)Kλ (x − t) dt = Φ(x − y)Kλ (y) dy −∞

−∞

and since, by 1.48(q), the mapping R → V M O(R), y → Φy is continuous, kλ Φ can be written as a Bochner integral: ( ∞ kλ Φ = Φy Kλ (y) dy. −∞

This implies that kλ Φ ∈ V M O(R), whence, by Lemma 3.18, kλ ϕ ∈ V M O. Furthermore, we also have ( ∞ Φ − kλ Φ = (Φ − Φy )Kλ (y) dy −∞

3.2 Asymptotic Multiplicativity

and thus

(

Φ − kλ Φ∗ ≤ (

∞ −∞

127

Φ − Φy ∗ Kλ (y) dy

= |y|<δ

( Φ − Φy ∗ Kλ (y) dy +

|y|>δ

Φ − Φy ∗ Kλ (y) dy.

Given any ε > 0 there is, by virtue of 1.48(q), a δ > 0 such that ( ( ε ∞ ε Φ − Φy ∗ Kλ (y) dy ≤ Kλ (y) dy = , 2 2 |y|<δ −∞ and having chosen this δ, there is a λ > 0 such that ( ( ε Φ − Φy ∗ Kλ (y) dy ≤ 2Φ∗ Kλ (y) dy ≤ 2 |y|>δ |y|>δ whenever λ > λ . Application of Lemma 3.18 completes the proof.

 

3.23. Theorem. Every approximate identity is asymptotically multiplicative on the pair (QC, L∞ ). Proof. Let ϕ ∈ QC and a ∈ L∞ , denote the periodic extensions by Φ and A, and note that Φ ∈ L∞ (R) ∩ V M O(R) (Theorem 3.17 and Lemma 3.18) and A ∈ L∞ (R). As in the proof of Proposition 3.20 we see that sup |(kλ ΦA)(x) − (kλ Φ)(x)(kλ A)(x)| ≤ A∞ sup kλ,x (|Φ − kλ,x Φ|). (3.1