Eigenvalues and S-Numbers (Cambridge Studies in Advanced Mathematics 13)

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 13 EDITORIAL BOARD D. J. H. GAR LING D. GORENSTEIN T. TOM DIECK P. WALTERS E...

Author: Albrecht Pietsch

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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 13 EDITORIAL BOARD D. J. H. GAR LING D. GORENSTEIN T. TOM DIECK P. WALTERS

Eigenvalues and s-Numbers

Eigenvalues and s -Numbers ALBRECHT PIETSCH Friedrich Schiller University, Jena, GDR

The right of the Cm ■ erstty of Cambridge to print ami sell all manner of books was granted 19 Heart. 1 Ill in 1534. The t imersity has printed and published continuously since 1584.

C:=2

CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON NEW YORK NEW ROCHELLE MELBOURNE SYDNEY

Published in the socialist countries Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig Published in the non-socialist countries Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 32 East 57th Street, New York, NY 10022, USA 10, Stamford Road, Oakleigh, Melbourne 3166, Australia ® Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig 1987 Licensed edition for Cambridge University Press, 1987 Printed in the German Democratic Republic

Library of Congress cataloguing in publication data Pietsch, A. (Albrecht) Eigenvalues and s-numbers. (Cambridge studies in advanced mathematics 13) Bibliography: p. Includes index. 1. Operator theory. 2. Banach spaces. 3. s-numbers. 4. Eigenvalues. 1. Title. 11. Series. QA329.P5 1987 515.7'24 86-829

British Library cataloguing in publication data Pietsch, Albrecht Eigenvalues and s-numbers. — (Cambridge studies in advanced mathematics) 1. Matrices 2. Eigenvalues I. Title 512.9'434 QA193 ISBN 0-521-32532-3

Preface The idea to write a monograph about Eigenvalue distributions arose almost immediately after I had finished the manuscript of Operator ideals. It was my intention to show how the abstract theory of operators on Banach spaces can successfully be applied in classical analysis. Fortunately enough, several important open problems were solved by W. B. Johnson, H. Kelnig, B. Maurey and J. R. Retherford in 1977 and presented at a conference held in Leipzig. Since that time the underlying techniques have been drastically improved so that now most of the proofs are streamlined (quoted from a letter of H. Kiinig). Many unnamed colleagues, friends and pupils had helped in the production of this work. In particular, I would like to thank H. Jarchow (Zürich) and D. J. H. Garling (Cambridge) for their critical remarks. The latter has also tried to revise my "Queen's English". I am especially grateful to the co-workers of my seminar for stimulating discussions of the manuscript as well as for reading parts of the galleyproofs. During the writing of this book I had the opportunity to visit several mathematical departments and to use their libraries. Among others, I am indebted to the Steklov Mathematical Institute in Moscow, the Banach Center in Warsaw, the Sonderforschungsbereich 72 in Bonn and to the Forschungsinstitut für Mathematik in Zürich. Last, but not least, I obtained a great deal of support from my home university in Jena. I am very grateful to friends and institutions for providing an excellent Xerox service. Furthermore, I wish to express my special thanks to Mrs. G. Girlich for very careful work in producing the typescript of this book. Many classical treatises on integral equations, written by D. Hilbert, M. Bôcher, G. Kowalewski, E. Hellinger, O. Toeplitz and F. Smithies, were published either in Cambridge or in Leipzig. Therefore I regard it as a good omen that this monograph will be jointly issued by publishing houses located in these cities. Moreover, in my opinion such undertakings are valuable contributions by scientists and editors to the realization of a peaceful coexistence of mankind. Obviously, it is a hard job to be married to a mathematician. Such human beings quite often sit in a chair and do nothing, while the housework remains undone. In gratitude for her sympathetic understanding this book is dedicated TO MY WIFE. Jena (GDR), July 1985 Albrecht Pietsch

Contents Introduction

9

Preliminaries

12

12 12 12 14 14 15

A. A.1. A.2. A.3. A.4. A.5.

Operators on finite dimensional linear spaces Finite dimensional linear spaces Operators and matrices Traces Determinants Eigenvalues

B. B.1. B.2. B.3. B.4.

Spaces and operators Operators on quasi-Banach spaces Operators on Banach spaces Duality Finite operators on Banach spaces

16 16 18 19 20

C. C.1. C.2. C.3. CA.

Sequence and function spaces Classical sequence spaces Direct sums of Banach spaces Classical function spaces The metric extension property

22 22 23 23 25

D. D.1. D.2. D.3.

Operator ideals Quasi-Banach operator ideals Some examples of operator ideals Extension of operator ideals

25 25 28 28

E. E.1. E.2. E.3.

Tensor products Algebraic tensor products Banach tensor products Tensor stability of operator ideals

29 29 30 30

F. F.1. E.2. F.3. F.4.

Interpolation theory Intermediate spaces Interpolation methods Real interpolation Interpolation of quasi-Banach operator ideals

31 31 31 32 33

G. G.1. G.2. G.3.

Inequalities The inequality of means The Khintchine inequality Asymptotic estimates

33 33 34 36

Chapter 1. Absolutely summing operators

37

1.1. 12. 1.3. 1.4. 1.5. 1.6. 1.7.

Summable sequences Absolutely (r, s)-summing operators Absolutely r-summing operators Hilbert-Schmidt operators Absolutely 2-summing operators Diagonal operators Nuclear operators

37 41 45 54 57 60 64

Contents

Chapter 2. s-Numbers 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2 3. 2.8. 2.9. 2.10. 2.11.

Lorentz sequence spaces Axiomatic theory of s-numbers Approximation numbers Gel'fand and Weyl numbers Kolmogorov and Chang numbers Hilbert numbers Absolutely (r, 2)-summing operators Generalized approximation numbers Diagonal operators Relationships between various s-numbers Schatten-von Neumann operators

Chapter 3. Eigenvalues 3.1.

3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9.

135

Related operators

149 151 154 156 158 160 164

The eigenvalue type of operator ideals Eigenvalues of Schatten-von Neumann operators Eigenvalues of s-type operators Eigenvalues of absolutely summing operators Eigenvalues of nuclear operators The eigenvalue type of sums of operator ideals

Fredholm resolvents Traces

Determinants Fredholm denominators Regularized Fredholm denominators The relationship between traces and determinants Traces and determinants of nuclear operators Entire functions

Examples of finite matrices Examples of infinite matrices Hille-Tamarkin matrices Besov matrices Traces and determinants of matrices

Chapter 6. Integral operators 6.1. 6.2. 6.3. 6.4.

73 79 83 90 95 96 97 100 107 115 118

135

Chapter 5. Matrix operators 5.1. 5.2. 5.3. 5.4. 5.5.

73

The Riesz decomposition Riesz operators

Chapter 4. Traces and determinants 4.1. 4. 2 . 4.3. 4.4. 4.5. 4.6. 4.7. 4.8.

7

Continuous kernels Hille-Tamarkin kernels Weakly singular kernels Besov kernels

138

167 168 170 185 195 200 206 210 221 227 227 229 230 232 236 242 242 2 44 248 252

8

Contents

6.5. 6.6.

Traces and determinants of kernels

Fourier coefficients

263 272

Chapter 7. Historical survey

279

Classical background Spaces Operators

280 286 289 295 299 10 2 304

7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7.

Eigenvalues

Determinants Traces

Applications

Appendix

312

Open problems

1 22

Epilogue

324

Bibliography

325

Index

1 54

List of special symbols

358

Introduction At the turn of this century, I. Fredholm created the determinant theory of integral operators. Subsequently, D. Hilbert developed the theory of bilinear forms in infinitely many unknowns. In 1918, F. Riesz published his famous paper on compact operators (vollstetige Transformationen) which was based on these ideas. In particular, he proved that such operators have at most a countable set of eigenvalues which, arranged in a sequence, tend to zero. Nothing was said about the rate of this convergence. On the other hand, I. Schur had already observed in 1909 that the eigenvalue sequence of an integral operator induced by a continuous kernel is square summable. This fact indicates that something got lost within the framework of Riesz's theory. The following problem therefore arises: Find conditions on an operator T which guarantee that the eigen value sequence (4(T)) belongs to a certain subset of co , such as 1,. with 0 < r < cc . In the context of integral operators such conditions were given in terms of integrability and differentiability properties of the underlying kernel. Roughly speaking, the following rule of thumb holds: The smoother the kernel, the quicker the convergence. Corresponding results for abstract operators were found relatively late. The decisive theorem is due to H. Weyl (1949). Using s-numbers, he proved that, for every compact operator T on a complex Hilbert space,

(s„(T)) e

implies (2.„(T)) E 4 .

The first result in the Banach space setting goes back to A. Grothendieck (1955) who showed that the eigenvalue sequence of every nuclear operator is square summable. The extension of Weyl's theorem from Hilbert to Banach spaces was an open problem for many years. The real breakthrough came at the end of the seventies when H. K6nig (1978) proved that

(an(T)) E 4 implies (2.„(T))

Here a,1(T) := inf fiT — L : rank (L)

n}

is the n-th approximation number. Afterwards, based on the theory of operator ideals, many striking results on eigenvalue distributions of abstract operators on Banach spaces were established and successfully applied to various types of integral operators. This demonstrates in a convincing way how tools from modern operator theory yield new results in classical analysis which could not be proved by oldfashioned techniques. The basic concept in the classical approach to the theory of eigenvalue distribution was that of a Fredholm denominator. This is an entire function (V, T) whose zeros are related to the eigenvalues of the operator T by the formula X = — 1. Then T)i

M exp

IM

implies 12„(T)1 5 cn- ' ir.

Another important problem, first raised by J. von Neumann (1932), is the following: Characterize those operators which admit a trace.

10

Introduction

The classical connection between traces and eigenvalues is given by the formula Co

trace (T) = n=1

which only makes sense in the case when the sequence (A „(T)) is absolutely summable. We further note that the Taylor coefficients of 6(4" , T) can be expressed in terms of traces (Plemelj's formulas). It is the intention of this monograph to give an extensive presentation of results connected with the problems mentioned above. The following table provides a rough impression of the interplay of the most important concepts: Abstract theory

absolutely summing operators

s-numbers of operators

Riesz operators

entire functions

Lorentz sequence spaces

eigenvalue distributions

determinants

traces

110.-

Applications

classical sequence spaces

matrix operators]

classical function spaces

Sobolev and Besov I function spaces

integral operators

trigonometric series The highlights of this monograph are the eigenvalue theorems for Weyl operators and for absolutely r-summing operators as well as their applications to integral operators induced by kernels of Hille-Tamarkin or Besov type. A nearly geodesic

Introduction

11

path to these results may be found by studying only those paragraphs indicated by (*). This means that the book contains a sub-book (including less than one third of the material treated in Chapters 1-6) which can serve as a starting point for first reading or as the basis of a seminar. Another feature of this monograph is an extensive historical survey, which is - given in the final chapter. In addition, most of the results are provided with references in the text. In a sense this treatise is a supplement to my monograph Operator ideals. Those readers interested to learn more about the abstract background should consult that book. On the other hand, I have carefully tried to keep this presentation selfcontained. The material is organized as follows: The smallest units are the paragraphs. These are denoted by symbols consisting of' three natural numbers. For instance, 3.6.2 refers to the eigenvalue theorem stated in Chapter 3, Section 6, Paragraph 2. Every paragraph contains precisely one definition, theorem, proposition, lemma or example. Sometimes a remark is added.

Preliminaries In this introductory chapter we summarize some well-known definitions and explain certain terminology used throughout this treatise. Furthermore, we list various results which are employed without proofs. It is assumed that the reader already has a basic knowledge in the theory of Banach spaces, linear and multilinear algebra, general topology and measure theory. An elementary background from complex analysis is also needed. This means that the greater part of the book can be understood by any average graduate student. We further assume some knowledge in interpolation theory (real method). In order to read the last but one chapter it is desirable, but not necessary, to be well-grounded in the theory of Sobolev and Besov spaces.

A.

Operators on finite dimensional linear spaces

In this section we summarize some basic results from linear algebra which are the background and model for all of the following considerations. Proofs may be found, for example, in the textbook of P. R. Halmos (HAL). All linear spaces are considered over the complex field. A.1.

Finite dimensional linear spaces

A.1.1. We denote by 1(n) the n-dimensional linear space of all complex-valued vectors x = (" 1 , The algebraic operations are defined coordinatewise.

A.1.2. Let (u 1 , un) be any basis of a finite dimensional linear space E. Then every element x e E admits a unique representation

i tt i with

E

x

i

=

E C,

and 11

—> E

U:

i=

defines an isomorphism between 1(n) and E. A.2.

Operators and matrices

A.2.1. By an operator T on a finite dimensional linear space E we always mean a linear map from E into itself.

A.2.2.

(au) be any (n, n)-matrix. Then

Let M

M

0P

7i)

(1

puni)

defines an operator on 1(n). In this way we get a one-to-one correspondence between (n, n)-matrices and operators on 1(n). A.2.3. Let T be an operator on a finite dimensional linear space E. Then, for un), there exists a so-called representing matrix M = (,uu) every fixed basis (u1 ,

A. Operators on finite dimensional linear spaces

13

determined by Tu = E i poi for j = 1, i=1

n.

With the operator Mop induced by this matrix, we have the diagram

U-1

U >1(n)

,

Mop

where U is the isomorphism defined in A.1.2. A.2.4. The representing matrix M of a given operator T depends on the underlying u„). If N is the representing matrix with respect to another basis basis (u 1 , (y 1 , v„), then there exists an invertible matrix X such that N = X-1111X. The matrix X = is determined by Vj =

E i=1

Note that the identity operator, denoted by In or I, is always represented by the unit matrix 1(n) = (b We now emphazise the extremely important fact that, by a clever choice of the underlying basis, it can be arranged that the representing matrix of a given operator takes a nice and handy form. For every operator T on a finite dimensional linear space E there exists a basis (u„ ...,un) such that the representing matrix has upper triangular form: A.2.5.

( 11 •• ••

M=

0

ju n „

This means that p i; = 0 whenever i > j. In the particular case when E is equipped with a scalar product we may even find an orthogonal basis (Schur basis) with this property. A.2.6. The preceding result can be improved further. For every operator T on a finite dimensional linear space E there exists a basis (u 1 , . • . 14 such that the representing matrix has the Jordan canonical form: )

0 M = (Mi • 0

Mk) .

The basic Jordan blocks Mt, are matrices, where a complex number ,u, is repeated on the principal diagonal, with ones just above this diagonal and zeros everywhere else:

O) 1 Ph

14

Preliminaries

A.3.

Traces

A.3.1. The trace of an (n, n)-matrix M = () is defined by 71

trace (M) :=

1=1

In the sequel we use the standard properties of this quantity without further explanation. The commutation formula trace (XM) = trace (MX),

which holds for arbitrary (n, n)-matrices M and X, is of particular interest. A.3.2. Let T be an operator on a finite dimensional space. If M and N are representing matrices with respect to different bases, then there exists an invertible matrix X such that N = X-1 MX. Hence trace (N) = trace (X -1 MX) = trace (XX -1 M) = trace (M).

Thus all representing matrices have the same trace, and this common value is defined to be the trace of the operator T. A.4.

Determinants

A.4.1.

The determinant of an (n, n)-matrix M = GO is defined by det (M) := E sign

(27) ■ - r- ln(1) • • •

where the sum is taken over all permutations n of 1 , n}, and sign (n) denotes the signature of 7r. In the sequel we use the standard properties of this quantity without further explanation. The multiplication formula {

det (MN) = det (M) det (N),

which holds for arbitrary (n, /i)-matrices M and N, is of particular interest. A.4.2. If the matrix M = () has triangular form, then det (M) =••

A.4.3. A.4.4.

pi'

•••

[inn •

A matrix M is invertible if and only if det(M)

O.

Let T be an operator on a finite dimensional linear space. If M and N are representing matrices with respect to different bases, then there exists an invertible matrix X such that N = X-1 MX. Hence det (N) = det (X -1 MX) = det (X-1 ) det (M) det (X) = det (M).

Thus all representing matrices have the same determinant, and this common value is defined to be the determinant of the operator T. A.4.5. We now establish Hadamard's inequality which is one of the main tools in proving the existence of determinants of infinite matrices and kernels. Lemma (J. Hadamard 1893: a). Let M = (ttii). Then n

Idet

(*I

n

\

H (E !Nil')

i=1 j= 1

1/2 •

15

A. Operators on finite dimensional linear spaces

Proof. If det (M) = 0, then we have nothing to prove. Therefore it may be assumed that det (M) O. Then the rows for i = 1,

...„(t in)

Mi :=

n

are linearly independent. Applying Schmidt's orthonormalization procedure with respect to the canonical scalar product defined on 1(n), we obtain the vectors Yri =

(nhi 5 • • •

for h = 1,

Thin)

n

which can be viewed as the rows of a unitary matrix Y. Obviously, mi = E h=

for i = 1,

;„ y„

1

n,

where X = ( ih) is a triangular matrix. It follows from (y i„ Hence )1 / 2

n IiiI

n

j=1

i

1/2

E Inu1 2 )

E Itto1 2

=

(

= 6I k that eih = ,

and we obtain Idet

idet (M)I

A.5.

(XY)I = Ida

po det (Y)I

12

n 1$„1

1=1

12

1=

(

n

j=1

1/2

Laii1 2 ) • 1

Eigenvalues

A.5.1. A complex number 2 0 is an eigenvalue of the operator T if there exists E E such that Tx = 20x

arid x o.

R em a rk. The eigenvalues of a matrix M are defined to be the eigenvalues of the operator M0 induced by M on 1(n). A.5.2. Note that 2 0 is an eigenvalue of T if and only if 2,0 I — T fails to be invertible. Hence the eigenvalues coincide with the zeros of the characteristic polynomial

yr(2, T) := det (2./ — T) = An + oc 1 2" +

+ oc„_,2 + oc„.

The coefficients are given by the formula —

Ak :=

it- •

"

i,=1

...

y

ik ,

det

where the right-hand sum is taken over all principal (k, k)-minors of any representing matrix M A.5.3. The algebraic multiplicity of an eigenvalue is defined to be its order as a zero of 7r(2, T). If every eigenvalue is counted according to this multiplicity, then we get a set (2, , 2„) arranged in any order we please. A.5.4. Writing the characteristic polynomial as a product of linear factors yields the determinant formula: det (A/ — T) = fT (2 i=1

—

16

Preliminaries

Equating the coefficients of A' 1 , we obtain the trace formula: trace (T) =

E

Ai .

A.5.5.

Let M = (au) be any triangular matrix representing an operator T. Then it follows from det (AI — T) = 11 (A — p, that the principal diagonal (A 1 , • • • An)•

B.

(11 11 3 • • •3 !Inn)

coincides with the set of eigenvalues

Spaces and operators

Some basic knowledge on Banach space theory is an indispensable assumption for reading this treatise. We use (DUN) and (TAY) as standard references. Throughout the monograph H denotes a complex Hilbert space.

B.1.

Operators on quasi-Banach spaces

B.1.1.

By a quasi-norm M defined on a linear space E we mean a real-valued function with the following properties: (1)

Let x e E. Then II xII = 0 if and only if x = o.

(2)

11.x. + yll

(3)

IAxII =

for x, y E E. cE[11x11 + 1 2 1 11x11 for x e E and A e C.

Here

CE

1 is a constant.

In order to indicate the underlying linear space E, we sometimes replace II xII by the more specific symbol IIx I Em. Formula (2) is called the quasi-triangle inequality. If CE = 1, then I • m is said to be a norm. In this case, (2) passes into the wellknown triangle inequality. B.1.2. Every quasi-norm II •11 given on a linear space E induces a metrizable topology such that the algebraic operations are continuous. A fundamental system of neighbourhoods of the zero element is formed by the subsets EU with E > 0, where U:=

e E: 11,0

1}

is the closed unit ball. B.1.3. A quasi-Banach space is a linear space E equipped with a quasi-norm 1 . 1 which becomes complete with respect to the associated metrizable topology. This means that all Cauchy sequences are convergent. In the most important case when 11 • 11 is a norm we call E a Banach space. B.1.4. Quasi-norms 1 . 111 and M 112 defined on a linear space E are said to be equivalent if they induce the same topology. This happens if and only if there exists a constant c > 0 such that

11X111C MXM 2

and

114 2

CIIX111 for all x e E.

17

B. Spaces and operators

Let E be a linear space with a quasi-norm 11. 1 E11. If > 0, then eE denotes the same linear space endowed, however, with the equivalent quasi-norm

11x 1 eE11 := eux Eli

for all

x e E.

B.1.5. By a p-norm (0 < p 1) given on a linear space E we mean a quasi-norm satisfying the p-triangle inequality:

Ix + Y IV

for all x, y e E.

11x11 P

If so, then the quasi-triangle inequality holds with a q-norm for 0 < q

CE

= 2 1 /P-1 . Every p-norm is also

d(x, y) := II x — YII P for all x, y e E,

we see that every p-norm defines a metric on E. B.1.6. A linear space E is complete with respect to the metrizable topology induced by a p-norm if and only if the condition Co

E IlxillP < co i=1 implies that the series formed by the elements x l , x2 , ... GE converges in E. B.1.7. A quasi-Banach space is called a p-Banach space if its quasi-norm is even a p-norm. B.1.8. Let 11 11 be any quasi-norm on a linear space E such that

Ix + y -5. cE[Ilx11 + IlY11]

for all x, y e E.

If p is determined by 2 1 /P-1 = C E , then there exists on E an equivalent p-norm. This means that every quasi-Banach space can be viewed as a p-Banach space for all sufficiently small exponents p. B.1.9. Let E and F be quasi-Banach spaces. By an operator T from E into F we mean a bounded linear map. Then the quantity

11711 := sup { Tx1I : x e is finite. In order to indicate the underlying linear spaces E and F, we sometimes replace 11711 by the more specific symbol 11T: E Fil. B.1.10. Suppose that E0 and E1 are quasi-Banach spaces such that E0 is a linear subset of E1 in the purely algebraic sense and that 1x1 E1 11 < CXEoll Then the identity map from

for all

x e Eo .

E0 into E 1 is called an embedding operator.

We say that a quasi-Banach space E lies between E0 and E1 if

/0 E0

>E

El ,

where 10 and I are embedding operators.

18

Preliminaries

B. 2.

Operators on Banach spaces

From now on, unless the contrary is explicitly stated, E, F and G denote complex Banach spaces with the closed unit balls U, V and W, respectively. B.2.1. Let 2(E, F) denote the set of all operators from E into F. Defining the algebraic operations in the canonical way and using the ordinary operator norm

n := sup 111Tx11 : x E we see that 2 (E, F) becomes a Banach space. To simplify notation, we write 2(E) instead of 2(E, E). Furthermore, 2:= U 2(E, F) E,F stands for the class of all operators acting between arbitrary Banach spaces B.2.2. For every operator T e 2(E, F) we define the null space N(T) := ftx e E : Tx = ol and the range M(T) := 1Tx : x e El . Both of these subsets are linear, and. N(T) is always closed. B.2.3. The identity operator of a Banach spaces E is denoted by ./E • 1f there is no risk of confusion, then we simply use the symbol I. B.2.4. We call ST: x Tx—* STx the product of the operators T e 2(E, F) and S e t(F, G). Observe that ST e 2(E, G) and 11ST11 115'11 11T11. The m-th power Tm of TE (E) is inductively defined by T': = TinT and T°: = I. B.2.5. An operator T e 2(E, F) is invertible if there exists X e 2(F, E) such that XT = IE and TX = IF. If so, then the inverse operator X, usually denoted by T -1 , is uniquely determined. According to the bounded inverse theorem (DUN, 11.2.2) and (TAY, 1V.5.8), an operator T E 2(E, F) is invertible if and only if it is one-to-one and onto: N(T) = {o} and M(T) = F. An invertible operator T E 2(E, F) is also referred to as an isomorphism between E and F. If IITII = 11T-1 11 = 1, then T is said to be a metric isomorphism.

B.2.6. An injection Je 2(E, F) is a one-to-one operator with closed range. In this case, Ilxilf := 11.1x11 for all x e E defines an equivalent norm on E. If, in addition, we have 11'11 = 11.11,,, then J is said to be a metric injection. B.2.7. By a subspace M of a Banach space E we mean a closed linear subset. Obviously, M becomes a Banach space with respect to the norm obtained by restriction. The canonical injection from M into E is denoted by J.

B. Spaces and operators

B.2.8.

19

A surjection Q e t(E, F) is an operator which maps E onto F. In this case,

:= inf { xII :

IIYIIQ

x e E, Qx = y}

for all y e F

defines an equivalent norm on F. If, in addition, we have II = II 'IQ then Q is said to be a metric surjection. B.2.9. Let N be any subspace of a Banach space E. Then the quotient space EIN consists of all equivalence classes x + N. It turns out that EIN becomes a Banach space with respect to the norm Ix +

:= inf [11x + yll : y e

N).

The canonical surjection from E onto EIN is denoted by Q. B.2.10. A projection P e t(E) is an operator such that P2 = P. B.2.11. A Banach space E is the direct sum of the subspaces M and N if E = M + N and M r N = {0}. In this case, we write E = M 0 N. The above conditions mean that every element x e E admits a unique decomposition x = .0 + y with u e M and y e N. Then P e t(E) defined by Px: u is called the projection of E onto M along N. Conversely, every projection P e t(E) determines a direct sum decomposition E = M(P) N(P); see (DUN, VI.3.1) and (TAY, IV.12.2). B.2.12. Let M be any subspace of a Hilbert space

:=

{X E

H : (x, y) = 0

for all y e

H. Then

MI

is called the orthogonal complement of M. It turns out that H = M 0 Mi. The projection P from H onto M along M-L is said to be the orthogonal projection from H onto M. Note that IIPI = 1 whenever M 4= {o}. B.3.

Duality

B.3.1. By a functional a defined on a Banach space E we mean a bounded linear form. The value of the functional a at the element x is denoted by <x, a>. B.3.2. Let E' denote the set of all functionals on a Banach space algebraic operations in the canonical way and using the norm

E. Defining

the

,

110 := sup {Kx, a>1 : x e

we see that E` becomes a Banach space. We refer to E' as the dual Banach space. The closed unit ball of E' is denoted by U°. B.3.3. The bidual Banach space E" is defined to be the dual of E'. Assigning to every element x e E the functional

: a —> <x, a>, we obtain the evaluation operator because

from E into E" which is a metric injection,

a E U° }. The dual operator T' e 2(F', E') of T e t(E, F) is determined by <x, T'b> = for all x e E and b e F'. sup 11<x, a>I :

B.3.4.

KE

Observe that IIT 1 11 = 11111.

20

Preliminaries

B.3.5. The bidual operator T" e t(E", F") is defined to be the dual of T' e t(F', E'). The following diagram holds:

F KE

KF

4'

E"

T"

F"

B.3.6. The concepts of an injection and a surjection are dual to each other. This means that J is an injection if and only if J' is a surjection, Q is a surjection if and only if Q' is an injection.

B.3.7. The dual of a Hilbert space can be described as follows. Every element y e H determines a functional y* : x

(x, y)

on H such that I v * = way. Hence

:y

CH

. Moreover, all functionals a e H' can be obtained in this

y*

yields a conjugate-linear one-to-one correspondence between H and H'. B.3.8. Let H and K be Hilbert spaces. The adjoint operator T* e t(K, H) of T e t(H, K) is determined by

(x,T*y) = (Tx, y) for all x e H and y e K. The relationship between adjoint and dual operators is described by the diagram T*

H

CK

CH

K'

T

H' '

B.4.

Finite operators on Banach spaces

B.4.1.

The dimension of a finite dimensional linear space M is denoted by dim (M).

B.4.2. Let N be a linear subset of a linear space E such that the quotient space N is finite dimensional. Then the codimension of N in E is defined by the formula codim (N) := dim (EIN). B.4.3.

Let a e E' and y e F. Then

a0y:x--<x,a>y yields an operator from E into F. Note that lia 0 yil = Ilall Ily11. B.4.4. An operator Te e(E, F) is said to be finite if there exists a so-called finite representation

ai 0 yi ,

T= i=i

B. Spaces and operators

where a 1 ,

y„ E F. This means that

a„ e E' and y,

Tx =

21

<x, ai> yi for all x c E.

We define rank (T) to be the smallest number of summands which are required in the above representation. Note that rank (T) = codim (N(T)) = dim (M(T)).

B.4.5. It easily follows from A.2.5 that every finite operator T e 2(E) admits a representation ■1

T=

E

i=1

ai 0 x i

such that the matrix (<xi , ai>) has upper triangular form: <xi , aj> = 0

i > j.

if

B.4.6. We prove here an auxiliary result about finite operators factoring through a Hilbert space H. Lemma. Suppose that the finite operator T e 2(E, F) can be written in the form T = YA, where A e tl(E, H) and Y E t(H, F). Then there exists an orthogonal projection P e 2(H) such that T = YPA

rank (P) = rank (T).

and

Proof. Let P denote the orthogonal projection from H onto M := Y-1 [M(T)]

N(Y)±.

Given x c E, we consider the decomposition Ax = fo + f with fo e N(Y)

and f e N(Y)± .

Since Tx = YAx = Yf, we have fE Y -1 [M(T)]. Therefore fe M. This proves that Y maps M onto M(T). Moreover, the restriction of Y to M is one-to-one. Hence rank (P) = dim (M) = dim (M(T)) = rank (T). Furthermore, it follows from M(P) N(Y)- that Ax — f = f o e N(Y) M(P)± = N(P). Hence PAx = Pf = f, which in turn implies that Tx = Yf = YPAx forai! x e E.

B.4.7. A Banach space E has the approximation property if, given any precompact subset K of E and E > 0, there exists a finite operator L E 2(E) such that lix — Lx11

8

for all x

E

K.

Roughly speaking, the overwhelming majority of all popular Banach spaces enjoy this property. However, there are also prominent counterexamples: 2 (12 ); see A. Szankowski (1981) and 4.7.6 (Remark).

22

Preliminaries

C.

Sequence and function spaces

We only require some elementary facts on classical sequence and function spaces. Concerning the measure-theoretic background the reader is referred to P. R.Halmos's textbook (HAM). An introduction to the theory of Lorentz sequence spaces is given in Section 2.1. Some basic properties of Sobolev and Besov spaces are summarized in Section 6.4. C.1.

Classical sequence spaces

C.1.1. In the following we consider complex-valued families x = ($ i) indexed by the elements of an arbitrary set I. In the case when this set is finite or countably infinite we speak of vectors and sequences, respectively. All algebraic operations concerning families are defined coordinatewise. C.1.2. Given any index set I, we denote by the collection of its finite subsets. C.1.3. The set of all bounded complex-valued families x = ($i) on a given index set I is denoted by /(/). It turns out that /,„(/) is a Banach space under the norm

l Ix I ioe(1 )11 := sup {Il : iel}. C.1.4. We say that a family x = ($ i) vanishes at infinity if, given e > 0, there exists i e 5?3.(I) such that

e whenever

i i.

The subspace of 1,0 (l) formed by these families is denoted by co(/). C.1.5. A complex-valued family ($ i) with i e I is said to be summable if there exists a complex number a with the following property: Given 8 > 0, we can find io e .(/) such that

cr —

$i

s whenever i c .?`"(I) and i

When this is so, then Cr is called the sum of ($i), and we write

Note that the families

and (lei l) are simultaneously summable or not.

C.1.6. Let 0 < r < co. We denote by MI) the collection of all complex-valued families x = ($) such that

x 441 := is finite. Such families are called r-summable. Note that 4(I) is a Banach space if 1 r < co and an r-Banach space if 0 < r < 1. The quasi-triangle inequality (Minkowski's inequality) holds with the constant Cr : = max (2 1/r-1 , 1). C.1.7. Let 4: = {1, ..., n} and N : = {1, 2, ...}. To simplify notation, we write 1,.(n) and /r instead of 4(4) and 4(N), respectively. The closed unit balls of these spaces are denoted by Ur(n) and U,.. Furthermore, the symbol co(N) is replaced by c0.

23

C. Sequence and function spaces

Let 0 < p,q < co and 1/r = 1/p + 1/q. In the case of complex-valued families Holder's inequality states that

C.1.8.

l$P7iirr ir -5

Inti qr

leer

for x = (e i) E lp(i) and y = (27) e /q(/). C.1.9. Given r with 1 < r < co, the dual exponent r' is defined by 1/r + 1/r' = 1. Furthermore, the exponents 1 and co are considered to be dual to each other. C.1.10. Let 1 r < co. Then, for every family y = (7)i) E 4,(I), the rule

x = ($1) <x, Y>

:=

t

defines a functional on 4(4 In this way the dual Banach space /,.(/)' can be identified with 4.,(/). Analogously, we have co(/)' = 4(4 C.2.

Direct sums of Banach spaces

C.2.1. Let 1 r cc. Given any family of Banach spaces E with i e I, the direct SU m[1,.(I), Ei] consists of all families (x 1) such that x i e E and e /,.(/). Note that Ed is a Banach space with respect to the norm

II(xi) I [4(/), E for 1

:= I

ary l"

r < co. If r = cc, then we let := sup { II(xi) I [G(/),

e /} -

I n the case of the countable index set IN we use the shortened symbol R. The direct sum of finitely many Banach spaces E1 ,...,E„ is denoted by E1 0 ... 0 E. Here it is unnecessary to indicate the exponent r, since all linear spaces [4.(L), coincide, while the corresponding norms are equivalent. C.2.2. For any fixed index k e I we define the following operators: The canonical injection Jk transforms every element xe Ek into the family (x i) with xk := x and xi := o for i * k. The canonical surjection Qk assigns to every family (x 1) its k-th coordinate xk Then ,

Jk

Ek -->

Ur(I),

Qk

is the identity operator of Ek and Pk := JkQk defines a projection on [4(1), C.3.

Classical function spaces

C.3.1.

In the following we consider complex-valued functions! defined on a set X, which is assumed to be either a compact Hausdorff space or a a-finite measure space. All algebraic operations concerning functions are defined pointwise.

C.3.2. The set of all continuous complex-valued functions f defined on a compact Hausdorff space X is denoted by C(X). As is well-known, C(X) becomes a Banach space under the norm fi C11 := sup tif($)1 : e

.

In particular, we write C(0, 1) if X is the unit interval [0, 1].

24

Preliminaries

C.3.3. By a measure space (X„ u) we mean a set X together with a measure ,u which is defined on a a-algebra formed by subsets of X. Once and for all, we assume (X, ,u) to be a-finite. The most important example is the unit interval [0, 1] equipped with the Lebesgue measure. The discrete situation treated in the previous section appears if we deal with the counting measure on a countable index set. C.3.4. Unless the contrary is explicitly stated, we identify measurable complexvalued functions f and g which are equal almost everywhere: g( ) } = 0. C.3.5. Let L oe (X„ u) denote the set of all (equivalence classes of) measurable complex-valued functions f which are essentially bounded on X. This means that 0 with ,u{$ E X f($)1 = 0. Setting there exists a constant c :

fi Loe := ess-sup {if()1 : e X} := inf c, we get a norm which turns L oo (X, y) into a Banach space. C.3.6. Let 0 < r < op. We denote by 4.(X„ u) the collection of all (equivalence classes of) measurable complex-valued hinctions f such that ) 1 /r

if I 411 =

If(e)Ir 41(0

l we simply speak is finite. Such functions are called r-integrable. In the case r 1 r < oc and an of integrable functions. Note that L,.(X, ,u) is a Banach space if r-Banach space if 0 < r < 1. The quasi-triangle inequality (Minkowski's inequality) holds with the constant c,. := max (2 1 /r-1 , 1). C.3.7. To simplify notation, we often write L r instead of L r(X, ,u). If the underlying measure space is the unit interval equipped with the Lebesgue measure, then we use the symbol L,.(0, 1). C.3.8. Let 0 < p,q < co and 1/r = 1/p + 1/q. In the case of complex-valued functions Holder's inequality (HAY, p. 140) states that (fx iM)i r dilt())

1r

(.1,1 f($)I P d itt ()) 1 P

a

g()1 11

for fe L p(X, ,u) and g e 4(X, ,u). r < co. Then, for every function g e L,.,(X, ,u), the rule C.3.9. Let 1 f
f f(e) g( ) di(E)

defines a functional on L,.(X, ,u). In this way the dual Banach space Z ia, ,ur can be identified with 4.,(X, ft). C.3.10. Finally, we state Jessen's inequality.

Let K be any complex-valued function defined and measurable on the product (X x Y, ,u x y) of the measure spaces (X, ,u) and (Y, y). If 0 < p q < co, then

f [f (

Y

X

n)IP d/2()]" dv(77)) 11q 5_ (fX

[f

Y

dvoid" Cit0)) 1 1P

•

Remark. The assertion follows immediately by applying Minkowski's inequality for integrals (HAY, p. 148) with the exponent r := 'gip to I K($, n)IP; see (HAY, pp. 31 and 150) and B. Jessen (1931).

D. Operator ideals

C.4.

25

The metric extension property

C.4.1.

A Banach space F has the metric extension property if, given any metric injection J E 2(E, E0), every operator T e 2(E, F) admits a norm-preserving extension T0 c 2(E0 , F). This means that T= ToJ and 11T0 11 = MTh. C.4.2. A Banach space possesses the metric extension property if and only if it is metrically isomorphic to a Banach space C(X), the compact Hausdorff space X being totally disconnected; see M. Hasumi (1958). The most elementary examples are the Banach spaces /0„(/) on an arbitrary index set I; see also 1.5.4 (Remark). C.4.3. Let E be any Ban ach space. Given x e E, the equation f(a) := <x, a> for all a e U° defines a family j; c G(U°). Setting J Ex := fx , we obtain a metric injection from E into lop(U°). In this way every Banach space can be canonically embedded into a Banach space with the metric extension property. D.

Operator ideals

The theory of operator ideals is extensively presented in (PIE). We use this monograph as a standard reference. A concise introduction to this subject is given in (JAR). Operator ideals on Hilbert spaces are treated in (001-1), (RIN) and (SCE). We also refer to (DUN, XI.6 and XI.9), (REE) and (SIM).

D.1.

Quasi-Banach operator ideals

D.1.1.

Suppose that, for every pair of Banach spaces E and F, we are given a subset 11(E, F) of 2(E, F). The class := U lt(E, F) E,F

is said to be an operator ideal if the following conditions are satisfied: ci (i) y E 91(E, F) for a c E' and y e F. (0,) (2) S + T e W(E, F) for S, T E 91(E, F). YTX (3) , F0) for Xe 2,(E0 , E), T e 91(E, F), Y c 2(F, F0). Since (0 3 ) implies that AT e It(E, F) for T e ?t(E, F) and 2 e C, every component ?t(E, F) is a linear subset of t(E, F). To simplify notation, we write 14E) instead of 91(E, E).

D.1.2.

A function II. I /III which assigns to every operator Te ?I a non-negative ?pi is called a quasi-norm on the operator ideal 91 if it has the following number II properties: (Q1) Ila CD Y = Ilall IIYII for aeE' and yeF.

(Q2) (Q3)

n

jjS + T Here cw

cal S I W11 + IT Will

for S, Te W(E, F).

1 is a constant.

j YTX ?CI Y e 2(F, F0).

11T I WI 11XII

for

X e 2(E0 , E), T E 91(E, F),

26

Preliminaries

In order to indicate the underlying Banach spaces E and F, we sometimes replace by the more specific symbol 'IT: E F tI TI Since (Q 3) implies that IATIS = I ITI

for

T E It(E, F) and A e C,

the function 11. IC' indeed yields a quasi-norm in the sense of B.1.1. In the case cia = 1 we simply speak of a norm. Remark. By using (Q 2 ) and (Q 3 ) only, it follows that = clIall IIYII

Ila 0 Y

for

a e E' and y E F,

where c := Ilk I /111. Hence property (Q 1 ) affects the normalization c = 1. D.1.3. Let II' be a quasi-norm on an operator ideal ?t. Then, as shown in (PIE 6.1.4), we have ilTII

forai! T e ?t(E, F).

Tj

D.1.4. On a given operator ideal ?t there exist various quasi-norms, good ones and bad ones. However, a quasi-norm II.I 9111 really fits if all components lt(E, F) are complete with respect to the induced metrizable topology. If so, then ?..1 is said to be a quasi-Banach operator ideal. In the particular case when we can find even a norm having this property, then ?1 is called a Banach operator ideal. It turns out that the quasi-norm of a quasi-Banach operator ideal is unique up to equivalence; see (PIE, 6.1.8). Therefore we may refer to a quasi-Banach operator deal without specifying its quasi-norm explicitly. D.1.5. An operator ideal 91 is closed if all components It(E, F) are closed linear subsets of ,t(E, F). This means that ?i becomes a Banach operator ideal by using the ordinary operator norm. D.1.6. A p-norm (0 < p 1) defined on an operator ideal ?,1 is a quasi-norm which, in addition to (Q 1 ) and (Q 3), satisfies the p-triangle inequality: IS + TI We

1 IS I

+ 11T ?Jr for

S, Te 11(E, F).

If so, then (Q 2 ) holds with c.52( := D.1.7. By a p-Banach operator ideal we mean an operator ideal equipped with a p-norm such that all components ?RE, F) are complete. D.1.8.

Let II

tII be any quasi-norm on an operator ideal ?t such that

< c[lIS I IS + TI —

+ 11T I 9111]

for

S, TE 11(E, F).

if p is determined by 2 11 P-1 = cw , then there exists on ?if an equivalent p-norm; see (PIE, 6.2.5). This means that every quasi-Banach operator ideal can be viewed as a p-Banach operator ideal for all sufficiently small exponents p. be quasi-Banach operator ideals with ?t $. Then, by the D.1.9. Let ?I and closed graph theorem, there exists a constant c 1 such that cIITI tI for all

T e 14E, F).

For a direct proof we refer to (PIE, 6.1.6).

D. Operator ideals

27

0 .1.10. An operator T e t(E, G) belongs to the product 8 o?I of the operator ideals and $ if it can be written in the form T = BA, where A e It(E, F) and B e $(F, G). Here F is any suitable Banach space. Obviously, $ o ?-4 is an operator ideal. Given quasi-norms 11 - 1 WI and 11' 1 $11, we put

tn

111'1 $ o

:= inf {11B $11 11A 1

the infimum being taken over all possible factorizations. In this way $ o/I becomes a quasi-Banach operator ideal whenever ?4 and $ are complete; see (PIE, 7.1.2). The m-th power of an operator ideal % is inductively defined by 91m+ 1 : = /rt. /I and ?1° : = 2. 0 .1.11. An operator Te t(E, F) belongs to the sum 9.4 + *3 of the operator ideals lt and $ if it can be written in the form T = A + B, where A e 14E, F) and B e $(E, F). Obviously, ?.4 + $ is an operator ideal. Given quasi-norms 11' 1 ?ill and II' 1 $11, we put

IT I

+ $11 := inf {IIA

tM + 11/3 1 $II } ,

the infimum being taken over all possible decompositions. In this way 1( + becomes a quasi-Banach operator ideal whenever ?if and $ are complete.

0.1.12. For every operator ideal ?,4 the dual operator ideal W is defined by

91'(E, F) := {T e t(E, F) : T' e W(F', E')} . Given any quasi-norm I %II, we put

I1T1

:= 1IT' 1 W11.

In this way W becomes a quasi-Banach operator ideal whenever % is complete. 0 .1.13. A quasi-Banach operator ideal ?I is approximative if ,R(E, F), the set of finite operators, is dense in every component WE, F). Let ?-1 be any quasi-Banach operator ideal. An operator Te ?1(E, F) is said to be 14-approximable if there exists a sequence of operators T„ e R'(E, F) such that

lim

—

?CI = 0.

The class of these operators is the so-called approximative kernel V a) of ?I. Obviously, F) in W(E, F). Therefore, restricting lt(a) (E, F) can be viewed as the closed hull of the quasi-norm of ?4 to Va) , we get an approximative quasi-Banach operator ideal. 0 .1.14. A quasi-Banach operator ideal N is said to be injective if, given any metric

injection J e t(F, Fo), Te t(E, F) and JT e ?t(E, F0) imply Te W(E, F) and

I TI ?Ili = IIJTI (II.

This property means that the answer to the question whether or not an operator Te t(E, F) belongs to the ideal ?.4 is independent of the size of the target space F. D.1.15. A quasi-Banach operator ideal L is said to be surjective if, given any metric surjection Q e 2(E0 , E),

T e t(E, F) and TQ

F) imply Te

F) and II T

Note that injectivity and surjectivity are dual concepts.

I II = lITQ

28

Preliminaries

D.2.

Some examples of operator ideals

D.2.1. Recall that, according to B.4.4, an operator T e t(E, F) is finite if it can be written in the form 11

T=

E

1=1

ai y i ,

where a1 , a„e E' and Yi, ...,y„ e F. The set of these operators is denoted by O(E, F). It easily turns out that O is the smallest operator ideal. D.2.2. An operator T e t(E, F) is said to be approximable if there exists a sequence of operators T„ e O(E, F) such that urn 11T — T,11 = O. 11

The set of these operators is denoted by (4(E, F) or 2. (a)(E, F). Obviously, S is the smallest closed operator ideal. Note that 04 = W. However, CO is neither injective nor surjective. D.2.3. An operator T e t(E, F) is compact if it maps the closed unit ball of E into a precompact subset of F. The set of these operators is denoted by SI(E, F). As is well-known, Pi is a closed operator ideal; see (DUN, VI.5.4) and (TAY, V.7.1 and V.7.2). Note that g = SI'. Moreover, St is injective and surjective, but it fails to be approximative. D.2.4. We call Te t(E, F) a Hilbert operator if it factors through a Hilbert space. This means that T = BA, where A e t(E, H) and B e t(H, F). The set of these operators is denoted by S,(E, F). For T E :40(E, F) we define :=inf {11/311 MC, the infimum being taken over all possible factorizations. It can easily be shown that k) is a Banach operator ideal ; see (PIE, 6.6.2). Note that = Moreover, is injective and surjective. D.3.

Extensions of operator ideals

D.3.1.

The concept of an operator ideal ?.1 := U ?1(E, F) E,F

also makes sense if E and F range only over a certain subclass of Banach spaces. Of particular interest is the case ?.1 := U ?i(H, K), H,K where H and K are arbitrary Hilbert spaces. D.3.2. Let ?t be an operator ideal on the class of Hilbert spaces. An operator ideal ?Jo defined on the class of all Banach spaces is said to be an extension of ?.1 if 110 (H, K) = ?I( U, K)

for all H and K.

29

E. Tensor products

In the case when 9Io and 21 are quasi-Banach operator ideals such that the corresponding quasi-norms coincide on all components W0 (11, K) = It(H, K) we speak of a metric extension.

D.3.3. Every operator ideal t defined on the class of Hilbert spaces admits a smallest and a largest extension which, in general, differ from each other considerably; see (PIE, 15.6).

E.

Tensor products

From the theory of tensor products we only need the basic definitions and some elementary facts which are to be found in (DIL), (KOT) and (JAR).

E.1.

Algebraic tensor products

E.1.1. Let E(E, F) denote the Banach space of all bounded bilinear forms z defined on the cartesian product E' x F'. The norm is given by

Ilz l ell := sup Ilz(a, b)I : aE U° , b

V°},

where U° and V° are the closed unit balls of E' and F', respectively. E.1.2. The algebraic tensor product of the Banach spaces E and F is the linear subset E F of E(E, F) spanned by all bounded bilinear functionals x

y (a, b) <x, a>

with x E E and y e F. This means that every element zeEOF admits a representation

(*)

Z =E

i=1

Xi

yi .

Alternatively, E F may be viewed as the collection of all formal sums (*), with the understanding that

A(x 0 y) = (Ax) 0 y = x 0 (Ay), (x l + x2)0 Y xi Y + x2 CD Y, x C) (y i + Y2) = X 0 Yt + x 0 Y2. E.1.3.

Let S e

E0) and TE t(F, F0). Given z c E(E, F), then

zo : (ao , 10—> z(S'a o ,Tb o) defines a bounded bilinear functional z o E e(E0 , F0). Setting E(S,T)z := z o , we obtain an operator e(S, T)e t(e(E, F), E(E0, F0)) with Ile(S, T )i ! = I Sll iiTll . Observe that e(S, T)(x C) y) = Sx C) Ty.

E.1.4. The algebraic tensor product of the operators S and T is the linear map S T from E Ø F into E0 0 Fo induced by s(S, T).

30

Preliminaries

E.2.

Banach tensor products

A non-negative function a: z z I all defined on all algebraic tensor products E C) F is said to be a tensor norm if it has the following properties: E.2.1.

x® yxI = I Ix 11 11Y11 for x e E and y e F. u+ + v ocil for u,yeE0 F. T) z ocII IISM ITII Ilz I 04 for S e t(E, E0), T E II(S

(TN1) (TN2) (TN 3 )

t(F, F0 ) and

ZEE® F.

Next we give the most important examples. There exists a schmallest tensor norm e which is defined by

E.2.2.

Ell := sup tiz(a, b)I : a E U ° , b e There exists a largest tensor norm y which is defined by := inf

the infimum being taken over all representations z = E xi y i . 1=1

F obtained by completing E C) F with respect to E.2.3. The Banach space E a given tensor norm cc is called the a-tensor product of the Banach spaces E and F. E.2.4. Let S e 2(E, E0 ) and T e 2(F, F0). Given any tensor norm a, then the continuous extension S C) j e (E C) 0,F , E0 C) ef 0) of S C) T is said to be the a-tensor product of the operators S and T. E.2.5. If H and K are Hilbert spaces, then there exists a scalar product on the algebraic tensor product H K which is uniquely determined by the following

property: (x 1 0 Yi, x2 0 y2) = (x 1 , x 2 ) (Yi, 3' 2) for x l , x2 e H and Yi, y2 e K.

The corresponding tensor norm (defined on the class of Hilbert spaces only) is denoted by a. E.2.6. To illustrate the preceding definition, we note that the continuous and linear extension of the map ($i) 0 ( Li) s a metric isomorphism between 12 (1 ) (D 0.12 (J) and 12 (1 x J).

E.3.

Tensor stability of operator ideals

E.3.1.

An operator ideal ?I is said to be stable with respect to a tensor norm oc if SE 11(E, E0 )

and

T e lt(F, F0)

imply

S

j e /It(E „F, E0

F0).

From the theoretical viewpoint the following consequence of the closed graph theorem is very important. In all concrete cases, however, this statement is a by-product obtained free of charge within the proof of tensor stability.

E.3.2.

31

F. Interpolation theory

Let It be a quasi-Banach operator ideal which is stable with respect to a tensor 1 such that norm a. Then there exists a constant c

1 11 1 F.

for Se 21(E, E0) and

clIS I ¶11T I

TE

F0 ).

Interpolation theory

In order to prove some fundamental theorems in their optimal version, we need a few results from interpolation theory (real method) which are to be found in (BER) and (TRI); see also (BUB). F.1.

Intermediate spaces

F.1.1. A quasi-Banach interpolation couple (E0 , E1) consists of two quasi-Banach spaces E0 and E1 which are continuously embedded into a Hausdorff topological linear space X. F.1.2. With every quasi-Banach interpolation couple (E0 , E1 ) we may associate the quasi-Banach spaces E0 n E1 and E0 + E, endowed with the quasi-norms I Ix E0 n E1 11 := max {1Ix I E0 11 , xI E 1 11}

and !Ix Eo

:= inf {Ix() E0 + Ix1 I E111 xo + x1 =

respectively. Then we have the following diagram, where the arrows indicate embedding operators:

E0 +

R emark. From now on the quasi-Banach space E0 + E1 can play the role of the Hausdorff topological linear space X which is only required to fix the mutual position of E0 and El . F.1.3. A quasi-Banach space E is said to be intermediate with respect to the quasi: Banach interpolation couple (E0 , E1 ) if it lies between E0 n E1 and E0 + Eo El

F.2.

>E

>E0 + El .

Interpolation methods

F.2.1. By an interpolation method we mean a functor P which assigns to every quasi-Banach interpolation couple (E0 , E1 ) an intermediate quasi-Banach space (E0 , E1 )e, having the so-called interpolation property: Let (E0 , E1 ) and (F0 , F1 ) be arbitrary quasi-Banach interpolation couples. If an operator T e $(E0 + E1 , F0 + F1) transforms E0 into Fo and E1 into F1 , then it induces an operator from (E0 ,E,),z, into (F0 , F1 )0 . In this case, there exists a constant c 1 such that I T: (E0 , E1)„, —> (F0 , F1)0 1

c max {T : E0 —> F0 11,

: El

-+ F1

}.

32

Preliminaries

F.2.2. Let 0
F.3.

Real interpolation

F.3.1. Given any quasi-Banach interpolation couple (E0 , E1 ), for x e E0 + E t and 2 > 0, the Peetre K-functional is defined by

K(r, x, E0 , El ) := inf nIxo I Eoll + TIlxi I E1II : x o + x 1 = x} . If there is no risk of confusion, then we write K(r, x) instead of K(r, x, E0 , Et ). Obviously, K( r , x) yields a one-parameter family of equivalent quasi-norms on E0 + E1 which, for fixed x c E0 + E 1 , depend continuously on T.

F.3.2. We now define the so-called real interpolation method, where the adjective "real" refers to the nature of the construction and not to the underlying quasiBanach spaces which (as always throughout this book) are supposed to be complex. Let 0 < 0 < 1 and 0 < w < co. For every quasi-Banach interpolation couple (E0 , E1 ) we denote by (E0 , E1 ) 0 ,,„ the collection of all elements x e E0 + L., such that the expression 11 w

II x I (Eo, E1)0 II := ( i IT -eK(r, x)l w d— rr o is finite. In the limiting case w = co the same is assumed for II x 1 (Eo, E1)13.. II := sup r-eK(T, x). ..,

It can easily be seen that (E0 , E1 )9,,„ becomes a quasi-Banach space with respect to the quasi-norm just defined. Furthermore, as shown in (BER, 3.1.2 and 3.11.2) and (TRI, 1.3.3), the functor

(E0 , E1) -+ (E0 , E1 )0 ,,, defines an interpolation method of type O.

F.3.3. The real interpolation method possesses the reiteration property: Let 0 < E0 < Et < 1, 0 < 0 < 1 and 0 < wo , w 1 , w _-_ cc. If we have L' = (1 - 0) E0 + 0 E t , then

((E0 , E1 ) 4 ,0 , (E0 , El),E l ov i )0 , w = (E0 5

E 1)Z,W .

Proofs of this formula are given in (BER, 3.5.3 and 3.11.5) and (TRI, 1.10.2).

F.3.4. We now describe the most important example which was the starting point of interpolation theory. For this purpose, let (X, ,a) be a measure space. Then the function spaces 4.(X, tt) with 0 < r..-_ co are continuously embedded into the complete metric linear space L o(X,,u) formed by all (equivalence classes of) measurable complex-valued functions. In this way (Lro(X, it), Lri(X, eu)) can be viewed as a quasi-Banach interpolation couple.

33

G. Inequalities

Let 0 < ro
(L„(X,

Lr l (X, itt))o,r = L r(X, eu

).

The quasi-norms on both sides are equivalent. F.3.5. Finally, we state Lyapunov's inequality which is closely related to the preceding result; see (HAY, pp. 27 and 146). Let 0 < ro
11/. 1 Lrll 5 Ilf I Lro l i—° Ilf I Lr i ll° F.4.

for all f e L,.. (X, tt) Lr i (X, 1 2).

Interpolation of quasi-Banach operator ideals

F.4.1. Let ?to and ?.1 1 be quasi-Banach operator ideals. Then for arbitrary Banach spaces E and F, the components WO(E, F) and W i (E, F) are continuously embedded in t(E, F). This means that (110(E, F), 111 (E, F)) is a quasi-Banach interpolation couple. Hence, given any interpolation functor 0, we may define (t0 ,

:= U (Ito(E, F), It 1(E, n)o.

It easily follows that (tf0 ,11,)4, is a quasi-Banach operator ideal.

G.

Inequalities

We use the classical treatise of G. H. Hardy /J. E. Littlewood / G. P6lya (HAY) as a standard reference on inequalities. For the convenience of the reader a proof of Khintchine's inequality is given in Section G.2. Further important inequalities are already stated above:

Hadamard's inequality (A.4.5), Minkowski's inequality (C.1.6), (C.3.6), Holder's inequality (C.1.8), (C.3.8), Lyapunov's inequality (F.3.5), Jessen's inequality (C.3.10). G.1.

The inequality of means

G.1.1.

The inequality of means (HAY, pp. 16-21) states that lln(1

n

H

k= 1

G.1.2. result:

n

n

for oc 1

E

,

O.

k =1

For completeness, we mention the following supplement of the preceding )1/n

n

k=1

ock

1 ,"

)1/p

= iim (— Nk P p-to n k=1

for

Note that 1

n

\ 1/p

(--' z ai,P)

n

k=1

llq

(1 1

n

if 0 < p
E

k =1

)

34

Preliminaries

G.2.

The Khintchine inequality

G.2.1.

The Rademacher functions ek with k = 1, 2, ... are defined by if 0._. _. r < 1.

k(r) := sign (sin 2krcr)

This means that

h—1

ek(r) = (-1)h+1 if

< 2" <

2k

h 2k

h = 1, ..., 2k.

and

G.2.2. Obviously, (ek ) is an orthonormal sequence in L2 (0, 1). We have, however, a much stronger property. Lemma. Let p i , ... , p„ e NI and k 1 , ... , kn e IN, where k, < ... < kn . Then I

f eiP,(r) ... 0

1 if all pi are even, 0 otherwise.

ef:(r) d'r =

Proof. Every factor of the integrand for which pi is even can be cancelled, since it equals 1 almost everywhere. Thus we may assume that p„ is odd. Divide (0,1) into intervals of length 2 -kn. Then the factors e r,f(r) with i < n are constant on the union of the (2h — 1)-th and the 2h-th interval. On the other hand, e r,n(r) takes the values +I and —1, respectively. This implies that the integral over each of these pairs of intervals vanishes and so does the integral over (0,1). G.2.3. We are now in a position to establish Khintchine's inequality which is one of the basic tools in the theory of absolutely summing operators. Lemma (A. Ya. Khintchine 1923). If 0 < r < co, then there exist constants a,. and br such that a„I (i I 2 )

f E $keker) o k=1

k=1

11r

r

n

C 1 T)

n

br

\ 1/2

E 1$k1 2)

(

k1 =

for all finite families of complex numbers $ 1 , ..., „. Proof. Since the Rademacher functions are orthonormal, Parseval's equation states that 2)1/2

n

E

k= 1

e 0 k(r)

(IT

n

=

\ 1/2

E lek1 2 ) 1

(k

=

•

Hence the above inequality holds by taking a,. = 1

if 2 -._ ._ r < co

and

b,. = 1 if 0 < r _-_ 2.

Next we treat the special case in which r = 2s with s = 1, 2, ... is an even number. ..., be real. Applying the n-nomial formula, by G.2.2, we obtain Let 1

2s

n

E k=1

kek(r)

dr =

E

(2s)! pso!... (2s„)! leil'i

35

G. Inequalities

and n

s

16,1 2 )

!(k =1

=

S

L

S1

1$.12,„

!

...S

.11

where the right-hand sums are taken over all combinations (s 1 , + s„ = s. Comparing the n-nomial coefficients, we see that s1 + )112s

2s

k= 1

o

(.;

s„) such that

1/2 b2s (kLpic1 2 )

$kek(r)

where b25

:= max

1/2s

s„!

[ (2s)! s 1 ! s! (2s 1 )!

(2s,,)!

Next it follows from

2m

(2m) 1. ni!

b2 5

[(2S)S

(2m)m

that

< s 1/2

This proves the right-hand Khintchine inequality for even exponents and real coefficients. The complex case can be treated by splitting into real and imaginary parts. Next, given an arbitrary exponent r with 2 < r < co, we choose s E NJ such that r 2s. Then the required inequality holds with b,. := b25 . It remains to verify the left-hand Khintchine inequality for 0 < r < 2. To this end, the parameter 0 is defined by

1/2 = (1 — 0)1r + 0/8. Applying Lyapunov's inequality and the preceding results, we get )1/2

in

1$k1 2

=1

(1 = 0.o •

eke k(r)

(f

0 k1 =

f

0 k= I

kei,(r)

ekek(r)

)(1-0)1r 1 dr 0 k= 1

•

1

(

k= 1

ekek(r)

)(1-0)I r

)0/2

b8 1E' 16,1

dr

2

k=1

Hence n

E led 2 1c=1

)112

1 G 21/(1-0)(f 0

k= 1 4:kok(r)

Remark. From now on we assume that the constants a,. and b,. are chosen best possible. Although these optimal values are of sporting interest only, they have been computed by the efforts of several mathematicians. The final result is due to U. Haagerup (1982) who also described the historical development.

36

Preliminaries

As observed by B. Tomaszewski (see S. J. Szarek 1976) the values of a,. and b,. are the same for the real and complex case.

G.3.

Asymptotic estimates

G.3.1.

Let (a„) and (i3n) be non-negative real-valued sequences. Then

,, for n = 1, 2, ...,

a„ -<13„ means that a„

where the constant c > 0 may depend on various parameters but not on the index n. We write

fl„ if oc„

an

ig n and /9,,

oc„.

G.3.2.

The following examples illustrate the above definition: Let —cia
1 2"&N(h + 1)fi

2(k + 1)P

if a < 0

and co

2ha(h

013

21c( k + 1)0

if a > O.

(

This implies that

1 ne+'( 1 + log in

n°`+ 1 (1 + log n)

if oc < 0

and co m=n+1 M cx+1

G.3.3.

(1 + log m) 0

na'(1 + log O P

if oc > O.

Next we state a consequence of Stirling's formula (TIT, 1.87):

n"+112 exp (—n).

n!

In this connection we refer to the elementary estimate fl

n!

exp (e) for all

which, by taking e

n!

>0

n, implies that

n" exp (—n).

CHAPTER 1

Absolutely summing operators This chapter contains an introduction to the theory of absolutely summing operators. We mainly concentrate on such results as are relevant for later applications within the theory of s-numbers and eigenvalues. We begin by summarizing some elementary facts about absolutely and weakly r-summable families in Banach spaces. Next the basic properties of absolutely s r < oo. These operators constitute (r, s)-summing operators are established, 1 a Banach ideal, denoted by Vr, s • Of special importance are the operator ideals 4,;,. := 4%.,, and In particular, we stress the outstanding role of V2, which has become an indispensable tool not only for the purpose of this monograph but also for Banach space theory as a whole. Among other circumstances this is due to the fact that the absolutely 2-summing operators share many properties with the famous Hilbert-Schmidt operators which have proved to be quite useful in the Hilbert space setting. We also deal with the powers of the operator ideal V, . Diagonal operators from 1, into / a are investigated. On the one hand, these operators yield instructive examples for exploring the relationship between different operator ideals. On the other hand, they play an important part within the theory of eigenvalue distributions of operators induced by infinite matrices. In the last section we treat ideals of p-nuclear and (r, 2)-nuclear operators for 2, respectively. 0 < p < 1 and 0 < r

1.1.

Summable sequences

Throughout this section we assume that 1 r < co. 1.1.1.* Let I be any index set, and denote by the collection of its finite subsets. 1.1.2.* An E-valued family (x i) with i e I is said to be summable if there exists an element s E E with the following property: Given E > 0, we can find i o G 'W) such that

— E xi

s whenever i

E " (I)

and i

j0 .

When this is so, then s is called the sum of (x1), and we write

s =E

Xi .

1.1.3.* An E-valued family (x i) with i e I is said to be absolutely r-summable if ( xi II) e 4(4 The set of these families is denoted by [IP), E]. For (x i) e E] we define 11(xf) I UP),

:=

If there is no risk of confusion, then we use the shortened symbol 11(x1) 411. Moreover, we write [4., E] instead of [4(N), E]. 1.1.4.* We state the following fact without proof. Proposition. [4(4 E] is a Banach space.

▪ 38

1.

Absolutely summing operators

1.1.5.* The next result is also straightforward. Proposition. Let T E 2(E, F). Then T(I) : (x i) -4 (Tx) defines an operator from [4(4 E] into [1,.(I), F]. Moreover, we have

11(1'4 J 411 —5_ T II(xi) I 411 for (xi) G Ur(i), E]. 1.1.6.* We now state an important formula which is proved in (BER, 5.6.1) and (TRI, 1.18.1). In order to include the limiting case, we let [1(I), E] denote the Banach space of all bounded E-valued families (x i) with i e I; see C.1.3. Interpolation proposition (J. L. Lions /J. Peetre 1964). Let 1 -.5_ ro < r 1 and 0 < 0 < 1. If 1/r = (1 — 0)/r 0 + elri , then ([1ro(I),

Ur,(1), EDed. = [1,(1), E].

The norms on both sides are equivalent. 1.1.7.* An E-valued family (x i) with i e I is said to be weakly r-summable if (<xi , a>) c 1,.(I) for all a c E'. The set of these families is denoted by [wr(I), E]. For (x 1) e [wr(l), E] we define

:= sup /LE i<xi, Orr'. : a c U° ), t‘ where U° is the closed unit ball of E'. If there is no risk of confusion, then we use the shortened symbol 11(x1) I wril. Moreover, we always write [W,., E] instead of [wr(N), Remark. It follows from the principle of uniform boundedness, (DUN, 11.3.21) and (TAY, IV.1.2), that 1 (xi) I W r il is indeed finite. 1.1.8.* We state the following fact without proof. Proposition. [wr(I), E] is a Banach space. 1.1.9.* The next result is an analogue of 1.1.5. Proposition. Let T c 2(E, F). Then 11(xi) [wr(/),

T(I): (x i) -4 (Tx i ) defines an operator from [wr(I), E] into [wr(I), F]. Moreover, we have

ii(Txi) J wril 11Th i wril for (Xi) G [MO), E]. Proof. Let U° and V° denote the closed unit balls of E' and F', respectively. Then 11(7'4 wrI1 = sup

Il ry ir : b V°)

= sup {(1<x 1 , ▪ sup

: b E V°)

Kxi, a>I 1t a G IITM U °)

= J T 11(X i) I WA • 1.1.10.* Obviously, every absolutely r-summable family is necessarily weakly r-summable. Proposition. [4(I), E] _s [wr(I), E].

39

1.1. Summable sequences

Remark. The famous Dvoretzky-Rogers theorem states that, for an infinite index set I, equality holds only in finite dimensional Banach spaces. Thus there is indeed a significant difference between the two concepts of r-summability. This fact has initiated the theory of absolutely r-summing operators which is the main subject of this chapter. 1.1.11.* Next we mention a classical result. Example. Every orthonormal family (u i) in a Hilbert space is weakly 2-summable, and we have 11(u 1 ) I w2I1 = 1 . Proof. The conclusion can be deduced from Bessel's inequality:

11x11 2

E ( ui, x)1 2

for all x e H.

1.1.12.* It follows immediately from the definition of a weakly r-summable family (x i) that : a --+ ((X 1 , a>) defines an operator from E' into 4(I) with I1(xi)op

--+ 41)11 = 11(x i) 1 wrIl

We now show that, for 1 < r < co, there exists X e i(4 , (I), E) such that (x 1)0 Lemma (R. S. Phillips 1940). Let 1 < r < oo. Given (Xi) E [w,.(1),

X : (o c i)

X'.

E oc ix i

defines an operator from 4.,(/) into E such that x i = Xe i for i e I. This correspondence yields a metric isomorphism between [w,.(I), E] and 2( 4. (I), E). Proof. Obviously, the expression X(oc i) := E A ix, makes sense for those families (N) which have a finite number of non-vanishing coordinates. Then it follows from KX(N1), a>I =

E

a>1

(Elociry lr' /E i<x i , a>irrir k

for a e E' that

11x(x1)11 5 11(xi) 1 WrIl 11(c‘i) 1

4111.

Thus we have already obtained a continuous linear map on a dense linear subset of le(I). Take its continuous extension, and observe that X(x 1) is given by the same formula for arbitrary families (oc i) e 4.,(/). Clearly, x i = Xei and 11x11 11(x1) I w,.11. Conversely, we show that every operator X e e(1,.,(I), E) can be obtained in this way. Note that (e1) is weakly r-summable in 4.,(/) and Ike 1 ) 1 W . = 1. Now it follows from 1.1.9 that (X 1) := (Xe i) is a weakly r-summable family with 11(x1) I 14',11 1.1.13.* As a corollary of the Hahn-Banach theorem we have ,

11x11 = sup {1<x, a>1 : a e U° 1

for all x e E.

For various purposes it is desirable that 11x11 can be obtained by taking the righthand supremum only over a certain subset A of U° , the smaller the better. This observation leads to the following definition.

1. Absolutely summing operators

40

A subset A of E' is said to be norming if

114 = sup {1<x, a>1 : a e Al for all x e E. For example, given any compact Hausdorff space X, the collection of all Dirac evaluation functionals 64 :f f($) with $ e Xis a norming subset of C(X)'. Identifying c5 and $, we denote this subset by X. Moreover, for every Banach space E, the image of U with respect to the evaluation operator K E defined in B.3.3 is a forming subset of E" 1.1.14.* The following lemma shows how the concept just introduced enters the theory of weakly r-summable families. X„ E E. Lemma. Suppose that A is any norming subset of E'. Let x l , Then =1 1<xi , a) Ir) sup {( iE

: ae

A} = sup { E

N iX 1

i= I

if 1 < r < co and sup

(

i =1

1<xi, a)1 : a e A)

E

= sup{

cc ix i

i=1

1.1.15.* Taking into account that the right-hand expressions in the preceding lemma do not depend on A, we are led to the following equation. Lemma. Suppose that A is any norming subset of E'. Let x 1 , , x„ e E. Then 1/r fl . : ae 11(Xi) = sup { (E 1<xi , a)1r)Al i= 1.1.16.

Finally, we illustrate the previous results in some concrete Banach spaces. E C(X), where X is any compact Hausdorff space. Example. Let f,

Then \ 11

f n

11(fi) W = su p

Ifi()1r)

r

:

e X} .

Proof. The assertion follows from the fact mentioned in 1.1.13 that X can be viewed as a norming subset of C(X)'. 1.1.17. Example. Let f1, , f G 0,(X , ,u), where (X, ,u) is any measure space. Then )11r : E X} . iEi (or 11(A) I WA = ess-sup ((ift • •

Proof. Given a = that iEocifg) l

oc n) e U,.,(n), by 1.1.14, there exists a null set N(a) such

11(fi) wrIl

for all $ e

X\ N(a) .

Choose any countable dense subset A of Ur,(n), and denote by N the union of all N(a) with a e A. Applying a continuity argument, we obtain

E=

aifi(0

11(f) w r

for all $ e X N,

1.2. Absolutely (r, s)-summing operators

41

where the null set N no longer depends on a e U,.,(n). Hence 1/r

n

i= 1

li(f)

Ifg)r)

for almost all e X.

I Wr II

This proves that 1/r

ess-sup

i=

ligr)

: e X}

wril

The reverse inequality is obvious. 1.2.

Absolutely (r, s)-summing operators r < Go.

Throughout this section we assume that 1 .1 s

1.2.1.* An operator T e F) is called absolutely (r, s)-summing if there exists a constant c 0 such that 11s hr C sup {(/ Kxi, a>ls) : a e U°) 11TX (i i=1 i= 1

x n e E. The set of these operators is denoted

for every finite family of elements by *,(E, F). For Te $,(E, F) we define := inf c,

IIT

the infimum being taken over all constants c holds.

0 for which the above inequality

1.2.2.* The term "absolutely (r, s)-summing" is derived from the fact that such operators send weakly s-summable families into absolutely r-summable ones. More precisely, we have the following criterion.

Proposition (A. Pietsch 1967). Let I be any infinite index set. An operator T c e(E, F) is absolutely (r, s)-summing if and only if T(I) : (x i)

(Tx)

defines an operator from [ws(I), E] into [/P), F]. When this is so, then

IT I 4

T.S

II = IT (I) : [ws(I), E] -+ [1,.(I), Fill .

Proof. Suppose that T e r, s(E, F) and

iL 11Txdry/r k

T

sup f lE1<xi,

for all i e rA(I). Passing to the limit as i

II(Tvi) I MD,

(xi) e [ws(I), E]. Then we

II T I

I yields

IIIII(xi) I [ 1444

This proves that T(I) : [ws(I), E]

Ur(I), Fill

The reverse inequality is obvious.

IT I

: ae

have

1. Absolutely summing operators

42

We now suppose that T e 2.(E, F) fails to be absolutely (r, s)-summing. Then, for h = 1, 2, ..., we can find xhi, •• xhh„ e E such that

E

TXhkil r

sup ( E

k =1

Put xhk := o if k > because CO

and

1 k=1 11h •

Kxhk,

a e U°) < 2- h.

a>ls

Then the double sequence (xhk) is weakly s-summable, cc

CO

1<X, a>15

E 2- " = 1 for ac U° . h=1

h=1 k=1

On the other hand, we conclude from

+

+

11TxhiclIr

E

h=1 k=1

that (Tx hk) cannot be absolutely r-summable. Therefore TON x N) fails to map [ws(N x N), E] into [/,.(N x N), F]. To finish the proof, we observe that every infinite index set I contains a copy of N x N as a subset. Hence the family (xhk) can be extended to a family (x i) with i e I by putting x i := o whenever i N x N. Thus there exists (x i) e [w(/), E] such, that (Tx) [1,.(I), F]. 1.2.3.* We now establish the basic result of this section. Theorem (S. Kwapien 1968). 13, is an injective Banach operator ideal. Proof. Let Te

F). By definition, we have 1Ir x

for x 1 ,

4*; r ,s11

SUP

((iE1

I<X i

1Is

a> Is)

:

a e U0)

x„ e E. Taking n = 1, we obtain IITxII III' I 9A, ,sil sup fl<x, a>I : a e U°1 for x e E.

This proves that IITI 43,•,sIl. Next we consider an operator of the form ao 0 yo with ao e E' and yo e F. Then it follows from 1Ir

n

E II(ao 0 Yo)

(

i=1

xi M r)

lls

n

_- (E 1<xi, ao>l s)

Voll

i=1

1Is

n

a>is) Kx i , 6 Ilaoll Ilyoll sup {(E

: a c U° 1

i

that

Ilao 0 Yo I Vr.sII -- Ilaoll IIY011Hence a() 0

yo e *r, s(E, F) and II ao 0 Yo I J.,SII = Ilaoll 11Y0l1

We now assume that

F). Then

S, T

(n

(n

I S (

i=1

HIS

T) x $r ,s11

)1Ir

E IIsxiIIr

i=1 1 IT

r ,s111

SUP

{(i KX

i =1

)1/r

(n

+ E 11Txtlir i=1

O s)

:

a e U° )

1.2. Absolutely (r, s)-summing operators

43

for x i , ..., X,, E E. Therefore Te 43r.s(E, F) and I S + iiT Vr.sii • If Xe t(E0 , E), T e *r, s (E, F) and YE 2(F, F0), then we deduce from 1.1.5 and 1.1.9 that S

and

YTX e 43,4E0 , F0)

1 YTX

I r,s

III

Vr,sll

II XII •

Finally, let (Tk) be a Cauchy sequence in 43,4E, F). Then it follows from

Tk lI

11Th —

11Th — Tk

I

*r,s11

that (Tk) has a limit Te t(E, F) with respect to the operator norm. Given e > we choose ko such that

11Th — Tk I 43r,s11 This means that

e for h > k

ko • lls

1/r

E sup {(E Kxi, a>r)

(E 11(Th — Tic) x il1r)

i=1

for x i

,x„ E E. Letting h n1

I

Tk)

i=1

: a e U° 1

Go, we obtain 1/r 1Is fl : a E U° ). E sup (( E 1<xi, a>Is ) i=1

Hence T — Tk C 43,4E, F) and

I IT — Tk

I r,s 11

e for k k o .

Thus T is also the limit of (Tk) with respect to the norm of V, ,s . The injectivity of the Banach operator ideal Vr , s is obvious. Remark. The assertion could be deduced by identifying Vr ,s(E, F) with a subspace o ' 2(Devs , El, [1r , F]) as described in 1.2.2. 1.2.4.* We now investigate how the operator ideals r , s depend on the parameters r and s. First a trivial inclusion is stated. s, so ro ri < co. Then Proposition (S. Kwapiefi 1968). Let 1 Vri ,si . Moreover, we have ro,so

43 ro ,so l for all

TE Vro,s0(E, F).

1.2.5.* The next result turns out to be slightly more complicated. ro r i < co and 1 Proposition (S. Kwapiefi 1968). Let 1 so . Moreover, we have If 1/ • — 1/r 1 = 1/s0 — 1/s 1 , then 913r0,s0 IT I

Vro»,0 11

I

s i < co.

for all Te 440 ,,o(E, F).

Proof. Define 1/r := 1/r 0 — Ur i and 1/s := 1/s0 — lisp By assumption, we obtain r = s. Let T e 4.4r0»so(E, F), x 1 , x„ e E and x 1 , E C. Then Holder's inequality yields llro

II 5_ T

E IlociTxill")

i=1

)1 Iso Vro,soll

sup

((

11,s 1 5-- T

I

Vr o ,so ll

sup

((i.1

1<xi ,a>1 11)

: a e U° 1

E 1ls°

i=1

1/s

: a e U° )

iocii s)

•

44

1. Absolutely summing operators

Taking oc i := I Tx 1 1, we conclude that

E 11(xiTxillr° = E

= E 11Tx1llri•

i=1

i=1

i=1

Thus fl

i/ri

i=1

ilir

11 7' 1

44,0 1

sup {(

i=1

\ 1 15 1

1<x i ,a>lsi)

: a e U°).

This proves that

F) and

. ro,so I 1.2.6. The following formula is very important. In order to include the limiting case, we let := 2. Interpolation theorem (H. Keinig 1978). Let 1 s ro < r 1 cc and 0 < 0 < 1. If 1/r = (1 — 0)/r0 + 0/r 1 , then Te

(4; 0,S 43.71,00,r

Proof. Given (x i)

E

II T

IT

r,s•

[w„ E], we define by X(T) := (Tx i ) an operator

X: 2(E, F) —> [w „ F]. Note that X: ro,s(E, F) —> [1,0 , F] and X: V ri,s(E, F) —> ['fri, F].

The interpolation property now implies that X: (Vro,s(E, F), 4511. 1,5(E, F))0,r

(Uro ,

F], Uri ,

F])0 , ,..

Hence the assertion follows from 1.1.6.

1.2.7.* In order to prove the following result, we need a criterion which is also of interest in its own right. Lemma. An operator Te t(E, F) is absolutely (r, 2)-summing if and only F) for all X E 2(12 E). When this is so, then if TX e suP {liT X I

IIT I 1r,2

r, 2 : X e 2.(/2 , E), II XII .5_

.

Proof. It follows from 1.1.12 that, given (x i) c [w2, E ] , 00

AT:

(oc,)—›

E i=1

defines an operator from 12 into E with II XII = w2II. If (e1) denotes the standard basis of /2 , then 11(e1) I w211 = 1, by 1.1.11. Moreover, xi = Xe i . We now assume that T e t(E, F) has the property stated above. Writing IIT j

2110 := sup {11TX I 3r,2 11 X e 2(1 2 , E), 11)(11

1},

we obtain

II(Tx) I 411 = II(Txei) I 411

liTX I

r .211 11(e i) I WI

liT I 43r ,211 o

i) I v211.

1.3. Absolutely r-summing operators

45

Hence

Te

F) and II TI r , 211

ITI

Vr,2 0 •

The converse is obvious. 1.2.8.* We now establish an important inequality. The following proof is due to S. Kwapieri (unpublished). Lemma. Let T e 2(E, F). Then n. n"r lITII whenever rank (T) 11 7'1 4r,211 x,„ e 12 (n), and denote the closed unit ball of 12 (n) by U2 (n). Proof. Let x 1 , Obviously, 1/2

( ln

I; IX

1

12(012)

( n =

E E 1(x,

j=1 i=1

2

ej)I )

1/ 2

/fl

_. n 1/2

1/2

nt

sup {(E 1 1(xi , a)I 2 )

: a e U2 (n)}.

Therefore IV: 12(n) —> 12(n) I V 2ii —. n il 2 . In the case 2 < r < c o , we define 0 by 1/r = (1 — 0)/2 + 0/co. Then (171

)

1/r

E Ilxi I 12(n)11r i=I

) 1-0/ 2 (

5

E Ilxi I /20011 2

imaxIlxi / 2 (n)) ° . k

Note that 1/2

nz

max Ilxi /2 (01 -5. sup ((

: a e U2 (n)}.

1(x, 4 2)

Combining the preceding inequalities, we obtain 1/r 1/2 ( in n(1-0)/2 sup ((E 1(xi , a)1 2 ) : e U2 (n)} . E Ilxi I /2(n)11r) i=1 i= This shows that

n Ifr r,211 We now treat an arbitrary operator TE t(E, F) with rank (T) n. Given X e 2(/2 , E), we put N: N(T X). Consider the factorization TX = SQ, where S e t(12IN,F) is the operator induced by TX and Q denotes the canonical surjection from /2 onto 12 IN. Note that II SI I = 11E11. Since 12IN is a Hilbert space with n, dim(1 2 /N) = codim(N) = rank(TX) rank(T) 12(n) ---* 12(n) I

it follows from the preceding result that I TX I

Vr,2

I 5_ IISII

12/N I

r,2

I M QII

n'

TI II XII

This yields the assertion, by 1.2.7. 1.3.

Absolutely r summing operators -

r < cc. Throughout this section we assume that 1 1.3.1.* An operator Te t(E, F) is called absolutely r-summing if there exists a

46

1. Absolutely summing operators

constant c

0 such that llr

I n

E

1Ir

TXi r)

C

sup { (E '<x i ,

i=1

a>1) : a e U°)

i= 1

x. e E. The set of these operators is denoted for every finite family of elements x 1 , r = 1 special case we simply speak of an absolutely summing by 43,.(E, F). In the operator, and write 4.4(E, F) instead of 43 1 (E, F). For T e r (E, F) we define

11T1 V,.11 := inf c,

the infimum being taken over all constants c

0 for which the above inequality

holds. 1.3.2.* Taking into account that r = Vr,r we may specialize some results proved in the preceding section. For example, 1.2.3 reads as follows. Theorem (A. Pietsch 1967). §‘,. is an injective Banach operator ideal. 1.3.3.* The next statement can be derived from 1.2.5. 5

ri < ci. Then 43,..

ro Proposition (A. Pietsch 1967). Let 1 Moreover, we have 117'1 V„Il 5_ 111'144 ° 1 for all T e *ro(E, F).

Remark. From Example 1.6.3, it can easily be seen that the above inclusion is strict whenever ro < r1 . 1.3.4.* We now take a first step towards the main theorem of this section. Proposition (A. Persson /A. Pietsch 1969). Let T e WE, F). If there exists an absolutely r-summable sequence (ak) in E' such that cX

TxII

(E Kx, k=1

tor )

1Ir

for all x e E,

then co

F)

Te

x n eE we have 11r n co

Proof. For x1 , (E i=1

II

and 11T 43,11

EE

II (

i= 1 k=1

< k=1

(k

E

hr

1

Ilakilr) •

\ 1/r

=

1<xi,ak>lr)

Wa r)

/ co

E

k=1 i=1

1Ir{n

sup ( E Kxt i=1

n

,

\11r

1<xi,ak>lr)

or)

1/r

:

a e U°1

Remark. By a diagonal procedure, it can easily be seen that every operator with the above property is compact. On the other hand, in view of Examples 1.3.8 and 1.6.4, there are absolutely r-summing operators which fail to be compact. Hence the inequality \11r co for all x e E IITxII 5_ E Kx, ak>lr) (k=1

must be modified in order to obtain a necessary and sufficient condition. This can be achieved by using an integral instead of the infinite sum: f 1<x, a>r cl,a(a)) 1ir for all x e E. 11Tx11

.3. Absolutely r-summing operators

47

1.3.5.* We now establish the announced criterion which dominates the whole theory of absolutely r-summing operators. To this end, some additional notation is required. First of all let P l(/) denote the linear space of all bounded real-valued families on a given index set I. Furthermore, if A is any bounded subset of a dual Banach space E', then we assign to every element x e E the bounded complex-valued family fx (<x, a>) with a e A. Domination theorem (A. Pietsch 1967). Let A be any norming subset of E'. An operator T e t(E, F) is absolutely r-summing if and only if there exist a positive normalized linear form m on Irr'(A) and a constant c 0 such that (*)

liTxli 5

CM('

fx 1 r

) 11r

for all x e E.

When this is so, then 11 1'1

rl = inf c,

where the infimum is taken over all constants c the required properties can be found. Proof. Assume that Te

0 for which a linear form m with

F), and let c := IIT I

Define

s(f) := inf {sup [f(a) + Cr E 1<xi, a>lr : a e A] — E =

i =1

the infimum being taken over all finite families of elements x,, xn e E. Easy computations show that s is a positive homogeneous and subadditive functional on the real linear space Irri(A): and f e

Pri(4),

(1)

s(4) = ils(f) for À> 0

(2)

s(f + g) __< s(f) + s(g) for f, g e IrŒe,a(A).

Moreover, we have inf [f(a) : a e A] (3)

s(f) <_ sup [f(a): a e A].

The Hahn-Banach theorem, (DUN, 11.3.10) and (TAY, 1.10.4), yields a linear form m such that

m(f) s(f) 1f f

for all Je Pr i(A).

0, then it follows from (3) that

m(—f) __< s(—f)

sup [—f(a) : a e A]

0.

Hence the linear form m is positive. Moreover,

m(— 1) s(-1) = —1

and m(1) s(1) = 1

imply m(1) = 1. Thus the linear form in is normalized. Finally, we observe that crifx in( _ cr fir) <

r)

sup [— crifx(a)lr + Therefore, because c

a>ir : a e Al — Txli r .

I T J ril and f(a) = <x, a>, we have indeed

11Tx11 5_ IIT I V,11 maillrY ir for

all

x e E.

48

1. Absolutely summing operators

Conversely, for

x1 ,

x„ e E, it follows from (*) that n

11r

11r

(E

C ( i1 M(Ifx1 19)

)11r

illf)cil")

= C(Tn(

i =1 1/r C

: a e /1).

sup ((I Kx i ,

Hence T e 43r(E, F)

and

11T I

.

Remark. We stress the fact that, for Te r(E, F), the inequality CO can be satisfied 0 equal to IIT vr il. with a constant c 1.3.6.* Next we rephrase the domination theorem in a measure theoretic setting. For this purpose, some preliminaries are needed. Let CI(X) denote the linear space of all continuous real-valued functions defined on a given compact Hausdorff space X. Then the Riesz representation theorem, (DUN, IV.6.3) and (HAM, p. 247), asserts that every positive linear form in on Creal(X) can be written as an integral:

m(f) = J f() du() for all Je Crea'(X) where it/ is a (regular) Borel measure on X. The underlying a-algebra is generated by the open subsets of X. We know from the Alaoglu-Bourbaki theorem, (DUN, V.4.2) and (TAY, III.10.2), that the closed unit ball U° of any dual Banach space E' is compact in the weak topology generated by E. The defining semi-norms are given by

pm (a) := sup {I<x,

xc MI

for a e

,

where M ranges over all finite subsets of E. Domination theorem (measure theoretic version). Let A be any norming subset of E' which is compact in the weak topology generated by E. An operator T e WE, F) is absolutely r-summing if and only if there exist a normalized Borel 0 such that measure p on A and a constant c

c(J Kx, \A When this is so, then IITxII

IITI

ditt(a)) 1/r for all x e E.

= 1111 C

0 for which a measure p with where the infimum is taken over all constants c the required properties can be found. Proof. If T e 2.(E, F) is absolutely r-summing, then we choose a positive normalized linear functional m on lrceo al(A) as described in the preceding paragraph. Next we restrict this functional to the subspace C"al (A). Therefore, according to the introductory remarks, there exists a normalized Borel measure itt such that

f(a) dp(a) for all f e C"a l (A) .

m(f) = A

Obviously, p has the required property. The sufficiency of the above condition follows in the same way as in the previous proof.

49

1.3. Absolutely r-summing operators

1.3.7. Finally, we state the domination theorem in a special but rather illustrative situation. Proposition. Let X be any compact Hausdorff space. An operator T e 2(C(X), F) is absolutely r-summing if and only if there exists a finite Borel measure ,u on X such that IlTf 11 _< ( f I f()ir dp()) 11 r for all f e C(X). When this is so, then V I 43r I I

=

inf p(X) 1 /r,

where the infimum is taken over all possible measures 1u. 1.3.8. We now consider some important examples which are the prototypes of absolutely r-summing operators. Example (A. Pietsch 1967). Let it be a normalized Borel measure on any compact Hausdorff space X. Then the canonical operator / from C(X) into L,.(X, ,u) is absolutely r-summing, and .4 I WI = 1. Proof. From 1.3.7 we deduce that /E

r(C, Lr). Moreover,

11/: C-4 Lrli Ili: C --* Lr i 4 3 r11 5_ 1. 1.3.9. Example (A. Persson/A. Pietsch 1969). Let (X, ,u) be any normalized measuie space. Then the embedding operator / from L oo (X, ,u) into L,.(X, ,u) is absolutely r-summing, and 1

III: Lco -4 4 i 43 ,11 = 1. Proof. From 1.1.17 we easily see that Je 43r(L,,,, Lr). Moreover,

1

1. 11/:Loo -- 411 11/:Loo -- Lr I 43 ril Remark. If (X, it) is an arbitrary measure space, then it can be shown by the same method that the multiplication operator Mt

:M) -4 t(0 f()

associated with any function t e LAX, it) is absolutely r-summing from L OE,(X,,a) into LAX, it), and

11Mt : Loo -- 4 I 43r11 = 11f111rll ; see (PIE, 17.3.8). For the special case of diagonal operators we refer to 1.6.3. 1.3.10. Next we establish a very important result. Multiplication theorem (A. Pietsch 1967). Let 1 -_ p, q,r < co. If we have + 1/q = 1/r, then 43„ . $i, g. 43,.. More precisely, lip IIST I 43r11 5. IIS I 1411 IIT I $q ii Proof. We are going to show that llr n E1 IISTxiii n) (

5_ Ils

for all T e 434(E, F) and S e 43p (F, G).

n 1/.. I 4411 II T I WI sup ((El i<xi , a>lr) : a E U°)

50

1. Absolutely summing operators

x n e E. for every finite family of elements x 1 , By 1.3.6, there exists a normalized Borel measure II on U° such that 1<x, a>lq cit4(a)) 1/q for all x e E.

T1

IITxI

Put 1Ir : a e Uo}

=1 1<x i , a>11.) sup {(E

:

and ei := (f '<x 1 , a>1" cl,a(a)) 1Iq (Jo

Without loss of generality, it may be assumed that Tx i o. Then it follows that e , > 0, and we can define x? := x i . Choose any finite family of complex numbers oc„ such that

E loc•IP'

1.

Applying the threefold Holder inequality with the exponents r', q and p, for a E U° , we get (f ocix9 x? a)

i= 1

E locilPq" tail" eii1<xi, a>1"1<xi, a>lr1P

1'

i =1 11r,

<

n

11P

Orr (E i<xi, a>!')

i

(I= \ 119

(n

P.T q l <Xi, 019

1.111 •

i= 1

Hence

▪ IIT 43q11

q

uo

y

f <x1 , a>r cl/(a)) (1 0

T 434,11 (i

▪ IT

dtt(a)

1/q

qii 1-112 •

By Lemma 1.1.14, this means that 1Ip

sup {(K Tx?, b> P)

: b e V°)

11T I 43,1 11

1=1

Therefore 1IP

(Ei l STX ?Ii P)

5-

I S I Vi,11 117'1

1391I er1P

$rIP

51

1.3. Absolutely r-summing operators

Finally, another application of Holder's inequality yields \1Ir

n

leisTx?11)

= (n

lip

IISI

11T

)119

Ils I 4.;p11 IT I *4711 './P+ria

E et°

IlsTAIP)

(E i.1

llr

n

I STxi 117 I

f=1

WI sup 1<x; { (E1 ,

I

Or)1

1.1r :

ae1

°).

This proves that

STE 43,.(E, G) and 11ST I 4,;,11

!IS I Vp11 I T I Vq11 •

1.3.11.* The following stability property is quite useful for the theory of eigenvalue distributions. Theorem (J. R. Holub 1970/74). The Banach operator ideal 43, is stable with respect to the tensor norm E. More precisely,

11.5 I VrIl IIT I 4411

(--5

Proof. The domination theorem on U° and V° such that

1.3.6

e(E Eo)

for S

and T e 44(F, Fo).

yields normalized Borel measures

Il

(f6

Kx, a>lr diu(a)) 1 ir for all x e E

I TYII 5_ I T I $ril

f

Ky, b>lr dv(b)r for all y e F,

11Sx11

11S

it

and y

and

respectively. Let

x1 ,

x„ E E and 3,1 ,

E <Sxi, ao>

bo>

i=1

E

F. If Hao II

1

and

11b0 M --.15

1, then

E Xj
S( i=1

( n

E xi
i=1

r

)1Ir d p(a) )11r

T (±1 <x 1 , a> y i) ,u(a) d

5_ IS I *Al IITI rM f [

uo vo i= 1

<xi , a>

y ir d v(bdd,u(a)

This means that

ii(S

nz

I Ell

for all z e E F. Since E extension, we obtain

11S I $ril

Ci eF is the

f

\u0

iz(a, b)Ird it(a)d v(b)) 1 fr Vo

closed hull of E F in E(E, F), by continuous

11(S(j eT) z I Eli 5_ IIS I VrIl I T 4rII ( Ç f jz(a, U° 1;°

du(a) d1(b)) 1ir

52

1. Absolutely summing operators

for all zeE0 c F. From the definition of the tensor norm s we infer that U° x vo a forming subset of 8(E x F)'. Hence, by the elementary part of the domination is theorem, STE Vr(EC),F, Eck) F0) and 11 ,90J' I VrIl Il S IrII JJT I VrIl 1.3.12. In the following we need some information about the local structure of the classical Banach function spaces. Lemma. Given fl , • • • e Lr(X, ,u) and e> 0, there exist Q e 2(4, ir(m)) and Je 2.(4.(M), L r) such that IIQII 1, VII 1 and Ilfi — JQf, I 411-5. 8 for

i = 1, .,n.

Proof. Choose simple functions .5' 1 , ..., s G Lr(X, p) such that n. Ilfi — si 1 Lrll 8/2 for i= 1, Write ,,

Si

=1

where h 1 , h, are characteristic functions of pairwise disjoint measurable subsets X„ ..., X,,, of X. We now define the required operators by ,71

112

Q:=I p(XJ)- "r'hi 0 ei and .1" := y p(Xi)-- 1/rei 0 hi . Obviously,

11Q1i

1. Moreover, it follows from JQhi = h., that

1 and 11.111

= si . Hence

Ilfi —

Jafi

—

1 411

si

I Lr11 + IJQ51 — JQf1 I 411

21Ifi —

si

I 411 5.

•

This completes the proof. 1.3.13. Let I denote the embedding operator from 1 2 into co . Then I' and I" can be identified with the embedding operators from 1 1 into 12 and from 12 into lc° , respectively. As proved in 1.6.4, we haver c , 12). On the other hand, I 0 43r(12 co) and I" 0 43,(12, 1,,). This observation shows that the property of being absolutely r-summing neither goes over from an operator to its dual nor conversely. The situation becomes more favourable for operators acting between special Banach spaces. Proposition (A. Persson 1969, P. Saphar 1970, J. S. Cohen 1973). Let , E') implies T E r(E, Lr) and II T I 3,.11 T e t(E, L r). Then T' E I1T' I 43,0. Proof. We first consider an operator T e t(E, l(m)). Then, for X E E, \ 1/r

ni

IITX I 4 141 = kE= 1Ilr)

/m

\ 1/r

<x, rek>lr) • = (1 kE =1

Hence it follows from 1.3.4 that \

/

I T I Vr11

1/r

1 r I rii ii(e r) I wrIl = IIT' I 43,11 •

E

k =1

We now treat the general case when T' e V r(11, E'). Given x 1 , x„ c E and e > 0, by the preceding lemma, there exist Q e t(Lr,lr(m)) and Je 2(4(05 Lr) such that IIQII 1 and 1, VII

1 Tx — .1QTx t L r Il

8 E 1 ±

11Tx

L 11 r

for

i

1,

n.

Since QTE 2(E, /r(m)), from the first part of the proof we know that

IIQT I rii

!WIT

lir I WI

1.3. Absolutely r-summing operators

53

Moreover, since

11.1QTx1 I 411 + 11 Tx — JQTx i 1 Lr11

IlTxi I 411

IIQTxi I 1r(m)11 + 1 + e e 11Tx1 I Lr11

we have

II(Txi) I 411 5_ (1 + 011 (QTxi) I 441 5_ (1 + e) IIQT I *AI 11(xi) I wrIl

This proves that Te 4.) and II T I ,.11 I IT' I 1.3.14. We now turn to the very special case in which the operators act from an arbitrary Banach space into a Hilbert space. Here we use the Rademacher functions defined in G.2.1. Moreover, a,. and b,. are the constants appearing in Khint,

chine's inequality G.2.3. Lemma (A. Pietsch 1972: c). Let T e t(E, 12 (m)). Then arlIT

(f E T'ekek(r)

I 43,41

k1

0

Proof. For x 1 ,

1Ir

r

in

5_ brIIT' I WI.

dT)

x„ c E, by Khintchine's inequality, we have

I /2(m)IIr k=1

y

fr

m I
<xi,

ek>,2niir

Tek eker)> T'ekek(r)>

k=I 11r

E rekek(r)

d'r)

1/r

sup {(

k =1

a>I)

i=1

:

a e t./ 0 ).

This proves the left-hand inequality. The domination theorem yields a normalized Borel measure it on U2(m), the closed unit ball of 12 (m), such that 11 7" J V,11 Ky, b>ir dy(y)yr for all b c 12 (m). \ U2 (m)

We now obtain the right-hand inequality as follows:

E T'ekek(r)

(0

k=1

) 1Ir

Fri

-5- 111'1 43,1I (1 f

dit(y)

O U2 (m)

1Ir

1

= Ilr I 43r1I

f

f

E
ek(x)

dt. dy(y))

U2(m) 0 k=1

5_ br lii" I 43,1

f

in

(1 KY, ek>I 2)

u2(m) k=1

)1/,-

rf2

diu(y)

brIIT' I VA!

54

1. Absolutely summing operators

This completes the proof. Remark. In place of Rademacher functions one may use independent Gaussian variables. 1.3.15. Proposition (S. Kwapien 1970). Let Te 2(E, H), where H is a Hilbert space. , E') implies Te V s(E, H) for 1 Then T' e r, s < oo Proof. In view of 1.3.3, it is enough to treat the case 1 sr < oo. Given xl , e E, we denote the linear span of Tx ,, Tx„ by M. Let Q e (H, M) be the orthogonal projection from H onto M. Identifying the finite dimensional Hilbert space M with 12(m), we may apply the preceding lemma to the operator QT. Hence l ls

in

E

asliQT Vs11

k =1

r (QTY e9.rck(r) d)

in

k =1

Since

Ho!

(QT)' e ic(T)

brii(QT)' r

r11 •

) 1/r

1, it follows that a's-l brilr I Vril

IIQT We now obtain II(Txi)

= ll(QTxi)

1 /1,11

a: 1 14 11T' 4.;rII 11(x t) I

wsll •

This proves that Te 4155 (E, H)

and jT Vsli 5. a; l brilr j WI •

1.3.16. Applying the preceding proposition twice, we arrive at a result which is, at first sight, very surprising. Theorem (A. Pelczynski 1967, A. Pietsch 1967). Let H and K be Hilbert spaces. Then all components ,.(H, K) with 1 r < oo coincide, the norms j I 43,11 being equivalent. Remark. The best possible constants in the estimates between II• I V2 IL and ii• 43,11 were determined by D. J. H. Garling (1970).

1.4.

Hilbert Schmidt operators -

Throughout this section H and K are complex Hilbert spaces. 1.4.1.* First we state an immediate consequence of Parseval's equation. Lemma (R. Schatten/J. von Neumann 1946). Let (u i) with i E I and (vi) with j e J be orthonormal bases of H and K, respectively. Then, for every operator T e 2(E, F), we have

E IlTuill 2 = Er

J

v)1 2 = E E Kui, T*vi)I 2 = E .1

where the sums may or may not be finite. 1.4.2.* We call Te 2(E, F) a Hilbert-Schmidt operator if (II Tu i ll 2) is summable for some orthonormal basis (iii ) of H. The set of these operators is denoted by Z(I/, K).

1.4. Hilbert-Schmidt operators

55

The preceding lemma implies that, for T e Z(H, K), the expression ell :=1lTu i ll 2) 1 / 2 does not depend on the special choice of (u1). 1.4.3.* The following theorem is well-known. We leave its straightforward proof as an exercise to the reader. Theorem (R. Schatten/J. von Neumann 1946). Z is a Banach operator ideal on the class of Hilbert spaces. Furthermore, given any orthonormal basis (u1) of H, the formula (S, T) : = E (sui , Tut) defines an inner product on Z(H, K). Remark. Thus Z is indeed a Hilbert operator ideal which means, by definition, that its norm can be derived from an inner product. 1.4.4.* The next result states that all Hilbert-Schmidt operators are approximable. Proposition (E. Schmidt 1907: a). The Banach operator ideal Z is approxi mative. Pro of. Let (ui) with i e / be any orthonormal basis of H. Given T e Z(H, K) and E > 0, we choose i e Ti.(I) such that (E

E.

Define the orthogonal projection

Pi := u C) u t which has finite rank. Then

JjT — TPi t

= (E IlTuill 2) 112 /\i

8*

1.4.5.* The concept of a Hilbert-Schmidt operator has proved to be very useful in treating various problems in Hilbert spaces. The following theorem explains why absolutely 2-summing operators can be considered as a fairly good substitute in the Banach space setting. Theorem (A. Pietsch 1967). An operator Te t(H, K) is absolutely 2-summing if and only if it is Hilbert-Schmidt. If so, then we have 11 T I *2 11 = tITI eli. Proof. Let Te 2 (11, K). If (ut) with i ci is any orthonormal basis of H, then 1.1.11 and 1.2.2 imply that Tu1ll 2) 112 < IT 1 43211. T e E(H, K) and In order to verify the converse, we consider in K any orthonormal basis (v i) with j e J. Let x 1 , x„ e H. Using Parseval's equation, we now obtain / n

i=1

\ 1/2

lInci11 2 )

\

n

=

i=1

E 1(Tx1, 01 2) J

1/2

=

E kxi, T*v)1 2)

J i=1

1/2 5-- (

IIT* Vf11 2 )

\ 1/2

n

1/2 SUP

a)1 2)

: a E U).

56

1. Absolutely summing operators

Consequently, T e 432(H, K)

and

11 T I V211 — _ ( 11 T *VJ li 2) 112 •

1.4.6.* A matrix M =(,ali) is said to be of Hilbert-Schmidt type if it belongs to 12(1 x J). Then Mop :

0'0 -- (.

ttipb)

defines an operator from 12(J) into 12 (1). 1.4.7.* Using the preceding definition, we may rephrase Theorem 1.4.5 in terms of matrices. Proposition. An operator from 12(J) into 12(1) is absolutely 2-summing if and only if it is induced by a Hilbert-Schmidt matrix M = (4uu). If so, then we have ittij1 2)" 2 .

IIM-op I 43211 = (,

1.4.8.* We now consider integral operators on Hilbert function spaces. For this purpose, let (X, ,u) and (Y, y) be os-finite measure spaces. A kernel K is said to be of Hilbert-Schmidt type if it belongs to L 2 (X x Y„u x y). Then K., : g(n) -4 f K($, 71) g('i) dy(n) defines an operator from L2(Y, y) into L2(X, itt). 1.4.9.* We now establish a counterpart of 1.4.7. Proposition. An operator from L2 (Y, y) into L 2 (X, ,u) is absolutely 2-summing if and only if it is induced by a Hilbert-Schmidt kernel K. If so, then we have ilKop

I 13211 = ( f f 114$, /01 2 dp($) dv(n))' 2 . \x Y

Pro of (sketch). Let (u1) with i e 1 and (vi) with j c J be orthonormal bases of L 2 (X, ,u) and L2 (Y, y), respectively. Observe that the functions ui ($) vi(Ti) form an orthonormal basis of L 2 (X x Y, 1,1 x y). If T e V2(L2(Y, v), L2(X, su)) , then

(X E 1(Tvi, ui)1 2r2 — IIT I 43211 \I

J

is finite. Hence we may define the kernel

K($, ID : =

(Tv i , u i) u g) v i(n) ,

where the right-hand sum converges in mean square. Note that K is of HilbertSchmidt type and

I iqe , 0 2 dtt($) dv(7/)) 112 = II TI 43 2Il . (if \x Y Moreover, K induces the given operator T. The converse is obvious.

57

1.5. Absolutely 2-summing operators

1.4.10.* We conclude this section with an elementary observation; see B.3.8. Proposition (R. Schatten/J. von Neumann 1946). The following are equivalent for T E t(H, K): T e (H, K), T* e (K, H), T' E (K', H'). The corresponding norms coincide.

1.5.

Absolutely 2 summing operators -

1.5.1.* In the case r = 2 the domination theorem leads to another result which has far-reaching consequences. Factorization theorem (A. Pietsch 1967). Every operator T c * 2(E, F) admits a factorization

A

/03 (1)

>H

Y

such that the following conditions are satisfied: (1) / is an index set. (2) H is a Hilbert space. A e t(E, L(/)) and 11A11 = 1, Ye t(H, F) and (3) (4) S e V 2 (1,0(1), H) and IISI 4211 = 1 .

11)7 11 = 11T I 43A

In particular, if T is finite, then it can be arranged that dim (H) = rank (T). Proof. Take U° as the underlying index set. Applying the domination theorem 1.3.5, we find a positive normalized linear form m on l'( U °) such that

117'4

IIT I V2I1 m(f 2) 1/2

for all x

E

E.

Obviously, in can be extended canonically to the complex Banach space /0,(U°). Form the quotient space 1,0(U°)IN, where

N := Ife 1„(U°): m(f 2) = 01, and denote by f := f + N the equivalence class determined by f. Then

(f, g) := m(fg*)

for f, g e lco(U°)

yields a well-defined inner product on /(U °)/N, and the desired Hilbert space can be constructed by completion. Next we define the operators A, S and Y. Put Ax := fx , where f(a) := <x, a> for a c U° . Let S be the canonical operator from /09(U°) into H given by Sf := f. By analogy with Examples 1.3.8 and 1.3.9 we see that

Se $2(1.3 (U°), H) and

11S I V211 = 1.

The definition of Y is slightly more involved. Observe first that fx , Tx, = Tx2 , because

11Tx 1 — Tx2 1

11 7'1 4;211

— f, 2 1 2)" for x„ x2 E E.

= fx2 implies

58

1. Absolutely summing operators

Hence the map Yo which sends fx into Tx is well-defined on M(SA). It follows from

IIT I 4 2I1 moL( 2) 1 /2 „ ((T( 13211 IV; I HII

1117(3411 = 11Tx11

IIT I that Yo is continuous: Yo Thus there exists a continuous extension Yo on Ho := M(SA). Next we consider the orthogonal projection Q c e(H, H0) from H onto Ho , and set Yf := Y oQ f for f e H. Then

I Yll

(IT( *211 and YSAx = YSL = YoQfv = Yofx=

I Yo I 1101

We already know that I1A = 1 and IIS

IIT I 33211 = I YSA

4 211 = 1.

Tx.

Furthermore,

I Yll 115'1 43211 11All

I Yll

((T( V2I1

yields 11 Yil = V211. Hence T = YSA is the desired factorization. Finally, we observe that M(T) M(Y) g M(T). Thus, in the case when T has finite rank, it follows that M(Y) = M(T). Passing from H to HIN (Y) we find a factorization through a rank (T)-dimensional Hilbert space; see also B.4.6.

1.5.2.* If /co (U 0) is replaced by C(U°), then the preceding proof becomes more transparent, since the abstract construction of the Hilbert space H turns into the classical procedure which relates L 2 (U°, 4u) to C(U°); see 1.3.6. Factorization theorem (measure theoretic version). Every operator Te V 2 (E, F) admits a factorization

>F

h. y

A

C(X) such that the following conditions are satisfied:

(1)

X is a compact Hausdorff space.

(2)

,u, is a normalized Borel measure on X. A e t(E, C(X))

and I

= 1,

Ye t.1 (L 2 (X,

F) and II YI; =

T V2

(3) / is the canonical operator from C(X) into L 2 (X,,u). (4) 1.5.3.* The preceding factorization theorems imply a useful multiplication formula. V2. The norms on both sides are equal. Proposition. 3; 2 = 1.5.4.* Absolutely 2-summing operators have another remarkable property. Extension theorem. Let J e t(E, E0) be any metric injection. Then every operator T e 432 (E, F) admits an extension To e V 2(E0 , F) which preserves the V2norm. This means that T = T0.1 and 117'0 I 434 = IIT I 43211. Pr oof. Consider a factorization T = YSA as described in 1.5.1. Since icc (/) possesses the metric extension property, there exists A o e 2,(E0 , /JO such that A = A 0.1 and I! Aoil = 1. Then T0 := YSA 0 is the extension we are looking for.

59

1.5. Absolutely 2-summing operators

Remark. The idea of the preceding proof is due to G. Jameson. Previously, one had used a factorization of the form

A

A

y

L (X , la) —> L 2 (X , it) which follows immediately from 1.5.2. However, to verify the metric extension ,a) is by no means as easy as in the case property of the complex Banach space of /0,(/); see (DIL, pp. 228 and 230) and (IKON, 1.c.2). 1.5.5. We now establish an extremely important lemma due to M. I. Kadets/ M. G. Snobar (1971). The original proof was based on a result of F. John (1948) concerning ellipsoids of minimal volume containing the closed unit ball of a given finite dimensional Banach space. The following approach goes back to S. Kwapien. Lemma. Let M be any n-dimensional subspace of E. Then there exists a n 1/2. projection P e 2(E) such that M(P) = M and VII 111/2. Thus, by 1.5.4, there exists Proof. It follows from 1.2.8 that 11/m I 43211 T e t(E, M) such that 4 2 1i n'i2 . = Im and 11711 Then P := JfifT is the projection we are looking for. 1.5.6. In the remainder of this section we investigate the operator ideals (432)"' with in = 1, 2, ... For this purpose, the following auxiliary result is needed. Lemma. Suppose that the quasi-Banach operator ideal /1 is p-normed. Then 'It V, is q-normed, where 1/q = 1/p + 1/2. Proof. Given S,T e It o V2 (E, F) and E > 0, we choose factorizations S = XA and T = YB such that Ac 2 (E, M), Xe 11(M , F), B e V2(E, N), Y e ?1(N, F),

A I $ 211

+

X IQ

+

1 IS I IS

° 43 2111q/ 2

1 13 I 211

+

° 4 211]"

Y1 ?CI

+

IT

° 432111q/ 2 ° AP I P •

Consider the direct sum M N endowed with the euclidean norm. The corresponding injections and surjections are denoted by m , N and QM, Q N, respectively. Then, for x 1 . x„ e E, we have \ 1/2 1/2 2 .1 NB) 1113x i11 x11 (E (11Ax i11 2 + E VMA + 2 )) 1=1 i= 1 fl112 5- (MA 14.3 211 2 ± 11B 211 2 ) " 2 SUP ((E Kx a>1 2 ) : e U°}. fl

(

i=1

Therefore

IIJMA + JNB1 V211 (MA 1 '41 211 2 + 1IB I $211 2) 112 ▪ (1 + E) /2 V2P) 1/2 ° V2li q + Moreover,

)1/P

GI X I 4UII + II YI WII P IIXQm + YQN I %II ▪ (1 + (IISI SU ° V211'1 + I TI SU ° 4;211'1 ) 11P •

60

1. Absolutely summing operators

Combining these inequalities, from S + T = (XQm + YQ N)(Jm A + JN B) we deduce that

I XQm + YQN I 1111 11.1mA

11S + T1 1- t ° 4,; 2ii (1 +

(II S I

°

2Il q + llTJ t°

+ JNBIV2II

211'1 ) 11g •

Letting E —+ 0 yields the g-triangle inequality. 1.5.7. Using the preceding lemma, we get the following result by induction. Proposition. The quasi-Banach operator ideal (43 2 )"' is 2/(m + 1)-normed. 1.5.8. Proposition. The quasi-Banach operator ideal (432)m is injective. Proof. The assertion follows from 1.3.2 and the trivial fact that the product of injective quasi-Banach operator ideals is injective, as well. 1.5.9. Proposition. The quasi-Banach operator ideal (432)" 1 +' is approximative. Proof. Obviously, it is enough to treat the case m = 1. From 1.5.1 we know that every operator T e ( 32) 2 (E, F) admits , a factorization T = Y SA such that A e 432 (E, H), S c V 2 (11, K) and Ye 2(K, F), where H and K are Hilbert spaces. Recall that 4 2 (H, K) = e(H, K), by 1.4.5. Since 2(H, K) is approximative, there exists a sequence of operators Sk e .R(H, K) converging to S with respect to the HilbertSchmidt norm. Finally, we see from YSA — YS k A I (V 2) 2 II Yll 11 s - sk 1 *211 I IA I 43 211 that the finite operators Tk := YSkA converge to T = YSA in ( 2) 2 (E, F). Remark. As shown in 1.6.4, we have J e V2 (l 1 , 12 ). Since IS(/ 1 , 12 ), the Banach operator ideal V 2 fails to be approximative.

1.5.10. Proposition. (C) "+ ' (V2r. Proof. By definition, every operator Te (1)"'(E, F) can be written as a product T = T.+1 ... T1 of operators Tk e Ek), where E0 = E and Em+1 = F. Next we choose factorizations T = A kXk such that Xk e 2 (EL, Hk) and Ak e 2(Hk , Eic_ 1 ), where H1 , ..., H177+1 are Hilbert spaces. In view of 1.3.15, it now follows from XkA k+i E 432 (Hk+i Hk) that A'k+1 X/, e V 2 (1//c , H ÷ ). Observe that this conclusion could also be inferred from 1.4.5 and 1.4.10. Hence KFT = THKE = Kni•i(A m i ÷iXh„) • • • (21'2X ) A;KE E (43 2) rn (E, F").

Finally, the injectivity of

1•6.

(

4 2) Pn Yields

T

(C)" (E, F).

Diagonal operators

p, q Throughout this section we assume that 1 oo. Moreover, ek denotes either the k-th unit sequence or the k-th coordinate functional. 1.6.1.* In the following we are concerned with diagonal operators Dr:($k)

(rk$k)

induced by complex-valued sequences t = (TO and acting from 1,„ into /a . Without 0. If 1 loss of generality, it may be assumed that Tk co, then the embedding operator from /i, into /a is denoted by I.

1.6. Diagonal operators

61

1.6.2.* First we state an immediate consequence of Holder's inequality. Recall that $÷ := max ($, 0) for $ e R. Lemma. Let 1/r = (1/q — 1/p) +. Then D, e 2(/„, /0 if and only if t e /,.. 1.6.3.* Next a basic example, related to 1.3.9, is treated. q < co. Then Example (A. Persson / A. Pietsch 1969). Let 1 D, e 13q(1,, , /0 if and only if t e 1. Proof. Assume that t e lg . Since liptx I 411

k=1

Kx, rkek>l'1

for all

X G laD

it follows from 1.3.4 that D, e /0 and 11D, : /co 4 I Vgli It 411. The converse is obvious, by 1.6.2. 1.6.4.* We now give an application of the domination theorem in which the underlying Borel measure can be described explicitly. Example (A. Grothendieck 1956: b). Je V( 11, 12). Proof. Let r(r) := (ei(r)), where e , denotes the i-th Rademacher function. The function r defined in this way maps the interval [0, 1] into the closed unit ball UOE, of lc° . If tc is the image of the Lebesgue measure, then Khintchine's inequality G.2.3 yields „ 1,2 dr aF 1 f E <-1c, ei> Ilx 1 1211 = ( E Kx, ei>1 2 )

i =1

iI

=ai 1 f Kx, r(r)>1 dr = ai 1 f 1<x, 01 d(a)

uo,

for all x e / 1 . This completes the proof, by 1.3.6. Remark. The example under consideration can be deduced from the famous Grothendieck inequality which implies that all operators from l into 1 2 are absolutely summing. More precisely, there exists a constant CG 1 such that IIT I l call for all T E t(l1 , 1 2). For proofs of this fundamental theorem the reader is referred to (DIE, p. 181), (LIN, Vol. 2.b.6) and (PIE, 22.4.4). The best possible value of the so-called Grothendieck constant CG is still unknown. 1.6.5.* The following result is classical. if 1 p 2. Example (W. Orlicz 1933). l e 432, 1 (1„, xn E l it follows from Jessen's and Khintchine's inequalities Proof. For x 1 , that 1 )1/2 )2/1 oo n / E 11Xi i=1

I

lp11 2

)

( i=I E (E I<Xi, ei>IP j=1

n PI 2 11P < _1 = ap i(E (E e i<X , f>1 2) C))

J=1 i =1 11P (1 I P = a-1 E x( ) lp! dr) P r 0

Thus we have lE 43 2 , 1 (4„ 4,), by .1.14.

.

co

1

<xi , ei >e,(7-)

P

)11P

dr

J= 1 o i=l a; 1 sup ( E

i= 1

i x t lp :

1). i=1

62

1. Absolutely summing operators

1.6.6.* We now give a more general example which includes both 1.6.4 and 1.6.5 as special cases. Example (G. Bennett 1973, B. Carl 1974). Let 1 p q 2. Then Je vs, i (/ /0, where 1/s = 1/p — 1/q + 1/2. Proof. We first assume that q = 2. If p = 1 andp = 2, then the assertion is proved in 1.6.4 and 1.6.5, respectively. Thus it suffices to deal with the case 1 < p < 2. Define 1/u : = p — 1 and 1/v := 2(p — 1)/p. Then 1/u' = 2 — p and 1 1v' = (2 — p)/p.

Let x 1 ,

xeI. Note that n

i=1

lixi 11211P =

cc

1, 1< Xi, e>1 2 11Xi 1 1211 1)-2

1=1 j=1 co

n

= j=1 1=1 1<x1, ei>1 2/u Kxi ei > , 21tv I lx i 1 Applying Holder's inequality twice, we obtain co

1 1211 P 5- E

i= 1

E 1<xi, ej)1

2

1 luf

)11u

1<x1, e)1 2 11Xt 1 1211 -1 )

j=1 i=1

(

co =-.-

(n

(Z

)p/2)1/v / co / n \P\ 1 1v' ( Z (E 1<x, ej>1 2 11Xi 1 1211 -1 ) ) Z '<X i , ej>1 • n

2

j=1 i=1

j=1 i=1

The two factors in the last expression are now treated separately. First, by Khintchine's inequality G.2.3, it follows that 2) p/2 ) 1/p (co I 00 n n 61;) 1 Z 1 Z <Xi, ej> eier) P d 'r 1112 Z Z( I<Xi ) ei>I /

(j=1 i=1

j=

1 b 1=1

x(r) i =1

a;1 sup { = a; 1 sup { i

1<xi , a>I: a e Up ,).

=

Next, the second term is estimated. Using Khintchine's inequality once again, for all a = (oc) e l, we have co

E

j=1

i= n

=

1

'<Xi, ej>I2 11Xi

co

ei>1 Ilxi 1

i=1 j=1 n

I 1211 -1

1/2

co

1=1 j=1 1

le(j<Xi, ej>1 2

1 12

1 -1

co

1/2

1=1

I<X1, ej)1 2 11Xi 1 1 211 -2 I

cc

f E

j=1

dr

5_ ai l

n

f E 1<x1, ar(r)>1 dr,

o i=

63

1.6. Diagonal operators

where ar(r) = (ociei(z)). This implies that oo (n

)P)11P

ail sup ( E i<xi , a>1:ae Upl.

E 1<xi, e:;>1 2 Ilxi 11211 -1 ( .J=1 E i=1

i=1

Combining the preceding inequalities, we obtain

( co

p12)11 p v

n

1=1 i=1

(Co n

1< xi, ei>1) 2

ari, 1 I v ai l I sup {

(

j=1 i= 1

1<x, ei>I 2 IIxi 1 12 ( 1 -1

) p)

' <x, a>I : a c i =1

Hence /e *p, Alp 1 2)• Finally, we treat the case 1:_.p
1/q = (1 — 0)/2 + Then it follows from 1/s = 1/p — 1/q + 1/2 that

lis = (1 — 0)Ip + 0/2. As already shown,

/E

12)

and le 432 , 1(///

1p)

•

Since

i=1

(n

11s

(n

Ilxi I /,711s) -- (

)1Is

E i=i

112 ( 1 1 ' Ilxi I /pills

n

) ( 1 - 0)/ P (n

E IIXi I 121I P

i=1

• 0/2

il:1 1 X i I 11)11 2)

for x 1 , ..., x„ e 1,„ we have (IL 4, .Hence /e 43 1 0

41

c,111

I(I:

4, .- 1 2 I

43p,111 1.-° III:4 -4

i,, 1 V2,111 0 .

le).

1.6.7.* Next we state an immediate consequence of the preceding result which is crucial for later applications

Example (G. Bennett 1973, B. Carl 1974). Let 1__-< p._._q.co. Then If E $r ,2(ip , 47), where

(1/p — 1/q if 1 __- . 1 , q _. 2, 1/r : = 1/p — 1/2 if 1 -. p. 2 q< cc, 0 if 2 -.13-.q.- oo . Proof. If 1 p-_q. 2, then the assertion follows from 1.2.5 and 1.6.6. If I __-.13___ 2 __-< q_co, then we conclude from /e 13,,, 1 (/p , 12) that /e 13„, 1 (4„ 4). Hence JE 13,.,2(ip , /q) with 1/r = 1/p — 1/2. The case 2 __-
64

1. Absolutely summing operators

1.6.8. Example. D E (4.3 2)m (12 , l) if and only if t e 12/(7, +1) . Proof. If t = ('rk) c 121(n _ 1) , then s := erki ( n+ 1)) e 12 . Now it follows from D,c 432 (12 , 12) and D, e 2,(/2 , /1 ) that D, = D,DTE (432) m(12, 11). By 1.6.4, the embedding operator from /1 into 12 is absolutely 2-summing. Hence D, e (432)T2, / 1) implies that D, G (132) "1 (/2 12). Since D, has the eigenvalue sequence t, we may infer from 3.7.3 that t e 121(.+1). 1.6.9. Example. D, 2 )"' (l oo , 12 ) if and only if t E /20„. Proof. If t = (l-k) e /2/,„ then s: = (vj in') e /2 . Now it follows from D, e , 12 ) and D, c 42(/2, 12) that D, = Dr-1 D, c (13 2)m(loo , 12 ). Conversely, D, e (V 2 )"' (loo , 12 ) implies D,c (132)m(1 2 , 12). Since D, has the eigenvalue sequence t, we may infer from 3.7.3 that t e 121 „,• 1.6.10. Example. D, e (V2)m (/, 12 ) if and only if t e 1 21(fl ,_ 1) . Proof. In view of 1.6.4, every bounded sequence induces an absolutely 2-summing operator from /1 into 12 . Therefore the case m = 1 is trivial. If t = (7-k) e /2/(n - 1) , then s := (41( "-1)) c /2 . Now it follows from D, e 4 2 (/2 , 1 2 ) that D, = Dr 1 e (432) m-1 (1 2 /2). Hence D, E (V2) m (i1 , 12), by 1.6.4. Let D,e(432)m(1 1 , 1 2 ). Since Ds e t(/2 , / 1 ) for all s = (crk)e 12 , we obtain immediately At = D,D, E (42)"(l1, /J. Hence, as in the previous proofs, we may conclude that st e 121 .. This implies that t e 1 2/(m-1). 1.7.

Nuclear operators

Throughout this section we assume that 0 < p 1 and 0 < r 2. 1.7.1. An operator T e 2(E, F) is said to be p-nuclear if there exists a so-called p-nuclear representation

T=

CO

a,, ®y,, 1=1 such that al , a2 , • • • G 1„. The set of these operators Yi, Y2, • • • G F and is denoted by 91,(E, F). In the special case p = 1 we simply speak of a nuclear operator, and write *E, F) instead of 911 (E, F). For T e F) we define co lip T I 9411 := inf(E 1 aili P Iy the infimum being taken over all representations described above. 1.7.2. The following result can easily be verified (PIE, 6.3.2 and 6.7.2). Theorem (R. Oloff 1969). Itp is the smallest p-Banach operator ideal. 1.7.3. We now give a very important characterization which relates p-nuclear operators to diagonal operators (PIE, 18.1.3). Factorization theorem (GRO, Chap. II, p. 12). An operator TE t(E, F) is p-nuclear if and only if it admits a factorization

>F A

I /OE,

D,

>1 1

1.7. Nuclear operators

65

such that the following conditions are satisfied: (1)

Dt is a diagonal operator with t E 1„.

(2)

A e t(E, / co) and

Y

F).

When this is so, then

iITj9il = ml {IIY lit J 11,11 where the infimum is taken over all possible factorizations.

1.7.4.

We now provide an auxiliary result. Lemma. Assume that oo

and

Xi 0

1= 1

Given

E>

0, there exists (x i) such that oo

0
< 1, 1imo =

0

and

oo

(1 + 6)

ocT i T i i=1

x1 . i=1

Proof. Let 0 = jo < j1 < ... such that oo

E

E in +1

— 4" i=1

ri

for n = 1, 2, ...

Define oc i

=

2-n if in < i < in + i and

n = 0,1, ...

Then CO

00

j1

OCT 1

=

1=1

1

n=1

00

in+1

2,"

ri ±

ri

E

(1 +6

i„+1

CO

2- n)

n=1

E

ri.

i=1

This completes the proof. 1.7.5. Proposition. The quasi-Banach operator ideal 9i, is approximative. More precisely, we have 91, = ON, (4. Proof. Given Te 911,(E, F), we consider a factorization T = YD t A as described in 1.7.3. In view of 1.7.4, the sequence t G ip can be written in the form t = ysa such that s Eli, and a, y G co . Then it follows that T = (YD y)Ds(DaA), where DaA e (4(E,l) D, e 91„(/ l i ) and YD, c F). Hence T E CO 91r, o WE, F).

1.7.6. Next we mention a fundamental result which is well-known as Auerbach's lemma; see (BAN, p. 238) and (PIE, B.4.8). First proofs were published by A. E. Taylor (1947) and A. F. Ruston (1962). Lemma. Suppose that the Banach space E is n-dimensional. Then there exist x 1 , e E and a l , a, e E' such that 11x111 = 1,

1.7.7.

Ja1i1 = 1

and <xi , al) =

We now prove an analogue of 1.2.8. Lemma (A. F. Ruston 1962). Let T e 2(E, F). Then

IITI 91Ipli

n "P 11711 whenever rank (T)

n.

66

1.

Absolutely summing operators

Proof. We first treat the identity operator of an n-dimensional Banach space E. Choosing x 1 , x„ e E and al , ..., an e E' as in Auerbach's lemma yields the representation 1E =

ai

nlIP Hence 11 1 E 1 9411 Let T e WE, F) be any finite operator. Put M = M(T), and consider the canonical factorization T = &T0 , where To e 2(E, M) is the operator induced by T. Since 11 7'011 = 11T11 and dim (M) = rank (T), we get

IIT I 9411 5_ InI1 II4f 1 9'411 111'011 5 n"P lITII Remark. If dim (E) = n, then it follows from trace (IE) = n that 11/E 194 = n. However, for 0 < p < 1, the value of 11/E 1 51} ,...„11 depends on the geometric structure

of E. 1.7.8. An operator T e t(E, F) is said to be (r, 2)-nuclear if there exists a so-called (r, 2)-nuclear representation 00

T=

i= 1

ai C) y i

such that (ai) e [4, E'] and (yi) e [w2 , f ] . The set of these operators is denoted by 94,2(E, F). In the special case r = 2 we simply speak of a 2-nuclear operator, and write 912 (E, F) instead of 912 , 2(E, F). For T e 94 , 2(E, F) we define

IIT 191r, 211 := inf {Ikai) 1 411 II (Yi) I w2I1}, the infimum being taken over all representations described above. Remark. In (PIE) the (r, 2)-nuclear operators are called (r, 2, 1)-nuclear. 1.7.9. For a proof of the following result we refer to (PIE, 18.1.2). Theorem. 94, 2 is a p-Banach operator ideal, where 1/p = 1/r + 1/2. 1.7.10. Several good properties of (r, 2)-nuclear operators are due to the fact that they factor nicely through the Hilbert space 12 . Factorization theorem. An operator T E t(E, F) is (r, 2)-nuclear if and only

if it admits a factorization

Y

D, such that the following conditions are satisfied: D, is a diagonal operator with t e (1) (2) A e WE, lco ) and Y e W/2 , F). When this is so, then

IIT I 94,211 = inf {II Yll Ilt I 411 11A111, where the infimum is taken over all possible factorizations.

1.7. Nuclear operators

67

1.7.11. The next result is analogous to 1.7.5. Proposition. The quasi-Banach operator ideal 94, 2 is approximative. More precisely, we have 94, 2 = OS .94,2 . 63. 1.7.12. We now establish a stability property which turns out to be quite useful for the theory of eigenvalue distributions. Theorem (J. R. Holub 1970/74). The quasi-Banach operator ideal 94, 2 is stable with respect to the tensor norm E. More precisely,

IIS (VI 94., 2 Il

I'S I 94,2II IIT I 94 ,2Il

S e 91,., 2 (E, E0 ) and T e 94, 2 (F, F0).

for

Proof. Given 6 > 0, we consider representations 00

00

E ai 0 x i and

S=

i= 1

T= i= I

such that

Il(ai) I [1r, E111 -. IIS I 94,211 and II(b) I [1r, Fill 6 IIT I 9ir ,2II 11(xi) I [w 2. , Eal -. 1 + 6 and IV.) i [w2, Fdli 5_ 1 + 6. Obviously, - E F)'111 -5 II(ai) I li(ai 0 ki) I [4(N x N), (E 05

[4, Fill 11(ki) I Ur, Fill

5_ 11 ,9191,211117'191,21k Furthermore, for all (yo) e 12 (N x N), we have 00 E E Yuxi 0 Y.; co

1=1 j= 1

{

CO

CO

E E y ii<xi , a> : Ilall 1/2 ( 00 co 6.0 + 6)2 E E In1 2) • = sup

i=1 J=1

i= 1

-.- 1, Olt _< 1}

.J=1

Thus, in view of 1.1.14, the double sequence E0 -(5 c Fo and

is weakly 2-summable in

11(xi 0 yi) i iw2(N x N), E0 '(.5 Foill -' (1 + 6) 2 . Therefore co

S

00

(5, T = 1=1.1=1 EE

(ai 0 bi) 0 (x i 0 xi)

is an (r, 2)-nuclear operator. Moreover, letting 6 -+ 0 yields IS (-5

TI 91r ,2II -- I S 1 94,211 IIT I 91r.2II .

1.7.13. The following result is obvious. Proposition. If 1/p = 1/r + 1/2, then

Remark. Note that 91,(/2 ) * inclusion is strict.

94,2(12),

by 2.11.26 and 2.11.27. Therefore the above

1. Absolutely summing operators

68

1.7.14. Next we investigate the relationship between the Banach operator ideals $2 and 912 •

Ca) = 912 .

Theorem (A. Pietsch 1968).

The norms on both sides are

equal. Proof. Let T e 912 (E, F). Given e > 0, there exists a representation CC

T=

1=1

such that

(1 + e) IIT I 912II I1(ai) I 1211 Then, for x e E and b G F', we have KTx,b>l = E <x, i =1 This implies that

(E i= 1

\ 1/2 / co

/ co

a1>

i=I

)

11 1'4

1 ( yi) 1 w2I1 -5 1 .

and

ai>1 2)

i=i

\ 1/2

Ky,

•

112

for all x e E

ai>1 2

Hence, by 1.3.4, Te 43 2 (E, F) and

(IT I $211 5. 1Kai) I 1211 5. (1 + 8) IIT 1 91211.

Letting e --+ 0 yields I( 7' I 211 11T 191211 for all T E 912 (E, F). This conclusion can also be inferred from 1.7.10 and 1.6.3. We now assume that T F). By 1.5.2, there exists a factorization

a(E,

E

F

A

Y L 2 (X, ,u)

C(X)

such that 1011 = 11 7" I 211 and II YII = 1. Moreover, as remarked in 1.5.1, we may arrange that Y has finite rank. Let Y=Lfi®Yi i= be any representation. Given e > 0, we choose simple functions s 1 , that 171

E Ilfi — sil

ilyi

L2

1=1

5 1.

e.

Define Z:=

1=1

si o y i .

Note that and

II Y — ZII

1141

I Yil

Y — Z I 9111

Y— Z1 912Il Z

1

e.

E

s„, such

69

1.7. Nuclear operators

Write

where h 1 , h, are characteristic functions of pairwise disjoint measurable subsets , •.., X,, of X. If

:=

3 )- 11 2hi and

, . = tt(Xj)112

i=

1 NiiY"

then

Z E u 0 z; and

Zit./ = zi .

Since (ui) is an orthonormal family in L 2 (X,

11(Zj) 1 141211

11Z11 11(21i) 1 W211

it),

by 1.1.9 and 1.1.11, we have

1 + 8.

Recall that I denotes the canonical operator from C(X) into L 2 (X, 4u). Thus Tu., is the functional defined by u; on C(X), and we obtain 1/2 ( n )1/2 11(if tij) I / 211 = 1 1 uj 1 L1 1 2) = tt (X.i) P(X) 112 = 1 j=1 j= 1 In view of the preceding results, it follows from

ZI = E

0 z;

j= 1

that MZI

I 2 Il

1 + e.

Next we get MY/ 9121i -IIZAT 9/21i + !KY — Z)/ 912 1

I + 2e.

Therefore

I IT I 43 211 Letting

E

1 1' 1 91 211 = Y A191211

(1 + 201011-5. (1 + 28) 11T 1

211

0 yields

= 1ITl91211 for all T e(E F). I (TI Finally, we consider any operator TE 13?(E, F). Then, by definition, there exists a sequence of operators Tk E (E, F) converging to T with respect to the norm of 432 . We see from

11Th —

Tk

I 91 211 = 1711

Tk

I 211

that (Tk) is a Cauchy sequence in the Banach operator ideal 912 (E, F). Since T is the only possible limit, it follows that T e 91 2(E, F). Moreover, we have

11 1'1 91 211 = lim11Tk1 91211 = lirn iiTk I 43211 = I T J 43211 • 1.7.15. The following result can be traced back to the the famous "Resume of

70

1. Absolutely summing operators

A. Grothendieck (1956: b) which contains the statement that every operator of the form C L2 C L2 is nuclear. Theorem (H. Jarchow / R. Ott 1982). (432 ) 2 = 5) o 91. The quasi-norms on both sides are equal. Proof. We first assume that T e 912 (E, F) and S e 432 (F, G). Given e > 0, there exists a representation 00

T= E at C) y i 1

i=

such that

11(ai) 11211

(1 +

8)

11T 1 91211 and 1 Yi) I w211 (

Hence 00

ST= E ai Sy i

1=

is a nuclear operator with 00 ((ST ! 91 11 E !lad 11.Vill 5. (1 +

This proves that V 2 °2

11S I 43211 ((TI 91211.

91. By 1.7.11, we have 912 = 91 2 ° Hence it follows from 1.7.14 that 912 o 4; 2 ç 91 2 0

g

o 43 2 ç 91 2 0

Moreover, 1.4.4 and 1.4.5 imply that S) 0 432

ç 1 2°912 S) ç 43?

9t2 .

Combining these results, we deduce from 1.5.3 that o 91 • ° 2 ° V2 (432)2 = ° 3 2 ° ° 43 2 Carefully looking at the quasi-norms involved, we obtain

((TI 0 .9111 5 11T I (q32) 2 1 for all T e ( 4 2) 2 (E, F). Conversely, let T e o (E, F). Given s > 0, there exists a factorization

>F A le

D, 11

fi

Y >H

such that

11 Yll 11B11 lit I /ill 11A11 5 (1 + 8 )11T 1 Choose s e 12 with

52

Ds e 2(L, 12)

S)

.9111.

t. Then, by 1.6.3, we have

and Ds e C(/2, /1)•

Hence, in view of 1.3.13, it follows that

T = Y(BD s) DA e (132) 2 (E, F)

1.7. Nuclear operators

71

and

IIT 1 (132)2 1 I Yll Bps I 43211 lips I V2 I1 11All 111711 I BM Is 1 1211 2 11All (1 + 8) IIT 1 ° 91 1 This completes the proof. 1.7.16. The following (strict) inclusion is obvious. Proposition.

912/m,2

(13 2)m •

1.7.17. We conclude this section with a fundamental result; see 1.5.5.

Lemma (D. J. H. Garling / Y. Gordon 1971). Let N be any n-codimensional subspace of E. Then, given E > 0, there exists a projection P e t(E) such that N(P) = N and (1 + n 1/2 . Proof. It follows from 1.2.8 and 1.7.14 that

n1/2 . N I 91 211 = I .4 N 43 211 Thus, identifying EIN and (EIN)", we can find a representation co IE1N =

i= 1

a 0 sci

such that

1 (â 1) I [w2, (El N) /]11

1

and

II(-)Zi) I [12,

(1 + 0 1 / 212 1 /2 .

Choose xi e E such that Qfsrxi = fc i and

x

(1 + 0 1/2 IIkI.

Define CC

al 0

S:=

xi .

1=1

Then

11 (â 1 ) I [w2 , (E/N)']11

1

and

11(xi)1 [/2 ,

(1 +

0 12

imply that

11511 = IS Ils' 191211 ( 1 + 8) 11112 We now show that P := SO is the desired projection. To this end, observe that QS = I EI N. Therefore P 2 = S(Q E tvs)

= SQfsr =

Next we deduce from

= P and

OP = (Qgrs) Qfs; =

that N(P) = N(07) = N. This completes the proof. 1.7.18. Finally, we give an example of a non-trivial nuclear representation.

Lemma (A. Grothendieck 1956: c). III: 11 (m) 1.(11)194 = 1.

72

1. Absolutely summing operators

Proof. Note that I

= 2- m

e

e,

where the sum is taken over all vectors e = (e„...,em) with ek = +1. Therefore

l(m)

/00(m) J 9 Ili ••• 2 - m

Ie

11(0'11

e 1 .0(m)11

=1

The reverse inequality is obvious.

1.7.19. Example. D,E91(11 ,1,0) if and only if t e c0 . Proof. Assume that t = (ric ) G co , and let D„ denote the diagonal operator induced by the finite sequence tn := (x i , 0, ...). By the preceding lemma, we have

ILDm

—

D „I 91 11

Ilt —

tn /0011

whenever ni > n.

Thus (D„) is a Cauchy sequence in 91(/, , lco). Since D, is the only candidate for the limit, it follows that D,E 941 1, 100. Conversely, D, e 91(l 1 , „„) implies D, e , l,o). Hence t E co.

CHAPTER 2

s-Numbers This chapter is devoted to the theory of s-numbers and the operator ideals determined by them. First of all, we provide some basic facts about Lorentz sequence spaces. Next the concept of s-numbers is introduced axiomatically. It turns out that there exist many possibilities of assigning to every operator T a certain sequence of 0 which characterizes its degree of approximability numbers s 1 (T) s 2 (T) or compactness. In this way the quasi-Banach operator ideals e 171/ , --11-s n = { Te are obtained. In the following sections we treat several important examples of s-numbers. The simplest ones are the approximation numbers. Next we deal with the Gel'fand and Weyl numbers. Their dual counterparts, called Kolmogorov and Chang numbers, are introduced, as well. As the last example we mention the Hilbert numbers which are less interesting for later applications within the theory of eigenvalue distributions. Next we investigate the relationship between the operator ideals $r. 2 and 2. Generalized approximation numbers associated with an arbitrary quasi-Banach operator ideal are treated subsequently. As in the preceding, chapter, we also consider diagonal operators from 4, into /, which yield instructive examples. Furthermore, we establish inclusions between operator ideals e(ps) and to(: ) determined by different s-numbers. The last section is concerned with operators on Hilbert spaces. In this special situation the s-numbers are uniquely determined by their axiomatic properties. Therefore, by restricting ,44s) to the class of Hilbert spaces, for any choice of s-numbers we obtain one and the same class. These are the famous Schatten von Neumann ideals Zr.i, which have played an important role as the historical starting point of the whole theory. Conversely, given any operator ideal defined on the class of Hilbert spaces, we look for extensions to the class of all Banach spaces. In particular, is investigated. the largest extension of Some results about operators on Hilbert space are used in the earlier sections. For this reason, it is a good idea, for the first reading at least, to proceed to Section 11 immediately after Section 4 and Proposition 2.9.4. If not otherwise specified, we assume throughout this chapter that 0 < r < cc and 0 < w co. All proofs are carried out for 0 < w < co, while the obvious modifications in the case w = oo are left to the reader. Moreover, m and n denote arbitrary natural numbers.

e:(

(n)

2.1. Lorentz sequence spaces 2.1.1.* A complex-valued family x = ($ i) on a given index set / is said to be finite if it possesses only finitely many non-zero coordinates. The cardinality of I: i *O} is denoted by card (x). 2.1.2.* The n-th approximation number of x e /OE,(/) is defined by a n(x) :=inf {11x — u I 1, 0 (1)11 : u E /(/), card (u) < n} . In the particular case when x = ($) is a sequence such that led le2 1 0 we have an(x) = n iI. Therefore (a (x)) is often called the non-increasing rearrangement of x.

74

2. s-Numbers

Remark. Note that a(x) = inf {c ._-_ 0 : card (i E i: iii

c) < n} .

2.1.3.* Next we list some basic properties of the approximation numbers just defined. Proposition. (1) (2)

... 0 for x e /,,(/). I ix I /00(/)II = ai(x) . a m+ ,_ 1 (x + y) .-_ am (x) + a(y) for x, y e /00 (/).

(3)

a no _ n _ i (xy)._-< a ,,,(x) a n(y)

(4)

If card (x) < n, then a „(x) = O.

for x, y e /00 (/).

2.1.4.* The Lorentz space 4., w (/) consists of all complex-valued families x = ($i) such that (71 1/r-liw an(x E lw . For x E 4.,,,(/) we define ))

li x 1 4,w(i)II := Il(n lir-liwan(x)) i /A • More explicitly,

,i/w

'CO

Ilx I 4,w(/)11 := E

n=1

[11"r-"wa n(X)r)

if 0 < w < co and

11x1 4.,,,,(/)11 := sup {n 1 fran(x) : n e N}. To simplify notation, we write 4, w instead of 4,0%0. Remark. The above definition makes also sense for r = co. 2.1.5.* We state the following result without proof. Theorem. ir,(/) is a quasi-Banach spaces. 2.1.6.* In order to make subsequent considerations easier to understand, we formulate a tautology. Proposition. Let x e /OE,(/). Then XE

4,(/)

if and only if (a(x)) e 4.,,.

Moreover, I ix I ir,w(i)II = Il(an(x)) I ir,wIl • 2.1.7.* The next result is closely related to Hardy's inequality; see (HAY, p. 239). Proposition. Assume that 0 < p
k=1

Moreover, Il x 1 ir,w(.0IITean :=

1n ((— 1, ak(x)P 1/p n k=1 ) )

defines an equivalent quasi-norm on

2.1. Lorentz sequence spaces

75

Proof. It follows from a i (x) 1

ah (x)

1/p ak(x)P) .

n

E

n

an(x) that

k=1

Consequently, Ilx j 4,40II 5 Ilx I irov(i)Ulean .

Hence 1 — ((

n

11P

n

E

4.,„ implies

ah(x)P)

x

k1 =

To verify the converse, we define 1/u := 1/p + 1/w. Choose any number 2 with hr < e < 1/p. We deduce from kak(x)P

E

— h= 1

for k = 1,

ah (x)P

n

and Holder's inequality that

1/u

n

E (

ah(x)P)

n

)1/u ak(X) P) 1 lw r

E [k -e(ke- wak (x))

=

k=1

k=1

E ice"

n

)1/p

E Re-

iwak (x)lw

'

)11w (n

k=1

(k=1

E

)1/w

ah(x)P

h=1

Therefore, by G.3.2, n

11P

(i

ah (x)P)

-._. conl/P-e (

k=1

E [ke- liwak(x)r )

k= 1

Applying G.3.2 again, we finally obtain 00

IX I

4nl ov (I ean = (E [n 1/r-l/w n=1 .....„. co (

(1—

,1 w/r -1--w

n=1 (

E

E ah(x)P n k=1 [ i kwp-l ak

co

E

(x)w1) 11w

k=1

kive - "ak (xr

k=1

.5. c (

.

)11P1w)11w

n

03

= Co

1/w

[E rs=k 1.1w

1/W nwir-we--11 )

kw/r-"ak (x)w)11 =

cx

k=1

This shows that 11P

x E 4, w(/)

implies

(( n1

Remark. If 1 :5_p
k= 1

ak(x) P)

w

e ir,w •

co, then

II. I

4(i)11 ;nean is even a norm.

2.1.8.* The following result can be viewed as the limiting case of the preceding proposition as p

0; see G.1.2.

Proposition. Let x E /00 (4 Then lln X E 1r, w (1)

if and only if

fl ch(x)) ((n k=1

E 1r,w •

76

2. s-Numbers

Moreover,

) 1/n)

17

I lr ov(i)11 Glean

k=1

ak(x)

W

defines an equivalent quasi-norm on 4.,,(/). Proof. It follows from a1 (x) a „(x) _<

01_ 1 ak(x) )

an(x) that

1. In

•

Consequently,

4,(1) 1

I ir,w(i)IT'"n •

Hence 1/n

n

e

11 ak(x))

x e , „(1)

implies

((\ k= 1

Conversely, we know from the inequality of means that \1

fl

(k =1

/fl

1

1113

n

ak (x)P) .

(—

a(x))

n

k=1

Fix any p with 0 < p < r. Then, by 2.1.7, X 4. , „,(/) II(Tean <

11( i) mean

X1 lr ov( 1 )11 •

This shows that 1/n

n

x e 4., w(/) implies ( iii ak(x) (

k= 1

Remark. The implication just established can also be deduced from Carleman's inequality co

1,„

n

1-1

0(k)

=1 k = 1

e

cc

0c„

for a, ,

,

0;

n=1

see (HAY, p. 249).

2.1.9.* In the following it is shown that we only need to know the approximation numbers a „(x) on a certain subset of indices in order to decide whether or not a given family x e G(/) belongs to some Lorentz space. First we mention the case in which n runs through an arithmetic progression. For later applications it is enough to deal with the subset of odd numbers. Proposition. Let x e GM. Then

x e 4,,„(/) if' and only if (a2 „_1 (x)) e Moreover, Pc I ir,w(/)11 0 d := II(a2„-1(x)) defines an equivalent quasi-norm on

I

ir.,11

2.1. Lorentz sequence spaces

77

Proof. Note that 21 1r -1 /Wer - 1 / W

(2n — Wir-liw < (2 n)iir-1/w

and

(2n)iir - i / 7 < (2n — 1 )1/r-11w

< 111/r-1/w

if

r

if

r

w

w.

Therefore we have )

Ilx I

=

_<

1/w

( 17 t/ r-1 /wan (x) r

(

n=1

max

(21ir, 2 1/

) -cc L-, [n 1/r-1 'wa2nn=1

=

max (21/r,

21'w) II x I

11w i(x)r"

)

lr ,w (1)11odd

Thus (a2n-1(X)) 6 irov

implies (a(x))

an(x) that

Conversely, it follows from a 211 _ 1 (x) {n 1/r-1/w

E

) ifw

a2n-1(x)ry

[n i/r_ifwan(x )]w)

11w

=

Consequently, (a(x)) e 4 r ,„

implies

(a, n - i(X))

E lr ,w •

2.1.10.* Next we consider the case in which the index n runs through a geometric progression (qk) with k = 0, 1, ... and any fixed q = 2, 3, ... This result is closely related to Cauchy's condensation test. Proposition. Let x e /,,(/). Then XE

4, w(i)

if and only if (qk/raqk(x))e 4v .

Moreover, :=

I (gkirae(x)) I 1.11

defines an equivalent quasi-norm on Proof. Let uk =

It follows from

qk

< n < -k+11 J•

78

2. s-Numbers

that ir ,w(i)11 =E

'ix

[n ilr—liw an(X)] w

k = 0 Uk

E

k

\ 1/W

7w/r-1 g]a 0

k(x)w)

Uk

c (L [ q k 1 r a qk (x)]w) k=0

C I lx I

11W

ir,wwilrorn.

Consequently, kq klr aqk (x)) e 4,

implies x E To verify the converse, let Vk :=

{n : qk < n < qk+1 }

Then n wlr-1

(k 1 ) Iv/ r

We therefore obtain oo

Ilx I 1r, w(011:e°n1 = q E [ (k

11w

ictraqk(X)0 co

< (ci axr + E

[q (k+1)Irae+1(X)Jw

) 1/w

k =0 co

<

c (ai (x)" + E

k = 0 Vk

11w/r -1 1 aqi,4 i(X) w

OD

< c

)11w

/W

i (x)' + E E [nur-liwa„(x)]w k = 0 Vk

C 11X

I

lr ,w (1)11

Hence

implies (qk/raqk (x)) 2.1.11.* As an immediate consequence of the preceding proposition we see that the 2-parameter scale of Lorentz spaces is lexicographically ordered. XE

/r, w(/)

Proposition.

4.00„.(/) g 4.,,,,i(/) for 0 < ro < r 1 < co and arbitrary wo , w 1 , kwo(/)

4.,,,i (1) for arbitrary r and 0 < wo < w 1

cc.

2.1.12.* The following example shows that the inclusions just stated are strict.

If 0 < r < oo and a ...;

0, then we let 1

2kir(k + l )

• •.' 2k/r(k + l )

where the k-th term is repeated 2k-times.

2.2. Axiomatic theory of s-numbers

79

Applying 2.1.10 with q = 2, we see immediately that

if and only if am > 1

Cr. 6 ir,w

and cr,„ e

The same result holds for the sequences 1

4,, := ( n ilro ± log n) ' 2.1.13.* Let 1,,„(1) . 1(I) denote the collection of all complex-valued families x

which can be written in the form

x = (NA)

with

a = (N) e /„,(/) and b = (#i) e

Using this notation, we formulate a useful multiplication formula. Proposition. If 1/p ± 1/q = 1/r and 1/u + 1/v = 1 1w, then

1„,u( 1) . 4,„(/) = 4.,,,,,(/). 2.1.14.* Finally, we describe how the scale of Lorentz spaces behaves under real

interpolation; see (BER, 5.3.1) and (TRI, 1.18.3). Interpolation proposition (J. Peetre 1963). Let 0 < r0 < r1 < co, 0< wo , w i , w -._ co and 0 < 0 < 1. If 1/r = (1 — 0)1r0 + 0Ir1 , then /r 1 ,w 1(/))0, = 4.w(/)-

The quasi-norms on both sides are equivalent.

2.2.

Axiomatic theory of s-numbers

2.2.1.* A rule s : T .— (s„(T)), which assigns to every operator a scalar sequence, is said to be an s-scale if the following conditions are satisfied:

... > 0 for Te 2(E, F).

(S 1 )

IlTil = si(T) >,_ s2 (T)

(s 2

s ,,„+„_AS + T) -._ S m (S) + s(T) for S, TE t(E, F).

)

5_

I YU sn(T)11XII for X e 2(E0 , E), T e 2(E, F),

(S3)

s (YTX)

(S4)

If rank (T) < n, then s(T) = 0.

Y E 2(F, Fo).

s n(1 : 12(n) -- 12(n)) = 1. We call s(T) the n-th s-number of the operator T. In order to indicate the underlying Banach spaces E and F, we sometimes replace s „(T) by the more specific symbol s„(T : E --, F). (S5)

2.2.2.*

First of all we show that the converse of (S4) is true, as well. Lemma. If s(T) = 0, then rank (T) < n.

Proof. Assume that rank (T) .. n. Then there exist x 1 , ..., x„ e E such that Tx i , ..., Tx linearly independent. Next we choose b 1 , ..., bn e F' with

80

2. s-Numbers

=

Define X e t(1 2(n), E) and Be t(F, 1,(n)) by

et ® x i and B :=

X :=

bi ei .

Since BTX = I is the identity operator of 12(n), we have

1 = s„(/)

11/311 s „(T) 11X11

Hence s„(T) > 0. We now observe that s „(T) depends on T continuously. Lemma. Is(S) — s n(T)I 5_ IIS — TI1 for S, Te t(E, F). Proof. Taking in = 1, it follows from (S 2) that 2.2.3.*

sn(S)

IIS — TII + s„(T).

Hence

s „(S) — s„(T)

IIS — TII • Interchanging the roles of S and T, we obtain s(T) — s„(S)

IIT — SII •

This proves the desired inequality. 2.2.4.* Let s be any s-scale. An operator Te t(E, F) is said to be of s-type ,w if (s„(T)) e lr , w. The set of these operators is denoted by t(rs?„.(E, F). For T e ers,),,(E, F) we define IT e(rs,LII := II(sn(T)) J lr,wil • To simplify notation, the index w is omitted whenever w = r. Remark. The above definition makes also sense for r = cc . 2.2.5.* Theorem (A. Pietsch 1972: a) 2;.si is a quasi-Banach operator ideal. Proof. Let S, T e 2;.-5?(E, F). Then, by 2.1.9,

+ T)) I lr ,wii 5_ c o ll(s„(S + T)) I ir 04,11 °dd I la =011 01 k-1 I 's 2.- 1(S + + Iws„(T)) cuour-iiwsn(s)) l,Jj lour-ilwsn( n) la] = cill(sn(S)) I 4,„11 + 11(s,i(T)) 1 ir,w111. This proves that S + T e t(E, F) and ciIIS I 2 1VII + IIT I I IS + T e(rs, )wil Since the other ideal properties are evident, we immediately pass to the proof of completeness. Let (Tk) be a Cauchy sequence in t;.s,„(E, F). Then it follows from IITh — Tk tV,LII Tkil that (Tk) has a limit Te e(E, F) with respect to the operator norm. Given e > 0, we choose /c c, such that — Tk I ,t (rs,

I

c

for h, k

ko •

81

2.2. Axiomatic theory of s-numbers

This means that for m = 1, 2, ...

< 8

V {iI iir-liwS„(Th — Tiji w n=1 (i._,

Letting h -4 oo, by 2.2.3, we obtain '

111

(1: n=1

[n llr-11 w

) 1/w

-._

S„(T — TkAw

for m = 1, 2, ..

e

Hence T — Tk e

L(E,

F) and

I1T

—

Tk

I

ti•s,)wil --- e

for k --- ko •

Thus T is also the limit of (Tk) with respect to the quasi-norm of 2,„. 2.2.6.* We now establish an immediate consequence of the results stated in 2.1.7, 2.1.8, 2.1.9 and 2.1.10. Proposition. The expressions n )1/p) 11T i t mean • = ((--1-- E sk(T) P 4., n k=1 ,..?w ii p

IIT I , .t!p)s w 11 glean : =

((ili sk(T)) 1m)

4,w

with 0 < p < r ,

,

17, 1 tp:)wil odd := iiii(s2._1(T)) 1 4,,,11, I T i ei.s,L g ewn : = I (gkirSe(T)) I 4„ I with

q = 2, 3, ...

define equivalent quasi-norms on t(i5)w . Remark. Note that the first of the above quasi-norms fails to be normalized. 2.2.7.* The lexicographical order of Lorentz sequence spaces stated in 2.1.11 can be transferred to the associated operator ideals. Proposition.

.),,v. s 2;!i), )„, for 0 < ro < ri < co and arbitrary wo , w 1 , 2;::)„,0 _.q. t.s,),„ i for arbitrary r and 0 < wo < w i _-ç_ oo . 2.2.8.*

An s-scale s is called multiplicative if

s + _ 1 (ST) :5_ s ,(S) s n(T) for T e 2(E, F) and S e e(F, G). 2.2.9.* The following result is very important. Multiplication theorem (A. Pietsch 1972: a). Let s be any multiplicative s-scale. If 1/p + 1/q = 1/r and 1/u + 1/v = 1 1w, then 2(p.ou 0 i_1(: ), , 2 ,..,, ,,,.

Proof. Let T E ,ts,),,(E, F) and S e 2;,s?u(F, G). Then, by 2.1.9 and Holder's inequality,

82

2. s-Numbers

we have I I(Sn(ST)) 1r,wil =

C11 (,S2 n -l(ST)) ir,w11 C11(2 1/1.-1/w S2n-1(ST)) CII01 1/13-11'uSn(S) 12 114-11 '4(T)) I

cffn" P-11"sn(S))1 lull 11(n ilq-ilv5n(T))1 Ivii = cll(s (S)) I II(sn(T)) I This proves that ST E 2(E, G).

2.2.10.* Next we establish another useful formula. Interpolation theorem (H. K6nig 1978). Assume that 0 < r o < r, < co, 0 < w o , w 1 , w oo and 0 <0 <1. If hr = (1 — 0)1r° + Olr„ then ,64s,)w. ( 2s0),wo , 2(rsi),wi)o,w Proof. Given T e (2 00 (E, F),

F))00w , we consider any decomposition

T = To + T1 with To e 2;.so), 00 (E, F) and

T1 e 2(,.s?„(E, F).

Then it follows from s2 (T)

s(T) + s(T1 ) + nvro "r1 11 7-' 1 I 2'(rs1),00lli To I

s2 „_ 1 (T)

that s2 (T) < n - liroK(n l ir° -1 /r1, T), where K(r, T) denotes the K-functional with respect to the quasi-Banach couple F)). We now obtain (2, œ (E, F),

E

[2k( 1/r-1/r 0 )Kpk(1 /ro -1 Ir 1) k=0

k =0

oo

< co

21<' 1

Ef

[0.11r-11r0.45 1 Ir0 -1 I r

jj

k = 0 2k

do.

niw

a

ciD = c

[-t--19K ( r, T)]v

Hence T e 2(E, F). This proves that (2 s) (

2(s) e( rs,),w)e,w

) (2(4) ,co s,) .0)13,,v e 2(rs, .

2.2.11.* Let s be any s-scale. An operator T e e(E, F) is said to be of s-type co if (s ( T)) e co . The set of these operators is denoted by es) (E, F).

2.2.12.* Theorem (A. Pietsch 1974: a). t(s) is a closed operator ideal. Proof. Since the ideal properties of t(s) are obvious, we only verify its closedness. Let (Tk) be any sequence in t(s)(E, F) converging to Te 2(E, F) with respect to the operator norm. Given E > 0, we successively fix k and no such that I IT —

e

and

sn(Tk)

E

for n

no.

2.3. Approximation numbers

83

Taking in = 1 in (S 2), it follows that s(T) MT — Tk M + s„(T) F). Hence T E

2.3.

2E for n

no.

Approximation numbers

2.3.1.* The n-th approximation number of T E a„(T) := inf

T—

e(E, F) is defined by

: L E a(E, F), rank (L) < n} .

2.3.2.* We begin the investigation of these numbers with the following observation. Lemma. If dim (E) n, then agE) = 1. Proof. Assume that an(I E) < 1. Then there exists L E a(E) such that 11 1 E — L11 < 1 and rank (L) < n. Using Neumann's expansion, we see that L = 1E - (JE - L) must be invertible. Therefore dim (E) = rank (L) < n. 2.3.3.* Theorem (A. Pietsch 1972: a). The rule a : T --+ (an (T)) ,

which assigns to every operator the sequence of its approximation numbers, is an s-scale. Proof. The preceding lemma implies (S 5). The verification of the other properties is an easy exercise.

2.3.4.* The foregoing result can be improved as follows. Theorem (A. Pietsch 1972: a). The approximation numbers yield the largest s-scale. Proof. Let TE e(E, F), and fix any natural number n. Given e > 0, we choose L E (E, F) such that IT — L11

(1 + e) an (T)

and

rank (L) < n.

Then, for an arbitrary s-scale s, it follows that

„(T) Letting

8 -->

— Lll + s„(L) _< (1 + an(T).

0, we obtain s,1(T)

a„(T).

2.3.5.* The operators belonging to

are said to be of approximation type l.

2.3.6.* The next result is an immediate consequence of 2.2.5 and 2.3.3. Theorem (A. Pietsch 1963: b). e;.% is a quasi-Banach operator ideal.

2.3.7.

r < co fails to be a Banach operator We stress the fact that 2;.° ) with 1 ideal. The following proposition describes this surprising situation more precisely. r < co. Then p: = rl(r + 1)is the largest exponent for Proposition. Let 1 which there exists an equivalent p-norm on Proof. Carefully looking at the proof of 2.2.5, it can be seen that the original quasinorm of 2;.a) satisfies the quasi-triangle inequality with the constant c := 2 1 /r. Thus, by D.1.8, we may find an equivalent p-norm with 1/p := 1 + 1/r.

84

2. s-Numbers

In view of 2.9.10, we have if and only if t e /„.,..

e 2;.0(4„, /I)

1, it turns out that

Moreover, for 0 < q

D, e

, li )

if and only if t e /q .

Hence 91,,(/,„ , / 1 )

i!,.12) (1 0 , /1 )

whenever p < q.

By 1.7.2, this implies that there cannot exist any equivalent q-norm on Remark. Unfortunately, we do not know any concrete and non-sophisticated expression which yields a p-norm on 2.3.8.

We now establish an extremely important characterization.

Representation theorem (A. Pietsch 1980:b, 1981:a). Let q = 2, 3, ... An operator T e t(E, F) is of approximation type 'r ,„ if and only if it can be written in the form T=>J Tk k=o such that Tk e

a(E, F), rank (Tk)

qk

and (qiCk T kil) c l. Moreover,

inf 11(qk°. TO) I where the infimum is taken over all possible representations, defines an equivalent quasi-norm on Proof. If T e

F), then we choose Lk e 2a qk(T)

T — -411

and

a(E, F) such that

rank (L k) < qk.

In the special case when aqk(T) = 0, it follows from 2.2.2 that rank (T) < qk . Hence we may take Lk := T. Define To := 0,

:= 0

and

Tk4-1

Lk

—k+ 1

for k = 0, 1, ...

Then rank(Lk+ ,) + rank(Lk) < le+ 1 ±

rank(Tk+2)

qk < Il k+ 2

and 1' 11+211

TII + IIT — Lk!!

IlLk+1

4a,k(T).

By 2.1.10, the latter estimate implies that e wqr i. Finally, we have CO

T = iiM Lk == k

E Tk •

k =o

Additionally, it turns out that IIT J tVvIi rgep

'Kg". cll(eraqk(T)) I

= cJJTI 2 ZII:• -( .1

ecun

2.3. Approximation numbers

85

We now verify the sufficiency of the given condition. To this end, assume that Te t(E, F) admits a representation with the properties stated above. Then h -1

( h -1

rank ( \ k =O

5_Eqk< le • k=o

Fixp and 0 such that 0 < p <min (w, 1) and 0 < 0 < 1. Define s by 1/p = 1/s + 1/w. Applying Holder's inequality, we obtain T—E

Tk

75

k=0

_. .. I co q

co

co

h-1

aqh(T)

1bs

-OksIr )

kk = h

co

)1IP

EllTkilP E IlTkil_...-( k=h

k=h

co

z tekwir 1lT k liw) (lc= h 1Iw

..__ coq -Ohlr ( E q okwir k=h

1171

0

1/w

.

From this estimate it follows that Co

III T ç (a) ii rom = ( E gin vir aqh(nw

)11w

h= 0

cc

—<-_

(1-0)hwIr [ E q 0kwir IlT k ilw q k=h h= 0

])11w

(E

co

_ c ( v°2. [ E k q (i_ohwiri g ekivjr

2.,

0

k=0 h=0 1/w

co

E

(

liTkr) 11w

qk w'r

k=o

IITkllw)

•

By 2.1.10, this proves that T E ertA,(E, F) and

11 T i 24.??,II:e" -. clIT i 2;•%11 1.:13 • Remark. For every finite operator T e t(E, F) the representation constructed above has the property that Tk = 0 whenever k is sufficiently large. We stress the fact the quasi-norm introduced in the representation theorem fails to be normalized. 2.3.9. Proposition (A. Pietsch 1963:b, 1981:a). If 0 < w < oo, then the quasiBanach operator ideal 2 1 is approximative. Proof. Consider any representation CO

T = E Tk k =0

a

such that T,, E (E, F), rank (Tk) _.< 2k and a natural number n o with

co

(k = noE [2kir IlTa] )

( 2k

1/W -

e•

Now it follows that n

T—

E Tk I .e .c,iv

k=o

This completes the proof.

rep

<e 2 —

for n

770 .

Tk) 11 E 1„,.

Given s > 0, we choose

86

2. s-Numbers

Remark. In the case w = co, the closed hull of tors Te 2(E, F) with (niiran(T))e co .

F) in 2;.a,20 consists of all opera-

2.3.10. The representation theorem can be used to establish inclusions between 2;.al and various quasi-Banach operator ideals. Embedding theorem (A. Pietsch 1981:a). Let 0 < r < co. Assume that 'It is any w-Banach operator ideal (0 < w 1) such that cer 1171 whenever rank (T)

IIT I ?III

n,

where c > 0 is a constant. Then 2,;.% g_ i. Proof. Given T e 2;.!?„ (E, F), we choose a representation CO

T=

Tk k= 0

F), rank (Tk) < 2k, and (2k/ill Tk il) e l . Then

such that T k e

IlTk f c implies that the sequence (Tk) is absolutely w-summable in W. However, since W is complete with respect to the w-norm 1 %IL we obtain T e ?r(E, F). 2.3.11. We now apply the preceding embedding theorem in a special case. Proposition (A. Pietsch 1963:b). ,44(1) g_ sM, for 0 < p < 1. Proof. The assertion follows from 1.7.7 and 2.3.10.

Remark. Considering diagonal operators from f that the above inclusion is strict; see 2.9.10.

/I , it can easily be shown

2.3.12. Proposition (A. Pietsch 1963: b). The approximation numbers are multiplicative. Proof. Let Te t(E, F) and S e t(F, G). Fix any natural numbers in and n. Given E > 0, we can find B e (E, F) and A e (F, G) such that

(1 + (1 +

I IS — IT —

a ,„(S) and rank (A) < in, a n(T) and rank (B) < n.

It follows from rank (AT + (S — A) B)

rank (A) + rank (B) < m + n — 1

that

a, Letting

E —+

1 (ST)

If ST — (AT + SB — AB)II = IRIS — A) (T — a „ ( S) a „(T) 5_ 11S — All IT — B11 -5. (1 +

0, we arrive at the desired inequality.

2.3.13. For the operator ideals determined by the approximation numbers the inclusion stated in 2.2.9 becomes an equality.

Multiplication theorem (A. Pietsch 1980:b). 1/u + 1/v = 1/w, then 41(a) 0 p(a) = 0(a)

72,u

""q,v

"-r,w •

The quasi-norms on both sides are equivalent.

If

lip + 1/q = 1/r

and

2.3. Approximation numbers

87

Proof. Given R e ,e;.??„ (E, G), we consider a representation OD

R= >

Rk

k =0

such that Rk e(E, G), rank (Bk) <2k , and (2k1r liRk11) G 1„. Choose factorizations 2k. Bk = SkTk with Tk e e(E, Fk), Sk E t(Fk, G), IIRkII = IISkII dill! and dim (Fk) If 0 < w < co, it can be arranged that

kit

2" P S

and

= (2k " I R kl1) 1 "

2k1q11T k = ( 2k " R

•

Form the Banach space F := [12 , Fk ] as described in C.2.1. Let Jk e t(Fk , F) and Qk e t(F, Fk) denote the canonical injections and surjections, respectively. Observe that rank (SkQk) < 2k and rank (./k Tk) 2k, ( 2k1P IISkQk11) E lu and (2 kig lIATkil) € iv • Therefore the operators 00

S := SkQk and k =0

T :=

Ejk Tk

k=o

are of approximation type /, and / q ,„, respectively. The representation theorem implies that R = ST e 4a14 In the case u = y = w = c(') we choose the factorization R k = Sk Tk such that and ITk II = llRiV = ig .

2.3.14. The quasi-Banach operator ideals associated with the approximation numbers behave better with respect to interpolation than those corresponding to arbitrary s-numbers. Interpolation theorem (J. Peetre / G. Sparr 1972, H. Keinig 1978). Let 0 < ro < r, < co, 0 < 14) 0 , w 1 ,w co and 0 < 0 < 1. If 1/r = (1 — 0)fr0 + then

4a) wo

(4

1)0,w

The quasi-norms on both sides are equivalent. Proof. First we establish the formula (a) (t , 2)o, „ = 2P1)v for 0 < p < co, 0 < w co, 0 < 0 < 1 and 1/r = (1 — 0)Ip + Oloo. Let Te e(E,F), and fix a natural number n. Choose L e F) such that

a(E,

IT — LII Then, for k = 1, T — LII

2a(T) and rank (L) < n. n, we have 2ak(T) and ak(L) 5 IlL — TI! + ak(T) 3ak(T).

Hence

K(n"P , T) 5 I IL I t ; ( ) II k =1

Obviously, K(r,

ak(L)P)

rlITII.

1

—L +2(

II ak(T)P

k =1

)1p

111, 5( k =1

88

2. s-Numbers

From now on we assume that T e 2,1„(E, F). Using an elementary discretization argument, it follows from 2.1.7 that 1

CO

f Fr-°Ker,

0

19)

(1 —

1 _

f

0

1

+e1

W

E

[n _O/PK(n l/P,

fljw

T)Pv dr (n +

1)'IP

— nlIP

n=1

+ e2 E

w

[n-e/P-iiwiqnvP,

n=1

1

Co

< C3

f Er -6W(T, T)r co

1

-.= <

oo

E [71.lir-l/w n=1

)1Ipjw

n

E

n k =1

< oo .

ak (T)P

Hence Te (2i,a)(E, F), t(E, F))0 ,,. This proves that (4a), 2 e. w. )

In view of 2.2.10, we even have equality. Finally, the general case is settled by the reiteration property. Choose p such that 0 < p < ro
1/r0 = (1 — E0)Ip and

1 ir i = (1 — E1 )/p,

respectively. Then

F) = (4,a)(E, F), t(E, F))E., w . and

„, (E, F) = (25,0 (E, F), 2(E, F))z 1 , w1 . Applying F.3.3, we get

(2;.a2,„.(E, F), 2;! 1 (E, F)) 0, w = = (04)(E, F), 2(E, F))E0 , wo , (2(pa)(E, F), t(E, F))z, „ i)e, „ = (4a)(E, F), 2 (E, F))E , w

=

F)

with 1/r = (1 — E) lp and E = (1 — 0) E0 + 0E,. Therefore lfr = (1 — 0)1i. 0 +

2.3.15. As shown by H. Keinig (1984:a) the operator ideals 2;,a,2, fail to be stable with respect to the tensor norms e and v. We conjecture that this negative result even holds for all tensor norms. On the other hand, it turns out that they are almost stable in the following sense. Proposition (A. Pietsch 1982: b). Let 0 < q

(5„F0) for every tensor norm a.

Do.

89

2.3. Approximation numbers

Proof. First we assume that u = y = 1. Choose a natural number k such that k+ 1

. Then, putting m := 2k+' and n := 2k, we have ml/P < n P q according to 2.3.8, we consider representations k

—

s=

s, and

. Next,

T = E Ti j=0 1-0

such that rank (S 1) On" S

m 1 and e!1

and

n-', rank (Ti) (n11Ti e 11-

Let R o := 0 and Rh+i :=

c)„ Ti for

E S.

h = 0, 1, ...

Then

y

rank (R + 1 )

) = E rank (S i) rank (Ti)

rank (S i

i+j=h 09

h

<

< mh E

m inj < m h E j=0

h

—

j=o

2

<m1.

Moreover, it follows from inhiPIIRh-Fi I -5, E m(t+Dn'llsi 0, Till

E mi'PlIsill +j= h

11Tili

that 00

GO

00

E m("+ 1 "PIIRh+111

h=0

< co •

m'P E m"PlIsill E i=0

j=0

Since we have the representation co ST= E Rh, h=o

Theorem 2.3.8 implies that S 0,,T is of approximation type Next the general case is treated. Enlarging q a little bit if necessary), it can always be arranged that y = 1. Choose po , p i and 0 such that lip = (1 — 0)1p 0 + Olpi and q
)

5„ F, E0 6

(

F0

)

and T ® 2 :11), 1 (E, EC') -+ 1 (E 6, F, E0 Fo)Therefore, by 2.3.14, the interpolation property yields T® 2, (E, Eo) Fo). 4a,I(E 6„ F, Eo Re mark. In the above proof, the interpolation argument can be avoided at the cost of some more complicated computations; see A. Pietsch (1982:b).

90

2. s-Numbers

2.3.16.

As shown in (PIE, 11.7.4), we have

an(T) = an(r) for all T E R(E, F). This formula implies that the operators T and T' possess the same approximation type. Proposition (C. V. Hutten 1974). =

en

2.4.

Gel'fand and Weyl numbers

2.4.1.* An s-scale s is injective if, given any metric injection J E e(F, F0), s„(T) = s„(JT) for all T e e(E, F). 2.4.2.*

The n-th Gel'fand number of T E WE, F) is defined by c„(T) := inf 11T./f,1 11 : codim (M) < n}.

Recall that ./f,/ denotes the canonical injection from the subspace M into the Banach space E. Hence Lar. is the restriction of T to M. Remark. Observe that the above definition reflects the basic idea of the FischerCourant minimax principle; see (DUN, X.4.3). 2.4.3.*

The following result can easily be verified. Theorem (A. Pietsch 1972:a). The rule C : T (c„(T)),

which assigns to every operator the sequence of its Gel'fand numbers, is an injective

s-scale. The operators belonging to 2;.;?„ are said to be of Gel'fand type /r ,„. 2.4.5.* Theorem. t;.'„), is an injective quasi-Banach operator ideal.

2.4.4.*

The identity operator from I1 (m) into /2 (m) shows that the Gel'fand numbers may be strictly smaller than the approximation numbers. However, there are important situations in which equality holds. Proposition. Suppose that H is a Hilbert space. Then c„(T) = a(T) for all T e 2(H, F). 2.4.6.*

Proof. Fix any natural number n, and consider an arbitrary subspace M of H with codim (M) < n. Let P e 2(H) denote the orthogonal projection from H onto M, and define L : = T — TP. Since M N(L), it follows that

codim (M) < n. rank (L) = codim (N(L)) Therefore a„(T) 11T — L11 = IITPl = c„(T). In view of 2.3.4, we even have equality. This implies that an(T) The next result is along the same line. Proposition (A. Pietsch 1974:a). Suppose that F has the metric extension property. Then 2.4.7.

c(T) = an(T) for all T e 2(E, F).

2.4. Gerfand

and Weyl numbers

91

Proof. Fix any natural number n, and consider an arbitrary subspace M of E with codim (M) < n. By hypothesis, there exists a norm preserving extension To e 2(E,F) of T./ 1 . This means that Tog! = Tiff and 11T0 ITJ 1 I. Define L := T — T o . Since M g N(L), it follows that codim (M) < n.

rank (L) = codim (N(L)) Therefore

11T — L11 = tT 0 I = TJII.

a(T)

c„(T). In view of 2.3.4, we even have equality.

This implies that a(T) 2.4.8.

Proposition (A. Pietsch 1972:a). The Gel'fand numbers are multiplicative.

Proof. Let Te t(E, F) and S e t(F, G). Fix any natural numbers in and n. Given e > 0, we can find subspaces X and Y of E and F, respectively, such that (1 + e) c„(T)

ilT611 SJÇ

(1 +

C nz(S)

and codim (X) < n, and codim (Y) <

Define M := X n T-1 (Y). Then it follows from codim (M)

codim (X) + codim (Y) < m + n — 1

c 1 ,,_ 1 (ST)

IIS411 117 -411

that

(1 + 8)2 c ,„(S) c n(T).

Letting 8 -4 0, we arrive at the desired inequality. 2.4.9.

Next we state an immediate corollary of 2.2.9 and 2.4.8. Multiplication theorem. If 1/p + 1/q = l/r and 1/u + 1/v = 1/w, then t(c) 2(c) q,v "-p,u

2.4.10.* We now establish a useful criterion. Proposition (H. E. Lacey 1963). An operator Te 2(E, F) is compact if and only if for every s > 0 there exists a finite codimensional subspace M of E such that E

.

Proof. Assume first that Te g(E, F). Given e > 0, we choose Yi, that T(U){Yi ±

y„ e F such

.

J=

Next we pick b 1 ,

b„ e V° with

M := {x e E:
Kyi , b>I =

10 = 0 for

. Now it can easily be seen that

j= 1, ..., n}

is the desired finite codimensional subspace. To this end, let xe Mr\ U. Then there E, and it follows from exists some yi with I Tx —

= KYJ, bi>1

Ky; — Tx, b.i>i

ii

e

92

2. s-Numbers

that

11 7'4 Hence !IT/fill

x— y +

2 e.

2E.

We now assume that the above condition is satisfied. Fix E > 0, and take any finite codimensional subspace M of E such that fi TJfI e. Then E can be represented as a direct sum E = M N, where the subspace N is finite dimensional. Let Pm and P N denote the corresponding projections. Put := min (

T II

, i).

Since P N has finite rank, there exists x 1 ,

PN (U)

x i,, E U such that

U {P Nx i + i=1

Thus, given x e U, we may choose x i with IIPNx

Ix — xiii + 11 13 Nx — P Nxiil

Pmx — P MX i

6. Then 2 + 6 3.

Hence

1 Tx — Tx

— TP mxiii + ITP N X — TP Nxiii 4e. 3 1 7411 +

This proves that

T(U)

U {Tx t + 48V}

Since e > 0 is arbitrary, we have T e 4(E, F). 2.4.11.* In terms of Gel'fand numbers the preceding result reads as follows.

Theorem. t(c) = 2.4.12.* Next we show that the Gel'fand numbers of any operator T are determined by the Gel'fand numbers of its restrictions to all finite dimensional subspaces. This means that c(T) is a "local quantity". Proposition (A. Pietsch 1974: b).

c„(T) = sup {c„(T.64-): dim (M) < ool

for all Te 2(E, F).

Proof. First of all, we make the trivial observation that

c„(T/gf)

c(T)

for all subspaces M.

Therefore the right-hand supremum is less than or equal to c n(T). In order to prove equality, it is enough to consider the case when E is infinite dimensional and n 2. Given E > 0, in every finite dimensional subspace M with dim (M) n we choose a subspace N(M) such that fiTg(m) fi (1 + E) c„(T.61)

and

dim (MIN(M)) = n — 1.

2.4. Gel'fand and Weyl numbers

93

According to Auerbach's lemma 1.7.6, there exist i(M), • • • , n- i (M) e M N(M)

and

(M),

dn- i(M) G 1NM) /

such that

= 1,

= 1 and <Sci(M), eij(M)>

Taking appropriate representatives x i(M) e fc i(M) and extending to E the functionals induced by âi(M) on M, we can find

x i (M),

xn _ i (M) e M

and

a i (M), ..., a _ i (M) e E'

such that

Ix(M) II 5 2, Ila1(M)11 = 1

and

<xi(M), arn» =

as well as

<x, ai(M)> = 0 for all x e MM). Define the projection

1 P(M) : =Eai(M) , l 0 x i(M), n—

and note that its range is a complement of N(M) in M. We know from the Alaoglu-Bourbaki theorem, already mentioned in 1.3.6, that U° is compact in the weak topology generated by E. Furthermore, the collection of all finite dimensional subspaces M with dim (M) n is directed upwards. Therefore the generalized sequence of (n — 1)-tuples (ai (M), an_ 1 (M)) has a cluster point (a 1 , an_ i) E U° X ... X U° ; see (DUN, 1.7). We now define the subspace

N: Ix e E: <x, ai) = 0 for i = 1, Then it follows from

n — 1} .

n-1

P(M) x = E <x, ai(M)> xi(M) 1=1 that Ern 11P(M) x11 = 0

for all x e N.

Hence, for fixed x e N r■ U, there exists Mo such that x e Mo and 11T11 11P(M) x11

E

for all M

Mo

Note that x — P(M) x e N(M) whenever x e M. For M M o we now obtain

11E1{11

IIT(x

P(M)fiTP(M) xI

1174M)11 fix — P(M)x11 + IITM V (M ) + 21171 11P(m) xi! 5 (1 + c„(744) + 2e. This implies that

c(T) 5 117411 Letting e

(1 + e) sup {c„(T.gf) : dim (M) < co} + 2s.

0 completes the proof.

94

2. s-Numbers

2.4.13.* The n-th Weyl number of TE 2(E, F) is defined by x(T) := sup {a(TX) : X e 2(i2 ,

1} .

Remark. According to 2.4.6 and 2.4.12, the same numbers are obtained by taking the right-hand supremum over all (finite) operators Xe 2(H, E) with 11Xli 1, where H is an arbitrary infinite dimensional Hilbert space. 2.4.14.* Theorem (A. Pietsch 1980:c). The rule

x:T

(x „(T)) ,

which assigns to every operators the sequence of its Weyl numbers, is an injective s-scale. Proof. The injectivity follows from 2.4.3 and 2.4.6. The verification of the other properties is an easy exercise. 2.4.15.* The operators belonging to e4,x01, are said to be of Weyl type /r ,„,. 2.4.16.* Theorem. 2;.x,i)„ is an injective quasi-Banach operator ideal. 2.4.17.* Proposition (A. Pietsch 1980:c). The Weyl numbers are multiplicative. Proof. Let Te 2(E, F) and SE 2(F, G). Fix any natural numbers m and n. Given Xe 2(12 , E) and e> 0, we first choose B e (l2 , F) such that 11 TX — BI

( 1 + a„(TX) and

Secondly we pick A e

rank (B) < n.

, G) such that S(TX — B) — Al —< (1 + e) a (S(TX — B)) (l2

and

rank (A) < m.

It follows from rank (A + SB) rank (A) + rank (B) < in + n — 1 that a „,+„_1 (STX) IISTX — SB — AII (1 + e) a m (S(TX — (1 + e) x m (S) 11TX — - 13 11 < (1 + e)2 x 1(S) an(TX) (1 + 6)2 x m(S) x„(T) IXI. Hence X m+n-l(ST)

Letting e

5_ (1 + 8)2 x,n(S) xn(T).

0, we obtain the result.

2.4.18.* Next we state an immediate corollary of 2.2.9 and 2.4.17. Multiplication theorem. If lip + lfq = lir and 1/u + 1/v = 1/w, then 4oGic) o 2(x) c 4100

"wp,u

q,v = "wr,w•

2.4.19.* We now compare the Weyl numbers with the Gel'fand numbers. Proposition (A. Pietsch 1980:c). x „(T) en(?) for all T e 2(E, F). Proof. By 2.4.6, we have an (TX) = c(TX) 5_ cn(T)11Xli This implies the desired inequality.

for

X e 2(/2 , E).

2.5. Kolmogorov and Chang numbers

95

2.4.20.* Finally, in view of 2.4.13 (Remark), we may formulate the following analogue of 2.4.6. Proposition. Suppose that H is a Hilbert space. Then x„(T) = an(T)

for all T e 2(H, F).

2.5.

Kolmogorov and Chang numbers

2.5.1.

An s-scale s is surjective if, given any metric surjection Q E e(E0 , E), s „(T) = s„(TQ) for all T e 2(E, F).

2.5.2.

The n-th Kolmogorov number of T e t(E, F) is defined by d(T) := inf {!WJ. - 11: dim (N) < n} .

Recall that Q N denotes the canonical surjection from the Banach space F onto the quotient space FIN. 2.5.3.

The following result can easily be verified. Theorem (A. Pietsch 1972:a). The rule d: T- (d„(T)),

which assigns to every operator the sequence of its Kolmogorov numbers, is a surjective s-scale. 2.5.4.

The operators belonging to 24.‘1)„ are said to be of Kolmogorov type 1,

2.5.5. ideal.

Theorem (I. A. Novosel'skij 1964). 2;.11,),: is a surjective quasi-Banach operator

2.5.6.

As shown in (PIE, 11.7.7), we have c „(T) = d„(T') and d(T) = c „(T') for all T E A(E, F).

This means that the concepts of Gerfand and Kolmogorov numbers are dual to each other. Proposition (A. Pietsch 1974:a). 2;.; ), = ()'

and

2.5.7.

The following result is a dual version of 2.4.11. Theorem. Ç (d) =

2.5.8.

The n-th Chang number of Te t(E, F) is defined by MT) := sup {a„(17): Y e 2(F, 12),

2.5.9.

t.l .d,v)v = (2)'.

11.

Theorem (A. Pietsch 1980:c). The rule y : T -± (y,(T)),

which assigns to every operator the sequence of its Chang numbers, is a surjective s-scale. 2.5.10. The operators belonging to 2,9,2, are said to be of Chang type 2.5.11. Theorem. 2;.-':4 is a surjective quasi-Banach operator ideal.

96

2. s-Numbers

2.5.12.

We state, without proof, that

x(T) = y„(T') and y„(T) = x„(T') for all T e 2(E, F).

This means that the concepts of Weyl and Chang numbers are dual to each other. Proposition (A. Pietsch 1980:c). 24.-;};, (',0' and 2;.Y4 = 2.5.13.

The following result is dual to 2.4.19. Proposition. y(T) d(T) for all T e 2(E, F).

2.6.

Hilbert numbers

2.6.1.

The n-th Hilbert number of Te 2.(E, F) is defined by X e 2(12 , E), 11X11 11 .

hn(T) := sup (a„(YTX): 2.6.2.

Y e 2(F, 12), 1117 11

1i

The following result can easily be verified. Theorem (W. Bauhardt 1977). The rule

which assigns to every operator the sequence of its Hilbert numbers, is an s-scale. 2.6.3.

s-scale.

The preceding result can be improved as follows. Theorem (W. Bauhardt 1977). The Hilbert numbers yield the smallest

Pro of. Let T e 2(E, F). If X e 2(/2 , E) and Y e 2(F, 12 ), then YTX acts on 12 Hence, by 2.11.9, we have .

an(YTX) = s„(YTX)

11)11 sn(T)11X11 for every s-scale s. The definition of Hilbert numbers now implies that h(T) s n(T). 2.6.4. 2,6.5.

The operators belonging to 2;.ho) are said to be of Hilbert type 4.0%, • Theorem. tqh,a, is a quasi-Banach operator ideal.

As observed in 2.9.19 (Remark), the Hilbert numbers fail to be multiplicative. We have, however, the following property. Lemma. If T e 2(E, F) and S e 2(F, G), then 2.6.6.

hni+„_ 1 (ST)

hn,(S) x(T)

and h„,+„_1 (ST) y„,(S) h„(T).

Proof. Fix any natural numbers m and n. Let Xe £(1 2 , E) and Z e 2(G, 12). Given 8 > 0, we first choose B E (l2 , F) such that

IITX — B11

(1 + 8) a(TX)

and rank (B) < n.

Secondly we pick A e a(12, 12) such that

lIZS(TX — B) — 1111

(1 ± 8) an,(ZS(TX — B))

and rank (A) < m.

It follows from rank (A + ZSB) :5_ rank (A) + rank (B) <m + n — 1

2.7. Absolutely (r, 2)-summing operators

97

that

a„

All 1 (ZSTX) IIZ STX — ZSB — (1 + an,(ZS(TX — B)) (1 + s) IIZU h m (S) IITX — Bil hm (S) a(TX) + < (1 + )2 IIZII h m (S) x(T) 11X11 •

Hence

h,„+„_ 1 (ST)

(1 + e) 2 h„,(S) x(T).

Letting e —> 0, we obtain the left-hand inequality. The other one can be checked by dual considerations. Reasoning as in the proof of 2.2.9 yields some further multiplication 2.6.7. formulas. Theorem. If 1/p + 1/q = 1/r and 1/u + 1/v = 1 1 w, then 400) 0

p,u

2.6.8. result.

"wq,v

4-t(h )

m-r,w

t(Y) r,w• p,u t(h) q,v — 2(12)

and

In view of 2.10.7, we may state the following supplement of the preceding

Theorem. Suppose that 0 < p < 2 or 0 < q < 2. If 1/p + 1/q — 1/2 = 1/r and 1/u + 1/v = 1 1 w, then ( h) c 0(h) t(h) 0 t q,v p,u =^-r,w• 2.6.9.

Finally, we mention that

h(T) = h„(T') for all Te

e(E, F).

This formula implies that the operators T and T' possess the same Hilbert type. Proposition (W. Bauhardt 1977). 2 .hov ) = t(rh,41 / . (

2.7.

Absolutely (r, 2)-summing operators

Throughout this section H denotes a complex Hilbert space. 2.7.1.* The following lemma due to G. Pisier (see H. 1(ônig 1980:c) is very elementary but quite useful. Lemma. Let Te WH, F). If a„(T) > 0, then for every e > 0 there exists an x„) in H such that orthonormal family (x 1 ,

ak(T) (1 + 8) IlTxk ll for k = 1, , n. Proof. The required family can be constructed by induction. If xl , ..., n -1 have already been found, then we define the subspace M„ := {x e H: (x, x k) -- 0 for k = 1, Since codim (Ma) < n, it follows from 2.4.6 that

a(T) = c(T) Hence there exists x,, e M „ such that

a(T) < (1 + e) !Tx ,3 1 This completes the proof.

and llx,111 = 1.

n—1} .

98

2. s-Numbers

2.7.2.* Lemma (H. Keinig 1980:c). Let 2

r < cc. Then

)1Ir

(E ak(T) r

11T 43,211

k=1

for all Te 43r ,2(I

F).

Proof. We may assume that an(T) > O. Given E > 0, there exist xl , constructed in the preceding lemma. Note that

ak(T)

(1 + 8) 11Txkll

and

x,, as

11(xk) jw211 = 1.

Therefore llr

(E ak(T)r) 5 (1 ± e) k=1

k =1

= (1 + e) II(Txk)1411

(1 + 8 )11T 1 §1r,211 11(xk) 1 W211.

0, we obtain the desired inequality.

Letting s

2.7.3.* The next result is extremely important. Lemma (A. Pietsch 1980:c). Let 2 r < co. Then

F).

n ilrxn(T) 5_ I T 1 r,211 for all T e Proof. If X e e(12 , E), then 2.7.2 implies that nifran(TX)

(kE iak(TX))

11TX I 43,., 2 1 5_ T 1 43r,211 11X11,

and the definition of Weyl numbers yields the inequality we are looking for. 2.7.4.* We are now in a position to establish the basic result of this section.

Theorem (A. Pietsch 1980:c). Let 2

r < cc. Then

en

Vr,2 • Proof. From 1.2.8 we know that whenever rank (T)

iiT Vr,211

n.

Therefore the embedding theorem 2.3.10 implies that 2;.ad We now assume that T e F). Then

43r, 2

TX e 2;02, F) Vr.2(12, F) for all X c 2(1 2 , E). Hence T e 43r, 2(E, F), by 1.2.7. This proves the first inclusion. The second one is an immediate consequence of 2.7.3. Remark. The following examples show that, for r = 2, the parameters in the first inclusion are the best possible. From 1.6.8 and 2.9.10 we know that

D, e 432 (l2 , /1)

if and only if

te

and

D, e 2(l2 , 11)

if and only if

tE

•

Therefore

2Sx,W2,

43 2(l2, l i ) whenever 1 < w < cc.

2.7. Absolutely (r , 2)-summing operators

99

Let 1/p := 1/r + 1/2. Then Je V r, 2 (1„, 12 ), by 1.6.7. Moreover, it can be shown that x(/: lp "'Ir. Hence

t';))v(ip, 12) whenever 0 < w < oo .

1.;,..2(ip, /2)

Thus the second inclusion is sharp for 2 Finally, in view of 2q.),c ( 12) C

43r,2(/2)

r < cc.

C

both inclusions are strict.

2.7.5.

The preceding result can be improved by interpolation.

Theorem (A. Pietsch 1980:c). Let 2 < r < co. Then erx) 43r,2 2. Proof. Choose ro , r1 and O such that 1/r = (1 — 0)Ir 0 + Olri , 2 < ro < r < r1 < co and 0 < 0 < 1. Then it follows from 2.3.14, 2.7.4 and 1.2.6 that 41/4,2)0,r g e(ra) = (t4a0),I , We now assume that Te 2 (,?0(E, F). Then TX

e t;.") (/2 ,

g 4 r, 2 (12 , F) for all X e 2(l2 , E).

Hence T e r , 2(E, F), by 1.2.7.

2.7.6.

Proposition (H. K6nig 1980:c). Let 2 < r < cc. Then

tfra)(H)

= t (rx)(11, F)

r,2(11, F).

Pro of. From the preceding theorem we know that 2;.°)(H, F) g $2,(Tx)(11, F) g 13r ,2(11,

Furthermore, Lemma 2.7.2 yields r,2(14

g e;.°) (H, F).

Thus equality holds.

2.7.7. As an easy application of 2.7.4 we now prove a striking result for absolutely (r, 2)-summing operators. Multiplication theorem 1/p + 1/q > 1/s, then

(A. Pietsch 1980: c).

Let 2

p, q, s < co. If

44,2 ° 43.7,2 S; 4k,2.

Proof. Writing 1/r := 1/p + 1/q, we have r < s. Hence 2.7.4, 2.4.18 and 2.2.7 imply that VP, 2 ° 434 ,2

4V09 ° 2Wo

Remark. Let l/p + 1/q = 1/r and 2

43p,2 0 434,2(0

2 (l2) •

2):Co g 21;1 g 43s,2.

r < co. Then, as shown in 2.11.30,

100

2. s-Numbers

On the other hand, we know from 2.7.6 that

43,,Ai2) = Hence 2°

p,

2

q

43r, 2 •

This proves that, in the preceding theorem, the assumption 1/p + 1/q > 1/s is essential.

2.7.8.*

Finally, we prove an inequality which looks rather complicated. It has, however, far-reaching applications.

Lemma (A. Pietsch 1980:c). Let T e 2 (E, F) and S E 432 (F, G). Then 11(2n-1) \ r n [ 2n 1 11n 1 . exp LI XaST) I IS I V211 2 ) in x,(T) ] — ( 2n — 1 k=1 k= I -

Proof. Since the Weyl numbers are multiplicative, we have

x(S) x k(T) and x 2k (ST)

X 2k- 1(ST)

x(S)

X k+ i(T) .

Hence

n-

2n- 1

1-1

X2k_ i (ST) n X(ST) k=1 k=1

X(ST)

k=1

\

n -1

X(S)

2.

11 -

xk (S) k=I

n-1

xk(T) 11 xk+,(T)).

I From 2.7.3 we know that k 112 xk(S) k= 1

Write a := MS V211

xk(s))

n

t xk (S)

k =1

k =1

0.11

-

k =1

I

k=

<

a". Thus

1/2

exp (a).

_ [ n! (n — 1)1

Furthermore, since r

il/ Il

n

k=

I

xk(T)]

it follows that n

-

k= I

2

[ n

1

X(T)

1k=

1(2n-1)1 n

n

xk(T)1 < F1

xk , i(T) = 11711 - ' k=1

[

X k(T)

k=1

Combining these inequalities, we obtain the desired result.

2.8.

Generalized approximation numbers

2.8.1.* Let ?,1 be any quasi-Banach operator ideal. The n-th ?I-approximation number of T Elt(E, F) is defined by

an(T 2.8.2.

:= inf

T—L I

: L e(E F), rank (L) < n}.

The basic properties of these generalized approximation numbers are similar to those of the original ones. Recall that ot denotes the constant in the quasi-triangle inequality of the underlying quasi-norm M. I tII.

2.8. Generalized approximation numbers

101

Theorem. (1)

(2) (3) (4)

IITI VIII = ai (T I ?I) >, a2(T I ?I) 0 for Te VI(E,F). a1+n-1 (S + T VI) cga „,(S J?I) + a (TJ VI)] for S, T e 91(E, F). an(YTX1?1).11Yil a n(T W) 11X11 for Xe 2(E0 ,E), T e W(E , F), Y et(F , F0). If rank (T) < n, then a „(T 1?-1) = 0.

2.8.3. If the operator ideal ?I is endowed with an arbitrary quasi-norm, then it may happen that the n-th 91-approximation number does not depend continuously on the operator. However, we have the following result, analogous to 2.2.3.

Lemma. Let 91 be a p-Banach operator ideal. Then

lan(S J 9j)P — a (T I W)PI

IS — TJ 911V' for S, T e Vi(E, F). 2.8.4. Let 91 be any quasi-Banach operator ideal. An operator T e 9 (E, F) is said to be of 9I-approximation type if (a „(T 91)) e 1, . The set of these operators is denoted by 9.g.a),(E, F). For T e 9g(E, F) we define

Il T J *.`,1111 := il(an(T I 91)) I irov I Remark. Note that the definition of a „(T I It) depends on the underlying quasinorm. However, if two quasi-norms are equivalent, then corresponding inequalities hold for the associated 91-approximation numbers. This shows, in view of D.1.4, that 11;.a,, is uniquely determined by VC Therefore, without loss of generality, we may occasionally assume that 91 is equipped with a p-norm. 2.8.5. The following theorem can be obtained by straightforward modification of the proof given in 2.2.5.

Theorem. 91g.a0)„ is a quasi-Banach operator ideal. 2.8.6. Proposition. Let 91 and 13 be quasi-Banach operator ideals with 91 Then 9t.a?„

Proof. By D.1.9, there exists a constant c > 0 such that

I IT J

clIT I VIII

for all T e 91.

It easily follows that corresponding inequalities hold for the associated approximation numbers:

a(T I Z)

ca„(T 91)

for all T e 91.

This implies the desired inclusion. 2.8.7.

We now establish various multiplication formulas. Proposition. Let 91 and 93 be quasi-Banach operator ideals. Then

91 o V.a?,

(91 o 93 );.a2i, and 91(ra?„

g (91 o

Moreover,

111,,a1 0 Vol

(91 zqa})v

provided that 1/p + 1/.7 = 1/r and 1/u + 1/v

1/w.

102

2. s-Numbers

Proof. Note that,

for

Te $(E, F)

a „,+„_1 (ST I ?,1 $)

and Se ?t(F,G),

a „,(S ?I) a(T I VS).

Modifying the estimates in the proof of 2.2.9, we obtain the third inclusion. The other ones can be checked by putting ni := 1 or n 1 in the inequality stated above. 2.8.8. Next we formulate a generalization of 2.3.8 which can be proved in the same way. Representation theorem (A. Pietsch 1981:a). Let q = 2, 3, ... An operator Te 11(E, F) is of ?I-approximation type /,.,„, if and only if it can be written in the form co Tk T= k=0

such that T k e

a(E, F), rank (Tk)

qk and (erTk lI I WI) e l . Moreover,

T I 4p1;.??„1117 = inf ! ( ell k WM) 1w11 where the infimum is taken over all possible representations, defines an equivalent quasi-norm on The following statement is analogous to 2.3.9. Proposition. If 0 < w < co, then the quasi-Banach operator ideal is approximative. 2.8.9.

2.8.10. In the setting of generalized approximation numbers the inclusion stated in 2.3.10 can be refined significantly. Embedding theorem (A. Pietsch 1981:a). Let 0 < r, s < co and 1/p = 1/r + 1/s. Assume that ?.1 and are quasi-Banach operator ideals such that whenever rank (T) I I IT I where c> 0 is a constant. Then tqa4 .

n,

Moreover, V.(:4, g 21 provided that ?I is a w-Banach operator ideal (0 < w < 1). Proof. Given TE $;;;?,(E, F), we choose a representation CO

T=

k=o

such that Tk e

Tk

a(E, F), rank (Tk) < 2k and (2k/PII Tk

Vic

C 2k/11 1k

e l . Then ,

I $11

implies that (2" IIT k I 1111) e l . Hence T e Wla))1,(E, F). The proof of V.a,4, g ?I can be adapted from 2.3.10. 2.8.11. We now provide some auxiliary inequalities. Lemma. There exists a constant c> 0 such that

11T I Wr6,?vii

whenever rank (T)

n.

103

2.8. Generalized approximation numbers

Proof. By G.3.2, we can find c > 0 such that \

1/W

< crti lr.

E kw1r- 1 ) k =1

Hence \ 1/W

T ta,?„11 = E [kltr - vwak(T k =1

?or)

.5_ cn i I rlIT I

.

an(T I ?1;!:20) for T e t(E, F). Proof. Given e > 0, we choose X, Y e a(E, F) such that n ilra2n-1(T I It)

2.8.12. Lemma.

IT

— X I /111

(1 + E) a,i(T I t)

rank (X) < n,

and

+ a,,(T — X1 ?1) IT — X — Since rank (X + Y) < 2n — 1, it follows that

and rank (Y) < n.

n ifra(Ti 21) =< nIIT — X — YI iIj =< (1 + ni Iran(T (1 + E)2 a(T ( 1 + IIT — X I t!)0011 Letting E 0 yields the desired inequality. 2.8.13. We are now in a position to describe what happens when the procedure tao) is applied repeatedly. ,

Reiteration theorem (A. Pietsch 1981:a). Let 0 < r, s < co and

1/p = 1/r + 1/s. Then (j(a)\(a)v = W pOw (

Proof. Assume that Te WE, F) is of it 2.1.9, we deduce from

-approximation type is,. Then, in view of

n li ra2„-(T I VI) a„(T that T e F). Consequently, (*.a.2)11

(*.a)o ,,A,4

The reverse inclusion follows immediately from the embedding theorem and Lemma 2.8.11. 2.8.14. Lemma.

n 1r x2.-1(T)

ari(T I 2) for

T e 2(E, F).

Proof. Recall that for all X e

a(TX) = x(TX)

E).

Hence, by 2.8.12, we have n itr a

(TX)

a(T X

= a„(TX I 2)

a„(T I 2 ) and the definition of Weyl numbers yields the inequality we are looking for. 2.8.15. The following inclusion is an immediate consequence of the preceding lemma.

Theorem (A. Pietsch 1980:c). Let 0 < r, s < co and lip = hr + 1/s. Then (2, No Xa?,v

2(p,c1

104

2. s-Numbers

2.8.16. For later use, we prove a further inequality which could also be obtained as a corollary of 2.7.4 and 2.8.14. Lemma.

Proof. Given

8>

n ik x2.-1(T) an(T r,2) for T e *,., 2 (E, F). 0, we choose L e R(E, F) such that

(1 + 8) a „(TI — Li Vr,2II Then it follows from 2.7.3 that

n l /rx n (T —

n lir x2n-1(T) Letting e

r , 2)

and

II — L

rank (T) < n . , 211 5_ (1 + e) a „ (T I

. 2) .

0 yields the result.

2.8.17. In the remaining part of this section we deal with operators of *2-approximation type. Proposition (A. Pietsch 1981 :b). The quasi-Banach operator ideal (424,"„), is injective.

Proof. The assertion follows from the fact that, given any metric injection Je t(F, F0), we have ) for all T e *2(E, F) To verify this formula, we choose Lo e R(E, Fo) such that a (T I 43 2) = a „(JTI

(1 + a„(JT 1132) and rank (L0) < n. IIJT — Lo I 432I1 By 1.5.1, there exists a factorization JT — L o = Y 0A , where A c 43 2(E, Ho), Yo G 2(H0 , Fc,) and II Yo u lIA I 4;2 11 V2 . Let H be the inverse image of M(J) with respect to Yo , and denote by P the orthogonal projection from Ho onto H. Then

Yf := J-1 Y o Pf defines an operator from Ho into F such that .TY = YoP and II Yll L := — Y A . If x e N(L o), then it follows from YoAx

I Yo . Put

JTx that Ax e H. Therefore

JLx JTx — JYAx = JTx — Y o PAx = JTx — Y oAx = L ox = o Thus we have x e N(L). This proves that N(L 0) g N(L) . Hence rank (L)

rank (L0) < n.

We now obtain a „(T j i)

11)11 IIIIA I *211 I Yoll I IA I 43211 = IIJT — Lo I 43211 (1 + IIJT I V21. — L I 42II = II YA I *211

Letting e 0 yields a(T I ) cln(JT I The reverse inequality is obvious. 2.8.18. Proposition (A. Pietsch 1981: b). Let 0 < s < co and 1/r = 1/s + 1/2.

2.8. Generalized approximation numbers

105

Then

(v2 Proof. From 1.2.8 we know that

iiT I V211
n.

(4;20.

We now assume that T e 24.c?„(E, F). Take any metric injection J e t(F, F0), where F0 has the metric extension Property. It follows from 2.4.7 that JT e 2;1(E, Fo). Hence JT e (4;2)(sal(E, F0). Finally, the injectivity of the operator ideal implies that TE 4;2X(E, F). This completes the proof of the left-hand inclusion. Furthermore, by 2.7.4 and 2.8.15, we have

( Aa,1 g (tnYsaW ei•xl • Remark. By considering diagonal operators from 1 co into 12 , it can be shown that the left-hand inclusion is strict. We conjecture that the right-hand inclusion is strict, as well. 2.8.19.

Proposition.

(42)M,2011-2) g (v2r+'

Proof. In view of 1.2.8,

n. n n" IT I 211 whenever rank (T) Furthermore, we know from 1.5.7 that (*2)m+ 1 is 2/(m + 2)-normed. Thus 2.8.10 yields ilT I ( 4 2) m+ '

(*2V/L,21(„2+2)

($2) m+1 •

We now assume that T E (V 2 )m ."(E, F). Then, by 1.5.1, there exists a factorization

T= such that A c 9 2 (E, H,), Tk e V 2 (Hk , Hk+ ,) and Ye 2 (H„, +1 , F), where H 1 , are Hilbert spaces. In view of 2.7.2, we have T1, e 2(Hk , Hk+i). Therefore Te

(e)"'

$2(E, F)

H, +1

(*2)Wm(E, F),

by 2.8.7. This proves that (432) m+1

Remark. The following examples show that the parameters in the above inclusions are the best possible. In particular, it turns out that both inclusions are strict. From 1.6.8 and 2.9.14 we know that and

D, E ( 2)"1+ 1 (/2 ,

if and only if

t G /2/(n+2),2/(m+2)

Dr e (2)Sa(i2, 11)

if and only if

t e 12/(m+2).w-

Therefore (*2)Saim,w(i2, ii)

(432)m+1(12,

whenever w > 2/(m + 2).

106

2. s-Numbers

Furthermore, 1.6.10 and 2.9.16 state that D, c ( 4 2 )m -1-1 (11 , /2)

if and only if t- e- /-21m,2Im

an d

(V2)S1) (/1 , /2) if and only if t e

D, Hence

(4 )(a) whenever w < 2/m. (42) m+1 (ii /2) 2.8.20. The next result is an immediate consequence of 1.7.10 and 2.9.15. Proposition. Let 0 < r < 2 and 1/s = 1/r — 1/2. Then 91r,2

(43 2)(sad—

Remark. By 2.9.14, we have D, e (432)(sai. (12 , /1 )

if and only if t e

where 1/p = 1/s + 1. Moreover, it can be shown that D, c, 2 (12 , li)

if and only if t e /p ,

where 1/p = 1/s + 1/2. This proves that the inclusion 94,2(/2,

(4,1 2).V(/2, li)

is strict. 2.8.21. Finally, for later applications, we establish a multiplication formula which is analogous to 2.3.13.

Proposition (A. Pietsch 1981:b). If 1/p + 1/q = 1/r and 1/u + 1/v = 1/w, then

2,SV,

o

o

(13 2 )(qa,)„ = (432 ) 1.

Proof. The inclusion

epa,)u ° ° (V2 ) v) 4",)t) ° (V2)(q 2 g (432)gt follows immediately from 2.8.7. The converse can be checked by modifying the proof of 2.3.13. Given T e (V2)(E, F), we consider a representation co

T=k =0 Tk such that Tk G a(E, F), rank (Tk) < 2k, and (2 kir ilTk I V211) e /„). In view of 1.5.1, there exist factorizations Tk = YkAk with Ak e 2.(E, /2( 2k)), Yk 6 2(/2(2 k), F) and Ykti ilAk 43211= Tk I 4211. If 0 < w < oo, it can be arranged that

211)

2 '11 Ykll = (2'11 77k I V wiu and Form the Hilbert space H := [12 , /2 (2k)] as and Qk e 2(H, /2 (2k)) denote the canonical Observe that (2"11

e lu and

2 k/ g1lAk I 43 211 = (21c1r1171 V211)4iv" described in C.2.1. Let Jk e 2(12(2k), H) injections and surjections, respectively.

(2k /q11JkAk I 43 211) G 4,.

2.9.

Diagonal operators

107

Therefore the operators A :=E

Y := E YkQk and

JkAk

k =0

k=0

2 -approximation type l„, respectively. are of approximation type l„ and of Thus T = Y A is the factorization we are looking for. In the case w = oo the proof requires the same modification as in 2.3.13.

2.9.

Diagonal operators

Throughout this section we assume that 1 p, q 2.9.1.*

co.

In the following we are concerned with diagonal operators D

lAc)

induced by scalar sequences t = (rk) and acting from /„ into lq • To simplify the 0. formulation of the results, it is always assumed that ri r2 2.9.2.*

For later use, we define the operators

(ii

,

• • •

5 $M) -4 ($1, •••

.//2

01

0

, • • •)

and

Q in : Note that

,

•••

5 $M, $n1+1, •••) -4 ($11 *••,

4,(i77)

4,11 = 1

and

11Q ,„:

tti)•

-+ l(m)II = 1.

Moreover, Pm := J„,Q,,, is the projection given by

0, 0 , •••)2.9.3.*

Lemma. Let s be any s-scale. Then rms,(/: /p(m)

/q(m))

s„(D, : 4, -+

4).

Proof. We may assume that r„, > O. Then D m := Q„,D,J,„ is invertible, and we have 4,11 l( 111 )

l(m)1 = T7ni

The s(/ : /p(m)

lq(m)) 5. II : 1„(m) r-,-„ls„(Dr : 1,

11,(111)11 s „(1) „,: !(m) lq),

lq(m))

which implies the desired inequality. 2.9.4.*

We now show that, for any diagonal operator in /2 , all the s-numbers

coincide. Proposition (A. Pietsch 1974:a). Let s be any s-scale. Then

s „(D, : 12 -+ 12) = r. Proof. Since

11D, —

: /2

/2 11 = r„

and

rank (P,,_ 1) < n,

108

2. s-Numbers

we have IlD — D rPn _ i :12

12-+ 12)

12 11 + sn(DtP„-i :12— 12) =

Tr; •

On the other hand, applying Lemma 2.9.3 with m = n, we deduce from (5 5) that r„ = TnS„(/

.5.(Dt

12(n))

12(n)

12).

2.9.5.* In the case of approximation numbers the preceding formula remains true in all sequence spaces 1„. Proposition. an(Dt : I, -4 I„) = r„.

Remark. The same formula holds for the Gerfand and Kolmogorov numbers.

2.9.6.

To prove the next proposition, we need some preliminaries.

Lemma (V. D. Milman 1970). Let N be any subspace of l0(m) such that codim (N) < n. Then there exists a normalized vector e = (ek) e N with card {k : led = — n + 1. Proof. By the Krejn-Milman theorem (DUN, V.8.4), the convex compact subset

UN : = tx = ($k) e N :11x F ico(m)11

1}

contains an extremal vector e = (ek). Let K := space M :=

= 11, and define the sub-

= ( k ) e /(m) $k = 0 for k e K}

.•

Obviously, card (K) + dim (M) = in. Assume that card (K) < dim (N). Then we can find a normalized vector xe Mr) N. Note that := 1 — max {led : k K} > 0. Hence it follows from e + 6x e UN that e cannot be extremal. This contradiction dim (N) ni — n + J. shows that card (K)

2.9.7.

Lemma. Let l_q
(

)11(1

)11q

(n

E lekl q k=1 n+1

k=1 n

E 18kr)up

k=1

min(lE i I ,..., jen 1). Then

k=1

lekl q )

1/p •

lekl"

Proof. Put and /3 := E lekl q)

A := (E lekr k= 1

\ k =I

Of course, we may assume that iek/En+ i I

__ 1

\1. Iq

(n

)1Ip

n

implies

for k = 1, ..., n. Hence 148 „+11 P -?..1/3/8n+-11q.

8 n +i 4

0. Then

lEkie n+11P

1 I Ski I ts+1111

2.9. Diagonal operators

109

We now obtain (#q + 18,1+11a) 114

#

1 + len+110)"

(0‘," + 18„+1IP)"

a. (1 + 18„-i-1/ocIP)llP — a

/3 •

This completes the proof.

2.9.8.

We are now prepared to compute the approximation numbers of certain identity operators. Proposition (A. Pietsch 1974:a). Let 1 q

(1 : l(in) Proof. Define •••9

ra--1.

r:5

$1n)

09

•••9

.

9

0).

Et follows from rank (P n _ i ) < n that an(/: /p(m)

= (m — n + 1)vq - - lit

— Pn _ : i(m)

Vm))

To verify the converse, we assume that L e 2(1(m), l q(m))

rank (L) < n.

and

Let N be the null space of this operator. Since codim (N) = rank (L) < n, we may choose a vector e = (ek) E N according to Lemma 2.9.6. Then =

— L) e j I,i(m)11 5_ III — L : 1,(m)

1,,(m)ll Ile 1p(m)11.

Without loss of generality, it can be arranged that je l l application of Lemma 2.9.7 now yields

III

—

le ,,,1 . Repeated

Ile I Vm)II

L I l(m) -4 4(011

Ile i (

fl

)11q

E leklq

k=1 rn

)1Iq

c—n+1

E

k=1

leklq

(m—n+1

( kE ilEkiP)11p

E

k=1

) 1/n

lekIP

= (ni — n + 1)11 "/P whenever 1 ._q
1q(m))

(ni — n + 1) 11q -1119 .

Remark. The Gel'fand and Kolmogorov numbers of the identity operator from lp(m) onto 4(0 are given by the same formula; see (PIE, 11.11.4).

2.9.9.

The next lemma is basic for the subsequent considerations. Lemma. Let ?i be a quasi-Banach operator ideal such that in(m)

where a

14(m) I ?III -‹

Mc'

O. If 0 < r < oo and 1/s = 1/r + a, then t e l., implies D E 11.aot(ip , /q)

110

2. s-Numbers

Proof. Given t = (rn), we define tk = r n if 2k < n < 0 otherwise.

Tkn • —

(rkrt)

by

2k+1 ,

Note that co tk and

t =

2k.

card (tk)

k=o

Let Dk denote the diagonal operator induced by t,,. Then Co

D, =

E

rank (Dk)

Dk and

k =0

—

2k.

Moreover, lq

I

T2k I lp(2k)

1111

4(2k

) ?" < 2k.'r2k.

By 2.1.10, we have (2k/sr 2k) e 4. Hence

I 11 II) e

(29Dk : p

•

Thus, in view of 2.8.8, we obtain D, e /I;.Vv(/ 4). 2.9.10. We are now in a position to establish one of the main results of this section.

Example. Let 0 < r < oo, 1 - _qp...ç_oo and 1/s = lfr — lfp + lfq. Then D, elq)

if and only if t c

Proof. Since l(m)

4041 =

we see from 2.9.9 that te

k w implies

D, e ty„(/ 4).

In view of 2.9.3 and 2.9.8, we have (m — n + 1) 11q-11Pr. = rinan(1:1P(m) —) lq(m))

a„(D, :1p

In particular, taking m := 2n — 1 yields an

(D, : —)

Hence, by 2.1.9, D, e 24a0Wp 4)

implies

te

2.9.11. Next we state a counterpart of 2.9.8. Proposition (S. B. Stechkin 1954).

an(I :11 (0

12(m)) =

(m — n + 1) 112

for n = 1,

m.

q) •

2.9. Diagonal operators

111

Proof. Consider any operator L e 2(11(m), 12(m)) with rank (L) < n. Let PE 2(1 2(m)) be the orthogonal projection from 12(m) onto the orthogonal complement of M(L). This means that N(P) = M(L). Then E IIPek I /2(m)II 2 = IIP I 611 2 = rank (P) = codim (N(P)) = codim (M(L))

k=1

— n + 1.

Therefore IIP(/ — 4I1 = IIPIII = max {11Pek I 1 2(m)II : k = 1, ..., m} (rn — n + 1) 112

— LII

in

This implies that an (I : l(m)

(m

12(m))

—

n + 1 ) 112

We even have equality. For a proof of this fact, which is not used later, we refer to (PIE, 11.11.8). 2.9.12. Example. Let 0 < r < cc. Then D, e /2) if and only if t e Proof. We know from 2.9.4 that t e 4., implies D, e 12).

.

Hence D, e 12). In view of 2.9.3 and 2.9.11, we have (m

—

n + 1)112

Tin = rmanq: 1 1(m)

1 2(m)) 5 an(Dt: 11 -- 12).

In particular, taking m := 2n — 1, it follows that r2n-1

2112 an(Dt

11

ID*

Therefore, by 2.1.9, D, e

er%( 1 1, /2)

implies

te

2.9.13. The following counterpart of the preceding example shows that the component t1(/ 1 , /2) is considerably larger than t(li 12).

Example. Let 0 < r < 2 and 1/s = 1/r — 1/2. Then D E 2(1 1 , 12) if and only if t e Remark. The proof of this result is based on an asymptotic estimate of the Gel'fand numbers c„(/: 11 (m) 12(m)) due to B. S. Kashin (1977). 2.9.14. Example. Let 0 < r < cc and 1/s = 1/r + 1. Then D, e (432 ) v (/ 2, li) if and only if t e k w •

112

2. s-Numbers

Proof. By 1.2.8,

1 1(M)11 = m.

fn 1"2 11 1 : 1 2(0 IV: 12(n) —> /Am) I 12 11 Applying 2.9.9, we see that t E is , w implies D t e (13 2)V12,

Let 1/r 0 := 1/r + 1/2. Then, by 2.8.18, implies Di e e'o?w(1 2 11) D, e (132 )(l 2 As shown in 1.6.4, the embedding operator from / 1 into 12 is absolutely 2-summing. Hence Je 2Y20 (li , 12 ), by 2.7.4. Since 1/r = 1/r 0 + 1/2, it follows from 2.4.18 that D, e 2(/2, 12 ). Finally, 2.9.4 yields t E 4,w • 2.9.15. Example. Let 0 < r < oo and 1/s = 1/r + 1/2. Then e ( 42)(ra,Woc, 12)

if and only if t e

•

Proof. We may deduce from 1.3.8 that 1.(m)

12(m) I 43211 = in "

Applying 2.9.9, we see that

te/

implies D, e

(2),(/,

Conversely, e (432)W/00, 12 ) implies D e

ec,i1(i., 12).

Therefore it follows from 2.9.10 that t 2.9.16. Example. Let 0 < r < co. Then D, e (43 2)2(/1, /2)

if and only if t e 4,w•

Proof. By 2.9.4,

t E 4,, implies Dr e

/2).

Hence it follows from 1.6.4 and 2.8.7 that D, E ($2)(/1 9 /2)9 Conversely, D, e (432)(4(/1, /2 )

Therefore t e

implies D r e

12).

4, by 2.9.12.

2.9.17.* The "if" part of the following example is basic for applications in the theory of eigenvalue distributions of infinite matrices. Example (C. Lubitz 1982). Let 0 < r < max(q, 2) and {1/q — 1/2 if 1 q5 2 l/s = 1/r + 0 if 2 q cc. Then D, E eV,1(ix

4) if and only if t E

Proof. From 1.3.8 we know that 11/: /oe (m) -+ 4,(m) I 134 11 =

2.9. Diagonal

operators

113

Define p by 1/s = 1/p + 1/q. Then it follows from 2.9.9 that t e /s,„

implies Dt e (4.;,)(pa,(/0a ,

Moreover, by 1.3.3, 2.7.4 and 2.8.15, we have

er),c1 if 1

($q)gt

( 2 ),0)(pa,?„

(5] \(a) OE

( 2 v2. ypow g. 2(r.1

q

2

q

cc.

and This proves that Dr e 2

if 2

(1., 4).

For the convenience of the reader, we present one more proof of the preceding implication which avoids the use of generalized approximation numbers. Since 1/s = 1/p + 1/q, given any sequence t e 4, we may choose a factorization t = ab such that a e 4, and b e lg . Then it follows from 2.9.5 and 1.6.3 that

Da e 2(; )(/00 , /ao)

and

Db

e 4;,,(100 , 4).

Next, setting go := max ( q , 2), we conclude from 1.2.4 and 2.7.4 that

Dt e 2,(qx.) . . epa)(/. , 4). Thus, in view of the multiplication theorem 2.4.18, t e 1,

implies Dt e 2.),c),(/ 09 , /a) .

The general case when t e /s, w can now be treated by an obvious interpolation argument; see 2.1.14 and 2.2.10. Conversely, by 2.4.20, implies Dt e 2(12 , Q.

Dt e erx0)„(/, 4)

If 1 __ q _. 2, then it follows from 2.9.10 that t e /s with 1/s = 1/r — 1/2 + 1/q. If 2 q -._ co, then, turning to the dual operator, we conclude from Dr e 2 ;??,,(/2 , lœ ) that Dt e 2(/, , /2). Hence 2.9.12 yields t E

2.9.18. Example (C. Lubitz 1982). Let 0 < r < ro and 1/s = 1/r — lir o , where p q 2, 1/p — 1/q if 1 p _. 2 _. q _- o o , 1/r0 := ( 1/p — 1/2 if 1 if 2 - p _. q . o o . 0

Then if and only if t e i.

Dt e 2(/„, 4i)

Proof. It follows from 2.9.5 that t e /s , w implies Dt e

4,).

Furthermore, by 1.6.7 and 2.7.4, we have JE

ro , 2O p,

g

Thus, in view of the multiplication theorem 2.4.18,

tG

implies A e 2(r),c,)„(4„

114

2. s-Numbers

Next we prove the converse in the case 1

p q

2. By 2.9.8, we have

(m - n + 1)112 = an(I : 12(m) -> 11 (m)) = x n(I : 12(m) -> 11 (m)) -5 I: 12(m) -÷ 11,(71)11 xn(I:ip(m) ->lii(m)) III: 4(m) -> 11(m)II < m( 1 /p-1/ 2)-1-(1 -1/q)xn q l(in) q( M )) Applying 2.9.3 with m = 2n - 1 now yields :

--

l

•

n112 r211_ 1 _< (2n - 1) 1 /P- "+"xn(D, : I, ->

lq) •

Hence, by 2.1.9,

D, e 2 :c2(/„, 1g) implies t e To treat the remaining cases, we deduce from 2.9.11 that (m n + 1)1/2 = an (I : 1 i (m) .- 1 2(m)) m -

= xn(l : 12(m) -> 16,0 (m)) .- I: 1 2(m) -› l(m)II x n(I : 1p(m) -÷ 4(0)111 : lq(m) -- 1.(m) II . Applying 2.9.3 with m := 2n - 1 yields (2n - 1) 11P x„(D,:lp -> 1g)

n 1 / 2x21_ 1

if 1 _- p _- 2

•

and 11112 T 2n- 1

(2n - 1) 112 x n(D,:lp -> 1g) if 2 < p <

ŒJ.

Hence, by 2.1.9

D, e 2,.,x,),(/„, /q) implies t e 2.9.19. We conclude this section with two special results about Hilbert numbers Proposition (B. Carl / A. Pietsch 1978). VI: li -> li ) =z n'12 . Proof. According to the Grothendieck theorem, mentioned in 1.6.4 (Remark), there exists a constant c > 0 such that II Y I 13211 .- cll Yll for Ye 2(li , /2). Let X e t(l2 , /1). Then, by 2.7.2, we obtain

n 112an(YIX)__- _ IIYIX 1 13 2 11.__ II Y I 43211 11x11 5- cll Yll IIXII . This proves that h n(I: 11 -> li ) _.-_ cn-112 . Conversely, we have 1 = h n(I : 12(n) -> 12(n)) ..-..- III : 12(n) .- i i (n)II 12,(I : l(n) -> 11 (n)) III : l(n) -> 12(n)II = n"hn(I: 11 (n) -> 11 (n)) _-<_ n"hn(l: li -> li). Remark. The above result shows that the Hilbert numbers fail to be multiplicative.

115

2.10. Relationships between various s-numbers

2.9.20. Example. Let 0 < r < 2 and 1 = 1/r — 1/2. Then D, E

,

/1 )

if and only if t e

Proof. It follows from h„(I :1, -+ 11 ) 2.9.18,

-

en-112 that l e

/J. Furthermore, by

t e is, implies D e esx)„(1„ 11 ) . Thus, in view of 2.6.7, we obtain D, e erit( 11, /1)As shown at the end of the preceding Paragraph, we have n-1/2 5

h „(I : 11 (n)

1 1 (n)) .

Applying 2.9.3 with ni := n now yields n-1 / 2r„

h„(D, :11 -+ 11 ).

Hence implies t e

D, e

2.10.

Relationships between various s-numbers

2.10.1. In this section the approximation numbers, the Gel'fand, Kolmogorov, Weyl, Chang and Hilbert numbers are compared with each other. To begin with, we summarize some results already proved in 2.3.4, 2.4.19, 2.5.13 and 2.6.3. Theorem. Let T e t(E, F). Then h(T) x(T) c„(T) a(T)

and h(T) y(T) d(T) a n(T).

2.10.2. We now turn to the much more complicated problem of estimating the larger s-numbers by the smaller ones. Proposition (A. Pietsch 1974:a, C. V. Hutton /J. S. Morrell / J. R. Retherford 1976). a(T)

2n 1 i 2 c„(T) for all T e t(E, F).

Proof. Fix any natural number n. Given e > 0, we choose a subspace M of E such that (1 +

c(T)

and codim (M) < n.

By 1.7.17, there exists a projection F E e(E) with M = N(P) and PI Define L := TP. Then it follows from rank (L)

n1 /2 .

rank (P) = codim (M)
that

a „(T)

11T — Ll = liT — (1 + e) (1 + n112) c„(T).

11 7 AA 11 1 — 1) 1

Letting s -+ 0 yields the inequality we are looking for.

2.10.3.* The following lemma is a counterpart of 2.7.1. Lemma. Let T e t(E, F). If c „(T) > 0, then for every E > 0 there exist

116

2. s-Numbers

families (x 1 , ..., x„) in E and (b 1 , ••• b„) in F' such that IlxklI = 1, ilbk11 = 1, ck(T) (1 + e)il for k = 1, n and = 0 for k = 1,

E

n — 11.

Since codim (Mn) < n, it follows that

c„(T) Hence there exists x„ e M„ such that

c(T)

(1 + e)11Tx„ II and

Ilxnll = 1.

Next we choose b n F' with Ii = 11T xnil and proof.

= 1. This completes the

2.10.4.* Proposition (A. Pietsch 1980:c). J1 In c2_1 (T)

2en'12

k =1

x k(T)

for all T E .2 (E, F).

Proof. Given e > 0, we choose Xi ,

EE

and

b1 ,

E F'

according to the preceding lemma. Define X2 „_ 1 e 2(12 , E) and by 2n- 1

=

X2

B2 n -1

e 2(F, 12)

2n-1

ek xk and

B2 1

1

:=

k =1

b,,0 k =1

Then

iiX2n-1ii =< (2n — 1) 1 / 2 and

11-8211- 1 V2I1

ek .

(2n — 1) 1 / 2 .

Since the matrix

(‹Txt, bi>) = ((-13211-1TX2„-1ez, Ci)) has upper triangular form, it follows from 2.7.8 and 2.11.13 that 2n-1

1 1

E C2n-1(T)

[

,

11/(2n-1)

i I
=1

bk>ii

Ideq(B2n-iTX2n-le

e))'/(2"')

On- 1

E. k = I

exp (

1/(2n-1)

xk(B2n-1Tx2.-1)] lln

1

2n — 1 IlB2n-1

< 2en 1 / 2 [ n Xk(T) 11In k=1

Letting e -+ 0 completes the proof.

[

k =1

Xi‘V X2n-1)i

2.10. Relationships between various s-numbers

117

2.10.5. The next lemma is closely related to 2.7.1 and 2.10.3. Lemma. Let Te t(H, F). If a2 „_ 1 (T) > 0, then for every E > 0 there exist an orthonormal family (x1 , ..., x n) in H and a family (b 1 , ..., bn) in F' such that Ilbkll = 1,

and

a2k-i(T)

(1

= 0

+ e)iI for whenever 1

k = 1, ...,n

_-_i
n.

Proof. The required families can be constructed by induction. If x l , ..., xn_ 1 e H and b 1 , ..., b n _ 1 e F' have already been found, then we define the subspace Mn := ftx e H :(x, x k) = 0 and (Tx, bk > = 0 for k = 1, ...,n — 11.

Since codim (M n) < 2n — 1, it follows from 2.4.6 that

= c2n-1(T) IIMNI . Hence there exists xn e M„ such that a2_1(T) ._.- (1 + E) IlTxn 11 and Ilxn 11 = 1. Next we choose b„e F' with jI = IITxn d and Ilb n II = 1. This completes the proof.

2.10.6. Proposition. i lln

n

x2 „_ 1 (T) _<_._ n1 / 2 [ n hk (T) k =1

for all T e t(E, F).

Pro of. Let X e 2(12, E). Applying the preceding lemma to the operator TX e 2 (1 2 , F), given E > 0, we find

ui , ..., u„ e /2 and

b i , ..., b n e F' .

Define Un e 2(12, 12) and 13„ e t(F, 1 2 ) by n

U,, : = E ek 0 uk and k=1

n

B„ := E b,, 0 ek . k =1

Then

Il U. Il = 1 Since the matrix

and 11/3,11 < n'12 .

(
has upper triangular form, it follows from 2.11.13 that 1

I -I- 8

ll n

n

a2n-1 (TX) __< [

< _

[

[I

k=1

= idet(OnTXUnei, ei))I 'In

KTxuk,bk>1]

ri 1 ak(B„TxUn)

i 1/n

< n 1/2

Hence lln n X2 n - i(T) ._.' n1/2 []J hani I +E k= i

I

•

Letting e —* 0 yields the inequality we are looking for.

{k=1 ri hk(ni lin ii XII •

118

2. s-Numbers

2.10.7. From 2.10.1 we immediately obtain the following diagram, where the arrows point from the smaller operator ideals to the larger ones:

ekv Ys)

(a)

Furthermore, in view of 2.1.8 and 2.1.9, the inequalities proved in 2.10.2, 2.10.4 and 2.10.6 imply certain reverse inclusions. See also 2.3.16, 2.5.6, 2.5.12 and 2.6.9. Theorem. Let 0 < p < 2 and 1/q = 1/p — 1/2. Then 44(y) Q(h) co(h)w 46'1,,w =

(x) (c) 2P,w

, 2 ,0

^A-

(pii?v (d)

q,w,

44(d) "hqi,tv,

9(a)

^-p,w = ^.•- q,tv •

Remark. Combining the above inclusions, we obtain _Ç 2(qa,?i,

if

0 < p < 2/3

and

1/q = lip — 3/2.

However, it seems unlikely that this result is the best possible. Since

an(I: it -+ 1.) = 1 and

h„(I: 11

1/p — 1 is certainly necessary, and we conjecture that it is also

the condition 1/q sufficient.

2.10.8.* Finally, we state an easy consequence of 2.10.4 and 2.4.11. Theorem (A. Pietsch 1980:c). Every operator T E t(E, F) with (n"x„(T)) e co compact. is Remark. We see from x„(/: /, T with (n 112x(T)) e 1..

2.11. Schatten

—

/,) = n-l / 2 that there are non-compact operators

von Neumann operators

Throughout this section we assume that H and K are complex Hilbert spaces. 2.11.1.* In a finite dimensional Hilbert space every orthonormal family (x i) is of course finite. In this case, we can obtain an infinite sequence by defining the missing elements by x, := o. This idea leads to the following concept: We call (x i) an extended orthonormal sequence provided that (x o xf) = 0 if i 4z j and either Ilx 1 11 = 1

or ilx,11 = 0.

2.11. Schatten— von Neumann operators

119

If not otherwise stated, in the following (x i) and (yi) denote extended orthonormal sequences in Hand K, respectively. Furthermore, (e i) denotes the standard basis of /2 .

2.11.2.* The following result is an immediate consequence of Bessel's inequality. Lemma. Let (x i) and (yi) be extended orthonormal sequences. Then 00

U: =E xl` yi i=1

defines an operator from H into K with Ulf

1.

Remark. The right-hand series converges with respect to the so-called strong operator topology induced by the semi-norms p Af (T):= sup {II Tx11 : x e

where M ranges over all finite subsets of H.

2.11.3.* A positive number r is said to be a singular value of the operator Te t(H,K) if there exist normalized elements x e H and y e K such that Tx = Ty

and

T*y = Tx.

This is equivalent to say that r 2 is an eigenvalue of TT* and T*T. Note that T and T* have the same singular values.

2.11.4.* We call 00

T = Irixt (1) y i

1=

a Schmidt representation of the operator T E 2.(1/, K) if the following conditions are satisfied: (ri) e co . (1) (2)

(x i) and (y i) are extended orthonormal sequences.

(3)

Tx = E ri(x, x i) y i for all x E H. i=1

CC

o and y i * o, then it follows that (Tx i , y i) = r.. In the opposite case, If xi where at least one of the elements x i and y i is zero, the value of the corresponding coefficient ri does not have any effect. It is therefore natural to assume that r i = O. In this way we obtain

(4)

(Tx i , y i) = Ti for i = 1, 2, ...

The formula CO

T — T ixr y j=1

= sup Iri l i=n+1

i>n

implies that the partial sums converge to T with respect to the operator norm. r2 O. Note that A Schmidt representation is said to be monotonic if r i all positive coefficients ri are singular values of T.

120

2. s-Numbers

2.11.5.* Diagonalization theorem. Let CO

T=

TiXt

0

yi

i=1

be a Schmidt representation of T E t(H, K). If

E

X :=

C) x i and Y :=

E e? C) y i ,

i=i

then

>K

H

Y Î

X* iï X

Y*

i

12

D, where D, is the diagonal operator induced by t = (r1). This diagram means that

T = YD,X*

and

D, = Y*TX

2.11.6.* Next we show that the coefficients of a monotonic Schmidt representation are uniquely determined by the underlying operator. Proposition (D. Eh. Allakhverdiev 1957). Let CO

ET,4

T=

yi

j=1

be a monotonic Schmidt representation of T e t(H, K). Then

;,

=

T).

Proof. From the above theorem we know that T = YD,X* and D, Y*TX Hence

an(T)

IIYII an(Dt)IIX * I

a(D) 5 I Y* II a„(T) 11X11

a„(T).

Thus, by 2.9.4, we have a„(T) = a(D) = rn . 2.11.7.* Every operator admitting a Schmidt representation is of course approximable. In order to show the converse, we need the following lemma which guarantees the existence of singular values. Lemma. Let Te 04(H, K). If T * 0, then r : = TII is a singular value of T. Proof. Choose any sequence (xk) with

lim I Txkll = r and

Ilxk ll = 1.

Set r---1 Txk . Since T is compact, we may suppose that (yk) converges to some y c K. Note that

IlYll = lim IlYkll = T-1 liM

11TX1c11

= 1.

It follows from 2 T*Yk — Tx,, II 2 = II Ty

(T*y k ,..r..x.k)

(rxk ,T*y k) + r2 2r 2 — 2IITXkli2

121

2.11. Schatten — von Neumann operators

that (xk) tends to x := 2-1 T *y. Finally, passing to the limit, we obtain from ryk = Tx,, and IX,, = 1 that ry = Tx and PO = 1.

2.11.8.* We are now in a position to establish the most important tool of the theory of approximable operators on Hilbert spaces.

Representation theorem (E. Schmidt 1907:a). Every approximable operator T e 2(H, K) admits a monotonic Schmidt representation. Proof. We are going to construct the desired representation by induction. If x i := TM > 0, then by the preceding lemma there exist normalized elements x, e H and y, e K such that Tx, = riyi and T*y i = rixi •

Assume that we have already found a scalar family (r„ r,,_ 1) and orthonormal families (x 1 , x„_„) and ( v 1 •• yn-1) satisfying the following properties: ri > 0, Tx, = riy, and T*y, = rix,. Put 5

•5

n-1

:=

If r„ = 0,

T—

E ri xfc 0 y,

1=J

then n-1

T = rixt 0 y i . 1=1 n, we arrive at the desired Therefore, setting r i : = 0, x i : = o and yi := o for i representation. If r„ > 0, then Lemma 2.11.7, applied to the operator n-1

T„ := T — E 1=1

0 yi ,

yields normalized elements x i, E H and y„ e K such that

and T:y„ = For i = 1, , n — 1 we have Tnx„ = rny„

r„(xi , xn ) = (xi, T:Y„) =

= (Tx, — TiYi, yn)

Tn(Yi, y„) = (yi,Tnxn) =

= (T*Yi — riXi

0

and Hence the enlarged families (x 1 , as well. This implies that Tx„ = T„x„ and

T*y„ = T„*y„.

Thus we have

and T*y„ In the next step we put Tx„ = rny„

T— E ixr y, i= 1

x,,_ 1 , x„) and (y1,

5

xn) =

•••5 yn-15 yn)

O.

are orthogonal,

122

2 . s-Numbers

It follows from (Tx — ri(x, xi) yi, Y.) = (x,T* Y.) — rn(x, xn) = 0

for A. e H

that the elements Tx — E ri(x, x i) y i and

rn(x, x„) y„

are orthogonal. Applying Pythagoras's formula, we obtain 2

n

-

1

2

Tx — E ri (x, x i) y i + xn) yn l 2 = Tx — E ri (x, x i) y i i=1 i=1 This proves that rn+1 rn• If this process, starting with r i , X 1 and Yi, does not end after a finite number of steps, then we obtain infinite sequences (r1), (x i) and (y 1). In particular, it follows that r 1 r, > 0. Since (Tx i) contains a Cauchy subsequence, 11Tx 1 — Txj 1 2 = ri• + rl whenever i j implies that (r i) tends to zero. This means that

=0.

lirn T — r C) y i pi

Therefore CC

T = ri xt C) y i

i= is the Schmidt representation we are looking for.

2.11.9.* Next we show that the s-numbers of any operator between Hilbert spaces are uniquely determined by their axiomatic properties. Theorem (A. Pietsch 1974:a). There exists one and only one s-scale on the class of all operators acting between Hilbert spaces. Proof. We first treat the case when TE e(H, K) is approximable. Then, by the preceding theorem, there exists a monotonic Schmidt representation CO

T=

y rix? C) y i .

1=1

Reasoning as in the proof of 2.11.6, we see that s,1(T) = s(D) = r„ for any s-scale s. We now consider an arbitrary operator T e 2(1/, K). Then

c„(74) = s„(TJZ)

s(T)

for all finite dimensional subspaces M. Hence, by 2.3.4, 2.4.6 and 2.4.12, an(T) = c(T) = sup {c„(Tiff): dim (M) < co}

s „(T) a„(T).

2.11.10.* We now provide an auxiliary result; see 2.7.1, 2.10.3 and 2.10.5. Lemma. Let T e t(H, K). If a2 „_1 (T) > 0, then for every E > 0 there exists an orthonormal family (x1 , ..., xn) in H such that a2k-1(T)

(1 +

ilTxkil

and Txi , ..., Tx, are orthogonal.

for k = 1,

n,

2.11. Schatten — von Neumann operators

123

Proof. The required family can be constructed by induction. If x l , already been found, then we define the subspace M„ := {x e H: (x, xk ) = 0

x„, have

(Tx, Tx k) = 0 for k = 1, ..., n — 11.

and

Since codim (Ma) < 2n — 1, it follows from 2.4.6 that 11 7114,3 -

a2.-1(T) = c2„-t(T)

Hence there exists x„ e Mn such that

a2n-1(T)

Ilx„11 = 1.

( 1 + E) lTxI and

This completes the proof. 2.11.11.* We are now able to establish an extremely important result. Theorem (J. W. Calkin 1941). Let ?.1 be an operator ideal which does not contain the identity operator of /2 . Then all components ?I(H, K) consist of approximable operators exclusively. Proof. Assume that 91(H, K) contains a non-approximable operator T. Then := lim a„(T) > O. According to the proof of the preceding lemma, we can find an orthonormal sequence (x,) such that for k = 1, 2, ...

a2k-i(T) _LS 2 11Txkil

Moreover, the sequence (yk) given by yk := TxkfljTxkll is orthonormal, as well. Define X e 2.(12 , H) and Ye t(/2 , K) by X := E

et

xk and

Y := E e C) y k .

k=1

k=1

Then S := Y*TXe 24(12) is induced by the diagonal matrix ((Tx i ,yi)). Moreover, we see from 2(Tx k , yk ) = 2 1iTxkll

a2k--1(T)

eic >

that S is invertible. Therefore we have I = the proof.

S e 21 (12). This contradiction completes

2.11.12.* Lemma (Ky Fan 1951). Let TE t(H, K) and 1

E 1(Txk, Yk)Ir

k=1

E ak(T)r

k=1

for all orthonormal families (x 1 , ..., xn) and (Yi, • • Y,1). Proof. Define Xe t(12 , H) and Ye t(l2 , K) by X:

Y :=

E et C) xk and k=1

k=1

Take a monotonic Schmidt representation co Y *TX = E ahut 0 vh ,

h=1

eit Yk •

r < oo. Then

124

2. s-Numbers

where (uh) and (vh) are extended orthonormal sequences in 12 . From 2.11.6 we conclude that

crh = ah(Y*TX) a h(T).

In particular, we have

ah = 0

E

= k=1 E

1(Txk,

k=1

for h > n. It follows that 1(Y*Txek ,

E

=

ek)i

ah(ek, uh) (vh, ek)

k=1 h=1

h=1

E Kek, uh)1 k=1

h

2

E

k=1

E

Ilvh11

( n

)1/2

E

ah

11=1

h=1

1(vh, ek)I

2

2 )11

ah (T).

h=1

This proves the assertion for r = 1. Next we treat the case 1 < r < co. By Holder's inequality,

2 = E ahKek, uh)1 21r 1(ek, ahl(ek, u,)1 E h=1 h=1 r )1I

E

14 01

2/r/

(n

ahl(ek, uh)I 2

r

h=1

E

h=1

Kek,

uh )12

)1/r1

Hence rI2

h= 1

ahl(ek 111)1 2 )

71

(

)

E

1/2

uh)1 2

h=1

Analogously, we obtain )r/2 (h=

ahl(Vh,

ek)I 2

)1/2

n

E 41(vh, ek)1 2

h= I

Moreover, 1(Txk, y k) 1 =

E ol12(eh , uh ) alh 12 (vh , eh ) h=1

1/2

E ahl(ek, 1101 2 )= h=1

1/2

n h

1

ahl(yh, ek)1 2)

Combining the previous inequalities, we finally see that k= 1

KTxk, Yk)Ir n

1/2 ( n

(12

EE

IhKek, uh)1 2

k=1 h=1 (n n

EE

)

01,1(ek, Uh)I 2 1/2

Grrhl(vh, ek)12

h=1

)1/2

k=1 h=1

( n

)1/2

E E 01,1(v,„

k

2

1/2 arhIlVh11 2 )

h=1

n

k=1 h=1

n

< (E arhIlUh11 2 ) h=1

E

)1/2

E h=

Remark. The above inequality fails for 0 < r < 1.

ah (T)r

2.11. Schatten —von Neumann operators

125

2.11.13.* The following inequality is closely related to the preceding one. Lemma (Ky Fan 1949/50). Let TE 2(H, K). Then fl a(T)

Idet ((Tx i , yi))I

k=

for all orthonormal families (x 1 ,

x„) and (y i ,

y,).

Proof. Adopting the notation from 2.11.12, we have !I

(Tx i , yi) = (Y*TXe i , ei)

= h=1 E ah(ei, uh)(yh,

es).

This implies that det ((Tx i , ys)) = det ((e 1 , uh)) (kfil ah) det ((v h , es)) . By Hadamard's inequality A.4.5, Idet ((ei, uh))I .6_ 1

and idet ((v h , ei))1

1.

Hence Idet ((T x 1 , Y))I-5. ri ah h =1

n ah (T ) h=1

2.11.14.* The following characterization is taken from (RIN, p. 58). Theorem. An operator TE 2(H, K) is approximable if and only if ((Tx k , yk))c co for all extended orthonormal sequences (xk) and (yk). Proof. If T is non-approximable, then the orthonormal sequences (xk) and (yk) constructed in the proof of 2.11.11 have the property that 1(Txk, YOI

oc/2

for k = 1, 2, ...

Hence the above condition fails. Let Te (H, K), and assume that there exist extended orthonormal sequences (xk) and (yk) for which ((Tx k , yk)) does not tend to zero. Then, passing to subsequences (uk) and (vk), we may find some 6> 0 such that

j(Tuk , vk)I

e

for k = 1, 2, ...

Now it follows from Lemma 2.11.12 that I n y ak(T) —ni n k =I

—

n E 1 (Tu, voi k=1

e.

This is a contradiction, since the arithmetic means of the approximation numbers converge to zero.

2.11.15.* All operator ideals e„ determined by arbitrary s-scales coincide on the class of Hilbert spaces. We denote this common restriction by The operators belonging to this class are said to be of Schatten-von Neumann type /,.,„,. K) we define For T e IIT I Zr.w11 := IIT I To simplify notation, the index w is omitted whenever w = r.

126

2. s-Numbers

2.11.16.* The following criterion is an immediate consequence of 2.11.6 and 2.11.8. Theorem. An operator T e 2 (H, K) is of Schatten-von Neumann type 4,,, if and only if there exists a Schmidt representation CO

T= i

=

Ti xf` (r) y i

such that t = (ri) /,.,„. When this is so, then

TI ar 04, 11 = It 2.11.17.* The next result is straightforward, as well. Proposition. An operator T E t(H, K) is of Schatten-von Neumann type 12 if and only if it is Hilbert-Schmidt. When this is so, then [IT I 22 I = [ T I ll 2.11.18.* We now establish a characterization of Z,,, which is analogous to 2.11.14. Theorem (A. Pietsch 1971). Let 1 < r < cc. An operator T E 2(H, K) is of Schatten-von Neumann type 4, w if and only if ((Txk, Yk)) E lrov for all extended orthonormal sequence (x k) and (yk). Moreover,

:= sup

[ TI Zr

4,11 :(A-k), (Yk)}

YO) I

defines an equivalent quasi-norm on er,w • Proof. Assume that the above condition is satisfied. Then 2.11.14 implies that T is approximable. Take any Schmidt representation Co

T

i= 1

ri4 0 yi

Since (Tx k , yk) = rk we have t = (r k) E 4. 0„. Hence, by 2.11.16,

T E 2,., w(H, K)

and

11 T Zr ovil = [[ t I 404,11

[IT I 2,,w110.

To verify the converse, let T E r,w W, K). Given extended orthonormal sequences (xk) and (yk), we may assume that O.

1(Tx2, Y2)1

l(Txt, Yi)l

In view of 2.11.12, it follows that

n k=1

i(Txk

AA) I

- E ak(T))1

lr

n

k =1

Therefore 2.1.7 yields ((Txk

Yk))

G Iroi,

and

I ((Txk

Yk))

I 4,11

er,w11

The last result can also be obtained by interpolation techniques. Given extended orthonormal sequences (xk) and (yk), we consider the operator

D :2(H,

1„

which assigns to every operator T the sequence ((Txk, Yk)). We see from 2.11.12 that l. D: i (H ,

2.11. Schatten — von Neumann operators

127

Hence, by 2.3.14 and 2.1.14, the interpolation property yields D : Zr,,,(H,

2.11.19.* In the special case 1 -.r=w< co a refined version of the preceding characterization can be obtained via 2.11.12. r < co. An operator Te 2(H, K) is of Schatten-von Proposition. Let 1 Neumann type I,. if and only if ((Tx k , yk)) E 1r for all extended orthonormal sequences (x,) and (yk). Moreover, TI

,11 = sup {11((Txk, Yk)) I 1,11: (xk),(Yk)}

2.11.20. The description of

II. I 2,11

just given implies the following result.

Proposition (R. Schatten / J. von Neumann 1948). If 1 is a Banach operator ideal.

r < co, then 2,

Remark. It follows from 1.7.2 and 2.11.26 that Zr is an r-Banach operator ideal for 0 < r < 1; see also S. J. Rotfeld (1967). 2.11.21. Next we formulate an elementary observation; see 1.4.10. Proposition. The following are equivalent for Te t(H, K): T e Zr , w(11, K), T* e

r,„(K, H), T' E

H').

The corresponding quasi-norms coincide. 2.11.22. In the Hilbert space setting 2.3.15 can be improved. Proposition. The operator ideal Zr is stable with respect to the tensor norm a. More precisely,

Ils 6,7 7' I

= IS

I Z,11 I IT I SA

for SE r (H, Ho) and TE r (K, K0)-

Proof. Let S = E Gri xr 0 Ili and i=1

T= E

j=1

(1) vi

be Schmidt representations. Note that (x, C) y i) and (ui C) vi) are extended orthonormal sequences in HC)K and Ho O a Ko , respectively. Since 00 00 S 5 T = E E Griri(x, 0 yx 0 (ui C) i=1 j=1 and (S EL T)

(x1 0.0 =

v.i)

it follows from 2.11.16 that S

s

T e 2,.(1/

(5g K, Ho (5.. K0)

and

IS (-5„ TI

il =

= IIS I

f r( 1%1 X IN)11 = Rai) I 411

ITO I 411

Il ITI 2,11.

2.11.23. We now need three complex Hilbert spaces denoted by H, K and L.

128

2. s-Numbers

Proposition (A. Horn 1950). Let 1/p + 1/q = 1/r. Then I ST

i

IT egIl

for

Te

K)

and

S

E

ep(K, L).

Proof. From 2.2.9 we know that

clI S 1 Spll IITI egll, (*) IIST1 where c 1 is a constant. Note that this inequality holds for all operators of Schattenvon Neumann type 1,. acting between arbitrary Hilbert spaces. Consequently,

S) (T 6 7 T) I 2,11

II(S c Obviously,

(

(S

T) = ST

S)

ells 6,7 s I 411 I IT (5. T j Zpil

60. ST.

Applying the preceding proposition, we now obtain

CI1S I Zp11 2 I eq11 2 . IIST1 Zr11 2 This implies that (*) holds with the constant c 1 / 2 , as well. Hence, if c > 1 is choosen c l i2 . Thus c = 1. as small as possible, it follows that c 2.11.24. In the following we investigate the restriction of some operator ideals to the class of Hilbert spaces. On the other hand, we try to find extensions of the Schatten-von Neumann ideals to the class of all Banach spaces; see D.3. The first result along this line can be obtained from 1.4.5 and 2.11.17. Proposition. The Banach operator ideal 1 2 is a metric extension of 22.

r < Do Remark. We stress the fact that, by 1.3.16, all Banach ideals 90, with 1 are extensions of 22 • However, their norms coincide with 11. I Z2 1 only when r = 2. 2.11.25. Proposition (A. F. Ruston 1951: a). The Banach operator ideal 9Z is a metric extension of e i . Proof. Using a Schmidt representation CO

T= E ;Al 0 yi

i=1

of T e 2 1 (H, K), we deduce from CO

CO

.E ITil

1= 1

YILA

E

i=1

= 117 1 Il

that

T e 91(11, K) and

11T 194 < IIT I

.

We now assume that Te 91(11, K). Since T is approximable, there exists a monotonic Schmidt representation CO

T= E Tixp 0 y i . i=1

Furthermore, given E > 0, we choose a representation

T= E

v.,

2.11. Schatten — von Neumann

operators

129

such that

c, j

I

=1

Ilv; I

(1 +

8)

IIT I 9111.

Then Cr_

CC

CC

CC

a,(T) = E ri = E 1(Txi, yi)1

i=1

i=1 CC

T e 2 1 (H, K)

1(xi, u.)1 2

YJI1

j =1

i=1 j=1

)1/2

)1/2 (co

(co

E

Thus we have

E E 1(x1,ui)(vi,Y)1

i=1

j=1 i=1

CO

and II TI

(1 +

'III

E

yi)I 2

i=1

IIT1 91 11.

lT9lI.

2.11.26. The preceding result extends as follows. Proposition (R. Oloff 1969, 1972). The quasi-Banach operator ideal 91, is a metric extension of 2, for 0 < p Lç_ 1. Proof. Reasoning as in the previous proof, it follows from T e Z„(H, K) that

T 91,(1/, K) and IlT I 9 ll 1IT 2pII • We now assume that T E 94,(H, K). Let 1/r := 1/p — 1. Given

E > 0, by 1.7.3,

there exists a factorization

D,

Ds such that

111'11 IIY 1 1 211 Ils I 411 la 1 /211 11A11

(1 +

IIT 91 lI •

In view of 1.6.3 (Proof), 2.11.21, 2.11.23 and 2.11.24, we now obtain

T = (YD,) D s(Da A)

E

2

° Zr °

Z2(H, K) 2,(H, K)

and

IT I 2p1I 5 1117Dy I 2211 Ws I 11YII IIY Letting e

rll I1Da4 I 2II (1 + 011 1'19'41k 1211 11 ,5'1 411 Ila I / 211 114ll

0 completes the proof.

2.11.27. Proposition. The quasi-Banach operator ideal 94, 2 is a metric extension of 2,. for 0 < r 2. Proof. Using a Schmidt representation of Te r(H, K), it follows immediately that

T E 94, 2(H, K) and IIT I 9 4,211 5_- IITI

rIl•

130

2. s-Numbers

We now assume that T E 94,2(H, K). Let 1/p := 1/r — 1/2. Given 8 > 0, by 1.7.10, there exists a factorization

î

Y

such that (1 + IIT I 94,211. II Yll Ils I /A Ila I /211 Milli In view of 1.6.3 (Proof), 2.11.23 and 2.11.24, we now obtain T = YD,(D„A) E 2p 22 (H, K) ZAH, K)

and

IIT I ar II

II Yll 1112', I sp11 1l1)aA I e211 1 94,211 • (1 +

II Yll Ils I 4,11 lia I 1211 11All

0 completes the proof.

Letting

2.11.28. According to S. Kwapiefi (1968) the next result, which is a special case of 2.7.6, was first established by B. S. Mityagin (unpublished). Here we give a direct proof. Proposition. The Banach operator ideal 43 r, 2 is a metric extension of Zr for 2 r < o o Proof. Since the identity operator of /2 fails to be absolutely (r, 2)-summing, we conclude from 2.11.11 that all operators Te K) are approximable. Thus there exists a Schmidt representation T=

E

rixt (i) y i .

1=1

It follows from II(Txt) I 44

jT I 914,2II

w2II

as well as IlTxill

ITii = l(Txi,

and

Il(xi) j w2ll

that

11(ri) I 1, 1

IIT I 4r,211.

Hence

Te 2„(H, K) and

Ili' l Zr11

IITI L,211.

1

2.11. Schatten— von Neumann operators

131

Conversely, assume that T E 4(H, K). Given x1 , ..., x X E 2(12, H) with 11x11 = 11(x1) I w2I1 by

H, we define the operator

n

X :=

E et 0 xi .

Let

TX = E (bur 0 yi j =I

be a monotonic Schmidt representation. Applying Holder's inequality with respect to the exponents p := r/2 and p' := rl(r — 2) yields 11Tx1ll 2 = Iliwe1ll 2 = E all(ei, u)1 2 co

= E Gril(ei, u)1 21P j= 1 co

Kei, lij)1

21pf

)1IP (co 2 -5,- (E a.r1(ei, u;)I E ( ei, u.1)1 2 J --1

j=1

)2 Ir oo 2 -.- rWei (E C ) 101 j= 1

Since

cf.; = ai(TX), we obtain 1Ir ii(T)Ci) I 411 =Xili ( i IIT r)E

1=1

)

(n :_--.

co

I r )1

E cKet• uA 2

i= 1 j = 1

r )1I 1/r (oo <(a;) Za; i = Eai(TXY

T ll Zrii ii-Kil — IIT = IilX T i Zrii -"- Ilj

I Zrli 11(xi) i W211.

This proves that T e 43r,2(H, K)

and II TI Vr,211__... 11T I

erll .

2.11.29. Proposition (H. KOnig 1979, 1980: c). Let 2._.. p
F) g 41;) . 2(H, F) _Ç. 2(H, F) = i!,(H, F).

To verify the converse, given Te ,,(H, K), we consider the factorization T

132

2. s-Numbers

where 1/s := 1/p + 1/2. By 1.6.7 and 2.9.10, D, e 2(qa,;,02 , Is)

and

This proves that T E Vp,2

JE 4p,2(1s, 12). K).

°

2.11.30. We now formulate an immediate consequence of 2.7.6 and the preceding result. Proposition (H. K6nig 1980: c). Let 2 p < 00 , 2 < q < oo and 1/r = 1/p + 1/q. Then the operator ideal 4 p,2 ° 4q,2 is an extension of Z,.,q • 2.11.31. An operator T e 2(E, F) is said to be of weak Schatten-von Neumann type 'r,„ if YTX

2,,w(i2)

for all X E 2(12, E)

and

YE e(F, 1 2).

The set of these operators is denoted by V;;ewak(E, F). For T E Z;:v,ee(E, F) we define T

I

: = sup {it YTX

r,w

(Br

Xe 2,02 , E), 11 x11 YE t(F, 12), Y I

w

11 11•

To simplify notation, the index w is omitted whenever w = r. 2.11.32. Theorem (A. Pietsch 1970, 1976). "-weak r ,vv is a quasi-Banach operator ideal.

Proof. The ideal properties can be checked by standard techniques. Therefore we only show that the quasi-norm defined above is finite. Assume the contrary. Then there exist Xk E 2(12, H) and Yk e 2(F, 12) such that

I n11

1,

1

and

Yk TXk

4kk.

Form the Hilbert space H := {12 , 12 1 as described in C.2.1. Define Ge

X:

= k=1

2- kxkQk

and Y:=

k=I

2 -vkYk,

where .4 E 2(12 , II) and Qk e t(H, / 2) denote the canonical injections and surjections, respectively. Since H can be identified with 1 2 , it follows that YTX e r ,,,(H). We now deduce from XJk = 2- kXk and

QY = 2-k Yk

that k

YkTXk ar..11 = 11QkYTXJkll

1117TX I 2,11

for k = 1, 2, ... This contradiction completes the proof. 2.11.33. Proposition (A. Pietsch 1970, 1976). The operator idealVr7,,ak is the largest extension of Sr ,„,.

Proof. It follows from 2.11.11 that every operator T E K) is approximable. Hence, by 2.11.5 and 2.11.8, there exist t E c0 , X E 2(17, H) and YE 2(/2 , K) such that D, = Y*TX

and

T = YD,X*

2.11. Schatten— von Neumann

operators

133

From T e ZZTk(H, K) we conclude that D, e Z„,„(12). Therefore T e 6,.,,(H, K). This proves that 2 7,e: k(H,

r,w(11, K) The reverse inclusion is obvious. Thus 27,?k is indeed an extension of 6,.,,„• Finally, let ?.1 be any extension of Zr ,„- If T e W(E, F), then YTX e /4/ 2 ) = er.,„(/2) for all X E 2(12 , E) and YE t(F, /2 ) .

Hence T eeN;k (E, F). Thus we have Vt g 2.11.34. Proposition. C'ecg k = Proof. Let T e Zet (E , F). Then

nura„(YTX)

1 YTX I r,.011

II VII

rek Il

iiT J

for all X e 2(12 , E) and Y e t(F, 12 ). Hence

n lirhn (T) This proves that sweak

iiT j

v,etkil.

9(h)

— r,co " The reverse inclusion follows immediately from 2.11.33 and the fact that is an extension of . 2.11.35*. Proposition (H. K6nig 1979). Let 1 z weak

r < 2 and 1/s = 1 1 r — 1/2. Then

Proof. Assume that T e Zrak(E, F). Given (xi ) that

[w2 , E], we choose b, e F' such

= 117-xill and jb1jj-5_ 1.

Define X e 2(12 , E) and B e 2(F, 1) by X :=

el ® x, and B :=

b,® e,.

Moreover, let D, e 2('œ' 12) be the diagonal operator determined by an arbitrary sequence s = (a1) e 1 2 . Since = cri = (D,BTXe i , e,),

it follows from 2.11.19 that (cri ii TE 435, 2 (E, F), by 1.2.2.

i II)

E

I,. Hence (Tx,) E [la , F]. This proves that

2.11.36. In the case r = l the preceding result can essentially be improved. For a proof we refer to (PIE, 17.4.3 and 17.5.2). Proposition (S. Kwapiell 1972).

2

1'7 2 •

2.11.37. The following diagram illustrates the containment relationships between various extension of e i . The arrows point from the smaller operator ideals to the larger ones. The results are taken from 1.7.16, 2.3.11, 2.8.18, 2.8.20, 2.11.33 and 2.11.36. Concerning the inner hexagon we refer to 2.10.7.

134

2. s-Numbers

eweak

°

(1

(a)

3 2)2,1

î

(432' )?,)1

° 432'

Further extensions can be obtained by forming finite sums of the ideals listed above. This is indeed only a small part of the "ideal zoo" in the sense of B. Simon (SIM, p. 114), isn't it?

CHAPTER 3

Eigenvalues In this chapter we investigate the asymptotic behaviour of the eigenvalues of operators belonging to a given ideal. The background of this theory is the concept of a Riesz operator which can be introduced from very different points of view. We present here a geometric approach. It turns out that every Riesz operator TE t(E) possesses a countable set of eigenvalues. Apart from the case when this set is empty or finite, we get a sequence (2„(T)) tending to zero. It is our main goal to obtain information about the rate of this convergence which is measured in terms of Lorentz sequence spaces. Roughly speaking, an operator ideal It is said to be of eigenvalue type if (2„(T)) E 4.0,

for all operators T E it(E) and arbitrary Banach spaces E. By means of some effective tools such as the principle of related operators, the principle of iteration, the principle of uniform boundedness, the principle of tensor stability, and more especially the Weyl inequalities, we determine the best possible eigenvalue type of several concrete operator ideals. A survey of these results is given in 7.4.6.

3.1.

The Riesz decomposition Throughout this section A denotes an arbitrary linear map acting on a linear space E. All considerations are purely algebraic. 3.1.1.* Let Nk(A) := {x e E: Akx = o} and Mh(A) := ftgx : x e E}. Obviously, {o} = No (A) g_ N i (A)• • • and •• • g- M1(A) Mo(A) = E ..,_ Write N co (A) :=

Cj

k o

Nk(A)

Furthermore, define n(A) := dim [1■Ico (A)]

and

and

co (A) :

r(21

h° 0

Mh(A).

m(A) := codim[M (A)],

where the right-hand quantities may or may not be finite. Remark. To simplify notation, in the following proofs we write of Nk(A) and Mh(A), respectively.

Nk

and MI, instead

3.1.2.* If there exists an integer k such that N(A) = N k+ i(A),

then A is said to have finite ascent. The smallest such k is denoted by dpi(A).

136

3. Eigenvalues

3.1.3.*

We now show that Nk(A) is constant for k d N (A).

Lemma. Nk(A) = Nk +i (A) implies Nk+i(A) = Nk+2(A). Proof. Let x e Nk+2. Then AX E Nk±i , and the assumption says that Ax E Hence

Nk •

X E Nk+1

3.1.4.* Lemma. If A has finite ascent, then

for k = 0, 1, ...

Mh(A) N(A) = {o}

and h

d N (A).

Proof. Let y e Mh Nk. Because y e Mh, there exists x E E with y = Ah x. We now conclude from Ah+kx = Aky = o that x e Nh4-k. Hence xeNh . This implies that y = O. 3.1.5.* Lemma. If A has finite ascent, then n(A)

dN (A).

Proof. Note that k

dim (Nk/Nk _ i ) > 1 for

1, ..., dN (A).

Hence 00

n(A) = dim [N °AA)] =

E

dim (Nk /Nk _ i )

d N(A)

k=1

3.1.6.*

If there exists an integer h such that Mk(A) = MH-1(A),

then A is said to have finite descent. The smallest such h is denoted by dm (A). 3.1.7.*

We now show that Mh(A) is constant for h

dm (A).

Lemma. Mh (A) = Mh+i(A) implies Mk+i(A) = Mh i_2(A). Proof. Let y e Mh+1. Then there exists x E E with y = A" -F i x. By assumption, we can find xo e E such that Ahx = Ah+ i x o . Hence y = A"+ 1 x = A"+2 x 0 , which means that y e- --h+2. M 3.1.8.*

Lemma. If A has finite descent, then Mh(A) + Nk (A) = E for

Proof. Let x e E. Because Akx e Mk Ak+kx0. Set

A kx

xm := A kx0 and Then X m cM,, and x N e

and k

h = 0, 1, ... = Mh+k,

dm (A) .

we can choose x o e E such that

x N := x — A hxo •

Nk.

Thus x = xm + xN is a decomposition we are looking

for. 3.1.9.*

Lemma. If A has finite descent, then ni(A)

dm (A).

Proof. Note that dim (Mh_ i /Mh)

1

for h = 1,

dm (A) .

Hence

nn(A) = codim

1 co (A)] = h=1

dim (114-k-1/Mh)

dm (A)

137

3.1. The Riesz decomposition

3.1.10.* Proposition. If A has finite ascent and finite descent, then dN (A) = dm (A).

Proof. Put h := dN (A) and k := dm (A). By 3.1.8, every element x e E possesses a decomposition X

= xm + xN with

We now assume that x

x m e Mry+1

E Nk+1.

Xm = X — XN G Nk+i

and

x N E Nk •

Then it follows from Nk = Nk+i

that x m e Mh+1 n Nk+ . Thus 3.1.4 implies that xm = o. Hence x = x N e proves that Nk = Nk+1. Consequently, dN (A) h := dm (A). Next we assume that x e MI,. Then it follows from XN" ="" X XM E

Nk.

This

Mh+1 = Mh

that x N e Mry r Nk. Thus 3.1.4 implies that x N = o. Hence x = xm e Mh+1 . This proves that MI, = Mk+1 . Consequently, dm (A) h := dN (A)• 3.1.11.* For every linear map A having finite ascent and finite descent the common value of dN (A) and d m (A) is denoted by d(A). 3.1.12.* A linear subset M of a linear space E is invariant under a linear map A if X E M implies Ax e M. When this is so, then we speak of an A-invariant linear subset. 3.1.13.* A linear map A is nilpotent if there exists an exponent m with Tin = O. The smallest such nt is called the order of nilpotency. 3.1.14.* We are now prepared to establish the main result of this section.

Decomposition theorem (F. Riesz 1918, A. E. Taylor 1966). If the linear map A has finite ascent and finite descent, then the linear space E is the direct sum of the A-invariant linear subsets M „(A) and N co (A). Moreover, the following holds: (1) The restriction of A to M co (A) is invertible. (2) The restriction of A to A rco (A) is nilpotent of order d(A). Proof. Write d := d(A). Then

M c (A) = Md

and

N 0.,(A) =

Nd.

Now it follows from 3.1.4 and 3.1.8 that E is indeed the direct sum of M x (A) and N ,(A). Obviously, both linear subsets are invariant under A. Given y E M co (A), there exists xo e E such that y = A" x o . Hence y = Ax, where x := Ad x 0 E Md. This proves that A maps M Go (A) onto MJA). Furthermore, we know from 3.1.4 that Md n N 1 = {o} . Thus the restriction of A to M (A) is one-to-one. Lastly, we note that Adx = o

for all x e N .(A).

3.1.15.* Finally, we prove the result stated in A.2.5. under a very special assumption.

Lemma. If A is a nilpotent linear map on a finite dimensional linear space E, then there exists a basis (u 1 , un) such that the representing matrix M = (p u) has upper triangular form with zeros on the principal diagonal. This means that = 0 whenever i j.

138

3. Eigenvalues

Proof. We begin by choosing a basis (u11 , ..., ui „,) of N1 . Then we extend this to a basis of N2 by adding (u21, .• •, u2 ,2), and repeat the process. In this way we obtain a basis ( 4 11, • • •, u1n1; • • • ; udl, • • •, udnd)

of E, where d = d(A). The proof is finished if we number these elements according to their lexicographical order.

3.2.

Riesz operators

Further information about Riesz operators is to be found in (BAS), (CAR) and (DOW).

3.2.1.* An operator T e 2(E) is said to be iteratively compact if for every 8 > 0 there exist an exponent n and elements u„ ...,uk e E such that k T n (U) E U h=1

tun

+

WI ,

where U denotes the closed unit ball of the underlying Banach space E. Remark. It can be shown that an operator T E 2(E) is iteratively compact if and only if we can find an exponent n and a compact operator K e 42(E) such that IIT" — Kil < 1; see K. Yosida (1939) and A. Pietsch (1961).

3.2.2.* We now establish a pigeon-hole principle which is basic for the considerations that follow. Lemma. Suppose that Te 2(E) is iteratively compact. Let (x i) be any sequence in U. Then for every E > 0 there exist an exponent n and an infinite subset I of N such that liT nXi — T nXill

E

for all i,/ e /.

Proof. Choose n and u1 , ..., uk e E such that k T"(U)

8

g U {u i, ± — 2 h =1

U} .

Setting Ih := {i E

N:

E U} for h = 1, ..., k, Tux, e uh + --2-

we have ilT"x i —

T nxill

E

for all i, j E 4.

Furthermore, it follows from k Ii

U ih = N =1

that at least one of the sets 11 , ..•,

Ik

is infinite.

3.2.3.* Next we state the Riesz lemma which, in the following proofs, is the opponent of the preceding pigeon-hole principle.

3.2. Riesz operators

139

Lemma (F. Riesz 1918). Let M be a proper subspace of the Banach space E. Then for every e > 0 there exists an element xo e E such that 11x0 11 = 1 and

1 for all x E M.

11x0 — x11

1+

e

Proof. Take any yo E EVM, and set : = inf {IlYo — x11 : x e M} .

Since M is closed, it follows that

11Y0 — zoll

e > O. Next we choose

(1 +

zo e M such that

e

Define xo : = N' (Y 0 — z 0). Then, for every x e M, we have — (zo + eox)II —x = Therefore xo has the required properties. We now deal with operators of the form I — T, where I denotes the identity operator of the underlying Banach space. 3.2.4.*

Proposition. If TE 2(E) is iteratively compact, then all null spaces Nk(I — T) finite dimensional. are

Proof. Assume the contrary, and take the smallest k for which Nk(/ — T) is infinite dimensional. Applying the Riesz lemma with e = 1/3, we can inductively choose elements x i e Nk (/ T) such that Iix1 = 1 and

3/4 for all x e span (x1 , ..., xi _ i) + Nk_ 1 (/ — T)

11x1 — xii It follows from

x — Tx = (I + T + + Tn -1) (I — x that x — Tx

G

Nk- - T)

for all x e Nk(/ — T) and n = 1, 2, ...

Hence Tnx i — Tnxi e xi — x; + Nk _ 1 (/ — T),

which implies that

3/4 whenever i >j and n = 1, 2, ... On the other hand, by the pigeon-hole principle, there exist an exponent n and different indices i and j such that !Jinx' —

1/2. II Tx — TxIi This contradiction proves the assertion. 3.2.5.*

Proposition. If T E WE) is iteratively compact, then ! — T has finite ascent.

Proof. Assume the contrary. Applying the Riesz lemma with 8 = 1/3, we can choose xk E - T) such that iiXkii = 1 and 3/4 for all x e Nk _ 1 (/ — T). Ilxk — x11

140

3. Eigenvalues

It follows from

x — Tnx = (1+ T+ + Tn-1 ) (I — T) x that Xk

—

T nXk G

T) for n = 1, 2, ...

Hence

Taxi, — Tnx k e xi, — x,, + N, 1 (I — T) + Nk _ i (I — T), which implies that

3/4 whenever h > k and n = 1, 2, ... 11r5ch — Tnxk II On the other hand, by the pigeon-hole principle, there exist an exponent n and different indices h and k such that 1/2. Tnxh — nrk This contradiction proves the assertion. 3.2.6.* Lemma. Let TE 2(E) be iteratively compact. Then every bounded sequence (x i) for which ((I — T) x i) is convergent has a convergent subsequence.

Proof. Without loss of generality, we may assume that (x i) is contained in U. Given e > 0, by the pigoen-hole principle, there exist an exponent n and an infinite subset / such that

e for all i, j E l.

— It follows from

x = Tnx + (/ + T

+ Tn-1 ) (I — T) x

that

Il x i —xI5_ x — Txll + III + T +

+ T"-1-

— T) x i — (I

Thus, since ((/ — T) x i) is Cauchy, we can find an infinite subset

4

. of / such that

2e for all i,j e /0 .

—

Let (4) := (x i) and E„, := 2 - m -1 for in = 1, 2, ... Applying the preceding construction infinitely many times, we obtain sequences (xr) each of which is a subsequence of its predecessor (x 7n-1 ) and such that

- xr11

2 em

for all i and j.

Then the diagonal (xii) is the desired convergent subsequence, because

114 —

2- m whenever i, j

3.2.7.* Proposition. If T E 2(E) is iteratively compact, then all ranges M11(/ — T) are closed. Proof. Let y = lim y, where (y i) is contained in M,,(/ T). Set

el

:=

inf {PO : (/ — T)" x = y il ,

3.2. Riesz operators

141

and choose xi E E such that

(I — T)h X 1 = yi and IIxI Assuming that

e

2e,.

cc, we put

ui := e r 1 x

and

vi

lyi .

2, and (vi) tends to zero. Next, h-fold application of 3.2.6 yields a Then litti ll subsequence of (u i) which converges to some u c E. Passing to the limit, we see that (I — T) h ui = vi implies (I — T)h u = o. Hence (I — T' (x i — e iu) = y,. Thus, by the definition of ei, we have

el

lixi — e iull

or

1 tri

1.

(c

This contradiction shows that ) has a bounded subsequence. Applying 3.2.6 once again, we find a convergent subsequence of (x,). If x is the corresponding limit, then (I — T)hx = y. This proves that y E — T).

3.2.8.* Proposition. If T E 2(E) is iteratively compact, then I — T has finite descent. Proof. Assume the contrary. Applying the Riesz lemma with e = 1/3, we can choose yh E Mh(/ — T) such that ii Yil = 1 and 3/4

ilYn — Yii

for all Y E Mh+1(1 — T).

It follows from y — Ty = (I + T + + T'1 ) (I — T) y

that G M+1(1 — T)

Yh

for n = 1, 2, ...

Hence Tnyh T nyke yh yk Mh+i(I —

T) +

T),

which implies that liTnYh

T yk il

3/4

whenever h
n = 1, 2, •••

On the other hand, by the pigeon-hole principle, there exist an exponent n and different indices h and k such that T nYn — TnYk

1/2 .

This contradiction proves the assertion.

3.2.9.*

We are now in a position to establish the main result of this section.

Decomposition theorem (F. Riesz 1918, K. Yosida 1939, A. Pietsch 1961). Let T E t(E) be iteratively compact. Then the Banach space E is the direct sum of

142

3. Eigenvalues

the T-invariant subspaces Mo0 (/ — T) and N. (I — T), the latter being finite dimensional. Moreover, the following holds: (1) (2)

The restriction of I — T to M oo(/ — T) is invertible. The restriction of I — T to Nœ (I — T) is nilpotent of order d(l — T).

Proof. According to 3.1.14 and the previous results of this section we need to show only that the restriction of I — T to M.(/ — T) is continuously invertible. Of course, since M 00 (/ — T) is closed, this follows from the bounded inverse theorem. There is, however, a direct approach. Suppose that the inverse fails to be continuous. Then we can find normalized elements xiGM Q0 (1 — T) such that ((I — T) x i) tends to zero. Hence, by 3.2.6, there exists a subsequence converging to an element XE Moo — T). Passing to the limit, we see that 'k i ll = 1 implies 114 = 1. Furthermore, (I — T) x i o yields (I — T) x = o. Hence x e M(/ — T) r N 1 (1 — T). Thus, by 3.1.4, we have x = o. This contradiction proves the continuity. ('

3.2.10.* For every iteratively compact operator Te 2(E) we let P.(' — T) denote the projection from E onto N(I — T) along M 00 (/ — T). 3.2.11.* The decomposition theorem 3.2.9 can be rephrased as follows. Theorem. Let T e t(E) be iteratively compact. Then there exists a finite projection P e 2(E) commuting with T such that the operators

TM : = (I — P)T

and TN := PT

have the following properties:

(1) (2)

I — TM is invertible. P — N is nilpotent of order d(I — T).

Proof. Define P :=

— T).

3.2.12.* An operator Te 2(E) is said to be Riesz if for every 8 > 0 there exist an exponent n and elements 71 1 , uk G E such that

u tuh + enul.

h=1

3.2.13.* The following criterion relates the concepts defined in 3.2.1 and 3.2.12 with each other. Proposition. An operator TE 2(E) is Riesz if and only if T is iteratively compact for all C e C.

Proof. Assume that T is Riesz. Given 6 :=

C and e > 0, we put

min(s, 1) l+ lI

Then there exist n and u1 ,

Tn(U) g U Itlh h=1

uk e E such that 6nUI.

3.2. Riesz operators

s and ICI 6,

It follows from ICI (5 (T) n(U)

143

U {C"111, h=1

1 that ICI" 6"

8.

Hence

EU}

Thus CT is iteratively compact. In order to prove the converse, we let e> O. Take

:= max (-1, 1). Since CT is assumed to be iteratively compact, we find n and ul , that

uk e E such

(CT)" (U) _g_ U {u h + h =1

It follows from Ce that

1 and C

1 that Cne"-1

1. Hence C- "e < en , which implies

T(U) U {tc h + snU} h =1

Thus T is Riesz. 3.2.14.* In view of the preceding criterion, Theorem 3.2.9 implies the following basic result. Decomposition theorem. Let T e t(E) be a Riesz operator. Then, for every complex number C, the Banach space E is the direct sum of the T-invariant subspaces Moc,(/ + CT) and Noo (I + CT), the latter being finite dimensional. Moreover, the following holds: (1) (2)

The restriction of I + CT to Mo,(/ + CT) is invertible. The restriction of I + CT to Nco (I + (:"T) is nilpotent of order d(I + CT).

Remark. We stress the fact that, conversely, an operator T e 2(E) is Riesz if it has the above decomposition property for all C e C; see (BAS, 0.3.5). 3.2.15.* A complex number 20 is an eigenvalue of the operator Te t(E) if there exists x e E such that

Tx = A ox

and

x o.

Every element x * o such that (20 1 — T)k x = o for some exponent k is called a principal element of T associated with the eigenvalue 20 . If k = 1, then we have Tx = /lox, and x is said to be an eigenelement. The multiplicity of an eigenvalue 20 is defined by

n(41 — T) := dim [N.(20/ — T)], where the right-hand quantity may or may not be finite. Remark. In contrast to the algebraic multiplicity just defined, the dimension of the subspace N(201 — T) is called the geometric multiplicity of an eigenvalue 20 . 3.2.16.* First we establish a very elementary property.

144

3. Eigenvalues

Proposition. Eigenelements x 1 , xm of any operator TE 2(E) associated with distinct eigenvalues A I , A n, are linearly independent. Proof. Suppose that cx i x, + equation, we obtain

Alfoc i xi +

+ o

m = o. Applying Tk to both sides of this

+ Ac m x m = o for k = 0, 1, ..., in — 1.

Since Vandermonde's determinant

det

1 Am

1 Ai .

4 _1

= i>j H

— A.1)

4-1

does not vanish, we have ocix; = o. It follows from xi + o that oci = 0 forj = 1, ..., in.

3.2.17.* The next proposition contains the preceding result as a special case. Proposition. Principal elements x 1 , xm of any operator TE 2(E) associated with distinct eigenvalues 2, 1 , ..., A m are linearly independent.

+ mx m = o, and choose a natural number k Proof. Suppose that a 1 x 1 + such that (A i l. — T)k x i = o for i = 1, ..., m. Fix any A. Since the polynomials (Ai — A)" and H (A — A)k are relatively prime, there exist polynomials p and c/i i+j

which satisfy pi (A) (A — A)" + q;(2)11 (A — A)k = 1. i*j

Hence

pi(T)(2/ — T)k + q(T)J -1(A. iI — T)k = I . Obviously,

pi(T) (A iI — T)k (oc ixi) = o. Moreover, it follows from

ocixj = — E oc ixi j*.;

that

q(T)FI (A LI — T)k (oc ixi) = o. j*.; o, we have oci = 0 for j = 1, ..., This proves that ocix, = o. Since xj

3.2.18.* Let T E 2(E) be a Riesz operator. Then, as observed in 3.2.14, the operators / + a with G C enjoy many important properties. In connection with eigenvalues, however, it is more natural to consider the operators AI — T with A e C. The point is that we have to exclude the value A = 0. Then, setting = — 11A, it follows that Nk (AI — T) = N k(I +

and

MI,(A/ — T) = 1I Ii,(I +

.

In particular, by 3.2.4, the multiplicity of every eigenvalue A * 0 is finite.

3.2.19.* Theorem (F. Riesz 1918). Let TE 2(E) be a Riesz operator. Then, for every e is finite. > 0, the set of all eigenvalues A with IA1

3.2. Riesz operators

145

Proof. Suppose that T possesses a sequence of distinct eigenvalues A i , A2, ... such e. Take any sequence of associated eigenelements u1 , u2 , e E. It that i;tk I follows from 3.2.16 that the subspace Ek :=

uk)

span (u 1 ,

is k-dimensional, E0 := [o]. Applying the Riesz lemma with e = 1/3, we may choose elements xk e Ek such that IXkII = 1 and IXk

—

3/4

There exist coefficients Xk — Nkilk

e

for all x e

CCk

E,,_ 1

.

for which

Ek_ .

Hence T nXk —

Nk27,/,/k E

E,,

_

and

1

A'ktxk —

OCkAZUk

E Ek_ 1.

This implies that for n = 1, 2, ...

T nXk — 2 k11Xk E Ek_

Consequently, Tax i, — Tnx k e 2x, — A incxk + Eh_ ,

Ek_ ,

and we obtain

I Tnx h —

TXk

n

whenever h > k and

n = 1, 2, ...

On the other hand, applying the pigeon-hole principle to the operator e choose an exponent n and different indices h and k such that

—

T, we may

Tnx h — Tnx k li This contradiction completes the proof. 3.2.20.* With every Riesz operator Te 2(E) we associate the eigenvalue sequence ( (T)) defined in the following way: (1)

Every eigenvalue A o zf 0 is counted according to its multiplicity. This means that it occurs n(A 0I — T)-times, one after the other.

(2)

The eigenvalues are arranged in order of non-increasing magnitude:

(3)

O. 1 22(ni 1A1(T)1 In case there are distinct eigenvalues having the same modulus these can be written in any order we please. If T possesses less than n eigenvalues A + 0, then 2„(T) := O. Thus, even for finite operators, (2n(T)) is an infinite sequence.

3.2.21.* In order to prove the following theorem, we require an auxiliary result. Lemma. Let T e 2(E) and A + ft. Then Nco (),I — T) M œ (/2I — T). Proof. Given x e N„,(AI — T), there exists an exponent k such that (AI — T )l' x = o. Applying the binomial formula to [Q. — tt) I +

— T)rx = o

146

3. Eigenvalues

and dividing by (A — tt)k * 0, we obtain x = Cut commutes with T. Hence

x = (a/ — T)"S"x

—

T) Sx, where the operator S

for h = 1, 2, ...

This proves that x E M oo (u/ — T). 3.2.22.* We are now able to establish a generalization of the Jordan decomposition, well-known for linear maps on finite dimensional linear spaces; see A.2.6.

Theorem. Let TE 2(E) be a Riesz operator. Then, given any finite set of A„, 0, the Banach space E admits the decomposition distinct eigenvalues A 1 ,

E=

n mo,(21/ —

1=1

T) N(2 1 1 — T) ... N œ (A„,I — T).

Proof. Applying the Riesz decomposition to the operators A iI — T with i = 1, • • in, we may write every element x E E in the form 5

x = xm , i + x" with x m , i E Moo (A il — T)

and

xN,i

— T).

The preceding lemma implies that Xm

= Xm.i

X— i=1

EX N

,

E

Ma (A il — T)

i*j

for j = 1, ..., in. Hence

= xm + E

XN ,

i=l

is the desired decomposition. The uniqueness follows from 3.2.14 and 3.2.21. Remark. Note that every linear map T on a finite dimensional linear space E has at least one eigenvalue. Hence, if {A 1 , ..., A,„} denotes the set of all distinct eigenvalues (possibly including A = 0), then we have the classical Jordan decomposition

E = Noo (A i l — T)

C)

— T).

In the infinite dimensional case the situation is completely different, since then there are Riesz operators without any eigenvalue. As an important example we mention the integration operator

S : g(n) —> ./( ) : = f g(n) dn defined on C(0, 1) or 4(0, 1). 3.2.23.* We now provide a lemma which is an elementary but basic tool in the theory of eigenvalue distributions.

Lemma. Let Te 2(E) be a Riesz operator. If A n(T) * 0, then there exists an n-dimensional T-invariant subspace En such that the operator T„ E 2(E„) induced by T has precisely A l (T), A n(T) as its eigenvalues.

A,„} the set of distinct complex numbers appearing in Proof. Denote by {2. 1 , P. 1 (T), /VT». In particular, we let A„, : = A(T). Since A nd- — T is nilpotent on

3.2. Riesz operators

147

N,,(2„,1 — T), we may choose a basis (x„) of this subspace according to 3.1.15. Let in —1

k := n —

E n(A i l — T).

i=1

Then 1 _. k n(A,I — T), and m-1

En := E _A rco (A iI — T) + span (x i , ..., xk) 1=1

is the T-invariant subspace we are looking for.

3.2.24.* We now prove a special but very precise version of the famous spectral mapping theorem; see (DUN, VII.3.11) and (TAY, V.3.4). Theorem. An operator Te 2(E) is Riesz if and only if Tm is Riesz for some (every) exponent in. In this case, the eigenvalue sequences of T and Tm can be arranged in such a way that (A„(T) m) = (An(T m)) •

Proof. Given e> 0, we put 1

6 :=

e.

11 71m-1 11

If T is Riesz, then there exist n and u1 , ..., uk e E such that k T(U) g_ hU111,1h ± 6"Ul .

Since T mn (U) g IlTm-1 11" T(U),

it follows that

k (T m ) n(U) .-- U {11T m-1 11 " Uh ± 0 U} • h =1

This proves that T m is Riesz for all exponents m. Conversely, assume that T m is Riesz for some m. Given e > 0, we let 6 := Em. Then there exist n and /41 , ..., uk e E such that k

(Tm)" (U) ,E.. U {u„ + 6"U} . h =1

This means that k

Tmn(U) g

U {uh + ennU}.

— h =1

Hence T is Riesz, as well. The remaining part of the assertion follows from Nt„,(y1 — Tm) = N(A i l — T) 0 ... 0 N o,(2.,,,I — T),

where A I , ..., Am are the roots of

and y + O. We first note that (p/ — Tm) = (Arl I + ... +T'" 1 )(A il — T) A m = itt,

148

3. Eigenvalues

implies

for i = 1, ..., m. Next let A be any eigenvalue of the operator induced by T on the invariant subspace Nco (uI — T"). Take any associated eigenelement x. Then we may choose k such x = o, which shows that A must be a root that (4u/ — Ti11` x = o. Hence (IA — of Am = ,u. Thus, in view of 3.2.22 (Remark), Nco (al — Tm) has the Jordan decompoNoo(A,I — T) N oo (uI — Tm)

sition described above.

3.2.25. Next we generalize another classical result. Proposition (J. Schauder 1930). An operator T e t(E) is iteratively compact if and only if T' is iteratively compact. Proof. If T is supposed to be iteratively compact, then for every e > 0 there exist n uh e E such that

and u 1 ,

h

Tn(U) g U {u. + eU} i=1 We may assume that u, = Tx, with x i e U. Define Xe t(li (h), E) by h

e, 0 x i .

X:= i= 1

Since X'(T')n has finite rank, we can choose a1 ,

ak e U° such that

U {X'(T')na ; + eUco (h)}, =1 where U° and co (h) denote the closed unit balls of E' and 1(h), respectively. Note that X'a = (<x1, a>) for a G E'. Given a E U°, there exists ai with X'(T')"(U °)

1<x„ (TT a — (TT ai>1

8

for i = 1, ...,h.

(1)

Next we pick some x e U such that a — (T')" a3 1

(2)

2Kx, (r)" a — ( r)n ai>I.

Finally, we select x, which satisfies

Tnx — Tnx,11

e.

Combining (1), (2) and (3), we obtain

(3)

—

11(T')" a — (TT aI < 21<x, (T)n a — (T')nai> 1 = 2I1 ▪ 2(x 1 , (T') a — (T ai> 1 + 4IITnx —

This proves that (T

(U°) U t(T')nai + 6EU°I

Thus T' is iteratively compact. Conversely, if T' is iteratively compact, then so is T", and it follows from KFT" = (T")" KE that T is iteratively compact, as well.

6e.

3.3. Related operators

149

3.2.26. We conclude with an important theorem which is an easy corollary of the preceding proposition. Theorem (T. T. West 1966). An operator T e WE) is Riesz if and only if T' is Riesz. In this case, the eigenvalue sequences of T and T' can be arranged in such a way that

(A n (T)) = (2(T'))

.

Proof. The first part of the assertion follows from 3.2.13 and 3.2.25. To finish the proof, we observe that

N °JAI' — T') = {a e E' : = 0 for y G M Œ (A1" - T)} . Hence dim [N „JAI' — T')] = codim

°JAI — T)] = dim [N (AI — T)].

This means that T and T' have the same eigenvalues A + 0 with the same multiplicities.

3.3. Related operators 3.3.1.* We begin with some auxiliary results. Lemma (A. Pietsch 1963: a). Let A e t(F, E) and B e t(E, F). Then B[Nk(/E — AB)] = Nk (I F — BA) for k = 1, 2, ... Moreover, n(I E — AB) = n(IF — BA). Proof. We first treat the case k = 1. If x e N(I E — AB), then x = ABx. Hence Bx = BA(Bx), which means that Bx e N (IF — BA). Therefore B[N(/ E — AB)] g N(IF — BA).

Analogously, it follows from y G F - BA) that Ay e N(I E — AB). y = BAy, we have y G B[N(I E - AB)]. Consequently,

Since

N(IE — BA) g B[N(I E — AB)].

If k = 1, 2, ..., then (/E — AB)k = I E - AkB and (4. — BA)k = where

k

Ak :=

E

i=1

k

( -AB) - 'A =

i=1

IF

- BAk,

( k.)A(—BA)i - l. I

Thus, applying the result already verified to the operators Ak and B, we obtain B[Nk (I E — AB)] = Nk(IF — BA).

This implies that dim [Nk(/E — AB)] dim [Nk(/E — BA)]. Interchanging the roles of A and B, we see that in fact equality holds. Hence n(I E — AB) = n(IF — BA).

3.3.2.* Lemma. Let A e t(F, E) and B e 2(E, F). If AB e 2(E) is iteratively compact, then so is BA e t(F).

150

3. Eigenvalues

Proof. Given

E>

0, we put 1

(5 :=

8•

I IA II 11 13 11 Then there exist n and 14 1 , ..., uh e E such that k

(AB)n (U) g_ U {u h + (5U} . h=1

Note that A(V) g IlAil U

and B(U) g 011 V.

We now obtain (BA )' ." (V)

= B(AB)A(V) g IAII B(AB)n (U) k

k

{112411 Bit,, + E U { 1 ,4 11 Buh + 6 11A11 B(U)} -E U1,1

al .

h=1

Hence BA is iteratively compact. 3 • 3 • 3 • * Operators S G 2(E) and T e 2(F) are said to be related if there exist A e 2(F, E) and B e 2(E, F) such that S — AB and T= BA:

S E

E

13\-1 \ .›..11 .

F T

3 • 3 • 4 • * Combining the preceding lemmas with 3.2.13, we obtain an elementary but extremely powerful result. Principle of related operators (A. Pietsch 1963: a). Let SE 2(E) and T e 2(F) be related. If S is Riesz, then so is T. Moreover, both operators have the same non-zero eigenvalues with the same multiplicities.

3.3.5.

Finally, we give the simplest example of related operators. Proposition. Every finite operator T e 2(E) of the form

m

T = L al C) x i with a1 , ..., a m e E' and x 1 , ..., x m e E is related to the operator M. G 2 (1 2(m)) induced by the matrix M : = (<.x. i , a.>). Proof. Define A

m

A

E

2(E, 12 (m)) and Xe 2(12 (m), E) by

:= E ai C) el and i=1

in

X :=

L ei 0 xj . j=1

Then we have T = X A and = <xi , ai>. The latter formula means that AX is indeed induced by the matrix M := (<xi , ai>).

151

3.4. The eigenvalue type of operator ideals

3.4.

The eigenvalue type of operator ideals

3.4.1.* An operator ideal /1 is said to be of eigenvalue type 4,, if, for arbitrary Banach spaces E, all operators T e 11(E) are Riesz and (A(T)) e 4,„ . The collection of these operator ideals is denoted by E,.,. If r = w, then we simply write Er instead of We say that an operator ideal ?I has optimum eigenvalue type /,., if It e Es., but /I Er ,,,. whenever 0 < iv() < w Lç. co. Of course, such an "infimum" (with respect to the lexicographical order of Lorentz sequence spaces) must not exist. However, for numerous concrete operator ideals the best possible eigenvalue type is indeed attained.

3.4.2.* In this section we prove some general facts about the eigenvalue type of operator ideals. To begin with, we state an immediate consequence of 3.2.26. Proposition. Let /I be an operator ideal. Then 91 and /V have the same eigenvalue type.

3•4•3•* Principle of iteration (A. Pietsch 1982: a). An operator ideal It is of eigenvalue type 4,, if and only if /..lm is of eigenvalue type 4.1 „,,,,/ „, for some (every) exponent in. Proof. Given T e 9tm(E), we consider a factorization T=T

T15 where

Tk

e

Ek+i)

for k = 1, • • •, in

and E1 = Enz+1 = E. Form the direct sum F := [12 (m) , E k ] as described in C.2.1, and define S := J1 T mQ m + J2T1 Q1 + ••• + 1 inTm--1Qm-15 where Jk e 2(Ek , F) and Qk E 2(F, Ek) denote the canonical injections and surjections, respectively. Then S E /4F), and we have S(x i ,

•••

7

x,,_1, 'C M)

=

( T M,

T1X1, • • •, T in - 1 X M -

1)

for x 1 e E1 , x„,_ i e Em _ i , X, Em . Observe that E can be identified with the subspace M(./ 1) of F which is invariant under Sm. Moreover, the restriction of Sm to M(Ji ) coincides with T = T m ... T1 . This shows that (A n(T)) is a subsequence of (2„(Sm)). By the spectral mapping theorem 3.2.24, we have ((Sm)) = (A „(S)'n). Hence, if (2,(S)) e ir ,w, then (2 „(T)) e 4/m,,,/,n. This proves that

11 e Er , implies /V e The converse is obvious, by 3.2.24. 3.4.4.

In the following we look for upper bounds for all the operator ideals belonging to Er.w • Proposition. If /I is of eigenvalue type

then It g

Proof. Let T e /4/2). Since the identity operator of /2 fails to be Riesz, it follows from Calkin's theorem 2.11.11 that T must be approximable. Consequently, by 2.11.5 and 2.11.8, there exist X, Y e 2(12) such that D, = Y*TX and T = YD,X*, where the diagonal operator D, is induced by a non-increasing sequence t = (r.) E CO Note that A(D) = -t n . Since T e /4/2), we have D, e 91( 12 ) and therefore t This proves that D, e r,m7(12), which in turn yields T e

152

3. Eigenvalues

Using the fact thatZwr,ewak is the largest extension of 2,. desired result.

we finally obtain the

Remark. Note that fEr , w does not contain a largest operator ideal; see 3.9.1. 3.4.5.

The next result is along the same line.

Proposition (A. Pietsch 1982: a). If 1.1 is of eigenvalue type 1,, then 111 " whenever m r.

Proof. By 3.4.3, we have 91m e E l . Hence it follows from 3.4.4 and 2.11.35 that %in weak V2. 3.4.6.* We now establish one of the most important tools of the theory of ei nvalue distributions. Principle of uniform boundedness (A. Pietsch 1972: b). Let It be a quasiBanach operator ideal which is of eigenvalue type 4.w . Then there exists a constant c 1 such that

cff T 114 for all T Elf(E)

il(An(T)) I 4 ,wil

and arbitrary Banach spaces E. Proof. Assume that the assertion fails. Then we can find operators T,, e tt(Ek) such that

k and

11(A n(T k)) I 1r

II TkI

_ (2c2 )" for k = 1, 2, ...,

where cw denotes the constant in the quasi-triangle inequality of f. f?ilk Define E := [12 , Ek ] as described in C.2.1. Then it follows from CO

k=m+1

that

kTkQk

I IT

E

k=1

19-4

(2c/1) -m for n >

M

co T =

E JkTkQk

k=1

belongs to t(E). Because Tk = Qk TJk , we may identify (2(Tk)) with a subsequence of (2„(T)). This implies that k for k = 1, 2, ... IKAn(TO) J 4-0,11 II(An(T)) I 4,w11 This contradiction proves the existence of the desired constant. 3 • 4 • 7.* In order to prove the next result, we need an auxiliary result of purely algebraic character. Lemma. Let S e 2(E) and Te t(F). Then Tco [(AI E — S)] 0 N 0,[(juIF — T)] N co (AttI E 0 IF — S 0 T) for 2, it E C Proof. Given x e No0 (2/E — S) and y e F — T), there exist h and k such that ( AI E — Sr x = o and (ju/F — T) k y = o. Write in := h + k — 1. Then it follows from for i = h, m, (21E — S) 1 x = o (juIF — T)" y = o for i = 0, ..., h — 1

153

3.4. The eigenvalue type of operator ideals

and 2,1dEoF — S 0

T=

E — S) 0 (pl F) + S 0 (ul F — T)

that (A/ilE®F — S =

xOy =

( m.) (AI E — S) i Sm -ix 0 ( IttIF)i (ittiF — T) m-i y = 0.

I Hence xO ye Nco (A,u/ E /F —SOT). =0

3.4.8.* The following result is very surprising, since it connects two concepts which—at a first glance—have nothing in common with each other. Principle of tensor stability (A. Pietsch 1986). Suppose that the quasi-. Banach operator ideal $t is stable with respect to a tensor norm oc such that jT

6 ,TI S

cTJ t11 2 for all T e ?1(E)

and arbitrary Banach spaces E. If VI is of eigenvalue type /r , then (with the above constant c 1) we have di' I ?ill for all T e tt(E). 11(An(TD I 1,11 Proof. The preceding lemma implies that the double sequence (Ah(T) Ak(T)) can be identified with a subsequence of (2,„(T T )) . By the principle of uniform boundedness, there exists a constant Cr 1 such that crlIS I WM for all S e It(F) I 411 and all Banach spaces F. Taking F := E n=1

1 211(T)0 2/1. =

h=1 k=1 C'AT

E and S := T

I4(T) Aani r

60, T

rir

n=1

T, we obtain

lAn(T 6 -„

TO ".

crclIT

This yields 11(4(n) I 4.11 5- (cr0 1/2 I T I 1111 If Cr is chosen as small as possible, it follows that Cr

(ci.c) 1 / 2 . Thus Cr

C.

3 • 4 • 9 • * Next we show that, roughly speaking, for tensor stable quasi-Banach operator ideals the set of eigenvalue types I,. is closed. Theorem (A. Pietsch 1986). Suppose that the quasi-Banach operator ideal /I is stable with respect to a tensor norm oc such that

TI VI < CT 111112 for all T E QT(E) and arbitrary Banach spaces E. If ?it is of eigenvalue type /,. +E for all e > 0, then it is even of eigenvalue type I,.. Moreover, we have T

II(An(n)

clIT I %II

for all TE /1(E).

Proof. The preceding theorem implies that 11(An(n) I 4+611 :5- cjIT I 1111 for all T e 9t(E). Since the constant c does not depend on E, we may pass to the limit as

E

O.

154

3. Eigenvalues

3.5.

Eigenvalues of Schatten—von Neumann operators

Throughout this section we exclusively consider operators acting on a complex Hilbert space.

3.5.1.*

First of all, we prove the multiplicative Weyl inequality. Lemma (H. Weyl 1949). Let T e (H). Then

n a(T)

11 1),k(T)i k=1

for n = 1, 2, ...

k=1

Proof. Fix any natural number n for which 2(T) l= O. By 3.2.23, there exists an n-dimensional T-invariant subspace H„ such that the operator induced by T has precisely 2 1 (T), ..., A n(T) as its eigenvalues. Let (x„ x„) be any orthonormal basis of H„. Then it follows from 2.11.13 that

12,(T)1 =

H ak (T).

idet ((Tx i ,

k=

k=1

3.5.2.* In order to establish the additive Weyl inequality, we need some purely analytic results. —co and /3, 16 2 oc. Suppose cc 2 Lemma. Let oc, that —

E I3 k

>C k

k=1

for n = 1, 2, ...

k=1

Then n

pi

E exp (13k) _15 E exp (AO k =1 k=1

for n = 1, 2, ...

Proof. On the real line we define the functions =

— r),_

(CX.k

and

bner)

E

:=

—

k= 1

k=1

If 13, > r, then we put in := max {k e Z„ : fik > r}. Now it follows that b„(T)

If yields

= k= 1 ( t3k —

5_ k= 1

—

an(r)-

r, then b(r) = O. Thus we also have +

b „(r)

a:X.0. Integration by parts

CC

exp (") =J( —

exp (r) dr.

Therefore +

11

+

OC

• exp ( 3k) = j

k=1

This completes the proof. 3 • 5 • 3 • * Lemma. Let oc,

b,(r) exP

oc 2

(r) dr

H /3„

H 04 k=1

ciar) exP (t) dx =

- f3 0 and 13, >

77

k=1

00

for n = 1, 2. ...

k=1

exP

k) •

O. Suppose that

3.5. Eigenvalues of Schatten — von Neumann

operators

155

If 0 < r < cc, then

E /31;

k=1

k=1

°C kr

for n = 1, 2, ...

Proo f. Apply the preceding lemma to the sequences

(r log (ock)) 3.5.4.*

and

(r log (8)).

We now establish the additive Weyl inequality. Lemma (H. Weyl 1949). Let Te 6(H) and 0 < r < co. Then

E lAk k=1

(nr

ak (T)r for n = 1, 2, ... k=1

Proof. Obviously, the assertion follows immediately from 3.5.1 and 3.5.3. Since this result is the keystone for most of the further investigation, we give here r < co. Fix any natuanother proof which works, however, only for the case 1 which A n (T ) * O. By 3.2.23, there exists an n-dimensional T-invan for ral number riant subspace Hn such that the operator Tn e t(H„) induced by T has precisely x) as described n (T) as its eigenvalues. Consider any Schur basis (x 1 , Al(T), in A.2.5. This means that ((Tx, x i)) is a triangular matrix. Hence, by A.5.5, (Tx , , x,), (Tx„, x n) are the eigenvalues of T. Now it follows from 2.11.12 that

E 2k(T)I r = E i(Tx,„ k=1

k=1

E ak(T )r

k=1

3 • 5 • 5 • * We are now in a position to establish the main result of this section which was the starting point of the theory of eigenvalue distribution of abstract operators. Theorem (H. Weyl 1949). The operator ideal Z„ is of optimum eigenvalue

type

4. Moreover,

11(An(n) 1 411 .< 11T 1 2,11 Proof. The result follows from ,• (T )I r k1

for all T e

5 y ak(T ) r k=1

cc. by passing to the limit as n r < co. Note that the second proof of the preceding lemma requires that 1 In order to treat the case 0 < r < 1 with this method, too, we choose a natural number m such that m r 1. Then Z., is of eigenvalue type /.„, and the assertion can be inferred from the principle of iteration 3.4.3. Furthermore,

11(2.(T))1 411 < 171 2,11 for all Te Z r(11) follows by means of tensor stability arguments; see 2.11.22 and 3.4.8. ... r2 If D, e t(l2) is a diagonal operator induced by t = ( r„) with r then 2„(Dt) = r„ = an(D t). This shows that 2, cannot have any eigenvalue type better than 1,..

3.5.6.* result.

In view of 2.11.17, we obtain an important corollary of the preceding

156

3. Eigenvalues

Eigenvalue theorem for Hilbert-Schmidt operators (I. Schur 1909: a, T. Carleman 1921). The ideal of Hilbert-Schmidt operators is of optimum eigenvalue type 12 . Moreover,

IIT I ell for all T E Z(H).

(An(T)) I 12 1

3 • 5 • 7 • * The Weyl theorem 3.5.5 admits the following generalization. Eigenvalue theorem for Schatten-von Neumann operators. The operator ideal Z,.,, is of optimum eigenvalue type Proof. The assertion follows from 3.5.1 and 2.1.8, or 3.5.4 and 2.1.7. Diagonal operators on /2 show that this result is sharp.

3.6.

Eigenvalues of s type operators -

3.6.1.* The following substitute for the multiplicative Weyl inequality is the key to almost all results about eigenvalue distributions of operators on Banach spaces. Lemma (A. Pietsch 1980: c). Let Te t(E) be a Riesz operator. Then I n

e

IA2 7 -1(T)I

.,(T)] n. k=

1/n

for n = 1,2, ...

Proof. Fix any natural number n for which A2n-1(T) 0. By 3.2.23, we may choose a (2n — 1)-dimensional T-invariant subspace En such that the operator 2211 (T) as its eigenvalues. In view T t(E„) induced by T has precisely A l (T), of 1.2.8 and 1.5.1, the identity operator I„ of En admits a factorization I„ = XnA n , where

A n e t(E„, 12 (2n — 1)) and X„ e 2(12 (2n 1), E„) and —

ilAn V2Il 5 (217 — 1) 1 / 2 , 11X„ I = 1 .

Since T„ = X„A„T„ and S, := A n T„X„ are related, both operators have the same eigenvalues. Therefore 2n —1

FI

12L211--i(T)1 2 "-1

k=1

2n-1

12k(1)1 =

H

k =1

2n — 1

lAk(T,i)1 =

H

k =1

12k(s„)I.

By 2.4.20 or 2.11.9, the multiplicative Weyl inequality for S„ E 2 (12(2n — 1)) reads 2n —1

n 12k(so1 k=1

2n- 1 k =1

Xk(S n) •

Recall that the Weyl numbers are multiplicative. Hence, applying 2.7.8, we obtain lA2 n _1(T)1

1 • exp ( 2n —

[n IA N I *211 2)

n 1xkv„x-n)

1/n

▪ e[

x,(T„) k

1/n

1/n

e [kil =i xk(T)]

.

=

3.6.2.* Within the theory of abstract operators on Banach spaces, the following generalization of the classical Weyl theorem 3.5.5 is the most important result of this monograph.

3.6. Eigenvalues of s-type operators

157

Eigenvalue theorem for Weyl operators (A. Pietsch 1980: c). The operator ideal 2;.-,'`2 is of optimum eigenvalue type irov•

Proof. Choose any natural number m with 2m > r. Then it follows from 2.4.18 and 2.2.7 that

T e ,e(E) implies

r n e 2;.-g,„",„(E)

eSx)(E).

Thus T'n is compact, by 2.10.8. This proves that T is a Riesz operator. Finally, combining 3.6.1 with 2.1.8 and 2.1.9, we conclude that

(x„(T)) e 4,, implies

(2„(T)) E ir,w •

Diagonal operators on /2 show that this result is sharp. Remark. According to the principle of uniform boundedness, there exists a constant 1 such that Cr

00

1 lr

llr oo X 5_ Cr (E1 n(T )) 2 n E 1 1 2 n(T )1)

for all T e (,? ) (E)

(

and arbitrary Banach spaces E. Suppose that c,. is chosen as small as possible. Then the following estimate is known from H. Kiinig (1984:b): 1 hr / _-__ vC (m + 1) e elt"'r for m = 1, 2, ... r ._ NI (m + 1) e (1 + — m

However, it is an open problem whether or not Cr —* co as r —> 0. 3.6.3. It seems to be worth while to formulate the following corollary of the preceding result.

Theorem (H. KÉinig 1978). The operator ideals 2 eigenvalue type l,..

.ci .

4, and 2

1 are of optimum

3.6.4. Let 0 < w < oo. A quasi-Banach operator ideal /1 is said to be of uniform eigenvalue type /r,w if it is of eigenvalue type l,.,,, such that the series cc

E [niir - lhv

n=1

I An(T)11"

converge uniformly on every precompact subset of any component t(E). This means that for every e > 0 there exists n o such that i E rn i/r_i/.1 A n (T)liw )1 I w .15_ kn- no

E

whenever T e.

Remark. We conjecture that this condition is automatically satisfied for all quasiBanach operator ideals Vt e Er,„.. Using the concept just introduced, we now improve theorem 3.6.2. Proposition (H. Leiterer / A. Pietsch 1982). If 0 < w < oo, then the quasiBanach operator ideal e;.'`)„ is of uniform eigenvalue type /r , w . Proof. We know from 2.2.6 that 11 pilv)1.1w co 1 n 3.6.5.

T

r an := E ni/r_i/w I 2(1,?v1I r n=1

11 k=1

)

defines an equivalent quasi-norm on 21.,x,;, provided that 0 <

p < r.

158

3. Eigenvalues

Let be any precompact subset of 21;x2(E). Given E > 0, we choose a finite e-net T e 2(E). Then there exists n o such that of operators 7'1 ,

1

\n

_?.no

[ 11 1Ir–liw (—

w w< 11 En XaTiY)) 111)

E for

n k=1

i = 1, ...,

Next, for any TE, we pick Ti with II T — Tii (n

E xX(T)" )

1/p

:5_

2 11Pc,[( E

E.

Note that

xk (T - TOPY /p + ( ki , X k(T 'i)12)

11p ,

k =1

k =1

where cp := max (2 1 'P -1 , 1) is the constant in the quasi-triangle inequality of • I /p(n)II . Hence nl/r-l/w

no

[

- En xk (T)

1w )11piw)1

71 k = 1

< 211PcPcw

—

[n

1 ,k(T)p) 11p1 w) n k-=1

l/r-l/wF E

n L. n o

< 2 1 + 1 /Pcpcws. Applying the inequality of means, we deduce from 3.6.1 that 1

IÀ 2 n– 1(T)I

11P

n

xk(T)P) . e (— n k1 =

Therefore

(-\n E- n o [f ur-11W I 22,---1(nllwrw

< 2'+'/Pcpcwee.

Finally, reasoning as in 2.1.9, we obtain [nl ir-i lw I A n(T)Irriw < ce

for all T

,

2n o -1

where the constant c > 0 depends on p, r and w. This completes the proof. 3.6.6. Theorem. Let 0 < r < 2 and 1/s = 1/r — 1/2. Then the operator ideal 2, is of optimum eigenvalue type 2. 2,çh2 indeed has eigenvalue type Proof. By 2.10.7, we have Furthermore, Example 2.9.20 show; that this type is the best possible. 3.6.7. Theorem (H. Ki5nig 1979). Let 0 < r < 2 and 1/s = 1/r — 1/2. Then the ec k is of optimum eigenvalue type operator ideal Proof. The result follows from 2.11.34 and 3.6.6.

3.7.

Eigenvalues of absolutely summing operators

3.7.1*. To begin with, we establish the most elementary result about eigenvalue distributions of absolutely summing operators. Theorem (A. Pietsch 1963: a, 1967 and 1972: b). If 1 operator ideal 43,. is of optimum eigenvalue type 12 . Moreover, I 1 2II

IIT I WI

for all Te

r

2, then the

3.7. Eigenvalues of absolutely summing operators

159

Proof. We know from 13.3 that V,. g 432. Therefore it is enough to treat the operator ideal 132 . Let TE 9, 2 (E), and consider a factorization T = X A, where A c *2 (E, H), X e 2(H, E) and ll XII ll A I 4-$4 211 = 11T 4,4 211 • It follows from 1.4.5 that S : = AX is a Hilbert-Schmidt operator. Hence the principle of related operators combined with 3.5.6 yields ll(An(T» I /2II = II(An(S)) I /211 <=. S I

ll

IIA I4;21 IIXII = IT l

By 1.3.16 and 1.4.7, we have

D, c 43,(12) if and only if t E /2 . Consequently, /2 is indeed the best possible eigenvalue type of 3.7.2.*

Next we establish a profound counterpart of the preceding result.

Eigenvalue theorem for absolutely r-summing operators (W. B. Johnson / H. K6nig / B. Mauey / J. R. Retherford 1979). If 2 r < cc, then the operator ideal V, is of optimum eigenvalue type /,.. Moreover, for all TE ,.(E) TI Proof. By 1.2.4 and 2.7.4, we have Vt. 2 x,20 . Hence it follows from 3.6.2 Vr,2 that 43 r G Er . This implies that V, is of eigenvalue type for all c > 0.Furthermore, as stated in 1.3.11, the Banach operator ideal V,. is stable with respect to the tensor norm 8. Therefore, in view of 3.4.9, we obtain ll(AH(T))

IT I q,.ii

11(2.(T)) l /id

for all Te V r(E).

It follows immediately from 1.6.3 that

D,

V,.(k)

if and only if t e 'r.

Thus V, cannot have any eigenvalue type better than I, Remark. If r = 2m is an even number, then 1.3.10 implies that (V2 ni)'' g 4;2• Therefore the result can be deduced from 3.7.1 and the principle of iteration 3.4.3. 3.7.3.

The previous theorem admits the following generalization.

Theorem (W. B. Johnson / H. K6nig I B. Maurey / J. R. Retherford 1979, r1 , r < co and 1/r = 1/r 1 + + 1/r n1 . (Then the A. Pietsch 1986). Let 2 operator ideal 43,. 1 Vrn, is of optimum eigenvalue type 1, . Moreover, II( 2 ,,(n) I 411

°Vrm il for all T e

'IT I Vr i °

Proof. Reasoning as in the preceding paragraph, the assertion follows from

V r1 ° • • ° r-rm =

2(X) ri,co

0

ex) co

2.1c20

see 2.4.18. = r = 2, then the result can be deduced from 3.7.1 and the Remark. If r 1 = principle of iteration 3.4.3. 3.7.4. Theorem (A. Pietsch 1981: b). The operator ideal (43 2)(sa.4, is of optimum eigenvalue type 4.,,„ where 1/r = 1/s + 1/2.

160

3. Eigenvalues

Proof. We know from 2.8.18 that (V2 ) a„), tkc,),. Hence ($2)1,a,), has eigenvalue type Recall that 432(H) = tSa)(H) for all Hilbert spaces H. Thus the reiteration theorem 2.8.13 yields

(V2)(H) = This implies that 4,, is indeed the best possible eigenvalue type of (432). The theorem can be proved by a completely different technique. For this purpose, we consider arbitrary decompositions 1/s = 1/p + 1/q

and

1 1 w = 1/u + 1/v.

By 2.8.21, every operator Te (V2)(E) admits a factorization T = BA such that A E (432)(pa, E, H) (

and

B e 2(qa3,(H, E),

where H is a Hilbert space. Applying 2.8.7, we obtain S := AB e (432)(pa,)„ o eqa3,(1/) g (13 2)(H)

Since S and T are related, 3.3.4 and 3.5.7 imply ((T)) e 3.7.5.

4.

Using the preceding result, we now give another proof of 3.6.3. Theorem. The operator ideals 2;.a?„ and 2;.;) are of optimum eigenvalue

type Proof. Choose any natural number m such that 2m > r. Then, in view of 2.4.9 and 2.8.18, we have (9 '(a) (2)w) m

ilj;i„

where 1/s := ink — 1/2. Hence (2.;.col,) 1" e Eon,„/„„ and the' principle of iteration implies that 2 6:4, e E,., 3.7.6. Eigenvalue theorem for absolutely (r,2)-summing operators (H. Kiinig / J. R.

Retherford / N. Tomczak-Jaegerman 1980). Let 2 < r < c/o. Then the operator ideal 43r, 2 is of optimum eigenvalue type 4,co• Proof. It follows from 2.7.4 that Vi.. 2 g • Therefore 4 r ,2 is of eigenvalue type /,.. co , by 3.6.2. We now consider the function f,37:0 E L,,,,,e (27C) defined in 6.5.8. Then, as shown in the proof of 6.5.9, the corresponding convolution operator is absolutely (r, 2)summing on Lo9 (27t). Furthermore, we have (yk(f,3,70)) 1,..,,(1) for 0 < w < cc. This proves that the above result is sharp.

3.8.

Eigenvalues of nuclear operators

3.8.1.

We begin with a classical result. Eigenvalue theorem for nuclear operators (GRO, Chap. II, p. 17). The

operator ideal 91 is of optimum eigenvalue type 12. Moreover,

11T1 91 1 for all T e91(E). 11(An(T))1 /211 Proof. Obviously, 91 e [2 can be deduced immediately from 91 g V2 and 4:4 2 e E,. We prefer, however, to give a more direct proof of this important fact.

3.8. Eigenvalues of

Let T E 91(E) and

nuclear operators

161

8 > O. Then there exists

a representation

00

T = E ai 0 x e

i=t

such that =

!lad 1141 5_ (1 +

111'1%11.

Without loss of generality, we may assume that I1x1I1la1IJ. Define A E t(E, 12) and X E 2(12 E) by co co A := ai 0 ei and X := E 0 X. j=1

Note that / co co

\ 1/2

/ co

2

\ 1/2 ( co

\ 1/2

• 11Xj 1 ) E j=1 1=1 Since S := AX E 2 (/ 2) is induced by the matrix M := (<xi , ai>), it must be a HilbertSchmidt operator. Therefore the principle of related operators combined with 3.5.6 i=i

E 1<xi,ai>1 2)1

yields

Ils' I ell 5_ + IIT 1%11. 11(2,(T) ) 1/211 = 11(2n(s)) 1/211 The example given in the following paragraph shows that the operator ideal 91 cannot have any better eigenvalue type than /2 ; see also 6.5.11 and 6.6.12. The following construction is analogous to that carried out in 3.4.3. Example (R. J. Kaiser /J. R. Retherford 1983, 1984). Given (A n) E 12 with 1„) having precisely +2 1 , +22 , ... as * 0, there exists an operator T e 91(4

3.8.2.

its eigenvalues. Pro of. By 1.7.4, we can find a factorization. (A n) = (o „) with (oc) E co and

(j9n) E l2.

Let s := (4) and t := (IV . It follows from 1.7.3 and 1.7.19 that D, E 91(ii , 1„) and De e 91(1,0 , /1 ). Therefore T := .1„D,Q1 J1 D,Q„ defines a nuclear operator on /1 T(x, y) = (Dey, D,x)

lc° . Note that

for x e /1 and y e 1„.

In particular, we have = ±11413nen, ±ocnen)• Thus -1-2„ is an eigenvalue of T. TOnen,

Finally, an easy calculation shows that T cannot have any other eigenvalue than

+21, + 22, •••

162

3. Eigenvalues

Remark. Very recently G. Pisier (1983) constructed a sophisticated Banach space E on which all approximable operators are indeed nuclear. Using this fact, R. J. Kaiser / J. R. Retherford (1983) were able to find an operator T E 91(E) having precisely the prescribed eigenvalues A l , 22, ... provided that (A n) E 1 2 and A n * O. Since 91(H)

3.8.3.

= 2 1 (H),

11(2n(T))

we see from 3.5.5 that

I T 194 for all

TE

This fact indicates that the preceding theorem can be improved for nuclear operators acting on special Banach spaces. Let 2

Proposition (W. B. Johnson / H. K6nig / B. Maurey /J. R. Retherford 1979). p < co and 1/r = 1/p + 1/2. Then

IIT1 91 11 for all T e 91(Lp) . 11(2.(T)) 1 411 Proof. Given E > 0, we consider a factorization Lip

Li,

A A 11

X

le

such that

11x11 1Ix 1 12 II lai 1 211 11A11

(1 + 8)111'194 have XD x e 43;,(12 , Lp).

Since Dx e $;,(/2 , /1 ), we Therefore 1.3.13 implies that Hence XD x E 43p(l 2 La). Furthermore, we know from 1.6.3 that D a e 42(i,

T = (XD) (DaA) e 43p 43 2 (4) For the corresponding quasi-norms we obtain the estimate IT 1 Vp 0 43211

IIXDx I VpIl 11DaA I 43211 (1 + e) 11T 19111 .

11 ,111111x 1 1211 Ila I /21111A11

Applying 3.7.3 and letting e —> 0, we finally arrive at the desired result. Remark. A Banach space E is said to be 0-hilbertian, 0 < 0 < 1, if there exists a Banach interpolation couple (E0 , H), where H is a Hilbert space, such that E can be obtained by complex interpolation: E = (E0 , H)0 . For example, we have Li, = (Lœ , L 2)0 with 0 := 2Ip. If E possesses this property and 1/r = (1 + 0) 1 2, then T e91(E) implies (2,(T)) e 4, see G. Pisier (1979).

3.8.4.

We now state, without proof, a certain converse of the preceding result. Proposition (W. B. Johnson / H. K6nig I B. Maurey /J. R. Retherford 1979).

3.8. Eigenvalues of nuclear operators

163

Every Banach space E with the property that

(2,„(T)) e l

for all T E 91(E)

is isomorphic to a Hilbert space. Remark. Let 1 < r < 2. It would be interesting to characterize the class of those Banach spaces E such that

(2,„(T)) e Ir for all T E Partial results were obtained by H. Ktinig / J. R. Retherford / N. Tomczak-Jaegermann (1980). 3.8.5. Eigenvalue theorem for p-nuclear operators (H. Klinig 1977). Let 0 < p < 1 and 1/r = 1/p — 1/2. Then the operator ideal 94 is of optimum eigenvalue type lr,p . Proof. In view of 1.7.3, every operator T E 91„(E) factors through a diagonal operator Dt E e(/ 0„ /1 ) with t E I„. By 2.9.17, we have Dt E , /1 ). Hence T e 2(E). This proves that 94 is of eigenvalue type /,.,,„. Let 0 < w < p. Then, for the weighted Littlewood-Walsh matrix W,. with a : = 1/w defined in 5.2.3, we have (Wr,)op 6 94(4) and (An( Thus 91p e Er, p is the best possible result.

st

r ov •

3.8.6. Eigenvalue theorem for (r,2)-nuclear operators (P. Saphar 1966, B. Carl 1976). Let 0 < r 2. Then the operator ideal 94, 2 is of optimum eigenvalue type 'r' Moreover, II(An(T)) I IITI9tr,211 for all T e Proof. Define 1/p := 11r — 1/2. Given e > 0, by 1.7.10, there exists a factorization

X

A

10

,

Dt

such that

11X11 Is! 4,11 liai /211 11A11

(1 +

11T1 Ttr,211

By 1.6.3, we have Da e $2 (/,„ /2). Therefore it follows from 1.4.5 and 2.11.17 that D„AX 22 (l2 ). Consequently, by 2.11.23, S := DaAX-D, E 22 ° 2p(12) er( 12)• Furthermore, I IS I

rIl

IIDaAX I

2li lips I Zpl!

lIa 1 1211

I •II Ils I 1211 •

164

3. Eigenvalues

Next we deduce from the principle of related operators and 3.5.5 that

411 = 11(),,1(s)) 1 411 5_ Ils 1 erll

li(2 ,(T)) I

+

IIT 04,211.

Letting e --> 0 now yields the inequality we are looking for. Hence 9'4, 2 is of eigenvalue type 4. Finally, since 3 r , 2 (H) = Z,.(1/), this is the best possible result. 3.9.

The eigenvalue type of sums of operator ideals

Then Suppose that the operator ideals 9t and 3 have eigenvalue type we are interested to get some information about the eigenvalue type of tt + $. A complete answer to this problem is still open. However, as the following examples show, the leading index r may change significantly for the worse. As a consequence of this observation we see that the class Er does not contain a largest operator ideal. 3.9.1.

, v

3.9.2.

Lemma. If

+ e E, then It o

E-

-r/ 2, w/ 2 •

Proof. In view of the principle of iteration 3.4.3, the assertion follows from

3.9.3.

(w

(11

21 0

$).

We are now in a position to give the announced examples.

Theorem. The operator ideal ($ 2)m +

(415 Dm

is of optimum eigenvalue

type /4,(2m-1)• Proof. We know from 1.5.10 that (4)m 02) m

(C)my.

(2)fh_1. Hence

(* 2 )2 m-1

By 3.4.3 and 3.7.3, this implies that (32)' + (C )fl indeed has the eigenvalue type /41(2m-1) • Recall from 1.6.9 and 1.6.10 that Da E (4.Z1) m (11, /2)

if and only if a E

Db e (V2)" (i., /2)

if and only if b e 1 2/,•

and Consequently, (V ) m (/2 )

D, e

if and only if -t e- /-2/(2m-1)•

Thus, in view of Lemma 3.9.2, the sum ($ 2 )m + (C)"' cannot have any eigenvalue type better than 14/(2 nt _ 1) .

Theorem. Let 0 < r < 2 and 1/r0 = 1/r — 1/4. Then the operator ideal 3.9.4. ton + 22 is of optimum eigenvalue type 4.0,. Proof. Define 1/s: = 1/r — 1/2. Then 2/r 0 = 1/r + 1/s, and it follows from 2.4.18, 2.5.12 and 2.10.7 that 2(x) 0 2(x) r, w 0 2(x)

2(x)

r, w = r e(y)

; r, w = r r, w t(x) 0 ^(x) 9(Y) 0 e(x) (x)

e(Y) 0 e(Y) r, w r, w

t(y)

r 12, wf 2

(x)

9(x) r0 /2, wf2

z, r0/2, w/2 r w/ 2 V(x) = "- r0 / 2. 42 •

3.9. The eigenvalue type of sums of operator ideals

165

Hence -

c= t14)0( 2,w/2*

P

By 3.4.3 and 3.6.2, this implies that 2;.x)„ + 2;.):,)„ indeed has the eigenvalue type Recall from 2.9.17 and 2.9.18 that Da E tx,?„(/ 1 , /2)

if and only if a E

e 2x,4,(/co , /2)

if and only if b e

iro,„,•

and Db

Consequently, D, e tx?„ tn,(/2)

if and only if t e /ro/2,„,i2 .

Thus, in view of Lemma 3.9.2, the sum 2."0),, + tY)„ cannot have any eigenvalue type better than irosw r < co the optimum eigenvalue type of t(rx,), + t.32, is Remark. In the case 2 unknown.

3.9.5.

The following example is analogous to the preceding one.

Theorem. Let 1/r0 = 1/s + 1/4. Then the operator ideal (432)1,e:?, + (4,ZA,1201, is of optimum eigenvalue type 1,.00 ,. Proof. Let 1/r := 1/s + 1/2. Then 1/r 0 = 1/r — 1/4, and we obtain from 2.8.18 and 2.5.12 that

0432riv + (V ) cav

+ tYL.

This proves that the operator ideal under consideration indeed has the eigenvalue type lro,w• Reasoning as in the proof of 3.9.4, we may infer from Examples 2.9.15 and 2.9.16 that this result is the best possible.

3.9.6. r.2

Theorem. Let 0 < r < 2 and 1/r 0 = 1/r — 1/4. Then the operator ideal 94,2 is of optimum eigenvalue type 1ro,"

Proof. We deduce from 1.7.10, 2.9.17 and 2.5.12 that

tJ

+

r ,2

tY:r •

Hence 91 r,2 ± 94.2 has the eigenvalue type /ro ,rDefine 1/s := lir — 1/2. We state, without proof, that and

Da E Ter ,2( 11 12)

if and only if a E

Db G 91r, 2(/00 12)

if and only if b E

4,,

Consequently, Dr E 91r, 2 °

r, 2( 12)

if and only if t E 40 /2, ,,2

Thus, in view of Lemma 3.9.2, the sum 9 better than /r,,,r•

3.9.7.

, ,2 r,2

91-;,2 cannot have any eigenvalue type

We conclude with a general result due to B. Carl (unpublished).

166

3. Eigenvalues

Proposition. Let /I and $ be operator ideals. Then + $ E E2,.q„ for £ > O.

$ e Er implies

Proof. By the principle of iteration, we have m

E Erim .

It now follows from 3.4.4 that (91

13) 2 M

rit

Vri

7,• 2 :":7k

In view of 3.6.7, this implies that + $Y. ni E Es12m,x 5

where s is defined by 2m/s = mIr — 1/2.

Choosing m so large that 1/s = 1/2r — 1/4m > 11(2r + E)

we finally obtain + Q3 e E.,,,c

E 2r -Fe•

Remark. We conjecture that $ e Er implies ti + 8 e E5 ,

where 1/s := max (1/2r, 1/r — 1/4).

CHAPTER 4

Traces and determinants In this chapter we develop an axiomatic theory of traces and determinants defined on certain operator ideals. Having in mind the classical formulas trace (M) =

E A i and

i=i

det (AI

M) =11(2 — i=1

which hold for every (n, n)-matrix M, our main goal is to investigate the relationship between traces, determinants and eigenvalues. When passing to the limit as n oo, it turns out to be advantageous to replace the complex parameter A by = —1/2 (the minus sign is a matter of taste). The historical starting point was the theory of nuclear operators. In view of the finite dimensional setting, it seems natural to define the trace of a nuclear operator CO

T=

E ai

1=1

0 xi

by 00

-1-(T) =

E <xi , a.>.

i= 1

However, a counterexample constructed by P. Enflo (1973) shows that, in certain Banach spaces, this value depends on the underlying representation. In fact, the situation is even more confused than this. It turns out that on the Banach ideal of nuclear operators (formerly called trace class operators) there does not exist any continuous trace. Even when restricting our considerations to good-natured Banach spaces, including co or 1,, we cannot avoid ill-natured operators. Thus the operator ideal 91 is inappropriate for the purpose it was originally designed for. As a consequence we observe that a substantial trace theory holds only on smaller operator ideals such as tça) , (4; 2) 2 and ( 4 2 ) 2 . In particular, due to the fact that 91 is the smallest Banach operator ideal, we are necessarily led to the context of quasiBanach operator ideals. We begin this chapter with some preliminaries concerning the Fredholm resolvent F( , T) := T(I + of a Riesz operator T which is a meromorphic operator-valued function of the complex parameter Next an axiomatic theory of traces is established. We first deal with the ideal of finite operators. Subsequently, the trace obtained in this elementary case is extended continuously to larger quasi-Banach operator ideals. Every quasi-Banach operator ideal ?t admitting a continuous trace r is of eigenvalue type 1 2 . This means that such operator ideals are comparatively small. On the other hand, they are so large that the sum of the eigenvalue sequence (2 „(T)) need not converge for all operators T E 9t(E). However, if It is of eigenvalue type /1 , then it may happen that the trace formula holds:

co

r(T) =

E A(T).

n= 1

We give several examples of quasi-Banach operator ideals having a continuous trace with this property.

168

4. T races and determinants

By analogy with the theory of traces, we develop an axiomatic theory of determinants. Given any continuous determinant 6 defined on a quasi-Banach operator ideal ?I, the associated Fredholm denominator

6(C, T) := b(I + a') for all E C is an entire function whose zeros Co are related to the eigenvalues 2 0 of the operator T e It(E) via C = —1/Ao . Moreover, there exists an entire operator-valued function T) such that the Fredholm resolvent of T admits the representation

FG, T) =

,

T) for all

E

Using a regularization process, Fredholm denominators can also be obtained for operators T E t(E) possessing a power Tm e t(E). One of the basic results of this chapter is the fact that, on every quasi-Banach operator ideal, there exists a one-to-one correspondence between continuous traces and continuous determinants. Finally, we present the strange phenomena which can occur in the trace theory of nuclear operators on Banach spaces. It is further shown that all the trouble disappears for the ideal of (1,2)-nuclear operators. The last section is an appendix containing some basic results from the theory of entire functions.

4.1.

Fredholm resolvents

4.1.1. The Fredholm resolvent set e (T) of an operator TE t(E) consists of all complex numbers C for which I + a' is invertible. Then T) := T (I + a)-- '

for all C e e (T)

is called the Fredholm resolvent. Note that F(C, T) and T commute. Remark. The advantage of T(I + CT) --1 over (I + a').-1 is that, for any operator ideal ?I, we have T(I + e 11(E) whenever TE 9 (E). Furthermore, if t is endowed with a quasi-norm, then the Taylor and Laurent expansions considered below converge in the corresponding topology.

4.1.2.

First of all, we state an elementary criterion.

Lemma. A complex number C belongs to the Fredholm resolvent set of T e 2 (E) if and only if there exists F(C, T) e t(E) commuting with T such that

(1+

F(C, T) = T.

When this is so, then

(I + 4.1.3.

=I—

T)

and

T(I +

=

T).

The next statement is routine, as well. Proposition (RIE, p. 117). For every operator T e 2(E) the 2(E)-valued

function

—> F(C , T) is holomorphic on e(T). More precisely, given C0 e e(T), there exists e > 0 such that

169

4.1 . Fredholm resolvents

IC — CI <s implies C e 9(T), and we have the Taylor expansion 00

F(C, T) =

n =o

(— 1)" F(C0 ,

2+

—

.

4.1.4. Let T e t(E) be a Riesz operator. Then, according to the decomposition theorem 3.2.14, for every complex number we have

E = „(I +CT) CI N(' + CT). The projection from E onto N(/ + CT) along M(/ + CT) is denoted by P,(I + CT). 4.1.5.

We now consider the case in which the above decomposition is non-trivial.

Proposition (F. Riesz 1918). Let T e 2 (E) be a Riesz operator. Then 0 2(T) if and only if 20 = —1/C 0 is an eigenvalue of T. Proof. If 20 is not an eigenvalue, then N „(I + Co T) = {o}. Therefore I + Co T must be invertible on E = M „(I +C0 T). On the other hand, if 20 is an eigenvalue, then I + Co T fails to be one-to-one. 4.1.6.

We are now in a position to establish the main result of this section.

Theorem (A. F. Ruston 1954). For every Riesz operator Te t(E) the 2(E)-valued function

-+ F(C, T) is meromorphic on the entire complex plane with poles located on the complement of o(T). More precisely, given C o 0 p(T), there exists e> 0 such that 0 < IC — C01 < E implies C e 2(T), and we have the Laurent expansion

T) =

d 17=

X„Go, 1t—

n n =o

where the coefficients X2 (C 0 , T),

R(Co, T) (C —

Xd(Co , T) are nilpotent finite operators and

X,(Co , T) = P,(I + C0 T). Moreover, the order of the pole at C o is equal to d(I + C0 T). Proof. Applying 3.2.11 to the operator — Co T, we write

M := M,(/ +

and

N := N,(1- +

P := P,(I + :0 T).

Note that T = Tm + TN

with T := (I — P) T

and

TN := PT.

We further know that the operator S : = P(I + 0 T) is nilpotent of order d := d(I +C0 T). Define

:= P and X,, := [Co S(I + S +

+ Sd--1 )]"--1 for n = 2, 3, ...

Since Sci = 0, we have

Xcd = 0 for n = 2,

d and

X„ = 0

for n = d + 1, d +2, ...

170

4. Traces and determinants

Moreover, it follows from I = I — S = (I — S) (I + S + + Sd-1 )

and

P(I — S) = — CoT N

that

.

P = — CoT N(I + S + + Hence S= —T NX 2

.

This in turn implies the recurrence formula

T N X fl+l = T NX1 = —SX 1 — SX„ . Put d —

Co

for )"

Then (I + T

, T N) = (P +TN) F(, N) = [S + — o) T1] FG N) d-1 SA',, + T NX.+1 Sdird +ET

(C — Co

) fl

+—

= TN.

This proves that (I

±

N) F( ,

= TN. Since F(C, T N ) and TN commute, it follows from 4.1.2 that Re,- To is indeed the Fredholm resolvent of TN at On the other hand, we have o e (TM). Therefore, by 4.1.3, there exists e> 0 such that IC — Col <e implies e (TM ) and TN)

,

F(C, TM) =n;( -1 ) " F(0, TAX .° — Co) n • •

Since Tm T N = 0 and TN Tm = 0, we have TmF(,TN) = 0 and T N F(, TM) = 0. Hence [I + C(T M + TN)] [F( , TM) + F( , TN )] = T + TN.

This proves that

F( , T) := F( , TM) + F( ,

N)

is the Fredholm resolvent of T at whenever 0 <

I<

8.

Remarks. Observe that the principal part in the above Laurent expansion is determined by TN while the regular part depends only on TM . We stress the fact that, conversely, an operator T e t(E) is Riesz if it has a meromorphic Fredholm resolvent such that all residues are finite operators; see (CAR, 3.4) and (DOW, p. 78).

4.2.

Traces By a trace r defined on an operator ideal 94 we mean a function which

4.2.1. assigns to every operator T E /1(E) a complex number r(T) such that the following

171

4.2. Traces

conditions are satisfied: (T1) (T2) (T3) (T4)

r(a C) =-- <x, a> for a e E' and x e E. r(XT) = r(TX) for T e WE, F) and X e t(F, E). r(S + T) = r(S) + r(T) for S, T e W(E). r(.1.7') = 2T(T) for T e W(E) and 2 e C.

4.2.2.

We now define the trace of finite operators. Lemma. Let Te 171

trace (T) := E

1=1

a(E). Then

<xi , ai>

does not depend on the special choice of the representation

T => ai C) x i . i= Proof. Given any representation in

T=

E ai C) xi ,

i=1

we choose linearly independent elements yi ,

y„ e E such that

x„, e span (Yi, yn). Then there exists a representation X1 ,

T=

E bh 0 Yiz •

h=1

Writing Xi

E $107h,

h=1

we conclude that the functionals 1) 1 ,

b„ E E' are uniquely determined by

in

bh = E

$ihai

Therefore

E <x1 , ai > = E E $ih = • i=1 h=1 h=1

i=1

The final conclusion now follows from a simultaneous application of this construction to two arbitrary representations.

4.2.3. ideal

a.

Theorem (A. Pietsch 1981: b). There exists a unique trace on the operator

Proof. For every finite operator

T=

E ai ®

i= 1

172

4. Traces and determinants

we define 111

trace (T) :=

E <xi , ai > .

i=1

In the special case T = a 0 x we therefore have (T1)

trace (a 0 x) = <x, a>.

Let Te

a(E, F) and Xe t(F, E). If in

T=

E ai y i ,

1=1

then

XT =

=1

a,

Xy i and

TX = E X'a i y i . i=1

Hence nt

1• 1

trace (XT)

E

i=1

ai > and

trace (TX) = E . 1=1

This implies that (T2)

trace (XT) = trace (TX).

Properties (T 3) and (T4) are obvious. Finally, given any trace r, for every finite operator ni

T =E ai 0 x i = we have

r(T) =

i =1

-c(a, 0 x i) = E

1=1

<xi , ai > .

This proves the uniqueness. 4.2.4. A trace r defined on a quasi-Banach operator ideal 21 is said to be continuous if the function T T(T) has this property on all components 21(E). Then there exists a constant c 1 such that IT(T)I

CT I /CI for all Te 11(E)

and arbitrary Banach spaces E. Remark. By the technique employed in the proof of 3.4.6, it can easily be shown that a trace r is continuous if and only if X —> T(XT) yields a continuous function on t.,(F, E) for every fixed T e W(E, F). This means that r(XT) viewed as a bilinear form on 2(F, E) x W(E, F) is separately continuous in the variable X. We stress the point that this property can be used to define the concept of continuity for traces given on arbitrary operator ideals. 4.2.5.

Next we state an elementary but important result.

Trace extension theorem (Fiist version). Let It be an approximative quasiBanach operator ideal such that i trace (T)I < cTI 1-111 for all T e where c

1 is a constant. Then 21 admits a unique continuous trace.

4.2. Traces

173

Proof. As indicated in the name of the theorem the desired trace is obtained by continuous extension of the unique trace defined for finite operators. Remark. Suppose that /1 and are approximative quasi-Banach operator ideals with continuous traces c. and fl, respectively. Then it is unknown whether or not it follows that for all T E W(E) n

oc(T) = 13(T)

We are now in a position to give some non-trivial examples.

4.2.6.

Theorem (H. KEInig 1980: b). There exists a unique continous trace on the quasi-Banach operator ideal (V 2) 2 . Proof. Let T E

a(E). By 1.5.1, for every 8 > 0 we can find a factorization

such that

11x11 Ils I 43211 IIA I 1211 :6 ( 1 +

iiT I (V2) 2 11 H and K being Hilbert spaces. In view of B.4.6, it may be further arranged that S is finite. Consider a Schmidt representation =E

owe" 0 yi .

Then 1.4.5 implies that 11 ,5 1 4;211 = Therefore it follows from

=E

trace (T) = trace (XSA) = trace (AXS)

x i)

i =1

that \

1/2

n

\

1/2

1 0'11 2) E I AXYill 2 ) i=1 i= I S I 42Il IA I 13 2II 11 ,1(11 6 (1 + 8) 0, we obtain

trace (T)I 6 (

Letting

E

ltrace (T)I

li T I ( 1 2) 2 ii

IT

1 (V2) 2 11 -

for all T e

We know from 1.5.9 that (V2) 2 is approximative. Hence the extension theorem applies. Remark. The assertion could also be deduced from 3.7.3 (Remark) and 4.2.17. A stronger result is proved in 4.2.30.

174

4. Traces and determinants

4.2.7. Theorem (A. Pietsch 1981: b). There exists a unique continuous trace on the quasi-Banach operator ideal (432)143 . Proof. By 2.8.8, every operator T e (432)Sa,l(E) admits a representation co

E

T

Tk k =0

such that Tk e WE), rank (Tk) 2k and

E 2" I Tk I 43211 -5 clIT I ( 4 2)Sa,I I =o

Here c where

k

1 is a constant. Applying 1.5.1, we can find factorizations Ak

e 2(E, 12 (2k)),

Xk E

2(/2 (2k), E)

Tk = XkAlc,

and 11Xkll Ak1 43 211 = llTk I 43 211 •

It follows from

2k trace (A kXk) = E (A kXkei , e 1) i=1

that 2

'trace (AkX0I

"

i=1

IlAkxkeill

< 2" (

21‘ i=1

\ 1/2 < 2k/2iirr 1lAkXkei ll 2) ii-k ga

II

•

If T e g;(E), then the above representation possesses only a finite number of summands; see 2.3.8 (Remark). Therefore trace (T) = E trace (Tk) = E trace (XkA k) = E trace (AkX k) k= 0

k= 0

k= 0

Thus we obtain I trace (T)I < c T I 03

il

for all T e (E) .

We know from 2.8.9 that (43 2)Sa3 is approximative. Hence the extension theorem applies. Remark. The assertion could also be deduced from 3.7.4 and 4.2.17. A stronger result is proved in 4.2.31. 4.2.8. An operator Te 2(E, F) is called integral if there exists a constant c > 0 such that I trace (XT)1

cilX11

for all Xe

a(F, E).

The set of these operators is denoted by 5(E, F). For T e 5(E, F) we define

IIT I 511 := inf c, the infimum being taken over all constants c holds.

0 for which the above inequality

175

4.2. Traces

4.2.9.

The following result is straightforward; (PIE, 6.4.2.).

3 is

Theorem (GRO, Chap. I, p. 128).

a Banach operator ideal.

4.2.10. Next we prove an inclusion which can be found in the work of A. Grothendieck, at least implicitly. Proposition.

Proof. Let T e WE, F). Given .7c1

= (Tx, b i > and

x„ e E, we choose b 1 ,

b„ e F' such that

1.

11b1 11

Define 72

X :=E bi 0 x i , 1=1 and note that

11X11

Kx, 5

sup {

a>I: a E o}.

i= 1

Then it follows that 71

1= 1

11Txill = i=1

= r(TX)

IT! 511

Therefore we have T E *(E, F) and

I IT I 43 1

IIT I all.

Remark. The embedding operator from /1 into 12 shows that the inclusion is strict. 4.2.11. Proposition. The Banach operator ideal 3 is a metric extension of Z 1 .

Proof. Define the projection Pm e P ng 1 • • • m

,n+1, • • •)

a(i2

)

= ($ 1

by •••

in

CI

,

• • •) •

Then it follows from trace (P ) = m and

11P,2 11 = 1

that the identity operator of 12 fails to be integral. Hence, by Calkin's theorem 2.11.11, every operator T e 5(H, K) is approxim able. Thus there exists a monotonic Schmidt representation

ri xr y i .

T= 1 =1

Define n

Urn :=1: 31 0 x i . 1=1

We then have i=i

ri = trace (U mT)

IIT I

all lit/mil

I IT I 511

176

4. Traces and determinants

, K) implies co, we see that T e Te Z i (H, K) and ITI 5 T J. K) that Conversely, it easily follows from T e

Letting m

T e 5(H, K)

and

IT

4.2.12. We now show that Zs is an upper bound for all quasi-Banach operator ideals which admit a continuous trace. This proves that those ideals are comparatively small. We stress the fact that there does not exist any continuous trace on the Banach operator ideal itself; see 4.7.6. Theorem (A. Pietsch 1981: b). Let 11 be a quasi-Banach operator ideal with a continuous trace. Then 11 g Proof. By hypothesis, there exists a trace r such that /CI

Ir(S)1 Hence, whenever T e 'trace (XT)I

for all Se /t(E). F) and X e a(F, E), we have XT1

< oe 61 7' _

11)if 11

This proves that T E a(E, F) and

11T I

1

clIT I 1411 .

4.2.13. We have the following corollary of the previous results. Theorem. Let ti be a quasi-Banach operator ideal with a continuous trace. Then /I is of eigenvalue type /2 . In particular, all operators T e W(E) are Riesz.

4.2.14. Suppose that the operator ideal If is of eigenvalue type / 1 . Then we may define the spectral sum

2(T) := E12,(T)

for all T

Unfortunately, we do not know whether this expression always yields a trace. The critical point is additivity. If A has this property, then we say that the operator ideal ?d, admits a spectral trace. Furthermore, a trace r is called spectral if the trace formula holds: CO

r(T) = E 2,(T)

for all T e

n= 1

4.2.15. We now treat the most elementary example. Theorem. The operator ideal

a admits a spectral trace.

Proof. Given T e WE), we consider a representation

T=

E ai

0 xi

i=1

such that the matrix M := (<x; , ai)) has triangular form; see B.4.5. We know from 3.3.5 that the operators T and Mop are related. Thus both have the same non-zero

4.2. Traces

177

eigenvalues. Therefore CO

171

fli

<xi , ai > =I A n(M) =

trace (T) = 1 =1

n=1

E A n(T) 17

=1

This proves that the unique trace of is5' is spectral.

4.2.16. The following result is an immediate consequence of the principle of uniform boundedness 3.4.6. Lemma. Suppose that the quasi-Banach operator ideal /1 is of eigenvalue type 1. Then the spectral sum is continuous at the origin. Proof. There exists a constant c

1 such that

CO

12(T)I n= 1

12n(T)I

cll

I

for all

T e 14E).

4.2.17. The next theorem is closely related to 4.2.5. Trace extension theorem (Second version). Let /1 be an approximative quasiBanach operator ideal which is of eigenvalue type / 1 . Then 4 1 admits a unique continuous trace. Proof. By 4.2.15 and 4.2.16, there exists a constant c I trace (T)I

ell

VIII for all

Te

1 such that

a(E).

Remark. It is unknown whether or not the property of being spectral survives the extension procedure from (E) to /1(E).

4.2.18. For the following considerations some preliminaries are required. First of all, we state a lemma which is proved in (DUN, XI.9.5). Lemma. Let T, , T2, G t(E) and T e t(E) be Riesz operators such that (Tk) converges to T with respect to the operator norm. Then the eigenvalue sequences of these operators can be arranged in such a way that lim 1,(Tk) = A n(T)

for n = 1, 2, ...

Remark. Readers who accept this assertion without proof may proceed immediately to 4.2.23. As an alternative some results from perturbation theory can be used. These are proved in the following paragraphs.

4.2.19. Lemma. Suppose that I — T with T e t(E) is invertible. Put :

—

T) - ' II >0.

If Se t(E)

and ocilS —

< 1,

then I — S is invertible, as well. 'Moreover, we have oc 2 II S(/ — S) -1 — T(I — 1 — q MS— TM. Proof. Elementary manipulations show that CO

S(I — S) — T(I —

T) - ' = (I — S) --1 — (I — T)' = (I — T) - ' QEQ", k 0

178

4. Traces and determinants

where

Q := (S — T) (I — T) -1- and IIQII cS — q < 1. Therefore (I — Sr exists, and the required inequality is obvious. 4.2.20. A number e > 0 is said to be an admissible radius of an operator T e t(E) if the circle 11,1 = belongs to the Fredholm resolvent set e(T). Then we may define (the projection) co

(e, T) :=

1

+ a)-1. ici =e

the contour integral being taken in the positive direction.

4.2.21. Lemma. Suppose that e is an admissible radius of an operator T E 2(E) . Put

:= sup {11(/ + T) - '

> 0.

=

If then

e

S e 2(E) and ocellS — TII q <1, is an admissible radius of S, as well. Moreover, we have

IIP.(e, S) — P, T)II

1 — e q lIS

—

Proof. Applying Lemma 4.2.19, we obtain

11Pm(e, s) —

{lisv + < — 1 — q S TII -

7)11 6 e

sup

0‘

2,

—

+

= (21

4.2.22. Lemma. Let 91 be a quasi-Banach operator ideal. Suppose that 0, is an admissible radius of an operator T e t(E). Put

= e l > 0.

: = sup {11(/ + 'T) -1 1

If S e ?I(E) and NOS — T I 14 6 q < 1, then

II SP co (e, S) — TT' „9 (e, T) J 14cJJS — TI ?IL where c> 0 is a constant only depending on q,

e and T.

Proof. Note that

IIS WI

— T

+

ItII)

+ IIT WI)

Hence

(e, S) — cgIIS

1.111 7)11 + I IS — TJ ?III

IIP.(e, c9.1 [c%+ I IT WO 1eic-2e + lI(e 1)111 IS — T q ' ace S)

P

179

4.2. Traces

4.2.23. We are now prepared to establish a stronger version of Lemma 4.2.16.

Lemma. Suppose that the quasi-Banach operator ideal Vi is of uniform eigenvalue type /1 . Then the spectral sum is continuous everywhere.

Proof. We consider a sequence of operators T1 , T2, ...,

W(E) converging to

T E 9I(E). By hypothesis, given 8 > 0, there exists n o such that

E p,„(ni

s

E 12,(TOI

and

for k = 1, 2, ...

e

n>no

n>no

Hence

1 2.(Tk)

- MT)!

n=

- 2 „( T) 1 + 2E.

1 2 .( Tk )

Applying Lemma 4.2.18, we find k o such that 1A(Tk) — A(T)1 -.5_ 38 whenever

k0 .

k

Thus 1 is indeed continuous. Another proof, based on Lemmas 4.2.21 and 4.2.22, goes as follows. no . Let Fix any T-admissible radius e with Ep. :=

sup {11(/ + CT) -1 Il : CI = e l

and

q := 1/2.

Choose ko such that °cell Tk — T I All

q for

k

k0 .

Then Lemma 4.2.21 implies that e is Tk-admissible, as well. Since T is a Riesz operator, it follows from 4.1.6 and the residue theorem that , T) =

E P „,(1 +

,

gl <0

where the right-hand expression contains only a finite number of non-trivial sumis an eigenvalue, then we have mands. If 20 — A[TP 00 (1 + W')] = n(201 — T) 20 . Therefore A[T1 3 00 (e, T)] = This implies that

E

lan ie> 1

E 12,1(ni lAnle< 1 n:5-11 0 E 12(T)! + E 12n(n1 nde + E.

12(T) — 2[TP.(e, lAnle <1

n>no

Hence 12(T) — 2ITP je, TAI 15. 2e.

In the same way we obtain — AiTkP co(e, 71)11

28 for k

According to 4.2.16, there exists c o Il(S)1 -5 co 1151 WI for all S e

ko

1 such that

a(E)

(2)

180

4. Traces and determinants

Note that 2L is additive on R(E). Hence we see from 4.2.22 that

IA[TkP °AO, TkA — A[TP, c (e,TAI k o such that Thus we may choose k,

for

c coilTk — T

Tk)] — A[TI

e

for

k

k1 .

k k o. (3)

Combining inequalities (1), (2) and (3) yields

JA(Tk) — 2(T)1

58

for

k

k1 .

This completes the proof for everyone. Remark. We conjecture that the spectral sum is continuous on all quasi-Banach operator ideals which are of eigenvalue type i. This would mean that the uniformity assumption is either automatically satisfied or superfluous. 4.2.24. The main result of this section is now easily obtained. Spectral trace theorem (H. Leiterer / A. Pietsch 1982). Let be an approximative quasi-Banach operator ideal which is of uniform eigenvalue type /1 . Then Vi admits a spectral trace. Proof. We have just proved that the spectral sum A is continuous. On the other hand, by 4.2.17, there exists a unique continuous trace r. Since both functionals A and r coincide on .i, ( E), which is dense in t(E), they must coincide everywhere. Thus 2 is indeed a trace. 4.2.25. Since it is unknown whether or not the finite operators are dense in the quasi-Banach operator ideal tV) , the preceding theorem only applies to its approximative kernel. Theorem (H. Leiterer / A. Pietsch 1982). The operator ideal (t ) )(m) admits a spectral trace. 4.2.26. We have the following corollary of the previous result. Theorem (H. K6nig 1980: a). The operator ideals t.' (f ) and t(f) admit a spectral trace. Proof. We know from 2.10.7, 2.8.18 and 2.8.9 that (2(x))(a) . (la) ,S41,(ic) 2 )Sal ( Therefore the spectral trace can be restricted from (tY))(a) to the smaller operator ideals. Remark. Since t(ia) is approximative, the spectral sum yields the unique continuous trace on this quasi-Banach operator ideal. On the other hand, we do not know whether there exists more than one continuous trace on 4.2.27. A trace r defined on an operator ideal ?l is said to be nilpotent if r(T) = 0 for every nilpotent operator Te WE). 4.2.28. Proposition. The spectral trace is nilpotent. Proof. Note that a nilpotent operator cannot possess any non-zero eigenvalue. Therefore the spectral sum vanishes.

4.2. Traces

181

4.2.29. We now describe a useful method for transferring a trace from certain operator ideals to related ones.

Trace transfer theorem (A. Pietsch 1981: b). Let If and $ be quasi-Banach operator ideals. If a is a nilpotent continuous trace on /I nilpotent continuous trace T on $ ?I such that

-c(BA) = a(AB)

for all

A

E

?i(E, F) and

In particular, if a is spectral, then so is

$, then there exists a

B e 11(F, E).

T.

Pro of. Given any operator Te $ o VI(E), we consider two factorizations T = Bi A i and T = B2A 2 , where A 1 e ?I(E, F1 ), B1 e $(F1 , E)

and

A, e ?I(E, F2),

B2 E $(F2,

E).

Form the direct sum F := F1 0 F2 and let ./1 and Q1, .12 and Q2 denote the canonical injections and surjections, respectively. It follows from (Bi Q i + B2 Q2) (J2A 2 — Ji A i ) = B2 A 2 — Bi A l = 0

and (Bi Q i

—

B2Q2) VIA

+ J2 ,4 2) = Bi A i — B2A 2 = 0

that the operators 1 ) (BiQi + B2Q2)

:= W2A2 —

and J2A2) (BiQi

N2 :=

B2Q2)

are nilpotent of order 2. Note that + N 1 = J2A 2B 2 Q 2 + N2.

Hence cr(A i Bi ) = (VI A iBiQi) = a(J2A2B2Q2) = a(A2B2)•

This proves that every factorization of an operator T e o lt(E) in the form T = BA with A e ?I(E, F) and B e $(F, E) yields the same value a(AB). Hence we may define T(T) := a(AB). If x e E and a e E', then we have

a

x = (1 ® x) (a ® 1)

and

(a ® 1) (1 x) = <x, a> 1 ® 1.

Note that 1

1 is the identity operator of C, and a(1

(T1)

x) = <x, a> .

r(a

1) = 1. Therefore

Let Te $ 11(E, F) and X e E). Choose any factorization T = BA with A e ?I(E, G) and B e 13(G, F). Then we have 1-(XT) = T(XBA) = a(AXB)

This proves that

(T2)

-t- (XT) = r(TX).

and

r(TX) = T(BAX) = a(AXB).

182

4. Traces and determinants

Given T1 , T2 e Ti . 1.1(E), we consider factorizations T i = Bi A i and T2 --= B2A2, where A, e 14E, F,), B 1 e a3(F1 , E)

Foirn the direct sum F := F1

and

A2

e 14E, F2), B 2 e 13(F2 , E).

e F2 as desciibed above. Then

T1 + T2 = (B1Q1 + B2Q2) (J1 11 1 + J2A2)• Observe that the operators N 1 := J1 A 1 B2 Q2 and tent of order 2. Hence it follows from

N2 := J2A2B1Q1 are

nilpo-

(Ji A i + J2 A 2) (B I Q I + B2 Q 2) = J1 A I B,Q 1 + J2A 2 B2 Q 2 + N1 + N2 that a[(JiA I + J2A2) (BiQi + B2Q2)] = cr(fiAiBiQi) + ti(J2A2B2Q2) = a(A i B i ) + a(A2B2)• Thus we have (T3)

r(T1 + T2) = r(Ti ) + r(T2 ).

Property (T 4) is obvious. Since the trace a is assumed to be continuous, there exists a constant c that clIS1 11 . $11

for all

1 such

S e /.1 . $(F)

and arbitrary Banach spaces F. Given Te . It(E) and E > 0, we choose a factorization T = BA such that

A e 14E, F), B e $(F, E)

and

11B1 $1111A 1 W11 __ < ( 1 + 8)11T i $ ° ?(11.

Then it follows that

_ clIA 1r(T)1 = 1a(AB)1 _<_ cliAB Ill . $11 < _< (1 + e) cliT I $ . ?ilk

I WI IIB I VI

Letting s —* 0 yields that IT(T)1 ---cll T1 $ ° ttll

for all

T e $ . It(E).

We now assume that Te $ .14E) is nilpotent. Considering any factorization T . BA with A c 14E, F) and B e Ti(F, E), we conclude from Tm = 0 that (AB)m+ 1 = AT " B = O. Thus, since a is nilpotent, it follows that T(T) = a(AB) = O. Finally, the principle of related operators implies that

T(T) = a(AB) = l 2(AB) = E ),,(BA) = P11 n=1 n=I provided that a is spectral. 4.2.30. Applying the preceding theorem, we are now in a position to give some important examples.

Theorem (H. K6nig 1980: b). The operator ideal (3 2) 2 admits a spectral trace.

183

4.2. Traces

Proof. By 1.5.3, we have

$ := 5) 4; 2 t

(432) 2 = $ ?.1 with

and

It follows from 2.7.2 and 2.3.13 that

Thus the spectral trace existing on 2?3) can be transferred to (132) 2 . Remark. Since the quasi-Banach operator ideal (132)2 is approximative and of uniform eigenvalue type / 1 (H. K6nig 1980: b), the assertion could also be deduced from the spectral trace theorem 4.2.24.

4.2.31. Theorem (A. Pietsch 1981: b). The operator ideal ($2)14/3 admits a spectral trace. Proof. By 2.8.21, we have

(43 2 ) =

o /f

$ : = 2(4a.1 t

with

and

/I := (43 2)(4.

It follows from 2.8.7, 2.7.2 and 2.8.13 that

°

= (432)(4a3 °

(432 °

°

g ( 2MS.a3 = eia) .

Thus the spectral trace existing on eia) can be transferred to (432 ) Remark. Since the quasi-Banach operator ideal (432)S,a?i is approximative and of uniform eigenvalue type / 1 , the assertion could also be deduced from the spectral trace theorem 4.2.24.

4.2.32. With the help of the following proposition we can produce more examples of quasi-Banach operator ideals admitting a continuous trace. Proposition. Let r be a continuous trace on the quasi-Banach operator ideal It. Then r' (T) : = r(T') for all T e /1'(E) defines a continuous trace -r' on the dual quasi-Banach operator ideal ?V. In particular, if t 1S spectral, then so is f.

4.2.33. The next proposition yields examples of comparatively large quasi-Banach operator ideals admitting a trace. Proposition. Let 92 and traces a and /3, respectively. If

cc(T) = AT)

be quasi-Banach operator ideals with the continuous

for all T e 92 (E)n Qi(E),

then there exists a continuous trace r on 92 + $ such that

r(A + B) = cc(A) + AB)

for all A e ?t(E) and

B e ?i(E)

Remark. The above assumption is satisfied in the special case when the traces a and 13 are spectral.

184

4. Traces and determinants

4.2.34. Summarizing the previous results, we see that all quasi-Banach operator ideals occuring in the following diagram admit a spectral trace: (t(x))(a)

1.0)(a)

2 (

tO2

(V2Aa,1

(V2')1a,i ° \ /1

91 12

e(ic)

,2

to, (id)

\

0(a) Furthermore, in view of 4.2.33, there exists a continuous trace on each of the following quasi-Banach operator ideals:

(2(x))(a) (elqa) , (V2)2‘1,1 These traces, however, fail to be spectral. (*2) 2

(432) 2

( 4 2)21

4.2.35. Proposition. Every continuous trace defined on a quasi-Banach operator ideal t is spectral on (S) It) cl) and (It o

Proof. It follows from 4.2.11 and 4.2.12 that (0o4,.)oF.OooS)c t (f).

Thus the spectral trace existing on tfa) can be transferred to = o (Si o Since the given trace and the spectral trace are continuous, they must coincide on the approximative kernel of 0 0 W. The quasi-Banach operator ideal (/1 oOa) can be treated analogously. 4.2.36. Proposition (J. C. Engelbrecht / J. J. Grobler 1983). Every continuous trace defined on an approximative quasi-Banach operator ideal ?„1 is spectral on 112 .

Proof. It follows from that

oWya) 4.2.37. It is very likely that the above proposition remains true in the case of nonapproximative quasi-Banach operator ideals, as well. At present, we know only the following result.

Proposition. Every continuous trace defined on a quasi-Banach operator ideal t is spectral on 113 . Proof. It follows from and

that

?tot ot

12 ° 432 °

1,3 2 ° 0 (41) °

2 g-

g

°2°

° 10(a)

(a)

4.3. Determinants

185

4.2.38. All examples of quasi-Banach operator ideals, treated in this chapter, admit either none or exactly one continuous trace. Only recently, it was proved by N. J. Ka lton that this is not so in general. For a proof of the following result we refer to the Appendix of this book. Theorem. There exist quasi-Banach operator ideals admitting different continuous traces. 4.3.

Determinants

4.3.1. By a determinant 6 defined on an operator ideal we mean a function which T with T e II(E) a complex number assigns to every operator of the form IE 6(/ E + T) such that the following conditions are satisfied:

(D1) (D2) (D3)

(D4)

6(/E + a 0 x) = 1 + <x, a> for a e E' and x e E. F) and Xe 2 (F, E). 6(1 E + XT) = 6(IF + TX) for Te 6[(/E + S) (JE + = 6(1E + S) W E + T) for S, Te ?4(E). 6(/E + `CT) is an entire function for fixed T e

Remark. If there is no risk of confusion, then all identity operators denoted by I.

IE, IF, ...

are

4.3.2. In the sequel we need an elementary result from the classical theory of determinants. Lemma. Let X be an (m, n)-matrix and Y an (n, m)-matrix. Then

det (/(m) + XY) = det (I(n) + YX). Proof. It follows from

(1(m) X \ (1(m) 0 \ — Y 1(n)) k Y 1(n))

(Am) + XY X \ 0 I(n)1

(1(m) 0 \ (Am) X \ Y 1(n)) \ Y 1(n))

(Am) X k 0 1(n) + Y X

and

that the determinants of the right-hand (in + n, m + n)-matrices are equal. This implies the formula stated above, since

det

( 1(m) + XY X ) = det (1(m) + X Y) I(n) 0

and

det 4.3.3.

'1(m)

k

X

0 I(n) + Y X

= det (/(n) + Y X) .

Lemma. Let T e WE). Then

det (/ + T) := det (6 1., + <xi , ai>)

186

4. Traces and determinants

does not depend on the special choice of the representation

T= E ag 0 x i .

i= Proof. Given any representation T = E ai 0 x i , i= 1 we choose linearly independent elements y i ,

E such that

x l , ••, Xnz 6 sPan (Yi •• •, n) • Then there exists a representation

T = E bh 0 yi,. h=1

Writing

E

Xi

$ihYh,

h=1

we conclude that the functionals b 1 ,

=

i=1

b n E E' are uniquely determined by

$ihai •

Hence

<x„ a> =

in

h=1

$0,
ai>

and

=

E Cvh,

j=1

(), then we have det (b ii + <xi , ai>) = det (I(m) + XY)

If X := ($ ih) and Y: and det (

) = det

hk

(I(n) + YX).

Therefore the preceding lemma implies that det (6 i; + <x,, ai>) = det (âhk + )-

The final conclusion now follows from a simultaneous application of this construction to two arbitrary representations. 4.3.4.

We now establish a counterpart of 4.2.3. Theorem. There exists a unique determinant on the operator ideal

Proof. For every finite operator rn

T = E ai i=1

xi

we define det (/ + T) := det (6i; +

In the special case T = a (D 1 )

det (/ +

a

<xi , ai>).

x we therefore have

x) = 1 + <x, a>.

a.

187

4.3. Determinants

Let T e

a(E, F)

X e t(F, E). If

and

T = E ai y i , i=i

then m

in

i=1

1=1

XT = E ai 0 Xy i and TX = E X'a i y i . Hence

+ <Xy i , ai>)

det (I + XT) = det and

det (/ + TX) = det (b i +
(D 2)

det (/ + XT) = det (/ + TX) .

Given S, TE (E), we choose representations in

S=

i=1

ai x i and

bh

T=

yh.

h=1

Then

S) T = E ai 0 x i + E bh 0 (Yh + x p).

(I

S

i=l

h=1

p =1

Thus

det [(I + S) (I + T)] = det [/ + S + (I + S) T] is defined to be the determinant of the matrix (

+ E

(

yh, ai,) <xp, c13>

kk

+

+E

P=1

P= 1

(Xp 5 bk> )

which can be written in the form 0 ,

6„q

<xp, bic> )

0

,

qk

Next we observe that the determinants of the right-hand and left-hand factor are equal to det (/ + S) and det (I + T), respectively. This completes the proof of the multiplication formula

det [(/ + S) (I + T)] = det (/ + S) det (/ + T) . Property (D4) follows from the fact that det (/ + "T) is a polynomial in the

(D 3)

complex variable Finally, we know from B.4.5 that every operator T e

a(E) admits a representation

771

T=

a,

xi

such that the matrix (<xi , a.i ) ) has upper triangular form:

<xi , aj> = 0 if i > j.

188

4. Traces and determinants

Therefore

/ + T = (I + a l

x1) ... (/ + an, Thus, given any determinant 6, we have 6(I + T) = fl Ô(/ + ai 0 i=1

xi) =

xm).

fl (1 + (xi , ai>) = det (6 ij + <xi , ai>). i=1

This proves the uniqueness.

4.3.5. Let 6 be any determinant on an operator ideal ?I. Then property (D4) implies that the Gâteaux derivative 6' (T) := hm 6(1. + CT) exists for all T e ?I(E).

4.3.6.

Lemma. Let ô be a determinant on an operator ideal W. Then

IcY(T)1

sup {16(/ + CT) — 11 : ICI = 1}

for all Te W(E).

Proof. Since 6(1 + CT) is an entire function, we have the Taylor expansion

6(1 + CT) = 1 + E cv,i(T)

for all

C E C.

n=

Hence the residue theorem implies that

1 = x1 (T) = 27ri

6(1 + CT) — 1 C2

the contour integral being taken in the positive direction. The desired inequality is now obvious.

4.3.7. A determinant 6 defined on a quasi-Banach operator ideal tt is said to be continuous (at the origin) if the function T 6(1 + T) has this property on all components ?I(E). 4.3.8.

Next we prove an analogue of 4.2.12.

Theorem. Let /I be a quasi-Banach operator ideal with a determinant which is continuous at the origin. Then It 5. Proof. By hypothesis, there exist a determinant c5 and E > 0 such that

16(/ + S) — 1 I

1 if ISI1I

E.

Hence, in view of 4.3.6,

16,*(S)I

1

if

VI 9111

.

Restricting attention to finite operators, we have 6' (S) = trace (S). Hence !trace (S)1

e--1 11S /Ili

for all S e

and, reasoning as in the proof of 4.2.12, we arrive at the desired conclusion.

189

4.3. Determinants

4.3.9.

We have the following corollary of the preceding result. Theorem. Let 11 be a quasi-Banach operator ideal with a determinant which

is continuous at the origin. Then 94 is of eigenvalue type /2 . In particular, all operators T e It(E) are Riesz. 4.3.10. Proposition. Let (5 be a determinant on a quasi-Banach operator ideal 11. If (5 is continuous at the origin, then it is continuous everywhere. Proof. Fix S e ?I(E). If I + S is invertible, then the continuity at S follows from

6(1 + S + T) = (XI + S) 6(1 + (I + S)'T) for all T e W(E). Otherwise, in view of the preceding result, there exists a Riesz decomposition S = S m + SN such that I + SM is invertible and SNe(E). Then

6(1 + S + T) = 6(1 + Sm + T) 6(I + (1 + Sm + T)-- S N) for all T e 1(E) provided that 1171 11 (1 + Sm)" < 1. As already shown T 0 implies 6(1 + Sm +T) 6(1 + S 1). Furthermore, we know from 4.3.4 that

6(1 + (1+ S m + T)- -1 S N) = det (c5 i; + <(I + SM + T)" x i , ai>), where SN =

a Øxi

is any representation of SN • Obviously, the right-hand determinant viewed as a function of T is continuous at T = O. This completes the proof. 4.3.11. Proposition. Let (5 be a continuous determinant on a quasi-Banach operator deal QC Then I + T with T e /I(E) is invertible if and only if 6(1 + T) ± O. Proof. Suppose that I + T is invertible. Then the inverse can be written in the form (1+ T) -1 = 1+ S, where S E It(E). We now obtain 6(1 + S) 6(1 + T) = 6(1) = 1. Hence 6(1 + T) * O. In the case when I + T fails to be invertible we consider a Riesz decomposition T = T + TN. Since the restriction IN + QNTTN of I + T to N : INT(I + T) is nilpotent, it follows that

6(I + T N) = det

(I

+ NQ N T) = det (IN + Q N TJ N) = O.

Hence

6(I + T) = 6[(I + TM)(I + TN)] = 6(1 + TM) 6(1 + T N) = O . 4.3.12. For every operator T e(E) we define exp (T) := I +

E

1 -„

Tk.

k=1

Note that 1 n exp (T) = lim + — T) .

n

The following result is obvious.

190

4. Traces and determinants

Lemma. Let ?,t be any quasi-Banach operator ideal. If TE ?-1(E), then

I E ?1(E)

and ilexp (T) — I 111 11 -."5 IIT I exP (11/ - 11)4.3.13. Proposition. Let ô be a continuous determinant on a quasi-Banach operator exp (T) —

ideal ?I. Then 6[exp (S + T)] = 6[exp (S)] b[exp (T)]

for all S, T e W(E).

Proof. If the operators S and Tcommute, then the above formula follows immediately from exp (S + T) = exp (S) exp (T). The verification in the general case is more sophisticated.

Assume that n 211S + TIL Then _1

[I Define

I + —1 (S + T) is invertible, and we have

+ 71 1 (S + T)1

< 2.

U,, and R„ by the equations [I +

and

(S + T)] (/ + tin) = +

n

ST) (/ +

n

T)

I + R„= (I + U„)n.

Since

n2

[I + '(s + T)] un

= ST,

we have n2 11Un 11411

21IST I

Next we deduce from

R„

(n ) k=k

that R „ e Tt(E) and

0 as n

Hence 11R n 91 -11 6

co

nll U„ I 1II exP

UnD •

co. Note that

)1 6 RI +

Ti n = (1 +

)]

s) 6 (1 + T 1 /' [i 1 (45 Td ô [I + Un]) = 6 + (S + T)] 0(I +

+

6

=

Letting n

" nk

?Ill

liRn

n

-R

n) •

yields the formula we are looking for.

4.3.14. Lemma. Let O be a continuous determinant on a quasi-Banach operator ideal U. Then

b[exp (T)] = exp [6' (T)] for all Te W(E). Proof. We have

(5 +

1

1 1 = 1 + — (T) + (-0 , n

191

4.3. Determinants

1

as n —> oo. Write

where nw (- ) —> 0

1 1 n (5. (T) + (— )1 — [1 + — (5. (T)1 . n

„ : = E1

1, we get

Taking n so large that n

nn—k

len1

E k=1

-

n

—

k=1

k!

[l + -1n 4 (T)] in 1 . r 1 ± -16 (T A] n

co (I--

o

)

1)

exp (1 + 16.(T)1).

This proves that en 0 as n —> co. Finally, the desired formula follows from

RI

[1 + 1 (T) + (1)]" = + 1 T )1 [(5 (1 ± 1 = (1 + — (5. (T)) + e n

by passing to the limit as n —> co. 4.3.15. We are now in a position to establish the main result of this section.

Theorem. Let (5 be a continuous determinant on a quasi-Banach operator ideal ?I Then the Gâteaux derivative

(T) := lim b(I c-.4o defines a continuous trace.

— 1 for all T e 94(E)

Proof. If a e E' and x e E, then it follows from

6(1 +

x) = 1 + Xx, a>

that

(T1)

(5. (a ® x) = <x, a>.

Let T e QI(E, F) and X G t(F, E). Then (5(/E + -XT) = 6(IF + -TX) for all Ç e C. This implies that 6' (XT) = (5 . (TX). (T2) Given T e ?I(E) and 2, + 0, we have 6(1 + 2.T) — 1 6(1 + NI) — 1 — lim lim Hence

(T4)

(5. ( AT) = AS' (T) .

192

4. Traces and determinants

e C. Then, in view of 4.3.14, it follows from + a)] = Nexp (CS)] b[exp (a)]

Let S, Te ?I(E) and

Nexp that

exp [0. (S + T)] = exp

(S)] exp [6(T) ].

Since this formula holds for all ç e C, we may conclude that

(T3)

6. (S + T) = (S) + 6' (T) .

The inequality 4.3.6 yields the continuity of 6.

4.3.16. For every operator Te 2(E) with 11711 < 1 we define co (-1)k Tic.

log (1+ T) := —

k

k=1

Note that

exp [log (I + T)] = I + T. The following result is obvious. Lemma. Let Vi be any quasi-Banach operator ideal. If Te It(E) and 1 711 5_ q < 1, then 1 log (1+ T) e W(E) and Dog (I + T) ICI 5_ 1 — q IIT tICI •

4.3.17. Using the logarithm, we may rephrase Lemma 4.3.14 as follows. Lemma. Let 6 be a continuous determinant on a quasi-Banach operator ideal It. Then

6(1 + T) = exp (6. [log (I + T)]) for all T E It(E) provided that 11111 < 1.

4.3.18. Lemma. Let ô be a continuous determinant on a quasi-Banach operator If 0 < q < 1, then there exists c > 0 such that ideal

clIT I VI

16(1+ T) — 11 provided that I IT l

for all T e /t(E)

q.

Pro of. According to 4.3.15, we can find a constant co

161. (T)I

co T I tl I

1 such that

for all Te ?A(E).

Recall from Lemma 4.3.16 that

Il log

+ T) I WI

1

1 — q IIT I ?ill if

I IT I 1111

q

Hence

11(Y [log (1 + 1)]11 5_ 1 — q coIlT I

if

IT I

Note that

lexP

I

11 exP (I) forai!

eC.

*I 5_

q.

193

4.3. Determinants

q, then we have

Therefore, if IT I WI

là(/ + T) — 1

lexp (6'[log (1 + T)]) — 11

1

[ _ q co exp

1 — q

ce )] IIT

.

4.3.19. In what follows the derivatives of functions q(C) and T(C) are denoted by T (C ) and T(C), respectively. Proposition. Let 6 be a continuous determinant on a quasi-Banach operator ideal It. Suppose that the It(E)-valued function T(C) is defined on a domain of the complex plane. If T(C) is differentiable at a point Co, then so is the complex-valued function (AC) := 6[1 + TO]. In the particular case when 1 + T(Co) is invertible the derivative is given by

99• (Co) = (5. [T(C 0)

+ G. °W I ]

+ T(C0)].

Proof. By hypothesis, we have

= T(Co) + (C — Co) TV()) + 0(C, Co),

where

lim II 0(C, Co) I

_ 0

IC — Col

•

First of all, we treat the case when 1 + T(C0) is invertible. Then there exists

S(C, Co) := (/ + T(Co) + (C — Co) r(C0)) -1 0 (C, Co) provided that IC — Co l is sufficiently small. It now follows that 6[I + T(C)] = + 0) + (4' - C0) r(CO] +

Co)].

Obviously, the function

+

0) + G -

r (C0)1 =

T' (Co) (I + T(C 0 ))-1 ] 6[1 + T(C0)] = 6[1 + (7; — is differentiable at C o . Furthermore, by the preceding lemma, we have + S(C, Co)] — li < IIS(C, Co) IC — 001 IC — Col = This implies that hm

[. / +

S(C, Co)] — 1 = 0. — Co

Thus 6[1 + T(C)] is indeed differentiable at CO3 and the derivative can be obtained by applying the product formula. In the general case we consider a Riesz decomposition T(C o) Tm(Co) + TN(Co) such that I + TAf (C 0) is invertible and TN(CO) e R(E). If IC — Co l is sufficiently small, then

6[1 + TO] = 6[I + T(C) — T N(C0)] Using any representation TN(Co) = Ea t 0 xi , i=

+ (I + T(C) — N(C0)) -1 N(C 0)] •

194

4. Traces and determinants

by 4.3.4, we have

6[I + (I + T(C) — T NG0D -1 T N(C0)] = = det (6 i; + <(I + RC) — T N ( 0)) -1 x i , ai>).

This determinant is of course differentiable at C o . The differentiability of 6[I + T(C) — T N (C0)] follows from the first part of the proof, since 1 + T( - 0 ) — N G 0 ) is invertible. 4.3.20. It is worth specializing the preceding result in the following way. Proposition. Let 6 be a continuous determinant on a quasi-Banach operator ideal It. Then the Gâteaux derivative 6*(1+ T, S) := lim

6(1 + T + S) — 6(I + T)

exists for all operators S, T we have Fredholm's formula

In the particular case when I

T is invertible

60"(/ + T, S) 615(1 + Ty 9.

6(1 + T) 4.3.21. Suppose that the operator ideal /X is of eigenvalue type /1 . Then we may define the spectral product CO

n(1 + T) :=

(1 + A(T)) for

all T e tt(E).

n=

Unfortunately, we do not know whether this expression always yields a determinant. The critical point is multiplicativity. If n has this property, then we say that the operator ideal It admits a spectral determinant. Furthermore, a determinant 6 is called spectral, if the determinant formula holds: Co

6(1 + T) = fl (1 + A(T))

for all Te 1 ( E).

n=

4.3.22. We now treat the most elementary example. Theorem. The operator ideal a admits a spectral determinant. Proof. Given Te a(E), we consider a representation

T = E a, 0 xi 1=1

such that the matrix M := (<xj , ai>) has triangular form; see B.4.5. We know from 3.3.5 that the operators T and Mop are related. Thus both have the same non-zero eigenvalues. Therefore i= 1 ni

co

+ 2(M)) = fl(l + 2(T)). This proves that the unique determinant of a is spectral. n=1

ri=

4.3.23. Proposition. Let 6 be a continuous determinant on a quasi-Banach operator ideal. If 6 is spectral, then so is the associated continuous trace 6..

195

4.4. Fredholm denominators

Proof. If the quasi-Banach operator ideal It is of eigenvalue type /1 , then the spectral product is Gâteaux differentiable at the origin, and we have

= 2(T) for all T e /RE).

hm c--.o 4.4. 4.4.1.

Fredholm denominators Suppose that the Fredholm resolvent of a Riesz operator T e 2(E) is written

in the form F(, T) =

D(C, T)

forai!

e e(T),

where (5G, T) is an entire complex-valued function and D(C, T) is an entire 2(E)valued function. Then we call b(C , T) and D(, T) a Fredholm denominator and Fredholm numerator, respectively. Obviously, both functions are uniquely related to each other. Remark. Occasionally, D ( , T) is said to be a Fredholm minor of first order. Fredholm minors of higher order were defined by A. F. Ruston (1953, 1967). 4.4.2.

Proposition. An entire function b(C , T) is a Fredholm denominator of a

Riesz operator T e t(E) if and only if every point Co 0 e(T) is a zero of multiplicity greater or equal than d(.1" + C0 T). Proof. Let (SG, T) be any entire complex-valued function. Then

D(C, T) : = 6( , T) F(C, T) defines a holomorphic 2(E)-valued function on e (T). Of course, the isolated singularities of this function are removable if and only if the zeros of 6(C, T) annihilate the poles of F(C, T). 4.4.3.

Proposition. Let Te t(E) be a Riesz operator. Suppose that

for all

F(, T) = b(c, T)

e

where

b ( , T) =

ri(T) Cn

for all

eC

and CO

D(C , T) = >A (T) CI' for all n=

o

EC.

Then A „(T ) = E, ( - 1 ) n h och(T)T" --h+' for

n = 0,

h= 0

Proof. Note that (SG , T)T = D ( , T) (I + CT) for all

e e(T).

Substituting the power series and equating coefficients, we obtain

A o(T) = ao(T) T

and

A „(T) = o

This implies the formula we are looking for.

T — A _ i (T)T

for n= 1, 2,

•••

196

4. Traces and determinants

4.4.4. The existence of Fredholm denominators follows from the Weierstrass factorization theorem 4.8.4. This construction, however, is very unsatisfactory, since we need to know all the eigenvalues of the Riesz operator under consideration. For this reason, we are interested in finding a more direct approach. Theorem. Let 6 be a continuous determinant on a quasi-Banach operator ideal 94. If T e WE), then

T) = 6(1 + CT) for all

eC

defines an entire function with zeros at all points C o 0 e(T) and at these points only. Furthermore, the multiplicity of the zero Co is equal to the multiplicity of the eigenvalue 20 = —1/C0 . Proof. First of all, we note that 6(C , T) is holomorphic, by property (D 4). Furthermore, it follows from 4.3.9 that T is a Riesz operator. O. If E e(T), then I + C0 T is invertible. Hence, by 4.3.11, we have 6(C ° , T) If Co 0 e (T), then we consider the Riesz decomposition

E = M09(1 + Co T) N„(1 + 0 T) as described in 3.2.14. Let P := P(1 + C0 T), and define

TN := TP.

Tm := T(I — P) and It follows from T = Tm + (I +

TN

and TR,TN = 0 that

CT) = (1+ CTM) (1 +

N ).

Hence

6(C, T) = 6(C, TM) 6(C, TN ). Since the restriction of I + C0 T to N 03 (1 + C0 T) is nilpotent, we may choose a basis (x 1 , x„) according to 3.1.15. Next, taking the canonical extension of the corresponding coordinate functionals, we find a representation

P =a1 0 x i = a„ e E'. By construction, with a 1 ,

<xi + Co Txi , ai>

=0

if

i j.

Hence —C o = <xi , ai> = c5 ii if i j. This implies that the matrix () has triangular form. Therefore

6(1 + CT N) = det (bij + C(Txi, ai>) =1T (1 + C) = (1 — i=1

or •

Consequently, we obtain 6 (C, T) = (1 — g.0)" (V, TM). Since I + Co T m is invertible, it follows that ô( 0 , T1 ,1 ) of 6(C, T) with multiplicity n = n(I + 0 T). 4.4.5.

The entire function

6(1 + CT) forai! C e C

O. Thus C o is indeed a zero

197

4.4. Fredholm denominators

is said to be the Fredholm denominator of T e U(E) associated with the continuous determinant b. In what follows, viewing T as a fixed parameter, the derivative of (V, T) with respect to the complex variable is denoted by 6' (C, T). 4.4.6. The next result shows that every continuous determinant can be reconstructed from the continuous trace 6. defined in 4.3.15. Hence the correspondence (5 is one-to-one. Proposition. Let (5 be a continuous determinant on a quasi-Banach operator ideal 11. If T e ?I(E), then the associated Fredholm denominator (3 G- , T) satisfies the differential equation

[FG., TA

(5(C, T)

for all

e e(T)

under the initial condition (5(0, T) = 1. Proof. Apply 4.3.19 to the ?1(E)-valued function T ) := T. (

4.4.7. The preceding result suggests the hypothesis that Fredholm denominators can also be obtained from continuous traces. This is indeed so. Theorem (J. J. Grobler / H. Raubenheimer / P. van Eldik 1982). Let a continuous trace on a quasi-Banach operator ideal IL If T e .1(E,) then e e(T) T) := T[F(C, T)] for all

T.

be

defines a meromorphic function with simple poles at all points o o e(T) and at these points only. Furthermore, the residue of the pole is equal to the multiplicity of the eigenvalue A 0 = — Proof. First of all, we know from 4.2.13 that T is a Riesz operator. Secondly, by 4,1.3, the function T ( ", T) is holomorphic on According to 4.1.6, about every point o (T) we have a Laurent expansion RC,

nE1

X('

0 T)

v R ( • „ T) —

nt`o

if

0 < r—

< 8•

Note that all coefficients belong to 94E). Since the operators X2 ( "0 , T), Xd(" 0 , T) are finite and nilpotent, it follows from 4.2.15 and 4.2.28 that their traces vanish. Moreover, T[X X .0 TA = "r[1) co(' + 0 1)] = n(I + 0 T). Combining these observations, we obtain

r(C , T) = n(I + + T[Rgo, TA „ g

0

n=o

— CO' if 0 < r — < E.

Thus !,- 0 is indeed a simple pole of .r( ", T) with residue n(I + 0 T). 4.4.8.

We now recall some well-known facts from complex analysis.

Let (5 be an entire function with (5(0) = 1. If we have

6(

)

=

—

6 0 (C)

is any zero of multiplicity n, then

198

4. Traces and determinants

where 60 is an entire function such that 60(ç O) * O. By logarithmic differentation we obtain r(C)

6.

(0

b(C)

— Co

±

beg)

•

kg)

This shows that r is a meromorphic function with simple poles. Furthermore, all residues are natural numbers. Conversely, given any meromorphic function r with these properties, we may solve the differential equation

6. (0 (5(C)

r(C)

under the initial condition 6(0) = 1 provided that r is holomorphic at the origin. Then

6(0 = exp [ r(y) dy] , the integral being taken along any path from 0 to avoiding the poles of r. Of course, the value of the integral depends on the path chosen, but when we take the exponential all ambiguity disappears. If Co is a pole of T, then we write

r(C)

" + To(C), — Co

where ro is holomorphic in a neighbourhood of Co . We now see from

1

6(C) =

exp T o(y) dy]

that the solution â has indeed a zero of multiplicity n at Co • 4.4.9.

The entire function obtained by solving the differential equation

— r[FG, TA

for all

e i2(T)

under the initial condition 6(0, T) = 1 is said to be the Fredholm denominator of T e 11(E) associated with the continuous trace T. 4.4.10. Next we put the preceding construction into an explicit form. Theorem (J. Plemelj 1904, J. J. Grobler / H. Raubenheimer / P. van Eldik 1982). Let r be a continuous trace on a quasi-Banach operator ideal K. If T e ?I(E),

then the coefficients of the associated Fredholm denominator co

„(T) Cn for all

6(C, T) =

eC

n =0

and of the associated Fredholm numerator 00

D(C, T) =

E A n (T) Cn n =0

for all

eC

4.4. Fredholm denominators

199

are determined in the following way: For n = 0 we have (0)

(x 0(T) = 1

and

A 0(T) = T.

For n = 1, 2, ... we may use either the recurrence formulas (1( 0) (R1)

1 n(T) = — T[A n-- i(T)] n =

E (- 1)n-"oc11(T) 1

h =o

11

or Plemelfs formulas /v(T)

0 2

T(T 2) T(T)

1 det n!

(Po)

1

'• r(T) n — 1 r(T2) r(T)

r(Tn --1) T(T n— 2 ) r(Tn) r(Tn -1 ) 1

T(T)

0\ 2

(P 1 )

A n(T) := 1 et c1 n!

1

T(T2) -r(T) r(T) r(T"') Tr'+' T"

r(T) ... T 2

n T

I

•

Proof. Formula (R 1) is verified in 4.4.3. Moreover, it follows from 6(0, T) = 1 that x o(T) = 1. Hence A 0 (T) = Observe that (5. , T) = T[FG, TA = T[Dg, TA . Therefore co co no(n(T) = r[A n(T)] V i • n=1 n =o Equating coefficients, we obtain (Ro). Suppose that a n(T)and A n(T) are defined by the formulas (P 0 ) and (P 1 ), respectively. Expanding the determinants by the last rows yields 1 n-1 ( 1)"-h-1. och(T) r(T" -h) o(T) = — n h= 0 and

T.

A I (T) =

E (- 1) " h (T)

h=0

Using the latter formula for n — 1 instead of n, we obtain n—1

=h ( =0

1)

n-h- 1 (x h (T)

-

Hence

1 A.„(T) = — r[A n_ 1 (T)].

T (Tn—h)

200

4. Traces and determinants

This proves that 0 c n(T) and A ri (T) satisfy the recurrence formulas (R o) and (R 1 ) which determine the coefficients of (V, T) and D ( , T) uniquely.

Remark. For another proof of Plemelj's formula (Po) we refer to 4.5.11.

4.4.11. Proposition (J. C. Engelbrecht / J. J. Grobler 1983). Let r be the spectral trace defined on a quasi-Banach operator ideal W. If Te 11(E), then the associated Fredholm denominator is given by the formula 00

T)

= H (1 + .2,(T))

for all

e C.

n=1

Proof. Let M IITI < 1. Then it follows from

T) = that

•

(— ok-1

co

T)

i) k_i

e,-k-1,r(Tk).

Hence, in view of the spectral mapping theorem 3.2.24, we have co 00 .0 co • T ) = E (_ 7-k-1 An t Tk, E E (_ ck-lA n(nk " n=1 k=1

k=1 co

„(T) (1 + A(T)) -1 .

=

n=1

This implies that

T) = 12-c' 1 (1 + N,„(T)) n=1

is the unique solution of the differential equation

T) (SG" , T)

=

T)

under the initial condition 6(0, T) = J.

4.5.

Regularized Fredholm denominators

Throughout this section, when writing Tm e ?I(E), we tacitly assume that T E

4.5.1.

The entire function E„, is defined by

exp (1 1 (-11)k

1 + C,n( ) : = (1 +

V‘)

k =1

for m = 2, 3, ... Furthermore, let s i (0 :=

Lemma. There exists an entire function Eng.) = Vng9„,(0

for all

(p,n

E C.

Proof. Assume that M < 1. Then it follows from log (1 + =

a'

(-1)k k

k

such that

e(E).

201

4.5. Regularized Fredholm denominators

that co

1

e m(C) = exp (—

(

_ i)k ck)

_ 1

k=m

(_ oni + / Cm — in

This expansion shows that the entire function e m has a zero of multiplicity in at C = 0. Therefore the holomorphic function

Tm(C) := ç' -rn eni(C) defined for C 0 possesses a removable singularity at the origin. 4.5.2. We now define e»1(T) by substituting T e t(E) in place of C e C in the Taylor expansion pm-1

e

" rn

It turns out that e(T) behaves like T'n

Lemma. Let It be any operator ideal. Then T"' e 11(E) implies e»(T) e /1(E). Proof. Note that s ni(T) = Tincp ,n(T).

4.5.3. We are now in a position to carry out the basic construction of this section; see 4.4.4.

Theorem. Let 6 be a continuous determinant on a quasi-Banach operator E 144 then

ideal W. If Tm

&(C, T) : = 6 [(I +

exp ("1:E 11 (—knk CkTk)]

for all

eC

defines an entire function with zeros at all points Co 0 e (T) and at these points only. Furthermore, the multiplicity of the zero Co is equal to the multiplicity of the eigenvalue 20 = —1/C0. Proof. First of all, we observe that the definition

6„,(C, T) : = 6[1 + e„,(CT)] makes sense, because E„,(CT)e ?1(E). Moreover, the composition 6[1 + E ,„(,7,1)] is holomorphic, by 4.3.19. Next we deduce from 4.3.9 and 3.2.24 that T is a Riesz operator. If Co e e(T), then / + ?,- oT is invertible, and so is in -1 (_ 1)k ' c'ocTk). ± Co n exP k=1

Hence, by 4.3.11, we have 6 (Co , T) O. If Co e (T), then we consider the Riesz decomposition

+ Co n

E = Mco (I + Co T) CD N

as described in 3.2.14. Let P : = Pop(' + Co n, and define

T = T(I — P)

and

TN

:= TP

202

4. Traces and determinants

Using the Taylor expansion of s in , it easily follows from T = T + TN and Tm TN = TN Tm = 0 that

sni(a) = e n (T m )

e U3(T N )

and e ,(T M) e

A.) = o.

Therefore

,,,(CT) = [I + ,,,((,T m)]

+ m(CT NA •

This implies that T N). (5 mG, T) = ô mg, T M) As shown in the proof of 4.4.4, we have

6(I +

N) = (1 —

o) n •

Hence in—

1

_ k f .k 1101

6 rexP kE

6,(C, T) = —

TM)

Since the operators

exp

( m -1 (-1)k clocni) k=1 k

I + o Tm

and

are invertible, we obtain

m —1 [exp (k ] 1

(-1)k

k

k k

TA)

0 and 6 ?go, TM)

0.

Thus Co is indeed a zero of 6 „i(C , T) with multiplicity n(I + 0 T). 4.5.4.

Let m = 1, 2, ... If Tm e t(E), then the entire function

(_1\k , T) :=

[(I + CT) exp (mkE:

V'Tk)1 all for

eC

is said to be the regularized Fredholm denominator of order ni associated with the continuous determinant 6. 4.5.5. Proposition. Let 6 be the spectral determinant defined on a quasi-Banach operator ideal /I If T ni e 11(4 then the associated regularized Fredholm denominator of order m is given by the formula m-1 (_ ok co . k2„(T)k)] for all ,;" e C. {(1 + CA „(T)) exp E n,(C , T) = k=1 k n=1 Proof. Extending the spectral mapping theorem 3.2.24, we can show that the operator E ngT) has the eigenvalues 8 ni(C2n(T)). This proves that CO

, T) = 6[1 + ET)] = fI[1 + m(C2,(T))] • Remark. Another proof of this result follows from 4.5.12, 4.5.8 and 4.3.23. 4.5.6.

Let T e t(E) and m = 1, 2, ... Then

F

T) := (-1)"1-1 Cm -1 1' (I + CT)- '

for all

e e(T)

203

4.5. Regularized Fredholm denominators

is called the regularized Fredholm resolvent of order m. Note that F,,„(C, T) and T commute. 4.5.7. The following formula shows that the functions F( ' , T) and F„,(C , T) differ from each other only by a polynomial.

Lemma. Let T E t(E). Then ni —1

F

T) =

F(

* , T) + k=

(-1)k —iTk for all

e e(T).

Proof. In view of 4.1.2, the above formula follows from rn — 1

CT)[F

T) + E ( _ )k-1 T ic] = k =1

m -1 = [(—CT)m -1 + (I + T) (—a)k-11 T = T. k

=

4.5.8. The following result extends 4.4.6 to the context of regularized Fredholm denominators.

Proposition. Let 6 be a continuous determinant on a quasi-Banach operator ideal 41.1. If T in e t(E), then the associated regularized Fredholm denominator ã , T) satisfies the differential equation ,

= 6' [F

T)] for all

e e (T)

under the initial condition 6,7,(0, T) = 1. Proof. Apply 4.3.19 to the 91(E)-valued function T(C) : = m(a) and use the fact that (I

4.5.9.

+

=F

T) for all

e e (T).

We now generalize the construction carried out in 4.4.7.

Theorem (J. J. Grobler / H. Raubenheimer /. P. van Eldik 1982). Let w be a continuous trace on a quasi-Banach operator ideal It. If Tin e 11(E), then

,n(C, T) :=

T)] for all

e e(T)

defines a meromorphic function with simple poles at all points o e(T) and at these points only. Furthermore, the residue of the pole o is equal to the multiplicity of the eigenvalue A o = — Proof. By Lemma 4.5.7, the meromorphic functions F(ç, T) and F T) have the same poles with the same principal parts. However, F T) has the advantage that T)) makes sense, and the proof all of its values belong to ?t(E). Therefore T(F can be completed by reasoning as in 4.4.7. 4.5.10. Let m = 1, 2, ... If T " e ?I(E), then the entire function obtained by solving the differential equation

T)

— r[Fm(C, T)] for all

c e(T)

204

4. Traces and

determinants

under the initial condition 6„,(0, T) = 1 is said to be the regularized Fredholm denominator of order m associated with the continuous trace -6. 4.5.11. The next theorem is an analogue of 4.4.10. Theorem (J. J. Grobler / H. Raubenheimer / P. van Eldik 1982). Let r be a continuous trace on a quasi-Banach operator ideal tt. If Tm E WE), then the coefficients of the associated regularized Fredholm denominator 00

ng T) = E oc(nin) (T)n n=o are given by Plemelj's formula (

(Po)

for all

x(T) T(T 2) -

1 c4, 1)(T) := n! det

eC

1 r(T)

0 2 ▪•

n— 1 r(T2) •r(T)

r(Tr• i— 2 )

r(T)

T(T)

r(T 1 )

z(Tm -1 ) replaced by zeros. Note that 40(T) = 1. but with .r(T), Proof. Let ICI IT < 1. Then it follows from

F03(, T) k=m

that T)

oe

_ k

(_i)k-i ck- 1 ..1.(Tk)

= ni

Above we made the convention that r(T) co

T) _ E

= 0. Thus we may write

•k-1,1-(Tk)

k=1

Hence, in view of the differential equation 6;n(C, T) , T), — T) we have E

n(n)()

n= 1

Ih =0 ar(T) =LE

r

00

[ E (- 1 )"C"r(Tk)

k=1

•

Equating coefficients now yields n- 1

nam)(T) = E (— on- h -1 04zno(nr(Tn - h). h =o

This implies that ocr(T),

ocr)(T) can be obtained by solving the linear system

=

1 oc(r)(T) z(T) • oc (inO(T) —

T( T)

= r( T2)

2 • oc ( m)(T) •

c4m )(T) — r(T

2) • cx."7)(T) +

+ (-1)".-1 n • ocr(T) = .4T").

205

4.5. Regularized Fredholm denominators

Therefore, applying Cramer's rule, we see that cx;,m)(T) is given by Plemelj's formula (Po ). In particular, we have o(T) = 0

for

m — 1.

n = 1,

Remark. Replacing w(T), r(Tm -1 ) and T, Tm -1 by zeros, we may define the coefficients A r(,m)(T) via Plemelj's formula (P 1 ); see 4.4.10. Then co

D

T) =

n=0

Ar(T)"

for all C e C

is an entire ¶E)-valued function, and we have Fm (C , T) — D T) T)

for all C e

4.5.12. Proposition (J. C. Engelbrecht / J. J. Grobler 1983). Let w be the spectral trace defined on a quasi-Banach operator ideal ?I. If Tm e ?I(E), then the associated regularized Fredholm denominator of order m is given by the formula m-1 (— l)k k CkAn(T)k)] for all C e C. T) =(1 fi[ l ± CA n(T)) exP ( kE 1

Proof. Let M IT < 1. Then it follows from GO

1 Tk

F T) = E (— k = in

that m (c , T) = E (_ k =M

1) k_1

Hence, in view of the spectral mapping theorem 3.2.24, we have

E (1)'

T) =

k =m oe

00

2n(T k ) =

n=l

= E (-1)---' cm -1 2,(T)n(1 + n=1

00

CO

E E

(_ o k-1

-1

)k

n=1 k=m V.,(T)) -1 .

This implies that m-1 ( 6„1(C , T) = fT 1 + CA (T)) exp E k=1 k n=1 is the unique solution of the differential equation

ck2,z(T)k)1

[(

(5 (C, T) = m (, , T)

TX, T)

under the initial condition 6(0, T) = 1. 4.5.13.

From the preceding proposition we obtain an interesting corollary.

Theorem. Let r be a continuous trace on an approximative quasi-Banach operator ideal ?I. If Te W(E), then co 6 2(C,

= ri [(1 + n=1

(T)) exP (—CA(T))]

for all C e C.

4. Traces and determinants

206

Proof. As shown in 4.2.36, the restriction of

r

to ?V is spectral.

Remark. In view of 4.2.37, the approximability assumption can be dropped when T) with in 3. we consider

4.5.14. We now compare regularized Fredholm denominators of different orders with each other. It. If

T

Proposition. Let r be a continuous trace on a quasi-Banach operator ideal e /1(E) and in p q, then

T) exp [qi1 (—ki

(5G , T) =

" )k

‘7,-kr(Tk•)

for all

e C.

k =p

, T) and (g* , T) have the same zeros, there exists an entire function Pro of. Since T) such that T) exp (E(.

, T) =

In view of 6„(0, T) = 1 and 6,(0, T) = 1, we may assume that 6(0, T) = 0. By 4.5.10,

(5PG T) = -r[Fig , T)]

T)

and

= T[Fag", T)].

Furthermore, 4.5.7 implies that q-

1

T) = E (- ok 1 Tk

Tg", T) —

k =p

Hence T)

q —1

= E (_

)k

1 i(Tk)

k =p

and we obtain q-1

k =p

4.5.15. According to the previous results, regularized Fredholm denominators of order In are defined for all operators belonging to the following quasi-Banach operator ideals: t) .c) t(rx) t(ry) r

•, 2

)

if m

r,

if in > r and m even, if in > r> 2 and in even.

In the case when in is an even number, then (93„,)"1 admits a spectral trace. Consequently, the regularized Fredholm denominator ,„( " , T) exists for all operators T E m (E). We do not know whether this is so for in odd.

The relationship between traces and determinants 4.6. 4.6.1. We know from Theorem 4.3.15 that every continuous determinant ô defined on a quasi-Banach operator ideal ?I determines a continuous trace

(T) : = lim

for all

Te

207

4.6. The relationship between traces and determinants

Furthermore, by 4.4.6, the correspondence 6 > 6' is one-to-one. Thus the question arises whether every continuous trace can be obtained in this way. In the following we give an affirmative answer. —

Proposition. For every continuous trace r defined on a quasi-Banach operator ideal ?-1 there exists a continuous determinant 6 such that -r = 6' . Proof. Let TE 11(E), and recall from 4.4.7 that r[T(1 + 'T)-. 9 is a meromorphic function with simple poles, the residues being natural numbers. If I + T is invertible, then we define

6(1 + T) := exp

-r[T(/ + yT)--1 1dy),

where the integral is taken along any path 1' from 0 to 1 avoiding the poles of the Fredholm resolvent. Of course, the value of the integral depends on the path chosen, but when we take the exponential all ambiguity disappears. Next we show that the Gateaux derivative 6 . (1

s)

hill 6(1 + T + "S) — 6(1 + T)

exists for all S e ?t(E). To this end, let := sup Illy (I + 'T) -1 11:y e < 1, then

If II

> O.

6( 1 + T + S) = exp f r[(T + S) (I + yT + "yS)-1- ] dy). Easy manipulations yield the Taylor expansion

(T +

+ yT +

= T(I + yT)-1 + (I + yT)-1 S(I + yT) -1 +

converging uniformely on the disk 1•1 < 1. This implies that 6(1 + T + = 0. is differentiable at Furthermore, we obtain 6*(1 + T, S)

6(1 + T)

= j .4(1 + yT)-1 S(I + yT) -1 ] dy. 0

The obvious Taylor expansion

yS(I +

= yo S(I + yo T) - ' + S(1 + 70 n -2

shows that

r[yS(1 + y1') 1 ] is a primitive of r[S(I + yT)-2 ]. Hence

r[(1 + 7 T) -1 S(I + yT) -1 ] dy = f -r[S(/ + yT) -2 ] dy = (x[yS(I + yT) -1 ]):1) = r[S(I + T)-1].

yo) + • • •

208

4. Traces and determinants

Thus we have the Fredhohn formula

6' (I + T, S) = x[S(I + T) ']. 6(1 + T) In particular, if T = 0, then it follows that 6. (S) = In the case when I + T is non-invertible we define X/ + T) := 0. It remains to prove that the function T 6(I + T) is indeed a continuous determinant. Write 6(?;, T) := 6(1 + a). Then Fredholm's formula implies that T)

T)

E e(T).

= r[F(, T)] for all

Thus (5(e,-, T) is the Fredholm denominator associated with the continuous trace T. This means that (5(1 + a) is an entire function with zeros located on the complement of e (T). Hence condition (D4) is satisfied. In the case T a® x we have

T(I + yT) -1 =

a C) x 1 + y<x, a>

if

1 + y<x, a> * 0 .

Therefore it follows from

r

r(a 0

r

<„,c , a>

dy = log 1 + <x, a>) 1 + Y<x' a> dy j 1 + y<x, a>

that

(D1)

(5 (/ + a® x) = 1 + <x,a>.

Next let T e ?1(E, F) and X e t(F, E). Note that 1E + y XT and IF + yTX are simultaneously invertible or not. Since

XT(/ E + 2XT) -1 = X(IF + yTX) -1 T, we have

z[XT(I E + 7XT) -1 ] = z[TX(IF + yTX) -l ] • Hence (D2)

6(/E + XT) = 6(IF + TX).

Assume that S, TE W(E). If I + T is non-invertible, then N(I + T) 4z {o}. Hence (I + S) (I + T) is non-invertible, as well. Thus the multiplication formula

6[(I + S) (I + T)] = 6(1 + S) 6(1 + T) becomes trivial. Next we treat the case when / + T is invertible. Write

S, T) : 6[(I + S) (I + T)] = 6[I + T + S(1 + T)]. Applying Fredholm's formula, we obtain

S, T) = T[S(I + T) ((I + 6 ( ", S, T)

(I + T))-1 ] = z[F(ç, S)].

4.6. The relationship between traces and determinants

209

Hence the entire functions (SG, S, T) and 6 ( ', S) satisfy one and the same differential equation. This implies that

S, T) = c06(, S), where co is an integration constant. Putting ç = 0, we conclude that co = 6(0, S, T) = 6(1 + T). Finally, letting = 1 yields (D 3)

= 6(1 + S) 6(1 + T).

(5[(/ + S)(/ +

Since the trace r is assumed to be continuous, there exists a constant c that

1 such

forai! S e

S I ?III 1'0)1 Let 20T1?111 < 1. Then

1 2 for 0 y 1. 1 —T[ I — Hence, integrating along the straight line from 0 to 1, we obtain

II +

r[T(I + yT)-1 ] dy

chT

o

if

+ yT) -1 11 dy

2 cllTJ %II

1.

Since lexp () —1"1 exp (1'1)

for all

e C.

it follows that l(5(/ + T) — 11 = exp I r[T(I + yT)-1 ] dy) — 1

2ce11T I II.

Thus the determinant is continuous, by 4.3.10. Remark. Given any continuous trace r, the associated continuous determinant 6 could be defined by co

6(1 + T) = 1 +o(T) for all T E W(E). n=

where the coef ficients NAT), oc2 (T), ... are computed either recurrently or by Plemelj's formula; see 4.4.10. However, we are unaware of any direct proof of the convergence of the right-hand series. To obtain this, it would be necessary to verify that lim 4.6.2. define

104ni lln = 0.

Let Te 2,(E) be a Riesz operator such that / + T is invertible. Then we log (1 + T) := f T(1 + FT)' dy,

where the contour integral is taken along any path from 0 to 1 avoiding the poles of the Fredholm resolvent. Determinations of log (1 + T) obtained from different

210

4. Traces and determinants

paths differ from each other by summands of the form 27:i

E

.J=1

niPCo (' +

where n 1 , •••, nk are integers. In the case fl < 1 we get the principal branch of the logarithm by integrating along the straight line from 0 to 1. Then, as in 4.3.16, we have co

log (/ + T) = —

I k=1

( —1k )

Tk.

it

Using the logarithm just introduced, the preceding proposition can be rephrased as follows. Proposition. Let r be a continuous trace on a quasi-Banach operator ideal IL Then

(XI + T) :=

exper[log (I + T)]) 0

if I + T is invertible; otherwise

defines a continuous determinant on I. Remark. Note that r[log(/ + T)] is determined up to a summand 27in, where n is an integer. Hence exp (r[log(/ + T)]) is well-defined.

4.6.3.

We now summarize the results stated in 4.3.15 and 4.6.1.

Trace-determinant theorem. There exists a one-to-one correspondence between continuous traces and continuous determinants on every quasi-Banach operator ideal.

4.6.4.

Combining 4.3.23 and 4.4.11, we arrive at the following result.

Theorem. A quasi-Banach operator ideal admits a spectral determinant if and only if it admits a spectral trace.

4.6.5. In view of the preceding theorems, all quasi-Banach operator ideals listed in 4.2.34 admit a continuous determinant or even a spectral determinant. 4.7.

Traces and determinants of nuclear operators

4.7.1.

We begin with an analogue of Lemma 4.2.2.

Lemma (GRO, Chap. I, p. 165). Let T e 91(E). If the Banach space E has the approximation property, then 00

1-(T) :=

E

i =1

ai>

does not depend on the special choice of the nuclear representation T=

E

s=i

ai x i ,

4.7.

211

Traces and determinants of nuclear operators

and we have IITJ 9 II. 'TVA Proof. First of all, we show that co trace (TL) = E for all

LE

i= 1

a(E).

To this end, let

L=Eki C)yi be any finite representation. Then it follows from

TL = E b. C) Ty; ; =1 that trace (TL) =

> =

n

,

1 =1 co n

= E E <xi ,

J=1 1=1 co

to = E . i=1

i=1

Since E has the approximation property, given any precompact subset K, we can find LK e such that

a(E)

— LKx11 .5_ 1 for all x e K. By Lemma 1.7.4, there exists (x i) e co such that 00

Ecç 1 x1lI 1a111 < 00

and x i >0.

1=1

Obviously, x i : i = 1, 2, ...) K° is precompact. If e> 0, then for every precompact subset K with eK i = 1, 2, ... it follows that II xi — LKx i ll eaT 1 Ilxi II

Ko and

Consequently, CO

— trace (TL K)

lc°

= E <xi —

1=1

1=1

ai>

8 Ec° 0( i-1 1141 Ilai II i=

Note that the collection of all precompact subsets is directed upwards with respect to the set-theoretical inclusion. In this way (trace (TL K)) may be viewed as a general-. ized sequence. Letting e 0, the preceding estimate implies that 00

E <xi , ai> = lim trace (TL K). 1=1

212

4. Traces and determinants

Since the right-hand expression does not depend on the underlying nuclear representation, the same holds for 00 -c(T):=

<xi , at>.

Finally, we conclude from

'TVA that

t1 IlxIl dad

i=

IT (T)ITJ 91 1 .

Remark. Conversely, it can be shown that a Banach space E must have the approximation property if T(T) is well-defined for all operators Te91(E); see (DIL, pp. 239-240) and (GRO, Chap. I, p. 165). Theorem. There exists a unique continuous trace on the Banach operator ideal% restricted to the class of all Banach spaces with the approximation property.

4.7.2.

Proof. It can easily be seen that the function T r(T) defined in the preceding lemma is indeed a continuous trace. Furthermore, we know from 1.7.5 that the finite operators are dense in 91 (E). This proves the uniqueness. Next we provide some auxiliary results. Lemma (A. Grothendieck 1956: a). An operator Te t(E, 11 ) is nuclear if and only if the sequence (T'e s) is absolutely summable. When this is so, then 4.7.3.

11T 1 9111 = E 11T'eili 1=1

Proof. If (T'et) E [li , El, then T admits the nuclear representation 00 T'e t C) E T=

et ,

i=1

and we have CC

IT! 9111

E 11T'eill i= Conversely, assume that Te 91(E, 11). Given any nuclear representation

T = E 0 y,, J =1

it follows that Cfj

T'e t = E a<3, , e s > for

i = 1, 2,

and

lyII = iE =1

e 1 >1

for j = 1, 2, ...

4.7.

Traces and determinants of nuclear operators

213

Hence 00

00 00

E Ilreill -5_ E i=1

00

1141 KY.i, ei>I

I 11411, j=1 1=1

ei>I

III IIM. J =1

This implies that

E

(IT 194.

i=

4.7.4. The ideal of nuclear operators fails to be surjective. It has, however, the following property. Lemma. Let T e WE, /1 ). If there exists a surjection Q e 2(E0 , E) such that TQ e 1(E0 , /1), then T e 9I(E, /1). Proof. By the preceding criterion, TQ e 94E0 , 1 ) implies that (Q ' rei) G [Ii , Since Q' is an injection, we have (T'es) e [l, , E']. Hence T E 9(E, 11)Since the Banach space /1 has the approximation property, every operator T e91(1 1 ) possesses a well-defined trace. However, the following example shows that this trace behaves rather strangely. It was originally constructed by P. Enflo (1973). Simplified approaches are to be found in (LIN, Vol. I, 2.d.3) and (PIE, 10.4.5). 4.7.5.

Theorem. There exists an operator S e 91(/1 ) such that

r(S) = 1

and

S' = O.

Remark. The nilpotent operator S cannot possess any eigenvalue 2. 0 * 0. Thus it is impossible to compute the trace r(S) ---- 1 from the trivial eigenvalue sequence (0, 0, ...). 4.7.6. As an easy consequence of the preceding example we now establish the most unpleasant result of this monograph. Theorem (A. Pietsch 1981: b). There does not exist a continuous trace on the Banach operator ideal VI considered on the class of all Banach spaces. Proof. Consider an operator S e9(11 ) such that r(S) = 1 and S 2 = O. Denote by Q the canonical surjection from 11 onto the quotient space E : = 11 1 N(S). Furthermore, let T e t(E, /1 ) be the operator induced by S. Then S = TQ, and we have the diagram

0 It follows from S' = 0 that /t/(S) -N(S). Therefore QT = O. Moreover, Lemma 4.7.4 shows that T e 91(E, 10. We now assume that there exists a continuous trace a defined on all components of the operator ideal 91. Then

1 = r(S) = a(S) = cr(TQ) = a(QT) = a(0) = O. This contradiction completes the proof.

214

4. Traces and determinants

Remark. As a by-product of the preceding construction we see that the quotient space /,/N(S) cannot possess the approximation property.

4.7.7. Next we prove a counterpart of Lemma 4.7.1 which holds in arbitrary Banach spaces. Lemma. Let T E 911,2(E). Then CO

T(T) := E (x i , ai > i= 1

does not depend on the special choice of the (1, 2)-nuclear representation, CO

T = E a. C) x i 1=1

and we have

ir(ni '6 11T i911,2ii • Proof. Let CO

E a, C) xi

i=1

be any (1, 2)-nuclear representation of the zero operator. Then we define the operators A„, E 91(E, 1 2) and X„, E t(l2 , E) by

ai 0 Am := i=m+i

6' 1

and

X„,:=

E

i=m+ t

e,

xi .

Note that the finite operators

ai 0 x i ai 0 xi = — i=rt/-4-1 1=1 admit the factorizations T,„= —X,„A„,. Furthermore, by B.4.6, there exist orsuch that X„,21,,,= X„,P„,A m . We now obtain thogonal projections Pm e T,„:=

a(12)

111

= 'trace (P„,A „,Xm)I 1/2 : a e u°) . Kx 1 , a>12 )

E <x1, ai> = Itrace (Tm)1 = 'trace (X„,P„LA

i= I

Co

5- liP mAniX.1 91 11

i=m+i

Ilaill) sup {( i=

oo yields

Letting in CO

E

1=1

<xi , ai> = O.

This proves that r(T) is well-defined for all operators Te 911 ,4E). The inequality Ti 91 1 . 2 11 is obvious. Jr(T)1 The following statement is an immediate consequence of the preceding 4.7.8. considerations. Theorem. There exists a unique continuous trace on the quasi-Banach operator ideal 91 1 , 2 . Remark. We know from 1.7.16 and 2.8.20 that ($ 2 ) 1 . (V0 2 and 91 1 . 2 9I1,2

215

determinants of nuclear operators

4.7. Traces and

Therefore the assertion could also be obtained from 4.2.6 or 4.2.7. In view of 4.2.30 and 4.2.31 the continuous trace defined on 91 1 , 2 is indeed spectral. For a completely different proof of this fact we refer to 4.7.15. 4.7.9. According to 4.6.1, every continuous trace induces a continuous determinant. This result remains true if the underlying quasi-Banach operator ideal is considered on any subclass of Banach spaces. Hence we have the following corollary of 4.7.2. Theorem (A. F. Ruston 1951: a, A. Grothendieck 1956: a). There exists a unique continuous determinant on the Banach operator ideal 91 restricted to the class of all Banach spaces with the approximation property. 4.7.10. The continuous determinant defined on the restricted Banach operator ideal 91 can be described explicitely. To this end, we use the Taylor coefficients of the associated Fredholm denominator defined in 4.4.10. Lemma. Let T e 91(E). If the Banach space E has the approximation property, then

1 n!• 1 CO

<xii , aii> ... <xi,, ain>

CO

in- _-1

det

-

<xin , a • i >

<xin , (20

does not depend on the special choice of the nuclear representation CO

T = E ai 0 i=

xi ,

and we have n

n/2

I 91 1i n •

IN„(T)I -5_ -

n!

Proof. We consider the finite operators

T„,:= E ai

X.

i=1

Then

6(1 + T„,) = det (b u +

<xi ,

ai>).

According to a well-known formula from multilinear algebra, the right-hand determinant can be written in the form rn

1+

n=1

where Ifl

=

fl

.

ni

( (xii , ai,> ... <xii , ain>

E E det j1

i„= 1

<xin , aii> ... <xin , ain> . )

In view of 4.4.10, the coefficients c(T) are continuous functions of T. Thus, letting m -3 cc yields the formula we are looking for.

216

5. Traces and determinants

Given s > 0, we choose a nuclear representation Co

T = E ai 0 x i

1=1

such that Co

1

T 9111

E= 1141 —5_ (1 + 1

1 for i = 1, 2,

and

Then Hadamard's inequality A.4.5 implies that

<xii , aii >

<xi,, ain>

<xi,, ai,>

<xin , ain>

det Hence n

n /2

co

co

E n! 1 1 =1 i„=1

lan(T)I

\

ni 2 / or,

E Ilaill)

n!

i=1

(1 + e) n

nn / 2

T n! I I I

0, we obtain

Letting e

nni2

locn(T)I

n!

4.7.11. The next result follows immediately from 4.7.8 and 4.6.1. Theorem. There exists a unique continuous determinant on the quasi-

Banach operator ideal 91 1 , 2 . 4.7.12. We now prove a counterpart of Lemma 4.7.10 which holds in arbitrary Banach spaces. Lemma. Let T E 91 1 , 2 (E). Then

1

...

n. 11

(xii , ai1 >

co

°°

(xii , ain>

E det

in =1

ai,> <xin , ain> does not depend on the special choice of the (1,2)-nuclear representation co

T = E ai C)

i=1

and we have

lan(T)I

1

91 1,211 n

.

!

Proof. The formula for the coefficients oc(T) can be obtained in the same way as in 4.7.10. In order to verify the estimate, we choose a (1,2)-nuclear representation Co

T = E ai 0 x i

i=1

4.7. Traces and

determinants of nuclear operators

217

such that

E Ilaill 5 (1 + E) 11T I 91 1 , 2 11 and E Kxi , a>1 2 5 1 i=1 i=1 Then Hadamard's inequality A.4.5 implies that <xii , aii>

air)

det

n

•

•

(X1

H

1

--- 1 p

aiH>

,

/n

5. 44,11

1<xi„, ai >1

2

\

for ac U°.

1/2

)

Ilai11.

Hence

1 c° 104T)1 5_n71

Co

44,11 n

/ 00

E Ilaill) n! i=t Letting

Ilai„11 +

T

I 911,211".

0, we obtain 1

171911,21l n -

4.7.13. We now investigate the eigenvalue distribution of (1,2)-nuclear operators by the classical method based on the theory of determinants and entire functions. Theorem. The operator ideal 91 1 , 2 is of eigenvalue type l. Moreover,

11 (2„(T)) J111

11T 191 1 , 2 1 for all T e 91, , 2(E)

Proof. We consider the Fredholm denominator Co

c5(, T) := 1 + E n (T) r for all

e C.

n=1

In view of the estimate

icx.(T)1

1 — n!

911,211"

it follows that ao

kV,

1

1 + E n. 117'1%1,211n 1 - 1 1 ' < exp (11T1 911,2111)• n=1

Since the zeros of (5“-, T) and the eigenvalue of T are related by the formula =-- — 1 liln(T), we see from 4.8.6 that (A(T)) e . Hence (2„(T)) e li+, for all E > O. At this point the old-fashioned method terminates. However, using the recently discovered principle of tensor stability, we are able to deduce from 1.7.12 that all T E 911.2(E). II(ÀM) I ill I T 11,2 4.7.14. We now prove the main result of this section. Theorem (V. S. Lidskij 1959). The operator ideal 911 , 2 admits a spectral determinant.

218

determinants

4. Traces and

Proof. Let 7' E 911 , 2(4 and consider any (1,2)-nuclear representation CO

E ai x i

T=

i=1

such that CO

CO

E 1141 <

co

i=1

and

E 1<x„ a>1 2

1

i=1

for

a e U°.

It is known from the proof in 4.7.12 that ( <xii

, aii > ... <xii , ain

>

det

<xin , a1 > ...

ai.>

Hence, writing the coefficients cx „(T) in the form <x1 , , (20 ... <x11 , (20 n (T) = E det <<

<xi., a11>

<xin , a1n>

we obtain the estimate CO

TA

—5_ 1 +n=1 E (cvn(T)! co

CO

1 +E E11(1 + !tail! 1 -1)n= li l <...< in i= 1 Given tz > 0, we choose a natural number k and M> 0 such that

Etatt

i.---k +1

-5 la

and

II (1 + Ita1ll 1 .1)

1 =1

M exP (ittrI)-

Then IW ni

(1 + H 1=1

kid

M) i=k+1 ri (1 + ta1II r1)

M exp (1,C- 1) exp

_

M exp (a

This proves that (5(, T) satisfies condition (1) of 4.8.9. From 4.7.13 we know that condition (2) is fulfilled, as well. Therefore the Fredholm denominator C", T) admits the representation CO

CO

• (V-, T) = H (1 — n=1

= H (1 + .2,2(T))

for all c; e C.

n=1

4.7.15. We are now in a position to give the classical proof of the trace formula which especially applies to nuclear operators on a Hilbert space; see 4.7.8 (Remark). Theorem (V. B. Lidskij 1959). The operator ideal trace.

Proof. Let T e 91 1 , 2 (E). Comparing the coefficients of in

(5G- , T) = 1 +

co

E <xi , ai>) +

911,2

admits a spectral

4.7. Traces and

determinants of nuclear operators

219

and (5(C, T) = 1 + (E

n=1

).„(T)) +

we obtain r(T) :=

E <xi , ai > =

1=1

n=1

4.7.16. In order to prove the next theorem, an auxiliary result is required. 1. Then

Lemma. Let 0 < r

O.

for all e

exp (1 er)

1+ e Proof. We have

log (1 + &)

=

J

1 +

f er-1 (21 = —1r

de

er•

4.7.17. We are now prepared to estimate the growth of the Fredholm denominator of (r, 2)-nuclear operators. Proposition. Let Te 94, 2 (E) with 0 < r T)I

exp

II T I 94,21r

1. Then

) for all ,7; EC.

Proof. Given e > 0, we consider an (r, 2)-nuclear representation CO

T=

E ai

xi

i=1

such that

,

,1 /,

co

E

i= 1

!lair)

co

., --. (1 + e) 11 7 ' 191r,211

E 1<xi , OF

and

1

for

ae

1=1

As in the proof of 4.7.14 we obtain I (5(C, TA --

12-1 (1 + 1141 I -1).

1= 1

Therefore, by the preceding lemma, it follows that I, T)I

exp (-1

Mr)

r i= i

exp

(1 ± EY 11T I

91 r,21I r Mr

0 yields the estimate we are looking for.

Letting e

4.7.18. Next we establish a counterpart of Lemma 4.7.16.

Lemma. Let 1 (Ji for all e C

r

2. Then

e2)112 iexp and e > 0.

r 4 exP [ -T:(K1 r 2+

e2P2]

)*

U° .

4. Traces and determinants

220

Proof. Write = $ + in with $, e R. Then (11 + I2 +

2)1l2 ■ 12 _2

lexP (— where 6 2 := $ 2 + 71 2 + e 2 and —

e)

)

1 = (1 +

2$ + 0'2) 1 / 2 exp

+a. Note that the function

f(e) : = (1 + 2$ + 62) 1 / 2 exp (—$) defined for all $ —M1 + a2) attains its maximum at $0 := f(e) exp If a 2, then a2-r < 22-r. Therefore we have f(0 < exp (.12_ a2 ) 5_ exp

If a

(e)

1 cc2 Hence

exp ( 1:42).

—a. Thus the function f($) is decreasing on the interval [—a, oo).

2, then $0

Hence

f(e) f(- a) = — 1) exp (a) 5_ exp (26 —2) _< exp (— r42r ce) for all $

—a. This completes the proof.

2, then there does not exist any continuous trace on the operator ideal 1r,2. However, given T e 91,., 2 (E), we may consider the regularized Fredholm denominator

4.7.19. If 1 < r

(5 2 ( ", T) = 6[(I + T) exp (—a)] for all c C, where ô is the spectral determinant defined on (94, 2) 2 . In the case of Hilbert-Schmidt operators the following inequality goes back to T. Carleman (1921). Proposition. Let Te 91,., 2 (E) with 1 < r la2g, TA

4

exp

I 91r,21Ir

2. Then

for all

e C.

Proof. Given e > 0, we consider an (r, 2)-nuclear representation CO

T = a Ø xi i=1

such that co

(E \ i =1

(1

E Kxi , 01 2 + 017 191r,2,11 and 1=1

1

for a e

Write in

T := y ai 0 x i . 1=1 Then

2 G.,

= 6[(I + T exp (

= 6[1. +

exP [— .*T(Tni)]

in

= det

+

ai>) exP (—

J=1

ai>) -

4.8. Entire functions

221

It now follows from Hadamard's inequality and the preceding lemma that \ 1/2 in

in im

=

i=1

j=1 i= 1

fl exP j=1 1/2

{(11 +

aj>1 2 + E <xi, ai>1 2) (--lexP

a;>)1]

inn

m

exp

,__

+ C<xi, af>1 2)

11

2( . T,)1

14

rI 2

(i n l I <xi , c)1 2) IlajlI r Kr] = exp

exp

exp [(1 + E )r Passing to the limit as m -* co and

Mr]

I 94,211r 0, we now obtain the desired inequality.

E

4.7.20. Finally, we investigate the eigenvalue distribution of (r, 2)-nuclear operators by the classical method already used in 4.7.13. Theorem. The operator ideal 91, 2 is of eigenvalue type iiRi(T))1

4. Moreover,

411 < JTj91., 2 11 for all T e 94, 2(E).

Proof. By 4.7.17 and 4.7.19 we have exP (PIT 11,2li r Mr] if O

if 1

r

1

and 02(C, TA 5 exP (i(di" I

2,

where the constant y > 0 depends on the exponent r. Since the zeros of 6(C, T) and the eigenvalues of T are related by the formula C„ = —112,„(T), we see from 4.8.6 that (2,(T)) e 4, 03 . Hence (A(T)) e 4.4_, for all e > O. Thus, in view of 1.7.12 and 3.4.9, we have 11(An(T)) J l,.JJ-<- IIT I%r,2Il

4.8.

for all T e 94,2(E).

Entire functions

For the convenience of the reader we now prove some basic results from the theory of entire functions which are used throughout this chapter. Complete treatments are to be found in (BOA), (LEV) and (VAL), for example. The classical textbook of E. C. Titchmarsh (TIT) can be recommended, as well.

4.8.1. An entire function 95, is a complex-valued function defined and holomorphic in the complex plane. This means that about every point Co there exists a Taylor expansion CO

940 = En=0 c‘n(Co)(C — Co)". Here and in the following, if not otherwise specified, C denotes an arbitrary complex number. For simplicity, we always suppose that (140) = 1.

4.8.2.

A complex number C o is said to be a zero of the entire function ço if ço(C o) = 0,

222

4. Traces and determinants

and its multiplicity is defined by

n(Co) := min {n: Nn(Co)

O}.

The set of all zeros does not possess any finite cluster point. Therefore it may be arranged as a sequence (C,z) such that the following conditions are satisfied: (1) Every zero Co is counted according to its multiplicity. This means that it occurs n( 0)-times, one after the other. (2) The zeros are enumerated in order of non-decreasing magnitude:

ICil

00 •

1C2I

(3)

In case there are distinct zeros having the same modulus these can be written in any order we please. If ço has less than n zeros, then C„ := co.

4.8.3.

We repeat the definition of the primary factors m—1(

1

exp

e ,g) := (1 +

k=1

r

already given in 4.5.1. Note that e i( C) := Lemma. Let 0 < q < 1. Then there exists a constant c> 0 such that

— 77c, ICIV"

1E401

q.

if

Proof. Put

_E

oc

k = in

It follows from

II

1

1 in 1 — q

1 —1—q

that

je„,(01 = lexp 4.8.4.

(a) — 11

I exp (Ioc1)

11

exp

g

The following classical result is basic for the theory of entire functions.

Theorem (K. Weierstrass 1876). For every sequence of complex numbers + 0 such that C„ -± co there exist entire functions with zeros at these points and at these points only. A function having this property is given by the formula I n—1 1 co r Gg.n)k)1 , (W) 940 = fl [0 — / C.)exP n=1

where the non-negative integers m 1 , m 2 , ... are chosen such that c0

EiCICV n=1

< co for all CE C.

In view of Lemma 4.8.3, this condition guarantees the convergence of the above infinite product.

223

4.8. Entire functions

Of special interest is the case in which we can find a fixed exponent m such that CO

<

Eknl

n= 1

co.

Then (W) is called canonical Weierstrass product. In particular, if co < co, I IC n=

then we have

99(C) := 4.8.5.

n=

ci — cg-n)•

We now establish a fundamental inequality.

Proposition (J. L. W. V. Jensen 1899). Let C i , modulus less than e > O. Then

Cff, be the zeros of q) with

m

sup {IT(C)I = el . IC 11 ,n1 — Proof. Define the entire function v by ni 2

V(0 = (PG) e- m n = 1

_ CC: 7

-

n

•

Then we have

p ( 0) =

o '" Cl

•••

Cm

•

Observe that 6,2

CC: =

o

if

= P. -

Applying the principle of maximum modulus, we obtain

IV(0)1 — sup tIVOI ICI = 0.1 = sup {q) II =

.

This completes the proof. 4.8.6. If an entire function satisfies an estimate as stated in the following proposition, then the infimum of all admissible exponents r is said to be its order.

Theorem (J. Hadamard 1893: b, J. L. W. V. Jensen 1899). Suppose that

19)01 M exP (MCI') for all C e C, where M, ,u and r are positive constants. Then (C„-1 ) e Proof. Fix any natural number n. Put

e:=

m : = max {k: ICkl <e}. Then Jensen's inequality 4.8.5 yields e=

leCn i

and

lCn eIm IeCnI m-n = 1-11

leCnI m

< M exp (ttleC n r).

224

4. Traces and determinants

Hence n

log M +

This proves that n l/ri C I

1

(2,u)iire whenever n

2 log M.

4.8.7. We now express the condition assumed in the preceding theorem in terms of Taylor coefficients. Proposition (E. Lindel6f 1903, A. Pringsheim 1904). Let CO

q)(0 = E n=

o

for all eeC.

oc„ - n

Given any positive number r, the following are equivalent: (1) There exist M> 0 and ,u > 0 such that M exp (urn

19)(C)1 (2)

e C.

for all

There exist A > 0 and a > 0 such that for all n = 0, 1, ...

A(N/n)nir

loc n i

More precisely, it follows from (1) that (2) holds with oc := pre. Conversely, (2) implies (1) for all ,u > oc/re. Proof. By Cauchy's formula we have

r)

1 ( 949

= n

n+

ICI= e

Hence

M exp (pe r) e — .

Ia n ! Taking

:= (nl,ur)r, we get

M(ure/n)nir. 1Nn1 Therefore (1) implies (2). To prove the converse, we fix Put e : = II and choose the index m such that m 2rera < m + 1. Note that the maximum of the function f() := (^TRW' with 0 < < a) occurs at

:=

ocer/e.

Since f(N = exp (aerler), we have

ni

E

fi

(m + 1) A exp (xerfer).

n

n=0

Moreover,

loc„C"I

A(cxer/n)thir

A/2"

if n

2reroc

implies CO

E ocnCn

n=m + 1

A.

(2)

4.8. Entire functions

225

Combining (1) and (2), we obtain

IT()1

A [(2

+ 1) exP (octfier) + 1] .

Hence, whenever It > aler, condition (1) holds for all

M

sufficiently large.

4.8.8. In order to prove the following theorem, we need an inequality which goes back to E. Borel (1897). Its final form is due to C. Carathéodory who communicated his version to E. Landau (LAN, p. 89) in 1905; see also F. Schottky (1904). Proposition. Let 0 < < < co. Suppose that the complex-valued function y with y(0) = 0 is defined and holomorphic on a domain containing the disk 2 1 . Then 2(,20 sup { i/P()1 sup {Re v(0 : = Oo Proof. Put := sup {Re,():IçI = Applying the principle of maximum modulus to exp ( p), we obtain Re

y(e,-)

ix

whenever

In particular, we have 0 = Re y(0) a. Obviously, if N = 0, then everywhere. Therefore we may assume that oc. > 0. Then the function

v

vanishes

eivG)

T(0

— v(0) is holomorphic on a domain containing the disk ICI AG) := Re v() and fl(C) : = 1m v(). Then

fl(ç)2 6'4°2 IT(01 2 = (2/x — (X 40)2 + AO'

1

if

Let

ICI =

Hence the principle of maximum modulus implies that

1

whenever ICI

el.

Since

vG) —

ei + (=PG)

it follows that

2 'e° if = eo • el — eo This completes the proof of the desired inequality. IV(01

4.8.9.

The next result is closely related to Hadamard's factorization theorem. Theorem (E. Lindel6f 1905, VAL, p. 90). Suppose that the entire function cp with (p(0) = 1 has the following properties: (1) Given ,u > 0, there exists M > 0 such that Iço(C)1 M exp (4ttIO for all ?; e C. (2)

(,T 1 ) e It-

4. Traces and determinants

226

Then ço coincides with the canonical Weierstrass product

12° 1 (1 — C/Cn) •

n =1

Proof. Since q) and n have the same zeros, there is an entire function s such that

9)(C) = exp (8(C)) t(). Moreover, because ep(0) = 1 and a(0) = 1, we may assume that e (0) = O. C„, be the zeros of ço with IC„I 0 . Then 2, and let C I , Fix e Tng)

TT ( 1 — CIL) TM/ n=1

defines an entire function T„, not vanishing for ICI dromy theorem,

9. Hence, in view of the mono-

1p„,() := log q),,,(C) with v(0) = 0 is uniquely determined on some domain containing the disk ICI

11 — C/C n i

if Ii = 2e

1

and

9. Note that

n = 1, . . . , m.

Therefore, by the principle of maximum modulus, sup {19)401: ICI = el .< suP {19)401 = 2e} sup {IT(C)1 :II = This implies that sup {Re ip m(C) :

ICI =

m exp (2,uQ).

= sup {log 19)„,()1: ICI = e

24ue + log M.

}

We now apply the Borel-Carathéodory inequality 4.8.8 to y„„ where This yields el :=

e.

eo := 1 and

2

sup {IVin(C)1: ICI =

(2Pe ± 1°g 1/4).

Obviously,

Ilog (1 — C/COI

if jJ._<= 1

2 ICg',31

and

I 1 >e IT

2.

Furthermore, CO

E(C)

= y„,G) — n =E

log

(1 — C/ • „)

if

ICI E 1.

Combining the preceding observations, we obtain

sup {1401 Letting e

= 1}

e

2

cc

(2,tte + log M)

+ 2=

co yields sup {Is(C)1:

= 1} _< 4/z.

Since u > 0 is arbitrary, it follows that 40 = 0 for 17,1

1 and so for all (:" e C.

Remark. We know from 4.8.6 that property (1) implies (T,') e i. However, examples show that the slightly stronger assumption ( 1 ) e 11 is indeed necessary. See (BOA, 2.10.3) and (LEV, Chap. 1, § 11).

CHAPTER 5

Matrix operators In what follows we illustrate the abstract theory of eigenvalue distributions by applying this machinery to matrix operators on sequence spaces. Although the discrete case is interesting in its own right, our main goal is to prepare the reader for a better understanding of integral operators, which are treated in the next chapter. In order to show that the basic eigenvalue theorems are sharp, we first provide some examples of finite and infinite matrices. Next we deal with infinite matrices M = (ttu) such that i plq)I 1 p 00 r co

E1 E j= 1

< co.

[

(i=

-1

The main result of this chapter is the eigenvalue theorem for Besov matrices. Finally, we investigate traces and determinants of infinite matrices. For this purpose, the Dixon-von Koch matrices are of special importance. These are defined by the condition E sup I,u u l < co. i 1 j We also describe regularized Fredholm denominators of Hilbert-Schmidt matrices.

5.1.

Examples of finite matrices

5.1.1.

Let M = (sit u) be any (n, n)-matrix. Then the set of eigenvalues corresponding to the operator Mop induced on 1(n) is denoted by ("11 (M), ...,

5.1.2.

We call 10 ) 1(n) := ( • . 0 1

the unit matrix of order n.

5.1.3.

For every vector t = (r i , -c i

-r„) we define the diagonal matrix

0

1) ,(n) := ( • . ). 0 r„ Obviously, the eigenvalues of the induced operator are r 1 ,

5.1.4.

The shift matrix

S(n) :=

0 0 10 : :

01 0 0

1 00 00

00 10

•••,Tn•

228

5. Matrix operators

has ones in the upper right-hand corner and just below the principal diagonal. All the other coefficients are zero. Since S(n) 0 : t"

($ii /

9

5•••5

/ • • • / $11 - 1)

the induced operator is unitary on 12 (n). 5.1.5.

We call 00

0

Ti 0

0

S ,(n) := (] 0

0 0 r n _t 0

0 0

the weighted shift matrix associated with the vector t = det (A.1(n) — S t(n)) =

— ti

TO. Note that

r„.

Hence I 4 k(St(n))1 = Iri

for

Tn1

Furthermore, if 1-1

r„

k = 1,

0, then it follows from S ,(n) = S(n) 1),(n) that

12 (n)) = k for

sk (S t (n)„: 12(n)

n.

k = 1,

n.

This example shows that it is impossible to improve the relationship between eigenvalues and s-numbers of operators on Hilbert spaces which is given by the multiplicative Weyl inequality 3.5.1. 5.1.6.

Next we consider the Fourier matrix F(n) := n- ' 12 (exp

where h, k = 1,

(21-ci

n

— hk))

n. It follows from

F(n)* F(n) = F(n) F(n)* = 1(n) that F(n) is unitary. Hence =1 Furthermore, if 2

p

for

k = 1, ..., n.

co , then we have

11F(n)0„: 1(n) -÷ 1„(n) I %II < n 1 / 2 1-11'13 This example shows that the inequality 11(Ak(n) I 411

61 7'1 91 11

for all

can only hold under the assumption

1,/t.

T E 9Z(/) 1/2 + 1/p; see 3.8.3.

Remark. The matrix F(n) determines the Fourier transform on the cyclic group Z„ := {1, ...,n}.

229

5.2. Examples of infinite matrices

5.1.7.* The normalized Littlewood-Walsh matrices W(2") are inductively defined by 1_ f + W(2"), + W(2")) (+1, +1\ W(2) and := 1,_ w(2' 1) k+ W(2"), - W(2") :

Obviously, all coefficients take the values +2 -"/2 . It follows from2 +1, —1). and W(2") W(2") = 1(2") that W(2") is hermitian and unitary. Next we note that 2h trace (W(2")) = 0 implies E 5t 1 (W(2")) = O. 1=1 Hence W(2") has eigenvalues +1 and —1 each with multiplicity 2" - I . Finally, by interpolation and duality, we obtain from

=W(2")

12 (211)11 =

:12(2") and

I W(2")„,p : (2'1)

= 2" 2

that 11W(2")op /p(2`1) -÷ Ip(2h)ll

21'hil1P-1121

Remark. The matrices W(2") were first considered by J. Hadamard (1893) : a), J. L. Walsh (1923) and J. E. Littlewood (1930).

Examples of infinite matrices 5.2. 5.2.1.* Observe that :Tr : (11, M = 2" + i — 1 defines a one-to-one correspondence between P := {(h, i): h = 0, 1, ...; i = 1, ..., 211 }

and N := {1, 2, ...}.

In this way every sequence x = („,) can be viewed as a family of vectors x,, = (ehl), where $,,, := $„,. Analogously, every infinite matrix M = Ca„,„) splits into (2", 2k)blocks -AIhk = (Phi In what follows the converse procedure is applied in order to build infinite matrices from finite ones. 5.2.2.* First, for 0 < r < co and A- 0, we consider the diagonal matrices 0 1 1 2oh

.42")

p o '3. FurtherObviously, 1,.,„ induces an approximable operator on I , with 1 more, in view of 2.1.12, for the corresponding eigenvalue sequence we have (2.„(4,))'r, w

and (An(Ir,o))

if and only if cx w > 1

230

5. Matrix operators

5.2.3.* Next, for 0 < r < co Walsh matrices

and

1

,=

0, we define the weighted Littlewood-

0 \

( 1 Wr

a

2hirh + 1

W(2") 1 )

0

It follows from in(24 )11 < 2h1 1 12-1 /PI that K a, induces an approximable operator on l whenever 1/r > 11/2 — 1/p1. Furthermore, in view of 5.1.7 and 2.1.12, for the corresponding eigenvalue sequence we have W(2h)op lp(24)

(An(W,,a))

if and only if

E 1r,w

OGIV > 1

and (a( w. 0 )) e lr ,.

Finally, letting 0 < r (Wro)0p

2, it can easily be seen that

94(4)

if

lfp = 1/r + 1/2

Hille Tamarkin matrices 5.3. Throughout this section we assume that 1

and ap > 1.

-

p < co

and

1

q

co.

5.3.1.* Recall from 1.1.3 that [l,,, E] denotes the Banach space of all E-valued sequences such that (11x1 11) e 1, equipped with the norm 1/p • IlxillP II(xi) I Up, EH := )

5.3.2.*

i= 1 The following result is an immediate consequence of 1.3.4.

Proposition. Let (x i) e [1„, E]. Then (xi).p : a (<xi , a>) defines an absolutely p-summing operator from E' into lp . Moreover, al •

11(X i) U

11(xi)op

5 • 3 • 3 • * An infinite matrix M = (ai) is said to be of Hille-Tamarkin type [1,, 1,i] if the sequence-valued sequence formed by mi :=

belongs to [la ,

(UN.,

Pi2

4]. For

1

• ")

q < oo, this means that CO

IIMI

Up, 1

r CO

= (E 1. E I i =1 = 1

1 / \

Jl q

11P

)

is finite. In the limiting case q = co the same holds for the expression \ 11p CO Ilm I Up, /.111 := (E supipu lP) . i=1

j

231

5.3. Hille-Tamarkin matrices

In the case p = q = 2 we get the famous Hilbert-Schmidt matrices defined in 1.4.6. Furthermore it is worth mentioning that the Dixon-von Koch matrices obtained for p = 1 and q = co just represent the nuclear operators acting on /1 , see 5.5.2. 5 • 3 • 4 • * Given M [I„, 1,], it follows from 5.3.2 that (70—> defines an operator from (4)' into /,,. If 1/p + 1/q

1, then there exists

OD

(x, y> :=

Œmi for

x=

(ei) E

l

and y = (lb)

e 4.

Using this duality, we can embed l into (4)'. This implies that Mop acts on every intermediate Banach space E:

E

Moo

(10'

I

>lp ----> E.

For example, we may take E : = ir with p r q'. Since the associated eigenvalue sequence does not depend on the choice of E, it is simply denoted by (2.„(M)). 5.3.5.*

We now state the main result of this section.

Eigenvalue theorem for Hille-Tamarkin matrices (B. Carl 1982: b). Let 1/p + llq > 1 and

{11p 1/p + 1/q — 1/2

1/r :

if if

1 2

q q

2, co .

Then

M

E

[la , I„] implies (.1„(M)) E /,.,„.

This result is the best possible. Proof. Write 1/q

oo

Ti

ittijl q)

•

=

Then t = (T) e l , and it follows from 2.9.17 that D, e t;.;:),(/ /e). Moreover, we define the infinite matrix A = (oc ii) by setting au := Since A °, e 2(4,, /„,), we obtain the factorization _Mop : 1(1,

A o„

>1co

D,

4,.

This proves that Mop e 2(k). Hence the eigenvalue theorem 3.6.2 implies that (1„(M)) E Using, the diagonal matrices defined in 5.2.2, we have

e [/„, /„]

if and only if

ocp > 1

and

(2„(4,„)) e /,

if and only if

am > 1.

Similarly, employing the weighted Littlewood-Walsh matrices introduced in 5.2.3,

232

5. Matrix operators

and letting 1/r: = 1/p + 1/q — 1/2, it follows that [l„, /g] if and only if op > 1 and if and only if aw > 1 .

().„( W,.,))e /,.,

These examples show that (2(M)) e'r,p is indeed the best possible result which can be obtained for all matrices of Hille-Tamarkin type [1,, 1, 11 . 5.3.6.* Finally, we deal with the limiting case 1/p + 1/q = I.

Theorem (W. B. Johnson / H. K6nig / B. Maurey / J. R. Retherford 1979). M

E

,

lpd

implies (.1, n.(M11 -

G max(v, 2) •

This result is the best possible. Proof. We know from 5.3.2 that M induces an absolutely p-summing operator on Hence the assertion can be deduced from 3.7.1 and 3.7.2.

.

It follows from the theory of nuclear operators that there are matrices ME [/ 1 , such that (2„(M)) /2 ,, if 0 < w < 2. Passing from M = (au) to N = we may obtain matrices N e [1,„ lp d] with A n(N) = A(M). Therefore the conclusion p 2. Furthermore, (2.„(M)) e 12 cannot be improved for any exponent p with 1 the best for 2 < p < Jr: show that (A(M)) e //, is possible result diagonal operators

5.4.

Besov matrices

Throughout this section we assume that — oo < a, x < + co, 1 co. u, y and 1

p,

CID

5.4.1.* We define the Besov sequence space E] := [1,4 , 21 [1p(2"), E]]. This means that [b, u , E] consists of all E-valued sequences (x111) indexed by P := {(h, i): h = 0, 1, ...; j = 1, ..., 2h} such that 11(Xhi)

[u

=(

co

24

h=0

2 ( E Ilxhill' E [' i=

)11p]u)11u

is finite. In the cases p = co or u = oo the usual modifications are required. If E is the complex field, then we simply write

5.4.2.* We first establish an elementary inclusion. Proposition. Let

O

—co < a i ao < + cc,

1 /P0 = c1 + 1 /pr. Then

E] g

E].

Proof. Observe that 24

E IlxillPi)

(

i=I

1/Pi

2h

5_ 2ii( '/ Pi - l/P0 ) ( E

1 /Po

oo

and

5.4. Besov matrices

5 • 4.3.*

233

The next result follows immediately from (BER, 5.6.1) and (TRI, 1.18.2).

u0 , u l , u

Interpolation proposition. Let — oo < 0.0 < a i < oo, 1 and 0 < 0 < 1. If o = — 0) do + 01 1 , then

E],

oo

ED0 , u = [N,„,

The norms on both sides are equivalent. 5.4.4.*

Lemma. Let o. + r > (1 — 1/p — 1/q),. Then there exists oo 2h <x, y> : = E E $1 for x = ($11i) e b and y = nu )e bTq, , 01fii

(

, 1

.

h=0i=1

Using this duality, we can embed 137,,u into (b)'. Proof. Define 1/r := (1 — 1/p — 1/q) + . Then Holder's inequality implies that 24 2h ) 1/q ) 1/P ( 2h • 2" 1$hi1hil 1=1

i=1

Hence ce

Kx, y>I

i=1

2h

E

h =0 i= 1

1$hinhil

00

2" h(G

T I ir)

1 IP

[213ff ( 1i1 E 1$1 P) i

h=0 li(cr+

iir)

h=0

I 121 T(

11

2h =i =

) 1 /(

Inhir

±

11x 1 b;,u11 11Y 1 brq ,v11

Thus <x, y) defines a bounded bilinear form. 5.4.5.* The following result is related to 2.9.9. Lemma. Let It be a quasi-Banach operator ideal such that

VmY j IQ -<

4,(i71) where

Ok.

+ > cx, then 71 (a+z a). a „(I : bpa (b, )' O. If

-

-

Proof. Let Qa$/ii) :=

k2k)

be the surjection which maps every sequence ( 1„) into its k-th block. Then 1p(2k )11 = 2-ka and 11Q„ : bt

11Qk b

-

÷ 1,,(2k)11 = 2-kT

Denoting the identity operator from 4,(2k) onto 4(2k)' by from b,„ into (bqp,v)' can be written in the form

-= E

=0

4,

the embedding operator

Qk7kQk •

According to D.1.8, we may assume that ?..t is r-normed. If 2h < 71 < 211+1 , then it follows from (h - 1

rank

h-

Qk'ikQk) k=0

21` < 2"

= k=0

234

5. Matrix operators

that h-1

a a : b; u .- (b)'I tO -. III - E Vk/kQk I wII -..5 k =0 _. i C — (k=h

(k=ih IIQ;c4Qk I 1111r)

1 /r

\ 1 /r

2 -k(a+ T - a)r II I: l(2") (2k ) p 4(2kY I %I I r) 2 -h(o. - f r -a.)

(

5.4.6.*

k=h

The result just obtained can be used to estimate the Weyl numbers. Proposition (A. Pietsch 1980: d). Let a + r > 0 and 1/p + 1/q 1. Then n-e , x„(/ : b;,„, (brq ,„)')

where

litp + 1/q - 1 1/p - 1/2

o := a + r +

if if if

0

1 _. _ p _- q ' ._._ 2, 1 _._. p ._.. 2 .-_._ q' 2 p __ q' _-. oo .

oo ,

Proof. Put

c llp + 11q - 1 if 1 p q' 2, 1/r := 1/p - 1/2 if 1 ._ p ._ 2 _-_. q' .- oo , if 2 -. p 0 q' __ oo . In view of 1.6.7, we have Ili: l(ni) -- 4(Tn)' 1 *r,2Il 5_ c := III: 11,— 1; 1 Vr,211. Furthermore, 2.8.16 implies that x2 „_ 1 (I: b;, „ -- (b, 0)') _ 11-11r a n(I : b;, u .- (by q .v )' I V e , 2) • Finally, taking ?t := V r,, and OC := 0, the preceding lemma yields the desired estimate. 5 • 4 • 7 • * We now generalize 5.3.2. Proposition (A. Pietsch 1980: d). Let (xhi)e [b u , E] and r := max(p, u). Then

(<xhi , a>) (xhi)op : a defines an absolutely r-summing operator from E' into b; . Moreover, 11(xhi)op 1 Vr11=. < 11(xhi) 1 [b.œp,u, EJ Proof. Given a 1 , yields

a,, E E', repeated application of Jessen's inequality C.3.10

n

E= 1 11((xht, ai>) I b;,ull r)

1/r

2" n [œ (

=

EEE

j =1 h=0 i= 1 n co [2"

u/p] rIti}lIr

( 2"Ixhi, aj>I P) )plriulp)1Iu

E <2"6 Ixhi, aj>ir

h =0 1=1 j = 1

E 1<x, ai>1) 1/r 1=1 This shows that (xhi),,i, is absolutely r-summing. 11(Xhi)

[b;,u, E]lf sup {(

235

5.4. Besov matrices

5.4.8.* An infinite matrix M = (phi ,,,i) is said to be of Besov type [bp' b] sequence-valued sequence formed by (itthi 3 O 1; Ph1,1 1

the

'fli,12; • • •)

5

p belongs to [ba

qv], it follows from 5.4.7 that

5•4•9.* Given M e [b;5

21, —>( E E co

M0 P : 0'10

k =0 j=1

Phi,kjnkj)

defines an operator from (bc`q ,, )' into bp(' If o + r > (1 — 1/p — 1/q) + , then bpa is embedded into (bqr,„)'. This implies that M0 acts on every intermediate Banach space E. Since the associated eigenvalue sequence does not depend on the choice of E. it is simply denoted by (4(M)). 5.4.10.* We are now in a position to establish the main result of this chapter. Eigenvalue theorem for Besov matrices (A. Pietsch 1980: d). o- + r > (1 — 1/p — 1/q), a -nd 1/r := + +

1/P 1/p + 11q

—

if 1 if 2

1/2

q q

2, oo.

Then

Al e [b„, No)] implies (1, n (M)) e 4, u • This result is the best possible. Proof. First we assume that p g' and e : . 0, + T + l/ip + 1/q — 1 ( hp-1/2 0

u= if if if

1. Define 1 p g' ..__ 2, 2 1 -__ . p q' .. oo , p g' oo , 2

and s := max (p , 2). We know from 5.4.6 that x„(/: bc1,, 1 -- (14 , „)') < n- ° .

Hence (la, ,

E

.

Furthermore, by 5.4.7 and 2.7.4, Mop E 21, )co((b rq, vY, b, 1) . Thus, in view of the factorization M : (brq, v)' Mov ba13.

I (b

,

we conclude from the multiplication theorem 2.4.18 that Mop

G

((hrq ,

,

where llr = e + 1/s takes precisely the values given in the above theorem.

Let

236

5. Matrix operators

If p > q' and u = l, then we see from 5.4.2 that Melba ,i,

implies

A/ e [bV

lq" , bT7,„].

In this way the second case is reduced to the first one. So far we have shown that the map M --+ Mop , which assigns to every matrix the corresponding operator, acts as follows: OP : [hic;, , hr

--+ 2 ),o ((l)Tq,„)').

This result can be improved by interpolation. To this end, choose ao , d i and 0 such that

a = (1

—

0) ao + 0a, and — oc

cso a i < + co.

Since a + r > (1 — lfp — 1/q) + , it is further possible to satisfy the condition + r> (1 — 1/p — lfq) + . We now apply the formula ([b;? 1 , E] [b;1 , E ])e,u = [b(1,,„, E.] for E := kv . Next, replacing a by a ° and a l , we define r o and r i in the same way as r. Then it follows from 1/r = (1 — 0)1r0 + 01r, that (2(4) .0, ti$V00)e,u g Hence the interpolation property yields op : [b;, u , b'q ,v] --+ 2;:;`[;(0 q,v)')• 1. By the eigenvalue theorem 3.6.2, we therefore obtain (2,„(M)) e Using the diagonal matrices defined in 5.2.2, for 1/r = a' E [bpa , bqr ]

if and only if

r + 1/p, we have

Au > 1

and (2n(4,,)) G Ir , w if and only if

OCI'V > 1 .

Similarly, employing the weighted Littlewood-Walsh matrices introduced in 5.2.3, and letting 1/r = a' ± r ± 1/p + 1/q — 1/2, it follows that e [bp° ,u, br] if and only if

Oat > 1

and G lrov if and only if aw > 1.

These examples show that ()i.„(M))e 4,„ is indeed the best possible result which can be obtained for all matrices of Besov type [b„, Remark. The eigenvalue theorem for Hille-Tamarkin matrices can be deduced. from the preceding one by taking a = 0, = 0, u = p and y = q.

5.5. 5.5.1.

Traces and determinants of matrices To begin with, we restate 4.7.3 in the following form.

Proposition (A. Grothendieck 1956: a). Let (ai) E [11 , El. Then (a1)0p : x -+ (<x, ai>)

5.5. Traces and determinants of matrices

237

defines a nuclear operator from E into / 1. with (ai)op I 91 11 = 11 (ai) Furthermore, all operators T e 91(E, /1) can be obtained in this way. 5.5.2. Specializing the preceding result, we arrive at a characterization of the nuclear operators acting on 1 1 . Proposition. Every infinite matrix Me [4,

1.]

induces an operator

Mop E 91(0 with IIMOp 1 91 11 = JIvII

Furthermore, all operators T e 91(0 can be obtained in this way. 5.5.3. Since /1 has the approximation property, every operator Mo„ E possesses a well-defined trace which is simply denoted by r(M). Theorem. Let M =

e , /œ ] • Then

CO

r(M) = i=

5_ II I Ui ,

p H and ir(M)i

Proof. Using the bounded sequences

02

mi :=

i1

514

12

• • • )5

we have the canonical nuclear representation CO

M0p

= E m i ® et . i=1

Therefore "r(M) = 5.5.4.

i= 1

<et, mi> =

i=1

[Ili •

Given any infinite matrix M = in)

m il

2

•• • :=

we write si„ )

det

\J1, •• •, • • • Pinin

In particular, observe that MO = u. 5.5.5.

We now describe the canonical Fredholm denominator c5(, M) = 1 +

of an operator

Mop

induced by a Dixon-von Koch matrix M.

Theorem (H. von Koch 1901). Let M x(M)

=

1

cc

E 1 ••• 4E= 1 -14 11: i 1=

(ii

5 • • •5

ti, •••,

and Jcx

PIA

nni2

n!

M [l i, 1 « ] l ?' •

in)

=

u) E [l ,

(1

Then

238

5. Matrix operators

Proof. The formula for the coefficients of the Fredholm denominator could be deduced from 4.7.10. We prefer, however, to present here a direct approach. Setting r i := suplyo l and applying Hadamard's inequality A.4.5, we obtain i

(

in)

l9•••9

< nni2 "Gi l

il, • • • in

This implies that the series ,• • •,

E • E 111 n! ii = i in

x(M) :=

=1

.1

• • •

in) in

9

is absolutely convergent. Furthermore, nn /2/

1NnW)I

co

(

\n Ti)

n n/2

II Al I

lodli n •

It remains to show that the numbers a n (M) just defined are indeed the coefficients of the Fredholm denominator 6G", M). To this end, we introduce the infinite matrices A „(M) = (cc„,ii(M)) given by co 1 c° i7i7l • • • 9 i n ij(M) = n! 4 ••• E m ( =1

in =

J,

'1

in) .

9•••7

Obviously,

Expanding the determinant l

5•••7

J9 ti

•••9

,r

i

in

by its first column, we get the expression M

(

'

(_os m Ili,

•• •, in) m (i) • ••9

s=1

in

\il 9

il

9 • • • 9 iS-1

iS+1 • • • 7

12

• • • is,

15+1, •••, in

in)

M

Hence Nn,ijW) = Nn(M)

s= 1

fins,i1M),

where

:=

(

—

1)s

, n.

C° E in =1

E=1

; 1

( i9 .1

Substituting k, is , •• • , i„_ i in place of is , i-5+1 1

(m) = =—

E E ..• tz! k=z z i =1 z„_,=1 1

5

il 7 • • • 5

is —i, s+1, • • • 9 i

.2

5•••

iS 5

•••9

„

(

i

L,

iS+ 1 7 • • • 9

in

(i s)

./

yields

11

••,

in-1

• • •, in-1

(1(

kJ.)

x_ c°

n k=1

an-1, 1k(M) Yki •

In particular, it turns out that 13,,(M) does not depend on the index s. Thus we obtain oo

n, O(M) = n (M )

E Nn— 1, ik(M),ukj •

k=1

5.5. Traces and determinants of matrices

239

In terms of matrices, the preceding formula reads An (M) = cxn(M)M —

'LAMM".

Next, by induction, we see that A n(M) =

h =0

(— 1)'"N,1(M) M n—h+1 •

This proves that x(M) and A„(M) satisfy the recurrence formulas (R0) and (R 1 ) stated in 4.4.10. Hence oc„(M) is the n-th coefficient of the Fredholm denominator we are looking for. Furthermore, the operator induced by the matrix A ,i(M) is the n-th coefficient of the associated Fredholm numerator. According to the principle of related operators, the following result shows 5.5.6. that the coefficients oc„(M) obtained in the preceding theorem can be used to define Fredholm denominators for nuclear operators on arbitrary Banach spaces. Of course, if the approximation property fails, then this construction depends on the choice of the nuclear representation. Theorem (A. Pietsch 1963: a). Every operator Te 91(E) is related to an operator Mop e91(4) induced by an infinite matrix M e Proof. We consider a factorization

A

A

/Go

X

D,

as described in 1.7.3. Since D e 9(l, 11 ) is nuclear, we can find a Dixon-von Koch matrix M such that D,AX = Mop . 5.5.7.

Next we state an immediate corollary of 5.5.1. Proposition. Every infinite matrix M e [/ 1 , 12 1 induces an operator Mop

01(12)

Ilm I U1 , /2] 11

with liMop I 9 II

Remark. It easily follows from the results in Chapter 2 that an infinite matrix represents a nuclear operator on /2 if and only if it can be decomposed as a product of two Hilbert-Schmidt matrices. 5.5.8. We now show in a special case how the estimates for the coefficients of the Fredholm denominator improve when stronger conditions are placed on the type of the underlying infinite matrix.

Proposition. Let M = (pii) n n(1/2-11q)

n(M)I

—

.

1

n!

Proof. Write co

:=

i =1

) itq

ulqIt

•

I

4] with 2

E

,

q < co. Then

240

5. Matrix operators

Given j 1 , .• •, in, Holder's inequality yields 1/2

E I uji 12)

1/q

E

n1/ 2-1/q (k1

< n 172 ii gri -

(k = 1 1 k

Next, applying Hadamard's inequality A.4.5, we obtain (ii

•• • 9

••

in) < n(1/2— 11 ti) 7. . _n

• in

• • '

Hence n n(1/2-1/q)

locn(M)i -5

n!

f co

n(1/2_1/q)

n

I

rt

M IU

n!

)

Mil t' •

Remark. It follows from 4.8.7 and Stirling's formula that 6(c,-, AI) is an entire function of ordet r, where 1/r := 1/2 + 1/q. 5.5.9. To give explicit formulas for the coefficients of the regularized Fredholm denominators of infinite matrices, the following concept is required. For every (n, n)-matrix A = (au) we define the modified determinant of order m by (m)

y

det,„(A) :=

,t

sign (70 OCi,(1)

OG n ( n ),

where the sum is taken only over those permutations :Tr for which all invariant subsets have at least m points. For example, if m = 2, then the original definition of a determinant is modified by cancelling all summands corresponding to permutations with a fixed point. This means that det2 (A) is the ordinary determinant of that matrix obtained from A by writing zeros on the principal diagonal. Given any infinite matrix M = (), we put A ir

n

il

••• 5 • • .5

f

lail j1

in)

•—

flii . j.

•

in

/tin ..ii • • • Pin 5.5.10. Let Me [1„ , 1J. If 1/p + 1/q 1, then the induced operator belongs to (E) for every Banach space E lying between 1„ and (4)% Consequently, there exist regularized Fredholm denominators CO

i (,r)( M) M) = 1 ± lEcx

of all orders in greater than or equal to p. By analogy with a result of H. Poincaré (1910) we conjecture that the coefficients are given by the formula co =1 E 0‘,7

n! j1 =i

oo E m.

i„=1

•• •5 in i i • • *5 in ,

•

Furthermore, it should be possible to establish estimates for the modulus of A „( m)(M) which imply, by 4.8.6 and 4.8.7, that (2 n(M)) E

,

1 where 1/r := ( /p 1/p + 1/q — 1/2

if if

1

q

2,

2

q

oe

5.5. Traces and determinants of matrices

241

This would be the classical way for obtaining the eigenvalue type of the operator ideal 5.5.11. At present, the program envisaged in the preceding paragraph has only been carried out for Hilbert-Schmidt matrices; see (S MI, Chap. VI) and (ZAN, 9.16 and 9.17).

Theorem. Let

42) (M)

=

and

IC42) (14 )1

1

M = (au) E [12, 12

co

cc'

n! i=i

e n/2 --7/7 r

•••

E m2

i„=1

1.

Then

in) l j , • .•, in 5 • • '5

M I U2 12] I n •

Remark. We stress the fact that the above estimate follows from Carleman's inequality 4.7.19,

c C, exP (BM I [12, 1 2 1 2 11 2 ) for all by reasoning as in the proof of 4.8.7. No direct approach seems to be known. 02 ( , -M)1

]

CHAPTER 6

Integral operators It is our aim to demonstrate the great importance of the methods of abstract operator theory to applications within the theory of integral operators. First of all, we prove the classical Schur theorem on the eigenvalue distribution of operators induced by continuous kernels. Next we treat Hille-Tamarkin kernels. These are defined by the condition

(

lip 1-1 pig < oo . f [f I K(E , 71)1q dn] a oo

In this context we also deal with kernels which may have a weak singularity on the diagonal — n The main results of this chapter are the eigenvalue theorems for kernels possessing certain smoothness properties which are described in the language of Sobolev and Besov spaces. Almost all proofs are based on the following factorization technique. We consider appropriate Banach spaces E0 and E, such that the given kernel K defines an absolutely p-summing operator from Ei into E0 and such that E0 embeds into E1 . Then the induced operator K0 acts on every intermediate Banach space E: .

I Kop I E ---- >E1 -----> E0 ---- >E.

In view of the principle of related operators, the eigenvalues of K0 are the same for any choice of E. Furthermore, in all cases under consideration it turns out that the smoothness properties of K are completely reflected by the properties of the embedding operator /from E0 into E1 . This means that the special shape of the kernel in question is not essential. In particular, the basic results apply to convolution operators induced by periodic functions. Then, up to factor 27, the eigenvalues coincide with the Fourier coefficients. In this way we obtain various classical and modern theorems showing how the properties of a given function are inherited by the asymptotic behaviour of their Fourier coefficients. Moreover, convolution operators yield the most important examples for proving that a result obtained in the general setting the is best possible. This means that the fine index w in the conclusion (2.„(K)) c 4,, cannot be improved. Finally, we investigate traces and determinants of integral operators. The classical case of continuous kernels is of special importance. Unfortunately, apart from the theory of Hilbert-Schmidt operators presented in F. Smithies's monograph (SMI), almost nothing is known about regularized Fredholm denominators.

6.1.

Continuous kernels

6.1.1.* We denote by [C(X), E] the set of all continuous E-valued functions 0 defined on a compact Hausdorff space X. It is well-known that [C(X), E] becomes a Banach space under the norm 110 1 [C, E]ll : = sup {110(e)11: E

G

X} .

6.1.2.* We begin by stating an auxiliary result; see (KOT, 44.7.2).

6.1. Continuous kernels

243

Lemma. The set of all functions 0 of the form li

0() = E

i =1

x ifg)

for e X

with x 1 , ..., ..-x„ e E and f 1 , ...,f„ E C(X) is dense in [C(X), E]. 6.1.3.*

The following fact is now obvious. Proposition. Let 0 E [C(X), E]. Then

a —> <00, a> defines an approximable operator from E' into C(X). Moreover, 11 00pii = 11 0 1 [C, EH 6.1.4.* Let X and Y be compact Hausdorff spaces. A kernel K defined on X x Y is said to be of Fredholm type [C(X), C(Y)] if the function-valued function Kx : --4 lq$,*) belongs to [C(X), C(Y)J. Remark. Obviously, the property just described is nothing other than an awkward transcription of the fact that the kernel K is continuous on X x Y. However, this approach turns out to be quite useful in the following sections when the type of the kernels becomes more involved. 6.1.5.* Let K E [C(X), C(X)i, and consider any finite Borel measure p, defined on the compact Hausdorff space X. Then Kop : g(n) -4 1 K(, n g(n) clp(71) ir -

)

yields an approximable operator from L i (X, ii) into L o,(X„ u). This implies that Ko„ acts on every Banach space E lying between L œ (0, 1) and L 1 (0, 1): I Ko, I E —> L i (X, du) --> Loo(X, p) --> E.

Of course, Kop can also be viewed as an operator on C(X). Since the associated eigenvalue sequence does not depend on the choice of E, it is simply denoted by (4(10). 6.1.6.* Eigenvalue theorem for Fredholm kernels (I. Schur 1909: a). Let p be any finite Borel measure on a compact Hausdorff space X. Then K e [C(X), C(X)]

implies (2(10) E 12.

This result is the best possible. Pro of. We consider the factorization

I Kop Kop : Loe(X, p) —> 1. 1 (X, ,u) --> L,(X, ,a). By 1.3.9, Je

V i (L(X,

eu), L i (X, du)).

244

6. Integral operators

Hence Kw, is an absolutely 1-summing operator on L 00 (X „ u), and we see from Theorem 3.7.1 that (2,(K)) G '2. The convolution operator derived from Example 6.5.11 shows that this result cannot be improved. This observation also follows from 3.8.1 and 6.6.8.

6.2.

Hille-Tamarkin kernels

Throughout this section we assume that 1 _- _13
Further-

6.2.1.* An E-valued function 0 defined on X is said to be simple if it can be written in the form

(15() = Ei = 1xA($)

for almost all $

G

X,

h,, are characteristic functions of measurable subsets x n G E and h 1 , where x 1 , X„. Remark. In the following we only consider those simple E-valued functions which are obtained from subsets with finite measures. 6.2.2.* An E-valued function 0 defined on X is called measurable if there exists a sequence of simple functions 0„ with lim 110($) —

0

for almost all $ e X.

Next we recall an important criterion; see (DIL, p. 42), (DUN, 111.6.11) and (HIP, 3.5.3).

6.2.3.

Proposition (B. J. Pettis 1938). An E-valued function 0 is measurable if and only if the following conditions are satisfied: All complex-valued functions <00, a> with a G E' are measurable. (1) (2) There exists a subset X0 of measure zero such that 0(XW0) is separable. Remark. A function 0 having property (1) is said to be weakly measurable, while property (2) means that it is almost separably-valued.

6.2.4.* The following result is obvious. Lemma. For every measurable E-valued function 0 the real-valued function I10011 is measurable, as well.

6.2.5.* We denote by [1, „(X , p), E] the collection of all (equivalence classes of) measurable E-valued functions 0 such that 110 I [L, E] :=

110($)11P cl,u($)) 11P

is finite. Such functions are called absolutely p-integrable. We know that [Lp(X , ,u), E] is a Banach space; see (DUN, 111.6.6). Remark. Note that [L 1 (X, u), E] just consists of all Bochner integrable E-valued functions; see (DIL, p. 45).

6.2.6.* For a proof of the next result we refer to (DUN, 111.3.8). Lemma. The set of simple E-valued functions is dense in [L p(X , ,u), E].

6.2. Hille-Tamarkin kernels

245

6.2.7.* We now establish an analogue of 5.3.2. Proposition. Let 0 e [L„(X, tt), E]. Then

, a>

0op : a

defines an absolutely p-summing operator from E' into L„(X, ea). Moreover, 11 0ov I PII

IN) I {4,

Proof. Given al , ..., an G E', we have 11P

ai> Lpii P)

(X

i=1

n (

=

1/p

E

I Ko

1 p )1

(e), ai>IP (1/40

E

sup {(ii1 Kx, ai>1P)

f 110($)11P dy($))

1/p

:x

U}

6.2.8.* Let (X, ,a) and (Y, y) be measure spaces. A kernel K defined on X x Y is said to be of Hille-Tamarkin type [L,(X, ,a), L q(Y, 0] if the function-valued function

Kx :

K(,-)

belongs to [L,(X, JO, L.,(Y, '0]. For 1

r 11 K I [L„, L,]11 := (5 [ f

q < co, this implies that (2)]P/ Ti)Ig

is finite. In the limiting case q = oc the same holds for the expression

K I [Lp , L o ]ll :=

ess-sup o IK(e, ?Dr citt(0)11P

6.2.9. Let (X x Y, it x y) be the product of the measure spaces (X, a) and (Y, y). For 1 q < co, we denote by [[L„(X, L,(Y, IA the collection of all (equivalence classes of) measurable kernels K defined on X x Y such that the iterated integral

dv(n)rq d[1()) 1 /P vi

is finite. In the limiting case q = oc the same is assumed for the expression

11K I ELp, Looll I : =

(

less- su I.K($, 77)111 dit($)) 1/1'

Note that [Lp(X,,a), Lq(Y, v)1 is a Banach space; see (JOR, p. 169). In the case p = q = 2 we get the famous Hilbert-Schmidt kernels defined in 1.4.8. Remark. The formal difference from the concept introduced in the previous paragraph is that the kernel K satisfies a different measurability condition.

6.2.10. By a simple rectangular kernel K defined on X x Y we mean a function which can be written in the form

n) = z i=

n),

246

6. Integral operators

R„ are characteristic functions of measurable where ),„ ..., An E C and R 1 , rectangles X, x Y1 , ..., X x Y„. Remark. In the following we only consider those simple rectangular kernels which are obtained from rectangles with finite measure.

6.2.11. Lemma. Let 1 < p, q < Y)1' . dense in E Lp(X, ft),

co. Then the set of simple rectangular kernels is

Proof. Without loss of generality, we may assume that the underlying measures are finite. Furthermore, it can easily be seen that the linear span of all characteristic functions K corresponding to arbitrary measurable subsets Z of X x Y is dense. As is shown in (J6R, p. 163), given s > 0, there exist pairwise disjoint rectangles X, x Y1 , X„ x 17„ such that the measure of the symmetric difference

D := Z A U x R denote the corresponding characteristic functions. Then

is less than E. Let R 1 , K0 := K —

z Ri

is the characteristic function of D. q separately. First, applying Holder's We now treat the cases p q and p inequality with respect to the variable we obtain K—

R i I 114, L‘a =[ i= LY 11,1(X)

77)1'J dv(97)1PRI du())

"I'

\X

f 1K 0g, n)iri d,a($) dv(97)) 11q

11P-114

\i

Y

tt(X) 11P-11q

i

Secondly, an application of Holder's inequality with respect to the variable ?I yields K—

>R i IL, LJ 1=1

— xf [ 1 <

v(y)i/q-i/P

?Dig dv(27 )1"dp())1/1) (f vccg, ?AP dit( dv(n)) 11P )

\X

Y

n i / q -11 P E,liP .

V(

Thus every measurable characteristic function K can be approximated by simple rectangular kernels. This completes the proof.

6.2.12. We are now prepared to give a useful characterization of Hille-Tamarkin kernels.

p, q < cc. Then the map Proposition (I. Inglis; see J. Kupka 1980). Let 1 which assigns to every kernel K the function-valued function Kx :

K(e , -)

yields a metric isomorphism between

ELp(X, ,u), L a(Y , Y)I

and

[Lp(X ,

La( Y,

y)].

6.2. Hille-Tamarkin kernels

247

Proof. Without loss of generality, we may assume that the underlying measures are finite. Given K e EL,„(X, ea), L,(Y, v)], Fubini's theorem implies that

K($, .) e 4,(Y, v) for almost all $

E

X.

In view of the preceding lemma, there exists a sequence of simple rectangular kernels 1‘,7 with

111( — K„ I fL p ,L q Jj i

2.

Then it follows from

— Krg, .) I 4111 d,u($) < oo

i n= 1

[

that

lim 11K($, -) — K,,($, .) L,11 = 0 for almost all $ e X. Therefore the L,(Y, v)-valued function Kx is measurable. This proves that the map C Kx acts from L(X, ,u), L,(Y, 0] into [I,„(X, 14Y, v)]. Obviously, it is a metric injection. As observed in 6.2.6, the set of simple 4,(Y, v)-valued functions is dense in [L,„(X, ,u), L,(Y, '0]. However, each of these functions can be obtained from a kernel Ke ,u), LQ(Y, v). Thus the map K ---÷ KX is indeed onto. Remark. In the case q = oo there exists K e EL(0, 1), L(0, 1)] such that the L c,(0. 1)-valued function Kx fails to be measurable. For example, take

K($,

n) : = {

1 0

if 0 <9'1 6 . $.. 1, if 0

$

Then Kx is not almost separably-valued. 6.2.13.* Given K e K0 p :

g(n)

L,(X, s)], it follows from 6.2.7 that f K($, n

dp,(97)

)

defines an operator from I,,(X eu)' into Li,(X, tt). If the measure tt is finite and 1,Ip ± liq 1, then there exists

g> := f f($) g(0 du(E)

for f e L,„(X, tt)

and g e L4(X, tt) .

Using this duality, we can embed Lp(X, it) into L‘,(X, /2)% This implies that Kop acts on every intermediate Banach space E:

L,(X, ,u)'

Kop

E L„(X,

E.

For example, we may take E := L,.(X„ u) with q' r p. Since the associated eigenvalue sequence does not depend on the choice of E, it is simply denoted by (?1,(K)). In the case of an arbitrary measure we must assume that p = q' in order to ensure that Kop acts on Lp(X, ,u).

248

6. Integral operators

6.2.14.* Using the concept of a Hille-Tamarkin kernel, we now generalize 5.3.6. The proof is analogous. Theorem (W. B. Johnson / H. Kiinig / B. Maurey / J. R. Retherford 1979).

K [L,(X, ,u),

,u)]

implies (2„(K)) e

imax(p, 2) •

This result is the best possible. 6.2.15.* Next we establish a counterpart of 5.3.5. The notation q+ := max(q/, 2) proves to be quite convenient. Eigen,value theorem for Hille-Tamarkin kernels. Assume that the measure ,u is finite, and let 1/p + 1/q 1. Then

K e [4(X, 10,

,u)]

implies (2„(K)) e i

.

This result is the best possible. Proof. Since p

q', we have

L,(X, ,u)].

[I,„(X, 1u), L,(X, y)] g__

Hence it follows from the preceding theorem that (2„(K)) e In view of 6.5.20, Examples 6.5.8 and 6.5.11 show that this conclusion cannot be improved. .

Remark. If p = q = 2, then the assertion follows immediately from the SchurCarleman theorem 3.5.6. We further note that the case 1 p < 2 can be reduced to this classical result via the principle of related operators. However, the remaining case where 2 < p < co was open for a long time. It was only solved in 1979 with the help of the theory of absolutely p-summing operators. Another approach, given by H. K6nig / L. Weis (1983), uses interpolation techniques, a change of density argument and the Schatten-von Neumann operators. Preparatory work in this direction is due to M. Z. Solomyak (1970), G. E. Karadzhov (1972), J. A. Cochran / C. Oehring (1977), B. Russo (1977), J. J. F. Fournier / B. Russo (1977), W. B. Johnson / L. Jones (1978) and L. Weis (1982). No old-fashioned proof is known.

6.3.

Weakly singular kernels

Throughout this section we assume that 1 < r < oo and 1

w

co.

6.3.1. Let (X, )u) be a measure space. For any measurable complex-valued function f defined on X, the non-increasing rearrangement f * is given by

f*(T) := inf {c > 0 : ,u($ E X: if(01 > c)

r}

for r

0.

< CO, the Lorentz function space L,.,(X,,u) consists of all (equivalence For 1 measurable complex-valued functions f such that classes of) 00 ) 1/,„ Ilf I 40,11 := *(r)]"' 0

is finite. In the limiting case w = co the same is assumed for Ilf I Lr. crj:= sup

* CO •

249

6.3. Weakly singular kernels

In this way we obtain a linear space which is complete with respect to the quasinorm I Lr „,11. Since there exist equivalent norms, L,(X, ,u) even becomes a Banach

space. In the following we only consider the case when (X, tt) is the unit interval equipped with the Lebesgue measure. The corresponding Lorentz function space is denoted by Lr ,(0, 1). Remark. For further information we refer to (BER), (BUB), (STI) and (TRI).

We now provide an inequality of Lyapunov type. The special value of the constant cr indicated below turns out to be quite useful in the proof of 6.3.5. Lemma. Let 2 < r < oo and 0 <0 < I. If l/r = (1 — 0) 12 + 0/co, then I Lr,ill crIlf I L211' If I Lcoll° for all f L co (0, 1), 6.3.2.

where cr

)1/2

r

:= + (r —

2

If 0Pro f.
4111

) d.r oo

a

rlir-lf eq_c)

T I/r-Ip( r) d.r 0

th

a

1/2

co

a

<

r 11r-I dr

0 <

if I L00I1 +

a

21r-2 dr) If I L2Il

r )1/2 042-0/2r r— 2

I Looll

L 2 11

Setting

.

0

( Ilf I L211 2

Ilf I L Coll yields the result we are looking for. Example (H. Keinig / J. R. Retherford / N. Tomczak-Jaegermann 1980). If 6.3.3. 2 < r < co, then the embedding operator from L 00 (0, 1) into 4, 1 (0, 1) is absolutely (r, 2)-summing, and Lr,1

I

r+f

§kr,211

r\112 2)

—

Proof. Define 0 by 1/r = (1— 0)12 + 0/co. Let af1, • • • , fn 6 L„(0, 1). Then it follows from 1.3.9 and 6.3.2 that

i=1 [Ilfi I L211i —e

14.10

I L,011 0 0

1/r Cr

:5_

This proves the assertion.

=

11f1 I L2I1 2 ) L, -4 L21

( sup

lir

Ilfi I Lcollr

211 1— ' 1Kfi) w211

250

6.

Integral operators

6.3.4. We are now in a position to determine the asymptotic behaviour of the Weyl numbers of the embedding operator from Loo(0, 1) into 1). Example. Let 2 < r < oo

1 w cc. Then

and

n- ilr

-4

Proof. In view of 2.7.3 and 6.3.3, the upper estimate follows from 41. : L

X n (I: L c

Lr,w)

--4 Lr,i) 11 I: Lr , i --+

Consider the intervals X1 , ..., X,,, obtained by dividing [0, 1] by the points $, := ilm h, denote the corresponding characteristic with i = 1, ..., in — 1, and let h 1 , functions. Define

E

Q ui : = E hi ® e i ,

h i and

i =1

where e1 is the i-th unit vector in 1(m). Obviously, LA = 1 Furthermore, 1/in

J If($)1

hi>i

f f * er) d.r

d -5.:

1Irn T -11r drilf f d

4, 0,11

implies that

11Qm

Lr , oo

r'm-l/r*.

1 0°041

We now conclude from the diagram mi.

and 2.9.11 that (ni — n +

k

in

1)1/ 2

=a

: 11 ( m )

(

1.(m))

m)) = a „(I : 12(m)

= x n(i: 12 (m) -4 1.(m)) x : 1.(m) - ml1Qm11 x(I:L„, L,,,,) -

r'm 1 irx(1: L

,

—>

Setting in := 2n yields the lower estimate of x„(I: L Remark. In the case 1 < r < 2 x n(I :

-4 L1.,„,)>---,

and

: Lro, Lr,„„).

1 w oo we have

1.(m))

Lr,coll •

6.3. Weakly singular kernels

6.3.5.

251

We now treat the limiting case r = 2. Lemma (H. KI5nig 1980: d). x,(1 : L.,

L2,1) -‹

/ 1 + log n ) 112

Proof. For every exponent r with 2 < r < co it follows from 2.7.3 and 6.3.3 that

x(/:L -+ L 2 , 1 )-5: xn(/ :

L,., 1 ) 11/ : Lro. Lr,

1

I

L2 , 1 11

43r, 2 II <

r )1 1/2 • r— 2

+

Setting 1 1 + log n

r := 2 I-

we have 2 < r

3 and

1/2 — 1/r = (r — 2)/2r < r — 2 =

1 1 + log n •

Hence 1 1 n- = exp — log n) < exp (— — log n +

2

log n 1 + log n

<

We further note that

r+

1/2

r (

r— 2 )

3 + [3(1 + log n)] 1 / 2 .

Therefore

( 1 + log n 1) / 2

X nq : L„

L2,1)

C

6.3.6. A kernel K defined on the unit square [0, 1] x [0, 1] is said to be weakly r-singular if it can be written in the form

-

.13(, 77)

1$ -

if

where B is measurable and essentially bounded. Remark. Note that Kx : -4

,

is an essentially bounded L,,(0, 1)-valued function which, however, may fail to be measurable.

Let K be a weakly r-singular kernel. Then, in view of the duality between 6.3.7. Lr ,(0, 1) and Lr, JO, 1), : g(22)

f

g(7)) dr)

on every defines an operator from L r, JO, 1) into L„(0, 1). This implies that K0 eigenvalue , 1). Since the associated Banach space E lying between Loo (0, 1) and L r, it is simply denoted by (4K)). of E, choice on the sequence does not depend

252

6. Integral operators

6.3.8.

We are now in a position to establish the main result of this section.

Eigenvalue theorem for weakly singular kernels (H. K6nig / J. R. Retherford N. Tomczak-Jaegermann 1980, H. Keinig 1980: d). Suppose that K is weakly rsingular. Then

(2„(K)) E irI ç if 1 < r < 2 and (.2.„(K)) E 12 if 2 < r < oo . In the remaining case r = 2 we have

+ log n\ 1/2 . n

14/0 I -‹

All results are best possible. Proof. We consider the factorization :

L(0, 1)

L

,4 ,

1(0, 1)

Kop

> L(0, 1) .

If 1 < r < 2, then it follows from 6.3.4 that Je

2;:;),,(L Œ,(0, 1), 444 (0, 1)).

Hence, by the eigenvalue theorem 3.6.2, we have ().„(K)) E Let f,?,:j be the 2n-periodic function defined in 6.5.8. Then the associated convolution operator shows that this result is sharp. If 2 < r < co, then J e 13 2 (L,„(0, 1),

1)).

Therefore 3.7.1 implies that (2(K)) even for continuous kernels. In the case r = 2 the estimate lAn(K)I

(

E 12

. Note that this conclusion cannot be improved,

1 + log n)1!2

n

is an easy consequence of 6.3.5 and 3.6.1. A rather involved example constructed by H. Keinig (1980: d) shows that the logarithmic term is indeed necessary: see also (KON, 3.a.11-13).

6.4.

Besov kernels

The standard monographs on Sobolev and Besov spaces of scalar-valued functions are (ADA), (NIK) and (TRI). However, the theory of E-valued functions has rarely been treated. The following papers are the only ones (known to me) which deal with this case: P. Grisvard (1966), J. Wloka (1967), J. Kadlec / V. B. Korotkov (1968), G. Sparr (1974), M. Sh. Birman / M. Z. Solomyak (1977: a) and Z. Ciesielski / T. Figiel (1983). Moreover, the situation is even worse, since these authors have very often omitted proofs claiming that they can be adapted step by step from the scalar-valued setting. Thus, apart from (KON), we are not in a position to recommend any rigorous reference. On the other hand, it would be beyond the scope of this book to provide all necessary details. This section is therefore in striking contrast to the rest of the book. It presents the most beautiful applications of the abstract theory of eigenvalue

6.4. Besov kernels

253

distributions to integral operators, but requires a lot of blind confidence on the part of the reader. Nevertheless, I bet my mathematical reputation (but not my car!) that all the statements are correct. Throughout this section all functions under consideration are defined on the unit p, q co and 1 u, y interval [0, 1]. Furthermore, let a, r > 0, 1 co.

6.4.1.* For every E-valued function 0 and 0 r 1 we put := °( + - OW if 0

1-

and Ab() : = 0 otherwise. The m-th forward difference of 0 with step r is inductively defined by 4+ 1 0 = ATA,0 and A ,1 0 := Agi. Obviously, we have

A'-cnOW =

E (- 1)m -h (in) h

+ hr)

if

0

1-

h=0

u < oo, the Besov function space [141(0, 1), E] 6.4.2.* Let 0 < < m. For 1 is the collection of all (equivalence classes of) functions 0 e [4(0, 1), E] such that 1/14 1 0 1 [B, E ] 11 := 11 0 I [4, E]ll +

f [r-clIATO I [La

r

0

is finite. In the limiting case u = co the same is assumed for ,

11 0 I [B:,

:= 1101 [4, E]II + sup tr -alIATO

j [L i,, EH}

If p = co, then [L(0, 1), E] must be replaced by [C(0, 1), E]. At first sight, this definition seems to depend on the choice of m. However, it can be shown that we always get the same Banach space, the norms 11 . I [BP,T, being equivalent. Hence the shortened notation [N(0, 1), E] is justified. If E is the complex field, then we simply write N(0, 1).

6.4.3. In the case 0 < < 1, p = co, u = co and m = 1 the above definition yields the Lipschitz-Hiilder function space [C 6(0, 1), E] which consists of all E [C(0, 1), E] such that

11 0 I [Ce , is finite.

11 0(i) - 0(2)11 := (1 0 I [C,-E] 11 + sup 1$1 - e2ia $, $2,

6.4.4.* The following inclusion is obvious. p i po co. Then Proposition. Let 1

„(0, 1), E] g

[Br, i,u(0, 1), E].

6.4.5.* We denote by [C"1(0, 1), E] the set of all E-valued functions 0 which have derivatives

D"O e [C(0, 1), E]

for

h = 0,

m.

Note that the norm

11 0 I [Cm, EH

:=

max {11 D"O I [C,

turns [C"(0, 1), E] into a Banach space.

: h = 0, ..., m}

254

6. Integral operators

p < aD. The Sobolev function space [W;(0, 1), E] is the collection 6.4.6.* Let 1 of all (equivalence classes of) E-valued functions 0 which have weak derivatives D"O E [4(0, 1), E] for h = 0, m. It is well-known that [ins(0, 1), E] becomes a Banach space under the norm m1/p I [J4/fl,:= Eh=0 11 ,01`0 [Li„ ERP) .

Remark. Since [C 1(0, 1), E] is a dense linear subset of [W;(0, 1), E], the Sobolev function space [W;(0, 1), E] can be obtained as the completion of [Cm(0, 1), El with respect to the above norm. Defining [W(0, 1), E] in this way, we may avoid all reference to distribution theory. 6.4.7.*

For a proof of the following result we refer to (KON, 3.b.7). Proposition. Let 0
p < co

and [.g(0, 1), E] = ([C(0, 1), E], [Cm(0, 1), E ])o , u .

The norms on both sides are equivalent. 6.4.8.* In view of the reiteration property, we have the following corollary of the preceding formulas. Interpolation proposition. Let 0
[14,

([13au0(0, 1), E], [13;;u (0, 1), Eno u

(0 , 1),

uo , /4 1 , u

a) and

E].

The norms on both sides are equivalent. < y„ < 1 and ni = 1, 2, ... Recall that OE + := max (OE, 0). 6.4.9.* Let 0 < y i < By an E-valued spline of order in with knots y, , y„ we mean a function L' which can be written in the form n

in 1= 1

+

j=1

xin+;( -

j ) T-1

,

where xl , x„,,„ e E. This representation is unique. in 2 we have the following characterization: The E-valued function In the case E is (m - 2) times continuously differentiable on [0, 1], and the restrictions to the intervals (0, y i ), (y i , yi+i ), (y„, 1) are E-valued polynomials of degree at most (m - 1). An E-valued spline of order 1 is piecewise constant and left-continuous at the knots. Remark. For further information about real-valued splines we recommend L. Schumaker's monograph (SCU). 6.4.10.* From now on we use the dyadic knots 2i - 1 ?hi

2h+1

with

h = 0, 1, ...

and

i = 1, ..., 2".

6.4. Besov kernels

255

That is 0

Vol

h= 0

1 0

1

h =1

1 0

Y22

Y21

Y23

Y24

1

h= 2 etc. Let S„, be the set of all complex-valued polynomials of degree at most (n — 1). Furthermore, we denote by Sg; the set of all complex-valued splines of order in with knots y ; y hi , yhi . Note that dim (Sn) = n

and

dim (S) = m + 24 + i

—

1.

Therefore

Sn

S,

shni

,

ss,ni

is a sequence of linear subspaces of the Hilbert space L 2(0, 1) such that the dimension increases by one at every step. Next, by Schmidt's procedure, we construct the orthonormal Ciesielski system: soni' ,

S1 ,

,

,

,

,

Each of these functions belongs to the corresponding subspace S,, and siz, respectively. If 112 = 1, then we obtain the classical Haar system, while m = 2 yields the so-called Franklin system; see A. Haar (1910) and Ph. Franklin (1928). 6.4.11.* We are now prepared to describe an isomorphism between Besov function spaces and Besov sequence spaces which is basic in what follows. For a complete proof of this extremely deep result the reader is referred to the original papers of the Gdansk school. Another approach via interpolation theory is due to Ff. Triebel (1981).

We define the Ciesielski transform (N„)

e (405

where 1

1 :=

and

ff() sp.(0

ocTi :=f f() s) d. 6

The desired result now reads as follows: Theorem (S. Ropela 1976, Z. Ciesielski / T. Figiel 1983). Let ni> a + 1 Then C m yields an isomorphism between

B° '0 1)

and

ip(m)

e bap-ullp+1/2.

-

1/p.

256

6.

Integral operators

Remark. We conjecture that this statement remains true in the case of E-valued functions and sequences. 6.4.12.* The following result is analogous to 5.4.4. Lemma. Let a + r> lip + 11q — 1. Then there exists := f f($) AO

a

for f E BL(0, 1)

and

g e N(0, 1).

Using this duality, we can embed ./3(0, 1) into B(0, 1)'. Proof. In view of 1 > (1/p — a) + (1/q — r), we may choose p o and qc, with 1 > 1/p o + 1/q0 , 1/po > 1/p — a, 1/q0 > 1/q — T. Then it follows from the classical embedding theorem that /4,„(0, 1) Lp0(0, 1) and Bqr,0(0, 1) 40(0, 1). Finally, Holder's inequality yields the existence of the bounded bilinear form . 6.4.13.* We now describe how the embedding operator /B from ./3;04(0, 1) into /5(0, 1)' is related to embedding operators i n, and Ib acting between sequence spaces. Proposition. Let + r>llp + 11q — 1 and m> max (a + 1 — 11P,r + 1 — 11q). Then IB

/37(0, 1)

Br(0,1)'

cml ip(n) 0

bap_,»p+i/2 In, 0

e (brq-)/q4-1/2)'.

lb

Proof. The assertion is an immediate consequence of Parseval's formula applied to the Ciesielski system. More precisely, if C ,,,f = (cx„) (4) and Ca = (9 n) IC) (Ani), then 1 cc 21. m /3„ + E E equt. f($) g($) d = E oc,7 n=1

o

h=0i=1

Remark. In fact, it is superfluous to prove the existence of the embedding operator IB in advance, since it follows directly from the above diagram and Lemma 5.4.4. 6.4.14.* We now transfer Proposition 5.4.6 into the context of Besov function spaces. For another, self-contained proof we refer to (KON, 3.c.5 and 3.c.7). Proposition (A. Pietsch 1980: d). Let a + r> 1/p + 1 11q — 1 x(/: B;,„(0, 1) —>

1)')

n-Q,

where

e:= a +

7-

+ 1/2 (0 — 1/q 1 — llp — 1/q

if 1 _- _ p _-_ q' if 1 p 2 q' if 2 ._ p

2, q' cc , cc .

0. Then

6.4. Besov kernels

257

6.4.15. Adaptation to 6.2.7 yields the proof of the following result. Proposition. Let 0 E W;(0, 1), E]. Then Oop :

a --+ <00, a>

defines an absolutely p-summing operator from E' into W7,7(0, 1). Moreover,

11(Pop I WI MOI [Wr, E ] I • 6.4.16.* An analogous conclusion holds for Besov function spaces. Proposition (A. Pietsch 1980: d). Let 0 e [B;,.(0, 1), E] and r := max (p, u). Then Oop :

a --+ <00, a>

defines an absolutely r-summing operator from E' into Nu(0, 1). Moreover,

11(Pop 4411 -5_ 11 0 [B;,u, Eili • a, e E', Jessen's inequality C.3.10 yields Proof. Given a1 , ) 1Ir ( 1 1 Plr ) 1IP aiMP [ l<0(), aid • (I [ o i= Therefore 1.1r n1 11<°() ai> I 40 5_ HO I [Li, al (ai) [w„ • (i=

Applying this result to ATO, we obtain 1/r \i

Hence

=1

!IA-TO

IKATOO, ai> I Lpli r)

n 1 a i > IL ijr (I [TE E 6IKAT00, i=1 0 1 n f (E

[

0 i= 1 1

< [f o

[T-

f [4 , E H'Kai) I [Wr,

yi hr

A

tz_. r

110770(.), ai> I I, 13110

uir

dr

<

ru

, i ii.

II(ai) I [34'r, Elll .

En)" ---11rr

et-allATo i [Lp,

Finally, we conclude from the preceding inequalities that n

(

E 11<0( . ), ai> I B;,„Ii r)

1/r

-.

i=1

(1

.),ai > ai> 5_ (ii i ll<00, ai> I 410 hr/ + [L f [CI<(

dr

i=1 0

5_ (11 0 I [4, EIII + [s (r- alIAT0 j [4, air 0

. = 110 1 [B,u, EJII 11(ai) I {14),., This shows that I, is absolutely r-summing.

dr T /")

II(ai) I

[vvr,

258

6. Integral operators

Remark. The result just proved could be deduced immediately from Proposition 5.4.7 provided that the Ciesielski isomorphism would be available in the setting of E-valued functions and sequences. 6.4.17.* A kernel K defined on the unit square [0, 1] x [0, 1] is said to be of Besov type [X,14(0, 1), 45(0, 1)] if the function-valued function Kx :

K(e , -)

belongs to [N„(0, 1), At-m(0, 1)]. 6.4.18.* Given K e [N,,u(0, 1), B(0, 1)], it follows from 6.4.16 that

Kop : g(l)) —> f

n g ( ri) (in )

1)' into ./P(0, 1). Here g denotes a distribution. defines an operator from Bq' 1)'. This implies If + r > 1/p + 1/q — 1, then BL(0, 1) is embedded into A; on every intermediate Banach space E. Since the associated eigenvalue that K0 sequence does not depend on the choice of E, it is simply denoted by ().(K)). 6.4.19.*We now give the most important application of the abstract theory of eigenvalue distributions to integral operators. Recall that q+ := max(q', 2). Eigenvalue theorem for Besov kernels (A. Pietsch 1980: d). Suppose that 11q — 1 and cr + T > 11p 11r := a + r + 11q+. Then - (0, 1)] implies (21„(K)) e K e [B(0, 1),/3 4 This result is the best possible. q' and u = 1. Define Proof (isomorphic to 5.4.10). First we assume that p if 1 _. p q' .._. 2, cx: ) , if 1 p-. 2 q' e := Ci + "1" + CO1/2 — 1/q cx) , q' if 2 p 1 — llp — 1/q and s := max (p, 2). We know from 6.4.14 that . ,(0, 1)') -< ire. x na: B, 1 (0, 1) -3 B L Hence j (0, 1), K./m(0, 1)'). Je Furthermore, by 6.4.16 and 2.7.4,

Kop G 2 t2,0 (13; ,v(0 1 Y B72,1 Thus, in view of the factorization Kop

Kop :

(0

1

)).

.13 1 (0, 1)

>

1)',

we conclude from the multiplication theorem 2.4.18 that Kop e

1)') ,

where 1/r = g + 1/s takes precisely the values given in the above theorem.

6.4. Besov kernels

259

If p > q' and u = 1. Then we see from 6.4.4 that K e [N i (0, 1), BL(0, 1)]

implies K e [K, 1 (0, 1), BTL,(0, 1)].

In this way the second case is reduced to the first one. So far we have shown that the map K K op , which assigns to every kernel the corresponding operator, acts as follows: op : [B o (0, 1), X(0, 1)]

2!:20 (N(0, 1)').

This result can be improved by interpolation. To this end, choose ao , ai and 0 such that

--- ( 1 — 0) o + °Cr 1 and 0 < ao < a l < oo Since a + t > I /p + 1/q — 1, it is further possible to satisfy the condition Gro + > 1/p + 1/q — 1. We now apply the formula cr

([./V1 (0, 1), E], [B'1 (0, 1), E])0 14 = [B(0, 1), E] for E := 1=P q.,,(0, 1). Next replacing a by ao and al , we define ro and r 1 in the same way as r. Then it follows from 1/r = (1 — 0)/1. 0 + 011% that 2(x) ) (ow \ r0, co 2(x) r I , 0,u = r,u • Hence the interpolation property yields op : [B(0, 1), ./37 ,(0, 1)] -3

1)').

By the eigenvalue theorem 3.6.2, we therefore obtain (2,(K)) e In view of 6.5.24, Examples 6.5.15 and 6.5.16 show that this conclusion cannot be improved. See also Theorem 6.5.17. Remark. The concept of a Besov function space [11;', „(0, 1), E] can be extended to arbitrary real exponents a; see G. Sparr (1974). In this setting the above theorem remains true under the condition a + > (11p + 11q — 1),. Furthermore, we may replace the unit interval by any smoothly bounded domain of the d-dimensional euclidean space. If (a + >(1/p + 11 q — 1), then the conclusion holds for 1/r := (ci + T.)1c1 + 11q+ ; see Z. Ciesielski / T. Figiel (1983) and (KON, 3.d.5). 6.4.20. The n-th spline approximation number of order in of an E-valued function ç/i e [4(0, 1), E] is defined by an(0 [Lb , E ],n) := inf 110 — I [L„, the infimum being taken over all E-valued splines n—1

z(0 = iE= 1 Xi -1

if

n

if

n>m+1.

n7 + 1

and ge) =

L. X ie i-1

i=1

E

2h+i- n—nz

-

1-1 +

1 yhi)

6.4.21. We now investigate the relationship between approximation numbers of functions and of operators.

Lemma. Let 0 e [L (O, 1), E]. Then an(00p : E' —> L(0, 1) I Vp)

an(0 I [L„, E]„,).

260

6. Integral operators

Pr o of. The assertion is an immediate consequence of the fact that every E-valued spline E occuring in the definition of a„(0 I [L,, E]„,) induces an operator from E' into 4(0, 1) such that 11 00p — Eop I WI 5_ 11 0 — E I [Lp Eill and rank(E0 ) < n. Remark. The above inequality remains true for the smaller approximation numbers obtained from E-valued splines with free knots. 6.4.22. The following criterion shows that, just like the operator ideals 4" )„ the Besov function spaces [13(0, 1), E] can be characterized by means of certain approximation rates. For a proof of this result we refer to Z. Ciesielski / T. Figiel (1983). The scalar-valued case is treated in (SCU, 6.27 and 6.55).

Proposition.

Let

E [B(0, 1), E]

0 e [4(0, 1), E] and if and only if

in > o + 1 — 1/p. Then

(a(0I [L,, E],„)) e

6.4.23. A kernel K defined on the unit square [0, 1] x [0, 1] is said to be of BesovHille-Tamarkin type [B7,,„(0, 1), L,(0, 1)] if the function-valued function

Kx :

K(E, -)

belongs to [B(0, 1), 4(0, 1)]. Kernels of Hille-Tamarkin-Besov type [L„(0, 1), No(0, 1)] are defined analogously. 6.4.24. Eigenvalue theorem for Besov-Hille-Tamarkin kernels (A. Pietsch 1981: a). Let i> 1/p -1- 1/q — 1 and

1/r := ri + Then K e [BUO, 1), 4(0, 1)]

implies

(11.„(K)) e

This result is the best possible.

Proof. Since B, 1 (0, 1) c 4(0, 1) c g,,,,(0, 1), the assertion can be deduced from the extended version of the eigenvalue theorem 6.4.19 (Remark). We prefer, however, to give here a different proof which works under the stronger assumption that ljp + 1/q 1. Then

KE

u(0, 1), Lq(0, 1)].

It follows from 6.4.21 and 6.4.22 that K., e (13,.);.(27,3 ,,,(4.(0, 1)). Thus, by 1.2.4, 2.7.4 and 2.8.15, we have K., e 21(L„,(0, 1)), and the eigenvalue theorem for operators of Weyl type implies that R,(K)) e 4,„. In view of 6.5.22, Examples 6.5.15 and 6.5.16 show that this result cannot be improved. 6.4.25.

In order to establish a counterpart of the preceding theorem, an auxiliary

6.4. Besov kernels

261

result from interpolation theory is required. For a proof we refer to (KON, 3.d.1). See also (TRI, 1.18.4). Proposition (M. Sh. Birman J M. Z. Solomyak 1977: a). Let (E0 , E1 ) be any interpolation couple. If p w, then [4(0, 1), (E0 , Ei )e ,„1

([4,(0, 1), Ed, [L(0, 1), E1 ]) 0 ,.

6.4.26. In Theorem 6.4.19 the asymptotic behaviour of eigenvalues does not depend on the fine index v. We now describe a situation where this is not so. A more detailed proof of the following result is to be found in (KON, 3.d.7). Eigenvalue theorem for Hille-Tamarkin-Besov kernels (M. Sh. Birman / M. Z. Solomyak 1977: a, A. Pietsch 1981: a). Let r> 1/p + 1Ig - 1, p v and

1/r :=

Z

1/q+.

Then K e [4(0, 1), ../3(0, 1)]

implies

(2„(K)) E /,.

This result is the best possible. g'. Then, reasoning Proof. Without loss of generality, we may assume that p similarly as in the proof of 6.4.19, it follows that the map K K op, which assigns to every kernel the corresponding operator, acts as follows: op : [L(0, 1), N o,(0, 1)]

.I.,'.3,00.,(L„(0, 1)).

This result can be improved by interpolation. To this end, we apply 6.4.25 to the interpolation couple (B:)„. (0, 1), N 1 (0, 1)). The interpolation property now implies that op : [L(0, 1), A(0, 1 )1

0).

By the eigenvalue theorem 3.6.2, we therefore obtain (2„(10) 6 lr,v. In view of 6.5.23, Examples 6.5.15 and 6.5.16 show that this result cannot be improved. Remark. In the case p > v the precise version of the above eigenvalue theorem is v can unknown. However, if r> 1/v + 1/q - 1, then the required condition p always be satisfied by setting p := v. 6.4.27. A kernel K defined on the unit square [0, 1] x [0, 1] is said to be of Sobolev type [W'(0, 1), r,(0, 1)] if the function-valued function

Kx :

.)

belongs to [Wn;(0, 1), WZ(0, 1)]. p, q < cc. We denote by W7(0, 1), J/V(0, 1)I the collection of all 6.4.28. Let 1 (equivalence classes of) measurable kernels K defined on the unit square [0, 1] x [0, 1] which have weak partial derivatives DIMI,(2K E [L,(0, 1), L.(0, 1)1 for

h = 0, ..., in

and

k = 0,

n.

Obviously, W;(0, 1), W:(0, 1)1 becomes a Banach space under a suitable norm.

262

6.

Integral operators

6.4.29. In the context of Sobolev spaces Proposition 6.2.12 reads as follows. p, q < co. Then the map which assigns to every Proposition. Let 1 kernel K the function-valued function

Kx :

K($, .)

yields an isomorphism between E n1(0, 1), W'(O, 1)jj

[ Wpm(0, 1), W41(0, 1)] .

and

6.4.30. The interpolation formula stated in 6.4.25 admits the following generalization. Proposition. Let (E0 , E1) be any interpolation couple. If p

w, then

[W(0, 1), (E0 , E 1 )0,1 s ([ Wir(0, 1), E0 ], [W7(0, 1), 6.4.31. We conclude this section with a result which originates in the work of M. G. Krejn (1937). Eigenvalue theorem for Sobolev kernels. Let 1 hr := m + n + 11q+ and

p < co, 1
w := min (q, 2).

Then

K e [1,1P;(0, 1), W,7(0, 1)]

implies 0, n(K))

E

This result is the best possible.

Proof. Assume first that 1 < q < 2. Reasoning as in the proof of Theorem 6.4.19, we see from the diagram

Kop : W(0, 1) i

I

if

B7;: co (0, 1)

/

K.,

> W:(0, 1)' A I > /31,12 ,„(0, 1)'

W;(0, 1)

that Kt),E 2 0 ( W;(0, 1)). Therefore the map K K op , which assigns to every kernel the corresponding operator, acts as follows: op : [W;(0, 1), n (0, 1 ), __+ 2t (

1)).

This result can be improved by interpolation. Choose q0 , q, and 0 such that

1/q = (1 - 0)1q0 + 0/q 1 and

1 < qo < q < q1 < 2.

Then

W: (0, 1) = ( WZ(0, 1), W 1(0, 1)), q . Without loss of generality, we may assume that p 6.4.30 that

q. In this case, it follows from

[ W;(0, 1), W:(0, 1)] g ([ Wpin(0, 1), W;10 (0, 1)], [W;(0, 1), WIi1(0, 1)])e,,,

6.5. Fourier coefficients

263

Letting 1/r0 := m + n + 1/.7,,

and 1/r 1

in + n + 1/q1,

we have 1/r = (1 — 0)Ir0 + Olri . Thus (2(rVoo 2 V ,30)o, Hence the interpolation property yields op : [W17(0, 1), W(0, 1)] -+

W;(0, 1))•

By the eigenvalue theorem 3.6.2, we therefore obtain (2„(K)) e We now assume that 2 q < cc. Obviously, it is enough to treat the case 1 p q = 2. It follows from 6.4.15 and 2.7.2 that Kop E t(P(W(O, 1)% W(0, 1)) . Furthermore, with a := in + n we have lE 2(4),,,,( W;(0, 1), W3(0, 1)') . The multiplication theorem 2.4.18 implies that K;::q(Wp'n(0 , By the eigenvalue theorem 3.6.2, we therefore have (),„(K))e 4, 2' In view of 6.5.21, the examples described in 6.5.18 show that the conclusions obtained above cannot be improved.

6.5.

Fourier coefficients

Throughout this section we consider complex-valued 277-periodic functions on the real line. In order to indicate this fact, the underlying function spaces are denoted by C(27.), L,.(2-rc), L,. ,„(277), W(2t) and Bg(277). The required modifications in the definitions are left t3' the reader.

6.5.1. By a trigonometric polynomial of degree n we mean a function t which can be represented in the form t() = where y_„,

6.5.2.

E

kn

yk exp (i1c)

for all

[11,

y„ E C and ly_„1 + ly„I > O. If so, then we write deg(t) = n.

Every function fe L 1 (27) induces a convolution operator 2.7c : g(n) -4

f( —

71)

g(n)

on C(27:.) and 4(277) with 1 p cc; see (DUN, XT.3.1) and (ZYG, 11.1.15). If t is a trigonometric polynomial, then Cf,r, possesses finite rank. More precisely, we have rank (Cop) = 2deg(t) + 1. 6.5.3. 1 w

Recall that the Lorentz function spaces are defined for 1 < r < co and cc; see 6.3.1.

264

6.

Integral operators

Proposition. Let JE L,(277). Then the associated convolution operator into L(27). Ch acts from Proof. Given g e L,.,,‘(27), by (HAY, p. 278) and Holder's inequality, for almost all $ e R we have 27t

2n

f($ — n g(71) dn )

p( r) g *(r) dr f

<

27t = f r 1/r— 1 / wpqr) r 1 le

6

o 27t

. 5_ . (f

—

I /wt g* (r) dr y /w'

1/w 27r 27t

[r 1/r— 1 /wp(rAw dr

[r 1/r' — 1 /Wg7N (r• iw' )J

dr

This proves that

Ilf I Lr,w11 Ilg I L,/,., 11.

Cimg LQ011

6.5.4. For every function Je L 1 (27c) and k e Z the k-th Fourier coefficient is defined by

f

27t

y k( f)

=

1

f() exp (—ik$) d$.

6.5.5. We now describe the relationship between the Fourier coefficients of a function and the eigenvalues of the associated convolution operator. Proposition (T. Carleman 1918). Let f e L, (27). Then the eigenvalue sequence of Ctp can be obtained as the non-increasing rearrangement of (27 yk(f)). Proof. Define

ek W := exp (ik0 for all $ e R. Then it follows from 2.7c

2.rc

f .fi$ -

ek(n) d = f f(71) exP (ik$ — ikn) dri= 27c yk(f) ek()

hat 27 yk(f) is an eigenvalue of CI; with the associated eigenfunction ek . Next, if ;. denotes an arbitrary eigenvalue of Ca;/, , then for any associatd eigenfunction g e L 1 (27) we have 27t 27c

A(g, e k) = (Cg, ek) = 27c 27t =

f

o

-

g(27) exp ( ik0 dn d$ —

0

f($) g(n) exp

— ikn) d chi

0 0

'k(f) (g, ek) • = ek) (g, ek) = Hence A * 2t yk(f) implies (g, ek) = O. Since g * o, the eigenvalue must coincide with at least one of the coefficients 27 y k(f). Finally, we note that the multiplicity of any eigenvalue 0 is the dimension of the subspace

Nao(A, q p) = Ni (A, C-Sp) = span tek 27c y k(f) = A} . 6.5.6.

We now state one of the oldest results about Fourier coefficients.

6.5. Fourier coefficients

265

Theorem (B. Riemann 1854, H. Lebesgue 1903). fe L i (27) implies (y k (f e co (Z). Proof. We know from the classical Weierstrass approximation theorem that the set of trigonometric polynomials is dense in L 1 (27). This shows that C‘i, is an approximable operator on C(217). The assertion now follows from 3.2.19 and 6.5.5. 6.5.7. Theorem (M. A. Parseval 1805, F. W. Bessel 1828, E. Fischer 1907, F. Riesz 1907). ))

f E L2(270

implies

(Yk(f)) 6 /2(Z) •

This result is the best possible. Proof. Obviously, the above implication is an immediate consequence of Bessel's inequality: If 1L211 2

bik(f)I 2

.

We prefer, however, to employ the tools presented in this monograph. To this end, we consider the factorization > L2 (27)

L ,,(27)

L„(27) .

Then it follows from .1 e 9

2(Lco( 27

0 1-2(27))

that C.4 is an absolutely 2-summing operator on I(27c). Hence, in view of 3.7.1 and 6.5.5, we have (yk(f)) E 12(1). Conversely, by the classical Fischer-Riesz theorem, every sequence (y k ) E 12 ( 7L ) determines a function f e L2 (2n). This proves that the above result is sharp. 6.5.8.

Example. Define f()

1 k = 1 k i r (I ± log k)

where 1 < r < co and

0. Then

0■•

if and only if

grac, E 4-0,(27r)

exp

AM

>1

and ./;2:oc E

Li., „(270 .

Proof. The assertion follows immediately from the fact that g: is continuous at all points except for = 2r k, and that it behaves like , og) as

1 14: 1 11r 0 + 'log $1)"

0; see (ZYG, V.2.6).

6.5.9. Theorem (W. H. Young 1912, F. Hausdorff 1923, G. H. Hardy/J. E. Littlew co. Then wood 1931). Let 1 < r < 2 and 1 J

E L,.,„(277)

implies Olaf e

This result is the best possible.

266

6. Integral operators

Proof. We first treat the case w = co. Recall from 6.3.3 that

/

44., 2 (L oc (27), L,, , , 1 (27)) .

Therefore, in view of the factorization

L„,(27)

C-r,p : L(217)

and 2.7.4, we have

Cr,p 43,/. 2 (L,,,(27c)) This proves that the map f C‘p , which assigns to every function the associated convolution operator, acts as follows: op : L„, c,,(21-c) Next we choose ro , r, and 0 such that

1/r = (1 — 0)/r0 + 01r1 and

1 < ro < r < ri < 2.

Then the interpolation formulas

(40, 0,(277) Lri ,„(27c))0, „ = L,,,(27r) and imply that op : L r ,(27c) Consequently, by the eigenvalue theorem for operators of Weyl type and 6.5.5, we have (iik(f)) E 4,,„(1). Finally, the preceding example shows that this result cannot be improved. 6.5.10. The following construction goes back to H. S. Shapiro (1951), W. Rudin (1959) and J. J. F. Fournier (1974).

Lemma. Let yo , tn($)

2n E yk(tn)

k l

y„ e C. Then there exists a trigonometric polynomial exp (ikt)

such that Y2h(tn) = y h

for

h = 0, ..., n

and n

\ 1/2

E Iyh1 2 )

Iltn

h=o

•

Proof. By homogeneity, it is enough to treat the case in which

E

h=0

Iyhl 2 < 1.

Using the complex variable

= exp (it), we construct two kinds of polynomials.

6.5. Fourier coefficients

267

First let to (C)

and

:= yo C.

Proceeding inductively, we then define = C2h-i sh-10

shG)

and yi 2

th(Ç)

h-1

+ th-10 •

Obviously, 21'

2h

= E

and

k=1

k=

yk(th) C

k .

Furthermore, for the leading coefficients we have y 2h(sh) = 1

and /,h(t h) =

Yh•

Observe that th coincides with the 2h-th partial sum of the Taylor expansion of provided that h n. Hence =

Y2h(tn) = Y2h(th)

for

h = 0,

tn

n.

For arbitrary complex numbers y, d and r we have + TI 2 = ( 1 + i1 2 ) ( 0V + ir1 2)• Applying this formula to the defining equations of sh and th yields — 7*TI 2 +

ish()1 2 + 14(01 2 = ( 1. + IYh1 2 ) (Ish-1(01 2 + Ith--1()1 2 )• Hence

ls.(01 2 + 14()1 2 = h=0 (1 + Irh1 2) Thus

t„

h=o

Ivh1 2)

e•

(considered as a function of possesses all the desired properties.

Remark. If yo =

y„ = 1, then we have

yk (tn) = +1 for

6.5.11.

exP

k = I, ..., 2 n and

Il t,, I

2n/ 2 .

In view of 6.5.7,

fe C(27) implies (yk(f)) e /2 (Z). It was T. Carleman (1918) who made the surprising observation that this result cannot be improved. Example (S. Banach 1930). Given (yh) such that y 2h( = yh for h = 0, 1, ...

E 12,

there exists a function

JE C(27)

Proof. The sequence c = (yh) can be represented by an absolutely convergent series C

=

co

Ck, k=1

where c, = (ykh) are finite sequences. Choose n 1 < n 2 < ... such that vkh = 0

if

h>

268

6. Integral operators

By the preceding lemma, there exist trigonometric polynomials tnk with and

72h(t.k) = y, Therefore, setting

=

f

e 112 1Ick I /211.

114, I C11

CID

k=I

tnk

yields the desired function. Remark. Another result along these lines states that for (7k) e 12(Z) there exists e C(27t) with lyk(f)I lykl; see K. de Leeuw J Y. Katznelson / J.-P. Kahane (1977) and (KAS, 9.2.5). 6.5.12. The n-th trigonometric approximation number of a function f e 4(27.) is defined by (f I L,) := inf

— t I LA: deg (t) < n} ,

where t denotes a trigonometric polynomial. 6.5.13. The next result is analogous to 6.4.21. Lemma. Let fe 4.(217). Then a2 „(qp :

-4 L op)

an(f I Lr).

Proof. The assertion is an immediate consequence of the fact that every trigonometric polynomial t induces a convolution operator from L s.,(27-E) into L,(27,7) such that

I f — t I LAI

IIC‘p — G, p 11

rank (Q,p) = 2 deg (t) + 1.

and

6.5.14. For a proof of the following criterion we refer to (BUS, 4.2.2 and 4.2.3) and (SCI, 3.7.1). See also (NIK, Chap. 5) and (TIM, Chap. 5 and 6). Proposition (O. V. Besov 1961). Let f E 4.(27c). Then if and only if

f e 13?.,(21-c)

(an(f I 4))

E 1 1 /0,

6.5.15. Next we modify the example given in 6.5.8. Example. Define ID

fr2,; ,() = k=1

, k e ir (1 + log k)"

where 1 < r < co, 0
<

exp

1/r and oc

if and only if ocw > 1

and fro

e

Proof. Let 0 < . n sin

r, 00 •

n. Then it follows from 1

O. Then

and

1 sin —$ — 2 — TC

Iv.

269

6.5. Fourier coefficients

that the partial sums . n sin

n

--

r,,($) := E exp 0/(0 =

exp (i

k=1

sin 1 $ 2

n +1 ) $ 2

satisfy the inequality ir()i — We now write 1

with yk :=

f,,($) := E Yi. exP (ik0 k=n

10+111'0 +

log kr •

Summation by parts yields

—

E

Ifn()1 =

yk( rk ($) k=n

—vnrn- 1() +

rk--1(0)

co

E (Yk k=n

— yk+i)) 7

=

E (yk — vki-1) rk ()

k =n

277 yn•

Next, for in > n, it follows from G.3.2 that m2.7z 'fa)! E + If„,(01 (corn + 7-)Ym. k=n

Note that ifn(27c —

= if1()1. Using G.3.2 again, we now obtain rr

If. I (

an(fr2,Z. Lr)

2

)1Ir

= (2 f 1.fa($)1r 0 cc

nl(m— 1)

I

E

tit= n + 1

Tr.

Îm

fr

Ific

r

) d

+

f If

d$) 1

Tcln rr

co Cl

2

E in-2 (mym)r m=n+1

+

1/r

a) -5= c2

f

1 ne(1 + log nr

Therefore, by the preceding proposition, fr2j', belongs to the desired Besov space. The converse implication follows from Theorem 6.5.17. 6.5.16. Let t„ denote the trigonometric polynomial constructed in 6.5.10 (Remark) such that y k (t„) = +1 for

k = 1, ..., 2"

and

Mtn

Example. Define co 1 exp 02 1'0 t„($), g():= 2"(" 112)(n+ W nto where and

9. >

and

0

47,

0■

,

e B,„(27) e

.M0 , 00 (270.

O. Then if and only if xw> 1

I CM

< 2n/2

270

6. Integral operators

Moreover, gr,i;', has Ti times the Fourier coefficient +2- 1(e+ 112) (n + 1)-". Pro of Approximating g,,2 :ro, by its partial sums, we see from G.3.2 that

a2 k i (etc, I C)

1

E

n=k 2"(Q+112)(n

<

C = 21(e(k

n

1 1)°' •

Hence, by 2.1.10 and 6.5.14, the function gkil belongs to the desired Besov space. The converse implication follows from thé next theorem. Finally, we observe that the trigonometric polynomial exp (i 2" ) t„() has Fourier coefficients +1 if and only if 2" < k < 2n+ 1 . 6.5.17. Recall that r+ : = max(r', 2). Theorem (A. Pietsch 1980: a). Let

1/s := e + 1/r+. Then

f

implies

(7, k(f))

G lsov(Z)-

This result is the best possible. Proof. In view of 6.5.24, the theorem is a special case of 6.4.19. We prefer, however, to give here a direct proof. To this end, we consider the factorization C4'p

Grp : L (2,7)

> L (27) ---->

.

Note that

l e 3 ,.,(L .(27c) , L (27))

2;.-T> c0 co (27), 4(27)) .

Moreover, by 6.5.13 and 6.5.14, we have

C-cf,p e 2(1 %,,,(4.,(27), L(27)). This proves that E

Hence 3.6.2 and 6.5.5 imply (7)k(f)) E is ,„(Z). Finally, Examples 6.5.15 and 6.5.16 show that this result cannot be improved. 6.5.18. Theorem. Let 1 < r < cc, 1/s := I + 1/r+

and

w := min (r, 2).

Then

f e WA27)

implies

(y k (f)) e

This result is the best possible. Proof. As shown in the course of the previous proof, the map f C-4, which assigns to every function the associated convolution operator, acts as follows: op : B

,

09

(27) -±

6.5. Fourier coefficients

Since 141(2n)

271

B:.„(277), we also have

op : W,f(27t) We now deal with the case 1 < r < 2. Choose r 1 , r1 and 0 such that 1/r = (1 — 0)/r o + 011. 1 and

1 < ro < r < r <2.

Let 1/s0 := / + 1/4,

and

1/s 1 := 1 + 1/r1

Then

1/s = (1 — 0)/s 0 0/s 1 and

0 < s1 < s < so < cc.

In view of the interpolation formulas

(1310(270, W, 1 (27c))6 ,,. = W,f(27t) and oc ,

e(sx) .0)0, r

we obtain op : W(27t)

,t(sxJ(L c0 (277)).

Therefore, if I < r < 2, then Je W71.(270 implies (yk(f)) e is,r(Z). Taking any /-th primitive of the function fr2; with ocr > 1, we see that this result is sharp. In the case 2 r < oo it follows from Je W(27t) that f e W(2t) = B1,2(277). Hence, by the preceding theorem, we have (yk(f)) e /s , 2 (7L). Finally, using a result stated in 6.5.11 (Remark), for 0 < w < 2, we may construct a function Je 0(27r) such that (7k(f )) 6.5.19. In what follows we investigate how certain properties of a function f e L 1 (27-c) are inherited by the convolution kernel C,17) : = f( — r))

for

e R.

In this way we obtain examples which show that the results stated in 6.1.6, 6.2.15, 6.4.19 etc. are sharp. 6.5.20. The first observation along these lines is well-known; see (ZYG, II.1.11). Proposition. Let 1 :5_ q < co. Then

Je 4277)

implies

Cf e [C(27c), 4,(27t)].

Proof. The assertion is trivial for continuous functions. Since C(277) is dense in 4(27), it can easily be extended to all J e La(2n). Remark. The conclusion fails for q = cc. However, it holds when L co (27c) is replaced by C(27c). 6.5.21. In the context of Sobolev spaces the preceding result reads as follows; see (NIK, 5.6.5). Proposition. Let 1 :5_ q < co. Then

fe W(2-î) implies

Cf e [Cm(27), W,;(27c)].

272

6. Integral operators

6.5.22. Proposition. Let 1

q

co. Then

Cf G [B , (27t), 41(270] .

Je BL(2t) implies

Proof. We know from the previous propositions that the map f—* Cf , which assigns to every function the associated convolution kernel, acts as follows:

con : L„(27c) —> [C(277), L,i(27z)] and con : W(27t) —> [C(27), /427)].

Let a = Om. Then, by 6.4.7, the interpolation property yields con : 14, u(27r)

(27r), 4,(27c)] .

[B

6.5.23. The next result is taken from (NIK, 5.6.5). y
Proposition. Let 1

Je Ar70,(0, 1) implies

Cf e [C(0, 1), No,(0, 1)].

6.5.24. Proposition (A. Pietsch 1981: a). fe Be4:Eur(270

implies Cf e [./3%, 1 (27-c),

1 (27-c)].

Proof. As stated above, the map f-+ Cf, which assigns to every function the associated convolution kernel, acts as follows:

con : N 1 (2n) —> [C(27r), /3 1 (277)]. Analogously, we have con : Mr(2 -n) —> [Cm(27c), N, 1 (27c)].

Let a = Om. Then, by 6.4.7, the interpolation property yields con : BZ:(27c) —> [B1,„(2n), N 1 (27)]. This completes the proof. 6.6.

Traces and determinants of kernels

6.6.1.

To begin with, we generalize 5.5.1.

Proposition (A. Grothendieck 1956: a). Let 0 e[L i (X, ,u), E']. Then :x

defines a nuclear operator from E into L i (X, It) with 11 00p I %II = II

I [L1, E111.

Furthermore, all operators Te 91(E, L i (X,,u)) can be obtained in this way. Proof. We first assume that 0 is a simple function. Then there exists a representation

0() = E where a 1 ,

1

aihg)

e E' and

for almost all h„

e X,

are characteristic functions of pairwise disjoint

273

6.6. Traces and determinants of kernels

measurable subsets X1 ,

X„. In this case 00, has finite rank and

E Ilai I

1100, 19111

= 110 I [L1, PH

i=1

If 0 E [L i (X,y), E'] is arbitrary, then it can be approximated by a sequence of simple functions 0„. It follows from — (000i, I 9111

—

0„ I

[

Li

,

that the associated operators (0„) 0, form a Cauchy sequence in 91(E, L i (X„ u)). Since 91 is a Banach operator ideal, there must exist a nuclear limit which is nothing other than 00,. By continuity, we get 11 00p I 9111

11 0 I [Li ,

•

Finally, we show that every operator T e 91(E, L i (X, du)) can be obtained in this way. To this end, given e > 0, we consider a nuclear representation 00

T = ai f i

i=1

such that i

Then it follows from dit() < a)

i=1

that 00

0() :=

i=1

aifg)

is defined almost everywhere. Furthermore, we have 0 e [L i (X,

110 I [L1 Letting 8

Will Ilfi I L1 II -5 ( 1 +

E'] and

I 7' IV!.

0 yields I [Li ,

7' I 9tIl •

Obviously, T = 0„. Specializing the preceding proposition, we arrive at a characterization of the 6.6.2. nuclear operators acting on L i (X, itc). Proposition. Every kernel K e [L i (X, ,u), L „ (X , ,u)] induces an operator K, e 9(L1(X, eu)) with 11 Koi, I 9/IJ = 1K I [L 1 , LA11 • Furthermore, all operators T e91(L i (X, ,u)) can be obtained in this way. 6.6.3. Since L i (X, ,u) has the approximation property, every operator K„ e9Z(L i (X„u)) possesses a well-defined trace which is simply denoted by T(K).

274

6. Integral operators

By analogy with 5.5.3 it could be conjectured that this trace is given by the formula

r(T) = f K($, e) du(). However, as in the case X = [0, 1], it may happen that the diagonal = n is a zero set of X x X on which K can be changed arbitrarily. Therefore, unless additional assumptions are placed on the underlying kernel, the right-hand integral makes no sense. 6.6.4.

Next we deal with continuous kernels; see 6.1.5.

Proposition (A. Grothendieck 1956: a). Given any finite Borel measure ,u on a compact Hausdorff space X, every kernel K e [C(X), C(X)] induces an operator

Ko, E 91(C(X))

I Kop 1111 <=__

with

*X)

K [C, Cl I •

Proof. Let 0 e [L,(X,,u), E]. Then by an appropriate dualization of the proof in 6.6.1, we can show that

f 0(0 f(e) du()

0°P f(e)

defines a nuclear operator from C(X) into E. Hence, taking E : = C(X), the assertion follows from the fact that every continuous kernel belongs to [L i (X, C(X)]. Since C(X) has the approximation property, every operator Ko, e 91(C(X)) 6.6.5. induced by a continuous kernel possesses a well-defined trace which is simply denoted by r(K). Theorem. Suppose that kt is a finite Borel measure on a compact Hausdorff space X. Let K e [C(X), C(X)]. Then r(K) = f K(e, e) d,u()

ir(K)I 5 ,u(X) IK

and

I [C,

Proof. We first consider a degenerate kernel K. Then there exist continuous funcfn and g 1 , g„ such that tions f1,

K(e, n) =

gio

for all $,

e X.

Hence T(K)

=

f fi(0 gi(e) dy(e) = f K(e, e) dy(e).

l

=1 x

According to the Weierstrass approximation theorem, the set of degenerate kernels is dense in [C(X), C(X)]. Thus, by continuity, the desired formula holds for all continuous kernels. 6.6.6.

Given any kernel K, we write K

fl) krk, •-•, rin

( K($1,

ni)

1411,

i)

In particular, observe that K(f„) = K(e,

n.))

6.6. Traces and determinants of kernels

275

6.6.7.

At the end of this monograph, we establish the very result which was the historical starting point of the whole theory. We are going to describe the canonical Fredholm denominator CO

K) = 1 -}-Ea , i n(K)

of an operator Kop induced by a continuous kernel K. Replacing infinite sums by integrals, the proof given in 5.5.5 can be adapted to this classical setting. Theorem (I. Fredholm 1903). Suppose that kt is a finite Borel measure on a compact Hausdorff space X. Let K e [C(X), C(X)]. Then ocn(10 =

n.

••• X

K (! 1 ' • • • !fl) di4e 1) • • • clitt( n) X

and

1°4-K)1

n7112

n

IU(X) n

11K I

[C, C]" •

6.6.8.

The next result is a counterpart of 5.5.6. It shows that, by the principle of related operators, the classical construction of Fredholm denominators can be transferred to nuclear operators on arbitrary Banach spaces. Proposition. Every operator T e 91(E)is related to an operator Kop e 91(C(0,1)) induced by a continuous kernel K. Proof. In view of 1.7.4, there exists a representation CO

T = E r•a, C) x i 1=1

such that CO

ri > 0,

E ri = 1, lim Ia, = 0, lirn Ix,II = 0. 1=1 Divide [0, 1] into disjoint intervals Xf of length ri , and denote the corresponding characteristic functions by h i . Then the functions gi := 112 hi form an orthonormal system. Next, by a small perturbation, we pass to an orthonormal system of continuous trapezoidal functions fi such that

2r7 112 and fi ( ) = 0 for e [0, 1] 1X . Define A e t(E, C(0, 1)) and X E 2(C(0, 1), E) by f I C11

A := E Trai 0 f, and i=1

X := E T1/2.6® j=1

Then T = XA. Furthermore, the related operator S := AX is induced by the kernel co co

n) := i= 1

j= 1

4 124 12 <xi, ai> fi(e)fi(n).

Note that, for every point (e, n), the right-hand expression contains at most one non-vanishing summand. This observation implies that 71) — E 4/2r)/ 2

ai> fi().fi(n)1

4 sup

:i +j

nl.

276

6. Integral operators

Thus K must be continuous, since it is the uniform limit of a sequence of continuous functions. 6.6.9.

The following result is a by-product of the proof given in 6.4.24.

Proposition (W. F. Stinespring 1958, G. E. Karadzhov 1977). Every kernel K E [B13(0, 1), L 2 (0, 1)] induces an operator Kop c.91(L 2 (0, 1))

I [Bi'?, Ldli •

with W., I 9/II

Remark. Note that

[C6(0, 1), C(0, 1)] c [B 2/?(0, 1), L 2 (0, 1)]

if

1/2 < cr < 1 .

6.6.10. We now show how the estimates for the coefficients of the Fredholm denominator given in 6.6.7 improve when the kernel satisfies stronger smoothness conditions. Proposition (1. Fredholm 1903). Let K e [C 6(0, 1), C(0, 1)] with 0 < o < 1.

Then n(K)1

26" IIK I [Ce, C]".

n!

Proof. Using standard properties of determinants, we transform

KG

-1(( 1, 1) • • • lq$ 1 , n)

$ 1, - • -, E

5 • • • /

:= det n

-

' IC( n , 1) • • • K(, n) n — 1, then the coefficients of the new j-th column

in the following way. If j = 1, are defined by K(

I<(f

+ 1)

i5

for i = I, ..., n ,

—

while the n-th column remains unchanged:

:=

for

„)

i = 1,

n.

This yields n-1

11 lei —

n ) = det (u i)

K (!'

j=1

'515 •••5

Applying Hadamard's inequality A.4.5, we obtain K

(15 •••5

fl)

n-1

n"1 2

s15•••5

n i=1

-

[C-,

Since the left-hand expression is a symmetric function, we may arrange the variables 1. Then the inequality of means G.1.1 implies that such that 0 (n-1

y(n-1)

—

n-1

1 2 n• n _ _< —

6.6. Traces and determinants of kernels

Hence, in view of 2n(n —

277

we have

1)" /7",

< nn(1/2-0) 2anliK [Ce,

The desired estimate is now obvious. Rem ark. It follows from 4.8.7 and Stirling's formula that 6G, K) is an entire function of order r, where 1/r := o + 1/2.

6.6.11. An operator Te t(C(27T)) is said to be a multiplier if STS 1 = T for all

eR,

where

S:f(a) fOc + is the translation operator with step e.

6.6.12. Finally, we give a nice application of the previous results. Proposition (W. Bauhardt 1979). Every function f e C(27) induces a multiplier e 91( C(270) with IIC , I9uI1 = 27c

Ilf 1 C11

Furthermore, all multipliers T e 4N(C(27)) can be obtained in this way. More precisely, we have

1 T = C, with f(0 :=-—x(ST)

for all e e R.

Proof. The first part of the assertion follows from 6.6.4. Next we show that

f <Sf, ;a>

y> d

y>

,u>

for f, g e C(277) and it, V e C(27r)'. In the special case of Dirac measures, it = and y = 6/3 , this formula reduces to the obvious fact that 2n-

27r

f(a + g(13 —

=

n g(oc — )

6

Recall that the linear span of all Dirac measures is dense in C(27 )' with respect to the topology of uniform convergence on the precompact subsets of C(270. Thus the above equation holds in general. Assume that T e 91(C(20 is a multiplier. Take any nuclear representation

T=EktiOfi, i =1

278

6. Integral operators

where sa l kt2 ,

e C(27-c )' and fi f2,

e C(27c). Let g e C(27) and v e C(270'. Then

2n 1

=

27-c

f <S,ITS,7 1g, 2n

co

1

Ef

27c

v> <S,71g„ ui>

0 2n

1 co E f <Seft , ft i > <Seg, v> 27: i = 1

=

f r(ST)

<Seg, v> d.

o Hence 2n

1 Tg =__fr(ST)S'gd. 2rc

o

This proves that T coincides with the convolution operator C4, where the continuous function f is defined by

1 f(0 := — 27 r(ST)

for all

e R.

Furthermore,

If I c I = sup If( $)I

1 sup Ir(Si T)1

2.7c

27r

T

.

6.6.13. It follows from 6.5.17 that all functions belonging to 13?(27-0 have absolutely convergent Fourier series. In the context of nuclear operators this result, due to B. S. Stechkin (1951), reads as follows. Proposition. Every function JE .13113(2n) induces a multiplier Cjo.p

e 91(L2 (270)

with

C{,p I

M

ellf I B1311.

Remark. The nuclear multipliers on L2 (27-c) are characterized as those convolution operators Cf,/, such that f is the convolution product of two square integrable functions. According to G. H. Hardy/ J. E. Littlewood (1928: b) this criterion goes back to M. Riesz. Another necessary and sufficient condition in terms of modified approximation numbers was given by B. S. Stechkin (1955); see also (BAR, IX.8) and

(KAH, 11.3).

CHAPTER 7

Historical survey Besonders aber wiinschte ich meiner Arbeit dadurch einigen Werth zu verleihen, dass ich soviel ais mitiglich bis zu den Originalquellen vorzudringen suchte, urn die ersten Erfinder von Methoden und die ersten Entdecker von Lehrsatzen citiren zu kiinnen. Solche Citate sind nicht nur em n Opfer, welches die spatere Zeit den friihern Offenbarungen des Genius schuldet, sie bilden em n Stack Geschichte der Wissenschaft und laden zum Studium der hohen Werke em, aus denen die Wissenschaft aufgebaut ist, und in denen noch immer reiche Schâtze ungehoben ruhen. Richard Baltzer (1857) In preparing this lecture, the speaker has assumed that he is expected to talk on a subject in which he had some first-hand experience through his own work. And glancing back over the years he found that the one topic to which he has returned again and again is the problem of eigenvalues and eigenfunctions in its various ramifications. Hermann Weyl (1950)

The last chapter is devoted to the history of the subject under consideration and to those mathematicians who essentially contributed to the development of this theory. The following memorial list is ordered according to the dates of birth. We have included some famous names who were involved in the creation of matrix theory as well as those who laid the foundation of the functional analytical approach to eigenvalue problems. Leibniz, Gottfried Wilhelm Cramer, Gabriel Euler, Leonhard D'Alembert, Jean Baptiste le Rond Lagrange, Joseph Louis Laplace, Pierre Simon Gauss, Carl Friedrich Cauchy, Augustin Louis Jacobi, Carl Gustav Jacob Hamilton, William Rowan Grassmann, Hermann Gunther Sylvester, James Joseph Weierstrass, Karl Borchardt, Carl Wilhelm Cayley, Arthur Dedekind, Richard Jordan, Camille Hill, George William Frobenius, Georg Poincaré, Henri Goursat, Édouard Jean Baptiste Peano, Giuseppe

1646-1716 1704-1752 1707-1783 1717-1783 1736-1813 1749-1827 1777-1855 1789-1857 1804-1851 1805-1865 1809-1877 1814-1897 1815-1897 1817-1880 1821-1895 1831-1916 1838-1922 1838-1914 1849-1917 1854-1912 1858-1936 1858-1932

Hilbert, David Dixon, Alfred Cardew Hadamard, Jacques

Fredholm, Ivar Koch, Helge von Plemelj, Josip Schur, Issai Fischer, Ernst Schmidt, Erhard Riesz, Friedrich Toeplitz, Otto

Lalesco, Trajan Mercer, James Hellinger, Ernst Weyl, Hermann Courant, Richard Tamarkin, Jacob David Banach, Stefan

Carleman, Torsten Hille, Einar Schauder, Pawel Juliusz Neumann, Johann von

1862-1943 1865-1936 1865-1963 1866-1927 1870-1924 1873-1967 1875-1941 1875-1954 1876-1959 1880-1956 1881-1940 1882-1929 1883-1932 1883-1950 1885-1955 1888-1972 1888-1945 1892-1945 1892-1949 1894-1980 1899-1943 1903-1957

280

7. Historical survey

From my personal viewpoint mathematics should be considered as a whole, and priorities of single mathematicians are of secondary importance. Solely, the main streams of mathematical thought are worth considering, and these have indeed been controlled by the great minds. Let us describe one of those lines of development. At the end of the nineteenth century G. Mittag-Leffler, a student and friend of K. Weierstrass, reprinted G. W. Hill's paper on infinite determinants in Acta Mathematica, and he inspired H. von Koch and I. Fredholm to work on this subject. The latter created his famous determinant theory of integral operators in spring 1899 during a visit in Paris where he attended lectures of H. Poincaré, E. Picard and J. Hadamard. The decisive moment came when E. Holmgren, a visitor from Stockholm, reported I. Fredholm's results in D. Hilbert's seminar (1901) at Göttingen. In the words of H. Weyl, "Hilbert caught fire at once", and he focused his attention on this new subject. He also filled several of his famous pupils with enthusiasm: E. Schmidt, O. Toeplitz, E. Hellinger, H. Weyl, R. Courant, J. Plemelj from Yugoslavia and F. Riesz from Hungary. Due to Nazi barbarism, the Hilbert school was scattered to the four winds. As a consequence, R. Courant came to Cambridge in 1934 where he directed F. Smithies's attention to A. Hammerstein's work (1923) on integral equations. For his part F. Smithies had several students who essentially contributed to the development of spectral theory: Shih-Hsun Chang, A. F. Ruston, J. R. Ringrose and T. T. West. After his graduation in Cambridge, on his way back to China, Shih-Hsun Chang visited Princeton and showed his results to H. Weyl who had been interested in eigenvalue problems throughout his life. This triggered off H. Weyl's note in 1949. A few years earlier, J. von Neumann, whose way of mathematical thinking was formed by E. Schmidt and H. Weyl, had created the theory of operator ideals on Hilbert spaces. Among his pupils were J. W. Calkin and R. Schatten. The Swedish tradition was continued by T. Carleman. Finally, it should be mentioned that E. Hille, a former student of Stockholm university, visited Göttingen in 1927. The preliminary version of his important paper on characteristic values of integral equations, jointly written with J. D. Tamarkin, appeared in 1928.

7.1.

Classical background In dieser Idee der Analogie mit der analytischen Geornetrie und allgemeiner iiberhaupt in der Idee des Übergangs von algebraischen Tatsachen zu solchen der Analysis liegt der Sinn der Lehre von den Integralgleichungen. Ernst Hellinger, Otto Toeplitz (1927) That Hilbert space theory and elementary matrix theory are intimately associated came as a surprise to me and to many colleagues of my generation only after studying the two subjects separately. Paul Richard Halmos (1942)

For further reading we recommend the monographs of N. Bourbaki (BOU), M. J. Crowe (CRO), M. Kline (KLI), C. C. MacDuffee (MAC) and T. Muir (MUT) which contain extensive contributions to the history of linear algebra. Important aspects of the historical development of matrix theory are presented in a series of papers by T. Hawkins (1975, 1977: a, b).

7.1. Classical background

7.1.1.

281

Determinants Numerum bb— ac, a cuius indole proprietates formae (a, b, c) imprimis pendere, in sequentibus docebimus, determinantem huius formae vocabimus. Carl Friedrich Gauss (1801)

The concept of a determinant appeared for the first time in a letter from G. W. Leibniz to G. F. A. de l'Hospital (April 23, 1693). However, this idea got lost and was rediscovered by G. Cramer: Introduction à l'analyse des lignes courbes algébriques. Genève 1750. In the appendix of his famous treatise he established a formula for solving systems of linear equations which became well-known as "Cramer's rule". Further important contributions are due to E. Bézout, A. T. Vandermonde, J. L. Lagrange and P. S. Laplace. The latter used the word "résultant" and proved the expansion formula. After this pioneering period, it was A. L. Cauchy: Mémoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et de signes contraires par suite des transpositions opérées entre des variables qu'elles renferment. Journal École Polytechnique, Paris 1815,

who developed the theory as a whole. In particular, he adopted from C. F. Gauss: Disquisitiones arithmeticae. Leipzig 1801,

the name "determinant" and introduced the symbol S(+,a, The notation deq), used throughout this monograph, is due to E. Catalan (1846) and F. Joachimsthal (1849). Special credit goes to C. G. J. Jacobi who popularized determinants as a powerful mathematical tool and made several important contributions to this subject. Finally, it should be mentioned that K. Weierstrass (1886) gave an axiomatic approach to determinant theory based on the concept of an alternating multilinear functional. However, for the purpose of this book the multiplication formula, independently discovered by J. P. M. Binet (1813) and A. L. Cauchy (1815), is more relevant.

7.1.2.

Matrices Die Zusammensetzung der Transformationen hat bekanntlich die grate Analogie mit der Multiplication der Zahlen und soli auch durch das Zeichen dieser Operation angedeutet werden, wenn die Substitutions-Systeme selbst gleichsam ais selbstândige Graen in der Rechnung auftreten, nur daB man hier die Ordnung der Faktoren nicht vertauschen darf.

Gotthold Max Eisenstein (1852) The idea of a matrix precedes that of a determinant but historically the order was the reverse and this is why the basic properties of matrices were already clear by the time that matrices were introduced. Morris Kline (1972)

The word "matrix" was first used by J. J. Sylvester (1850) to denote a rectangular array of numbers. Such arrays or systems were considered much earlier. However, before the middle of the nineteenth century the main interest was directed towards the determinant as a value assigned to every square matrix. It is only due to A. Cayley: A memoir on the theory of matrices. Phil. Trans. Roy. Soc. London 1858

282

7. Historical survey

that the arrays themselves came into consideration. More important from the viewpoint of operator theory is the fact, already known to C. F. Gauss (1801), that matrices can be employed to describe linear substitutions as well as quadratic and bilinear forms. In this way the formal algebraic manipulations acquired some concrete meaning. 7.1.3.

Vector spaces Such step-sets could be added or subtracted, by adding or subtracting their component steps, each to or from its own corresponding step, as indicated by the double formula, (b 1 , ...,b„) ± (al , ...,a„) = (b1 ± al ,

b„ + an);

and a step-set could be multiplied by a number a, by the formula oc(ai , ..., an) = (aa i ,

aa„). William Rowan Hamilton (1853)

Zwei extensive Gr6ssen, die aus demselben System von Einheiten abgeleitet sind, addiren, heisst, ihre zu denselben Einheiten geh6rigen Ableitungszahlen addiren, das heisst,

E ae +El3e =E( + fi) e. Eine extensive Gr613e mit einer Zahl multipliciren heisst, ihre sA mmtlichen Ableitungszahlen mit dieser Zahl multipliciren, das heisst

oce fi = fi .Eoce = E(cxf3) e. Hermann Gunther Grassmann (1862)

The first rigorous approach to the concept of a complex number, by considering ordered couples of reals, was given by W. R. Hamilton (1837). Later on, he generalized this idea in his famous theory of quaternions (1853), observing that the same general view of algebra admitted easily, at least in thought, of an extension of this whole theory, not only from couples to triplets, but also from triplets to sets of moments, steps, and numbers. Similar and even more profound considerations are to be found in H. Grassmann: Die (lineale) Ausdehnungslehre. Leipzig 1844 und Berlin 1862,

an important work which was neglected for a very long time. As indicated by the term "hyper complex number", at this period the main goal was the construction of linear algebras in which there is also defined a multiplication such as in the real or complex field. -

7.1.4.

Eigenvalues It will be convenient to introduce here a notion, namely that of the latent roots of a matrix—latent in a somewhat similar sense as vapour may be said to be latent in water or smoke in a tobacco-leaf. James Joseph Sylvester (1883)

7.1. Classical background

283

Quod attinet ipsam ipsius r formationem, observo, si signo summatorio S amplectamur expressiones inter se diversas, quae permutatis indicibus 1, 2, 3, ..., n proveniunt, fieri:

r =E +

a1,1a2,2 ••• an,n —xSE ±a 1 ,02 , 2 } x 2 SE +a1,1a2,2 ••• an 20 2

- -

-

-

xn-2 SE a1,1a2,2

Qua in formula expressio SE ±a1,1a2,2 ••• anion

designat summam

n(n — 1) ... (n + 1 — m) 1. 2 ... m expressionum, quae e

E

±a 1ti a2v2

a mon

proveniunt, si in al,ia2.2 ••• a,,,,

,,,

loco indicum priorum simul ac posteriorum 1, 2, ..., ni scribimus omnibus modis, quibus fieri potest, in alios e numeris 1, 2, 3, ...,n. Carl Gustav Jacob Jacobi (1834)

In the course of applying the methods of analysis to the vibrating string problem and to the theory of secular perturbations, J. B. D'Alembert: Traité de dynamique. Paris 1743, J. L. Lagrange: Méchanique analytique. Paris 1788, J. P. Laplace: Traité de méchanique céleste. Paris 1799,

were led to the consideration of systems of linear differential equations with constant coef ficients. In this context the concept of an eigenvalue arose for the first time. Another equally important source of eigenvalue problems was the principal axis theorem of mechanics of rigid bodies and of the theory of quadratic surfaces. First results in this direction were already mentioned by L. Euler: Introductio in analysin infinitorum. Lausanne 1748. At the very beginning, the so-called "secular equation" n(2., M) = 0 of a matrix from M = (au) was obtained by eliminating the unknowns

E ttiA, = )4i

for

i = 1, ..., n

and n = 2, 3, 4, sometimes 5. It was A. L. Cauchy: Sur l'équation a l'aide de laquelle on détermine les inégalités séculaires des mouvements des planètes. Exer. de Math. 1829,

who identified and treated .7(2„ M) as the determinant of the matrix AI — M, the number of coordinates being arbitrary. A few years later,

284

7. Historical survey

C. G. J. Jacobi: De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares in alias binas transformandis, ... J. Reine Angew. Math. 1834,

expressed the coefficients of 742, _ 2n oc A'

on oc n

by means of principal minors. For example, oci= /1 11 + ••• + 4Unn• Nowadays, following A. L. Cauchy (1839), polynomial". The terminology for its zeros A, colourful: "characteristic numbers, latent roots, and most of the possible combinations of these term "eigenvalue" (German: Eigenwert) prevailed.

we call n(2, M) the "characteristic , A„ was and is multifarious and secular values", words. Curiously enough, the hybrid

The eigenvalues of a matrix M are precisely those complex numbers A o for which 20 1 — M fails to be invertible. G. Frobenius (1878) was the first who considered the resolvent (AI — M) --1- as a meromorphic function of the parameter A and obtained the Laurent expansions about the poles. 7.1.5.

Traces Unter der Spur der Zahl 0 verstehen wir die Summe aller mit ihr konjugierten Zahlen; wir bezeichnen diese offenbar rationale Zahl mit S(0); dann ist

S(0) = e1 , 1 +

e2 , 2

Richard Dedekind (1882)

The word "trace" (German: Spur) was introduced by R. Dedekind (1882) in the framework of algebraic number theory; see also R. Dedekind/ H. Weber (1882). As mentioned above, C. G. J. Jacobi (1834) observed that the sum of the diagonal elements of a matrix appears as a coefficient of its characteristic polynomial. The first explicit reference to the trace formula originates from C. W. Borchardt (1846) who even proved that trace (M") = A +

+2

for all natural exponents k. 7.1.6.

Canonical forms of matrices Le premier de ces problèmes n'a pas encore été abordé d notre connaissance; le deuxième a déjà été traité (dans le cas où n est pair), par M. Kronecker, et le troisième par M. Weierstrass; mais les solutions données par les éminents géomètres de Berlin sont incomplètes, en ce qu'ils ont laissé de côté certains cas exceptionnels, qui ne manquent pas d'intérêt. Leur analyse est en outre assez difficile A suivre, surtout celle de M. Weierstrass. Camille Jordan (1874)

7.1. Classical background

285

Die vorstehenden Entwickelungen zeigen, dass in der Jordanschen Abhandlung die L6sung des ersten Problems nicht eigentlich neu ist, wdhrend die des zweiten sich ais gdnzlich verfehlt und die des dritten ais durchaus unzuldnglich begründet erwiesen hat. Nimmt man hinzu, dass eben dieses dritte Problem in Wahrheit die beiden ersten ais besondere Fane umfasst, dass ferner dessen vollstandige L6sung einestheils unmittelbar aus der Weierstrass'schen Arbeit vom Jahre 1868 folgt und anderntheils mit leichter Milhe aus den Bemerkungen entnommen werden kann, welche ich damais daran angeschlossen habe, so ist wahrlich hinreichender Grund vorhanden, Hrn. Jordan „seine Resultate", soweit sie eben richtig sind, streitig zu machen. Aber nicht um dieses untergeordneten Zweckes willen bin ich hier und in meiner früheren Mittheilung auf die Jordanschen Arbeiten niiher eingegangen; es gait vielmehr die wirkliche Bedeutung der darin enthaltenen Methoden und Resultate zu ermitteln und ihre Beziehungen zu den vorher bekannten aufzukldren. Es war also nicht die Feststellung der Prioritdt, sondern die Feststellung der Wahrheit der eigentliche Zweck meiner Ausführungen, aber sie erfiillen nebenher auch die Bestimmung, es im Voraus zu rechtfertigen, wenn ich mich künftig der Racksichtnahme auf die bezüglichen Jordan'schen Publicationen enthalte. Leopold Kronecker (1874)

Beginning with the principal axis theorem, canonical forms played an important role in the theory of matrices and operators. Since the symmetric or hermitian case is not relevant for the purpose of this monograph, we focus our attention on the general situation. The main result was obtained almost simultaneously by K. Weierstrass : Zur Theorie der quadratischen und bilinearen Formen. Monats ber. Preuss. Akad. Wiss. Berlin 1868,

and C. Jordan: Traité des substitutions. Paris 1870.

While the eminent geometer from Berlin looked at matrices from the viewpoint of quadratic and bilinear forms, his French colleague thought of matrices in terms of linear substitutions. At least in functional analysis the latter interpretation is more important. Maybe this was the deciding factor in the naming of the classical canonical form of a matrix the "Jordan form". As a first step towards this result, triangular matrices should have been considered. However, there is only a paper of L. Fuchs (1866) in which this concept was used for solving systems of linear differential equations. Thus it was only I. Schur (1909: a) who proved that every matrix is unitarily equivalent to a triangular one. 7.1.7.

Axiomatic approach Un'operazione R, a eseguirsi su ogni ente a d'un sistema lineare A, dicesi distributiva, se il risultato dell'operazione R sull'ente a, che indicheremo con Ra, è pure un ente d'un sistema lineare, e sono verificate le identità R(a + a') = Ra + Ra', R(ma) = m(Ra),

286

7. Historical survey ove a e a' sono enti qualunque del sistema A, ed in un numero reale qualunque. Giuseppe Peano (1888)

The axiomatic definitions of linear spaces and linear mappings are due to G. Peano : Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann. Torino 1888.

However, for some decades these coordinate-free notions remained unused. It was only in 1918 that H. Weyl reestablished this concept in his excellent book "Raum, Zeit, Materie", where he gave an axiomatic definition of vector spaces in which the "Dimensionsaxiom" was clearly separated from the other ones. Surprisingly enough, up to the thirties, in all famous textbooks of algebra the so-called "vector spaces" were viewed as sets of n-tuples. Therefore, when writing his ingenious thesis, S. Banach (1922) was compelled to establish the definitions of all algebraic concepts himself. Thus, without exaggeration, it may be stated that linear algebra, in its present form, was developed as a by-product of functional analysis.

7.2.

Spaces Die geometrische Deutung der in diesem Kapitel entwickelten Begriffe und Theoreme verdanke ich Kowalewski. Sie tritt noch klarer hervor, wenn A(x) statt ais Funktion ais Vector in einem Raum von unendlich vielen Dimensionen definiert wird. Erhard Schmidt (1908)

The historical development of the theory of sequence and function spaces as well as of abstract topological linear spaces is extensively described in the books of N. Bourbaki (BOU), J. Dieudonné (DIU) and A. F. Monna (MON). We also refer to a paper of M. Bernkopf (1966). Concerning spaces of differentiable functions we quote J. Liitzen's "The prehistory of the theory of distributions" and S. M. Nikol'skij's monograph (NIK). Valuable comments on the history of function spaces and interpolation theory are to be found in a book of P. L. Butzer/ H. Berens (BUB).

7.2.1.

Sequence spaces Fortan ziehen wir durchweg nur solche Wertsysteme der unendlich vielen Variablen x 1 , x2 , ... in Betracht, die der Bedingung X21 + X22 + < • • =1

genügen. David Hilbert (1906) Considérons l'espace hilbertien ; nous y entendons l'ensemble des systèmes (xk) tels que E ixk l2 converge. Frédéric Riesz (1913)

In view of the fundamental contributions of D. Hilbert: Grundziige einer allgemeinen Theorie der linearen Integralgleichungen (Vierte Mitteilung). Giittinger Nachr. 1906,

7.2. Spaces

287

to the theory of linear equations in infinitely many unknowns, the sequence space 1 2 descendants are with good reason named after this anditsxomclyefd ingenious mathematician. Nevertheless, it should be emphasized that D. Hilbert himself mainly directed his attention towards the closed unit ball of /2 . It was due to the elegant work of E. Schmidt: Über die Aufasung linearer Gleichungen mit unendlich vie/en Unbekannten. Rend. Circ. Mat. Palermo 1908,

that the algebraical and topological structure of 1 2 came into consideration. A few years later F. Riesz: Les systèmes d'équations linéaires et une infinité d'inconnues. Paris 1913,

introduced lp for I < p < co. Finally, the one-parameter scale of classical sequence spaces was refined by employing a second parameter such that the resulting scale of defined even for 0 < p, u cc, is lexicographso-called Lorentz sequence spaces I ically ordered. This idea originates from a paper of G. H. Hardy/ J. E. Littlewood (1931) in which the non-increasing rearrangement of a sequence was introduced. Next G. G. Lorentz (1950) defined certain forefathers of the function spaces nowadays denoted by L(0, 1). Afterwards those spaces were built on arbitrary measure spaces; see R. O'Neil (1963) and R. A. Hunt (1966). The fact that Lorentz spaces can be obtained from the classical ones by real interpolation was observed by J. Peetre (1963).

7.2.2.

Function spaces ..TInnetinoe mnoroo6pa3He Bcex cymmnpyembix ctiyincumil x n) 'met-011nm B orpankrtennoil 06J1aCTI4 Q nee o6o6mein-me npon3Boxibie nopsigica 1, cymmnpyembie co cTerienbto p > 1, naaosem Cepreri Ilbsoang Co6onen (1950)

Due to the work of C. Arzelà, G. Ascoli and K. Weierstrass, around 1885 mathematicians became interested in statements not only referring to single continuous functions, but concerning certain collections of them. Later on, C(0, 1) was identified by F. Riesz (1909) as a complete normed linear space (modern terminology). Fortunately, at the beginning of this century H. Lebesgue established his famous integral. This cleared the way for the introduction of spaces of p-integrable functions. By the use of concepts such as absolute continuity and generalized differentiability it was possible to define spaces of differentiable functions. Finally, by means of LipschitzHolder conditions, spaces with fractional order of differentation were introduced. At present we know several equivalent approaches to the theory of so-called SobolevBesov spaces: —the classical definition via derivatives and Lipschitz-Holder conditions or by the completion of spaces of infinitely differentiable functions with respect to certain norms', —the characterization of the desired functions in terms of their degree of approximation, —the construction of function spaces by interpolation, —the decomposition method based on the Fourier transform, —approaches via singular integrals.

288

7. Historical survey

Since we are not in a position here to give a detailed account of the history of function spaces, the following list should be enough to create a rough impression. spaces

authors R. Riesz (1909) S. Banach (1922) J. Schauder (1934) F. Riesz (1907), E. Fischer (1907) F. Riesz (1910) O. Nikodym (1933) S. L. Sobolev (1936) S. Banach (1922) N. Aronszajn (1955) L. N. Slobodeckij (1958) G. H. Hardy/ J. E. Littlewood (1928: a), S. M. Nikol'skij (1951) O. V. Besov (1961), M. H. Taibleson (1964/66)

The chronological disorder between W: and Wy' is due to the fact that S. L. Sobolev treated functions in several variables, while S. Banach considered the 1-dimensional case only.

7.2.3. Banach spaces Thus the two pieces of work, Banach's and my own, came for a time to be known as the theory of Banach-Wiener spaces. For thirty-four years it has remained a popular direction of work. Although many papers have been written on it, only now is it beginning to develop its full effectiveness as a scientific method. For a short while I kept publishing a paper or two on this topic, but I gradually left the field. At present these spaces are quite justly named after Banach alone. Norbert Wiener (1956)

Based upon the work of F. Riesz (1918), the axiomatic theôry of Banach spaces was independently created by S. Banach: Sur les opérations dans les ensembles abstraits et leur application aux equations intégrales. Fundamenta Math. 1922,

and N. Wiener (1923). A particular highlight in the history of mathematics as a whole was the publication of S. Banach: Théorie des opérations linéaires. Warszawa 1932.

Since that time his celebrated monograph has been an indispensable tool for all those working in functional analysis. The concept of an abstract Hilbert space was designed by J. von Neumann (1929), relatively late. Quasi-Banach spaces and p-Banach spaces, as well as the connection between both notions, were defined and investigated by among others T. Aoki (1942), D. G. Bourgin (1943) and S. Rolewicz (1957).

289

7.3. Operators

7.2.4.

Interpolation theory A 0 et A 1 étant donnés, construire des espaces intermédiaires, de façon à ce que la propriété d'interpolation par rapport aux applications linéaires ait lieu.

Jacques-Louis Lions, Jaak Peetre (1964) Les A. ont été soutenus par l'Interpol

Based on the classical ideas of M. Riesz (1926) and G. O. Thorin (1938), interpolation theory was founded at the end of the fifties. The most important papers following this pioneer time are J. L. Lions / J. Peetre: Sur une class d'espaces d'interpolation. Inst. Hautes Études Sci. Pub!. Math. 1964,

and A. P. Calderon: Intermediate spaces and interpolation, the complex method. Studia Math. 1964.

Subsequently, interpolation theory has become a special branch of functional analysis with important and far-reaching applications. The final version of the real method, used throughout this book, is due to J. Peetre : Inletining till interpolation. F6reldsningar, Lund 1964-1966.

This powerful technique was first applied within the theory of operator ideals by H. Triebel (1967), A. Pietsch / H. Triebel (1968), R. Oloff (1970), J. Peetre / G. Spaar (1972) and H. K6nig (1978). 7.3.

Operators . Cette théorie mérite donc avec raison, aussi bien par sa valeur esthétique que par la portée de ses raisonnements (même abstraction faite de ses nombreuses applications) l'intérêt de plus en plus croissant que lui prêtent les mathématiciens. Aussi on ne s'étonnera pas à l'opinion de M. J. Hadamard, qui considère la théorie des opérations comme une des plus puissantes méthodes de recherche de la mathématique contemporaine.

Stefan Banach (1932)

For further reading we recommend J. Dieudonnes "History offunctional analysis". The monumental treatise of N. Dunford / J. Schwartz (DUN) contains a lot of historical information, as well. We also quote an excellent survey about spectral theory written by L. A. Steen (1973). 7.3.1.

Infinite matrices Diese beiden Sâtze kiinnen leicht durch eine Betrachtung, die der Eingangs dieses Paragraphen gefiihrten analogisch ist, streng nachgewiesen werden, indem man zeigt, dass, weil dieselben gelten far em n System linearer Gleichungen von endlicher Anzahl, und unverdndert Geltung behalten, wenn die Anzahl der Gleichungen urn eine endliche Zahl wachst, sie auch gelten miissen, wenn die Anzahl der Gleichungen und Unbekannten ins Unbegrenzte wdchst. Theodor K6tteritzsch (1870)

290

7. Historical survey

Es handelt sich dabei um eine Reihe von Konvergenzhilfssatzen, die im wesentlichen in der einen Tatsache gipfeln, daB man mit passender Vorsicht für unendliche Matrizen (Bilinearformen von unendlichvielen Veranderlichen) einen ebensolchen Ka lkül aufstellen kann, wie er sich seit Cayley und Frobenius in der Algebra der Bilinearformen von endlichvielen Verânderlichen eingebürgert hat. Ernst HeBinger, Otto Toeplitz (1910)

General infinite matrices were considered for the first time by T. Keitteritzsch (1870) when he tried to solve systems of linear equations in infinitely many unknowns, written as o In

co

Co

Eo pfoOn, xp =

9)„z

in his notation. Later on, this concept became basic for the theory of infinite determinants. D. Hilbert (1906: a) used infinite matrices to define bilinear forms in infinitely many variables and, finally, it was F. Riesz (1913) who stressed the fact that every infinite matrix induces an operator on a suitable sequence space. There does not exist any intrinsic description of those infinite matrices determining the operators on lp with 1 < p < co. Thus it was quite natural to take into account some classes of infinite matrices M = (au) which can be defined by simple conditions such as 00

CO

EE

i=1 i=1

< 00

(D. Hilbert 1906: a)

and CO

E

i=1

sup 1,u, ii l < co (H. von Koch 1901, A. C. Dixon 1902).

Both of these inequalities are included as special cases in a 2-parameter scale of conditions first introduced by E. Hille / J. D. Tamarkin (1934) in the setting of kernels; see also S. B6br (1921) and L. W. Cohen (1930). The definition of infinite matrices of Besov type [b„, kJ was given by A. Pietsch (1980: d). E-valued sequences came under consideration when I. M. Gel'fand (1938) and R. S. Phillips (1940) observed that these are especially suited to represent operators, for example, from / 1 into the Banach space E. 7.3.2.

Integral operators Why should one study integral operators? The traditional answers are that integral equations have important applications outside of mathematics, and that they are the proper extension to analysis of the concepts and methods of the classical algebraic theory of linear equations. A third possible answer is that the theory of integral operators is the source of all modern functional analysis and remains to this day a rich source of non-trivial examples. Since the main obstacle to progress in many parts of operator theory is the dearth of concrete examples whose properties can be explicitly determined, a systematic theory of integral operators offers new hope for new insights. Paul R. Halmos, Viakalathur Sh. Sunder (1978)

7.3. Operators

291 A glaring example of lack of perspective is given by the

Flellinger-Toeplitz article of 1927 in the Enzyklopddie der Math. Wiss., which gives undue emphasis to integral equations. Jean Alexandre Dieudonné (1981)

Following D. Hilbert (1904) a function K(,1)) defined for 0 be a "kernel" if it is used to determine an integral operator K0 : g(27)

$,

1 is said to

K(e, 97) An) dn .

As in the case of infinite matrices, certain classes of kernels proved to be of special importance. This statement is in particular true for the Hilbert-Schmidt kernels: f f IK($, 012 d$ d < oo .

o Generalizing this notion, E. Hile / J. D. Tamarkin (1934), F. Smithies (1937: b) and I. A. Itskovich (1948) considered kernels for which piq

iip

(I (f 1K($, 77)1(1 dn) d$ o o Such conditions have not only been put on the kernels themselves but also on their derivatives. This yields a large variety of different types of kernels which, from the viewpoint of the theory of function spaces, are defined as functions with mixed degrees of differentiability. The most important examples are the kernels of Besov type [B e , No] introduced by A. Pietsch (1980: d). We would also like to mention the so-called weakly singular integral operators which arise in potential theory and have played an essential role as basic examples for applications. The fact that certain operators on function spaces can be represented by means of E-valued functions was discovered by N. Dunford, B. J. Pettis, R. S. Phillips and I. M. Gel'fand between 1935 and 1940; see (DUN, Vol. I, pp. 542-551).

7.3.3.

Abstract operators Die in der Arbeit gemachte Einschrdnkung auf stetige Funktionen ist nicht von Belang. Der in den neueren Untersuchungen über diverse Funktionenrdume bewanderte Leser wird die allgemeinere Verwendbarkeit der Methode sofort erkennen. Friedrich Riesz (1918)

The concept of an abstract operator between Banach spaces is due to S. Banach (1922). In another paper (1929) he introduced dual Banach spaces and dual operators. However, even in his monograph there is no reference to the set 2(E, F) constituted by all operators acting from E into F. The definition of the product of two operators is also missing. Clearly, S. Banach must have been well acquainted with these notions. However, it was not until the work of I. M. Gel'fand (1941) that "Normierte Ringe" became attractive.

292

7. Historical survey

7.3.4.

Operator ideals on Hilbert spaces Since that time, I have had a great respect and use for a subject that may be called "the hard analysis of compact operators in Hilbert space". Barry Simon (1979)

In the late twenties, J. von Neumann (1929) stated the fact that the set of all operators acting on the separable infinite dimensional Hilbert space H can be viewed algebraically as a ring. Next, in a small but important paper (1937), he introduced r < aD for finite matrix operators. Ten years later the the norms li* with 1 generalization to the infinite setting was carried out by him jointly with one of his pupils, R. Schatten / J. von Neumann: The cross-space of linear transformations. Ann. of Math. 1946 and 1948.

These papers contain the basic facts from what is now called the theory of Schattenvon Neumann ideals. In the meantime, another one of his students, J. W. Calkin: Two-sided ideals and congruences in the ring of bounded operators in Hilbert space. Ann. of Math. 1941,

had investigated the ideal structure of 2(H) from an abstract viewpoint. The outcome was a famous theorem which asserts that (S(H) is the largest proper ideal in 2(H). Thus, at the end of the forties, the theory of operator ideals on the Hilbert space, mainly developed in Princeton, was ready for application to eigenvalue problems. Who should do it? Of course, H. Weyl (1949), who was also in Princeton since 1933. A masterly presentation of this subject was given by I. C. Gohberg / M. G. Krejn: Introduction to the theory of non-self-adjoint operators in Hilbert space (Russian). Moscow 1965.

7.3.5.

Operator ideals on Banach spaces Unangenehm ist mir nur immer Dedekinds Terminologie gewesen, welche aller Anschaulichkeit entbehrt. Er nennt diese Gesamtheiten Ideale. Er hdtte von „Realen" sprechen sollen. Felix Klein (1915)

Ideal theory in L(X), where X denotes a generic Banach space, is likely to be a zoo. Barry Simon (1979)

The theory of operator ideals on Banach spaces emerged as a by-product of the theory of tensor products which was created by R. Schatten: A theory of cross-spaces. Princeton 1950, A. F. Ruston: Direct products of Banach spaces and linear functional equations. Proc. London Math. Soc. 1951, A. Grothendieck: Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. Sao Paulo 1956.

Clearly, special ideals such as those of finite, approximable and compact operators had been considered much earlier. In the period from 1965 to 1975, the theory of

293

7.3. Operators

operator ideals developed into a separate discipline of functional analysis, which provides powerful methods for other branches of mathematics. Important applications have been made to the geometry of Banach spaces, to Brownian motion and, in particular, to eigenvalue problems. An extensive treatment was given by A. Pietsch: Operator ideals. Berlin 1978.

7.3.6. s-Numbers b, a Es sei K(s, t) eine far a s t b definierte reelle stetige Funktion, die nicht ais symmetrisch vorausgesetzt werden soli. Wenn dann die beiden reellen oder komplexen stetigen nicht identisch verschwindenden Funktionen T(s) und y(s) den Gleichungen a

.p(s) =

2 r K,s,

dt

b

2p(S) =

K(t, s)cp(t)dt

gentigen, so sollen sie ais em n Paar zum betreffenden Eigenwert 2. gehi5riger adjungierter Eigenfunktionen des Kernes K(s, t) bezeichnet werden. Erhard Schmidt (1907)

The concept of s-numbers was invented by E. Schmidt : Entwicklung willkürlicher Funktionen nach Systernen vorgeschriebener. Math. Ann. 1907,

when he established his famous representation for integral operators induced by arbitrary continuous kernels. However, contrary to modern terminology, he referred to these quantities as eigenvalues. Later on, F. Smithies (1937: a) used the term "singular value" which was finally changed into the present notation. For a long time the sequence of s-numbers of an operator acting on a Hilbert space had been defined as the sequence of the eigenvalues of the positive operator 171 := (T*T)' 12 . From the viewpoint of our present knowledge this approach is quite troublesome. However, it was not until 1957 that D. Eh. Allakhverdiev, in an exotic journal, proved that

s„(T) = inf

— LII : L

a(H), rank (L) < n} ;

E ,

see also M. Fiedler / V. Ptdk (1962). Only then did it become possible to extend this useful notion to operators acting on Banach spaces. This was carried out by A. Pietsch (1963: b) when he invented the so-called approximation numbers. Subsequently, it turned out that there exist various other possibilities for generalizing the s-numbers to the Banach space setting. Thus, based on A. N. Kolmogorov's notion of diameters, the concept of Kolmogorov numbers was defined by I. A. Novosel'skij (1964). The dual concept of Gel'fand numbers, first considered by H. Triebel (1970: a), originates from the work of V. M. Tikhomirov (1965) who was stimulated by a proposal of his famous teacher. In view of

c(T) = min {max [11Tx11 : x e M, 11x11

1]: codim (M) <

,

294

7. Historical survey

the definition of these numbers is closely related to a well-known minimax principle attributed to E. Fischer (1905) and R. Courant (1920). In order to systematize the theory of all these s-numbers, A. Pietsch (1972: a, 1974: a) developed an axiomatic approach. Later on, W. Bauhardt (1977) studied the so-called Hilbert numbers. A temporary keystone was laid by A. Pietsch (1980: c) when he introduced the Weyl numbers and their dual counterparts. According to a suggestion of J. R. Retherford, generalized s-numbers with respect to an arbitrary quasi-Banach operator ideal were considered for the first time by H. Keinig (1978). Nevertheless, it is worth while to stress the fact that already E. Schmidt (1907: a) had determined the best approximation of an integral operator by those of rank less than or equal to n with respect to the Hilbert-Schmidt norm (Approximationstheorem). Beginning with the Schatten-von Neumann classes 2,.(11) := {T e 2(H): (s„(T)) e many useful operator ideals have been defined by means of s-numbers. The fine index w, well-known from the theory of Lorentz sequence spaces, was introduced by H. Triebel (1967). Afterwards it was no problem to extend these constructions to operators acting on Banach spaces. The quasi-Banach operator ideals t(s) r, W •

{T e : (s,,(T))

have now become the indispensable background of the modern theory of eigenvalue distributions.

7.3.7.

Absolutely summing operators Soit u une application semi-intégrale droite d'un espace de Banach E dans un autre F, de norme semi-intégrale droite 1. Alors on peut trouver un espace compact muni d'une mesure positive p, de norme 1, une application linéaire cc de norme 1 de E dans Lu) et une application linéaire y de norme 1 de 1,2 (u) dans F, tels que l'on ait u = ypoc, où g9 est l'application identique de Lœ (a) dans L 2 (p). Alexandre Grothendieck (1955)

The theory of absolutely summing operators originates from a paper of W. Orlicz p 2 all summable sequences are absolutely (1933) who proved that in at,„ with 1 2-summable. Later on, A. Grothendieck: Produits tensoriels topologiques et espaces nucléaires. Memoirs Amer. Math. Soc. 1955,

characterized the so-called "applications semi-intégrales à droite" by the property that the image of every summable sequence is even absolutely summable. The main part of the theory of absolutely p-summing operators is due to A. Pietsch: Absolut p-summierende Abbildungen in nornzierten Rtiumen. Studia Math. 1967.

J. R. Holub (1970/74) established the s-tensor stability of the operator ideals 4;,.. Absolutely (r, s)-summing operators have been defined by B. S. Mityagin / A. Pelczyriski (1966). Important contributions are due to S. Kwapieri (1968).

7.4. Eigenvalues

7.3.8.

295

Nuclear operators (H A H)* represents the "trace class" of linear transformations on H, that is, all linear transformations on H which can be represented as a product of two linear transformations on H belonging to the Schmidt class.

Robert Schatten, Johann von Neumann (1948)

At the beginning of this century, E. Schmidt (1907: a) proved his famous representation theorem which states, in modern terms, that every approximable operator on a Hilbert space can be written in the form 00

T = E r,x;i: 0 1=1 where (x i) and (y i) are extended orthonormal sequences and (ri) e c0 . The trace class operators, defined by R. Schatten / J. von Neumann (1946), can be characterized by the property (ri) e I,. This means that the Schmidt series, formed by the 1-dimensional operators rix i * 0 y i , is absolutely summable. Based on this fact, A. F. Ruston (1951: a) and A. Grothendieck (1951) independently introduced the concept of a trace class or nuclear operator which also makes sense in the Banach space setting: CO

T

E ai 0 y i .

1=1

Modifying the assumption (11a 1 ii 1lYi E 1 1 which guarantees the absolute convergence of the right-hand series, several types of nuclear operators have been considered. These are, for example, the p-nuclear operators introduced by A. Grothendieck (GRO, Chap. II, p. 3) as "opérateurs de puissance p. ème sommable" and the (r,2)-nuclear operators first defined by P. Saphar (1966).

7.4.

Eigenvalues Die Werte 2 1 , A2, ... und die zugehdrigen Funktionen p i (s), q22 (s), sind wesentlich durch den Kern K(s, t) bestimmt ; ich habe sie Eigenwerte bzw. Eigenfunktionen des Kerns K(s, t) genannt. David Hilbert (1906)

There does not exist any book or paper devoted to the history of eigenvalue distributions of abstract operators. We may only refer to a report of H. Keonig (1981) which surveys the development around 1977, the year of the great revolution.

7.4.1.

Riesz theory F. Riesz's 1918 paper is one of the most beautiful ever written: it is entirely geometric in language and spirit, and so perfectly adapted to its goal that it has never been superseded and that Riesz's proofs can still be transcribed almost verbatim. Jean Alexandre Dieudonné (1981)

The first determinant-free treatments of linear equations in infinitely many unknowns were carried out by A. C. Dixon (1902), D. Hilbert (1906: a) and E. Schmidt

296

7. Historical survey

(1908). An analogous approach to the theory of integral equations is due to E. Schmidt (1907: b). However, the real birth of modern spectral theory of compact (non-self-adjoint) operators took place with the publication of F. Riesz: Mer lineare Funktionalgleichungen. Acta Math. 1918 (Hungarian version : Linearis fiiggvényegyenletekr61. 1916).

This basic branch of functional analysis was completed by J. Schauder (1930) who proved the missing statements about dual operators. In the following period it was observed that the typical results, established by F. Riesz, remain true even for noncompact operators which have some compact power. Finally, A. F. Ruston: Operators with a Fredholm theory. Journal London Math. Soc. 1954,

found necessary and sufficient conditions characterizing those operators which share all the nice properties with the compact ones. Among others he gave a useful criterion by means of the meromorphic nature of the Fredholm resolvent. The geometric definition of an iteratively compact operator, used as a starting point in this monograph, is due to A. Pietsch (1961). Later on, M. R. F. Smyth (BAS, p. 11) gave a characterization of Riesz operators in the same spirit. The fact that the Riesz properties of an operator are inherited by its dual and conversely was proved by T. T. West (1966). 7.4.2.

Related operators The latent roots of the product of two matrices, it may be added, are the same in whichever order the factors be taken. James Joseph Sylvester (1883)

One hundred years ago J. J. Sylvester (1883) observed that, for two matrices A and B, the products AB and BA have the same eigenvalues. This result extends to O. That fact was tacitly used by arbitrary linear mappings and eigenvalues A A. Grothendieck (GRO, Chap. II, p. 15) in order to determine the eienvalue type of the ideal of nuclear operators. The explicit formulation of the principle of related operators as well as several important applications are due to A. Pietsch (1963: a, 1972: b). 7.4.3. Eigenvalues and s-numbers I have so far been unable to establish any direct connection between the orders of magnitude of the eigenvalues and the singular values when the kernel is not symmetric. Frank Smithies (1937)

We now describe the history of those results concerning the asymptotic behaviour of eigenvalues which can be obtained by determinant-free methods. More precisely, we deal with the relationship between eigenvalues and s-numbers which gave rise to the title of this monograph. The first observation along this line was made by I. Schur: Mer die charakteristischen Wurzeln einer linearen Substitution mit einer Atzwendung auf die Theorie der Integralgleichungen. Math. Ann. 1909.

Using his theorem on the triangular form of matrices, he established the inequality 11

E 12,.(K) I 2

n=1

6

n)1 2 0

7.4. Eigenvalues

297

for arbitrary continuous kernels. Later on, T. Carleman (1921) proved the natural fact that this formula also holds for Hilbert-Schmidt kernels. This means that the operator ideal S 2 is of eigenvalue type /2 . There was then a long stagnation period until 1949, when H. Weyl: Inequalities between the two kinds of eigenvalues of a linear transformation. Proc. Nat.

Acad. Sci. U.S.A. 1949,

found his famous inequalities. This eminent mathematician, who was fascinated by eigenvalue problems again and again, became stimulated anew by the thesis of ShihHsun Chang (1949) in which the optimum eigenvalue type of r with 0 < r 2 was obtained via determinants. A preliminary version of this result, formulated, however, in terms of the order of Fredholm denominators had already been given by S. A. Gheorghiu (1927/28). Immediately after H. Weyl's paper there followed several publications devoted to this subject by Ky Fan (1949/50, 1951), G. 1361ya (1950), A. Horn (1950) and J. P. 0. Silberstein (1953); see also A. S. Markus (1964). Subsequently, there was again a long break until 1978 when, using approximation numbers, Gel'fand and Kolmogorov numbers, H. Keinig: Interpolation of operator ideals with an application to eigenvalue distribution problems. Math. Ann. 1978,

proved a modified Weyl inequality for operators on Banach spaces. A first step in this direction had been taken by A. S. Markus / V. I. Matsaev (1971). Finally, a more or less complete form of this theory is due to A. Pietsch : Weyl numbers and eigen values of operators in Banach spaces. Math. Ann. 1980.

7.4.4. Eigenvalues of nuclear operators Le déterminant de Fredholm d'un noyau de Fredholm dans E' E est d'ordre 2, et la suite de ses valeurs propres est carré sommable. Alexandre Grothendieck (1955)

Having in mind that every nuclear operator is related to an integral operator induced by a continuous kernel, we can claim that the eigenvalue distribution of nuclear operators was already determined by I. Fredholm (1903) and I. Schur (1909: a). The explicit statement of the fact that the operator ideal 91 has eigenvalue type /2 is to be found in the famous thesis of A. Grothendieck (GRO, Chap. II, pp. 13-17). According to an example given by T. Carleman (1918) this is the best possible result in general Banach spaces. For operators acting on the Hilbert function space, the same problem had already been treated by T. Lalesco (1915). He stated, but without a rigorous proof, that the eigenvalue sequence of any operator which is the product of two Hilbert-Schmidt operators must be absolutely summable. Nowadays it is known from the fundamental papers of W. B. Johnson / H. K6nig / B. Maurey / J. R. Retherford (1979) and H. K6nig / J. R. Retherford / N. Tomczak-Jaegermann (1980) that the worst possible asymptotic behaviour of the eigenvalues of a nuclear operator Te 91(E) heavily depends on the geometric structure of the underlying Banach space E. A complete clarification of this interplay is, however, still pending.

298

7. Historical survey

7.4.5. Eigenvalues of absolutely summing operators In the case of a matrix of complex numbers, the preceding proposition reads (p 2) itik prp,) ip I2 i(T)I") I /P ki

We do not know a simpler proof of this inequality exept for 2.

p =

..., Bernard Maurey, ... (1979)

The first result about eigenvalue distributions of absolutely summing operators is the famous Schur-Carleman inequality for Hilbert-Schmidt kernels. Using the principle of related operators, A. Pietsch (1963: a, 1967) proved that the operator ideal V, 2 has eigenvalue type 1 2 . The much more complicated case when r with 1 2 r < oo was solved by W. B. Johnson / H. Ki5nig / B. Maurey / J. R. Retherford: J. Funct. Anal. 1979.

Eigenualttes of p-summing and la-type

operators in Banach spaces.

Very recently A. Pietsch (1986) developed the tensor stability technique which yields a new and simpler approach. Further contributions to this subject, which is closely related to the theory of Hille-Tamarkin operators and to the Hausdorff-Young theorem, are due to M. Z. Solomyak (1970), G. E. Karadzhov (1972), J. A. Cochran / C. Gehring (1977), B. Russo (1977), J. J. F. Fournier / B. Russo (1977) and H. Kônig L. Weis (1983). The precise asymptotic behaviour of the eigenvalues of absolutely (r, 2)-summing operators was determined in a fundamental paper of H. Kbnig / J. R. Retherford / N. Tomczak-Jaegermann: On the eigenvalues operators and constants associated with normed spaces. J. Funct. Anal. 1980.

7.4.6.

of (p,2)-summing

Summary

The most important results concerning the optimum eigenvalue type of operator ideals as well as their discoverer are recorded in the following table. operator ideals

=

(22) 2

eigenvalue types

authors

12 t,

I. Schur (1909:a), T. Carlemann (1921)

9i

12

9Ip. o < p < 1

4,p 1/r

94,2

1/n+ 1/2

ir

12 (

02) m 2
1 2/m

Vr,2

Ir,co

0(x)

1r, w

W

'■•r,W

(V2) (s!?4,

w 1/;= 1/s+ 1/2

T. Lalesco (1915), S. A. Gheorghiu (1927/28) S. H. Chang (1949), H. Weyl (1949) A. Grothendieck (1955) A. Grothendieck (1955), H. Ktinig (1977) P. Saphar (1966) A. Pietsch (1963:a, 1967) W. B. Johnson et al. (1979) W. B. Johnson et al. (1979) H. Kiinig et al. (1980) H. Kbnig (1978) A. Pietsch (1980:c) B. Carl and H. Kizinig (unpublished), A. Pietsch (1981:a)

299

7.5. Determinants

7.5.

Determinants Most scientific discoveries are made when "their time is fulfilled"; sometimes, but seldom, a genius lifts the veil decades earlier than could have been expected. Fredholm's discovery has always seemed to me one that was long overdue when it carne. What could be more natural than the idea that a set of linear equations connected with a discrete set of mass points gives way to an integral equation when one passes to the limit of a continuum? Hermann Weyl (1944)

The history of determinants of infinite matrices and integral operators is extensively presented in the Enzyklopddie-Artikel by E. Hellinger / O. Toeplitz (HEL) and in a paper by M. Bernkopf (1968). A survey given by R. Sikorski (1961) contains several historical remarks about determinants of abstract operators on Banach spaces. We further quote a nicely written examination-paper of P. Mdurer (1981) which is unfortunately unpublished.

7.5.1.

Determinants of infinite matrices Pour que le déterminant des Aik et tous ses mineurs convergent absolument, il suffit que le produit des éléments diagonaux converge absolument et que les éléments non-diagonaux de chaque ligne soient moindres en valeur absolue que les termes d'une série donnée absolument convergente. Helge von Koch (1901)

Apart from an insufficient attempt of T. K6tteritzsch (1870), determinants of infinite matrices were investigated, for the first time, by the American astronomer and mathematician G. W. Hill (1877). Subsequently, H. Poincaré (1886) put these ideas into a rigorous form. Very extensive contributions to the theory of infinite determinants are due to H. von Koch who published a whole series of papers beginning in 1890. From the viewpoint assumed in this monograph his most important result concerns those matrices M = (IO for which CO

E i=1

sup !p u ' <

CX)

•

j

As we now know, this definition anticipated the concept of a nuclear operator acting on the sequence space 11 . Surprisingly enough, H. von Koch was not interested in eieenvalues, and, in contrast to his colleague I. Fredholm (1903), he failed to look at det (I + M) as an entire function of the complex variable Another peculiarity of H. von Koch's approach is that he never used Hadamard's inequality for verifying the convergence of infinite determinants. Clearly, at the very beginning of his work, this tool, established only in 1893, was not yet available. But later on it would have yielded significant improvements; see O. Szdsz (1912).

7.5.2.

Determinants of integral operators Si le déterminant Df d'une équation fonctionnelle de la forme q)(x) + f f(x, s) cp(s) ds

= v(x),

300

7. Historical survey

où f(x, s) et y(x) sont des fonctions finies et intégrables, est différent de zéro, il existe une et une seule fonction p(x) satisfaisant à cette équation. Cette fonction est donnée par l'équation: 92(x)

V(x)

Dpfe''' ) y(y) dy. o Ivar Fredholm (1903)

Certainly, Fredholm's determinant theory of integral operators induced by continuous kernels is the first milestone in the history of functional analysis. The fundamental paper I. Fredholm: Sur une classe d'équations fonctionelles. Acta Math. 1903,

contains so may new ideas that even nowadays it is worth while to have a look at this original treatise. The contributions of J. Plemelj: Zur Theorie der Fredhohnschen Funktionalgleichungen. Monatsh. Math. Phys. 1904,

are also extremely important. In particular, he expressed the coefficients of Fredholm's determinants by very elegant formulas which now bear his name. For applications to potential theory it turned out to be necessary to cover those kernels which have singularities on the diagonal. A first successful step in this direction was made by D. Hilbert (1904) who modified Fredholm's coefficients by writing zeros instead of the undefined values K(e, Later on, T. Carleman: Zur Theorie der linearen Integralgleichungen. Math. Z. 1921,

extended this approach to his famous determinant theory of Hilbert-Schmidt operators. By employing Plemelj's formulas, further improvements were made by F. Smithies (1941) who also wrote a fine textbook (SM1) on this subject. Regularized Fredholm denominators of order m > 2 were first considered by H. Poincaré (1910). 7.5.3.

Determinants of abstract operators It happens rather often in Mathematics that some problems are open for a long time and suddenly they are solved independently by several mathematicians at the same time. This has also happened in the case of the theory of determinants in Banach spaces. Roman Sikorski (1961)

As previously stated, the development of functional analysis has heavily been stimulated by Fredholm's determinant theory of integral operators (1903). Subsequently, it took only a few years until E. Schmidt (1907: b) and F. Riesz (1918) established the determinant-free approach. The fundamental work of the latter may also be considered as the starting point of abstract operator theory. Taking into account J. Schauder's contributions (1930), we are entitled to claim that the spectral theory of compact operators was more or less complete by the end of the twenties. A first representation is to be found in S. Banach's monograph (BAN). Afterwards

7.5. Determinants

301

for a long period, no progress was made towards an abstract version of determinant theory. It was only in the early fifties that A. F. Ruston: On the Fredholm theory of integral equations for operators belonging to the trace class of a general Banach space. Proc. London Math. Soc. (2) 1951,

T. LeZariski: The Fredholm theory of linear equations in Banach spaces. Studia Math. 1953, A. Grothendieck: La théorie de Fredholm. Bull. Soc. Math. France 1956,

almost simultaneously defined determinants for nuclear or integral operators on Banach spaces. Because of the vexed approximation problem, however, their theories were quite sophisticated. As a consequence, this subject has been treated in books exclusively in the Hilbert space setting. When P. Enflo (1973) discovered his famous counter-example to the approximation property, it became clear that, on arbitrary Banach spaces, a satisfactory determinant theory could be expected only for classes of operators strictly smaller than that of the nuclear ones. Contributions along this line are due to A. Grothendieck (GRO, Chap. II, pp. 18-19), A. Pietsch (PIE, 27.4.9) and H. K6nig (1980: a). Taking the work of A. D. Michal / R. S. Martin (1934), S. G. Mikhlin (1944), R. Sikorski (1953, 1961), S. T. Kuroda (1961), A. Pietsch (1963: a), J. J. Grobler / H. Raubenheimer / P. van Eldik (1982) and J. C. Engelbrecht / J. J. Grobler (1983) as a basis, A. Pietsch designed an axiomatic approach to determinant theory which is published for the first time in this monograph. Regularized Fredholm denominators were investigated by P. Saphar (1966), H. J. Brascamp (1969), B. Simon (1977) and H. K6nig (1980: b); see also (DUN, XI.9) and (GOH, IV.2). Concerning the history of holomorphic functions defined on Banach spaces we refer to the reports of A. E. Taylor (1970, 1971). 7.5.4. Eigenvalues and zeros of entire functions Herr Fredholm hat eine gewisse, durch K(s, t) eindeutig definierte ganze transzendente Funktion D(x) --= 1 +

+ d2 x 2 +

angegeben, die nur fill- die Eigenwerte A des Kerns K(s , t) verschwindet. 1st A eine n-fache Nullstelle von D(x), so heiBt n die Ordnung des Eigenwerts 2. Diese Zahl n hat noch folgende Bedeutung: es lassen sich n, aber nicht mehr ais n, linear unabhângige (stetige) Funktionen i (s), q)2(s), q(s) bestimmen, far die K(s, t)

o (t ) dt = c11(s) + • + c,11(s)

1

el

wird, wo die co,fl Konstanten sind.

Issai Schur (1909) Das Laguerresche Geschlecht dieser ganzen Funktion kann den Wert 2 nicht iibersteigen, vermutlich ist es niemals 2, sondern im allgemeinen gerade zu 1, bei durchwegs stetigem f(st) vielleicht sogar stets 0. Josip Plemelj (1904)

302

7. Historical survey D* (A) ist hiichstens vom Geschlechte Eins und besitzt die Produktdarstellung A

ao

D* (A) =

(1 -

e

v.1

Torsten Carleman (1921)

0 of an integral I. Fredholm (1903) had already observed that the eigenvalues A operator induced by a continuous kernel and the zeros ?.," of its canonical Fredholm denominator are related via the formula 24" = —1. The geometric interpretation of the order of those zeros as the dimension of the linear space spanned by the principal functions associated with A is due to J. Plemelj (1904), E. Goursat (1908) and I. Schur (1909: b). Thanks to the fundamental contributions of K. Weierstrass (1876), J. Hadamard (1893: b), E. Borel (1897), J. L. W. V. Jensen (1899), A. Pringsheim (1904) and E. Lindel6f (1903, 1905) the theory of entire functions was well-developed at the turn of the century. In particular, the interplay between the growth of the maximum modulus and the asymptotic behaviour of the zeros had become transparent. These results have been applied in order to get information about the eigenvalue distributions of integral operators determined by kernels with certain properties such as integrability, differentiability etc. This was done by T. Lalesco (1907, LAL), S. Mazurkiewicz (1915), T. Carleman (1917, 1921), A. O. Gel'fond (1957), D. W. Swann (1971) and, in particular, by E. Hille / J.D. Tamarkin: On the characteristic values of linear integral equations. Act. Math. 1931.

The powerful tool of determinants had also been used by S. A. Gheorghiu (1927/28) and Shih-Hsun Chang (1949) when they investigated the connection between the eigenvalues and the s-numbers of Hilbert-Schmidt integral operators.

7.6.

Traces

Bei aller Karze und Einfachheit unserer auf die Spur bezilglichen Betrachtungen sind dieselben mathematisch nicht einwandfrei. Wir haben nâmlich Reihen ohne Riicksicht auf ihre Konvergenz betrachtet, ineinander umgeformt (umsummiert) — kurzum alles getan, was man korrekterweise nicht tun soil. Zwar kommen derartige Nachldssigkeiten in der theoretischen Physik auch sonst vor, und die vorliegende wird in unseren quantenmechanischen Anwendungen kein Unheil anrichten — es muB aber doch festgestellt werden, daB es sich um eine Nachlâssigkeit handelt. Johann von Neumann (1932)

Concerning the history of trace theory, we refer to an excellent report given by J. R. Retherford (1980).

7.6.1.

Operator ideals with a trace It should be observed that, although to each AeBO y B* there corresponds a unique operator of the trace class, we have not proved in general that to each operator of the trace class there corresponds a unique element of B y B*. Anthony Francis Ruston (1951)

7.6. Traces

303 On notera que faute d'avoir résolu le problème de biunivocité, il n'est pas possible en général de parler de la trace de l'operateur dans E défini par un élément u de E' E. Alexandre Grothendieck (1955)

Clearly, expressions such as 1 CO E p li and J 1((, 1=1 appeared early in the history of infinite matrices and integral operators. Nevertheless, J. von Neumann (1932) was the first to deal with the abstract concept of a trace. At the very beginning he restricted himself to positive operators on a Hilbert space. In the subsequent period, jointly with his student R. Schatten (1946/48), he established a theory of so-called trace class operators which were originally defined as the product of two Hilbert-Schmidt operators. Later on, the approach via snumbers was used. Based on this work, A. F. Ruston (1951: a) and A. Grothendieck (1951) independently introduced the notion of a trace class or nuclear operator between Banach spaces. Their main goal was in fact the creation of a determinant theory. However, to this end a trace was required as well. Unfortunately, because of the negative answer to the approximation problem, such a well-defined trace cannot exist for arbitrary nuclear operators. Thus the extremely powerful concept of the trace class failed for the purpose it was originally designed for. In view of those difficulties A. Grothendieck (GRO, Chap. II, pp. 18-19) reduced the ideal of nuclear operators to that of 213 nuclear operators, and for this smaller class he got all the desired results. Later on, A. Pietsch (1963: b) showed that every operator of approximation type 11 , acting from an arbitrary Banach space into itself, possesses a welldefined trace. The axiomatic approach to traces which live on certain operator ideals is due to the same author (1981: b). -

7.6.2.

The trace formula IlycTb onepaTop C HMCT cneg. Torzia, xaxoB 61,1 HH 6bin opToHopmlipoHainibni 6a311c cpi(i= 1, 2, ...) B H, cripaHezaHHo paBeHCTBO CO

00

i=1

s= 1

E 2.s

— CO6CTBeHHb1e 3HatleHH51

C. BHxTop BopHcormq

JIITaCK141,1

(1959)

A classical theorem of J. Mercer (1909) states that 1 CO A n(K) K(, 0 de = n=1

for every positive definite continuous kernel. Later on, using Carleman's determinant formula (1921), the Czech mathematicians B. Hostinsky (1921) and J. Kaucky (1921) 71), showed that, for every Hilbert-Schmidt kernel 11

co 71)

n (10 2 .

K(7), 0 de d = n=1

304

7. Historical survey

It was not until the end of the fifties that V. B. Lidskij: Non-self-adjoint operators with a trace (Russian). Doklady Akad. Nauk SSSR 1959,

succeeded in proving the trace formula for trace class operators on an abstract Hilbert space. A few years earlier, A. Grothendieck (GRO, Chap. If, p. 19) had obtained this result for 2/3-nuclear operators on Banach spaces, and, referring to H. Weyl (1949), he also claimed that the canonical Fredholm denominator of every nuclear operator on a Hilbert space has genus zero (GRO, Chap. II, p. 13). This fact immediately implies the trace formula which he used without comment in another paper (1961). Both authors, A. Grothendieck and V. B. Lidskij, approached this problem indirectly, via determinant theory. All subsequent proofs, given by J. Weidmann (1965), (DUN, XI.9.19), (RIN, p. 139) and (REE, Vol. IV, p. 328), proceeded along the same line. Likewise employing this method, H. Kbnig: s-Numbers, eigenvalues and the trace theorem in Banach spaces. Studia Math. 1980,

verified the trace formula for those operators on a Banach space which have summable approximation numbers. In another paper (1980: b) he successfully treated the operator ideal (0 2)2 which, in the hidden form S)o 91, was already considered by A. Grothendieck (1961) and Chung-Wei Ha (1975). Determinant-free proofs of the trace formula are due to J. A. Erdos (1968, 1974) and S. C. Power (1983: a, b). Their methods apply, however, to the Hilbert space setting exclusively. The comparatively elementary approach presented in this monograph is taken from H. Leiterer / A. Pietsch (1982).

7.7.

Applications I ch schrieb diese Gleichungen nicht hin, ais ob sie etwa das Problem listen oder doch der Lbsung ndher führten, sie sollten nur em n Beispiel unter zahllosen sein, dafür, dass man bei Randproblemen der linearen partiellen Differentialgleichungen bestândig vor dieselbe Gattung von Aufgaben gestellt wird, welche jedoch, wie es scheint, far die heutige Analysis im Allgemeinen unaberwindliche Schwierigkeiten darbieten. Ich meine die zweckmdBig Integralgleichungen zu nennenden Aufgaben, welche darin bestehen, dass die zu bestimmende Function, ausser ihrem sonstigen Vorkommen, in ihnen unter bestimmten Integralen enthalten ist. Paul du Bois-Reymond (1888)

The standard reference for the history of integral equations and equations in infinitely many unknowns is the famous Enzyklopddie-Artikel of E. Hellinger / O. Toeplitz (HEL). Many historical remarks concerning eigenvalue distributions of integral operators are to be found in J. A. Cochran's monograph (COC). Excellent reports about s-numbers of integral operators on Hilbert spaces were written by M. Sh. Birman / M. Z. Solomyak (1974, 1977: a). Concerning the asymptotic behaviour of eigenvalues of differential operators we refer to C. Clark (1967) and M. Sh. Birman / M. Z. Solomyak (1977: b). A complete survey of results from the theory of diameters, obtained by Soviet mathematicians, is given in V. M. Tikhomirov's book (TIK). L. Schumaker (SCU) provides many valuable comments to the history of spline functions. The best reference about trigonometric series, also from the historical viewpoint, is still A. Zygmund's treatise (ZYG).

7.7. Applications

7.7.1.

305

Diagonal and embedding operators AHLIHO $1 npHute.rt K Bonpocam TeopHH BJ10)1CellH51 B C1331311 c ,ELasHo HHTepecoBaBummtt meHR Haesimit maccwiecxort Teoppm lip1161-114XeHH51 (1)yHmiHit 1-101111H0MaMM, npexae BCeB0 TplITOHOmeTpmecKmmil no.nHHomamH H MX HenepHowlectcHmkt aHastoramH - Hesimmit (1)yHKLIHAMII 3KCI-10}1e111.1HaSH,HOTO mina.

Cepreit

MHXarISIOBWIHIIKOJICKIIri

(1969)

p , then every bounded sequence (r k) determines a diagonal operator If 1 q acting from 1„ into /g . In the case 1 _- _.q<poo, it follows from classical results of O. Holder (1889) and E. Landau (1907) that the same statement holds if and only if (rk) E 1,. with 1/r = 1/q — 1/p. On the other hand, apart from the inclusion 4,(0, 1) c 4(0, 1) for 1 according to S. L. Sobolev: Applications of functional analysis in mathematical physics (Russian). Moscow 1950,

we have the embedding W;(0, 1) c 40,1) provided that a > 1/p — 1/q and 1 p
Employing this result, A. Pietsch (1980: d) obtained the diagram B(1,, p(0, 1) --->I4, q(0, 1Y

IA1,

,

pc

>

IT

where D, is the diagonal operator induced by (1c-cx) with a

= a + r + 1 — 1/p — 1/q > 0.

It now became transparent why certain properties of diagonal and embedding operators are analogous. In a less explicit form this relationship had already been described by H. Kiinig (1974) and V. E. Majorov (1975). 7.7.2. s-Numbers of diagonal operators riplmertem Alm npHmepa cne,uytoaudi 133511.1.1111Arl pe3y.ribTaT B. 14. Mauaesa: eclat M - KOIsiffIaKT B l , COCTOALLIHR 11 3 BCCX nocneAoBaTenE,HocTert x = {x ;} Taxmx, 15_ ei , re H ei < co , TO n-nonepegmix Komnax-ra M e; ei+i onpegensteTcH paBeHCTBOM CO

d„ (M)

ei. AnexcaHAp CemeHosm Mapxyc (1966)

306

7. Historical survey

The first asymptotic estimate for the approximation numbers of diagonal operators /1 was obtained by A. Pietsch (1963: b). In 1974, employing a method from l due to V. D. Mil'man (1970), the same author computed the precise values of these numbers not only for diagonal operators from 1„ into 1, but also from lp into /g The opposite case 1 -p
Nikolay Andreevich Kolmogorov (1936) reopemami4 BROACeH1451 nommatoT "KailecrBeliyTBeinK4em4R o He11pepbmnrocn4 1451I4 KOMMKTHOCTI4 B50Acern4R owioro Knacca W7, B Apyroti Taxoti Knacc. Hamen rJlaBHOl ABJD1eTCH "Kwurfeci-Bermoe" mccne,LtoBaime COOTBeTCTB3/1011114X oriepaTopoB BJ10>KeH1451, Kœropme, Boo6we roBopsi, IlpeACTaBJDIM C060P1 KOMIlaKTHble onepaTopm. 06b111110 110A -

Male"

Mxxaxn IllnemoBin Empmax, Muxayin 3axapompi CalomsiK (1974)

Until the discovery of the Ciesielski isomorphism, s-numbers of embedding operators had been estimated by direct methods, for example, by trigonometric or piecewise-polynomial approximation. There is a vast literature devoted to this problem. The most important contributions were made in the USSR. We quote, for example,

307

7.7. Applications

V. M. Tikhomirov : Diameters of sets in functional .spaces and the theory of best approximation (Russian). Uspehi Mat. Nauk 1960, M. Sh. Birman / M. Z. Solomyak Piecewise-polynomial approximations of functions of the classes W; (Russian). Mat. Sb. 1967, M. Z. Solomyak / V. M. Tikhomirov: Geometric characteristics of the embedding from plici; into C ( Russian). Izv. Vyssh. Uchebn. Zaved. Mat. 1967.

In contrast to the approach used in this monograph, the Soviet mathematicians take another viewpoint. They consider the closed unit ball of the first space as a precompact subset of the second one and look at its diameters. In this setting, however, the powerful concept of s-numbers gets lost.

7.7.4.

Eigenvalues of infinite matrices Man suche die linearen Gleichungen Wx k —

hr(k1) x i = 0

zu Ibsen; das ist nur mbglich far gewisse Werte des Parameters W, nâmlich W = W, wo W die Eigenwerte (Energiewerte) bedeuten. Max Born, Werner Heisenberg, Pascual Jordan (1926)

The literature concerned with eigenvalue distributions of operators induced by infinite matrices is very poor. Clearly, the Schur-Carleman inequality, proved for Hilbert-Schmidt kernels, could be rephrased in these terms. The main reason for the underdevelopment of this branch of functional analysis seems to be the fact that there was no need to deal with very special operators which are covered by the general spectral theory of operators acting on abstract Hilbert or Banach spaces. Nowadays, within the framework of a refined theory, matrix operators play an important role not only as simple patterns for the more complicated case of integral operators but also in their own right.

7.7.5. Eigenvalues of integral operators What can be said about the distribution of the characteristic values of the Fredholm integral equation y(x) = f K(x, ) y(4) a

on the basis of the general analytic properties of the kernel such as integrability, continuity, differentiability, analyticity and the like? K(x,)

Einar Hille, Jacob David Tamarkin (1931) The first observation concerning the asymptotic behaviour of eigenvalues of integral operators is implicitly contained in I. Fredholm's paper from 1903 in which he determined the order of (5G- , K) for continuous and Lipschitz-Holder continuous kernels. This result was turned into an explicit statement about eigenvalue distributions by T. Lalesco (LAL, p. 88). The most basic theorem, obtained by determinantfree methods, is due to I. Schur (1909: a) and T. Carleman (1921).

308

7. Historical survey

In order to get the desired information about the eigenvalues of integral operators the following techniques have been employed: - estimates of the growth of the Fredholm denominator, - the Fischer-Courant minimax principle, - estimates of the s-numbers. As shown in this monograph, the last method is by far the most powerful. In the course of proving those results for certain types of kernels the following stages of development have usually been passed through: for e > 0, n) e 4+6 - (2„) with the best possible fine index w. n) e 40„ The following list, where the exponent of convergence is given by 1/r := m + n + 1/q+ or 1/r := + r + 1/q+, respectively, yields a relatively complete historical survey. -

(2

-

(2

kernels

eigenvalues

authors

[C, C] [Cm, C]

12

I. Schur (1909:a) H. Weyl (1912), S. Mazurkiewicz (1915), A. O. Gel'fond (1957) I. Fredholm (1903), T. Lalesco (1912) T. Carleman (1921) W. B. Johnson et al. (1979) M. G. Krejn (1937), S. H. Chang (1952), I. C. Gohberg / M. G. Krejn (1965), V. I. Paraska (1965) E. Hille / J. D. Tamarkin (1931) E. Hille / J. D. Tamarkin (1931), F. Smithies (1937) M. Sh. Birman! M. Z. Solomyak (1967/69) A. Pietsch (1980:d)

[Ce, C] [L2 , L 2 ]

[L2 , [Lp , WiZ] [Lp ,

[ w:21 rBL, B4, vi

The above references do not mean that the authors have solved the problem for all possible choices of the parameters. Up to 1977 almost all investigations about s-numbers of integral operators have been carried out in the Hilbert space setting. The initiators were M. G. Krejn (1937) and F. Smithies (1937: a). Some results along this line may be found in the famous monograph of I. C. Gohberg / M. G. Krejn (GOH). The most important contributions are due to the Leningrad school: M. Sh. Birman / M. Z. Solomyak (1967/69, 1970, 1974, 1977: a), M. Z. Solomyak (1970), G. P. Kostometov / M. Z. Solomyak (1971), G. P. Kostometov (1974, 1977) and G. E. Karadzhov (1972, 1977). We also refer to the work of V. I. Paraska (1965), P. E. Sobolevskij (1967), H. Triebel (1967, 1970: b), J. Kadlec / V. B. Korotkov (1968), S. L. Blyumin / B. D. Kotlyar (1970), J. A. Cochran (1975, 1976), J. Weidmann (1975) and B. D. Kotlyar / T. N. Semirenko (1981). Various criteria of nuclearity were estabilished by Shih-Hsun Chang (1947), W. F. Stinespring (1958), J. Weidmann (1966), M. Sh. Birman / M. Z. Solomyak (1969), J. A. Cochran (1974, 1977) and G. E. Karadzhov (1977). The special case of Hankel operators was treated by J. S. Howland (1971) and V. V. Peller (1980). All authors just quoted are concerned with integral operators acting on a Hilbert function space. This technical assumption requires some additional conditions on

7.7. Applications

309

the underlying kernel which sometimes look quite unnatural. The trouble disappears in the Banach space setting. The main results in this new branch of the theory of eigenvalue distributions are due to W. B. Johnson / H. Kiinig / B. Maurey / J. R. Retherford (1979), H. K6nig / J. R. Retherford / N. Tomczak-Jaegermann (1980), H. Kiinig (1980: d) and A. Pietsch (1980: d). 7.7.6.

Eigenvalues of differential operators In einer vollkommen spiegelnden Huile lainnem sich stehende elektromagnetische Schwingungen ausbilden, anlich den Tiinen einer Orgelpfeife. Hierbei entsteht das mathematische Problem, zu beweisen, daB die Anzahl der gentigend hohen Obertiine zwischen n und n + dn unabhdngig von der Gestalt der Huile und nur ihrem Volumen proportional ist. Hendrik Antoon Lorentz (1910) In der vorliegenden Arbeit habe ich mir die Aufgabe gestellt, mit den Methoden der Integralgleichungstheorie folgenden Satz

zu beweisen: Schwingungsvorenge, deren Gesetzmnigkeit sich in einer linearen Differentialgleichung vom Typus der gewôhnlichen Schwingungsgleichung ausspricht, besitzen, unabhdrigig von der geometrischen Gestalt und physikalischen Beschaffenheit der Rdume, in denen sie sich abspielen, im Gebiet der hohen Schwingungszahlen aile wesentlich em nund dasselbe "Spektrum". Hermann Weyl (1912)

The interest in the asymptotic behaviour of eigenvalues goes back to a conjecture stated by H. A. Lorentz in Giittingen in 1910 when he gave a series of lectures "Über aile und neue Fragen der Physik". Only after a few months H. Weyl: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. 1912,

solved this problem by the method of integral equations. Later on, R. Courant : Cber die Eigenwerte bei den Differentialgleichungen dei- mathematischen Physik. Math. Z. 1920,

found another approach based on the minimax principle. Finally, the Tauberian techniques were introduced by T. Carleman: Über die asymptotische Verteilung der Eigenwerte partieller Driferentialgleichungen, Ber. Sdchs. Akad. Wiss. Leipzig 1936.

Since all problems concerning eigenvalue distributions of differential operators can be transformed, via the Green function, into those for integral operators, one could guess that both subjects are closely mixed up with each other; see S. Agmon (1965). Surprisingly enough, this is not so! Each of these branches has developed its own types of results and methods. Thus it remains a problem for future work to bring together these estranged relatives. 7.7.7.

Fourier coefficients Ob nun die Coefficienten der Reihe zuletzt unendlich klein werden, wird in vielen Mien nicht aus ihrem Ausdrucke durch bestimmte Integrale, sondern auf anderem Wege entschieden

310

7. Historical survey werden massen. Es verdient indess em n Fall hervorgehoben zu werden, wo sich dies unmittelbar aus der Natur der Function entscheiden lat, wenn nâmlich die Function f(x) durchgehends endlich bleibt und eine Integration zulaBt. Bernhard Riemann (1854)

Vor einiger Zeit haben Sie die Frage aufgeworfen, ob die Ordnung des zu einem stetigen Kern geheirigen FREDHOLMschen Nenners die obere Schranke 2 erreichen kann. Ich werde im folgenden em n Beispiel angeben, fiir das diese Grenze tatsdchlich erreicht wird. Das Beispiel Ungt ihrigens mit einer Konvergenzfrage betreffend die Fourierkoeffizienten einer stetigen Funktion zusammen, welche vielleicht auch an und far sich nicht ohne Interesse ist. Torsten Carleman (1918) in a letter to A. Wiman Many basic notions and results of the theory of functions have been obtained by mathematicians while working on trigonometric series.

Antoni Zygmund (1958)

The fact that the Fourier coefficients of a periodic function f coincide up to the factor 27 with the eigenvalues of the associated convolution operator, 2n C{,p :

g(n)

dri,

—

6 was realized rather late. The first remark concerning this connection is due to T. Carleman: Ober die Fourierkoeffizienten einer stetigen Funkt ion. Acta Math. 1918.

Thus it happened that corresponding results for eigenvalues and Fourier coefficients were proved independently and at quite different dates. For example, the precise asymptotic behaviour of the eigenvalues of a Lipschitz-I-161der continuous kernel was already known to I. Fredholm (1903), while S. N. Bernstein proved not until 1914 that all functions f e Ca(27) with d> 1/2 have absolutely and uniformly convergent Fourier series. Later on, the theory of trigonometric series became an extremely rich source for counterexamples in the theory of eigenvalue distributions. The following list sketches the historical development of the most important results related to the question of how the sequence of Fourier coefficients reflects the properties of the underlying function. functions

Fourier coefficients

authors

L1 L2

Co

B. Riemann (1854), H. Lebesgue (1903) F. Riesz (1907), E. Fischer (1907) S. N. Bernstein (1914) S. N. Bernstein (1934) O. Szdsz (1946), S. B. Stechkin (1951) W. H. Young (1912), F. Hausdorff (1923), G. H. Hardy! J. E. Littlewood (1931)

Cr > 1/2 Bg21 •

1 4, < r < 2

12 11 11 11 r

Is,W

/r'

H. Weyl (1917), 0. Szasz (1922, 1928), A. A. Konyuskov (1958), A. Pietsch (1980: a)

311

7.7. Applications

7.7.8.

Practical applications Teoria cum praxi. Gottfried Wilhelm Leibniz (1700) Le but unique de la science, c'est l'honeur de l'esprit humain. Carl Gustav Jacob Jacobi (1830) Mathematicians are people who devote their lives to what seems to me a wonderful kind of play. Constance Reid (1980) Though this be madness, yet there is method in't. William Shakespeare (1602)

As already indicated, results about the asymptotic behaviour of eigenvalues of differential and integral operators have some significance in physics. Nevertheless, at present we do not know any application in the real world. Hopefully, this will change in the future. On the other hand, mathematics in its own right is an important and indispensable part of human culture. Bearing in mind these facts, the following motto was used when writing the monograph in hand: If a mathematical theory has not yet proved useful, then it should at least be beautiful.

Appendix At the time when finishing the work on this monograph, I did only know examples of such quasi-Banach operator ideals which admit either none or at most one continuous trace. Luckily enough, this gap was filled by N. J. Kalton during the "International Conference on Banach Spaces and Classical Analysis" (Kent, Ohio, summer 1985). He constructed quasi-Banach operator ideals supporting a large amount of different continuous traces. This fact shed a new light on many aspects of the trace and determinant theory. In particular, some open problems have become much more urgent than before. In what follows we present a new approach to Kalton's example. His original paper "Unusual traces on operator ideals", essentially based on Hilbert space techniques, is going to appear in the journal "Mathematische Nachrichten".

(el) denotes a weight sequence which, by defi-

A.X.1. Throughout this appendix r = nition, satisfies the following condition: 00

(W1)

etc >

0

= 1.

and k=0

(W2)

There exists a constant c, > 1 such that

A.X.2.

Operators T0 , T1 , rank (Tk)

2'

e

and

a(E, F) are II Tk

.0

C k+i -

for

k = 0, 1, ...

said to constitute an r-decomposition if

cek for

k = 0, 1, ...

where c > 0 is a constant. Then we let II ( Tk) I 9NT II := sup

I I Tk I

Every r-decomposition (Tk) determines an operator co T = E Tk k=0

such that

IITII < EjTj k=0

1(TO Mid E ek= II(Tk) j )411 . k=o

Hence (Tk) is also called an r-decomposition of T e t(E, F). A.X.3. We denote by 9)1,(E, F) the collection of all operators T c t(E, F) admitting an r-decomposition CO

T=

E

Tk.

k=0

Let TI 9)1r11 = inf 11(TO I gr11-

the infimum being taken over all possible choices of (Tk). A.X.4.

The following result is obtained by standard techniques. Theorem. 9)1, is a quasi-Banach operator ideal.

Proof. Obviously, IITII

IIT I ad for T e 9N,(E, F).

313

Appendix

Moreover, if rank (T) = 1, then we obtain an r-decomposition (Tk) by putting Tk ekT. Hence

9NrII -5_ II(Tk) I )411 = IITII This proves that

y I %Id' = Ilall IIYII

aOye 9)1,(E, F) and

for a E E' and y

E

F.

Given S, T E $1,(E, F) and E > 0, we choose r-decompositions

= E Sk

S

E

T

and

k=0

k= 0

Tic

such that

II(sk) I 9)411

and

11 S 93rIl

(1 +

( 1 + ) IT I

(T k) I " ril

Define

R0 := 0

and

Rk+ : = Sk

Tk

for k = 0, 1, ...

Then rank (Sk ) + rank (Tk) < 2k+1

rank (Rk+i) and

II-Rk±i I -.5_ Ilskll + jjTkjj

[II(Sk) I

+ II(Tk) Hence (Rk ) is an r-decomposition of S + T with II(Rk) IVA

Cre k +1 •

MILS j Wird + jTI NrIli •

(1 +

0, we obtain

Letting E

S + T E 931,(E, F) and j S + T j 9)1,11

cr[IIS

+ IIT

-

If X e t(E0 , E), T E Mr(E, F) and Ye t(F, F0), then it follows immediately that

YTX E 91 r(E0 , F0 ) and

I YTX

j

I

I Yll I T19)11.11

So far we have shown that 1L is a quasi-normed operator ideal. In order to check its completeness, we consider a sequence of operators Ph) E 9311 (E, F) such that "JD

h=

Given

E

>

<

c'1!+ 1 IIT'")

co.

0, there exist r-decompositions CC

rh)

Thk such that II(Thk) j 93r11

(1 +

IIT

(")

Define

To := 0 and

T1+1 :=

Thk h+k=i

for

i = 0, 1,

Then rank (T1+1 )

E rank (T,,k) < E 2k < 2'+' h+k=i

k= 0

. . .

314

Appendix

and

Il TH-111

E 11Thk11

E

h+k=i

h+k=i

E II(Thk) I 9Xtll

h+k=i

II(T) I

Cr " ei+1

E cit!" IIT ( h) 1

(1 +

ei+i.

11=0

Hence (Ti) is an r-decomposition of 00 00 00 T:= = E E Thk = h=0

h=0 k=0

1 =0

with GO

11(T1) I 9)4 11 Letting E

(1 +

8

Cr

)

h=0

0, we obtain CO

F)

TE

and

II T

i

E 4 +1 II") I 9ntll •

h=0

Finally, given any Cauchy sequence (Ta) in sequence of natural numbers nh such that

I T,,, —

I

T,,

(20 - h-1-

for

TIME,

F), we choose an increasing

m, n > nh .

Let

Ph+1) := Tn„, — T„,. for

T ( ° ) := T„o and

h = 0, 1, ...

Then

z CO

T o)

h=0

I 9nr11

CrOlTh 0

I and + 1)

Hence CC

T :=E h T(h) c M r(E, F)

and

IT —

TrIk

I

=

I IT — h=0 E Ph) I 9Nr11

cc

C'tz+ 1 h=k+1

2 - k Cr•

Therefore the subsequence (TO converges to T with respect to the metrizable topology of 9)1,(E, F), and so does (Ta). Remark. Suppose that the weight sequence r = (03 satisfies the condition h=k

COek

for k = 0,1, ...

where c o > 1 is a constant. Then it can easily be seen that an operator Te e(E, belongs to 9J1,(E, F) if and only if there exists a constant c > 0 such that

a2k(T)

cek for

k = 0, 1, ...

315

Appendix

In the particular case

r=

(e2 -k I r) ,

where 00

0 •=[ L •

and

l 1

0 < r < co ,

k =0

it follows from 2.3.8 that 9N, consists of all operators having approximation type . In other words, 9N, = A.X.5.

Next we prove an auxiliary result.

Lemma. Let T E t(E). Then n 11Th whenever rank (T)

'trace (T)1

n.

Proof. Without loss of generality, we may assume that rank (T) = n. In this case, by Auerbach's lemma 1.7.6, there exist x 1 , „ E M(T) and al , a„ e E' such that

hIXhl = 1, lIa1 II = 1 and <xi , al> = (S ip Then it follows from

Tx = E x i for x e E that

T=

xi .

T'ai

Hence

E <xi , T'ai>

'trace (T)I = A.X.6.

i= 1

E

1=1

A weight sequence r =

n 111'11 .

(to is said to be strongly convergent if

co

2kek < co

.

k=0

A.X.7. Theorem. The quasi-Banach operator ideal 9)1, admits a continuous trace if and only if r = (ak) is strongly convergent. Proof. We consider first a strongly convergent weight sequence r = (ek). If T e 9X,(E), then for every r-decomposition

T

co Tk k=0

we have 00

CO

00

E 'trace (Tk)I k =0

E 241111

k=0

k

E 2kek11(71) I mr11. o

Hence the right-hand side in the following definition makes sense:

r(T):= E

k=0

trace (Tk).

316

Appendix

Of course, it must be shown that this quantity is uniquely determined. In order to verify this fact, we consider different r-decompositions 00 T E TV ) and T = Tk2) k=0

k=o

Next we define an r-decomposition (Tk) by putting To := 0

and

Tk±i

Tk i) —

Tk2) for

k = 0, 1, ...

Note that CO

0 and rank ( k=0

Tk)

2"'.

k=o

Thus the preceding lemma implies that E trace (TO

trace ( E Tk)i _<_ 2"+ 1

k=0

k=0

E

= 2n+ 1

Tk

2n1-1

k=n+1

E

k=n+1

2kek

Tk

k=n+ 1

IITII

11(n) I srll

Hence, letting n —› co , we obtain 00 00 E trace (Ti' )) — 1" trace ( n2)) k=0

k=0

E k=0 oo

.

.0D E trace (Tk) = O. k= o

This proves that the above definition is correct. Furthermore, it can easily be seen that r is indeed a continuous trace on 9)4. In particular, we have CO

IT(T)1 :6 k=o 2kek IIT I 9)11-11

for TE

(E).

Conversely, we assume that 9N, supports a continuous trace r. Then there exists 1 such that a constant c Ir(T)I 5._ cli T1 $1,11

for

T e Mt(E) .

Let

E

Pk,

k=0

where Pk denotes the orthogonal projection from /-, onto the subspace spanned by i < 2"+ 1 . We now conclude from the unit sequences e1 with 2" trace

(Dr))

In

=

E

and

for

in = 0, 1, ...

2ke k k=0

11,0(rin'i

<1

that 171

E 2kok < c k=0 —

Hence the weight sequence r = (eu) must be strongly convergent.

Appendix

317

Remark. We know from 4.2.26 that the quasi-Banach operator ideal t(in admits a unique continuous trace. Moreover, it follows from 2.3.8 that 9N, c 2(10 for every strongly convergent weight sequence r = Hence the desired continuous trace on 9)Z, can be obtained by restriction.

(ek).

A.X.8. Suppose that the quasi-Banach operator ideal ?.t supports different (continuous) traces r l and r,. Then r : = r 1 — T2 still possesses the properties (T2), (T3) and (T4), stated in 4.2.1, while (T 1 ) changes into

(T7)

r(a

x) = 0 for a E E' and

xE

E.

A function w satisfying these modified conditions is called a (continuous) zero trace. Of course, we are only interested in the non-trivial case when r does not vanish identically.

A.X.9. A strongly convergent weight sequence r = (ok) is said to be singular if there exists an increasing sequence of natural numbers n i such that CO

2ni k—n lim co

=

—

E

O.

2kek

k=ni

A.X.10.

We are now in a position to establish the main result of this appendix.

(0)

be a singular strongly convergent weight sequence. Theorem. Let r = Then 9)Z, admits a continuous non-trivial zero trace. Proof. First of all, we fix an ultrafilter 6// on N o := {0, 1, ...} such that

2n /2 k=11

urn qj

ek

_ O.

E 2ke k k=n

This condition is satisfied if 61/ contains all tails {n i , in A.X.9. If T 9)1,(E), then for every r-decomposition

171+ 1 , ...}

of the sequence described

co

T= E

Tk

k =0

we have

E

E Itrace (Tk)I

k=n

k=n

Tkil

E 2ke.k 11(T ) I 9)1r11.

k=n

Hence the right-hand side in the following definition makes sense: CO

E Tql (T)

:= lim

k =n

trace (Tk )

318

Appendix

Of course, it must be shown that this quantity is uniquely determined. In order to verify this fact, we consider different r-decompositions 00 T E T o) and T = k=o

k

k=0

Next we define an r-decomposition (Tk) by putting — Tk 2) for To := 0 and Tk±i := Note that co 2"+ 1 . E Tk= 0 and rank ( Tk

k = 0, 1,

)

k=o

k

As shown in A.X.7, we have 00

1, trace (Tk) = k=o

Thus Lemma A.X.5 implies that

E

E c° trace (Tk) ik=n+1

k

trace

(Tk)

r=0 00

< 2n+ 1

Tk

=

2"+'

k=0

k=n+1

co

2 n+1

E

E ek11(Tk) F 9)1,11.

k=n+1

Hence 00

E

trace (TV)

k=n

CO

E

E

trace (TO)

k=n

co

co

E

k=n

E

2"e k

trace (TO

k=n+1

E

2kek

k=n

k=n

2" ek

E

2"-"

k=n+1

k)

oo k=n

I

Mr II •

2"ek

Passing to the limit with respect to the ultrafilter V, we conclude that the above definition is correct. Furthermore, it can easily be seen that rq, is indeed a continuous zero trace on .9.Nr . In particular, we have IL TI TI,11

11-*(T)1

for

T e 9»,(E).

Finally, we define 00

D,

E

ekPk,

k=o

where Pk denotes the orthogonal projection from 12 onto the subspace spanned by the i < 2"+.1 • Then unit sequences e with 2" D, E M1(/2)

and Te(Dr) = 1.

Therefore Tow does not vanish identically.

319

Appendix

Example. Let

A.X.11.

ek := e2-k(k + 1)-1' , where co

and ia > 1.

2— k(k + 1) -- P

9 := [ k=0

Then r = (ek ) is a singular strongly convergent weight sequence. Proof. Obviously, co

(k + 1)- < cc.

2keic = k=0

k=0

We further observe that Do

2n

ek = 2ne k=n

2— k(k + 1) — A

2e(n + 1)-P

k=n

and co

c(n + 1)1_ IL.

(k + 1)-A

2kek = e k=n

k=n

Hence r = (ek) is singular.

A.X.12. Example. Let 0 < r < 1. Then there exists a singular strongly convergent c2-kir . weight sequence r = (ek) such that ek Proof. Fixa> 1, e > 0 and 0 = m o
Let n i — m i and

si :=

oc i := o2(si - ni )01 -1) and

Ai

ft k:m i

k < ni l

/5'i := e2s0 -1) ,

and B i : {k:n i

k< mi+1 . }

We now define {oc i2- k ek :=

13,2-kA

if if

k

Ai,

k e Bi .

Note that

and enii „ = Ai+12-1111+1 = i2—mi+iti • This means that the different pieces used in the definition of r = (ek) agree with each = cx42 - "i

13 1 2 - "itt

other. Since 2ek , 1

ek

for k e A i and 2mek+ = ek for k e Bi,

we have ek

2Pek+,

and 2ek+1

ek for k = 0, 1, ...

320

Appendix

Hence r = (ek) is indeed a weight sequence provided that 00 E 24 k k=o

1'

This condition can be satisfied by a proper choice of the parameter e . We now specialize the preceding construction by setting

p := 2Ir — 1, m i := 2(i + 1) i and n, := 2(i + 1) 2 — 1. Then

ai = 2i + 1

and

si = (i +1)2.

Note that

— 1 = 2(1/r — 1) = 2(ja — 1/r) > 0

If m i

and

2s1 — ni = 1.

k < ni , then log ek le = (si — ni)

— 1) — k = (s i — n 1) (It — 1) + k (1/ /. — 1) — k r

2

(si — n,)(,u — 1) + n i(lIr — 1) — kIr = (2s1 — n 1)(1Ir — 1) — klr = (1Ir — 1) — kir. k < mi+i : The same estimate holds for n, log (2ki (2 si(ct — 1) — ku = s1ca — 1) +

—

— kir

2

=_ si(a — 1) + ni(lIr — < = (2s, — n i)(iu — 1Ir) —

— kir = (kt — 11r) —

Hence

k = 0, 1, ...

ek < c 2-klr for

Moreover, 09

2ke k >

k=

2ke k = ociai = (2i + 1) 2mierni Ai

and CO

CO

2"li

ek k= 1711

2m1 E

2'ni - ke nit

= 21171 +1e„,i .

k=

is indeed a singular strongly convergent weight sequence with the Thus r = required property. (

A.X.13.

(2tc

)

Summarizing the preceding considerations, we obtain the following result.

Theorem. Let 0 < r < 1. Then there exists a quasi-Banach operator ideal 9)4 with different continuous traces which is contained in 2r( a.)00 . A.X.14. The counterpart of the concept of a zero trace is that of a one determinant. Suppose that the quasi-Banach operator ideal ?,1 supports different (continuous) determinants 6 1 and 62 . Then 6 := 61 162 still possesses the properties (D,), (D 3 ) and (D 4), stated in 4.3.1, while (D 1 ) changes into

(D?)

WE + a

x) = 1

for

a c E' and

x e E.

Appendix

321

A function 6 satisfying these modified conditions is called a (continuous) one determinant. Of course, we are only interested in the non-trivial case when (5 takes values different from one. Remark. Let O'' be the function which assigns to every operator 1-E T with T e %(E) the complex-conjugate of 6(/E + T). Then, with the above notation, the function 6*(4 + .:T)6 2(IE + CT) may fail to be holomorphic. This proves that the property (D 4) is independent of (D 1), (D 2) and (D3).

Open problems In what follows we present a collection of open problems, most of which have already been mentioned in the text. Hopefully, the reader will be stimulated to do his own research work in this interesting branch of functional analysis. The problems marked with (*) are of special importance. 0.P.1 (see 2.3.7). Find a non-sophisticated expression for a p-norm on lip = 1 + 1 fr and 0 < r < o

where

0.P.2 (see 4.2.25 and 4.2.26). Prove that the quasi-Banach operator ideals er and fail to be approximative.

tr)

0.P.3 (see 4.5.15). Show that (4; 3 ) 3 is approximative. 0.P.4 (see 2.10.7). Let 0 < p < 2 and 1/q = 1 /p — 1 , 2. Is it true that pc x q( a)w ? 0.13.5* (see 2.10.7). Let 0 < 41(h)

'•-pov

p

< 1 and 1:q =

1. Do we have

to(a) 9 '`wq,w •

0.P.6 (see 2.8.18). Let 0 < s < (43 2)(:),) OE tr(x,)„ is strict.

-JD

and

is + 1 /2. Prove that the inclusion

0.13.7* (see 3.6.1 and 3.6.2). Does there exist a constant c > 1 such that 126„(T)1

[

n xkg-)]

1112

for

n = 1, 2, ...

k=

and every operator T e t'(E), where E is any finite dimensional Banach space? 0.P.8 (see 3.6.7). Determine the optimum eigenvalue type of 0.P.9 (see 3.9.4). Find the optimum eigenvalue type of 0.P.10 (see 3.9.7). Suppose that + 2171 ? eigenvalue type of W I +

?f„

rw weak

for 0 < r <2.

+ 2 -Yi„) for 2

r < c o.

What can he said about the

0.P.11 (see 3.9.1). Find examples of operator ideals which are maximal elements in the partially ordered class 0.P.12 (see 3.8.4). Characterize those Banach spaces E for which T E T(E) implies where 0 < r < 2. (4(T)) e 0.P.13* (see 4.2.13). Construct a quasi-Banach operator ideal admitting a continuous trace and having the optimum eigenvalue type 1 2 . 0.P.14 (see 4.2.25 and 4.5.15). Does there exist a continuous (spectral) trace on the quasi-Banach operator ideals t(ix) and (V 3 ) 3 ? 0.P.15* (see 4.2.14). Must every operator ideal t e E 1 admit a spectral trace? 0.P.16 (see 4.2.5). Suppose that It and Qi are approximative quasi-Banach operator ideals supporting continuous traces Y. and fi, respectively. Is it true that (T) 13(T) for T e W(E) r $(E)? 0.P.17* (see 4.2.37). Let St be a quasi-Banach operator ideal with a continuous trace --r. Show that the restriction of -r on W2 is always spectral.

Open problems

323

0.P.18 (see 4.2.38 and A.X.13). Modify the definition of a continuous trace by adding such conditions that the new object is uniquely determined by its axiomatic properties. In the special case when the underlying quasi-Banach operator ideal is of eigenvalue type 1, one should obtain the spectral trace.

0.P.19 (see 4.2.14). Let Vt be a quasi-Banach operator ideal with a continuous trace T. Suppose that the operators S e It(E) and T e U(F) have the same non-zero eigenvalues. Does it follow that r(S) = 0.P.20 (see A.X.5). Let T e t(E). Prove that 'trace (T)I where c

cn 112 lIT 142 + 43f2.1

whenever rank (T)

n,

I is a constant.

0.P.21 (see 5.3.6 and 7.4.5). Give an elementary proof of the inequality / n \ 1/p n r n 1P1P'\11P E 'Poi/ ) Elitk(M)1P)E k =1

i =1

1 =1

which holds for every (n, n)-matrix M = (au) and 2 < p < cc. The limiting case p = 2 was already treated by I. Schur (1909: a).

0.P.22 (see 5.5.10). Let M [1„, 1„,] and 2 < p < ni (1/p + 1/q — 1),_ and 1/r = a + r + 1 / q+ . Find the best possible value of the fine index w in the following assertion:

K e K.(0, 1), FZ,,,,(0,1 )}

implies

(2.„(K)) E l.

Epilogue Finally, we list some subjects, related to eigenvalues and s-numbers, which are not treated in this monograph. First of all, we have to mention the theory of entropy numbers and their applications to eigenvalue distributions. The most striking result in this direction is Carl's inequality

12„(T)1

en(T)

for all T e

st(E).

Further important topics, originating from classical theorems on normal forms of matrices, are the invariant subspace problem and the problem to determine the closed linear span of all principal elements of a given operator. Concerning applications we have restricted most of the considerations to integral operators defined on function spaces over the unit interval. The extension of the results to bounded domains in the n-dimensional euclidean space does not cause any new phenomena provided that the boundary is sufficiently smooth. However, in the case of unbounded domains the situation may become rather involved. Here one deals with special types of kernels characterized by means of weight functions. Another interesting subject is the theory of integral operators induced by holomorphic kernels. We now give a small selection of references. Entropy numbers: (PIE, Chap. 12), B. Carl (1981: a, b, 1982: a, b, 1984), B. Carl / Th. Kuhn (1984), B. Carl / A. Pietsch (1977), B. Carl / H. Triebel (1980), Th. Ktihn (1984), B. S. Mityagin / A. Pelczynski (1966). Invariant subspaces and triangularform: (DOW, pp. 54-66), (DUN, XI.10.1), (RIN, pp. 165-179), (TAY, V.7.16), V. I. Lomonosov (1973), H. Radjavi / P. Rosenthal (1982), J. R. Ringrose (1962). Completeness of principal elements: (DUN, XI.6.29, XI.10.15), (00H, V.6.1, V.6.2), H. E. Benzinger (1973), H. Kanig (1980: b), A. S. Markus (1966), E. Reuter (1981). Weighted kernels: M. Sh. Birman / M. Z. Solomyak (1967/69, 1977: a, b), A. Pietsch (1983), H. Triebel (1970: b). Holomorphic kernels: A. O. Gel'fond (1931), E. 1-hil1e / J. D. Tamarkin (1931), H. Kiinig J S. Richter (1984).

Bibliography The bibliography is divided into two parts: A) Textbooks and monographs, B) Research papers. Books are quoted by a symbol consisting of three typical letters from the name(s) of their author(s). The other items are cited by the name(s) of their author(s) and the year of publication. The numbers at the end of each entry indicate the paragraphs in the text where this book or paper is quoted. For example, 7.0.0 and 7.1.0 refer to the introductions of Chapter 7 and Section 7.1, respectively. Textbooks and monographs

A)

Adams, R. A. Sobolev spaces. New York: Academic Press 1975. (ADA)

6.4.0. Baltzer, R. Theorie und Anwendungen der Determinanten. Leipzig 1857.

(BAL)

7.0.0. Banach, S. (BAN)

Théorie des opérations linéaires. Monografie Matematyczne, Warszawa 193 2 .

1.7.6, 7.2.3, 7.3.0, 7.5.3. Bari, N. K. (BAR) A treatise on trigonometric series. Moscow: Fizmatgiz 1961, and New York: Pergamon Press 1964.

6.6.13. Barnes, B. A.; Murphy, G. J.; Smyth, M. R. F.; West, T. T. Riesz and Fredholm theory in Banach algebras. Research Notes in Math. 67, Boston, London, Melbourne: Pitman 1982.

(BAS)

3.2.0, 3.2.14, 7.4.1. Bergh, J.; Liifstrbm, J. (BER)

Interpolation spaces. Berlin, Heidelberg, New York: Springer-Verlag 1976.

F.0.0, F.3.2, F.3.3, F.3.4, 1.1.6, 2.1.14, 5.4.3, 6.3.1. Boas, R. P., Jr. (BOA) Entire functions. New York: Academic Press 1954.

4.8.0, 4.8.9. Bôcher, M. (BOC) An introduction to the study of integral equations. Cambridge Univ. Press 1909. Preface. Bourbaki, N. (BO U) Eléments d'histoire des mathématiques. Paris: Hermann 1960.

7.1.0, 7.2.0. Butzer, P. L.; Berens, H. (BUB)

Semi-groups of operators and approximation. Berlin, Heidelberg, New York: Springer-

Verlag 1967. F.0.0, 6.3.1, 7.2.0. Butzer, P. L.; Scherer, K. (BUS)

Approximationsprozesse and Interpolationsmethoden. Mannheim, Zürich: Bibliographisches Inst. 1968.

6.5.14.

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(MON)

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267-269. 7.1.4, 7.4.2. Szankowski, A. (1981) B(H) does not have the approximation property. Acta Math. 147, 89-108. B.4.7. Szarek, S. J. (1976) On the best constants in the Khintchine inequality. Studia Math. 58, 197-208. G.2.3. Siasz, O. (1912) A végtelen determincinsok elméletéhez. Math. és Phys. Lapok 21, 224-295. 7.5.1. (1922) Ober den Konvergenzexponenten der Fourierschen Reihen gewisser Funktionenklassen. Sitzungsber. Bayrische Akad. Wiss. München, Math. Phys. Ki., 135-150. 7.7.7. (1928) Ober die Fourierschen Reihen gewisser Funktionenklassen. Math. Ann. 100, 530-536. 7.7.7. (1946) On the absolute convergence of trigonometric series. Ann. of Math. (2) 47, 213-220. 7.7.7. Taibleson, M. H. (1964/66) On the theory of Lipschitz spaces of distributions on Euclidean n-space. J. Math. Mech. 13, 407-479, 14, 821-839, and 15, 973-981. 7.2.2. Taylor, A. E. (1947) A geometric theorem and its application to biorthogonal systems. Bull. Amer. Math. Soc. 53, 614-616. 1.7.6. (1966) Theorems on ascent, descent, nullity and defect of linear operators. Math. Ann. 163, 18-49. 3.1 14. (1970) Historical notes on analyticity as a concept in functional analysis. Problems in analysis (Ed. R. C. Gunning), 325-343, Princeton Univ. Press. 7.5.3. (1971) Notes on the history of the uses of analyticity in operator theory. Amer. Math. Monthly 78, 331-342. 7.5.3. Thorin, G. O. (1938) An extension of a convexity theorem due to M. Riesz. Kungl. Fysiogr. Sdllsk. i Lund Ftirh. 8, 166-170. 7.2.4. Tichomirov, V. M. (1960) Diameters of sets in functional spaces and the theory of best approximation (Russian). Uspehi Mat. Nauk 15: 3, 81-120. 7.7.3. (1965) A remark on n-dimensional diameters of sets in Banach spaces (Russian). Uspehi Mat. Nauk 20: 1, 227-230. 7.3.6.

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Bibliography

Triebel, H. (1967) Ober die Verteilung der Approximationszahlen kompak ter Operatoren in Sobolev-BesovRdumen. Invent. Math. 4, 275-293. 7.2.4, 7.3.6, 7.7.5. (1970: a) Interpolationseigenschaften von Entropie- and Durchmesseridealen kompakter Operatoren. Studia Math. 34, 89-107. 7.3.6. (1970: b) Ober die Verteilung der Approximationszahlen von Integraloperatoren in Sobolev-BesorRdumen. J. Math. Mech. 19, 783-796. 7.7.5, Epilogue. (1981) Spline bases and spline representation in function spaces. Arch. Math. (Basel) 36, 348-359. 6.4.11. Walsh, J. L. (1923) A closed set of normal orthogonal functions. Amer. J. Math. 55, 5-24. 5.1.7.

Weidmann, J. (1965) Ein Satz über nukleare Operatoren im Hilbertrawn. Math. Ann. 158, 69-78. 7.6.2. Integraloperatoren der Spurklasse. Math. Ann. 163, 340-345. (1966) 7.7.5. (1975) Verteilung der Eigenwerte fiir eine Klasse von lntegraloperatoren in L 2 (cf, b). J. Reine Angew. Math. 276, 213-220. 7.7.5. Weierstrass, K. Zur Theorie der quadratischen and bilinearen Formen. Monatsber. Preuss. Akad. Wiss. (1868) Berlin, 310-338. 7.1.6. (1876) Zur Theorie der eindeutigen analytischen Funktionen. Abhandl. Preuss. Akad. Wiss . Berlin, 11-60. 4.8.4, 7.5.4. Zur Determinantentheorie (Vorlesungsausarbeitung von P. Gunther). Coll. Works III, (1886) 271-286. 7.1.1. Weis, L. (1982) Integral operators and changes of density. Indiana Univ. Math. J. 31, 83-96. 6.2.15. West, T. T. (1966) Riesz operators in Banach spaces. Proc. London Math. Soc. (3) 16, 131-140. 3.2.26, 7.4.1. Weyl, H. Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialglei(1912) chungen ( mit einer Anwendung auf die Theorie der Hohlratanstrahlung). Math. Ann. 71, 441-479. 7.7.5, 7.7.6. (1917) Bemerkungen zum Begriff des Differentialquotienten gebrochener Ordnung. Viertel j ah rschrift Naturforsch. Gesell. Zürich 62, 296-302. 7.7.7. (1944) David Hilbert and his mathematical work. Bull. Amer. Math. Soc. 50, 612-654. 7.5.0. (1949) Inequalities between the two kinds of eigenvalues of a linear transformation. Proc. Nat. Acad. Sci. USA 35, 408-411. 3.5.1, 3.5.4, 3.5.5, 7.3.4, 7.4.3, 7.4.6, 7.6.2. Ramifications, old and new, of the eigenvalue problem. Bull. Amer. Math. Soc. 56, 115-139 . (1950) 7.0.0.

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Wiener, N. (1923) Note on a paper of BC1170Ch. Fund. Math. 4, 136-143. 7.2.3. Wloka, J. (1967) Vektorwertige Sobolev-Slobodeckijsche Distributionen. Math , Z. 98, 303-318. 6.4.0. Yosida, K. (1939) Quasi-completely-continuous linear functional equations, Japan J. Math. 15, 297-301. 3.2.1, 3.2.9. Young, W. H. (1912) Sur la généralisation du théorème de Parseval. C. R. Acad. Sci. Paris 155, 30-33. 6.5.9, 7.7.7.

353

Index absolutely p-integrable function 6.2.5 — r-summable sequence (family) 1.1.3 — summing operator 1.3.1 — y-summing operator 1.3.1 (r, s)-summing operator 1.2.1 additive Weyl inequality 3.5.4 adjoint operator B.3.8 admissible radius 4.2.20 algebraic tensor product of Banach spaces E.1.2 — — — — operators E.1.4 almost separably E-valued function 6.2.3 approximable operator D.2.2 /1-approximable operator D.1.13 approximation numbers of functions 6.4.20, 6.5.12 — — — operators 2.3.1 — — — sequences (families) 2.1.2 — property B.4.7 — type 2.3.5 91-approximation numbers 2.8.1 — type 2.8.4 approximative kernel of a quasi-Banach operator ideal D.1.13 — quasi-Banach operator ideal D.1.13 ascent 3.1.2 Banach operator ideal D.1.4 — space B.1.3 p-Banach operator ideal D.1.7 — space B.1.7 Bennett-Carl example 1.6.6, 1.6.7 Besov function space 6.4.2 — sequence space 5.4.1 — type of a kernel 6.4.17 — — — — matrix 5.4.8 Besov-Hille-Tamarkin type of a kernel 6.4.23 bidual Banach space B.3.3 — operator B.3.5 Bochner integrable function 6.2.5

canonical injection B.2.7, C.2.2 — surjection B.2.9, C.2.2 — Weierstrass product 4.8.4 Chang numbers 2.5.8 — type 2.5.10 characteristic polynomial A.5.2 Ciesielski system 6.4.10 — transformation 6.4.11 classical function spaces C.3.2, C.3.5, C.3.6 — sequence spaces C.1.3, C.1.4, C.1.6 closed operator ideal D.1.5 — unit ball B.1.2 codimension B.4.2 compact operator D.2.3

continuous determinant 4.3.7 — trace 4.2.4 convolution kernel 6.5.19 — operator 6.5.2 decomposition theorem 3.1.14, 3.2.9, 3.2.14 descent 3.1.6 determinant defined on an operator ideal 4.3.1 — formula A.5.4, 4.3.21 — of a finite operator A.4.4, 4.3.3 — — — matrix A.4.1 diagonal matrix 5.1.3, 5.2.2 — operator 1.6.1, 2.9.1 diagonalization theorem 2.11.5 dimension B.4.1 direct sum B.2.11, C.2.1 Dixon-von Koch matrix 5.3.3 domination theorem 1.3.5, 1.3.6 dual Banach space B.3.2 — exponent C.1.9 — operator B.3.4 — — ideal D.1.12 eigenelement 3.2.15 eigenvalue of a matrix A.5.1 — — an operator A.5.1, 3.2.15 — sequence of a kernel 6.1.5, 6.2.13, 6.3.7, 6.4.18 — matrix 5.3.4, 5.4.9 —— — — — an operator 3.2.20 — theorem for absolutely r-summing operators 3.7.1, 3.7.2 — — — — (r, 2)-summing operators 3.7.6 — — — Besov kernels 6.4.19 — — — — matrices 5.4.10 — — — Besov-Hille-Tamarkin kernels 6.4.24 — — — Fredholm kernels 6.1.6 — Hilbert-Schmidt operators 3.5.6 — Hille-Tamarkin kernels 6.2.15 — — — Hille-Tamarkin matrices 5.3.5 — — — Hille-Tamarkin-Besov kernels 6.4.26 — — — nuclear operators 3.8.1 — — — p-nuclear operators 3.8.5 3.8.6 (r, 2)-nuclear operators —— — — — Schatten-von Neumann operators 3.5.7 Sobolev kernels 6.4.31 weakly singular kernels 6.3.8 — — Weyl operators 3.6.2 type 3.4.1 embedding operator B.1.10 — theorem 2.3.10, 2.8.10 entire function 4.8.1 equivalent quasi-norms B.1.4

Index evaluation operator B.3.3 exponential function of an operator 4.3.12 extended orthonormal sequence 2.11.1 extension of an operator ideal D.3.2 — property C.4.1 — theorem 1.5.4 factorization theorem for absolutely 2-summing operators 1.5.1, 1.5.2 — — — p-nuclear operators 1.7.3 — — — (r, 2)-nuclear operators 1.7.10 finite ascem 3.1.2 — descent 3.1.6 — operator B.4.4, D.2.1 — representation B.4.4 — sequence 2.1.1 forward difference 6.4.1 Fourier coefficient 6.5.4 — matrix 5.1.6 Franklin system 6.4.10 Fredholm denominator 4.4.1, 4.4.5, 4.4.9, 5.5.5, 6.6.7 — formula 4.3.20, 4.6.1 kernel 6.1.4 — numerator 4.4.1 — resolvent 4.1.1 — — set 4.1.1 functional B.3.1 Gâteaux derivative 4.3.5, 4.3.20 Gel'fand numbers 2.4.2 — type 2.4.4 Grothendieck example 1.6.4 — inequality (theorem) 1.6.4 Haar system 6.4.10 Hadamard inequality A.4.5 Hilbert numbers 2.6.1 — operator D.2.4 — type 2.6.4 Hilbert-Schmidt kernel 1.4.8 — matrix 1.4.6 — operator 1.4.2 H ille-Tamarkin type of a kernel 6.2.8 — — — — matrix 5.3.3 H ille-Tamarkin-Besov type of a kernel 6.4.23 Holder inequality C.1.8, C.3.8 identity operator A.2.4, B.2.3 inequality of means G.1.1 injection B.2.6 injective operator ideal D.1.14 — s-numbers 2.4.1 injectivity D.1.14 integrable function C.3.6 r-integrable function C.3.6 integral operator (induced by a kernel) 6.1.5, 6.2.13, 6.3.7, 6.4.18

355 — — (in the sense of Grothendieck) 4.2.8 intermediate space F.1.3 interpolation couple F.1.1 — functor F.2.1 — method F.2.1, F.3.2 — property F.2.1 interpolation theorem (proposition) for absolutely (r, s)-summing operators 1.2.6 — — — — approximation type operators 2.3.14 — — — — Besov function spaces 6.4.8 — — — — Besov sequence spaces 5.4.3 — — — — classical function spaces F.3.4 — — — classical sequence spaces 1.1.6 — — — — Lorentz sequence spaces 2.1.14 — — — — s-type operators 2.2.10 invariant subset 3.1.12 inverse operator B.2.5 invertible operator B.2.5 isomorphism B.2.5 iteratively compact operator 3.2.1 Jessen inequality C.3.10 Jordan canonical form A.2.6 — block A.2.6 K-functional F.3.1 kernel of Besov type 6.4.17 — — Besov-Hille-Tamarkin type 6.4.23 — — Fredholm type 6.1.4 — — Hilbert-Schmidt type 1.4.8 — — Hille-Tamarkin type 6.2.8 — — Hille-Tamarkin-Besov type 6.4.23 — — Sobolev type 6.4.27 Khintchine inequality G.2.3 Kolmogorov numbers 2.5.2 — type 2.5.4 Lipschitz-Milder function space 6.4.3 Littlewood-Walsh matrix 5.1.7, 5.2.3 logarithm of an operator 4.3.16, 4.6.2 Lorentz function space 6.3.1 — sequence space 2.1.4 Lyapunov inequality F.3.5 matrix of Besov type 5.4.8 — — Dixon-von Koch type 5.3.3 — — Hilbert-Schmidt type 1.4.6 — — Hille-Tamarkin type 5.3.3 measurable E-valued function 6.2.2 Minkowski inequality C.1.6, C.3.6 monotonic Schmidt representation 2.11.4 multiplication theorem for absolutely r-summing operators 1.3.10 r, 2)-summing operators 2.7.7 — — — approximation type operators 2.3.13 - — — s-type operators 2.2.9

356

Index

muliplication theorem for Weyl operators 2.4.18 multiplicative s-numbers 2.2.8 — Weyl inequality 3.5.1 multiplicity of an eigenvalue A.5.3, 3.2.15 — — a zero 4.8.2 multiplier 6.6.11

nilpotent operator 3.1.13 — trace 4.2.27 non-increasing rearrangement of a function 6.3.1 — — — — sequence 2.1.2 norm on a linear space B.1.1 — — an operator ideal D.1.2 p-norm on a linear space B.1.5 — — an operator ideal D.1.6 norming subset 1.1.13 nuclear operator 1.7.1 — representation 1.7.1 p-nuclear operator 1.7.1 — representation 1.7.1 (r, 2)-nuclear operator 1.7.8 — representation 1.7.8 null space B.2.2 operator A.2.1, B.1.9 — ideal D.1.1 — norm B.2.1 optimum eigenvalue type 3.4.1 order of an entire function 4.8.6 — — nilpotency 3.1.13 orthogonal complement B.2.12 — projection B.2.12 Peetre K-functional F.3.1 pigeon-hole principle 3.2.2 Plemelj formulas 4.4.10 power of an operator B.2.4 — — — operator ideal D.1.10 primary factor 4.8.3 principal element 3.2.15 principle of iteration 3.4.3 — — related operators 3.3.4 — — tensor stability 3.4.8 — — uniform boundedness 3.4.6 product of operators B.2.4 — — operator ideals D.1.10 projection B.2.10, B.2.11 quasi-Banach interpolation couple F.1.1 — operator ideal D.1.4 — space B.1.3 quasi-norm on a linear space B.1.1 — — an operator ideal D.1.2 quasi-triangle inequality B.1.1 quotient space B.2.9

Rademacher functions G.2.1 range B.2.2 rank B.4.4 real interpolation method F.3.2 regularized Fredholm denominator 4.5.4, 4.5.10 — — resolvent 4.5.6 reiteration property F.3.3 — theorem 2.8.13 related operators 3.3.3 representation theorem for approximation type operators 2.3.8, 2.8.8 — — — operators on Hilbert spaces 2.11.8 representing matrix A.2.3 resolvent 4.1.1 resolvent set 4.1.1 Riesz decomposition 3.2.14 — lemma 3.2.3 — operator 3.2.12 s-numbers 2.2.1 s-scale 2.2.1 s-type 2.2.4, 2.2.11 Schatten-von Neumann operator 2.11.15, 2 .11.31 — type 2.11.15, 2.11.31 Schmidt representation 2.11.4 Schur basis A.2.5 shift matrix 5.1.4, 5.1.5 simple E-valued function 6.2.1 — rectangular kernel 6.2.10 singular value 2.11.3 Sobolev function space 6.4.6 — type of a kernel 6.4.27 spectral determinant 4.3.21 — product 4.3.21 — sum 4.2.14 — trace 4.2.14 spectral trace theorem 4.2.24 spline 6.4.9 spline approximation numbers 6.4.20 Stirling formula G.3.3 subspace 8 .2.7 sum of operator ideals D.1.11 — — a sequence (family) C.1.5, 1.1.2 surnmable sequence (family) C.1.5, 1.1.2 r-summable sequence (family) C.1.6 surjection B.2.8 surjective operator ideal D.1.15 — s-numbers 2.5.1 surjectivity D.1.15

tensor norm E.2.1 — product of Banach spaces E.2.3 — — — operator ideals E.2.4 — stability E3.1 trace defined on an operator ideal 4.2.1

Index trace formula A.5.4, 4.2.14 — of a finite operator A.3.2, 4.2.2 — — — matrix A.3.1, 5.5.3 — — — kernel 6.6.3, 6.6.5 trace-determinant theorem 4.6.3 trace extension theorem 4.2.5, 4.2.17 — transfer theorem 4.2.29 triangle inequality B.1.1 p-triangle inequality B.1.5 triangular form A.2.5 trigonometric approximation numbers 6.5.12 — polynomial 6.5.1 type of a kernel 1.4.8, 6.2.8, 6.4.17, 6.4.23, 6.4.27 — — — matrix 1.4.6, 5.3.3, 5.4.8 — — an interpolation method F.2.2

357 uniform eigenvalue type 3.6.4 unit ball B.1.2 — matrix A.2.4, 5.1.2 weak Schatten-von Neumann operator 2.11.31 — — type 2.11.31 weakly measurable E-valued function 6.2.3 — singular kernel 6.3.6 — r-summable sequence (family) 1.1.7 Weierstrass factorization theorem 4.8.4 — product 4.8.4 Weyl inequality 3.5.1, 3.5.4 — numbers 2.4.13 — type 2.4.15 zero trace

A.X.8

List of special symbols Throughout this monograph we use the standard notations

N := f1,2, ...I, Z := {0, ± 1, ... } Z„ := {1, ..., /1}. Furthermore, [FR, and C denote the real and complex fields, respectively. The positive part of a real : = max ($, 0), and V` denotes the conjugate of a complex number number is defined by = 1, and we If 1 r 00, then the dual exponent r' is determined by the equation hr + write r+ := max (r', 2). Set-theoretic inclusions are denoted by , while C means that equality is excluded. The symbols -< and ;.--; are used to denote asymptotic estimates; see G.3.1. The letter c (without or with index) is reserved to denote numerical constants which may or may not depend on some fixed parameters. All operator diagrams are assumed to be commutative.

Banach and Hilbert spaces

1.

Unless the contrary is explicitly stated, E and F (without or with index) denote arbitrary complex Banach spaces, while H and K are complex Hilbert spaces. Elements are usually denoted by x and y, functionals by a and b.

B.1.1 I lx Ell B.3.3, C.4.3 fx <x, a> B.3.1 B.3.2 1011 x y E.1.2 E.2.1 13.3.7 y* B.3.2 E' E" B.3.3 U B.1.2 U° )3.3.2 e E„ C.2.1 • () Ix

2.

,

[1,(1), Ei ] C.2.1 B.2.1 e(E, F) e(E, F) E.1.1 E.1.2 E F E® F E.2.3 dim (M) B.4.1 codim (M) B.4.2 MI B.2.12 K(T, x, E0 , E1), x) (E0 , E1)0,w F.3.2 (E0 , Ei ) F.2.1

F.3.1

Operators Operators (which may or may not belong to special ideals) are mostly denoted by S and T.

B.1.9 IIT : E Fil B.3.8 T* B.3.4 T' B.3.5 T" B.2.5 T-1 Tm B.2.4 B.4.4 rank (T) 9(T) 4.1.1 4.1.1, 4.5.6 T) F , T), 3.2.20 /3,,,(T) ).(T) 4.2.14 4.3.21 n(I + T) 4.2.2, 4.2.1 trace (T), r(T) 4.4.7, 4.5.9 T), T) 4.3.3, 4.3.1 det (/ + T), d(I + T) 4.4.1, 4.4.5, 4.5.4 T) dR, T), ô(, 4.4.1, 4.5.11 T) DR, T), D 4.3.5 b. (T) 4.5.2 E ,i(T) (

o‘n(T), 04m(T) A(T), M,m ) (T)

4.4.3, 4.5.11 4.4.3, 4.5.11 M(T), Mry(T), Mœ (T) B.2.2, 3.1.1 N(T), Nk(T), N„,(T) B.2.2, 3.1.1 ni(T), n(T) 3.1.1. d(T), dm (T), div (T) 3.1.11, 3.1.6, 3.1.2 exp (T) 4.3.12 log (I + T) 4.3.16, 4.6.2 a 0 y, x* ( y B.4.3, B.3.7 A.2.4, B.2.3 I„ KE B.3.3 f J, B.2.7, C.2.2 • Qk 13 .2.9, C.2.2 ST B.2.4 e(S, T) E.1.3 S 0 T E.1.4 S T E.2.4 Dt 1.6.1, 2.9.1

List of special symbols

359

s-Numbers

3.

2.2.1 s „(T), s„(T : E F) 2.3.1, 2.8.1 a„(T), a„(T Ili) c,1 (T) 2.4.2 d(T)

h „(T) x„(T)

Yn(T)

2.6.1 2.4.13 2.5.8

2.5.2

Operator ideals

4.

Generic operator ideals are denoted by /I and IIT I It 11 , IIT : E —> F I ?III D.1.1 It(E, F) ?IV D.1.12

3

D.1.2

4.2.8 A` D.2.3 B.2.1 !..?, e(s) , t)4s) , tf.s4, 2.2.11, 2.2.4 A.X.3 91, 91p , 94,2 1.7.1, 1.7.8 i' 1 1 1.3.1, 1.4.2, 2.11.15 S, er , r , w esrveak , sweviak 2.11.31 3.4.1 Er, [E,,,„

D.1.10 91i R3 D.1.11 + D.1.10 11'n F.4.1 /1 1)0 D.1.13, 2.8.4 11(a) , 944) , 9il„ D.2.1 D.2.2 D.2.4 it)

5.

Sequences and matrices Complex-valued families x

and E-valued families (x,) are defined on an arbitrary index set

I. In the case when this set is finite or countably infinite we speak of vectors or sequences, respectively. As far as possible, the letters i, j, h, k and m, n are reserved to denote indices (in particular, natural numbers). Finite or infinite matrices are denoted by M = (4usj).

C.1.2 "-,(1) ek 1.6.0 card (x) 2.1.1 2.1.2 an(x) <x, y> C.1.10, 5.4.4 1.1.12 Mop A.2.2, 5.3.4 5.3.4 „(M) trace (M) A.3.1 A.4.1 det (M) r(M) 5.5.3 (x1)0 p

6.

6G. M), an(M) (

5 . 5. 5

i l 3 • • •3 in

5.5.4 • • •, in) 5.1.3 D(n) F(n) 5.1.6 1(n) A.2.4, 5.1.2 4 e, 5.2.2 S(n) 5.1.4 5.1.5 S r(n) W(2") 5.1.7 5.2.3 Wr,a

Spaces of sequences and matrices

A.1.1 C.1.4, C.1.7 co(/), co GM, I co , lœ (n) C.1.3, C.1.7 C.1.6, C.1.7 4.(1), 1,, 1,.(n)

I(n)

C.1.7 U„ Ur(n) 1.1.3 [I„ El [1r(n,

El

1.1.7 2.1.4 [by, w , El 5.4.1 bta , v ] 5.4.8 Up 4] 5.3.3 [wr(I),

r , w( 1), r ,w

360

List of special symbols

7.

Functions and kernels

Complex-valued functions f and E-valued functions 0 are defined either on a compact Hausdorff space X or on a ci -finite measure space (X, ,u). On the unit interval we always use the Lebesgue measure. Variables are usually denoted by and n By a kernel K we mean a function of two variables which is used to determine an integral operator. .

6.3.1

f*

6.5.4 ?k(J) 6.5.12 a„(fI L,.)

<.f, g> Cf C4, ek

C.3.9 6.5.19 6.5.2 G.2.1 6.5.8, 6.5.15 ,4,, OE 6.5.16

sk, s i

6.4.10

8.

Spaces of functions and kernels

C.3.2 C(X), C(0, 1) C.3.5, C.3.7 L cc,(X „u), L Œ,(0, 1) C.3.6, C.3.7 1) ',JO, L r(X, 6.3.1 4,w(X,Y),Lr ov( 0, 1) 6.1.1 [C(X), E] 6.2.5 E] [Lr(X, 6.4.5 [Cm(O, 1), E] [Ca(0, 1), El [ W/.(0, 1), E]

6.4.3 6.4.6

a„(0 I fL„ Elm) 6.4.20 Ar0, 6,710 6.4.1

6.1.3 Kx 6.1.4 Kop 6.1.5 6.1.5 2,(K) r(K) 6.6.3, 6.6.5 K), o(K) 6.6.7 p

K (" , • • • , no'

6.6.6

[13°.,„(0, 1), E] 6.4.2 [C(X), C(Y)] 6.1.4 [Lp(X, ,u), Y , i))] 6.2.8 ELp(X, ,u), Lq(Y, it)] 6.2.9

[W7,1(0, 1), WZ(0, 1)]

E vv (0 , 1), fr(0, 1)1 [B; . „(0, 1), BZ, p(0, 1)] [B;,,,(0, 1), Lq(0, 1)] [4(0, 1), B (0 , 1)]

6.4.27 6.4.28 6.4.17 6.4.23 6.4.23

The fact that a space consists of periodic functions is indicated by the symbol 217. For example, we write C(27) instead of C(0, 1).

Eigenvalues and S-Numbers (Cambridge Studies in Advanced Mathematics 13)

Eigenvalues and s-Numbers (Cambridge Studies in Advanced Mathematics 13)

Galois Theories (Cambridge Studies in Advanced Mathematics)

Tolerance Graphs (Cambridge Studies in Advanced Mathematics)

Additive Combinatorics (Cambridge Studies in Advanced Mathematics)

Random Graphs (Cambridge Studies in Advanced Mathematics)

Algebraic Homotopy (Cambridge Studies in Advanced Mathematics)

Additive Combinatorics (Cambridge Studies in Advanced Mathematics)

Galois Theories (Cambridge Studies in Advanced Mathematics)

Ergodic Theory (Cambridge Studies in Advanced Mathematics)

Galois Theories (Cambridge Studies in Advanced Mathematics)

Practical Foundations of Mathematics (Cambridge Studies in Advanced Mathematics)

Practical Foundations of Mathematics (Cambridge Studies in Advanced Mathematics)

Galois Groups and Fundamental Groups (Cambridge Studies in Advanced Mathematics)

Lévy Processes and Stochastic Calculus (Cambridge Studies in Advanced Mathematics)

Lévy Processes and Stochastic Calculus (Cambridge Studies in Advanced Mathematics)

Galois Groups and Fundamental Groups (Cambridge Studies in Advanced Mathematics)

Wavelets and Operators (Cambridge Studies in Advanced Mathematics)

Modular Forms and Galois Cohomology (Cambridge Studies in Advanced Mathematics)

Lévy Processes and Stochastic Calculus (Cambridge Studies in Advanced Mathematics)

Reflection Groups and Coxeter Groups (Cambridge Studies in Advanced Mathematics)

Real Analysis and Probability (Cambridge Studies in Advanced Mathematics)

Lévy Processes and Stochastic Calculus (Cambridge Studies in Advanced Mathematics)

Real Analysis and Probability (Cambridge Studies in Advanced Mathematics)

Models and Games (Cambridge Studies in Advanced Mathematics 132)

Spectral Theory and Differential Operators (Cambridge Studies in Advanced Mathematics)

Automorphic Forms and Representations (Cambridge Studies in Advanced Mathematics)

Ip Studies in Advanced Mathematics)

Ip Studies in Advanced Mathematics)

Cellular Structures in Topology (Cambridge Studies in Advanced Mathematics 19)

Commutative Ring Theory (Cambridge Studies in Advanced Mathematics 8)

Eigenvalues and S-Numbers (Cambridge Studies in Advanced Mathematics 13)

Eigenvalues and s-Numbers (Cambridge Studies in Advanced Mathematics 13)

Galois Theories (Cambridge Studies in Advanced Mathematics)

Tolerance Graphs (Cambridge Studies in Advanced Mathematics)

Additive Combinatorics (Cambridge Studies in Advanced Mathematics)

Random Graphs (Cambridge Studies in Advanced Mathematics)

Algebraic Homotopy (Cambridge Studies in Advanced Mathematics)

Additive Combinatorics (Cambridge Studies in Advanced Mathematics)

Galois Theories (Cambridge Studies in Advanced Mathematics)

Ergodic Theory (Cambridge Studies in Advanced Mathematics)

Galois Theories (Cambridge Studies in Advanced Mathematics)

Practical Foundations of Mathematics (Cambridge Studies in Advanced Mathematics)

Practical Foundations of Mathematics (Cambridge Studies in Advanced Mathematics)

Galois Groups and Fundamental Groups (Cambridge Studies in Advanced Mathematics)

Lévy Processes and Stochastic Calculus (Cambridge Studies in Advanced Mathematics)

Lévy Processes and Stochastic Calculus (Cambridge Studies in Advanced Mathematics)

Galois Groups and Fundamental Groups (Cambridge Studies in Advanced Mathematics)

Wavelets and Operators (Cambridge Studies in Advanced Mathematics)

Modular Forms and Galois Cohomology (Cambridge Studies in Advanced Mathematics)

Lévy Processes and Stochastic Calculus (Cambridge Studies in Advanced Mathematics)

Reflection Groups and Coxeter Groups (Cambridge Studies in Advanced Mathematics)

Real Analysis and Probability (Cambridge Studies in Advanced Mathematics)

Lévy Processes and Stochastic Calculus (Cambridge Studies in Advanced Mathematics)

Real Analysis and Probability (Cambridge Studies in Advanced Mathematics)

Models and Games (Cambridge Studies in Advanced Mathematics 132)

Spectral Theory and Differential Operators (Cambridge Studies in Advanced Mathematics)

Automorphic Forms and Representations (Cambridge Studies in Advanced Mathematics)

Ip Studies in Advanced Mathematics)

Ip Studies in Advanced Mathematics)

Cellular Structures in Topology (Cambridge Studies in Advanced Mathematics 19)

Commutative Ring Theory (Cambridge Studies in Advanced Mathematics 8)

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