Banach spaces for analysts

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 25 EDITORIAL BOARD D. J. H. GARLING , D. GORENSTEIN, T. TOM DIECK , P. WALTERS

Banach spaces for analysts

Banach Spaces For Analysts

P. Wojtaszczyk Institute of Mathematics, Polish Academy of Sciences

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CAMBRIDGE UNIVERSITY PRESS Cambridge

New York Port Chester Melbourne Sydney

Puhlished by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 100 1 1 , USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia ©Cambridge University Press 1991 First published 1991 Printed in Great Britain at the University Press, Cambridge British Library cataloguing in publication data

Wojtaszczyk, P. Banach spaces for analysts. 1 Banach spaces I. Title 515.732 Library of Congress cataloguing in publication data available

ISBN 0 521 35618 0 hardback

Contents ix

Preface

Part I. Introduction I.A. LB.

Functional analysis Examples of spaces and operators

3 9

Part II. Basic concepts of Banach space theory II. A.

II.B.

II. C.

II.D.

II. E.

Weak topologies Weak and weak* topologies, Mazur's, Alaoglu's and Goldstine's theorems, reflexive spaces Isomorphisms, bases, projections Banach-Mazur theorem, block-bases and selection of block basic subsequences, examples of bases: Haar, FaberSchauder and trigonometric systems, small perturbations of basic sequences, decomposition method Weak compactness Eberlein- S mulian theorem, weakly compact operators; their factorization and basic properties Convergence of series Definitions and characterizations of unconditionally and weakly unconditionally convergent series, Orlicz theorem and Orlicz property, unconditional bases; spaces without unconditional basis, the Haar system is an unconditional basis in Lp for 1 < p < oo Local properties Bounded approximation property and 71",>.-spaces, BanachMazur distance and distances between £; -spaces and spaces of trigonometric polynomials, good c-nets in Auerbach lemma, principle of local reflexivity

27

35

49

57

69

Bx,

Part III. Selected topics liLA.

Lp-spaces; type and cotype Sobolev spaces ( 1 < p < oo ) and Bergman spaces (1 � p < oo ) are isomorphic to Lp-spaces, complemented subspaces of lp, stable laws and embeddings of lp into

83

vi

Contents

Lq , 0 < q � p � 2, type and cotype, Kahane's inequality, types and cotypes of Lp-spaces, Carleman theorem, Banach-Saks theorem. III.B Projection constants Leo is injective, extension and projection constants, Lewis's theorem on projections in Lp, Kadec-Snobar theorem, projection constants of various spaces of polynomials, existence of an inner function on the ball in ([d , polynomial bases in C( 'l') , projective tensor products and dual treatment of extensions of operators III.C. L1 { p,)-spaces Semi-embeddings and the non-existence of semiembeddings of L1 [0, 1) into Co, Menchoff theorem, Lyons theorem characterizing zero sets for measures with Fourier transforms tending to zero, Schur theorem that weakly compact sets in £1 are compact, uniform integrability and characterization of weakly compact sets in L1 ( p,) , weak sequential completeness of L1 (p,) , finite representability of £1 in a Banach space; connections with type, reflexive subspaces of L1 ( p,) , Nevanlinna theorem on unimodular functions in cosets of Leo/Heo III .D. C(K)-spaces M-ideals and best approximation, Heo +Cis a closed algebra, {Heo + C) /Heo is an M-ideal in Leo/Heo, Michael-PelczyD.ski linear extension theorem, Milutin theorem, Franklin system - construction and basic properties, Lipa ( 11') is isomorphic to leo, Dunford-Pettis property for rich subspaces of C(K), in particular for ball algebras and Sobolev spaces III. E. The disc algebra Rudin-Carleson theorem, interpolating sequences, interpolation of continuous functions by functions with finite Dirichlet integral, lacunary Fourier coefficients of functions in the disc algebra, projections in and infinite divisibility of existence of basis III.F. Absolutely summing and related operators Absolutely summing operators, Grothendieck's and Pietsch's theorems, unconditional bases in L1-spaces, local units in L1 (G) , Grothendieck 's inequality, polynomially bounded but not power-bounded operators, p-nuclear and p-integral operators, trace duality, characterization

A,

111

131

151

181

A

199

Contents

III.G.

III.H.

III.I.

vii

of Sidon sets, extrapolation inequalities for spaces of all p-absolutely summing operators, Grothendieck-Maurey theorem, p-absolutely summing operators and cotype 2 Schatten-von Neumann classes Approximation numbers of operators between Banach spaces, Weyl's inequality, Schatten-von Neumann classes and eigenvalues, Hilbert-Schmidt operators, Fourier coefficients of Holder functions, summability of eigenvalues of p-absolutely summing operators between Banach spaces with applications Factorization theorems Nikishin's theorem and some applications, Maurey's factorization of operators into Lp, reflexive subspaces of L 1 , connections between factorization and p-absolutely summing operators, structure of series unconditionally convergent in measure, dilation theorem and connections with orthogonal series, Menchoff-Rademacher theorem on almost everywhere convergence, more about Fourier coefficients of Holder functions, some multipliers into Bergman spaces Absolutely summing operators on the disc algebra Proper uniform algebras are uncomplemented, Bourgain's analytic projections onto Hp(ll.d>.), extrapolation inequalities between p-absolutely summing and qintegral norms for operators on has cotype 2 and the Grothendieck theorem holds for extensions of @ finite rank operators on is closed in C (Y) @C (Y) , reflexive subspaces of Ll/ H1 . extrapolation inequalities for the Riesz projection, extension of operators from reflexive subspaces of Ll / H1 into Hoc, interpolation for certain hi-analytic functions

237

257

291

A, A* A*, A, A A

Hints for the exercises List of symbols References Index

323 341 345 377

P reface Banach space theory became a recognised part of the mathematical scene with the appearance of Banach [1932). From its birth it maintained close ties with the rest of analysis. It turned out that Banach space theory offered powerful tools to other branches of analysis. The most useful ones are duality theory for spaces and operators, results about infinite dimensional convexity and results connected with the Baire category theorem, most notably the closed graph theorem. These powerful gen­ eral concepts are now well known among analysts and appear in almost every textbook on functional analysis or real variable theory. They were already well understood in the forties and fifties, and at that time it seemed to many that Banach space theory was dead or at least rele­ gated to an obscure corner of science. However, this was not the case. The sixties, and especially the seventies and eighties, saw an enormous eruption of activity in the field. Old problems were solved and, more importantly, new problems and new ties with the rest of mathematics emerged. Also, new and powerful methods and directions of research appeared. For those in the field progress seemed to be constantly accelerating, but to those outside it may have looked as if the theory was dying again. Probably one of the reasons for this perception was that this research activity (maybe because it was so dynamic ) unfortunately did not pro­ duce many books on the subject, and those that did appear were usually devoted to a special topic. There are two notable exceptions to this state­ ment, Beauzamy [1982) and Lindenstrauss-Tzafriri [1977, 1979). There are old favourites, still beautiful and in good shape like Banach [1932), Dunford-Schwartz [1958) and Day [1958 and newer versions) , but nat­ urally they do not present the more recent results. So the newcomer to the field, after learning what was generally known thirty years ago, had a choice of either starting on more specialized books or turning to Beauzamy or Lindenstrauss-Tzafriri. Now I like both these books and the present one is not intended to replace either of them. Beauzamy is a nice, easily readable introduction and Lindenstrauss-Tzafriri, although difficult in places for the novice, has every mark of a classic, especially if the long-promised volumes III and IV are added to the first two. But both these books have one feature in common: they study Banach spaces for their own sake and their own beauty. In this way they are great for

X

Preface

the future specialist who is already under the spell of Banach spaces. However, there are some mathematicians (even a majority) who are not convinced that Banach space theory is the most enchanting branch of mathematics, and although I am not one of them, I understand them. In fact, Banach spaces are not only beautiful and interesting, but also useful. This point is not, in my opinion, made clear in Lindenstrauss-Tzafriri or Beauzamy. The methods created in Banach space theory since the late 60's can be applied in other areas of analysis. Hence somebody interested primarily in harmonic analysis, functions of a complex variable, orthonormal series, approximation theory or proba­ bility theory can find it useful to know some Banach space theory. This book is directed towards such a person. Ideally I think of it as a text­ book for a graduate course for students who have already learned some functional analysis and are interested in analysis or in some area of it. I also hope that a mature analyst may read it, or some of it, as part of the ongoing education process which is an important part of the life of any active mathematician. I hope too that Banach space specialists will find some portions of the book interesting because they present some applications of the subject I was not aware of. Let me digress a bit and comment on the possible profit a classical analyst might derive from Banach space theory. I do not claim that Banach space theory can solve all his important problems. But it may help him to see the problem in a new light which makes it easier to isolate essential features. He can also use general theorems and techniques in the solution. The general framework can also suggest interesting new problems. To mention one example, the power of duality methods makes it a standard procedure to try to find the dual of any Banach space considered, a problem that could not even arise without this more general framework. Another example is the corona theorem. It is now a major part of the theory of analytic functions, but its origin lies in the attempt to get some understanding of the maximal ideal space H00 , a question which is unthinkable without the general theory of Banach algebras. Mathematics and each of its parts lives and grows on the exchange of ideas, between mathematicians, between various branches of math­ ematics and between mathematics and other areas of human activity. Banach space theory is no exception to this rule. It utilises ideas and techniques from other fields and in turn provides other branches of math­ ematics with some of its own. In this respect this book is one-sided. It concentrates heavily on ideas and results that have a clear potential to be useful in other branches of analysis.

Preface

xi

Formally the book is divided into three parts, numbered by roman numerals. Each of these parts is divided into chapters, distinguished by capital letters. Each chapter is divided into sections with arabic num­ bers. Each such section contains at most one Theorem, Proposition or Lemma. The Theorem appearing in section II. B. 7 is later referred to as Theorem II.B.7, or within the same chapter just as Theorem 7 (or Lemma or Proposition as the case may be) . Part I has an introduc­ tory chapter. It contains well known and some less well known results (without proofs) that will be used later. Chapter LA contains basic results from general functional analysis and Chapter I.B discusses the examples of Banach spaces that are considered later and quotes some analytical results about these spaces. The main function of Part I is to establish notation and conventions and to serve as a refresher and refer­ ence for the background material needed later on. Part II is essentially an introduction to the language and basic techniques of Banach space theory. The real heart of the matter is Part III, where a selection of topics is studied in depth. The reader can get an idea of the contents of each chapter from the Table of Contents, so I will not attempt a more detailed description here. Also, each chapter of Parts II and III starts with a short description of its contents, so the reader can find additional details there. Each chapter concludes with Notes and Remarks containing biblio­ graphical data and comments about generalizations, ext ensions or appli­ cations of the results presented in the chapter. I have tried really hard to find the correct reference and credits. On the other hand, I have not conducted a full scale historical investigation into the origin and devel­ opment of each idea and result. I would like to offer sincere apologies for any omissions and inaccuracies in this respect. In the main text theorems are only given names if it is common usage. The absence of a name in the text or of a credit in the Notes and Remarks does not mean that the result is due to the author. It means either that the result is so well known that I judged it to be folklore or simply an omission on my part. Each chapter of Parts I and II contains exercises. These exercises range from routine to very hard and I have not given any indication of their di fficulty. There is a hint for each exercise located in the special chapter at the end of the book. These hints range from almost complete solutions to the reference only. I have tried to point out, if possible, where the solution of an exercise can be found in the literature and to give proper credit. There are also some repetitions in the exercises. I have simply put the same or similar problems into different chapters if

xii

Preface

they fit well into the material. This should be useful for those read­ ers (the majority?) who read only selected parts of the book. It also indicates different approaches to the same question. The bibliography contains only the works actually referred to some­ where in the book. I have made no attempt to make it complete. Also, it does not include any data about translations (this is particularly impor­ tant with respect to publications in Russian) or reprints and republishing (it happens sometimes that an East European book is published origi­ nally in English and later republished without any changes by a West European or American company) . If the work appeared in Russian, this is indicated in the references and the author 's English translation of the title is given. This translation should be close to the one used in Mathematical Reviews, but need not be identical. As with most mathematics books, it will be unusual for a reader to read this book from beginning to end. It is also unnecessary. The reader interested in a particular theorem or chapter should start right there. The choice of material in the book reflects my philosophy, taste and, last but not least, knowledge and ability. Here I would like to mention some subjects which really should have been included but which were not because of limitations of space and time and (probably most impor­ tant) my poor understanding of them. The first is the deep connections between Banach spaces, descriptive set theory and the classical theory of sets of uniqueness for trigonometric series (see Kechris-Louveau [1987] and Lyons [1989] ) . The second is the applications of the study of finite dimensional spaces to problems on convex bodies. This in turn has ap­ plications to harmonic analysis, number theory and other subjects. This whole area is currently one of frantic activity and is undergoing constant and fascinating changes. Probably anything I could write about it now would be outdated by the time this book reaches the reader. As an in­ troduction to this area I suggest Pisier [P] , Milman-Schechtman [1986] or Milman [1986] . The next subject which I regrettably had to omit is the connection between Banach space theory and probability theory. Actually probabilistic methods underly much of the current research in Banach spaces. This shows even in this book despite my unfortunate lack of knowledge of probability theory. The study of probability in Ba­ nach spaces is developing too: see Linde [1983] . The last subject I would like to have included is the general area of vector valued functions and operators acting on such functions. There is considerable activity in this area as well. As an introduction to it I suggest Burkholder [1986] and Pisier [P 1] .

Preface

xiii

While writing the book I received helpful advice from many math­ ematicians. I am grateful to all of them for their time, advice and help. First of all I would like to express my profound gratitude to my teacher and colleague, Prof. Aleksander PelczyD.ski. He convinced me that I should write the book in the first place and offered plenty of advice on what it should contain. Much of the time I did not follow his advice, but the effort needed to refute his arguments helped greatly to clarify my own ideas. I would like to thank Prof. Keith Ball and Prof. Ben Garling for reading large parts of the manuscript and providing numerous and invaluable pieces of advice on language and presentation. The following other mathematicians helped me greatly by generously offering their ad­ vice, knowledge and insight: Dan Amir, Sheldon Axler, Don Burkholder, Joe Cima, John Fournier, Ben Garling, Nassif Ghossoub, Yehoram Gordon, Pawel Hitczenko, Bill Johnson, Serguey Kislyakov, Boris Kashin, Stanislaw Kwapien, Elton Lacey, Joram Lindenstrauss, Niels Nielsen, Gilles Pisier, Richard Rochberg and Walter Schachermayer. Some work on the book was done during visits to St. John's College Cambridge; Monash University, Clayton; University of British Columbia, Vancouver; Odense University and Texas A &M University, College Station. I would like to thank Ben Garling, Ala Sterna-Karwat, Nassif Ghossoub, Niels Nielsen and Elton Lacey for arranging these vis­ its and for being great hosts, each in his or her unique way. Clearly most of the actual work was done in Warsaw where my own Mathemat­ ical Institute of the Polish Academy of Sciences gave me the freedom to do the job and an excellent library to help me. The final version was written, and all typing done, at Texas A&M University. I would like sincerely to thank Robin Campbell for her unique ability to transform my scratchy and unreadable (sometimes even to myself) handwriting into the beautifully typeset text. Finally I would like to express my deep gratitude to my wife Anna and my children Ola and Kuba for all the love, support and distractions they generously provided over the years. Without their presence (and at suitable times their absence!) writing this book would have been much more di fficult.

Part I Introduction This introductory part contains background material. It is not intended to be a course in any subject. It is simply a collection of definitions and facts given without proof (with one exception) . We provide references to works which contain detailed exposition, full proofs, examples and motivation. In a sense this whole part is a quick reference guide to re­ sults which will be used in the later parts. The introduction is divided into two chapters, Chapter LA describing what we will need from gen­ eral functional analysis and Chapter I.B which contains results about concrete spaces and operators. Since I.A is really a review of a standard course in functional analysis, references are given only at the end of the chapter. In I.B we give references after each paragraph. The references given in this part are usually to standard textbooks and monographs, not to original works. If a particular result or subject cannot be easily located using the table of contents or index we try to provide more detailed information (sections or pages) . Sometimes we formulate a result in a form which is more convenient to us but different from the one given in the reference. Usually in such a case it is easy to derive our formulation from the one given in the reference.

I.A. Functional analysis 1. A linear topological space X is a linear space over the real or complex numbers endowed with a topology T such that the map (x, y) �--+ x + y is continuous from (X, r ) x (X, r ) into {X, r ) and the map (t, x) �--+ tx is continuous from R x X (or CC x X) into X. Such a topology is fully described by a basis of neighbourhoods of 0. A subset V c X is called convex if whenever x �, x2 E V then the whole interval ax1 + { 1 - a)x2 for 0 ::; a ::; 1 is in V. A linear topological space is called locally convex if it has a basis of convex neighbourhoods of 0. A functional on X is a continuous linear map from X into scalars. The set of all functionals on X will be denoted X* , and called the dual space. A linear operator (or just operator) T: X --+ Y {where X and Y are linear topological spaces) is a continuous linear map. A subspace of X will always (unless explicitly stated otherwise) denote a closed linear subspace. Given a set V C X by span V we denote the closure of the set of all linear combinations of elements from V (i.e. the subspace of X spanned by V).

2.

A linear topological space X i s called an F-space i f its topology is given by a metric p such that p(x, y) = p(x-y, 0) and X is complete with respect to this metric. A quasi-norm on a linear space X is a function q from X into the nonnegative reals satisfying (a)

q(x) = 0 if and only if x = 0,

q(Ax) = I Ai q(x) for all scalars A and all x E X, (c) there exists a constant Cx such that q(x + y) ::; Cx ( q(x) + q(y)) for all x, y E X.

{b)

One easily checks that each quasi-norm defines a linear topology on X. The basis of neighbourhoods of the point x consists of 'balls' around x , i.e. sets {y E X: q(x-y) < e }, e > 0. A linear space X with a quasi-norm will be called a quasi-normed space. A very important special case of quasi-norm is a norm. This is a quasi-norm on X for which the constant in (c) above equals 1 (one gets from {b) that � 1 ) . The usual notation for the norm of x is ll x ll . Every norm 11 · 11 on X defines a metric p (x, Y ) = ll x - Yll · A Banach space X is a linear space equipped with a norm II · II and such that X is complete with respect to the metric p. Clearly every Banach space is an F-space. The symbol will denote

Cx

Cx

Bx

4

l.A.

Functional analysis §3.

the closed unit ball of X, i.e. Bx = {x E X : l l x l l $ 1 } . Its interior is an open convex set, so every Banach space is locally convex. This need not be true for general quasi-norms. Note also that a subspace of a Banach space {resp. quasi-normed space) is a Banach space (resp. quasi-normed space) . We simply have to restrict the norm {resp. quasi-norm) . Let X and Y be two Banach spaces and let T: X -+ Y be a linear map. Then T is continuous if and only if II TII = sup{ II Tx l l : l l x l l $ 1 , x E X} < oo. The quantity II Til is a norm on the linear space L(X, Y) of all operators from X into Y. The space L(X, Y) with the above defined norm is a Banach space. Unless otherwise indicated the convergence of operators will be understood in this norm.

3.

In particular, for a Banach space X the space X* is also a Banach space with the norm of a functional x* E X* defined as llx* ll = sup{ lx* (x) l : x E X, l l x l l $ 1 } .

4.

Let X and Y b e Banach spaces and let T: X -+ Y be a linear operator such that T(Bx ) contains some open ball in Y. Then T(X) = Y and there exists a positive number r such that T(Bx ) :::> r · y = {y E Y: II YII < r} .

5 . Open Mapping Theorem.

B

6. Closed Graph Theorem. Suppose that T: X -+ Y is a linear map (not assumed to be continuous, but defined everywhere on X) from an F-space X into an F-space Y. Assume that { (x, Tx) : x E X} C X x Y is closed in the product topology. Then T is continuous. 7. Banach-Steinhaus Theorem. Suppose (T-y ) -yer is a family of linear operators from a Banach space X into a Banach space Y. As­ sume that for every x E X we have sup{II T-yxii='Y E r} < oo. Then sup{ II T'YII ='Y E r} < oo . In particular we get that the pointwise limit of a sequence of linear operators {if it exists everywhere) is a linear operator. Let X be a linear space over the real numbers (without any topology) and let Y C X be a linear subspace. Assume also that we have a function p: X -+ R such that p( x + y) $ p(x) + p(y) for all x, y E X and p(tx) = tp(x) for all x E X and t E R, t � 0. Assume moreover that we have a linear map f: Y -+ R such that J(y) $ p(y) for all y E Y. Then there exists a linear map F: X -+ R such that F I Y = f and -p( -x) $ F (x) :::; p (x) for all x E X .

8. Hahn-Banach Theorem.

l.A. Functional analysis §9.

5

This general algebraic theorem has many very important special cases. Some of them are stated below in 9-11. Note that despite the fact that 8 is true only for real spaces, the consequences listed below are true also for spaces over the complex scalars. Taking in 8 p(y) = IIYII we obtain: If X is a Banach space and Y C X is a subspace and if y* E Y* , then there exists x* E X* such that ll x* II = IIY * I and x* I Y = y* . In particular we get 9.

ll x ll = sup{ l x * (x) l : x * E X* , ll x* ll � 1 } . A judicious choice of p yields also the following: If X is a locally convex space and B C X are disjoint closed convex sets with being compact, then there exists a continuous linear functional I on X and a real number a such that Re I < a and 10.

Re

I(B)

> a.

A

A,

(A)

In particular we see that X* separates the points of X.

A, I(A)

We also have the following version of 10 . If X is a locally convex space and B c X are disjoint convex sets with open, then there exists a continuous linear functional I on X and a real number a such that Re < a and Re I(B) 2: a. 11.

A

12. If T: X --+ Y is a linear operator then it induces an operator T* : Y* --+ X* , called the adjoint (or dual) operator and defined as T* (y* ) (x) = y* (Tx). One easily checks that II T II = li T* II · 13. If T: X --+ Y is onto then T* is 1-1 and there exists a constant c > 0 such that II T*y* ll 2: cii Y * II for all y* E Y*.

If T: X --+ Y is such that there exists c > 0 such that II Tx ll for all x E X then T* maps Y* onto X*. 14.

2:

cllx ll

15. A linear map T: X --+ Y (X, Y Banach spaces) is compact if the set T( Bx) is compact in the norm topology of Y. One easily checks that each compact map is automatically continuous. The set of all compact operators from X into Y, denoted K(X, Y), is a subspace (i.e. closed and linear) of L(X, Y) . Also an operator T: X --+ Y is compact if and only if T* : Y* --+ X* is compact. It is easy to see that if T E K(X, Y) and 81 E L(ZI. X) and 82 E L(Y, Z2 ) then S2 TS1 E K(Z1 . Z2 ) .

I.A. Functional analysis § 1 6.

6

An operator T: X -+ X is called power-compact if Tn is compact for some n E N. 16.

An operator T: X -+ Y is called invertible if there exists an 17. operator S: Y -+ X {usually we write r - 1 instead of S) such that ST = idx and TS = idy. By idx (or idy) we mean the identity operator on X (or on Y), i.e. idx (x) = x for all x E X. It is important that we need both conditions ST = idx and TS = idy; one of them is not enough. If X is a complex Banach space (i.e. it is a linear space over the complex numbers) and T: X -+ X is a linear operator, then the spectrum of T, denoted u(T) , is the set of ali A E cr such that ( >-.idx - T) is not an invertible operator. The set u(T) is a non-empty, compact subset of cr. A number ).. E cr is called an eigenvalue of T if there exists a vector x E X, x -=F 0, called an eigenvector associated with the eigenvalue ).. , such that Tx = >-.x . Clearly each eigenvalue of T belongs to u(T) . With each eigenvalue ).. we associate its spectral manifold E>.. = E>.. ( T) = Un>l ker( >-.i dx - T) n . Clearly E>.. is an increasing union of subspaces of X. The number dim E>.. (possibly oo ) is called the multiplicity of the eigenvalue .>-.. 18.

19. If an operator T: X -+ Y is power-compact then u(T) is finite or consists of a sequence of points tending to zero together with zero itself. Every point ).. E a(T) , except possibly zero, is an eigenvalue of T of finite multiplicity. 20. An operator P: X -+ X is called a projection if P2 = P. Then P ( X ) is a closed subspace of X and Px = x for x E P(X ) . A subspace

Y C X which equals P(X) for some projection P: X -+ X is called complemented.

21. Let V be a convex set in a locally convex topological vector space X. A point v E V is called an extreme point if it is not in the interior of any closed interval contained in V, i.e. if v 1 , v2 E V and v = av1 + (1 - a)v2 with 0 < a < 1 then v = v 1 = v2 . If is any subset of X then the convex hull of denoted conv equals

A A, A, n n n x E X: x f;ajaj with f;aj = f; ia31 1 { and a3 E A for 1, 2, . .. EN · =

=

j

=

, n, n

}

I.A. Functional analysis §22.

7

22. Krein-Milman Theorem. If V is a compact, convex subset of a locally convex, topological vector space, then V equals the closure of the convex hull of its extreme points. In particular this theorem implies that each convex, compact subset of a locally convex, topological vector space has an extreme point. References. Everything said above can be found in most textbooks on

functional analysis. In particular everything can be found in Dunford­ Schwartz [1958] or Edwards [1965] and everything except power-compact operators can be found in Rudin [1973] .

I.B. Examples of spaces and operators 1. Whenever we consider a measure space {0, E, J.L) we assume that the measure J.L is complete and that there are no atoms of infinite measure. We say that the measure space (0, E, J.L) is separable if there exists a countable family of sets C E such that the smallest complete is E. The following characterizes separable u-field containing measure spaces. Suppose that {0, E, J.L) is a separable non-atomic measure space, with J.L a positive measure and J.L (O) = 1 (i.e. J.L is a probability measure) . Then (0, E, J.L ) is isomorphic to the unit interval (0, 1] with the Lebesgue measure. This can be found in Halmos (1950] §41 . A general classification theorem for arbitrary measure spaces is due to Maharam (1942] and can also be found in Lacey (1974] § 14.

(Aj)� (Ai)�1 1

J

For any measure space {0, E, J.L) with J.L positive we define Lp(O, E, J.L) , 0 < p :5 oo, to be the space of E-measurable functions {more precisely of classes of functions where we identify functions which are equal J.L-a.e.) such that Jn l f(w ) I P dJ.L (w ) < oo. For p = oo we mean supess l f(w ) l < oo. Usually the u-field E is clear so we will use the notation Lp (O, J.L) and when either the measure or the set are clear from the context we will suppress them also. We will use the notation I f li P = 0 such that

H c r is called a Sidon set if there

for every sequence of scalars ( a-y ) -yEH· The details can be found e.g. in Rudin [1962a] .

15.

The Dirichlet kernel is defined as

n

'Dn(O) =

L

k= - n

e ik9 ,

n = 0, 1 , 2, . . .

Jt E M(11') and Jt = L::�: P,(k)eik!J then 'Dn * Jl = k L;":_nP,(k)ei !J. This means that the convolution with the Dirichlet ker­

If we have

nel realizes the partial sum projection with respect to the Fourier series. We have r I'Dn(O) Idm(O) = 42 log(n + 1) + o( 1 ) .

I�r

1r

Note that this equals the norm of the operator f 1-t f * 'Dn in the spaces £1 ( 11') and C(11') . This operator will sometimes be called the Dirichlet projection. For details see Zygmund [1968] , Katznelson [1968] p. 50, Kashin­ Saakian [1984] or any book on trigonometric series.

16.

The Fejer kernel is defined as

Fn(O)

1 = - n+ 1

) ik!J. t vk(O) = tn ( 1 - J!1._ n+ 1 e

k=O

k=-

One checks that Fn(O) ?: 0 for n = 0, 1 , 2, . . . and so J1r IFn(O) Idm(O) = JyFn(O)dm(O) = 1 . If f E L p( 11') , 1 � p < oo, or f E C(11') in the case p = oo, then II/- f * Fnllp ---+ 0 as n ---+ oo. The norm of the Fejer operator, f 1-t f * Fn, is 1 in spaces Lp ( 11') , 1 � p � oo . The details can be found in Zygmund [1968] , Katznelson [1968] p. 1 2, Kashin-Saakian [1984] or in any book on trigonometric series.

17. The de la VallOO-Poussin kernel is defined as Vn = 2F2n-l -Fn-l· Clearly IIVnll1 � 3 and II/-Vn* fliP---+ 0 as n---+ oo for f E L p( 11') , 1 � p < oo, or f E C(11'). This follows immediately from 16. The nice

17

I.B. Examples of spaces and operators § 1 8.

property of the de la VallOO-Poussin kernel is that for f = L�n ajeii 9 we have Vn * f = f. All this follows easily from properties of the Fejer kernels. The details are in Zygmund [1968] and Katznelson [1968] .

18.

The Poisson kernel is defined as

+oo - oo

( 1 - r2 ) Pr(8) = :�:::> l i l eijfJ = (1 2r - cos 8 + r2 )

for

0 < r < 1.

Clearly Pr (8) � 0 and J'lf Pr(8)dm(8) = 1 for 0 < r < 1. As for the Fejer kernel we have II/ - f * Pr ll p - 0 as r - 1 for f E Lp(1l'), 1 � p < oo , or f E C(1l'). The important feature of the Poisson kernel is that for J.L E M(1l'), Pr*J.L can be treated as a function on D by h(rei 9 ) = Pr*J.L( 8) . Such a function is always harmonic i n D. Basic properties of the Poisson kernel can be found in Zygmund [1968] , Katznelson [1968] , Hoffmann [1962] , Duren [1970] , Koosis [1980] and in many other places.

19.

The Hardy spaces Hp(D),

{

Hp(D) = f(z): f(z)

11/IIH, =

0 < p � oo , are defined as

is analytic in

D and

)

�

1 2� f(rei9 ) P d8 " 1 l ��� 271"

( 1


-oo. function. Let h E L 1 It is known that for f E Hp ( ) , 0 < p � oo, 1!1 satisfies these assumptions. We take the harmonic conjugate of log h(t) and form the analytic function H(z) = exp(log h + ilog h) .

1I'.

(1I'),

1I'.

1I'

One checks that limr-+1 H(rei 6 ) exists almost everywhere and has mod­ ulus h on Applying this to l f(ei 6 ) 1 , for f E Hp (ID) we get the outer function F(z). This function is in Hp (D) and en Y we have l f(ei 6 ) 1 = I F(ei6 ) 1 a.e. Thus we can write every f E Hv (D) aB f = B · F · l where B ( z) is the Blaschke product formed from the zeros of f (counting multiplicity) and F is an outer function with I F(ei 6 ) 1 = l f (e i6 ) 1 a.e. on The remaining factor I can be defined simply as f · (B ·F)- 1 . It is an analytic function with ll l lloo = 1 and I I(ei 6 ) 1 = 1 a.e. on T. This factor is called a singular inner function (if it is not a constant) . In general an inner function is a function g in H00 ( ) such that l g (ei 6 i = 1 a.e. on Thus the Blaschke product is also an inner function, and a singular inner function is a zero-free inner function (non-constant). The singular inner functions have the representation

1I'.

1I'.

1I'

1I'.

1 I(z) = exp - 271'

!11' -.e•i-6 -d + p, (O) - 1r

z

e6-z

where p, is a positive, singular measure on [ - 11' , 11' ] . The a)ove described factorization f = B · F · I is called the canonical factcrization. The

21

I. B. Examples of spaces and operators §24.

outer functions provide the tool to build analytic functions in Hp(D) with given modulus on 1l'. This will be used extensively in III.I. Detailed proofs of the canonical factorization can be found in Hoff­ mann [1962] , Duren [19 70] , Koosis [1980] and Garnett [1981] .

24. The following inequality due to R.E.A.C. Paley will be very useful. Suppose that f = E:=o an z n E H1 ( D) . Then

Since this inequality will be used later to prove Grothendieck's The­ orem III.F.7. which is the basis for many results presented in chapters III.F -I, I have decided to present sketches of the proofs here.

Proof A. (Paley [1933] )

It follows from 23 that one can write

1 1 f(z) = (E:=o bn zn ) (E:=o en zn ) where fE:=o lbn l 2 ) 1 2 (E:=o l en l 2 ) 1 2 = ll f ll 1 · From this we infer that a2 k = L s ,r: s+ r= 2 k c8 br, so

00

00

� 4 L lcs l 2 L lbs l 2 = 4 11 ! 11 � s=O s=O Proof B. (Smith [1983] ) define inductively

Let

o:k

=

10- 1 a2 k (E� 1 l a 2• l 2 ) - 1 / 2

91 = o: 1e2i9 k k 9k = 21 o:k e2 i9 + (1 - lo:k l 2 )9k - 1 - 21 o:k e - 2 i9 9k2 - 1 · -

and

22

-

I. B. Examples of spaces and operators §25.

D we have I� + (1 - I W)a1

I

a;b � 1 we see that IIYk I � 1. One checks that for l � k we have Yk (2 ) = (1 - la1 l 2 )(1 - l a2 l 2 ) · · · (1 ln1 - 1 l 2 )a 1 and Yk (s) = 0 for s � 0 and s =F 21 . Since L� 1 1ak l 2 � 100- 1 this product is bounded away from 0, so 2 .. ll f ll 1 � 2 Yk f k 2 = L (1 - l a1 1 ) · · · (1 - l a 1 1 l 2 ) a1 a 2 t 1=1

Since for a, b

E

� �1 l

�c Since this holds for all

(t, I a,. 1 ')"'

k we have the claim.

We will also use the following inequality due to Hardy. If f = then I: :'= o � 1r ll f l h · The proof can be found in Duren [1970) p . 48, Hoffmann [1962) p. 70, Katznelson [1968] p. 91. The inequalities of Paley and Hardy fail for £ 1 ('1'). This is one possible way to show that the Riesz projection is not continuous on

25.

l: :'=o anzn E

H1(D)

(�a+!)

£ 1 ('1').

A(D)

D

The disc algebra is the space of all functions analytic in which admit a continuous extension to D. We can identify with the subspace span{ e i n9 }n�o C C('l'). This is the space of all functions f E C( ) such that j(n) = 0 for n = - 1, -2, . . . . It will be denoted by The symbols Ao(D) and Ao('l') will denote the subspace of A(D) consisting of functions vanishing at 0 E D. The annihilator of C C(T) is described as follows.

26.

T A(T). A('I')

F.-M. Riesz Theorem. If J.L E M('l') and JT fdJ..L = 0 for all f E A('l') then J..L is absolutely continuous with respect to the Lebesgue measure. Consequently df..L = hdm with h E

Hf('I').

The proof can be found in Hoffmann and Katznelson [1968) p. 89.

[1980]

D, Hp (D) Hp (D),

[1962] ,

Duren

[1970) ,

Koosis

and z E then the map f �--+ f(z) defines a If f E linear functional on 0 < p � oo. The norm of this functional is estimated in the following simple inequality

27.

I.B. Examples of spaces and operators §28.

The proof is in Duren

[1970]

23

p. 36.

V E ([n is an open set, then we can define the Bergman space Bv (V), 0 < p as the space of all holomorphic functions in Lp (V ) , where on V we have 2n-dimensional Lebesgue measure. We will be mostly interested in the case V = D or sometimes V = IBn and V = Dn . In all those cases Bv (V) is complete. Also the point evaluations are continuous and we have inequalities of the form: for every compact set K V there exists a constant C = CK such that l f (z) l C l f l v for z E K. We will also discuss the ball algebra A ( IBd ) which is the space If

28.

:::;

oo

:S:

C

of all functions analytic in IBd C ([d which have a continuous extension to IBd. The polydisc algebra A(Dd ) is the space of functions analytic in Dd which admit a continuous extension to Dd . The space A(Dd ) can d) . be naturally identified with a subspace of A good reference is Rudin [1980] where the case IBn (so in particular D) is discussed in detail. Also Axler [1988] discusses in detail. For the polydisc the standard reference is Rudin [1969] .

C(1I'

f

Bp (D) (K

f

, p) , we say 29. If is a scalar valued function on a metric space that satisfies the Holder condition of order a, 0 < a :::; 1 (for a = 1 the term Lipschitz condition is also used) if

k K

l f (k) l 'Po:U) Lipo:(K,

Clearly for every E the quantity is a norm and + the corresponding Banach space of all functions satisfying the Holder condition of order a is denoted by p) . We will consider only being or [0,1] with the natural metric. One can also replace the function in the definition by a more general function.

1I',R. o: t-

K

Now we want to discuss the Sobolev norms which measure the smoothness of functions of several variables. The theory is usually done in a much more general setting, but we restrict our attention to functions on s = 1, 2, . . . . Let = 0, 1, 2, . . . be the set of all multiindices A= such that the j ' s are nonnegative integers and p A) = z::; = l :S: Each A E defines a partial derivative denoted OA . We define the Sobolev space for k � 0, s � 1, to be the space of all functions on such that oA is continuous for all A E The norm which makes into a Banach space can be defined as

30.

1I'8 , (i1. i2 , . . . , is ) ij k.

D(k), k i D(k) k 8 C (1I' ) ys k 8 f C (1I' )

11 ! 11 � = sup { i i &Aflloo : A E D(k) } .

(

D(k).

24

I.B. Examples of spaces and operators §31 .

If we want to extend this definition for p < oo we encounter some techni­ cal problems, so to avoid them, we first define the norm on functions as

11 / 1 1 ;

= ( AED(k) L I 8A f l � )

c=

1

p.

By the Sobolev space w; (1fB ) we understand the completion of the functions under this norm. Quite often we will treat as a subspace of where V is the disjoint union of copies of11' 8 • (Here the modulus signs represent cardinality.) The natural isometric injection is defined by the rule that on the torus 1rs which is a subset of V with the index E is simply We can also treat as a subspace of E) , the space of continuous, E-valued functions on 11'8 where E is the Banach space The natural injection maps � The same operations work also for w; ( 11'8 ) and we can treat w; (11'8 ) as a subspace of Lv (V) (V the same as above) or a subspace of Lp ( 11's ; The standard reference for the theory of Sobolev spaces is Adams [1975] . It is also presented in Stein [1970] and Stein-Weiss [1971] .

c=

Ck (1fB)

C(V) I D (k) l j: Ck (11'8 ) ---+ C(V) j(f) A D(k) 8A f· kC (1fB) 8 C(11' ; t'�(k) l . f (8A f(t)) AED(k)· f1D(k) l ) . c=

31. We have defined w; ( 11'8 ) as the completion of the functions under a certain norm. Nevertheless it is interesting to know up to what degree the elements of w; ( 11'8 ) are really functions. This is answered by the so-called embedding theorems. We will use the simplest one.

c=

functions ex­ The identity on tends to a continuous, linear operator from w; ( 11'8 ) into w; - m ( 11'8 ) where 0 :=:; m :=:; and mp < s and p :=:; q :=:; (s �:T.v ) . In particular w; ( 11'8 ) embeds into Lq ( 11'8 ) if p :S q :S s �lv .

Sobolev Embedding Theorem.

k

( )

The proof can be found in Adams [1975] or Stein-Weiss [1971] .

ein1 91 ein2 92 eins 9s

32. We will also use the general multiplier theorem on 11'8 • Let us recall that the characters of the group 11'8 are when ( n t , . . . , n 8 ) E 7L8 • A bounded function m(�) , for � E 7L8 , defines an operator on £2 ( 11'8 ) , called a multiplier and defined by •

.

• .

The following theorem describes sufficient conditions for a multiplier to act on Lp (1I'8 ) , 1 < p < oo and to be of weak type (1-1) .

25

I. B. Examples of spaces and operators §32.

Multiplier Theorem. (1) r.p is bounded, (2) for every multiindex with

k

>

� we have

Suppose that r.p is a function on

R8 such that

A E D(k) (the description of D(k) is in 30)

nr sup R2 A 1 8Am(x) l dx < oo O 1 is of weak type (1-1) (see Folland-Stein [1982] Th. 3.37). Also the proof in Garcia-Cuerva-Rubio de Francia [1985] pp. 210-14 gives it.

Hp ,P

Lq ,

Part II Basic concepts of Banach space Theory II.A. Weak Topologies In this chapter we discuss two topologies which are weaker than the norm topology. They play an important role, both in the general theory and in applications. These topologies allow easy use of compactness arguments, and so they are helpful in existence proofs. This is particularly true of the w* -topology which can be defined on any dual space. We also introduce and study the class of reflexive spaces.

(

1. On each Banach space X there exists a weak topology {or a X, X * )­ topology or w-topology) . For each point xo E X its basis of neighbour­ hoods is defined as

U(xo; c, xi , . . . , x�) = { x e X: l xj (x) - xj (xo) l < c

for

j = 1,

. . . ,n

}

where xi , . . . , x� is an arbitrary finite set in X* and c is an arbitrary positive number. Obviously this defines a locally convex topology on X. A sequence (xn )�=l C X such that for some x E X, Xn --+ x in the a X, X* )-topology is said to be weakly convergent. We write this convergence as Xn � X. This is clearly equivalent to the condition that for every x* E X* the sequence x* (xn) --+ x* (x) as n --+ oo. A sequence (xn )�=l C X such that for every x* E X* the scalar sequence x* (xn) is convergent is said to be weakly Cauchy.

(

Xn = {1 , . . . , 1, 0, 0, . . . ) . Then Xn ..___..., n is weakly Cauchy because for every x* = (�j )� 1 E c0 = £ 1 we have x* (xn) = E;=l �j --+ E�1 �j as n --+ oo . On the other hand {xn )�=l is not weakly convergent, because {1, 1, 1, . . . ) fj CQ . 2 Examples.

{a)

Let X

= eo

and

times

Usually it is quite difficult to describe the weak topology o n the whole space. Quite often it is easier to describe this topology when restricted to the unit ball of X.

28

II.A. Weak Topologies §3.

B

(b) On the unit ball of eo the a(co , t'l )-topology coincides with the topology of pointwise (i.e. coordinatewise) convergence (i.e. the Ty­ chonoff topology on n:=l [-1, 1] restricted to the unit ball of eo ) . Clearly every open set in the topology of pointwise convergence is open in the a(eo , t'1 )-topology. Conversely, given a neighbourhood n U(xo; c, xi , . . . , x�) with ll xo ll � 1 we fix N such that L �N l xi (k) i < 1e0 for j = 1, . . . , n and put M = m i = l , ... , n ll xj ll . One checks that {x E eo : ll x ll � 1 and l x(k) - xo(k) l < 1 C:M for k = 1, . . . , N } C n U ( x o ; c, xi , . . . , x �). On the other hand the a( eo , t'1 )-topology and the topology of pointwise convergence do not coincide on the whole space Co · To see this let us consider Xn = 2 n en for n = 1, 2, . . . . Clearly E t'1 we have Xn converges pointwise to zero, but for x* = n- 2

B

a.x

B

( ) �= l

X n ¢ U(O; 1, x*).

For more examples see Exercises

1,2,3.

3 Lemma. A weakly Cauchy sequence is norm-bounded. Proof: For a given weakly Cauchy sequence (x n )�=l C X we define an operator T: X* ---+ c by T(x* ) = (x* ( xn ))�=l · By the closed graph theorem I.A.6, the operator T is continuous so IIT II = supn llxn ll < oo.a 4 Theorem. (Mazur)

If A is a convex set in X then the norm closure r

A of A equals its a(X, X*)

closure-

.

Clearly A c r . On the other hand if X E r\A then by the Hahn-Banach theorem I.A.lO there exists x* E X such that x* (x) > sup{x* (a): a E A } = sup{x* (a): a E A } . This implies that • x ¢ r.

Proof:

Suppose Xn � x. Then, by Theorem 4 for each j we have conv{xn } �=j · Thus we have the following useful

5.

xE

Corollary. If Xn� x then there exists a sequence of convex combinations

Yi = E��] AkXk such

we

that II Yi

- xll

----. 0.

a

6. On a dual space X* can introduce the w* -topology or a(X* , X)­ topology. The basis of neighbourhoods of a point x 0 E X* is given by

U(xo; c, X b

, , .

.

j = 1, . . . , n} an arbitrary

{x* E X* : I x * (xi ) - x(;(xi ) l < c for Xn is an arbitrary finite subset of X and c is

•

, Xn) =

where x 1 positive number. Clearly this defines a locally convex topology on .

•

•

X* .

29

II. A. Weak Topologies § 7.

7 Examples. (a) On £00 = li the w*-topology restricted to the unit ball coincides with the topology of pointwise convergence. The argument is the same as in Example 2b) .

(b) On lp and Lp [O, 1] for 1 < p < topology coincide.

oo

the w-topology and the w*­

8 Remark. Let T: X --+ Y be a continuous linear operator. Then T is continuous from (X, a( X, X* ) ) into (Y, a(Y, Y* ) ) and T* is continuous II from (Y* , a(Y* , Y)) into (X* , a(X* , X)). 9. The great usefulness of the w* -topology stems from the fact that it allows us to use compactness. We have Theorem. (AJaoglu)

The closed unit ball Bx · of X* is a(X * , X)­

compact.

Proof: Let Kt for t � 0 denote the set of scalars of absolute value � t. We define a map cp: Bx · --+ P = H�: e x Kll x ll by cp(x* ) = {x* (x) }x eX · By the Tychonoff theorem P is a compact space when equipped with the product topology. The very definitions of w* -topology and product topology give that cp is a homeomorphic embedding. The proof is finished once we show that cp(Bx · ) is closed in P. If Po = Po (x) E P\cp(Bx · ) then either

(a) for some Xt , X 2 such that Xt

=

>.x 2 we have Po (xt ) =f. APo (x 2 )

or (b) for some X1 , X 2 , x3 such that Xt + x2 Po (x3) ·

=

X3 we have po (xt ) +Po (X 2 ) =f.

In case (a) we define U = {p E P: Jp(x t ) - Po (xt ) l < 8 and Jp(x 2 ) - Po (x 2 ) 1 < 8} where 8 = ! I Po (Xt ) - >.po (x 2 ) J . Case (b) is treated analogously. In both cases we get an open set U C P with Po E U and U n cp(Bx · ) = 0. Thus cp(Bx · ) is closed. II Let X be a Banach space. There exists a canonical embedding i: X --+ X** given by i(x) (x* ) = x* (x) . When X = eo one easily sees that i: co --+ £00 is the identity map. For X = f.p, 1 < p < oo we have X** = X and i is the identity. Thus the following is not very surprising. 10.

Proposition.

The map

i: X --+ X**

is a linear isometry.

II.A. Weak Topologies § 1 1 .

30

Proof:

We have i(ax1 + ,8x 2 ) (x * )

x * ( ax1 + .8x 2 ) = ax * (xi ) + ,8x * (x 2 ) = [ai(x1 ) + ,8i(x 2 )] (x* ) . =

Since this holds for arbitrary scalars a, ,8 and arbitrary x1 , x 2 E X and x* E X* we see that i is a linear map. From the Hahn-Banach theorem I.A.9 we get, for x E X, , ll i(x) ll

=

l xsup• l $ 1 l i(x) (x * ) l l xsup• l $ 1 l x* (x) l =

thus i is an isometry.

=

ll x ll ; •

Quite often we identify X with i(X) and treat X simply as a sub­ space of X** . Note that this identification identifies the a(X, X*)­ topology with the a( X** , X* )-topology restricted to i(X). In other words i is a homeomorphism of (X, I!( X, X*)) onto i(X) with the a( X** , X* )-topology. 11. One can view the a( X, X*)-topology on X as the weakest topology such that all norm-continuous linear functionals are still continuous. Analogously the a( X* , X)-topology is the weakest topology which makes all functionals in i(X) C X** continuous. It is a useful fact that these are all the functionals which are continuous in those topologies. We have the following

(a) The space of all continuous linear Eunctionals on (X, a( X, X*)) equals X* .

Proposition.

(b) The space of all continuous linear Eunctionals on (X* , a( X* , X)) equals X.

Proof: Part (a) is easy since the a(X, X* )-topology is weaker than the norm topology. The proof of (b) is as follows: Let r.p be any linear funct­ ional on X* continuous in a(X* , X) . Then {x* E X* : l r.p(x* ) l < 1 } :::> {x* E X * : l xj (x* ) l < c j = 1 , 2, . . . , n} for some c > 0 and some x1 , . . . , Xj E X*. The application of the following algebraic lemma com­ pletes the proof. , ( c ) . Bx with the a ( X, X* ) -topology is homeomorphic to Bx·· with the a ( X** , X* ) -topology which is compact by Theorem 9. ( c ) => ( a) . We get that i ( Bx ) is a compact subset of X** with the a ( X** , X* ) -topology. Thus by the above Theorem 13 we have i ( Bx ) = Bx•• , so also i ( X ) = X** . ( d ) => ( a) . Obvious. ( a) => ( d ) . If Y is a norm-closed subspace of X then the Hahn­ Banach theorem yields that Y is a ( X, X* ) -closed. Thus By is a ( X, X* ) ­ compact. But on Y the a ( X, X* ) -topology coincides with a ( Y, Y* ) so the implication (( c ) => ( a)) gives that Y is reflexive. ( a) => ( b ) . If X is reflexive then the a ( X*, X ) and a ( X* , X** ) ­ topologies coincide on X* , so Theorem 9 gives that Bx· is a ( X* , X** ) ­ compact. Thus ( c ) => ( a) gives that X* is reflexive. ( b ) => ( a) . If X* is reflexive then using the implication ( a) => ( b ) we get that X** is reflexive so by ( a) => ( d ) X is reflexive. ( e ) {:} ( a) . We have to note that the dual of a quotient space of X is a a subspace of X* and use a previous equivalences. Proof:

We would like to conclude this chapter with the following useful observation. 15.

Proposition. able.

( a)

If X is separable then (Bx· , a (X* , X)) is metriz­

If X* is separable then (Bx , a (X, X* )) is metrizable. Proof: ( a) Let (xn)�=l be a dense set in Bx . We define a metric on Bx· by p(xi , x;) = nL=l T n l xi (xn) - x;(xn)i. Clearly p is a a ( X*, X ) -continuous metric. Let U = {x* E Bx· : i x *(x)­ < c-} for a fixed x E X and x0 E X*. Changing c we can assume x0(x)i x E Bx . If we take such that Xn is close enough to x then U ::J {x* E Bx· : ix* (xn) - x0 (xn) i < �} ::J {x* E Bx· : p(x0 x* ) < � } . ( b)

00

n

,

33

II.A. Weak Topologies §Notes.

Taking finite intersections we infer that every a(X * , X)-open set in Bx· contains a ball in metric p, so p defines the a( X * , X)-topology on Bx· . (b) is an immediate consequence of (a) and the remarks made after Proposition 10. a Notes and

remarks.

Weak convergence of sequences in some special instances was used at the beginning of the century by D. Hilbert and M. Riesz. The general theory is presented in Banach [1932] . Banach, however, did not use the notion of weak topology, he relied exclusively on sequential arguments. This resulted in unnecessary separability assumptions in many theorems. The weak topology defined by weak neighbourhoods appears (for algebras of operators on Hilbert spaces) already in von Neumann [1930] . In the thirties it became clear that general topological notions are needed in order to establish non-separable theorems. Theorem 9 for separable X is in Banach [1932] and the general form was announced by several authors with the proof first published by Alaoglu [1940] . Theorem 4 and Corollary 5 are from Mazur [1933] and Theorem 13 was first proved in Goldstine [1938] . The notion of reflexive space is implicit in Banach [1932] . Reflexive spaces, under the name 'regular spaces', were an object of intensive study in the late thirties and early forties. Theorem 1 4 summarizes some of this work. The implication ( c) =? ( a) for separable X is in Banach [1932] chapter XI Th. 13. It is interesting to note that he did not realize the converse, despite having a sequential version of Theorem 9. The equivalence ( a ) n , hn (t) · hm(t) equals either zero or a constant times hm . { b) Let FN , N = 2i + k, j = 0, 1, 2, . . . , k = 0, 1, . . . , 2i - 1, be the space of functions which are constant on intervals of the family Proof.

n

= 1, 2, . . and that for .

SN = {(s2 - i - I , {s + 1)2 - i - 1 ): s = 0, 1 . . . , 2k + 1 , and (s2 -i , (s + 1)2 - i ) s = k + 1, . . . , 2i - 1}. Since h n E FN for n = 0, 1, . . . , N and dim FN = N + 1 we infer that span {hn ) �=O = FN . This description implies that span {hn ) �=O is dense in Lp [O , 1] , 1 :::; p < oo. One also easily sees that the projection PN ( f)

=

L I I I - 1 I f(t) dtxl

IESN

I

is a partial sum projection. Applying the Holder inequality we have

p 1

I PN !I I � = L i I f (t) dt i I XI :::; L I I I 1 -P I J I P/q I I J I P = L I I J I P = 1 ! 1 � So I PN I = 1 for N = 0, 1, 2, . . . and Proposition 8 completes the proof. P IESN

IESN

a

I I I -p

0

I

I

IESN I

40

II.B. Isomorphisms, Bases, Projections § 1 1 .

Let us consider the trigonometric system (e i n9 )��- oo · We know (see I.B.20) that the Riesz projection 'R( f) = E n > o j(n)ei nB is bounded in Lp (T), 1 < p < oo. Since the operator of �ultiplication by e i nB , In (f)(O) = einB f(O), is an isometry from Lp (T) into Lp ('I') we infer that for -oo < N < M < oo, the map SN,M = IM (I - 'R)IN - M'RL N is a bounded operator on Lp (T), 1 < p < oo. One checks that SN ,M(f) = 11.

E nM= N f(n)em9 . A

•

Trigonometric polynomials are dense in Lp (T) so Proposition 8 gives that the characters (ei n B )��- oo ordered 0, 1, - 1, 2, - 2, . . . form a basis in Lp (T), 1 < p < oo . The Faber-Schauder basis in C[O, 1] . Actually we will describe a whole class of bases in terms of partial sum projections. Let (tj )� 1 be a sequence of distinct points of [0,1] with t 1 = 0, t 2 = 1 and such that {tj }� 1 = [0, 1] . We define projections in C[O, 1] as 12.

Pt (f)(t) =

/ (0),

PN (f) is a piecewise linear function with nodes

at tj , j = 1, 2, . . . , N and PN (f)(ti ) = f(tj), j = 1, 2, . . . , N. Clearly the projections PN , N = 1, 2, . . . satisfy conditions (a) , (b) , (c) of Proposition 8. The classical Faber-Schauder system is defined as r.p 1 ( t) = 1, r.pn (t) = ll hn - 2 11 1 1 J; hn - 2 (s)ds for n = 2, 3, . . . , where (hj )�0 are the Haar functions. The Faber-Schauder system corresponds to the sequence (tj)� 1 being naturally ordered dyadic points, i.e. 0 , 1, 1/2, 1/4, 3/4,

1/8, 3/8, . . . .

13. A system (x n )�=l C X is called a basic sequence if it is a basis in its closed linear span. Obviously every subsequence of a basis is a basic sequence. Also if (x n ) is a basic sequence in X and i: X --+ Y is an isomorphic embedding then (i( xn )) is a basic sequence in Y. We can also note that the Haar system is a basic sequence in £00 [0, 1] . Two basic sequences (x n )�=l in X and ( Yn )�=l in Y are called equivalent if the series E:= l an Xn converges if and only if E:=l an Yn converges. This is equivalent to saying that there exists an isomorphism be­ tween span( xn)�=l and span(yn)�= l which sends onto Yn for n = 1, 2, 3, . . . .

Xn

Now we will present two useful ways to find basic sequences. 14. One is a small-perturbation argument and the other involves the so­ called block-basic sequences. They are of fundamental importance in

41

II.B. Isomorphisms, Ba.ses, Projections § 1 5.

Banach space theory and in many applications. Let us start with the perturbation arguments. They are ba.sed on the following well known

U: X -+ X and l idx - U l < 1. Then U is invertible. The series L::'=0 (idx -U) n is absolutely convergent and defines

Lemma. Let Proof:

u- 1 •

a The following proposition sums up the most useful perturbation arguments.

(xn)�= 1

(x�)�= 1

x�(xm) 8n,m · (Yn )�= 1

15 Proposition. Let be a ba.sic sequence in X and let be its biorthogonal functionals, i.e. = (a) C X is a sequence of vectors such that is a ba.sic sequence equiv­ = < 1 , then alent to is complemented by a projection with (b) If, moreover, span < is complemented in X. then span y

If (Yn)�=1 2::::'= 1 l xn - Ynl l l x� l 8 (xn)�= 1 . (xn)�= 1 1 ( n)�= 1 I PI 8

T:

(xn)�= 1

P

-+

T(x)

T(xn) Yn· 2::::'= 1 x�(x) Yn· (1) l x - T(x) l :S nL= 1 l x�(x)l l Yn - Xn l :S 8 l x l we infer that (1 - 8) l x l ::; I T (x) l ::; (1 + 8) l x l so T is an isomorphic embedding. This shows that ( Yn)�= 1 is a ba.sic sequence equivalent to (xn)�= 1 · (b) Let us define A: X -+ X by A idx - P + TP where T is defined above. By (1) I I - A l = l i P - TP I ::; I P I 8 < 1, so Lemma 14 yields that A is invertible. One ea.sily checks that APA - 1 is a projection Proof:

(a) Let us define Obviously

=

span Since

X by

00

=

onto span(yn)�= 1 ·

(xn)�= 1

a

Un (an)

anXn (nk ) (uk )/:'= 1

16. If is a ba.sic sequence in X then a sequence of non­ zero vectors of the form = I:�:�\ for a strictly increasing a sequence of scalars is called a block-ba.sic sequence of integers and is a ba.sic sequence and sequence (block ba.sis for short) . Clearly

be(uk ) :S bc(xn) ·

17 Proposition. Let biorthogonal functionals

(xn)�= 1 be a ba.sis in X with l xn l 1 and (x�)�= 1 . Let (zk )� 1 be a sequence in X such =

42

Il.B. Isomorphisms, Bases, Projections § 1 9.

that

(2) l zkl l :2: 8 > 0 for k = 1, 2 ,3, . . . (3) k-+limoo x�(zk ) = 0 for n = 1, 2 , 3, . . . Then there exists a subsequence (zn . ) which is equivalent to a block basic sequence of (xn)· Proof: Let us put K = bc(xn) and let Pn denote the partial sum projections of the basis (xn) · Fix < 3 /4 8(K + 1) - 1 . Inductively we choose two increasing sequences of integers n8 and N8 in such a way that (4) I PN. - 1 (zn.) l < 'f/4- s , 8 (5) l zn. - PN.(Zn.) l < 'f/4- • We start with n1 = 1 and N1 chosen to satisfy (5). This is possible since Pn(x) - x as n for x E X. Having n and N8 for = 1, 2 , 3, . . . we pick nr+l so big that (3) ensures that8 (4) holds. Next we find Nr+l so big that (5) holds for Zn r+ 1 • The sequence U 8 = PN.Zn. ­ PN _ zn. is a block-basic sequence. We can find biorthogonal functionals u: .such1 that l 2u: l - :5s 2K I us l - 1 :5 2K(8 - 'f//4) - 1 (Corollary 7). Since l zn. - us I :5 .,.,4 we infer from Proposition 15 (a) that (zn. ) is the "'

oo

, r

s

a

desired subsequence.

Corollary. Every weakly null sequence has a basic subsequence.

(xn)�= 1 in X with l xn l = 1

(xn) 4

1]

Proof: Since span is separable we can assume that X is separable as well. Using Theorem we can treat X as a subspace of C[O, which has a basis ( 12) . Now the corollary follows from Proposition 17. a

1, . . . Xn

Ej= 1 x :L:j= 1 1, . . . , Xn (xj)j= 1

Xn

be subspaces of a Banach space X. We say that X 19. Let X is a direct sum of Xj 's and write X = Xi or X = X1 E9 · · · E9 if every E X has a unique representation = Xj with Xj E Xj . If we are given Banach spaces X we can form their direct sum X= Xi as follows: X is the set of n-tuples of elements with Xj E Xi , with coordinatewise algebraic operations and product topology. We can introduce a norm in X in many equivalent ways. The most = ( i Y. common are lp-sums, 1 :5 p :5 oo , where I

x :L:j= 1

Ej=1 I x l i P 1 The direct sum with such a norm will be denoted ( Ej= 1 Xj ) . P (xi ) l i P

43

II.B. Isomorphisms, Bases, Projections §20.

20. The following proposition states the easy but fundamental prop­ erties of these constructions:

(a) The space X is a direct sum of its subspaces X1 and X2 if and only if there exists a projection P: X -+ X such that X1 = P(X) and X2 = ker P. {b) IfT; : X; -+ }j for j = 1 , . . . , n, then we can define an operator

Proposition.

by the formula T((x; ) j= 1 ) = (T; (x; )) j=l · (c) If in (b) every T; is an isomorphism then T is.

X2

{b) and (c) are obvious. For (a) note that for x = x1 + x2 with X1 E X1 and E X2 we define P(x) = X 1 . P is continuous by the closed graph theorem. Conversely given a projection P we write x = Px + (x - Px), and Px E P{X) and (x - Px) E ker P. The a decomposition is unique.

Proof: Parts

Quite often we will need infinite direct sums. Since a basis provides an infinite direct sum decomposition of a space into one-dimensional subspaces, we see that, unlike the finite case, the form of the norm on the whole sum is crucial. In practice we will consider only lp-sums. They are formed as follows: Given a sequence of Banach spaces {X; )� 1 we put for 1 � p < oo 21.

00

00

l.

( J�=l xi) p = { (x;)� l : x; E X; and ll (x;) II P = ( J�=l l x; II P) " < oo } . We also define

( jtx i) =l

and

00

=

{{x; )� 1 : x; E X; and

l (x; ) l oo = s�J p l x;l l < oo}

= { (x;)� l E ( fx;) ( Jfx;) l x;l l - o } . =l J =l 0

00

:

In all the above sums the algebraic operations are performed coordi­ natewise. It is routine to check that all the above sums are Banach spaces.

44

Il.B. Isomorphisms, Bases, Projections §22.

Very similarly to the case of lp spaces one checks that

( L:xi 00 ) ( I:00 x; ) , ( jL:xi 00=1 ) 0 ( jI:00=1 x; ) 1 if *

*

p

j= 1

=

=

1 � p < oo

q

j= 1

1

1

and - + - = 1 . p

q

In particular if the X; 's are all reflexive and 1 < p < oo then E�1 X is reflexive.

i)

(

P

p�

(

(

(a) Lp [O, 1] is isometric to E:= 1 Lp [O, 1]

22 Examples. 00

(

(b) lp is isometric to E:= 1 lp

)

)

P

for 1 � p �

oo

)

P

for 1 �

and co is isometric

to E:= 1 co . 0 (c) C(�) is isomorphic to E:= 1 C(�) where � is the Cantor 0 set. To see (a) is relatively easy. We put An = [(n + 1) - 1 , n - 1 ] and identifying f E Lp [O, 1] with the sequence ( !IAn)�= 1 we see Lp [O, 1] = E:= 1 Lp(An )) p. Since Lp(An ) � Lp [O, 1] we have the claim. (b) is analogous. In order to prove (c) we start with the same idea. Let us fix an increasing sequence tn /' 1 such that t1 < 0, t 2 > 0 and � n (tn , tn+ l ) = � n is homeomorphic to �. Identifying the n-th summand of E:= 1 C(�) with C( �n ) we can identify E:= 1 C(�) with 0 0 the subspace Co (�) = { ! E C(�) : / (1) = 0}. So we have to prove that Co (�) ....., C(�). This follows from the following

(

)

(

)

(

23 Lemma. morphic to

Co .

(

)

The space C(�) contains a complemented subspace iso­

With �n as above we put en = sup �n· The map P f = E:= 1 [/ (en) - / (1)]x.6., is a projection from C(�) onto a subspace of

Proof:

Co (�) of functions constant on each �n· One easily checks that this a subspace is isometric to CO · Now we can write C(�) ....., Z $ co . Since C0 (�) has codimension 1 in C(�) Proposition 3 gives that Co (�) is isomorphic to Z $ V where

45

Il.B. Isomorpliisms, Bases, Projections §24.

V C eo , V we have

=

{(ai )� 1 E eo: a1 Co (a)

=

0}. Since V is clearly isometric to

"' z a1 "' z a1 "' v

eo

C(a).

Co

a

24. Arguments like that used in 22 . ( c ) are used quite often in proving that two Banach spaces are isomorphic. In the above case it is possible to write an explicit formula for the isomorphism, but in more complicated cases it is practically impossible. The following theorem is the most common variant of the 'decomposition method' which is the elaboration of such arguments.

X Y

X Y Y X. X X) P X "' Y. Let us write X "' Z $ Y and Y "' V $ X. Then

Theorem. Let and be two Banach spaces such that is iso­ morphic to a complemented subspace of and is isomorphic to a complemented subspace of Assume moreover that is isomorphic to E:'= 1 for some p with 1 � p � oo or 0. Then

(

Proof:

and also

X "' ( � x)P "' ( � za1Y) P "' ( � z) P $ ( � y)P "' ( � z)P $ ( � y) P $Y "'X$Y. So X "' Y. One easily checks that each step is justisfied.

Notes

and Remarks.

a

Much of the material in this chapter is folklore of Banach space theory. The notion of isomorphic embedding appeared in the early 30's, and var­ ious cases of embedding and impossibility of embedding were studied by Banach and his collaborators. Those results are summarized in Banach [1932] where in particular Theorem 4 is proved and in Banach-Mazur [1933] . The comparison between Banach [1931] and Banach [1932] shows the crystallization of the notion. The general notion of a Schauder basis was introduced in Schauder [1927] . By now the subject has grown enormously. An encyclopaedic

46

ll.B. Isomorphisms, Bases, Projections §Exercises

treatment is in Singer [1970] and [1981] and a very good, concise pre­ sentation in Lindenstrauss-Tzafriri [1977] . The Haar system, introduced in Haar [1910] , is one of the most important orthonormal systems (see Kashin-Saakian [1984] ). It is also a prototype of the general notion of martingale difference sequence. Haar studied his system in order to show that the expansion with respect to this system of every f E C[O, 1] is uniformly convergent {Exercise 7{a) ) . Proposition l O. (b) was proved by Schauder [1928] . The classical Faber-Schauder system was introduced and shown to be a basis in C[O, 1] in Faber [1910] . The general case was independently treated in Schauder [1927] . The notion of a basic sequence and a block basis emerged in the late 50's in the work of Pelczyiiski and Bessaga. The main ideas of Proposition 1 5 are in Krein-Milman-Rutman [1940] . They were rediscovered and applied in Bessaga-PelczyD.ski [1958] . This paper contains also Proposition 1 7 and Corollary 1 8. Direct sums appear already in Banach [1932] . Theorem 24 is taken from PelczyD.ski [1960] but germs of such ideas are due to Borsuk (see Borsuk [1933] and Banach [1932] chap XI. §7) .

Exercises 1.

Show that, if X1 and X2 are closed subspaces of a Banach space X , both of codimension n , then xl x2 . A norm 1 1 1 · 1 1 1 on a Banach space {X, 11 · 11) is called an equivalent norm if for some C > c > 0 we have cllxll ::;; l l l x l l l ::;; C ll x ll for all .....,

2.

x E X.

3.

(a) Suppose Y c X and d ( Y, Z) < c {here d ( Y, Z) is the Banach­ Mazur distance defined in II.E.6) for some Banach space Z. Construct an equivalent norm 1 1 1 · 1 1 1 on X such that c- 1 ll x ll ::;; l l l x l l l ::;; ll x ll for all x E X and {Y, 1 1 1 · 1 1 1) is isometric to Z. (b) For f E C[O, 1] put I I Ifi l l = ll f ll oo + ll f ll2 · Show that I l l · I l l is an equivalent norm on C[O, 1] and £ 1 is not isometric to any subspace of (C[O, 1] , 1 1 1 · 1 1 1 ) . (c) A norm is strictly convex if for all x, y, with x f. y and ll x ll = II Y II = 1 we have ll x + Yll < 2. Show that every separable Banach space has an equivalent strictly convex norm. {d) If r is a set of continuum cardinality then ioo (r) does not have any equivalent strictly convex norm. Suppose that (xn)�= l is a basis in X. Show that the biorthogonal functionals (x;'.)�= l form a basic sequence in X* . This need not be a basis (even if X* is separable) .

Il.B. Isomorphisms, Bases, Projections §Exercises

47

C X be a bounded sequence which is not norm rela­ tively compact. Show that there are sequences nk and mk such that (xn,. - Xm ,. ) � 1 is a basic sequence.

4. Let (xn)�= l 5.

Suppose X and Y are subspaces of a Banach space Z, such that X n Y = {0}. Show that, if the algebraic sum X + Y is not closed in Z, then there exist a basic sequence (xn)�= l C X and a basic sequence ( Yn )�= l C Y such that ll xn ll = llYn I = 1 for n = 1, 2, . . . and ll xn - Yn ll < 4 - n for n = 1 , 2, . . . . In particular X and Y have isomorphic subspaces.

6.

Prove the following. (a) The space C[O, 1] has a basis { !n )�= l consisting of polynomials. {b) There exists a sequence of polynomials which is a basis in all Lv [O, 1] , 1 � p < oo.

7.

(a) Show that for I E C[O, 1] the Haar-Fourier expansion converges uniformly. {b) Construct a basis in C( �) .

8.

Let (cpn(t))�= l be the Faber-Schauder system. Show that I = L:�= l an'Pn E LipQ [O, 1] , 0 < a < 1 if and only if l an l = O{n - Q ), so LipQ [O, 1] is isomorphic to £00 •

9.

Show the following isomorphisms: (a) Hp (T) "' Lp(T) for 1 < p < oo;

{b) Lipl [O, 1] "' Loo [0, 1] ; (c) C k [O, 1] "' C[O, 1] for k = 1, 2, 3, . . . ; {d) W;' [O, 1] "' Lv [O, 1] for k = 1, 2, 3, . . . and 1 � p < oo.

10. Suppose S C � is a closed subset. Prove the following.

(a) If S has continuum cardinality then C(S) "' C( � ) . (b) If S is countable then C(S) is not isomorphic to C{�). 11. Show the following isomorphisms. (a) C[O, 1] "' {EC [O , 1] )o ;

{b) C [O , 1 ] EB eo "' C[O , 1] ;

(c) Lv [O, 1] EB lv "' Lv [O, 1] .

48

II.B. Isomorphisms, Bases, Projections §Exercises

12. Prove the following. (a) Every isometric embedding of fp into Lp (J.L) (J.L-arbitrary mea­ sure) , 1 � p < oo , p =F 2, maps unit vectors onto disjointly supported, norm-one functions. (b) Every subspace of Lp (J.L) isometric to ip , 1 � plemented.

p


0 and every N E N there exists a finite sequence such that = ei)..i xi ll < e for every choice of ei 1 >-.i l = 1 and II with l ei l = 1.

"L-f=j.M

(>-.j )f=+ff + "L-f=NM

"L.f=+,_M Aj

14. Let E be a a-algebra of subsets of n and let J.L be a probability measure on E. Let E1 be a sub-a-algebra of E. Prove the following. (a) For every I E £1 (E, J.L) there exists a unique function PI which is E 1 -measurable, such that for every B E E 1 we have J8 l dJ.L = f8 P ldJ.L. (b) P is a linear map. (c) If I � 0 then PI > 0.

(d) P is a .norm-one projection in Lp(E, J.L) for 1 � p � (e) If g E L oo (Et . J.L) and I E Lt (E, J.L) then P(gf)

=

oo.

gPI .

15. Prove the following. (a) Every separable Banach space X is a quotient of £1 .

(b) If r is a set of continuum cardinality then i 1 (r) is isometric to a subspace of £00 •

(c) If r is a set of continuum cardinality then lp (r), 1 < and eo(r) are not isomorphic to any subspace of £00 •

p


0 and increasing se­ quences (n k )k'= 1 and (Ns )':: o with No = 0 such that 11 2: :,;,iJ. +1 Xnk I � c for k = 1, 2, 3, . . . . Let m8 be an increasing enumeration of N\{n k }f: 1 . One checks that for the permutation a defined by a

the series 2:::"= 1 Xu (n) diverges.

3. There is also a related notion of weakly unconditionally convergent series. The series 2:::"= 1 Xn is said to be weakly unconditionally conver­ gent if for every functional x* E X* the scalar series 2:� 1 x* (xn) is unconditionally convergent. Actually the name 'weakly unconditionally convergent' is a bit misleading, because such series need not converge (even weakly) . As an example take 2:::"= 1 en in More generally, the series l: ::"= o fn in the space C(K) , K compact, is weakly unconditionally convergent if and only if there is a such that

co.

c

Il.D. Convergence Of Series §4.

59

c < oo for every k E K. This can be easily checked using the Riesz representation theorem (I.B. l l ) .

E::'= o l/n (k) l :5

4 Proposition. For a series E;:'= 1 Xn in a Banach space X the follow­ ing conditions

are

equivalent:

(a) the series E ;:'= 1 X n is weakly unconditionally convergent; (b) there is a constant C such that

eo

L l x* (xn ) l :5 C ll x* ll for every x* E X*; n= l (c) there exists a constant C such that for every ( tn ) �= l E leo

(d) there exists an operator T: eo -+ X such that T(e n ) = X n · (a)::::} ( b) . We define S: X* -+ l 1 by S(x*) = (x* (x n )). The closed graph theorem implies that S is continuous, thus (b) holds. (b)::::} ( c) . For every N and every ( tn)�= l E leo we have Proof:

I nt= l tnXn I =

l ( nt= l tnxn ) I N I nL= l tnx* (xn ) l :5 ll (tn) lleo

S�p x •

ll x 11 9

= sup

ll x* ll 9

:5 C ll (t n ) lleo ·

sup

ll x * ll 9

eo

L l x* (xn) l

n= l

(c)::::} ( d) . Obvious. (d)::::} ( a) . For x• E X* we have 00

eo eo L l x * (xn) l = L lx * (Ten ) l = L I T* (x * ) (en ) l . n= l n =l n= l Since T* (x*) E l1 = c0 this is finite.

a

Il.D. Convergence Of Series §5.

60

5. Now we will investigate weakly unconditionally convergent series which are not convergent. It turns out that the example we gave in 3 is a canonical one. More precisely we have

I::'= t Xn P t r t P2 r2

Suppose that is a non-convergent weakly un­ conditionally convergent series. Then there exist increasing sequences of integers with < < < < · · · such that the vectors

Proposition.

Pk , Tk

(1) form a basic sequence equivalent to the unit vector basis in CQ .

I::'= t Xn does not converge we can find sequences (Pk )k';. (rk ) k= l as above such that the vectors (uk ) k= l given by (1) satisfy1 l uk l � c > 0 for some c. From Proposition 4(d) we infer that uk �O. Corollary II.B. 18 yields a subsequence (which we will still denote u k ) which is basic. Since (u k ) k= l is basic, there exists a constant C such that for all finite sequences ( t k ) we have

Proof: Since

and

On the other hand Proposition 4(d) yields a constant

I for all

(tk ) E

Co ·

�::> k k ukl l :::; C l (tk ) l oo

This completes the proof.

C1 such that a

We would like now to investigate the unconditionally convergent series in Lp , 1 :::; p :::; 2. We have the following 6.

I::'= t Xn

converges uncondition­ Theorem. (Orlicz) . If the series ally in Lv (n, J..L ) with 1 :::; p :::; 2 and J..L is a probability measure then

I::'= t l xn l 2 < oo.

Let (rn(t))�=l denote the Rademacher functions. It fol­ lows from Propositkn 4( e) that there exists a constant such that

Proof:

II. D. Convergence Of Series § 7.

sup N

61

SUPte[o,1J I z:::= 1 rn (t)xn l v C. Thus we have 1 N 1 N p CP � I I n�- 1 rn(t)xn l pdt = I I I n�- 1 Tn(t)xn(w) I PdJL(w )dt 1 N = I I I nL= 1 rn (t)xn(w) I PdtdJL(w). �

0

0

n

n

{2)

0

The Khintchine inequality I.B.8 and the fact that p � 2 yield

CP � K: I ( nt= 1 l xn(w )1 2 ) P/2 dJL(w) N K = : I ( nL= 1 (l xn(w) IP)� ) 2 dJL(w ) n

n

E

{3)

� K: ( n�- 1 ( I l xn(w) IPdJL(w) ) ) 2 N = K: ( � l xn l ; ) 2 �

N

P

n

E

E

•

Since N was arbitrary the theorem follows.

a

The important technical feature of the above proof is the use of the Rademacher functions to represent all possible choices of signs, each occurring with the same probability. This will be used extensively in the sequel.

Remark.

We will discuss this type of question in more detail in Let us note that what we actually proved is the following inequal­ ity: There exists a constant such that 7

liLA.

C

{4)

(xn)�= 1

C Lv (f!, J.L) , 1 � p � 2, and J.L a probability measure. for all finite A general Banach space X in which (4) holds is said to have the Orlicz property.

Il.D. Convergence Of Series §8.

62

(x�)�= l

(xn)�=l I:::'= t x�(x)xn

with biorthogonal functionals 8. A Schauder basis in a Banach space X (see II.B.5) is called an unconditional basis if for every E X the series converges unconditionally. A basis which is not unconditional is called conditional. Clearly the unit vectors in lp , 1 :5 p < oo, or in co form an unconditional basis. One easily checks that if is an unconditional basis in X then is an unconditional basic sequence in

x

(xn)�= l

9

Proposition.

tional basis in

Lp

(x�)�= l

x·.

()

The trigonometric system T only when p = 2.

(ein9 )��- oo is an uncondi­

2( )

Proof: The trigonometric system, being complete and orthonormal is clearly an unconditional basis in L 1l' . By duality it is enough to consider only the case 1 :5 p < 2. Suppose that the series represents a function I E Lp T and converges unconditionally. Then Theorem 6 gives < oo, thus I E L 2 (1l') . •

() 2 I:�: l an l

I:�: aneinB

In view of Proposition 9 the question arises whether Lp , p =/:. 2, has an unconditional basis at all. The answer is positive for 1 < p < oo and negative for p = 1. We have the following. The space Lt [O, 1] does not embed into any space with an unconditional basis.

10

Theorem.

The proof of this theorem relies on the following general 11 Proposition. A block basic sequence of an unconditional basis is a an unconditional basic sequence.

Proof of Theorem 10.

(rn(t))�= l be the sequence of Radx E Lt [O, 1]

Let emacher functions. Note that for every

x · Tn�O

as n -+ oo

(5)

(6) l x + xrn l t l x l t as n -+ Assume that £1 [0, 1] embeds into a Banach space Y with an uncondi­ tional basis ( Yn)�= t· Starting with Xt = 1 we define

and

-+

oo .

Il.D. Convergence Of Series § 1 2.

63

where kn increases so fast that

the sequence (x n )�= l considered in Y is equivalent to a block-basic sequence of the basis ( Yn )�= l ·

{8)

Condition {8) can be ensured using {5) and arguments like those in II.B. 17. Condition {7) follows from (6) because x1 + · · · + Xn - 1 = (x1 + · · · + Xn - 2 ) + (x1 + · · · + Xn - 2 )rkn - l . Conditions (7) , (8) and Proposition 1 1 show that for some constant C (depending on the norm of the embedding of £ 1 (0, 1] into Y) and for every choice of e n = ± 1 and for every N II

But this and (7) clearly contradict the inequality {4) .

12 Corollary. The space C(O , 1] does not embed into any space with an unconditional basis.

Proof: It is a special case of II.B.4 that £ 1 (0, 1] embeds into C(O , 1] , II so the proof follows from Theorem 10.

The existence of an unconditional basis in the spaces Lp[O, 1] , 1 < p < oo, is contained in the following theorem, which actually gives more precise information. 13 Theorem. numbers with

Let 1 < p < oo. If ( a k )k:,0 and (bk )k:, 0 l bk l ::5 lak l for k = 0, 1 , 2, then for all n � 0

are

complex

(9) where (h k ) k= O is the Haar system and p* =

ma:x.(p, pf(p - 1)).

Let us recall that the Haar system was defined in II.B.9. Proof: One easily checks using duality that it is enough to consider only the case 2 < p < oo (when p* = p) . Let v: ([ x ([ --+ 1R. be defined as

v(x, y) = I Y I P - (p - 1)P ix i P.

( 10)

Il.D. Convergence Of Series § 1 4.

64

The crucial part of the proof is the following 14 Lemma.

There exists a function then

a, b, x, y E CC and l b l � I a I

u: CC

x

CC

-+

R.

such that if

v(x , y ) � u (x, y ), u(x, y ) u( -x, -y ), u (O, 0) 0, u(x + a , y + b) + u(x - a, y - b) � 2u(x, y ).

= =

= E �= O

{11) {12) {13) {14)

==E�=O =

Assuming this lemma for a moment we complete the proof as fol­ lows. Put fn bk h k . Assume also that ak h k and Yn (h k )k:, 0 is normalized so that ll hk ll oo 1 for k 0,1, . . . . This clearly does not influence the theorem but some formulas are a bit shorter. Then by (10) and {11)

= I v(fn (t), gn (t))dt � I u(fn (t), gn(t))dt. {15) Since both fn - 1 and Yn - 1 are constant on the In = supp hn we get from {14) 11 Yn ll � - {p - 1) P II fn ll � "

1

I0 u(fn(t), gn(t))dt = (0,1I)\1.. uUn- 1 (t), gn- 1 (t))dt + I u{!n - 1 + an , 9n - 1 + bn)dt { h,. >O } + I uUn - 1 - an , Yn - 1 - bn)dt { h,. < O} � I u Un - 9n - d dt [0 , 1 ] \1.. + � I [u{ !n - 1 + an , Yn - 1 + bn ) b

+ uUn - 1 - an , Yn - 1 - bn)]dt

::::::; J uUn- 1 , 9n-1 ) dt 1

0

(16)

II.D. Convergence Of Series § 14.

and thus inductively

llYn II � - (p - 1) P II fn ll � �

65

J u(fo , 9o)dt = u(ao, bo)

= 21 (u(ao, bo) + u(-ao, -bo)) � u(O, O) = o. a

Proof of Lemma 14.

The desired function u (x, y ) is given by

( )

p 1 u(x, y ) = av( l x l + lyi ) P - 1 ( IYI - (p- 1) l x l ) where ap = p 1 - p1 - . (17) Conditions (12) and (13) are clearly satisfied. To prove (11), by homo­ geneity it is enough to assume l x l + I YI = 1. Letting l x l = s, (11) reduces then to the inequality

F(s) = ap( l -ps) - (1 - s) P + (p - 1) P sP � 0 for 0 � s � 1 and p � 2. (18)

One checks by a direct computation that

F(O) > 0, F(1) > 0, F

G) = F' G) = 0, F" ( � ) > 0

and F" has only one zero in [0, 1] . This is enough to see that (18) holds. To prove (14) it is enough to assume that x and a are linearly indepen­ dent over IR and the same is true for y and b. Under this assumption the function G(t) = u(x + ta, y + tb) is infinitely differentiable. A routine but rather tedious computation gives

G"(O) = av { - p(p - 1)( 1al 2 - I W ) ( I x l + lyi ) P - 2 (19) - p(p - 2) [ l b l 2 - Re < 1 1 ' b > 2 ] 1YI - 1 ( 1xl + lyi ) P- 1 - p(p - 1)(p - 2) [Re < 1 1 ' a > +Re < I I ' b >] 2 1 x l ( l x l + lyi ) P- 3 }

� :

�

Since l b l � l a l , the Cauchy-Schwarz inequality gives G"(O) � 0. Since this holds for all x, y , a and b with l b l � Ia I we see that in general G"(t) � 0. This clearly implies (14). a Notes and remarks.

The interplay between conditional and unconditional, i.e. absolute con­ vergence for scalar series, was already the subject of research in the

II.D. Convergence Of Series §Exercises

66

nineteenth century, and probably even earlier. Thus, with the emer­ gence of a theory of Banach spaces, investigation of series in Banach spaces became an important topic. In this section we only scratch the surface of this vast area. We will present further development of these ideas in some of the subsequent chapters, most notably in liLA and III.F. The fundamental early paper on the subject is Orlicz [1933] . It contains our Theorem 2 (actually it contains the proof of (d) 0. (Ac­ tually it is a 1r1-space, but this requires more care.) This can be proved directly. The alternative argument is to note that C(K)** = L00 (t-L) for some (usually non-a-finite) measure 1-L · From this and the principle of local reflexivity (Theorem 15) we infer that there exists an increasing net (Xoy) of finite dimensional subspaces of C(K) such that U X-y is dense

71

II.E. Local Properties §6.

x

in C{K) and d(X..,. , ta:,m ., ) � 1 + c. {The notion d(X, Y) is defined in 6. ) It follows from the Hahn-Banach theorem that there is a projection P..,.: C(K) �X..,. with li P..,. II � 1 + c. 6. Since all n-dimensional spaces are isomorphic, in order to investigate finite dimensional spaces more precisely we will use a more quantitative notion. Let X and Y be two isomorphic Banach spaces. The Banach-Mazur distance between X and Y denoted as d( X, Y) is defined as

d(X, Y) = inf{ II T II · II T - 1 1 1 : T: X�Y is an isomorphism} .

{4)

If the spaces X and Y are not isomorphic we set d(X, Y) = oo . The Banach-Mazur distance has the following, almost obvious, properties: (5) d(X, Y) � d(X, Z) · d(Y, Z); if X and Y are isometric then d(X, Y)

=

1.

{6)

Thus the Banach-Mazur distance is not a metric in the geometrical sense, but its logarithm is. We follow, however, the long established custom, and use the definition {4) and the name 'Banach-Mazur distance'. We have the following converse to {6). 7

Proposition. If X is finite dimensional and d(X, Y) = 1

isometric to Y.

then

X is

Proof: Let us take Tn : X ---+ Y such that II Tn ii · II T; 1 II -+ 1. Multiplying Tn by an appropriate scalar we can assume II Tn ll = 1 for n = 1, 2, . . . . Since L(X, Y) and L(Y, X) are finite dimensional Banach spaces, passing to a subsequence we can assume II Tn - T il 0 as n ---+ oo and II T; 1 S ll 0 as n ---+ oo for some T E L(X, Y) and S E L(Y, X). Obviously T is invertible and r- 1 = s. Moreover II T II = II S II = 1 so T is the --+

--+

desired isometry.

II

8. As an easy example of a computation of the Banach-Mazur distance we offer the following

Proposition. For every n = 1, 2, 3, . . . and p such that 1 � p � oo have d(l;, .ey) = nl t - ! 1 .

we

Proof. Note that for reflexive spaces X and Y, d(X, Y) = d( X* , Y* ) ; thus it i s enough to consider 2 � p � oo . T he upper estimate i s obvious

72

ll.E. Local Properties §9.

T:id: l�l�. T I 1. n; = I t, ±ei 1 2 � I t, ±Tei 1 2 � j I t, rj (t)Tei 1 2dt = Jt l t, ri (t)Tei (k) l 2dt = t t, l rei (k) l 2 = l rej t, r Tei 2 n ;-! I T-111 � n !-; . f: T n 1 p if we take £;: -+ operator £;: -+ II.D. { 4) we get

In order to check the lower estimate take any with l i Analogously as in the proof of �

s�p

s�p

Thus i�f ii 3

•

and

ll �

of trigonometric polynomials of order Let us consider the space with Lp-norm. More precisely the elements of Tf:, � � oo, are trigonometric polynomials of the form 9.

We have

Theorem.

There is a constant C such that

d(Tn•P lp2n+l ) - p-p--2 1 · = e'· 0 there exists an embedding i: X ---+ t� with (1 - c) ll x ll :5 ll i(x) ll :5 ll x ll where N :5 1 �e if X is a real space and N :5 1 �e if X is a complex space.

13 Proposition.

every c

( r

( fn

Let (xj )f= 1 be an c-net in Bx· given by Proposition 10. We define i: X ---+ t� by i(x) = (xj (x))f= l · Given x E X let x* E Bx· be such that ll x ll = x* (x), and let j be such that ll xj - x* ll < c. Then

Proof:

l xj (x) l = l x* (x) + xj (x) - x* (x) l � l x* (x) l - l (xj - x*)(x) l � (1 - c) ll x ll . a Thus ll i(x) ll � (1 - c) ll x ll . Obviously ll i ll :5 1.

Remark: This result is a finite dimensional or 'local' version of Theo­ rem II.B.4. Actually without an estimate for N this result easily follows from Theorem II.B.4. 14. Now we will return to the interplay between local and global properties of Banach spaces. We start with the important Theorem. (Principle of local reflexivity.) Let X be a Banach space and let E c X** and F C X* be finite dimensional subspaces. Given c > 0 there exists an operator T: E ---+ X such that

I I T II · II T - l I T(E) II :5 1 + c, T J E n X = id, f(Te) = e(f) for all f E F and e E E.

(11)

(12) ( 13)

77

Il.E. Local Properties § 1 5.

This theorem asserts that finite dimensional subspaces of X** are basically the same as finite dimensional subspaces of X. The proof relies on the following

and let X** .

A;

(A; )f= 1

be bounded, norm-open convex subsets of X denote the norm interior of the a( X** , X * )- clos ure of A; in

15 Lemma. Let

N N If n;=1 A; =f. 0 then n;=1 A; =f. 0. (b) If we have a map T: X Y with Y space then T** C n_f= 1 A; ) = T(nf= 1 A;). -

(a)

--+

Proof:

a finite dimensional Banach

(a) Let XN be the direct sum of N copies of X. The set

is a bounded, norm-open, convex subset of XN . If n.f=, l A; = 0 then A n V = 0 where V = {(x;)f= 1 E XN : x; = x1 for j = 1 , 2, . . . , N} . Let

A- -- {( X;**) jN= l

and

{( X;** )jN= l

E X** N ·. X;** E

3,

A- ·

X;**

J. -

1 , ... , N}

= x 1** £or J = 1 , . . . , N} . If A n V = 0 then there exists
such that

=

E X** N

· •

.

N

a > (3 > a} see that nf= 1 ij =f. 0 but nf= 1 Aj is empty. This contradicts (a) and Let

so proves (b) .

we

a

II.E. Local Properties § 1 5.

78

Let dim E = n and dim E n X = n - k. Fix a biorthogonal system (xj* , ej ) j=1 in E x E* such that span(xj* ) j= k+ l = E n X, and ll xj* ll = 1. The identity id: E � X** can be written as id ( e) = Ej=1 ej ( e )xj* . We want to find X t , . . . , Xk in X such that the map T: E � X defined as T (e) = E7=l ej (x)xi + E�+ l ej (e)xj* will have the desired properties. Property ( 12) is satisfied with this definition. Let Z be the direct sum of k copies of X and let 6 > 0 be a small number. Fix the following finite sets: { /i }� 1 is a basis in F, {xj }� 1 c Bx· is such that for every e E E ll e ll $; (1 + 6) sup{ l xj (e) l : j = 1, . . . , R } , { ei }_f= 1 i s a 6-net i n BE. We have n "'""' A, jr X r** e3. - L..J

Proof of Theorem 14.

r= l

•

Now we form the following subsets of Z: Cj

= { (x s )==l : and ll x s ll

I t, A

� Xs +

< 1 + 6,

s

t

l

,\�x : • < I l ei II 1 8 = 1, . . . k}, j = 1, . . , N. .

Those subsets of Z are norm-open, bounded and convex. Since (x: • )==l E n_f=, 1 Ci C Z** (where - has the same meaning as in Lemma 15) Lemma 15 gives (x s )== l E nf=1 Ci =f. 0 . Let us consider an operator s : z � RM · k $ R R · k (or into ccM · k $ (CR · k ) , defined as

From Lemma 15 we infer that there exists (x 8 )==l such that (x s ) != t E and

N

n=l Ci

j

S((x s)== l ) = S ** ((x : • )==l ) ·

With this choice of (x8 )== l we clearly get (13). Also we get ixj ( Te) i II Te ll ;::: j =sup l , ,R . . .

=

sup ixj (e) i ;::: ( 1 + 6 ) - 1 ll e ll . j = l , ,R ...

( 14)

79

II.E. Local Properties § 1 5.

Given e E BE let us fix have II Tei ll :5 ll e; ll · So

II Te ll

:5

ei

with

II Tei ll + II T(e - e;) ll

A very crude estimate for

:5

li e - ei ll

:5

6.

Since

(x s )!=1

E

C;

li e; II + II T II · 6 :5 ll e ll + 6 + 6II T II .

li T II is ( 1 + 6 ) :E7=1 ll ej II 6 is small enough ( 1 1 ) follows from ( 14 ) and (15 ) .

:5

2 :E 7= l

we

(15 )

ll ej II , so if a

15. We say that a Banach space X is finitely representable in a Banach space Y if there exists a constant C such that for every finite dimensional subspace X1 c X there exists Y1 c Y with d(X1 , YI ) :5 C. In other words X is finitely representable in Y if finite dimensional subspaces of X are subspaces of Y. Note that Proposition 12 yields that every Banach space X is finitely representable in c0 • From Proposition 12(a ) and Example 5(a) we get that Lp (O, ft ) is finitely representable in lp for 1 :5 p :5 oo. On the other hand from II.D. (4) we easily get that lp for p > 2 is not finitely representable in any Lq(O, ft ) for 1 :5 q :5 2. Theorem 14 shows that X * * is finitely representable in X . Notes and remarks. The general idea of approximation of a separable Banach space by finite dimensional ones is quite old and was around in Lw6w in the thirties. Banach ( 1932] asked the question if every separable Banach space has a basis. A notion of approximation property (a. p. ) , the concept even weaker than b.a.p. , was also invented then. We say that a Banach space X has the approximation property if for every norm-compact set K c X and for every e > 0 there exists a finite dimensional operator T: X � X such that sup { II Tk - k ll : k E K} :5 e. The first deep study of a.p. and b.a.p. is contained in Grothendieck ( 1955] . The notion of 71'A-space emerged in the sixties (see Lindenstrauss ( 1964] and Michael-Pelczynski ( 1967] ) . The real breakthrough in the study of approximation proper­ ties come with Enflo's ( 1973] example of a Banach space without a.p. Many examples differentiating various approximation properties have been produced later. We refer the interested reader to Lindenstrauss­ Tzafriri ( 1977] and ( 1979] , Pisier ( 1986] and Szarek ( 1987] . All this is a very fascinating subject but beyond the scope of our book. As a first result of the local theory of Banach spaces one can consider the following well known fact proved in Jordan-von Neumann ( 1935] . Ac­ tually the analogous three dimensional characterisation was given earlier by Frechet [1935] .

80

II.E. Local Properties §Exercises

A Banach space X is isometric to a Hilbert space if and only if for all x, y E X . Note that this result implies that X is isometric to a Hilbert space if and only if every two-dimensional subspace of X is isometric to a Hilbert space. For an isomorphic version of this see Exercise 9(b) . The local theory of Banach spaces gathered momentum in the sixties with the study of L1-preduals (see Lacey [1974] ) and p-absolutely summing operators (III. F) . Today it is a vast subject having connections with operator theory, harmonic analysis, geometry of convex bodies etc. Some of it will be presented later. For a more detailed presentation of different aspects of the theory, the reader should consult Milman-Schechtman [1986] or Tomczak-Jaegermann [1989] or Beauzamy [1985] . The notion of Banach-Mazur distance is in Banach [1932] . Theorem 9 is a classical result of Marcinkiewicz [1937a] (see also Zygmund [1968] chapter X §7) . Proposition 8 is an easy special case of a result of Gurarii -Kadec-Macaev [1965] . The Auerbach lemma is mentioned without proof in Banach [1932] . The principle of local reflexivity ( Theorem 14) was proved in Lindenstrauss-Rosenthal [1969] . Exercises

1.

A reai Banach space X is called uniformly convex if there exists a function c,o (e) > 0 for e > 0 (called the modulus of convexity) such that if x, y E X, l l x l l = IIYII = 1 and l l x - Y ll > 2e then II (xty) I $ 1 - c,o(e) (draw the picture) . (a) Let X be uniformly convex and let x• E X* , llx* ll that diam{x E X: l l x l l $ 1 and x* (x)

=

1. Show

> 1 - e:} --+ 0 as e --+ 0.

(b) Show that uniformly convex spaces are reflexive. (c) Show that Lp [O, 1 ] , 1

< p < oo,

is uniformly convex.

(d) Show that if L:;::"=1 Xn converges unconditionally in X then n= l (e) Suppose that X is a uniformly convex space and that Y C X is a closed subspace. Show that for every x E X, there exists a unique y E Y such that ll x - Y l = dist(x, Y) .

i

81

Il.E. Local Properties §Exercises

2.

Let X be a complex Banach space. We say that X is complexly uniformly convex if there exists a function cp( e) > 0 for e > 0 (called the complex modulus of convexity) such that if y E X with IIYII � f: and llx + ei 6 y li :5 1 for all () then ll xll :5 1 - cp(e).

x,

(a) Show that

L 1 [0, 1]

is complexly uniformly convex.

(b) Show that if cp is a complex modulus of convexity of X and the L: ::'= 1 Xn converges unconditionally in X then

L: ::'= 1 cp( ll xn ll )
0 for every open set U and such that J.L(UI ) < oo for some open set U1 . Translation invariant means that J.L(A + x ) = J.L(A) for every x e X.

7.

Suppose that T: X - Y. Let (Ya)aer be a net of finite dimensional subspaces of Y, ordered by inclusion and such that Uaer Ya = Y. Assume that there is a C such that for each a E r there is an operator Sa : Ya ---+ X such that l i Sa I :5 and TSa = idy"' . Show that T * (Y * ) is complemented in X* .

onto

c

8.

(a) Show that £2 is isomorphic to a complemented subspace of (b)

( L: ::'= 1 £�) 00 . Let Mn denote the set of all n-dimensional Banach spaces (up

to isometry, i.e. we identify isometric spaces). Show that for every n E N the set Mn with the Banach-Mazur distance (or rather, its logarithm) is a compact space.

82

Il.E. Local Properties §Exercises

(c) Let (Bk )� 1 be a sequence of finite dimensional spaces such that {Bk } � 1 n Mn is dense in Mn for all n E lN. Show that for every separable Banach space X the space (:�:::.: ;:'= 1 Bk) 00 contains a complemented copy of X* . 9.

(a) Show that if X is a Banach space finitely representable in a uniformly convex Banach space Y (Exercise 1 ) , then X has an equivalent uniformly convex norm. (b) Show that if a Banach space X is finitely representable in £2 then X is isomorphic to a Hilbert space.

10. A Banach space X has the uniform approximation property ( u.a.p.) if there exist a constant C and a function cp( e, n) , n E lN, e > 0, such that for all X 1 , . . . , Xn E X there exists an operator T: X --+ X such that IITx; II :::; e l l x; II for j = 1 , 2, . . . , n and liT II :::; C and dim T ( X ) :::; cp ( e , n) . Show that Lp [O, 1] , 1 :::; p :::; oo, have the

x2 , -

x;

u.a.p.

1 1 . Show that

£� is isometric to a subspace of £00 but is not isometric to any subspace of co .

Part III Selected Topics III . A Lp- Spaces; Type And Cotype. In this chapter we investigate the Lp (JL)-spaces, 1 < p < oo . We start by proving the isomorphisms of some natural spaces to spaces Lp (JL) . We show that the Sobolev space W� ("11'2 ) is isomorphic to Lp ("11'2 ) for 1 < p < oo and that the Bergman space Bp (D) is isomorphic to lp for 1 $ p < oo . Along the way we prove a useful criterion for the boundedness of integral operators on Lp(f!, JL) (Proposition 9). Later we borrow from probability theory and show the existence and basic properties of stable laws. These provide isometric embeddings of lp into Lq , q $ p $ 2. We continue the line of thought started in II.D.6 and introduce the general notion of type and cotype of a Banach space. In order to study these notions efficiently we prove the vector valued generalization of Khintchine inequality (Kahane's inequality) . A gener­ alization of a classical result of Carleman from the theory of orthonormal series is also presented. We conclude this chapter with the Banach-Saks theorem and its generalizations to almost everywhere convergence. 1.

We start this chapter with some general observations.

A separable space Lp (f!, JL) , 1

to one of the following spaces: e; , n

Proposition.

t;)p, n = 1 , 2, . . . , (Lp (O, 1]

E9

lp)p o

=

$

p


p and if X has type p it has also type q for q < p. (d) The smallest constant for which ( 13 ) holds for a given space X is called the cotype p constant of X and is denoted Cp(X). Similarly we define the type p constant of X, denoted by Tp(X). The following vector valued generalization of the Khintchine in­ equality is a fundamental tool for investigating types and cotypes. 18 Theorem. (Kahane's inequality) There exist constants Cp , 1 � p < oo such that for every Banach space X the inequality

holds for every finite sequence

(xi ) J= l

C

X.

The proof follows from the following distributional inequality. Let V(t) = II E.i=o ri (t)xi ll · {t: V(t) > 2a} l l � 4 l {t: V(t) > aW .

19 Proposition. we have

Proof: For k � n let us put the following sets:

Then for every

Vk (t) = II E7=o ri (t)xi ll

n

0

and let us define

Am = {t: Vk (t) � a , k = 0, . . . , m - 1 , Vm (t) > a}, n A = U Am = {t: sup Vk (t) > a}, k m= l B = {t: V(t) > a}, C = {t: V(t) > 2a }, Cm = { t : I I L rj (t) xj l l > a } . J =m

a>

III.A Lp-Spaces; Type And Cotype § 1 9.

96

Let us write Ej=o Tj (t)xi = E;: o Tj (t)xj + Ej= m + l rj (t)xj = a(t) + b(t). It follows from properties of the Rademacher functions that b(t) is symmetric on every set where a(t) is constant. Since ll x ll � max( ll x + Yll , ll x - Yll ) for every x, y E X, we see that at least on half of the set Am we have ll a ll � ll a + b ll · Thus

(16) Analogously

(17) We put Am = (A m n{t: rm (t) = 1})U (A m n{t: rm (t) = - 1}) = A;\:. uA;;;, and Cm = (Cm n {t: Tm (t) = 1}) u (Cm n {t: rm (t) = - 1}) = c� u c;;;, . The independence of the Rademacher functions gives

and Since

l A� n C� I = 2 I A� I · I C� I l A;;;, n C;;;, l = 2 I A;;;, I · I C;;;, I . l A;\:. I = l A;;;, I = ! I Am l and I C� I = I C;;;, I = ! I Cm l we have (18)

Since obviously

C C B C A,

from

(18), (17)

and

(16) we get

I C I � L I Am n C I � L I Am n Cm l = L I Am i iCm l � sup i Cm l · L I Am l � 2 I B I · I A I � 4 I B I 2 m

•

Proof of Theorem 18. The left hand side inequality is obvious, while the right hand side inequality is a standard passage from a distribu­ 1 tional inequality to an integral one. We can assume J0 V(t)dt = 1, so l {t: V(t) > 8} 1 � �- Applying Proposition 19 inductively for k = 1, 2, 3, . . we obtain .

This gives

1

k . 8) P · l {t: 2 k - l . 8 < V(t) � 2 k . 8} 1 J0 V(t)Pdt � 8 + kL(2 =l k � 8 + L (2 k . 8) P . T 2 = c:. lc = l 00

00

a

97

III.A Lp-Spaces; Type And Cotype §20.

20 Remarks. ( a) An obvious and immediate consequence of Theorem 18 is that in (13) and ( 14) instead of J II Erj Xj I we can use (J II Erj Xj ll q ) � for every q, 1 � q < oo. (b) For some applications the magnitude of the constant Cp is im­ portant. Our proof, as can be easily verified, gives Cp � C p for p ;::::: 2. The correct order of magnitude is Cp � C.,jP (see e.g. Milman­ Schechtman [1986] ) .

·

The following repeats arguments from II.D.6.

21.

Proposition.

If X has cotype p and E:: 1 X n is an unconditionally

convergent series in X, then

Proof.

E:: 1 ll xn li P < oo.

We have for every N

(� ) 1 I I n�- 1 rn (t)xn l dt N

.!

ll xn ll p

P

N

� Cp (X)

0

� Cp (X) s�p

I n�- 1 rn (t)xn l N

� C. a

22. A part of the relation between type and cotype is explained by the following. Proposition. If X has type p then X* has cotype q , where � + �

x1 , . . . , X n E X and xi , . . . , x� E X* 1 n n n xi (x i ) = ri (t)xi ri (t)x i dt •= 1 •= 1 •= 1 0

Proof:

For arbitrary

1.

we have

I ( ?:

?:

�

�

Since

=

) ( ?: ) ] I t•-1 ri (t)xi 1 · 1 t•-1 ri (t)xi l dt 2 ! 2 ! r (t)x r dt (t)x i l ) ( } I t•- 1 i ( } I t•- 1 i i l ) . 0

0

0

( 19)

98

III.A

Lp-Spaces; Type Cotype And

§23.

from (19) we get

1

!

( ?:t=1 l xill q) ; (Ji i ?=t=n1 ri(t)xi11 2) 2 { ( } I tt= 1 ri(t)xill 2) ! : tt=1 l xiiiP } n

�

0

· sup

� 1 ·

0

a

Using Theorem 18 we get the claim. 23.

The types and cotypes of Lp-spaces are as follows.

The space Lp O, J.L , ) ( cotype L00(0, J.L)

Theorem. max(2, p) .

1 �p�

oo,

is of type

min(2, p) and

(x;) Lp (O, J.L)

Proof: Clearly has type 1 and cotype exactly like in II.D.6 we obtain for every c

For 1 � p
1. In the case p > 1 different bases have been constructed in Matelievic-Pavlovic [1984] . Proposition 7 and Corollary 1 6 address special cases of the follow­ ing question: for what p, q is the space 1] or isomorphic to a subspace of 1] ? Under the name of linear dimension this was stud­ ied already in Banach [1932] and Banach-Mazur [1933] . Today the full answer is known. It is summarized in the Table 1 . From the results given in this section the interested reader can easily deduce all these facts except the case 2 < p < q < oo, which is due to Kadec-PelczyD.ski [1962] (see Exercises 3 and 4) . The answers are the same if we replace by 1] . This follows from Proposition 2 and the following

Lp[O ,

£1 [0,

L1[0 , LI[O ,

Bp(D)

fp

Bp(D)

Bp(D),

fp-

Lp[O ,

Lq[O ,

lp

fp Lp[O ,

:S:

Then there exists a measure subspace of oo.

Lp (J.t) .

:S:

J.t such that X is isomorphicfp, to1 a

Proposition. Let X be a Banach space finitely representable in

p

105

III.A Lp-Spaces; Type And Cotype §Notes.

Thble 1.

fp\Lq

q=1

1 0.

Theorem.

fn E Tk(n)

Proof: Let Fn = span ( /k ) k= l · Since ( /n ) ';'= 1 is a basis for 0(11') , II.B.6 and Theorem 5 show that .X(Fn ) :5 C for some constant C, n = 1, 2, . . . . Using (4) we obtain �--:---:---

.X(Tk(n) ' C(1l') ) + 1 2k(n) + 1 - n + 1. .X(Fn , C(1l') ) + 1 :5 vf Thus Theorem 22 yields log n :5 log 2k(n) + 1 :5 C J2k(n) + 1 - n so k(n) �

i + c log2 n.

a

We would like to conclude this chapter with a more abstract 25. treatment of the extension of operators. This is connected with the

125

III.B. Projection Constants §26.

concept of tensor product. Actually we will consider only one tensor product. We assume that the reader is familiar with the algebraic notion of tensor product. Definition. If X and Y are Banach spaces then their projective tensor product X ® Y is a completion of the algebraic tensor product X ® Y with respect to the norm

l � xi ® Yi l N

J =l

II

= inf

{ I) xj ii iiYj ll : � xi ® Yi = � xj ® yj } . (13) N

3

3

J =l

26 Lemma. The dual space of X ® Y can be isometrically identified with L(X, Y* ) . Proof: The duality which we use is basically the trace duality ( compare with III.F. 16) . Explicitly it is given as

j

j

for T: X --+ Y* and I:j Xj ® Yi E X ® Y. One checks that this definition is independent of the representation of the tensor. Since

I (T, � xi ® Yi ) l � L I T(xi )(Yi ) l � II T II · � ll xi ii · IIYi ll 3

3

3

we infer from (13) that every T E L(X, Y* ) induces a linear functional on X ® Y of norm � II T II . Considering the elementary tensors we get

II T II = sup{ I (Tx)( y) l : IIYII = ll x ll = 1} = sup{ I (T, x ® y) l : ll x ii · IIYII = 1 } � sup (T, � xi ® Yj) : � Xj ® Yj

{I

3

l �

3

L � 1}.

This shows that each T E L(X, Y* ) gives a functional in (X ® Y)* of norm l iT II so L(X, Y* ) is isometrically a subspace of (X ® Y) * . Now suppose that we are given cp E (X ® Y)*. Note that for a fixed x E X, cpx (Y ) = cp (x ® y ) is a linear functional on Y with norm � ll cp ll - One easily checks that the map x ...... cpx is a linear operator from X into Y* . a This shows that L(X, Y*) = (X®Y ) * .

126

III.B. Projection Constants §27.

27 Corollary. Suppose Y conditions are equivalent:

C

X and Z* are given. The following

(a) every operator T: Y --+ Z* extends to an operator T: X --+ Z* ; (b) the map r: L(X, Z* ) --+ L(Y, Z*) given by r(T)

=

T I Y is onto;

(c) the identity map i: Y ® Z --+ x ® z is an isomorphic embedding. Proof: This is a routine duality (see I.A. 13 and 14) and the observation

that i*

=

a

r.

28. Now we will formulate the proposition which is the dual version of Theorem 2. Before we do so we will define the space £1 (S1, f.£, Y) of Bochner integrable Y-valued functions (Y is a Banach space) . First we consider the step functions of the form f(t) = Ej= 1 YiXA; (t) where Yi E Y, j = 1, . . . , N and (Aj ) f= 1 are disjoint subsets of S1 with f.£(Aj ) < oo for j = 1, 2, . . . , N. For such f we put

11/11

=

I: 11Yi ll f.£(Aj ) = In 11/(t) ll df.£(t) . i= l N

A Y-valued function f(t) is Bochner integrable if there exists a sequence (/n)':'= l of step functions, as above, such that Jn 1 1 /(t) - fn (t) ll df.£(t) --+ 0 as n --+ oo . The space of all Y-valued, Bochner integrable functions (when we iden­ tify functions equal f.£-a.e.) is denoted by £1 (S1, f.£, Y) . One easily checks that it is a Banach space. This description makes the following proposition easy to believe. Proposition. If (S1, f.£) is a u-finite measure space and Y is a Banach space then

We define a map cp: Y ® £1 (f.£) --+ £1 (S1, f.£, Y) by the formula cp( Ef= l Yi ® /i ) = L:f= 1 /j (t) · Yi · One notes that this map is well defined and

Proof:

127

III.B. Projection Constants §Notes.

so ll

c log n for some positive c. 4. Show the analogue of Theorem 24 for £1 (11') and A(11'). 1 1 5. (a) Show that if 2 $ p $ oo then cnii $ .X(t;) $ nii for some c > 0. (b) Show that lp, 1 < p $ 2, is not isomorphic to a complemented 1. 2.

Show that

£00

is isomorphic to

•

•

•

subspace of any £1 (p,).

.X(£1). 7. Show that, if X is separable and Y C X is isomorphic to co, then Y is complemented in X. 8. (a) Show that ioo /co contains a subspace isometric to co(r) , where r has continuum cardinality. (b) Show that co C £00 is not complemented. (c) Show that if X C £00 and X rv c0 then X is not complemented 6.

Estimate

in i00 •

9.

10.

Suppose that p (z) is a polynomial of degree N on CCd , d 2:: 1 such that < 1. Show that there exists an inner function cp on md such that if cp = E:=o 'Pn with 'Pn a polynomial homogeneous of degree n, then p = E �= l 'Pn· Show also that for every r < 1 and every e > 0 this inner function cp can be constructed in such a way that sup{ l cp(z) - p(z) l : l z l < r} < e. For d > 1 let v c md be the set { (zb . . . ' Xd): l zi l $ d - ! j = ' 1 , . . , d}. Show that there exists an inner function cp on md such that l cp(z) l < � for z E V.

II P IIHoo(IBd)

.

11.

If f(z) is holomorphic on md , then its radial derivative Rf (z) is defined as Rf ( ) = E�= l Show that if the d-th radial d derivative of J , R f , is in H1 (md) then f E A(md) .

z

(zi 8£t> ) .

130

III.B. Projection Constants §Exercises

£2 defined T( f ) = T: A ( JBd ) , where (Pk ) are the polynomials con­ =l structed in Proposition 18, is onto. 13. A function f(z) holomorphic in ID is called a Bloch function if sup 1/ '(z) l · (1 - l z l 2 ) < oo .

12 .

Show that the operator

( fsd f( ( )P2n ( ( )da( ( ) )

�

---+

as

zEID

( a) Show that L::= l anz 2 n is a Bloch function if and only if ( an ) ;::='= l E ioo . (b) Find a Bloch function which does not have radial limits a. e. on the circle 11'.

14.

For a function analytic in 1Bd and ( E Sd we define a slice function fc; (z) analytic in ID by fc; (z) = f(z · ( ) . We say that a function f analytic in 1Bd is Bloch if

Find a Bloch function in 1Bd which does not have radial limits a-a.e. on Sd.

III. C . L1 (JL)-Spaces This chapter discusses some topics connected with £1 ( JL ) -spaces. We start with the general notion of semi-embedding and investigate semi­ embeddings of £1 ( JL ) into various Banach spaces. This is applied to the class M0 (11' ) of all measures such that p,(n) --+ 0 as n --+ oo. We prove the classical Menchoff theorem that there are singular such measures and a theorem of Lyons characterising zero sets for the class Mo (ll' ) . Next we describe relatively weakly compact sets in £1 ( Schur's theorem ) and in general £1 ( JL ) -spaces for a probability measure JL ( the Dunford­ Pettis theorem ) . We also discuss the connection between type and finite representability of £ 1 . Some characterizations of reflexive subspaces of £1 are given. We conclude with some results connected with the classical result of Nevanlinna about cosets F + H00 (11') C £00 (11') . We have already seen that properties of £1 spaces differ from properties of Lp-spaces for 1 < p < oo. In particular, £1 being non­ reflexive, its unit ball is not weakly compact. Actually more is true.

1.

Proposition. If JL is a non-atomic measure and T: £1 ( JL ) --+ X is a 1-1, weakly compact linear operator then T(BL1 (p.) ) is not norm-closed.

Since BL1 (p.) has no extreme points and T is 1- 1 T(BL 1 (p.) ) also has no extreme points. The Krein-Milman theorem I.A.22 implies that T(BL1 (p.) ) is not weakly compact, thus it cannot be norm-closed.a

Proof:

2.

Let us introduce the following

Definition. A 1-1 linear operator T: embedding if T(Bx ) is closed in Y.

X

--+

Y is called a semi­

Clearly every isomorphic embedding is a semi-embedding. Also if X is reflexive then every 1-1 map T: X --+ Y is a semi-embedding. Proposition 1 clearly says that there is no semi-embedding from £ 1 [0 , 1] into any reflexive space.

3. The following is a general, topological observation showing that r-1 I T(Bx ) has at least one point of continuity. We formulate it for Banach spaces only, because this is the way we will use it.

132

III. C.

L 1 (p,)-Spaces §4

Proposition. Let X and Y be separable Banach spaces and let be a semi-embedding. Then there exists an x E X, ll x ll = 1 such that if (x n )�= I C X, ll xn I ::; 1, n = 1, 2, . . . and Tx n ---. Tx then

T: X ---. Y Xn ___. X.

Since T is a semi-embedding, the image of every closed ball in X is closed in Y. Let us fix c: > 0 and a relatively open set V C T(Bx). Let us fix a sequence (vj ) � 1 C Bx such that we can write Bx = U;: 1 B(vj , c:) n Bx . Applying the Baire category theorem to the covering of V by closed sets T ( B ( vj , c:) n Bx) we get the following statement:

Proof:

> 0 and for every relatively open set V C T(Bx) there exists a non-empty relatively open set U C V such (1) 1 that diam T (U) < c:. for every c:

Applying (1) inductively we find a sequence of relatively open sets (Un )�= I in T(Bx) such that Un :J Un + I :J Un +I and diam T - 1 (Un ) < � and 0 � ul . Clearly n�=l T - 1 Un consists of exactly one point Xo . The a desired X equals .

I �� I

4.

Applying Proposition 3 for

X = L 1 [0, 1]

we get

Lemma. If T: L 1 [0, 1] ---. X is a semi-embedding, then there exists an f E L 1 [0, 1] with II f II = 1 and a number 8 > 0 such that for all 1 real-valued functions r.p with J0 r.p(t)dt = 0 and I'PI = 1, p,-a.e. we have

II T( rp f) ll

�

8.

From Proposition 3 we infer that there exists an f E L 1 [0 , 1] , 11 !11 = 1 and 8 > 0 such that II ! - gi l < � whenever IIYII ::; 1 and li T f - Tg ll < 8. Given r.p above we take '1/J = r.p or 'ljJ = -r.p so that we have J(1 + '1/J) I f l dp, ::; 1. For g = (1 + '1/J)f we have IIYII ::; 1 and IIY - ! II = 11 '1/Jf ll = 1, thus 8 < li T/ - Tg ll · But II T( rpf ) ll = li T/ - Tg ll

Proof:

as

so the claim follows.

5.

a

Now we are ready to prove

Theorem. Let p, be an atom-free, separable measure. There is no semi-embedding from L 1 (p,) into Co · We see from III.A. 1 that it is enough to consider LI [O, 1] only. Assume to the contrary that T: £ 1 [0, 1 J ---. eo is a semi-embedding, and

Proof:

III. C.

L 1 (p,)-Spaces §6.

133

II T II = 1. Take f E L 1 [0 , 1] and 8 > 0 as in Lemma 4. Fix an integer N > 2 /8 , and let A� . . . . , A N be disjoint sets such that fA - 1/1 = j;; . 1 For each n , n = 1 , 2, . . . , N let rj denote a sequence of Rademacher-like functions on An , i.e.

l r'J I = X An ' rj = 0,

J

= 1 , . . . , N, j = 1 , 2, . . . , n = 1, . . . , N, j = 1, 2, . . . , n

rj f �O as j ---+ oo ,

for every

n, n

= 1, 2, . . . , N.

(2)

(3) (4)

Using the standard 'gliding hump' argument (cf. proof of II.B. 17) con­ dition (4) yields numbers j (n) , n = 1 , . . . , N, such that

But (2) , (3) and Lemma 4 give

which is impossible. This contradiction proves the theorem.

a

6. The spirit of the above results is that it is rather difficult to put a weaker, linear topology on the unit ball of L 1 [0, 1] and make it compact or even complete. The intuition is that somehow we always have to add some singular measures in the completion. The following classical theorem of Menchoff is only an example of this. Theorem. (Menchoff) . Let G b e a compact, infinite, metrizable abelian group with dual group r. Let M0 (G) c M(G) denote the set of all measures v with C'(-y) E eo(r). Then Mo is a non-separable closed band in M(G) . Since the Fourier transform A: M(G) ---+ l00 (r) is continous and since Mo = (A - 1 )(c0(r)), we see that M0 is a closed linear subspace. Also for v E Mo and p = :L: ..,. eA a..,. 'Y with finite A c r one easily checks that p · v E Mo. Since Mo is closed this gives that Mo is a band. If M0 is separable, then there exists a positive measure p, E M (G) such that Mo = L1 (p,) . One easily checks that p, has no atoms. By Theorem 5 (B M0 )A is not closed in co ( r} , i.e. there exist o!(-y) E c0(r)\(BM0 )A

Proof:

134

III. C.

and fLn E such that a in of Since 1 11-LII � 1 and contradicts the choice of a.

BMo {!Ln} �=1·

P,n -+

P,

L1(IL)-Spaces § 7.

eo(r).

= a

Let fL be an w*-cluster point we infer that fL E This

BMo·

a

Note that the group structure plays almost no role in the above argument.

7. Given a class of measures it is natural to seek its zero sets, i.e. sets of measure zero with respect to every measure in the class. We will discuss zero sets for M0 (11'). This is a small part of the classical branch of the theory of Fourier series or harmonic analysis on more gen­ eral groups. In order to proceed we need some definitions. We say that a sequence C 11' has an asymptotic distribution if and only if N 8x n converges a(M(11') , C(T)) to some measure v E M(11'). Let

(xn )�=1 1 N- n=L:1 us recall that 8x denote the Dirac measure concentrated at the point x. 1 Since for every m E we have zl(m) lim N -+oo ( N- L::= 1 8x n ) " (m) 1 lim N -+oo ( N- L::=1 exp ( - im x n ) ) we see that the sequence (x n )�= 1 N 11' has an asymptotic distribution if and only if Nlim k L: exp( - im xn)) ( -+oo =1 n exists for every m E Z. A Borel subset E C 11' is called a Weyl set if there exists an increasing sequence of integers (nk)k:, 1 such that for ev­ ery x E E the sequence (nk · x)f= 1 has asymptotic distribution with 7l

=

the corresponding measure measure.

8.

Vx

=

C

an

different from the normalized Lebesgue

The following theorem characterizes

M0 (11').

The measure fL is in Mo(11') if an d only if fL(E) every Weyl set E C 11'.

Theorem.

=

0

for

Throughout the whole proof it is sufficient to consider only positive measures (see Theorem 6). =>. Let fL be a non-zero measure in Mo 11' and let E be a Weyl set with the corresponding sequence For m E Z, m =/= 0 we put

Proof:

c

m

(t) =

{

(nk)k:, 1 .

1

lim N-+ oo N

. 0,

()

t k =1 exp ( - imnk t)

,

t E E, t (/_ E .

(5)

III. C.

£1 (p,)-Spaces §9.

For a Borel subset

135

F c E we have (6)

Mo(ll') is a band (Theorem 6) p, I F E Mo(ll') so (6) gives JF Cm (t)dp,(t) = 0. Since F was an arbitrary Borel subset of E we infer that cm (t) = 0, p,-a.e. for m =/= 0. But E is a Weyl set, thus for every t E E there is an m =I= 0 such that em (t) =I= 0. This shows that p,(E) = 0. ¢=: . Let us take p, fJ. M0(11') and let us fix a sequence of integers n k � oo as k - oo (or n k � -oo as k - oo ) such that jL(n k ) � Since

a

=I= 0 as k -

oo.

Applying Theorem III.A.29 to the family of sequences

{exp(-imn k t) }f= 1 in L2 (11', dp,) we get a further subsequence (n�)� 1 such that JJV (t) = N - 1 L�= 1 exp( - imn�t) converges p,-a.e for each m E 7l. Let us put E = {t E 11' : N-+oo lim JJV(t) exists for each m E 7l and lim j].{t) is not zero}. N-+oo

This is a Weyl set. We have

Thus

a

p,(E) > 0.

9. Our goal now is to characterize weakly compact sets in L 1 (p,)­ spaces. This characterization has many further applications (see III. C.19 or III.H. l O among others) and generalizations (see e.g. III.D.31 ) . It also nicely connects the general functional analytic notions with measure­ theoretical concepts. We start with the distinctive special case of the space £ 1 . Theorem. (Schur) . For a bounded subset H ditions are equivalent:

c

£1

the following con­

III. C. L 1 ( !-l ) -Spaces § 1 0.

136 (a) (b) (c)

H is relatively compact; H is relatively weakly compact; there is no sequence (an )�= 1 C H which is a basic sequence equiv­ alent to the unit vector basis of € 1 . The proof clearly follows from the following.

10 Lemma. If H C € 1 is a bounded subset, not relatively compact, then there exists a basic sequence (an)�= 1 C H equivalent to the unit vector basis of € 1 . We find {bn}�= 1 C H such that I I bn II :::; C, n = 1 , 2, . . . for some C and ll bn - bm ll 2: 8, for m #- n and {j > 0. A standard diagonal procedure gives a subsequence { bnJ � 1 such that bn1 (k) ___, b(k) as j -+ oo, for every k = 1 , 2, . . . . Clearly b E € 1 and l l bn1 - b ll 2: 8, j = 1 , 2, . . . . II.B.17 gives a further subsequence (call it also bnJ such that (bnj - b)� 1 is equivalent to a block-basic sequence, thus to the unit vector basis in € 1 . Let Y = span{(bnj - b) }� 1 . Omitting if necessary a finite number of j's we can assume that b � Y. Then

Proof:

I :L:O �j bnj ll = II �:::j:a� (bnj - b) + < :L:O �j )b ll 2: K L iaJ I , thus

{bnJ � 1

2:

K l l :L:O �j (bnj - b) ll II

is the desired sequence.

11. Now we will discuss relatively weakly compact sets in L 1 ( !-l ) for a general probability measure 1-l· Our main tool will be the notion of uniform integrability. Definition. A subset H C L 1 (1-l) is called uniformly integrable if for every c: > 0 there exists an TJ > 0 such that sup

{ i l f l d!-L : !-L(A)

:::;

TJ,

fEH

}

:::;

c:.

(7)

If 1-l is an atom-free probability measure then every uniformly in­ tegrable set in L1 (!-l) is norm-bounded. This follows from (7) and the observation that for every f E L1 (!-l) there exists a set A with !-l(A) = * such that fA l f ld!-L 2: n - 1 J l f ldl-l. In the other direction let us observe

III. C. Ll (IL)-Spaces §12.

137

that every one-element set, and thus every finite set, is uniformly in­ tegrable. To see this put A n = {t : l f(t) l > n } . Since IL( A n ) -+ 0, the Lebesgue dominated convergence theorem gives l f l diL --+ 0 as

JAn

n -+ oo .

12. The next theorem gives the promised characterization of relatively weakly compact sets in £ 1 (IL) . This is the main result of this chapter. It says that basically there is only one reason for a bounded set not to be relatively weakly compact. In this sense it is similar to Proposition II.D.5 and also to Theorem 9. Note also the equivalence of finite and infinite conditions. This will be investigated later. Theorem. Let IL be a probability measure and let H be a bounded subset of Ll (IL) · The following conditions are equivalent: ( a) H is not relatively weakly compact in Ll (IL) i ( b) H is not uniformly integrable; ( c ) there exists an that

e > 0 and a sequence of disjoint sets (An) ;:::>= 1 n =

( d ) · there exists a basic sequence vector basis in e 1 ;

( Jn );:::>= 1

such

1 , 2, . . .

C H equivalent to the unit

( e ) there exists an e > 0 such that for every integer N there exist N disjoint sets A1 , . . . , A N such that sup

{ in l f l diL : f E } H

�

e,

n =

1 , 2 . . . , N;

( f ) there exists a constant K such that for every integer N there exist it , . . . , fN C H, K -equivalent to the unit vector basis in if . Proof:

The proof will consist of the following implications:

(b)

/ ( a) � (e)

(d ) � ( f)

�(e) /

III. C.

138

Lt (J-t)-Spaces § 1 2.

with the implications marked * being obvious. (a) -+ (b) . Suppose H C Lt (J-t) is uniformly integrable, thus bounded, i.e. for f E H we have 11!11 � M. Given an integer n we write every function f E H as f = r + fn = f · X{ l f l �n } + f · X{ l f l 0. Put hn = (JAn l fn l dJ-t ) - t fn . X An and cpn = sgn fn · XAn · Clearly Y = span{hn }�= t is isometric to it and P(f) = I: :'= t J f cpndw hn is a projection from Lt (J-t) onto Y. One easily sees that P ({ fn }�=t ) is not relatively compact in norm. From Theorem 9 we get a subsequence Un; } � t such that { P C fn; H � t is equivalent to the unit vector basis in it . but this implies that Un; } � t itself is equivalent to the unit vector basis in it . (f) -+ (e) . We can assume II !J II � 1 , j = 1 , . . . , N and thus K - t I:f=t l aj l � J I I:f= t a3 f3 l for all sequences of scalars ( aj ) f=t · Let r3 (t) be, as usual, the Rademacher functions. We have

III. C. L t ( J.L )-Spaces §13.

139

l

( �N l aj fi l ) dJ.L j N :::; ( J mr l aj fi i dJ.L) ( J � iaj fi l dJ.L) :::; (! mr l aj fi l dJ.L) ( �N l aj l )

:::; c mr l aj fi l ) �

·

2

l

2

l

(8) l

2

l

2

2

•

Thus (9) and in particular ( 10) Let B8, s = 1 , 2, . . . , N, be disjoint sets such that (some of them can be empty) . From (10) we get

11/s ll :::; 1 we JB. l fs l dJ.L � 2 k2 ·

Since

(maxj lfi i ) I Bs = l fs l

infer that for at least ( 2Kllf- t ) indices

s

we have a

13. Remarks. (a) If we keep track of the constants in the proof of (f) --+ (e) we get the following statement: If (JJ ) f= 1 C L1 (J.L) are such that K-

1

� N

l aj l :::;

JI� N

l � l aj l

aj fi dJ.L :::;

N

for all scalars(aJ ) f= 1

then for every 8 < 1 there exists a subset A C { 1 , . . . , N } and disjoint sets {Aj } jE A such that

where for every 8, r.p0 (N)

----+

oo

as N

.......

oo .

III. C.

140

Lt (11)-Spaces § 1 4.

(b) If 11 is an arbitrary measure on n and H c Lt (0, 11) is weakly relatively compact then there exists a set nl c n of a-finite measure 11 such that all functions from H are supported on 0 1 . Thus when dealing with relatively weakly compact sets in £ 1 (11) we can restrict our attention to a-finite 11 · But this case, as we know (II.B.2(c) ) , easily reduces to the probability measure 11 ·

14 Corollary. (Steinhaus) . The space Lt (l1) , 11 arbitrary, is weakly sequentially complete, i.e., weakly Cauchy sequences are weakly conver­ gent. Proof: Since every sequence is supported on a a-finite set it is enough to consider a probability measure 11 (Remark 13(b) ) . A weakly Cauchy sequence which is not weakly convergent is not relatively weakly com­ pact, so from Theorem 12 it has a subsequence equivalent to the unit vector basis in ft , thus not weakly Cauchy. This contradiction proves the corollary. II 15. We have seen in Theorem 12 the interesting interplay between global notions like weak compactness and the local concept of finite representability of £ 1 . We want now to discuss the finite representability of £ 1 in a general Banach space X. We start with an interesting lemma about finite dimensional isomorphs of if . Lemma. Let X be an N-dimensional Banach space with d ( X , if ) = a. Then there exists a subspace X1 C X with dim X1 = [ VN] and

d(Xt , € 1[ v'NJ ) :::; yr;:.a.

Proof: scalars

Let us fix a basis we have

( a3) f= 1

(x3) f= 1

in

X such that for every sequence of

Let us also fix [ VN] disjoint subsets A s c { 1 , 2, . . . , N} each with car­ dinality [ VN] . For each s = 1 , 2, . . . , [ VN] we define

:

III. C. L1 (11)-Spaces § 1 6.

141

If for some s we have ds � )a than xl = span{Xj j E A s is a good choice. On the other hand if for all s we have d8 < )a then we fix Ys = LjeA . fr.jXj such that IIYs ll < )a and L jeA. l ai l = 1 for s =

1 , 2, . . . ,

so for X1

=

[vr.:r] . 1v

For every sequence of scalars

span { Ys

}

[v'NJ we have

(f3s ) s = l

•

r;;.

} s[v'N= l ] we get d(Xt , i1[v'N] ) � y a .

16 Theorem. Let X be an infinite dimensional Banach space. The following conditions are equivalent: (a) X does not have type p for any p

> 1; and every e > 0 there exist norm-one vectors

(b) for every n = 1 , 2, . . . Xt , . . . , X n in X such that

(c) i 1 is finitely representable in X ; (d) for every n = 1 , 2, . . . and every e Xn , e C X with d ( Xn , e , ii' ) � 1 + e.

>

0 there exists a subspace

This is a remarkable theorem. We will use it in III.I to study some questions about the disc algebra. Note that it contains the passage from a probabilistic context of the definition of type to the purely determin­ istic situation described in (c) and (d) . It is also nice because it tells us that certain abstract things (like (a) ) happen only due to the presence of a very 'concrete' subspaces, namely il 's. For the proof of this theorem we introduce constants 'Yn (X) , n = 1, 2, . . . , defined by the formula 'Yn (X)

{ ')' : ( / I t

2 ri (t)x i dt

=

inf

�

'/' Vn ( t llxi ll 2) ' for all

( x i )� 1 C

X} ·

l )

�

(11)

III. C. L 1 (J-t)-Spaces § 1 7.

142

Note that 'Yn (X) :5 1 for n = 1 , 2, . . . . The following lemma really explains some consequences of condition (a) of Theorem 16. 17 Lemma. (a) The constants 'Yn (X) are submultiplicative, 'Yn· k (X) :5 'Yn (X) · 'Yk (X) for all n, k E N. (b) If 'Yn(X) < 1 for some n then X has type p for some p > 1 .

(xj ) j;:1

(a) Fix integers such that

For

0, . . . , (k

Proof:

s =

- 1)

n

and

k,

a number

e>

i.e.

0 and a sequence

define

( s+ l ) n r/Js (O) = L Tj (O)xj . j = s· n + l For every (} we have

and integrating over (} we get

k- l

'Y� ( X ) k L

J

ll ¢s (O) II 2 d0 s=O (s+ l ) n k-l ( X ) 'Y k (X)n :5 'Y� L ll xi ll 2 L � j = s· n + l s=O k ·n 2 = 'Y�(X) · 'Y� (X) · k · n L ll xi ll • j=l :5

( 13)

e was arbitrary, comparing (12) and (13) we get 'Yn· k (X) < 'Yk (X)'Yn (X). ( b ) Fix q > 1 such that 'Yn(X) = n - l / q' where * + f, = 1 . Observe also that it follows directly from ( 1 1 ) that ('Yn (X) Jn)�= l is an increasing Since

III. C. L ( !1 ) -Spaces § 1 7.

1

143

(xj )j� 1 2::7= 1 I xi l i P = s+l ) /p :::; l xJ I :::; n - s/P } . ( A = n s 1 I As l :::; n8+ . 2 ! I ( ( J I t- l rj (t)xj l 2 dt) ! ::; � L rj ( t )xj l dt) / s-0 J E As J :::; L I'I A.I ( X ) JiA:T ( _L l xi l 1 2 ) 2 s=O J EAs 1 :::; L l'n • +l (X) vns +l (n8 + n - 2 s iP )! j=O - � ) s+l ns+l n - s/p (n :::; L s=O :::; n 1 - .!. s=O ns ( l - .!._1 ) This shows that X is of type for every

sequence. Take 1

< p < q.

For an arbitrary finite sequence with 1 (we can put in some additional zeros to have the right length) we define sets of indices {j : Clearly We have k

!

00

00

·

00

"\"' � 00

•'

p

•'

< oo .

v

a

p < q.

From Lemma 15 we see that (c){:}( d) . Also obviously both (c) and (d) imply (a) . (a)=>(b) . From Lemma 17 we see that l'n 1 for = 1, 2, 3 . . . . This means that for every Xn in such that = 1 , 2, . . . and every e > 0 there are vectors

Proof of Theorem 16.

(X) =

n

n

x1, , •

n)

•

•

X

l xi l , j =

If e is very small (depending on we see from (14) that 1 , 2, has to be practically constant so has the property for every and every e > 0 there exist vectors such that 1 for j 1, XI , , Xn E with

. . . , n,

n

X I xi I = (1 - e)n :::; (J I j=lI:n rj (t)xJ I 2 ) 2 . •

.

.

1

X = . . . ,n

Since obviously

2 dt) ! 1 ( )x t j j J r ( I� :::; (T n

"�2�ll t€jXj l 2

+ (1 - 2 - n )

( t 11 x1 11 r ) !

(15)

144

III. C. Lt (!-L)-Spaces § 1 8.

we infer from (15) (make c: very small) that X satisfies (b) . (b)=>(c). For each sequence T/ = (c:i )' J= 1 with C:j = ±1 there exists a functional x; E X* such that ll x; ll = 1 and E;= l x; (c:i xi ) > n - c: . Since I xi I = 1, an elementary computation shows that for every T/ and j, we have l x; (c:i xi ) - 1 1 < J&. Using this, for any sequence of numbers (ai ) J= 1 with Ej= 1 l ai l = 1 we obtain

a

18. As an application of our previous considerations we have the following useful Corollary. Let X be a closed subspace of Lt (!-L) · The following con­ ditions are equivalent:

(a) X is reflexive;

(b) X has type p for some p > 1;

(c) X does not contain a subspace isomorphic t o f 1 ; (d) f 1 is not finitely representable in X. Since each of the conditions holds for X if and only if it holds for every separable subspace of X we can assume that 1-L is a probability measure (see III.A.2). Now the corollary immediately follows a from Theorem 12 and Theorem 16 (see also II.A.14) . Proof:

19.

We wish to conclude this section with the proof of the following.

Theorem.

H00 (1I') } < 1. 0 such that

Suppose Fo E L00 (1I') is such that inf { I I Fo + h ll oo : h E Then there exists an F E Fo + Hoo (1I') and h E Ht (1I') , h =/=

F · h = lhl

a. e . on

1I' .

(16)

The proof of this theorem is a nice application of Theorem 12. More­ over the following lemma is relevant to some questions which will be discussed in Chapter III . E .

I45

Ill. C. £1 (J-L)-Spaces §20.

20 Lemma. If {En } is a sequence of measurable subsets of '][' such that I En l --+ 0, then there is a sequence {gn } of functions in H00 such that

(a) supess{ l9n (t) l : t E En } ----+ 0 as n --+ oo , (b) 9n (O) = (211") - 1 fv 9n (t)dt ----+ I as n --+ oo , (c) IYn l + I I - 9n l :5 I + en where lim en = 0. Fix numbers An such that An ----+ oo as n --+ oo and An iEn l ----+ 0 as n --+ oo. Let In be the Poisson integral of AnXEn + iAnXEn (XEn is the harmonic conjugate of XEn , I.B.22) . Clearly In is an analytic function on D taking values in the right half plane. Since the map z 1--+ ( l ! z ) maps the right half plane onto the disc Proof:

( I 7) we get that hn (z) = ( 1 + /n ( z ) ) maps D into the disc given by (I7) . Since ln (O) = A n iEn l we get hn (O) ----+ I as n --+ oo . Also supess{ l hn (t) l : t E En } :5 supess{ =

I : t E En } (I + Re ln (t))

I ----+ 0 (I + An)

as

n --+ oo .

Now observe that the map z --+ z6 compresses the disc (I7) into the ellipse l w l + I I - w l :5 I + e(6) where e(6) ----+ 0 as 6 --+ 0. All the above yields that for some sequence On --+ 0 slowly enough the functions a 9n = h�n satisfy (a) , (b) and (c) . Proof of Theorem 19. Put

Since the unit ball in H00 is a(L00 , £1 )-compact this supremum is at­ tained at some F E F0 + H00 • Clearly I � dist(F, H�,) = inf{ II F - h ll : h E Hoo , h(O) = 0}. If II F - h lloo < I for some h E H! we see that for small e's the function F - h + c: is an admissible I in (IS) giving a larger mean than F. This shows that dist(F, H! ) = 1 . Since Hi = Leo/ H! we get by duality

146

III. C. L1 (J-t)-Spaces §20.

Fix a sequence (hn)�= l in H1 with ll hn ll � 1 and

(19) If (hn)�= l has a weakly convergent subsequence (hnk )k:: 1 we put h w- lim hnk E H1 (Y) . Now (19) gives

=

so in particular h =i 0. Since II F IIoo � 1 and ll h lh � 1 we get (16). We complete the proof by showing that the assumption that (hn)�= l has no weakly convergent subsequence leads to a contradic­ tion. If (hn )�= l has no weakly convergent subsequence Theorem II.C.3 (Eberlein-Smulian) and Theorem 12 give sets (En) C 1I' with I En l -+ 0 such that (20) hn (e i8)d0 > {3 > 0 2

1

� � Ln

at least for a subsequence of (hn) · Now let 9n and C:n be given by Lemma 20 and put (1 - 9n)hn . Hn - 9nhn and Kn = 1 + C:n 1 + C:n We infer from Lemma 20(c) that II Hn l l 1 + II Kn l l 1 � ll hn ll < 1. Also since C:n -+ 0, (19) gives _

1 1'1m 1 - n--+oo 2 71'

1 1211" 0

F

hn 1 + C:n

--

I

(2 1 )

Thus the limit in the middle exists and equals 1. From (20) and Lemma 20 (a) we get limn II Kn l h � {3 > 0 so (21) yields (2 2 ) Note that Lemma 20 (b) gives Kn (O) ----+ 0 as n -+ oo. The dual­ ity relation (HP)* = L 00 / Hoo and ( 22 ) give dist(F, H00) > 1. But

III. C. L1 (t-L)-Spaces §21 .

147

dist(F, Hoo ) = dist(Fo , H00 ) < 1. This contradiction shows that a (h n );:"= 1 actually has a weakly convergent subsequence.

21 Corollary. Let (z3 ) �1 C 1D be such that 2:::3 (1 - l zJ I ) < oo . For every f E H00 with ll f ll oo < 1 there exists an inner function r.p with r.p ( zj ) = f(zj), j = 1, 2, 3, . . . . Let B be the Blaschke product with zeros (z3 ) �1 . We de­ fine Fo E L00 (1r) by Fo = B f and apply Theorem 19. We obtain a unimodular F = F0 + g for some g E H00 • Since on the circle '][' we have BF = f + Bg, we see that BF is a boundary value of an analytic function. This function is clearly inner and satisfies BF(z3 ) = f (z3 ) for a j = 1, 2, . . . . Thus BF is the desired r.p.

Proof:

Notes and remarks. The fact that there is no weaker topology on L1 (f..L ) , f..L-atom-free, with some compactness properties is well established. Our Proposition 1 only states the easiest fact of this type. A more detailed study of semi­ embeddings is in Bourgain-Rosenthal [1983] . This paper contains our Theorem 5 and the proof of Theorem 6. The first proof of Theorem 6 was given in Menchoff [1916] . Much more detailed information on supports of measures in Mo(G) can be found in Varopoulos [1966] . The following fact was shown in Pigno-Saeki [1973] .

Theorem A. lE f..L is a measure in M(G) such that then f..L E L 1 ( G) .

f..L * Mo(G) c L1 ( G )

Theorem 8 is taken from Lyons [1985] . It completes a long line of investigations in the theory of Fourier series and solves problems going back to Rajchman in the 20 ' s. The non-trivial implication (b)::::} (a) in Theorem 9 is due to Schur [1921] in the language of summability methods. Banach spaces where this implication holds are nowadays said to satisfy the Schur property. The fundamental Theorem 12 is usually called the Dunford-Pettis theorem. They established the equivalence between (a) and (b) and successfully used it in their papers Dunford [1939] and Dunford-Pettis [1940] . The relevance of condition (d) was realized in Kadec-Pelczynski [1962] . This theorem is by now classical and various versions of it with many different applications are presented in Dunford-Schwartz [1958] , Diestel-Uhl [1977] , Kopp [1984] . This last book shows the fundamental importance of this theorem in probability

III. C. L1 ( J.L )-Spaces §Exercises

148

theory. Corollary 14 is a classical theorem of Steinhaus [1919] . We will discuss important generalizations of these facts in the next chapter. Lemma 1 5 is a well known finite dimensional version of a result of James [1964] (see also Exercise 9) . The important Theorem 1 6 was proved by Pisier [1974] . We basically reproduce his proof here with the changes necessary to obtain the result for complex spaces as well. This theorem was the beginning of the study of connections between type and cotype on one side and geometry on the other side. By now the subject has grown enormously. A presentation of this is contained in the beautiful monograph Milman-Schechtman [1986] . Let us only note the direct generalization of Theorem 1 6 proved by Maurey-Pisier [1976] .

Theorem B. Let X be an infinite dimensional Banach space and let Px = sup{p: X has type p} , qx = inf{q: X has cotype q}.

Then fpx and lqx are finitely representable in X .

The connection between the reflexivity of subspaces of L1 ( J..L ) and conditions like (b) in Corollary 1 8 was recognized in the important paper Rosenthal [1973] . Theorem 1 9 and Lemma 20 is due to Garnett [1977] (see also Gar­ nett [1981] ) . The first version of Lemma 20, with a more complicated proof, was discovered by Ravin [1973] . Various variants, usually referred to as the Ravin lemma, are known. The main idea is to show that on small sets there are analytic functions almost peaking on them. Theorem 111.1.9 presents a very elaborate version of this idea. Corollary 21 is an easy special case of a classical theorem of Nevanlinna. For more details on such matters the reader should consult Garnett [1981] , in particular chapter IV.4.

Exercises

1.

Suppose that T : c0 ---+ X is a semi-embedding. Show that T is an embedding.

2.

Suppose X is a Banach space and X* is separable. Show that X* does not contain a subspace isomorphic to L1 [0, 1] .

3.

Construct a sequence which does not have the asymptotic distribu­ tion.

III. C.

4.

L 1 (J.L)-Spaces §Exercises

149

Suppose that J.L is an arbitrary measure on f! and that H c £ 1 (J.L) is a relatively weakly compact subset. Show that there exists a subset V C f! of a-finite measure such that for every f E H we have suppf C V.

5.

Suppose (f!, J.L) is a probability measure space and K is a compact space. Show that, if T: £ 1 (J.L) - M ( K) is a continuous linear opera­ tor, then there exists 11 E M(K) , 11 � 0 such that T(L 1 (J.L)) C £ 1 ( 11 ) .

6.

Show that H C £1 (J.L) is relatively norm-compact if and only if it is relatively weakly compact and relatively compact with respect to convergence in measure.

7.

Find a weakly compact set H c £ 1 [0, 1] such that there is no func­ tion cp � 1 , cp finite almost everywhere, such that {!: f · cp E H} C Lp [O, 1] for some p > 1.

8.

Suppose that Un)':'= 1 is a sequence in H1 (1l') and that fn � f and 11/n ll - II/II as n - oo for some f E H1 (1l'). Show that 11/n - /II - 0 as n - oo .

9.

Suppose X is isomorphic to £ 1 . Show that for every c > 0 there exists an infinite dimensional subspace X 1 c X with d (X 1 . £ 1 ) �

1 + c.

10. Show that if £ 1 is not finitely representable in finitely representable in X* .

X,

then

£1

is not

1 1 . Suppose that the sets (En ) in Lemma 20 are closed. Show that the functions (gn ) can be chosen in the disc algebra A.

12. Show that if H c £1 [0, 1] is not uniformly integrable then there exists a sequence (cp n)':'= 1 C C[O, 1] such that E:'= 1 I 'Pn(t) l � 1 1 and lim sup n--+oo sup hEH I f0 'Pn(t)h (t)dt l > 0.

13. Suppose that T: X - £ 1 is a non-compact operator. Show that there exists a complemented subspace X1 and T I X1 is an isomorphic embedding.

c

X such that X1 "' £ 1

III.D. C(K)- S paces We start this chapter with the general notion of an M-ideal and show that for every element one can find a best approximation to it in any M-ideal. We discuss the space Hoo + C and show that every function f E L00(1l') has a best approximation in H00 + C. We prove the linear extension theorem of Michael and Pelczyfiski and the Milutin theorem that all spaces C(K) for K compact, metric and uncountable are iso­ morphic. We present the construction of the periodic Franklin system and prove its basic properties. We investigate its behaviour in Lp (1r) , C(1l') and Lip0 (1l'). We show that Lip0(1r) "' £00 • Then we investigate weakly compact sets in duals of C k (1r8 ) and A(1Bd ) and show that they have properties similar to weakly compact sets in £ 1 The unifying framework for this study is provided by the concept of a rich subspace of C(K, E). We also introduce and study the concepts of the Dunford­ Pettis property and the Pelczyfiski property.

(J.L) .

1. Our subject in this chapter is the spaces of the form C(K) where K is a compact space. This includes also spaces L00 ( J.L ) since they can be realized as C(K) (I.B.lO) . This fact is a standard result in the theory of Banach algebras. It shows the importance of the multiplicative structure existing in C(K)-spaces. Given a closed subset S C K put C(K; S) = {! E C(K) : / I S = 0 } . This is clearly a closed subspace and actually an ideal in the algebra C(K) . Proposition. Every closed ideal in C(K) is of the form C(K; S) for some closed S C K. Given a non-trivial closed ideal I C C(K) we define 81 = Since every I E I is non-invertible we get , - 1 (0) =F 0. Given It and h in I we see that 1/1 1 2 + 1 12 1 2 = !d1 + !d2 E I which gives that the family {f- 1 (0) } J E I is a family with the finite intersection property so its intersection 81 is not empty. Thus C(K; 81) is a proper ideal containing I. Note that if f E I then f E I. To see this let g E C(K) be such that g = f/1 on the set {k E K: 1/1 2:: c } and IIYII � 1. Such a g exists by the Tietze extension theorem. Then g · f E I and IIY · f - fl loo � 2c. Since c was arbitrary and I was closed we get f E I . Note also that for any two points k1 =F k2 in K\81 there is an

Proof:

n! E l ,- 1 (0).

152

III.D. C(K)-Spaces §2.

f E I with f (kl ) =F f(k2 ) =F 0. Thus the Stone-Weierstrass theorem a I.B.12 yields that I = C(K; SI ) . The notion of an ideal is not a linear concept but the following concept generalizes the above considerations.

2.

Definition. A subspace M of a Banach space X is called an M-ideal if there exists a projection E from X* onto Ml. = {x* E X*: x* I M = 0 } such that for every x* E X* we have

ll x * ll = II Ex * ll + ll x * - Ex * II · One checks that every subspace of C(K) of the form C(K; S) is an M-ideal. The projection E of a measure p, is given by E(p,) = p, I S. Other examples of M-ideals are given in Exercises 1 and 2 and also in Theorem 8.

3 Proposition. Let M be an M-ideal in X and let open balls B(xt , rt ) = B1 and B(x 2 , r2 ) = B2 be such that B 1 nB2 =F 0, B 1 nM =F 0 and B2 n M =F 0. Then B 1 n B2 n M =F 0. Proof: Consider B 1 x B2 C X E9 X and let a = {(m, m) : m E M} C X E9X. If the conclusion does not hold then (B 1 x B2 ) na = 0. Since B 1 and B2 are open we can diminish r1 and r2 a bit in such a way that the assumptions are still satisfied and there exists cp = ('Pt . cp2 ) E X* E9 X*

such that

for some positive c. Passing to biduals we get from Goldstine's theorem II.A.13 that a** = { ( m ** ' m **) : m ** E M** } is disjoint from Bi* X B2* = B(x1 , rt ) x B(x 2 , r2 ) where the balls are in X** this time. This translates back to the statement that there are two open balls Bi* , B2* in X** such that Bi* n M** =F 0, B2* n M** =F 0, Bi* n B2* =F 0 and Bi* n B2* n M** = 0. But X** = (M** E9 Z) oo (since M is an M-ideal) a so this is clearly impossible. Using this proposition we will obtain the following useful

4.

If M is an M -ideal in X, then the quotient map X/M maps the closed unit ball in X onto the closed unit

Theorem. q:

X

--+

153

III.D. C(K)-Spaces §5. ball in X/M, or equivalently, for every M : ll x - m il = dist(x, M) } =1 0.

xE

X the set

PM(x)

=

{m E

Let us note that the open mapping theorem yields that every quo­ tient map maps the open unit ball onto the open unit ball. For the closed unit ball this is generally false. As an example take map T: £ 1 ---+ £2 de­ fined by T(e n) = Xn where ( xn) �= 1 is a sequence dense in the open unit ball of £2 . Clearly T maps the closed unit ball of £ 1 onto the open unit ball of £2 . Fix x E X\M and let d = dist(x, M) . Given c > 0 there exists m 1 E M such that ll x - m 1 ll < d + c. The open balls B(mt . �c) and B(x, d + !c) satisfy the assumptions of Proposition 3 so there exists m2 E M such that ll m 1 - m2 ll :::; �c and ll x - m2 ll < d + !c. Now the balls B(m2 , ( i} 2 c) and B(x, d + (!) 2 c) satisfy the assumptions of Proposition 3, so we get m3 E M such that ll m2 - m3 ll < ( � ) 2 c and ll x - m3 11 < d + (!) 3 c. Continuing in this man­ ner we find a sequence (m k )k"= 1 in M such that ll m k - m k + l ll < ( � ) k c and ll x - m k + l ll < d + ( ! ) k c. This sequence clearly converges to a limit m E M and ll x - m il :::; d. a

Proof of the theorem.

Remark.

Actually the above argument gives the following statement.

For every m 1 cM with

m E PM ( x) such that

5.

ll m 1 - x ll < d + c there exists li m - m 1 ll :::; 3c.

(1)

Using (1) we obtain the following improvement of Theorem 4.

Proposition. If M is an M-ideal in X, then for every x set PM (x) algebraically spans M.

E X\M the

Proof: For every m E M with ll m ll < d = dist(x, M) we will show that m = z - v for some z, v E PM (x) . Since PM (x + m) = m + PM (x) we have to show that PM (x) n PM (x + m) =1 0. Assume to the contrary that PM (x) n PM (X + m) = 0. Then

dist(PM(x), PM(x + 2m)) � l i m II ·

From Proposition 3 we get that for every

c > 0 there exists a point

me E B(x, d + c) n B(x + 2m, d + c) n M.

(2)

154

III.D. C(K)-Spaces §6.

From (1) we get m 1 E PM (x) such that ll m 1 - me ll $ 3c and m2 E PM (x+2m) such that ll m2 - me ll ::5 3c, so dist(PM (x), PM (x+2m) ) $ 6c. a This contradicts (2) if we choose c < lo II m il · This proposition shows that the best approximation by elements of an M-ideal is always possible and in a very non-unique way. A concrete application of this fact will be given in Corollary 9.

6. Let us introduce now the space H00 + C which has many important applications in operator theory and function theory. To be more precise Hoo + C is the subspace of L00 {1r) algebraically spanned by C(1l') and boundary values of H00 • Our study of this space is based on the following general Lemma. Suppose Y and Z are closed subspaces of a Banach space X and suppose that there is a family � of uniformly bounded operators from X into X such that (a) every A in � maps X into Y, (b) every A in � maps Z into Z, (c) Eo� every y E Y and c

> 0 there exists A E � such that ll y- Ay ll < c.

Then the algebraic sum Y + Z is closed. Let x E Y + Z. We can find a sequence X n E Y + Z such that E�= l Xn = x and ll xn ll $ 2 - n for n > 1. Every Xn, n = 1, 2, . . . can be written as X n = Yn + Zn with Yn E Y and Zn E Z. Using (c) we can find An, n = 1, 2, . . . , in � such that llYn - A n Yn I < 2 - n . Let us put Yn = Yn - An Yn + AnXn and Zn = Zn - An Zn · Properties (a) and (b) show that Yn E Y and Zn E Z. Since � is uniformly bounded we see that II:Yn ll ::5 c 2 - n . Since X n = Yn + Zn we have ll zn ll ::5 c2 - n . This gives x = E�=l Xn = E�=l Yn + E�=l Zn · But Y and Z are complete a SO X is really in Y + Z.

Proof:

7 Corollary. The algebraic sum H00 + C is closed. It is also a Banach algebra. Proof: We apply Lemma 6 with X = L00 (1l') , Y = C(Y) , Z = H00 (Y) and � the family of Fejer operators. This shows that H00 + C is closed.

155

III.D. C(K)-Spaces §8.

In order to show that H00 + C is a Banach algebra it is enough to show that h · I E H00 + C for h E H00 and I E C(11') . Since H00 + C is closed it is enough to consider only trigonometric polynomials I, but then co N h . I = L anei n6 . L bnein6

n= O

-N

( nt0 anein() ) . ( f.N bn ein() ) + ( f an e in() ) ( f. bn ein () ) . N+l -N =

The first summand is in C(1r) and the second is in H00 , so H00 + C is an algebra. a 8 Theorem.

L oo/Hoo .

The quotient space (Hoo + C)/Hoo is an M-ideal in

:

Let us identify L00 (11') with C(M) where M = M (L00 (1r)) is the space of all non-zero linear and multiplicative functionals on L00 (11') (I.B.lO) . Then (L oo /Hoo )* = H� = {JL E M (M) f ldJL = 0 for all I E H00 } . The annihilator of (Hoo + C)/Hoo in (L00 /H00)* can be identified with

Proof:

(Hoo + C)l. = {JL E M (M)

:J

ldJL = 0 for all I E H00 + C} .

Let m denote the Lebesgue measure considered as a measure on M . From the generalized F.-M. Riesz theorem (see Garnett [1981] V.4.4 or Gamelin [1969] 1I.7.9) we get (3) where Msing denotes the space of all measures on M singular with re­ spect to m. Suppose that JL E (Msing n H� ) . Let a = J e - i 6 dJL. Clearly e- ie JL - am E H� and from (3) we get e - ie JL - am = lm + v. But both v and e - i e JL are singular with respect to m so -a = I and since I E H� we get a = 0. This means that e- i e JL E H� . Repeating this we get that J ldJL = 0 for I E u:=l e - i n() . Hoo which is dense in Hoo + c, so JL E (Hco + C)l. . Since no I E L1 (m) n H! can annihilate H00 + C we get that (Hoo + C).L = Msi ng n H! c H! = (Leo / H00) * . From(3) a we see that (Hoo + C)/Hoo is an M-ideal in L00 /H00 •

156

III.D. C(K)-Spaces §9.

For every f E Leo there exists h E Heo + C such that II J - h ll eo = inf{ II J - Yll : g E Heo + C}.

9 Corollary.

From Theorems 4 and 8 and the definition of the quotient norm we infer that given f E Leo there exists a g E Heo + C such that dist(f - g, Heo ) = dist(f, Heo + C) . Since balls from Heo are w*­ compact in Leo we obtain that there exists an h1 E Heo such that ll f-g- ht ll = dist(f -g, Heo ) = dist(f, Heo +C) . The function h = g +h t is in Heo + C, so it is the desired best approximant. Proof:

10. Now we want to address the problem of linear extensions. If K is a compact space and S C K a closed subset then for every f E C(S) there exists a g E C(K) such that IIYII = II I II and gi S = f. This is easy to see directly but can also be derived from Theorem 4 since the map g --+ gi S is a quotient map from C(K) onto C(S) = C(K)/C(K; S) . The question is when there exists a linear map u: C(S) --+ C(K) such that u(f) I S = f. In some cases such a map obviously exists (see II.B.4) and in some cases it does not exist (see Exercise III.B. 8). Because of applications to the disc algebra (III.E.3) we will study this question in some generality. Let T be a topological space and let S be a closed subset. Suppose we are given linear subspaces E c C(S) and H c C ( T ) . We say that u: E --+ H is a linear extension operator if u(f) I S = f for all f E E. Clearly, in order to be able to talk about such operators we need H I S :::> E. Actually we will always assume that H I S = E. We will denote the set of such extension operators A(E, H) . If a linear extension u E A(E, H) exists, then the operator P(h) = h - u(h i S) defines a projection in H with ImP = {h E H: hi S = 0} and kerF = u(E) � E. 11 Definition. The pair ( E, H) as above has the bounded extension property if there exists a constant C such that for every c > 0 and every open set W :::> S and for every f E E there exists g E H such that IIYII ::::; C II J II , g i S = f and l g(t) l ::::; c ll f ll for t E T\W.

Let us start with two easy observations. 12 Lenuna. If G c C(S) is a finite dimensional subspace and u E A (G, C(T)) then for every c > 0 there exists an open set W :::> S such that lu(g) (t) l ::::; ( 1 + c) II Y II for g E G and t E W.

157

III.D. C(K)-Spaces §13.

Proof: Since the closed unit ball of G is compact and u is continuous one checks that cp(t) = sup{ l u(g) (t) l : g E G, IIYII :5 1 } is a continuous function. Since cp(t) :5 1 for t E S the lemma follows. a 13 Lemma. If ( E, H) ha.s the bounded extension property and G C E is a finite dimensional subspace, then there exists a constant C such that for every open set W :::) S and every c > 0 there exists u E A(G, H) with ll u ll < C and l u(g) (t) l :5 c iiYII for t E T\W. Proof: We choose an algebraic ba.sis in G and extend each function separately using Definition 1 1 . This yields the desired operator u. a 14 Proposition. Let F C G be finite dimensional subspaces of E. Assume that 11': G�F is a projection with 11 11' 11 :5 1 . Given c > 0 and u E A(F, H) there exists u E A(G, H) such that u i F u and

=

ll u ll :5 max(1, ll u ll ) + c.

Remark. The mysterious expression max(1 , ll u ll ) is justified by the fact that we allow F = {0} and u = 0.

VI

Let us put F1 = ker 11'. We start with an arbitrary open set wl :::) s and from Lemma 13 we get E A(Fl , H) with ll vl ll :5 c and l v1 (f) (t) l :5 � c ll f ll for t E T\W1 . We define u1 E A(G, H) a.s u1 (g) = u(7r(g)) + v1 (9 - 11' (g) ) . Now Lemma 12 gives an open set W2 , S c W2 c W1 such that l ul (g) (t) l :5 (1 + i ) IIYII for g E G and t E w2 . Using Lemma 13 we get v2 E A(F1 , H) such that ll v2 ll :5 C and l v2 (f) (t) l :5 ! c ll f ll for t E T\W2 . We define u2 E A(G, H) a.s u2 (g) = u(7rg) + v2 (9 - 7r(g)) . Repeating this procedure N times we obtain a decreasing sequence of open sets W1 :::) W2 :::) :::) WN + 1 :::) 8 and a sequence of extensions ui = u o 11' + vi o (id - 11') E A(G, H) , j = 1, 2, . . . , N, such that ui i F = u, j = 1, 2, . . . , N and Proof:

·

·

·

for

t E Wi+ l •

for t E T\Wi , otherwise.

}

(4)

The desired extension u is defined a.s u = N- 1 L: f= 1 Uj Obviously u i F = u. From (4) we see that for any given t E T, we have ·

III.D.

158

C(K)-Spaces § 1 5.

l ui (g)(t) l > max(1, ll u ll ) + � for at most one index j, so we obtain that a ll u ll � max(1, ll u ll ) + c, provided N was big enough. The same argument gives for every e E E and every c > 0 and every open set W :J S an extension h E H such that ll h ll � 2 ll e ll and l h(t) l � c for t E T\W. Remark.

15 Corollary. If ( E, H) has the bounded extension property and E is a separable 1r1 -space then there exists a linear extension operator

u : E ---+ H.

In E we have an increasing sequence of finite dimensional subspaces En and a sequence of projections ?rn : E � En with ll 1rn ll = 1. Using Proposition 14 with en such that Ec n < oo we get a sequence of extensions Un E A(En , H) with un i Ek = Uk for k < n and supn ll un ll < oo. Since UEn = E we infer that u(f) = limn-+ oo Un ( f ) extends to a a well defined linear extension operator on E. Proof:

16. Being a 1r1-space is a rather restrictive condition. It is difficult however to modify the above proof using the weaker approximation con­ dition. Nevertheless the following theorem is true, albeit with a rather roundabout proof. Theorem. Let T be a topological space and let S C T be a closed subset. Let E C C (S ) and H C C(T) be closed linear subspaces and let (E, H) have the bounded extension property. Assume that E is separable and has the bounded approximation property. Then there exists a linear extension operator u: E ---+ H.

We will deduce Theorem 16 from Corollary 15 applied to a properly enlarged space. Let w = { 1, 2, 3, . . . } U { oo } be the one-point compactification of the natural numbers, and let S = S x w C T = T x w. Let Tn : E ---+ E, n = 1 , 2, . . . , be a sequence of finite dimensional operators with Tn (e) ---+ e for e E E. Denote En = span U�= l Tk (E) . We define E C C ( S) by Proof:

E

We also define

!oo E E, fn E En , n = 1, 2, . . . , and fn ---+ foo as n ---+ oo } .

= { ( f')'hEw : H=

{( f"Y ) "YEw : f"Y E H for 'Y E w

and fn ---+ foo as n ---+ 00 }.

(5)

159

III.D. C(K)-Spaces § 1 7.

One easily sees that E is a closed subspace of C(S) and ii is a closed subspace of C( T ) . Claim.

The pair (E, H) has the bounded extension property.

Proof of the claim. If W C T is an open set containing S then there exists an open set W00 C T, such that W00 :J S and W00 x w :J S. Given ( /-y)-yEw E E and c > 0 we can find 9oo E H such that 9oo i S = foo and l 9oo (t) l < � for t E T\W00 • Using the remark after Proposition 14 we can find hn E H with ll hn ll :5 2 ll foo - fn ll , hn i S = foo - fn and l hn(t) l < � for t E T\Woo. Since ll foo - fn ll --+ 0 as n --+ oo we also have ll hn ll --+ 0 as n --+ oo . The desired extension (g-y)-yEw is defined by 9n = 9oo - hn·

Returning to the proof of the theorem let us observe that E is a 1r1-space. To see this we define En = {( /-y)-yEw E E: fk = fn for - onto k 2: n } . The projections Pn: E-+En are defined by Pn (( /-y)-yEw) = ( JI , h , . . . , fn, fn, . . . , fn)· Obviously II Pn ll = 1 and UEn is dense in E. From Corollary 15 we get a linear extension operator u: E --+ ii. We define

u:

E --+ H by

u ( f ) = u(Tl (f), T2 (f), . . . ' !) I T

X

{ 00 }.

a

17 Corollary. If S c T and S is compact metric and T is normal then there exists a linear extension operator C(S) --+ C(T) .

u:

Proof: Since S is compact metrizable the space C(S) is separable. The

Tietze extension theorem implies that (C(S) , C(T)) has the bounded extension property. Since C(S) has the bounded approximation property a (see II.E.5(c)) the corollary follows. 18. The above corollary exhibits many complemented subspaces of C(K)-spaces. One more, different and very important example is pro­ vided by the following. Proposition. {Milutin) Let yN denote the countable product of circles. The space C(A) contians a 1-complemented copy of C(1l'N ) .

Let us identify A with { - 1 , 1}1N . By p we mean the classical Cantor map from A onto 1l', i.e. p((c3)� 1 ) = 21r E� 1 ! ( 1 + c3 ) 2 - j . It is an easy and well known fact that there exists a map 7': '][' --+ A such

Proof:

160

III.D. C(K)-Spaces § 1 9.

that jYf = and f is measurable and continuous on Y\D where D is a countable set of dyadic points. Since � is homeomorphic to � N we infer (take products) that there exists a continuous map p: ��yiN and a measurable map yN ---+ � such that p = and T is continuous on YIN\Doc where Doc has measure 0. Note that both � and yiN have a ·natural group structure, so we can perform algebraic operations. We define the isometric embedding C ( � x �) � C � as a {3 = f(p( a) + p(f3 )). We define a norm-1 map C ( � x �) ---+ C(YN ) by

idy

,

r idyN

r:

I(f)( , )

I : C (YN ) ---+

/1fN

Q:

()

Q (g)(t) = g(r(s) , r(t - s))ds . To see that Q (g) E C (YIN ) let us take a sequence tn E YIN, tn ---+ t . Then g(r(s) , r(tn - s)) ---+ g(r(s) , r(t - s)) as n ---+ oo for all s E YN \(U := l (t n - Doc ) (t - Doc ) Doc ) , i.e. for almost all s E By the Lebesgue dominated convergence theorem Q (g)(tn) = JTN g(r(s) , r(tn-s))ds ---+ JTN g(r(s), r(t-s))ds = Q (g)(t) as n ---+ oo, so Q (g) is continuous. Since QI(f)(t) = lT{ N f(p( r(s)) + p(r(t - s)))ds = }yN{ f(t)ds = f(t) yiN .

U

U

a

the proposition follows.

This proposition should be compared with Exercise 4, which indi­ cates that some ingenious embedding is needed. 19.

result.

The above proposition allows us to prove the following surprising

Theorem. {Milutin) For every compact, metric, uncountable space K, the space C(K) is isomorphic to C � .

()

Proof: As is well known, every uncountable, compact metric space K contains a subset K1 homeomorphic to � (see Kuratowski [1968] or Semadeni [1971] ) . So by Corollary 17 the space C(K) contains a com­ plemented subspace isomorphic to C(�) . It is also elementary and well

known (cf. Kuratowski [1968] ch.4§4l.VI.) that every compact metric

161

Ill.D. C(K)-Spaces §20.

space is homeomorphic to a subset of yiN so in particular we obtain from Corollary 17 that C(K) is isomorphic to a complemented subspace of C(YN ) . This and Proposition 18 yield that C(K) is isomorphic to a complemented subspace of C(�) . Since C(�) ,...., (EC(�))o Theorem a II.B.24 gives the claim. This theorem in particular implies that every C(K)-space for K compact, metric, and uncountable has a Schauder basis. Also, such a C(K)-space is isomorphic to (EC(K)) 0 • 20. Our aim now is to present in some detail the orthonormal Franklin system. Usually it is constructed on [0, 1] . We will present the detailed construction on the circle (i.e. we will construct the periodic Franklin system). This will be useful for some of the future applications, in particular Theorem III.E. 17. The reader interested in [� ,) ]--Should be able to repeat the construction in this case witho¢ -any difficulty. We will identify the circle Y with the interval [0,1). For an integer n = 2 k + j, with k = 0, 1, 2, . . . and 0 � j < 2 k we_"'define tn = and we put to = 0. The Franklin system Un) ':'= o is an orthonormal system of real valued, continuous, piecewise linear functions such that In has nodes at points t3 , j = 0, 1, . . . , n, for n = 0, 1, 2, . . . . This definition specifies In up to the sign. Let Fn = span{IJ }J::;n · Clearly Fn is the space of all continuous, piecewise linear functions with nodes at { t3 }J::;; n · For a fixed n we will denote by (s3) 'J=o the increasing renumbering of (tj ) 'J=o • i.e. 0 = so < s1 < · · · < Sn = 1. Let Zn + l denote the group of integers 0, 1 , . . . , n with addition mod (n + 1). The natural group invariant distance p( · , · ) on Zn + l is defined as p(k, l) = min( l k - l l , in + 1 + l - ki, I n + 1 + k - l i ) , for k, l = 0, 1, . . . , n. We define also (for fixed n) the 'tent' functions Aj , j E Zn + l by the conditions A3 E Fn , A3 (s3 ) = 1 and A3 (s k ) = 0 for k =f. j. Let us also note that 1 (6) � dist(s3 , s3+ 1 ) � � for j E Zn + l · n 2n 21. Our main goal now is to establish the following technical proposi­ tion. It describes the behaviour of an individual Franklin function and of the integral kernel of the partial sum projection. This proposition will allow us to investigate the properties of the Franklin-Fourier series E ':= 0 (/, In) In for I in various classes of functions. Proposition. There exist constants C e very n = 0, 1, 2 . . . , we have


1. We say that the matrix (aij ) i ,jEZn + I is m-banded if a ij = 0 for p ( i , j) > m. One easily sees that the inverse

of an m-banded matrix (if it exists at all) need not be banded. The

163

III.D. C(K)-Spaces §24.

following proposition shows that something remains. The entries of the inverse of an m-banded matrix are exponentially small far away from the diagonal. Proposition. Let A = (aij ) i , j EZn + l be an m-banded invertible matrix with II A II � 1 and II A- 1 11 � C where the norm is understood as the operator norm on There exist numbers K = K(C, m) and q = q(C, m) , 0 < q < 1 such that

i2(Zn+l) ·

l bij l

�

KqP (i ,j )

where A - 1 =

(bij ) i ,jE Zn + l '

(11)

The proof of this proposition uses the following easy approximation fact.

f(x)

n

< a < b and let = � · Then a is an algebraic polynomial of degree l < n } Kqn + 1 for some K = K(a, b) and q = q(a, b) < 1 . 24 Lemma.

inf{ ll /

-

Let 0

P ll c [a , b] :

p

Let = ( a� b) . Then � converges in C(a, b] . We have

� 00


n 1 there are subsets A, B C { 1 , . . . , n } such that

Since there is a only finite number of subsets in { 1 , . . . , n 1 } we find sets A and B such that (25) holds for this pair of subsets A, B on the set of

the form (24) . Thus (after renumbering) we can assume that there are vectors (xi, . . . , x�) C X and (ki, . . . , k�) C K for n > n 1 , satisfying (19)-(22) . We put xi = k-:1 where s = max A. In the second step we analogously find n 2 > n 1 . and z2 = I A I - 1 Lj e A xj2 I B I - 1 L j eB xj2 such that J i z2 id ( iJ.Lj i + a) < c 2 for all (after passing to appropriate subset as before) n, j with n > n 2 , j = 1, . . . , n, and such that J i z2 ida is so small that dist((1 l z 1 i )z2 , X) < c 2 . We put x2 = k-: 2 where s = max A. Continuing in this manner we find sequences ( zj ) � 1 C X and ( xj ) � 1 C K such that

-

-

�6 -

(this follows from (21) and (22)) , i xj ( zj ) i > 1 kdist ( IJ (1 i zi i )zk , X < c k , j=1 i zi i idJ.Lk i < C"j for k > j where J.Lk is the Hahn-Banach extension of xA; .

J

)

(26) (27) (28)

Using (26) , (27) and (28) we construct the desired weakly uncondition­ a ally convergent series exactly like in the proof of (e)::::} ( d) . 33. Now we wish to cast the above considerations into a more general context. Let us introduce the following concepts. We say that a Banach space X has the Dunford-Pettis property (for short DP) if for every

III.D. C ( K ) -Spaces §34.

172

Xn�O in X and x� �O in X* we have x� (xn) --+ 0. Clearly if X* has DP then also X has DP. We say that a Banach space X has the Pelczynski property (for short P) if for every subset K c X* that is not relatively weakly-compact there exists a weakly unconditionally convergent series E �= l Xn in X such that infn supx* E K x* (xn) > 0. Clearly Theorem 31 shows that every rich subspace of C(S, E) has DP and P. Also any L 1 (J.L) space has DP. 34.

We have the following, rather routine

Proposition. equivalent:

The following conditions on the Banach space X are

(a) X has the Dunford-Pettis property; (b) every weakly compact operator T: X

__.

Y transforms weakly

Cauchy sequences into norm Cauchy sequences; __. c0 transforms weakly Cauchy sequences into norm Cauchy sequences.

(c) every weakly compact operator T: X

(a)=> (b) Passing to differences it is enough to show that II Txn ll __. 0 for every Xn�O. But if Xn�O and II Txn l l ;::: 8 > 0 for n = 1, 2, . . . , then we can take y� E Y* with IIY� II = 1 such that w* y�T(xn) ;::: 8. Passing to a subsequence we can assume that y� --+y* for some y* E Y* . But y* (Txn) __. 0 so we can replace y� by y� - y* and additionally assume that y� � O. But T* is weakly compact (see II.C.6(b)) so T* (y� ) �O. This contradicts (a) since T* (y�) (xn) = y� (Txn) ;::: 8. (b)=>(c) . Obvious. (c)=>(a). Let us take x� E X* such that x� �O and define an operator T: X __. co by T(x) = ( x� (x) ) :'= l · Clearly T** (x**) = (x** (x�))�= l E eo so T is weakly compact by II.C.6(c) . Applying (c) we get that for Xn �O in X, II Txn ll __. 0 so in particular x� (xn) __. 0. a Proof:

35 Proposition.

Suppose X has the Pelczyriski property. Then

(a) X* is weakly sequentially complete, (b) for every operator T: X __. Y that is not weakly compact there exists a subspace xl c X such that xl Co and T I Xl is an "'

isomorphic embedding.

If K is a subset of X* that is not relatively weakly com­ pact then there exists a sequence {x� }�=l C K which is not relatively

Proof:

III.D. C(K)-Spaces §Notes.

173

weakly compact (see II.C.3) . Thus there exists a weakly unconditionally convergent series :L:;:: 1 X k in X and a subsequence ( x � k )� 1 such that l x� k (x k ) l > 6 > 0 for k = 1 , 2, . . . . From II.D.5 we see that we can additionally assume that ( x k )� 1 is equivalent to the unit vector basis of Co · In order to prove (a) we take K = {x � } �= l where x � is weakly Cauchy but not weakly convergent. Let T: c0 --+ X be defined by T(e k ) = X k . Then T* : X* --+ i 1 and one easily sees that T* ( x� k ) has no norm Cauchy subsequence, so by III.C.9 T* (x� k ) has no weak Cauchy subsequence. This contradicts the fact that (x�)�= l was weakly Cauchy. In order to prove (b) let us put K = T* (By. ) . Then X1 = span (x n )�= l is clearly isomorphic to Co and for x = L�= l anXn we have

so T I X1 is an isomorphic embedding.

•

Notes and remarks.

The C(K)-spaces are among the most widely used Banach spaces. They are also the easiest examples of Banach algebras. As usual in this book we discuss the multiplicative structure only so far as it relates to the linear structure. Thus the well known Proposition 1 serves as an in­ troduction to the concept of an M-ideal. This concept was introduced by Alfsen-Effros [1972] where Theorem 4 is also proved. Our proof is a modification of proofs given in Behrends [1979] and Lima [1982] . A similar proof and many applications of the theorem can be found in Gamelin-Marshal-Younis-Zame [1985] . We will present some other ap­ plications of the concept of an M-ideal in III.E. The space Hoo + C was introduced into analysis in the sixties by D. Sarason and A. Devinatz. Corollary 7 is due to Sarason but the simple proof presented here is from Rudin [1975] ; a similar proof is given in Garnett [1980] . The analysis of the particular example H00 + C led to the general theory of Douglas algebras, i.e. closed algebras X such that H00 C X C L00 • H00 + C is the smallest such algebra. It is a deep theorem of Marshall and Chang that each such X is the smallest algebra generated by H00 and complex conjugates of certain inner functions. For detailed information about all this we refer to Garnett [1980] or Sarason [1979] . Corollary 9 was proved by complicated operator-theoretical argu­ ments by Axler-Berg-Jewell-Shields [1979] . The simple proof given here

174

III.D. C(K)-Spaces §Notes.

is due to Luecking [1980] . This started the investigation of M-ideals in Douglas algebras and other spaces connected with function theory. We refer the interested reader to Gamelin-Marshal-Younis-Zame [1985] and to the references quoted there. There are also important applications of the concept of M-ideal to the theory of C* -algebras; see Choi-Effros [1977] or Alfsen-Effros [1972] . It seems that the first linear extension theorem was proved by Bor­ suk [1933] where he established a version of our Corollary 1 7 and used it to show that C [O, 1] "' ( I: C [O, 1] )0. Borsuk's argument was different and together with later improvements by Dugundji [1951] it gives the following Theorem A. (Borsuk-Dugundji) . Let S be a closed non-empty sub­ set of a metric space T and let X be a normed vector space. Then there exists a linear extension operator C(S, X) ----+ C(T, X) such that = 1 and for every g in C(S, X) the values of the function (g) belong to the convex hull of the set g(S) . Our Corollary 15 was proved by Michael-Pelczynski [1967] . Actually

llull

u:

u

it was proved with the additional information that llull = 1. We decided not to present this improvement because we are mainly interested in isomorphic theory and our goal is Theorem 1 6. This theorem was proved by Ryll-Nardzewski (unpublished) and the proof we follow here was later given in Pelczynski-Wojtaszczyk [1971] . Davie [1976] used Proposition 18 in his discussion of classification of operators on Hilbert space. It is his version of the proof that we present. It should be stressed, however, that questions of linear extensions are not limited to spaces with the sup­ norm. There is an extensive literature on the existence and non-existence of linear extensions when other norms are involved, in particular Sobolev or Besov norms; see Stein [1970] or Triebel [1978] and the references quoted there. Proposition 18 and Theorem 1 9 are due to A.A. Milutin. He proved them in his Candidate of Sciences dissertation presented to the Moscow State University in 1952. Those results were only published in Milutin [1966] . These are important results. Some reasons for this opinion are as follows. (a) Results of an isomorphic nature, once established for one 'sim­ ple' space K, like K = � or K = 11' are valid for C(K) with more complicated compact, uncountable metric spaces K. One modest exam­ ple of this is the comment made after Theorem 1 9. (b) This is an important step in the programme of isomorphic clas­ sification of C(K)-spaces. For separable spaces C(K) , i.e. K-compact,

III.D. C(K)-Spaces §Notes.

175

metric such classification is known. For countable, metric compact spaces this was done in Bessaga-Pelczynski [1960] . The situation for non-separable spaces is more complicated and only partial results are known. (c ) The Milutin theorem shows that for most important compact sets K the multiplicative structure of C(K) has nothing to do with its linear-topological structure. This contrasts sharply with the isometric situation. Namely we have the following. Theorem B. If d(C(S) , C(K)) < 2 then S is homeomorphic to K, so in fact there exists a linear isometry i: C(K) � C(S) preserving the multiplication.

This theorem under the assumption that C(K) and C(S) are ac­ tually isometric ( with the a description of the isometries ) was proved for metric K and S in Banach [1932] and for general K and S in Stone [1937] . This version was given independently by Amir [1965] and Cam­ bern [1967] . The Franklin system was introduced in Franklin [1928] , where The­ orem 25 was proved. We follow an approach developed by Ciesielski and Domsta in order to deal with systems of more general spline functions; see Ciesielski-Domsta [1972] . The Franklin system itself was earlier in­ vestigated in detail in Ciesielski [1963] , [1966] where the fundamental Proposition 21 was proved. Proposition 23 was proved by Demko [1977] . The very ingenious proof presented here was given in Demko-Moss-Smith

[1984].

Theorem 2 7 was first proved in Ciesielski

[1960] using the Faber­

Schauder system and later in Ciesielski [1963] using the Franklin system. The Franklin system is one of the most important orthonormal systems (see Kashin-Saakian [1984]). The Dunford-Pettis property as defined in 3 3 was explicitly defined by Grothendieck [1953] who undertook an extensive study of this and related properties. He was directly influenced by the important paper Dunford-Pettis [1940] where among other things it was proved that ev­ ery weakly compact operator T: £1 [0, 1] - X maps weakly compact sets into norm-compact sets. Our Proposition 34 and much more can be found in Grothendieck [1953] . The PelczyD.ski property (obviously under a different name, property V) appeared first in Pelczynski [1962] , where he showed that C(K) has P. The class of rich subspaces of C(K, E) ap­ peared in Bourgain [ 1984b] . In Bourgain [ 1983] and [ 1984b] our Theorem 31 and Examples 30 were proved. Theorem 31 for the particular case

176

III.D. C(K)-Spaces §Exercises

of the disc algebra was shown earlier by Delbaen [1977] and Kislyakov [1975] (independently and almost simultaneously) . Clearly Theorem 31 when applied to C(K) gives information about subsets of L 1 ( ll) . This information is akin to that given in Theo­ rem III. C. 12. One can derive Theorem III.C.12 from Theorem 31 but even then many of the measure-theoretical arguments from the proof of III.C.12 have to remain. We have chosen to present a separate proof of Theorem III. C. 12 because it is an important theorem and the direct argument is relatively simple. It stresses the important notion of uni­ form integrability which cannot appear explicitly in the more general Theorem 31 .

The paper Diestel [1980] contains a nice survey and exposition of the Dunford-Pettis property and related topics. It does not however, discuss Theorem 31 . Exercises

1.

Show that the space of compact operators on lp , 1 < p < M-ideal in L(lp ) ·

2.

Show that, if (E, H) has the bounded extension property then H0 = {! E H: ! I S = 0} is an M-ideal in H. (The notation agrees with

oo ,

is an

10.)

3.

Show that every C(K)-space, K compact, is a 1r1-space.

4.

Let cp: �- [0, 1] be the classical Cantor map, i.e. if � = { - 1, 1} N then cp(ei ) = E� 1 (ei + 1) - l - j . Let I"': C[O, 1] - C(�) be given by Icp (f) = f o cp . Show that Icp (C[O, 1] ) is uncomplemented in C(�) .

5.

Find two non-homeomorphic, compact metric spaces K1 and K2 such that d( C(KI ), C(K2 ) ) = 2.

6.

A matrix (aj k )j, k � o is called a Toeplitz matrix if aj, k

onto

=

cp(j - k).

(a) Show that a Toeplitz matrix is a matrix of an operator on l2 if and only if cp(s) = j ( s ) , s = 0, ± 1, ±2, . . . for some f E Loo (Y) . (b) A matrix (bj k )j, k �O is a Schur multiplier if for every matrix (mj k )j, k �o of a linear operator on l2 the matrix (bi k · mj k )j, k �O is a matrix of a linear operator on l2 . Show that the Toeplitz matrix is a Schur multiplier if and only if cp(s) = [l,( s ) , s = 0, ± 1 , ±2, . . for some measure ll on 1r.

177

III.D. C(K)-Spaces §Exercises

(c) Show that the main triangle projection, i.e. the map (aj k ) j, k ?. O - (bj k ) j, k ?_ O where b.

{

if j � k, aj J k - 0k otherwise,

is unbounded on L (£2 ) . 7.

Show that if (n k )%"= 1 is a lacunary sequence of integers and the Fourier series 2: �= 1 a k ein k B represents a bounded function, then 2::'= 1 J a k I < oo.

8.

(Korovkin theorem) . Suppose that Tn: C [O, 1] - C[O, 1] is a se­ quence of linear operators such that J I Tn ll - 1 as n - oo and Tn (P) - p as n - oo for every quadratic polynomial p. Show that Tn (f) - f in norm for every function f E C[O, 1] .

9.

For f E C[O, 1] we put Bn (f)(x) = I: �= O f(�)( ! )x k (1 - x) n - k . The operators Bn are called Bernstein operators. (a) Show that each Bn is a linear, norm-1 operator. (b) Show that for f E C[O, 1] we have Bn (f) - f uniformly.

10. For s > 0 we define

X8 = {f(z): f is analytic in ID and J / (z) J (1 - Jz l ) 8 E L00 (ID) } and X� = {! E X8 : J / (z) J = o (1 - J z l ) - 8 }. Show that Xs rv £00 , X� rv C() and (X�)** = X8 • 11. Show that there exists a function f E A(IBd ), d � 1, such that fmd I Rf(z) Jdv(z) = oo where R is the radial derivative (see Exercise III.B. ll) and v is the Lebesgue measure on IBd .

12. A sequence of finite dimensional Banach spaces (Xn )�= 1 is called a sequence of big subspaces of £! if there exist constants C and a such that for each n there exists a subspace Xn C f!n such that d(Xn , Xn) :S: C and Nn :S: a dim Xn . (a) Let T;;" C C(Y) be the space of trigonometric polynomials of degree :S: n, n = 1, 2, . . . . Show that (T;;" )�= 1 is a sequence of big subspaces of £! . (b) Let W;;" (IBd ) C C(IBd ) be the space of all polynomials homoge­ nous of degree n. Show that for every d = 2, 3, . . the sequence (W;;" ( 1Bd ) ) �= 1 is a sequence of big subspaces of £! . .

178

Ill.D. C(K)-Spaces §Exercises

(c) Show that, if E C £! , dim E = n and 0 < 8 < 1, then there exist an integer k > 2 � and a subspace G c E, dim G = k, such that d(G, £� ) :::; �i��� ·

13. Construct the system of quadratic splines analogous to the Franklin system. More precisely, construct an orthonormal system of func­ tions ( gn ) �= O c L 2 (Y) such that each g� is a continuous, piece­ wise linear function with nodes at points t3 , j = 0, 1, . . . , n for n = 0, 1, 2, . . . , . (The points t3 are defined in 20.)

14. Let I c '][' be an interval. The function a 1 ( t ) is defined as a1 (t) =

{

0 if t ¢ I, I I I - 1 if t is in the left hand half of I, - I I I - 1 if t is in the right hand half of I.

We define the space B as the space of ail functions f(t) which admit a representation f(t) = >.o + I:: :'= 1 >.na ln (t) for some se­ quence of intervals ( Jn ) �= 1 and some sequence of scalars (>. n ) �= O with I:: :'= o l >.n l < oo. Then inf :L:: :'= o l >.n l over all representations of f is the norm denoted by II/Il B · (a) Show that B is a Banach space. (b) Show that f E B if and only if

� (n + 1) - ! l l f(t)fn(t)dt l

< oo

where Un ) �= O is the Franklin system.

15. The space A. (the Zygmund class) is defined as the space of all functions in C (Y) such that 11 /11

* -

_

sup

{I f(x - h) + f(x + h) - 2f(x) .· x E '][' h > o } I h ,

< oo .

Show that f E A. if and only if I J1f f(x)fn (x)dx l = O(n - � ) , where ( /n) �= O is the Franklin system.

16. Show that the Franklin system is not an unconditional basic se­ quence in Lip1 (Y). 17. Let (!n ) ':'= be the Franklin system and let f E L 1 ('][') . Show that 1

the series "2.:'=0 (.f, fn ) fn converges almost everywhere.

179

III.D. C(K)-Spaces §Exercises

R

18. Let A = 7L U �'ll - and let us consider the subspace V c L 2 ( ) consisting of all continuous, piecewise linear functions on with nodes at the points of A. Let r be a function which is continuous, piecewise linear with nodes at Ao = A U { � } and orthogonal to V. Assume also that ll r ll2 = 1 .

R

( a) Show that

l r(x) l

::; Cq l x l for some C

> 0 and q < 1. ( b ) Show that the family of functions {2i 1 2 r(2i x - k)}(j, k )EZxZ is a complete orthonormal system in L 2 (R). 19. Suppose that X is a separable Banach space with the Dunford­ Pettis property. Assume also that X "' Y* for some Banach space Y. Show that X has the Schur property, i.e. weakly convergent sequences converge in norm. It follows that Ld H1 and L1 [0, 1] are not isomorphic to a dual space. 20. Show that, spaces L1 (J.L) and C(K) do not have complemented, infinite dimensional, reflexive subspaces. 21. Find a Banach space X with the Dunford-Pettis property, such that X* does not have it.

22. Show that if X is a rich subspace of C(K) , then X* has the Dunford­ Pettis property.

23. Show that every operator 24.

space, is weakly compact.

T: £00 --+ X for X a separable Banach

Show that the identity operator id: C k ('l£'8 ) --+ c k - 1 (Y8 ) , k � 1, s � 1 is compact. 25 . Show that C 1 (Y2 ) * = M EB F where M is isomorphic to M(Y) and F is separable.

26. Let us consider H00 (S) + C(S) where S is the unit sphere in ([d , d > 1 . Show that H00 (S) + C(S) is a closed subalgebra.

27. Let us consider the algebraic sum H00 (1l"" ) + C(11"" ) . Show that it is closed in L00 (1l"" ) but is not a subalgebra if n > 1.

III.E. The Disc Algebra

First we study some interpolation problems in the disc algebra A. We describe those subsets V c ID for which we have A I V = C(V) . We also describe the sets V c 11' such that every f E C(V) can be extended to a function F E A with finite Dirichlet integral. We also show that every £2 sequence is a sequence of lacunary Fourier coefficients of a function in A. Next we show that A rv (EA)o. We show that A is not isomorphic to any C(K)-space. We present the construction of a Schauder basis in A and give different isomorphic representations of H00 • 1. Let us recall that the disc algebra A is the space of all functions continuous for l z l $ 1 and analytic for l z l < 1, equipped with the norm II/II = sup l z l 9 1 /(z) l. The maximal modulus principlen easily implies 1 1 / 11 = sup l z l = l 1/( z) l so A can be identified with span{e i B }n�o C C(T) . The disc algebra appears prominently in many parts of analysis; it is the canonical example of a uniform algebra, the von Neumann inequality (see III.F. (25)) gives the functional calculus on A for every contraction on a Hilbert space, etc. It is also a very interesting Banach space. From the F.-M. Riesz theorem I.B.26 we get immediately that

A* = C(11') * /AJ.. = LI/Jtt EB1 Ms

(1)

where M8 denote the space of measures on 11' which are singular with respect to the Lebesgue measure. 2. We will discuss some interpolation results for the disc algebra and other related spaces. More precisely, given a space X of analytic functions on ID we look for sets V c ID (or even V c ID if elements of X extend naturally to 11') such that the restrictions X IV fill the space C(V) or £00 (V). We want to impose minimal, sensible conditions. The above description (1) of A* easily yields the following result. Proposition. (Rudin-Carleson) Let Ll = Ll c 11' be a set of Lebesgue measure 0 and let cp E C(11') be a strictly positive function with cpi Ll = 1 . Then for every f E C(Ll) there exists a g E A such that giLl = f and lg(O) I $ 11 /11 cp( O) for all 0 E 11'. ·

III.E.

182

The Disc Algebra §3.

Let us introduce an equivalent norm on A by ll g ll "' = Proof: sup{ l g( O ) I cp- 1 (0) : 9 E 1I'} and let r: (A, I · II "') ---+ C(Ll) be the re­ striction operator r(g ) = gi Ll. Since Ll has Lebesgue measure 0 and cpl .!l = 1 we infer that r* is an isometric embedding, so r is a quotient map. Note that r* assigns to the measure JL on Ll the same measure treated as a measure on 1I' and considered as a functional on A. Since (ker r) .L = r* (C(Ll) * ) we infer that ker r is an M-ideal. Thus from The­ orem III.D.4 we get that there exists g E (A, I · II "') with g i Ll = I and a ll gii 'P � II I II , i.e. l g( O ) I � cp( O ) II I II · Note that I Ll l = 0 is also a necessary condition for A I Ll = C(Ll) for a closed set Ll c 1I'. If I Ll l > 0 then the restriction r: A ---+ C(Ll) is 1-1 (It is known that I E Hp (1I') cannot vanish on a set of positive measure. This fact is hidden in the canonical factorisation and the form of an outer function; see I.B.23) and it easily follows from Proposition 2 that it is not an isomorphism. Thus it cannot be onto. 3.

From this proposition and Theorem III.D. 16 we get

Corollary. If Ll c 1I' has Lebesgue measure 0 then there exists a linear extension operator u: C(Ll) ---+ A. a

Interpolating sequences in the open disc are also of considerable in­ terest. First we consider them for the space H00 • A sequence (>.n)�=O C D is called interpolating if the map I t-+ (f(>.n))�=O transforms H00 onto l00 • Obviously (see I.B.23) any interpolating sequence has to sat­ isfy the Blaschke condition :E:= 0 (1 - l >.n l ) < oo. Thus we can form the Blaschke products B(z) = TI:= o Mn (z) and Bn (z) = B(z)/Mn (z) where Mn (z) = l >.n l >.; 1 (>.n - z) ( 1 - Anz) - 1 • The following gives some information about interpolating sequences. 4.

Theorem. The following conditions on the sequence (>.n)�=O C D are equivalent:

(a) (>.n)�=O is an interpolating sequence; (b) infn ;::: o I Bn (>.n) l

=

8 > 0;

(c) there exists a bounded linear map T: loo ---+ Hoo such that T(e) (>. k ) = ek for k = 0, 1 , 2, and any e = (en ) E loo Note that condition (c) gives a linear lifting to the m ap I t-+ (!(>.n))�=O ·

1 83

III.E. The Disc Algebra §4.

(a)=*(b) . The open mapping theorem yields a constant C such that for each n = 0, 1, 2, . . . there exists In E Hoo with Illn il ::; C and ln (>.j ) = 6ni · From the canonical factorization I.B.23 we get that In = cpn · Bn with cpn E Hoo and ll cpn ll = llln ll · This gives condition (b) . (b) =*(c) . Let us assume 0 < l >.o l ::; l >. 1 l ::; . . . . We define Proof:

(2) where an(z) = L k > n (1 + X k z) (1 - X k z) - 1 (1 - l >. k l 2 ). Clearly ¢n (>. k f= 6n,k so the operator T( en ) = L n > O en¢n satisfies (c) provided it is continuous. This will follow from

L: l¢n(z) l ::; c{j for l z l n�O


n

I kfmII Mk (>.n) l ::; 2 - 4 log 6 = Kl) .

Using (4) , (b) and (5) we get

L l¢n (z) l ::; L [Rean (z) - Rean+l (z)] 6 - 1 exp Kli exp( - Rean(z )) n�O n�O ::; c{j L exp(Rean (z) - Rean+ l (z)) - 1 . exp - Re an(z) n�O = c{j L [exp ( - Rean+ l (z)) - exp( -Rean(z)) ::; c{j . n �O

[

(c)=*(a) . is obvious .

]

]

a

184

III.E. The Disc Algebra §5.

5. If (.Xn)�= O C ][) is an interpolating sequence then it is quite difficult to describe the set {(f(.Xn))�=o= f E A } C C00 • This set clearly depends on the topological structure of the closure of (.X n)�= O in 1D. One can easily construct examples (see Exercise 6) where this set is not closed in C00 • The following theorem characterizes sets of 'free' interpolation for the disc algebra. Theorem. Let V c 1D be a closed subset. Then A I V = C(V) if and only if I V n lr l = 0 and V n 1D is an interpolating sequence.

If A I V = C(V) then A I (V n lr) = C(V n lr) so by the obser­ vation made after Proposition 2 we have I V n lr l = 0. Clearly V n 1D is countable and there exists a constant C such that for every finite subset {.A 1 , . . . , .An} C V n 1D and any numbers (a 1 , . . . , an ) there exists f E A such that 1 1 /11 � C max 1 :::; j :::; n l ni l and f(.Xj ) = Ctj . By the standard normal family argument we get that V n ][) is an interpolating sequence. Conversely let I v n lr l = 0 and v n ][) be an interpolating sequence. Let us write A = A0 EB A 1 where A0 = {! E A: !I V n 1[' = 0 } and A 1 = u (C(V n lr)) where u is a linear extension operator from C(V nlr) into A (cf. Corollary 3). Let us also split C(V) = C0 EB C 1 where C0 = {! E C(V): f i V n lr = 0 } and C 1 = A 1 1 V. In order to prove the theorem it is enough to show that A0 1 V = C0 • Let V n 1D = {.An}n;:: o with I.Ao l � I .A1 I � I .X 2 I � . . . . Let (rf>n)n;:: o be given by (2) . It is known (and easily follows from the standard proof that the Blaschke product converges) that each Bn(z), n = 0, 1, 2, . . . is continuous on 1D\(V n lr). Also the an(z) , n = 0, 1 , 2, are continuous on 1D\(V n lr) so all the (rf>n)n;:: o are continuous on 1D\(V n lr) . Let 'Pn E A be such that II 'Pn ll � 2, 'Pn i V n 1[' = 0 and 'Pn(.Xn) = 1 for n = 0, 1, 2, . . . . The existence of such 'Pn 's easily follows from Proposition 2. It follows from (3) that for 1/Jn = 'Pn · r/>n , n = 0, 1, 2, . . . we have Proof:

L 1 1/Jn (z) l n;:: o

�

C for l z l


O f(.Xn)'I/Jn · Since a F E A and F I V = f the theorem is proved. 6. Let us consider now a more specialized interpolation result. Before we proceed we have to recall the notion of a Dirichlet integral (Dirichlet norm) which is instrumental in proving the Dirichlet principle by vari­ ational methods. We restrict our attention to analytic functions only.

III.E. The Disc Algebra §6.

185

We define the Dirichlet space

{

D = f(z): f(z) is analytic for l z l < 1 and

(L

l f'(z) l 2 dv(z)

)

1

2


o ](n)z n E D if and only if L n >o n l ](n) i 2 < oo. We are interested in D n A or, more precisely the sets E C T for which we have D n A l E = C ( E ) . Clearly E has to have measure zero but this is not enough. The proper condition involves the notion of capacity which we will now recall. For a closed set E c 1I' and a Borel measure f..L on E we define the energy of f..L as

2 e(f..L ) = r r log -- df..L ( x)df..L ( y), jEjE 1 X - y 1

(6)

where this integral is understood as the Lebesgue integral on ExE with respect to the measure f..LXf..L . Since for x = e i6 and y = ei"', l x - Yl = 2 1 sin ! ( (J - cp) I we can write the integral (6) as

r r log . �- df..L ( fJ)df..L ( cp) . J.11:J.1f I sm 7I Writing log I sin ! O I - l rv I: � : 'YneinB we get e(f..L ) = I: � : 'Yn i J1 (n) i 2 . Integration by parts yields 'Yn = 1 �1 + o ( 1 � 1 ) . Moreover one can show that 'Yn � 0. This gives e(f..L ) � 0 for every Borel measure f..L on 1I' and (7) e(f..L ) < oo if and only if :L: l n i - 1 1M n) l 2 < oo. n ; 0 because the Lebesgue measure restricted to E has finite energy. Now we are read to state: Theorem. Let E only if 7(E) = 0.

c

1I' be a closed set. Then ( D n A ) I E

For the proof we will need two lemmas.

=

C ( E ) if and

Ill. E.

186 7 Lermna. Let J-t E M(Y) and £(J-t) < E C '][' with 'Y(E) = 0 we have J-t(E) = 0.

oo.

The Disc Algebra § 7.

Then for every closed set

Proof: It is enough to check for positive J-t only. Now if J-t(E) > 0, then, since 'Y(E) = 0 we have oo

2 = £( �-t i E) = }[ }[ log ---1 d�-t(x)d�-t( Y ) E E 1x - y 2 $ [ [ log --- d�-t(x)dJ-t( Y ) = £(J-t) < oo j.lf j'Jf 1 X - Y 1

a

which is absurd. 8 Lermna. Let J-t be a measure in M(T) . � < "'+ oo

Lm= - oo , n ;o!'O

Proof:

(

lnl

)

IE L:�= 1 ( I P. 0 such that Proof of the Theorem.

inf{ II (O, J-t) - (S, v) ! l : (S, v) E X l.. } ;::: CII JL II for all J-t E M(E) . (9)

Ill.E. The Disc Algebra §9.

187

Assume now that 7(E) = 0. Since (e inB , einB ) E X for n = 0, 1, 2, . . . we get that for (S, v) E XJ. we have S(ei n6) + v(-n) = 0 for n = 0, 1 , 2, . . . . For S E D* we get easily that L::= l n- 1 I S(e inB ) j 2 < oo so for {S, v) e Xl. we have L::= l n- 1 l v{-n) l 2 < oo. From Lemma 8 and {7) we get e(v) < oo. From Lemma 7 we get v(E) = 0. Thus for (S, v) E XJ. and p, E M(E) we have 11 {0, p,) - (S, v) ll = II S IIv • + lip, - vii � ll llll so by {9) r is onto. Assume now 7(E) > 0. From {7) we get that for every p, E M(ll') with £(p,) < oo the functional Sp. defined on D as Sp. { / ) = L n > o f(n)[l,(n) is continuous and II Sp. l l v · � Ce(p,) ! . In particular if e(p,) < oo, then ( -Sp. , p,) E X 1. . Since 7(E) > 0 there exists a probability measure p, on E with £(p,) < oo. Let cpn E L00 {p,) be a sequence of functions such that lcpn l = 1 p,-a.e. and cpn ---+ 0 as n ---+ oo in the a(L00 (p,) , £ 1 (p,))-topology. From {6) we infer that e(cpn Jl ) ---+ 0 as n ---+ oo. So inf{ II {O, cpnP,)) - {S, v) ll : (S, v) E XJ. } � II {O, cpn Jl ) - (-S'Pn l-' • cpn Jl) ll = II S'Pn P. II D• � C£( cpn Jl ) ! ---+ 0 as n ---+ 00 . This shows that {9) is violated, so r is not onto.

a

9. In the previous sections we have been mostly interested in interpo­ lation taking into account values of the function. The other very natural and important problem is to impose some conditions on Fourier coeffi­ cients. The following result is interesting in itself and will be used in Chapter III.F. Theorem. Let (n k )k:: 1 be a sequence of positive integers such that n k+ l � 2nk for all k in N and let (vk )k:: 1 E £2 . Then there exists g E A with g(z) = L::= 0 g(n)z n and g(n k ) = Vk and IIY II oo � C ll (v k ) ll 2 ·

Since (g(n ) ) n�o E £2 for every g E A one cannot relax the condi­ tion on the sequence (v k )k:: 1 . Thus the theorem can be rephrased as { f (n k ) : f E A} = f2 . Let us consider only numbers z with l z l = 1 . Assume also = 1 . We define inductively two sequences of polynomials 9k (z) and h k (z) , k � 0 as follows : Proof:

L:%': 1 l vk l 2

9o (z)

=

vo z n o

and ho ( z )

=

1

(10)

188

Ill.E. The Disc Algebra § 1 0.

and for k > 0 we put

9k (z) = 9k - l (z) + Vk zn k h k - l (z), hk (z) = h k - l (z) - 'ihzn k 9k - l (z).

{11)

Using the elementary identity

b, v we inductively obtain that k = {1 + l vi l 2 ) � C. (zW (zW + h IT I Yk l k j =O

valid for all complex numbers

a,

{12)

We also obtain inductively that

nk 0 9k = 'L, §k (j)zi and hk = 'L, h k (j)zi . j� �-� Since n k + l � 2n k we infer that there is no cancellation of Fourier co­ efficients in {11). In particular we get h k (O) = 1 for all k and thus Yk (ns ) = V8 for s � k. Thus {12) and the open mapping theorem I.A.5 give the claim.

a

10. We have seen projections in A whose image is a C(K)-space. Now we will investigate projections whose image is isomorphic to A. This will lead to the proof of Theorem 12. Given a positive number c and an interval I in 11' we say that a function I E A is c-supported on I if ll (t) l � c lllll for t E 11'\f. We say that a subspace X C A is c-supported on I if every I E X is c-supported on I. We have the following. Proposition. For every c > 0 and an interval I C 11' there exists a subspace X C A and a projection P: A�X such that

(a) X is c-supported on I, (b) d(A, X) � 1 + c,

( c ) P 1 = 0 and II P II � 1 + c,

(d) for every g E A, t5 -supported on 11'\I we have II Pgll � (c + t5 ) II Y II ·

189

Ill.E. The Disc Algebra § 1 1 .

This is an interplay between averaging projections and con­ formal mappings. Every conformal map cp: [) --+ [) induces an isometry I'P : A --+ A defined by I'P ( f ) = f o cp. Suppose the proposition holds for some e > 0 and some interval I. Then it holds for the same e > 0 and any other interval I1 · To see this let us take a conformal map cp such that cp(ft ) = I. Then I'P (X) is e-supported on I and I'PPI'P- 1 is a projection onto I'P (X) and (a)-(d) hold. Let Proof:

(13) This is a norm-1 projection and ImQ n � A. For a positive number � < ! we find a function F E A, with II F II � 1 such that

27r I F(e '"(J ) - 1 1 < � for n

-

< (J < 21r - - and F(1)

27r n

= 0.

Such a function is easy to construct using conformal mappings as before. Observe that for f E ImQn we have and

11/11 ;;::: II F . /II ;;::: (1 - � ) II/II II Q n (F · f) - /II = II Qn C f · (F - 1)) 11

(14)

� 2 � 11 / 11 ·

(15)

Let I be such that I F(t) l < � for Y\f. From (14) we infer that X = F · ImQn is a closed subspace of A, ( � )-supported on I, with d(X, A) � (1 - �)- 1 . The condition (15) shows that Q n i X is an isomor­ phism between X and ImQ n with II (Qn i X) - 1 11 � (1 - 2 �) 1 . We define P = (Qn i X) - 1 Q n . This is a projection onto X with II P II � (1 - 2�) - 1 . Since Qn 1 = 0 P(1) = 0 as well. If g is a function in A which is 8-supported on Y\f then -

If � was chosen small enough and

��

n

big enough we see that (a)-(d) a

1 1 Proposition. The space A contains a complemented subspace iso­ morphic to (I: A)o .

190

Ill.E. The Disc Algebra § 1 2.

Proof: Let us take a sequence of disjoint closed intervals {In )�= l in '][' and a sequence of positive numbers (en ) �= l with L:: :'= l en = e < 0. 1. Using Proposition 10 we construct subspaces Xn C A, en-supported on In for n = 1 , 2, . . . , and projections Pn from A onto Xn satisfying (a)-(d). It is routine to show that for Xn E Xn , n = 1, 2, . . .

(1 - 2e) sup ll xn ll :5

I�

l

Xn :5 (1 + e) sup ll xn ll

(16)

so X = span{Xn} �= l "' (EA)o . Let R(f) = L:: :'= l Pnf · In order to show that R is continuous it is enough to check that for every f E A we have II Pn /11 -+ 0. Fix tn E In. Since f is uniformly continuous in [) we get that f - f(tn) is On-supported on ll'\In for some On -+ 0. From Proposition 10 (c) and (d) we get II Pn/11 = II Pn (f - f(tn)) ll :5 (on + en) II / II , and this yields the continuity of R. Clearly R: A -+ X. For x = L:: :'= l Xn E X with Xn E Xn and ll x ll = 1 we define hn = L:;;:: l ,n # X k · Since Rx - x = L:: :'= l Pnhn we get from (16) that II Rx - x ll :5 (1 + e) supn II Pnhn ll · Once more using (16) we get ll xn ll :5 (1 - 2e) - 1 for 1 n = 1, 2, . . . so ll hn ll :5 (1 + e) (1 - 2c) - . Since each Xn is en-supported on In we get that for t E In we have l hn (t) l :::; e(1 - 2c) - 1 . Proposition 10 (d) gives II Pn (hn) ll :5 [en + e(1 - 2e) - 1 J II hn ll so we conclude that II Rx - x ll < 0.8. This shows that R I X is an isomorphism of X (see II.B. 14) and one checks that (R I X)- 1 is a continuous projection from A onto X. a 12. The decomposition method (see II.B.21) and Proposition 11 yield immediately Theorem. The disc algebra A is isomorphic to its infinite eo-sum.D 1 3 Remark. Since the projections constructed in Proposition 10 have the property that P* (Lt fHt ) C Lt fH1 we easily see that (ELI/H1 ) 1 "' Ltf H1 and passing to the duals we get (EHoo)oo "' Hoo . 14. Most of the results in this chapter show the analogy between A and C(K)-spaces. We would like to point out however that A is not a C(K)-space. To see this we need to observe that every C(K)-space has the fol­ lowing extension property:

Ill.E. The Disc Algebra § 15.

191

There exists a constant C such that for every Banach space X, every subspace Y of it and every finite rank operator T: Y ---+ C(K) there exists an operator T: X ---+ C(K) such that T I Y = T and II T II :5 C II T II · This follows {for any C > 1 ) directly from II.E.5{c) and III.B.2 (or the Hahn-Banach theorem) . Let I = 'E::N aneinB be a trigonometric polynomial. The operator Tt: A ---+ A defined by Tt (g ) = I* g is a finite dimensional operator and II Tt ll :5 11 1 11 1 · The operator Tt has a unique rotation invariant extension Tt: C(11') ---+ A which is given by Tt (g ) = 'RI * g where 'R is the Riesz projection (see I.B.20). The standard averaging argument shows that for any extension T we have II T II � II Tt ll · But II Tt ll = II 'RI II 1 · Since the Riesz projection is unbounded on £ 1 (11') (see I.B.20 and 25) we see that A does not have the above property (*). Since if Z has (*) its complemented subspaces also have (*) we get Theorem. The disc algebra A is not isomorphic to any complemented subspace of any C(K)-space.

Despite this theorem and some other striking differences between and C(K)-spaces, which we will exploit e.g. in III.F.7, the general impression is that A is quite similar to a C{K)-space. Actually the idea of comparing A to C(K) underlies most of the results about the disc algebra A presented in this book. A

15. Another proof of the above theorem, which is different, although similar in spirit, follows from the Lozinski-Kharshiladze theorem III.B.22 and the following Proposition. Let T� denote the space of all trigonometric polyno­ mials of degree :5 n with the sup-norm. There exists a sequence of projections (Pn)�= 1 in the disc algebra such that

{a) II Pn ll :5 C,

{b) d(Im Pn , T�) :5 C for some constant C and all n = 1,2,3, . . . . Proof: The proof depends on the properties of the Fejer kernels; see I.B. 16. Let An = span{ 1 , z, . . . , z 2n ) C A. We define i n : A n ---+ A EBoo A

192

Ill.E. The Disc Algebra § 1 6.

) (

(

by in L:: �:o a k z k L:: �: o a k z k , L:: �: o a k z 2 n - k We also define ¢n: A ffi A -+ A n by

)

.

Clearly I l i n I :5 1.

Using the properties of Fejer operators once more we get II 1J

then we can write

1

h f(y) fo"N cot � · (KN (x - t, y ) - KN (x + t, y) ) dtdy + h f(y) [ l AN,x (t)KN (t, y)dt - AN,x (y) ] dy ( 19 ) + h f(y)AN,x ( y )dy = + /2 +

SN ( / )(x) =

h

h

For every y , KN (x, y) is a piecewise linear function with nodes at least 2� apart so Proposition II.D.21 (b) yields the estimate for the slope

194

Ill.E. The Disc Algebra §18.

which gives

-k cot � · I KN (x - t, y ) - KN (X + t, y ) l dtdy 1 h N � 11 /lloo h 1 cot � · C(N + 1) 2 q Ndi s t ( x, y ) tdtdy � C ll/lloo (N + 1) h q Ndi s t ( x, y ) dy � C ll/ll oo ·

h �

11 /lloo ·

1

(20)

The estimate

(21) follows immediately from the known properties of the trigonometric con­ jugation operator; see Zygmund [1968] p. 92 or Katznelson [1968] Corol­ lary 111.2.6. In order to estimate 12 we write

where PN is the N-th partial sum projection with respect to the Franklin system. Let 'PN,x be a piecewise linear function in C(ll') with nodes at to, tt , . . . , t N such that 'P N,x (tj ) = AN,x (tj ) for j = 0, 1, . . . , N. One checks (draw the picture) that (23) Since the Franklin system is a basis in £1 (ll') (see III.D.26) from(23) we get

From (20), (21), (22) and (23) , (17) follows immediately.

a

18. Now we want to give different isomorphic representations of H00 • Having different isomorphic representations of an important space is generally useful because each representation carries with it different in­ tuitions, and even the possibility of using different analytical tools. Let Ui ) f'=. o be the basis in A which exists by Theorem 17 and let H� = span{ /j };::;n · Let us recall also that An = span(1, . . . , z 2 n ) C A (see 15) .

III.E. The Disc Algebra §Notes. Theorem. The spaces morphic.

( I:�= 1 H;;, ) 00 , ( I:�= 1 An) 00 and H00

195 are iso­

First note that ( I:�= 1 H;;, ) 00 is isomorphic to its infinite £00-sum. This follows from II.B.24. A standard perturbation argument (see II.E.12) shows that for certain A each H;;, is A-isomorphic to a A­ complemented subspace of A k ( n ) and analogously it follows from Propo­ sition 15 that each An is A-isomorphic to a A-complemented subspace of H'j, n ) . This yields that ( 2:: �= 1 H;;, ) 00 is isomorphic to a comple­ mented subspace of ( 2:: �= 1 An) 00 and also ( 2:: �= 1 An) 00 is isomorphic to a complemented subspace of ( 2:: �= 1 H;;, ) 00 so our first observation and II.B.24 give ( 2:: �= 1 H;;, ) oo rv ( 2:: �= 1 An) 00 • Remark 13 yields 00 that ( L H;;, ) oo is isomorphic to a complemented subspace of H00 • Proof:

n= 1

To complete the proof we need to show that ( 2:: �= 1 An) 00 contains a complemented copy of H00 • Let :Fn be the Fejer kernel and define i: Hoo ---+ ( 2:: �= 1 An) 00 by i(f) = (f * :Fn )�= 1 . The properties of the Fejer kernel (see I.B. 16) give that i is an isometry. To define the projec­ tion onto i(Hoo ) we use a compactness argument. Let B denote the unit ball of H00 with a(H00 , LI/ H00 )-topology, and let Bn be the unit ball in An · We define maps 1fn: f1�= 1 Bn ---+ B by 1fn(h1 , h2 , . . . ) = hn. Since the space of all maps from I1�= 1 Bn into B is compact we take 7f to be a cluster point of {7rn} �= 1 . One checks that 7f is homogenous and ad­ ditive so it extends to a continuous linear map 1r: ( 2:: �= 1 An ) 00 ---+ H00 • Moreover

This shows that i1r is a norm-one projection onto i(H00 ) .

a

Notes and remarks. As noted in 1 the disc algebra is an important space. It is a prototypic

uniform algebra, so much information about it can be found in Gamelin [1969] , Hoffman [1962] or Garnett [1981] . The closely related space H00 is even more fascinating; the whole book Garnett [1981] deals with it. A more Banach space oriented exposition is in Pelczynski [1977] . The connection with operator theory hinted in 1 is presented in detail in the beautiful lectures of Nikolski [1980] . The theory of peak sets and peak-interpolation sets is a well developed topic in uniform algebra theory; see Gamelin [1969] 11. 12. Our Proposition 2 is a prototype of

196

Ill.E. The Disc Algebra §Notes.

this theory. It was proved by Rudin [1956] and Carleson [1957] . The appeal to Theorem III.D.4 can be avoided but it saves some calculations. The problem of characterizing interpolating sequences for H00 was an object of very intense study in the late 50's; Hoffman's book [1962] contains a nice presentation of these early results. By now it has grown into a vast area (see Garnett [1981] ) . The beautiful and simple proof of (b)=>(c) in Theorem 4 is due to Peter Jones (see Gorin-Hruscov­ Vinogradov [1981] ) . Our Theorem 5 is a particular case of results in Casazza-Pengra­ Sundberg [1980] where complemented ideals in A are fully described. The description of ideals in A is contained in Hoffman [1962] . This says that every closed ideal I in A is of the form AK · F where K C '][' is a closed subset of Lebesgue measure 0 and AK = {! E A: JI K = 0} and F is an inner function such that F - 1 (0) n ll' C K and the measure determining the singular part of F is supported on K. The result of Casazza-Pengra-Sundberg [1980] asserts that I is com­ plemented in A if and only if F is a Blaschke product whose zeros form an interpolating sequence. In paragraph 6 we gave a crash course in ele­ mentary potential theory for subsets of ll'. Chapter III of Kahane-Salem [1963] contains everything we state and use. Theorem 6 is one of the re­ sults contained in Hruscov-Peller [1982] . Our presentation follows Koosis [1981] . The direct proof of Theorem 9 is taken from Fournier [1974] . Much more general theorems are proved in Vinogradov [1970] . In partic­ ular he has shown that given ( v k ) � 1 E £2 there exists f(z) = E�o anz n such that a2 k = Vk and f(z) is holomorphic in G and continuous in G where G is any region in cr with smooth boundary and aG n ll' contains an interval. Theorem 12 and its proof are taken from Wojtaszczyk [1979] . There are many proofs that A is not isomorphic to any C(K)-space. We will see some more in 111.1. The fact was first observed by Pelczynski [1964a] with basically the same proof as the one given in 14. The argument was extended to the context of ordered groups by Rosenthal [1966] . Proposition 15 is an unpublished observation of J. Bourgain and A. Pelczynski. The question if A has a Schauder basis was asked by Banach [1932] . The answer was given by Bockariov [1974] . The use of the Franklin system in the construction was rather unexpected. Theorem 18 is a rather routine consequence of previous results. It was observed in Wojtaszczyk [1979a] . It shows in particular that H00 is isomorphic to the second dual of a Banach space. Note that with the natural duality we have H00 9:! (LI /H1 ) * . This is the unique isometric predual of H00 (see Ando [1978] and Wojtaszczyk [1979a] ) .

197

III.E. The Disc Algebra §Exercises.

The space Ltf H1 in its turn is not isomorphic to a dual Banach space; this was noted in Pelczyilski (1977] and Wojtaszczyk (1979a] ; see Exercise III.D. 19. Exercises

1.

Let � c T be a compact set of measure zero and let (nk)k:: 1 be a lacunary sequence of natural numbers. Given f E C(�) and a sequence (ak) E £2 show that there exists h E A ( D) such that h i � = f and h(nk) = ak for k = 1, 2, . . . .

Suppose that x * E A* . Show that there exists only one measure on T such that ll�tll = ll x * ll and �ti A = x * . 3. Let V c A* be a relatively weakly compact subset. For each E V let v E M(T) be its norm-preserving extension (see Exercise 2). Show that V = { v : E V} is relatively weakly compact in M(T). 4. (a) Let f E D (the Dirichlet space) . Show that f induces a func­ tional on H1 (T), i.e. we have an inequality I Jy g(e i9 )f(e i9 )dO I ::;; cf . II Yil t . (b) Show that there are unbounded functions in D. 5. The matrix ( a ii ) i ,j�O is called a Hankel matrix if Cl! ij = cp( i + j) for 2.

v

v

some cp. An operator on £2 is called a Hankel operator if its matrix in the natural unit vector basis (ej ) � 0 is a Hankel matrix. (a) Show that L 00 (T)/ H! is isometric to the space of all Hankel operators.

(b) Show that C(T)/Ao is isometric to the space of all compact Hankel operators. (c) Show that for every Hankel operator T there exists a best ap­ proximation by a compact Hankel operator.

6.

(a) Suppose that (zk)k:: 1 C D is such that d�1"1:��)1 ) < c < 1 for k = 1, 2, 3, . . . . Show that (zk)k:: 1 is an interpolating sequence. Show that if (zk)i:'= t are positive real numbers then the above condition is also necessary for (zk)i:'= 1 to be interpolating. (b) Find an interpolating sequence (A n)�= l C D such that { (/(An))�= 1 : / E A } is not closed in £00 •

7.

Let IPr = {z E CI:: : r ::;; l z l ::;; r - 1 } for 0 < r < 1 and let A(IPr ) denote the space of all functions continuous in IPr and analytic in the interior.

198

III.E. The Disc Algebra §Exercises.

(a) Show that A(1Pr) contains a !-complemented isometric copy of

A(D) .

8.

(b) Show that A(D) contains a complemented copy of A(JPr) with the constants independent of r. (c) Every f E A( 1Pr ) can be written as L:: � : anz n . Show that the map Pr ( L:: � : anz n ) = L:: := o anzn is a projection from A(JPr) onto A(r - 1 D). Show that sup l< r< l II Pr ll = oo. (a) Show that f E Hoo (D) is a Blaschke product if and only if 1/ (z) l ::; 1 and limr-+1 f:_1r log lf (rei9 ) l d0 = 0. (b) Suppose f E H00 (D) is an inner function. Show that for every p with 0 < p < 1 the functions

- peicp wcp (z) = 1 f(z) - pe- icp f(z)

9.

are Blaschke products for almost all cp, 0 < cp ::; 21r. (c) Show that every f E H00 (D) is a limit in the topology of uniform convergence on compact subsets of D, of a sequence of finite Blaschke products. (d) Show that the closed unit ball in A is the closed convex hull of finite Blaschke products. (e) (von Neumann inequality) . Let T: H -+ H (H a Hilbert space) be a contraction, i.e. II T II ::; 1. Show that for every polynomial p (z) we have ll p (T) II ::5 sup zEID lp (z) l . (a) Show that if P: A -+ A is a norm-one, finite dimensional projection with dim /m P > 1 then Im P* c {J.q.t .l m} where m is the Lebesgue measure on Y.

(b) Show that the disc algebra is not a 1r1-space. 10. (a) Suppose f E L00 ('U') . Show that there exists 9 E H00 ( 'U') such that I I/ - 9 11 = inf{ l l / - h ll : h E Hoo (Y) }. (b) Suppose f E C(Y) . Show that dist(f, H00 ) = dist(f, A) . (c) Suppose that f E C(Y) . Show that there exists only one 9 E H00 ('U') such that I I/ - 9 11 = dist(f, Hoo ), and that 1 9 - /I = const. (d) Show that there exists f E C (Y) such that its best approximation in H00 (Y) , i.e. 9 E H00 (Y) such that II/ - 9 11 dist(f, H00 ) , is not continuous.

III.F. Absolutely Summing And Related Operators .

We discuss in detail p-absolutely summing operators. The Pietsch fac­ torization theorem, which is basic to this theory, is proved. The funda­ mental Grothendieck theorem is proved in its three most useful forms. Later we improve it and show the Grothendieck-Maurey theorem, that every operator from any £ 1 -space into a Hilbert space is p-absolutely summing for all p > 0. We present the trace duality and show that the p' -nuclear norm is dual to the p-absolutely summing norm. We also introduce and discuss p-integral operators. We show the connection be­ tween cotype 2 and the coincidence of classes of p-absolutely summing operators for various p's. The extrapolation result for p-absolutely sum­ ming operators is proved. We apply Grothendieck's theorem to exhibit examples of power bounded but not polynomially bounded operators on a Hilbert space and to give some estimates for the norm of a polynomial of a power-bounded operator. We also present many applications to harmonic analysis: we construct good local units in L 1 { G ) , we prove the classical Orlicz-Paley-Sidon theorem and give some characterizations of Sidon sets. 1. In this chapter we will discuss several important classes of operators, namely p-absolutely summing, p-integral and p-nuclear operators. All these classes have some ideal properties so we will introduce the general concept of an operator ideal. We are given an operator ideal if for each pair of Banach spaces X, Y we have a class of operators J(X, Y) such that { 1 ) J(X, Y) is a linear subspace {not necessarily closed) of L(X, Y)

containing all finite rank operators,

E J(X, Y), A E L(Z, X) and B E L(Y, V) that BTA E I(Z, V) for all Banach spaces X, Y, Z, V and all operators A, B.

{2) if T

An operator ideal is a Banach ( quasi-Banach) operator ideal if on each I(X, Y) we have a norm {quasi-norm) i such that {3) (J{X, Y ) , i) is complete for each X, Y

(4) i(BTA) ::; II B I !i(T) II A II whenever the composition makes sense

III.F Absolutely Summing And Related Operators §2.

200

(5) for every rank-one operator where = · y.

T(x) x*(x)

T: X -+ Y we have i(T) l x *l l I Y I , =

·

Actually the reader has already encountered some examples of Ba­ nach operator ideals. Compact operators (see I A.15) and weakly com­ pact operators (see II.C.6) form Banach operator ideals with the opera­ tor norm. .

T: X -+ xi X

2. An operator Y is p-absolutely summing, 0 < p � oo (we write lip ( Y)), if there exists a constant C < oo such that for all finite sequences ( ) J=l C we have

T E X,

n

.!

n

!.

( � 1 Txi 1 P) p � C sup { ( � i x* (xi ) I P) p= x* E X*, I x* l � 1 } . (6) We define the p-summing norm of an operator T by 7rp(T) inf{C: (6) holds for all (xj )J=l X,n 1, 2, . . . } . (7) c

=

=

Let us observe that for p = 1 the condition (6) is equivalent to the fact that transforms weakly unconditionally convergent series into absolutely convergent series.

T

3.

We have the following

Theorem. For 0 < p � oo the p-absolutely summing operators form a quasi-Banach (Banach if 1 � p � oo) operator ideal when considered with the p-absolutely summing norm 7rp ( · ) .

The proof of this theorem consists of routine verification of condia tions (1)-(5) and is omitted.

X,

II00(X,

L(X,

Y) = One easily checks that for all Y we have Y) and = For p < oo the situation is less trivial. The iden­ tity id: £2 £2 is not p-absolutely summing for any p < oo. To see that condition (6) fails it is enough to consider finite orthonormal sets. The canonical example of a p-absolutely summing map is given as fol­ lows: Let /.L be a probability Borel measure on a compact space K. Let 4.

1r00 (T) I T I . -+

lii.F Absolutely Summing And Related Operators §5.

idp: C(K) --+ Lp(K, f..L ) be the formal identity. Then for have Trp(i dp) = 1 . Simply we have

�

(:�� f; l /; (k)IP)p { ( t, I j l ) · n

= sup

201 1�

p < oo

.l

/;dv ' ; ' v

E M(K),

I

we

(8) v ii = 1

}

so Trp(idp) � 1 , but taking the one element family consisting of a con­ stant function we see that Trp(idp) = 1 . A slight but useful variation of this example is the map f �---+ f g defined as a map from L oo (f..L ) into Lp(f..L ) (clearly g E Lp(f..L ) ). Here we do not assume that f..L is a proba­ bility measure; it can be arbitrary. The same calculation as in (8) gives 'lrp( / 1--+ f . g ) =

I YI p·

5. The following proposition describes some formal but useful proper­ ties of p-absolutely summing operators.

I

I

Proposition. ( a) lE T E IIp(X , Y), 0 < p < oo and X1 c X is a closed subspace then T X 1 E Ilp(Xt . Y) and 7rp( T X t ) � 7rp(T) . (b ) lE T E IIp(X, Y), 0 < p < oo and Y1 C Y is a closed subspace and T (X) C Y1 then T E Ilp(X, Yt ) and the norms of T in both spaces are the same. ( c ) If ( X. J'YEr is a net of subspaces of X directed by inclusion such that U "Y H X"Y is dense in X and T: X --+ Y then Trp (T) = sup"Y Trp (TIX"Y ) for 0 < p � oo . ..

Parts ( a) and ( b ) are obvious from the definition. Part (c ) requires a simple approximation argument (see II.E. 12) and is omitted.a

Proof:

Now we will give some very important examples of p-summing maps. Actually these are one operator acting between different spaces. Later on we will call this operator the Paley operator. 6.

Proposition. For f

E L 1 (Y)

let P(f) = ( / (2 n

) ':=t ·

( a) P: A --+ £2 is 1-absolutely summing and onto. ( b ) P : C(Y) --+ £2 is p-absolutely summing for 1 < p < oo and onto.

202

III.F Absolutely Summing And Related Operators § 7.

(c) P: C(Y) -+ co is 1-absolutely summing. It is clear that P is continuous in all the cases (a) , (b) and (c) . That P is onto in cases (a) and (b) follows directly from Theorem III.E.9. In order to see (a) let us factor

Proof:

where P1 is the Paley projection, i.e. the operator P acting on H1 (Y) . It follows from Paley's inequality I.B.24 that P1 is continuous. We see from Proposition 5 (a) , (b) and from 4 that id is !-absolutely summing so P is also (see ( 4)). To see (b) we consider the factorization

where Pl ( L�: J(n)e inB ) = ( } (2 n ))':= l · The operator P1 is bounded by the remark after Proposition III. A. 7 so P is p-absolutely summing by (4) and 4. For (c) we use the factorization a

The above proposition easily yields the fundamental theorem of 7. Grothendieck.

Theorem. (Grothendieck) . Every operator T: L1 (/.L) -+ H, where H is a Hilbert space, is 1-absolutely summing.

Remark. It follows from the closed graph theorem or from the proof given below that there exists a constant K such that 1r1 (T) � KIITII for all T: £1 (t.L ) -+ H. The smallest such constant is called the Grothendieck constant and denoted by Kc .

III.F Absolutely Summing And Related Operators §8.

Let us start with an operator T: 6 ( a) we have a commutative diagram Proof:

i1 i2. --+

203

Using Proposition

where

0 with 7rp(T) = 1 . Suppose that there is a probability measure J..L on (0, 1 ] satisfying (10) , so we have

-

1 1 1 J(s)ds r � c J l f(s) I PdJ..L ( s)

for all 1 E c [o, 1] .

(14)

One checks that if ( 14) holds for some measure J..L it also holds for the part of it which is absolutely continuous with respect to the Lebesgue

206

III.F Absolutely Summing And Related Operators §9.

measure, f..Lc · But f..Lc is a non-zero, non-atomic measure and (14) says that T induces a non-zero linear functional on Lp( [O, 1] , f..Lc ) · Since p < 1 this is impossible; see I.B.4. From Pietsch's theorem we can derive some properties of 9. p-absolutely summing operators. Corollary. (a) If O < p < q < oo then llp(X, Y) c ll q ( X, Y) for all Banach spaces X, Y. Also 7rp(· ) 2: 7rq (·). (b) Every p-absolutely summing operator 0 < p < oo is weakly compact and maps relatively weakly compact sets onto norm-pre-comp­ act sets. (c) If X is a subspace of X1 and T E flp (X, Y) with p 2: 2, then T extends to an operator T1 E fl 2 ( XI , Y).

(a) and (b) follow directly from (10') and the fact that Xp is q-absolutely summing and satisfies (b) . For (c) ob­ serve that by (a) we can assume p = 2. Then use (10') and the fact that every subspace in a Hilbert space is complemented. Proof:

id: i(x)

-+

All this may seem rather trivial and quite abstract. In order to convince the reader that these are important and powerful concepts, before investigating them any further, let us give some applications of the results already obtained. 10.

Theorem. Let X be a complemented subspace of L 1 (J.L) . Assume also that ( xn, x�)�= l is a normalized unconditional basis in X. Then there exists a constant C such that L::= l l x�(x) l :::; C ll x ll for all x E X, so (xn)�= l is equivalent to the unit vector basis in £1 . Proof: Since X has cotype 2 (see III.A.21-23) or by the Orlicz theorem (see II.D.6) we get that the map T: X -+ £2 defined by T(x) = (x� (x))�= l is continuous. Let P: L 1 (J.L)�X be a projection. By Theorem 7 the operator TP is !-absolutely summing so T = TP I X is also !-absolutely summing. Thus for every finite sequence of scalars (an);{= 1 we have

N N L l an l = L II T(anXn) ll n= l n= l = 1r 1 (T) �up

:S

N 1r 1 ( T ) sup L l x * (anxn) l ll x * ll:9 n = l

l

sup x *

ll x * ll 9 lc: n l = l

( nt= l Enanxn ) I

III.F Absolutely Summing And Related Operators § 1 1 .

207 a

Let us note some special instances of this theorem (a) Since £1 [0, 1] is not isomorphic to £ 1 (see III.A.5 and III.A.7) we get a new proof that £ 1 [0, 1] does not have an unconditional basis (see II.D.lO) . (b) The spaces Lp [O, 1] and fp, 1 < p :::; 2, are not isomorphic to complemented subspaces of L 1 [0, 1] , although they are isomorphic to subspaces of L1 [0, 1] (see Notes and remarks to III. A) . The same conclusion can be easily derived from III.D.34(b) (see Exercise III.D.20) . Note however the important difference between these two arguments. Using III.D.34(b) we get no information about the complementation of e; in £ 1 [0, 1] . A careful reading of the above proof gives that if P is a projection from L 1 (f..l ) onto a subspace isometric to e; then II P II :::: Cn 1 - i , 1 :::; p :::; 2. This reflects the difference between the global, infinite dimensional approach and local, more quantitative one. (c) £ 1 has only one unconditional basis in the sense that every normalized unconditional basis in £ 1 is equivalent to the unit vector basis. Let Wf (lf2 ), l 1 , denote the Sobolev space (see I.B.30) . 2: By the Sobolev embedding Theorem I.B.30 the identity operator 1 2 2 id: Wf (1r ) -+ wJ (1f ) is continuous. Note that wJ - 1 (1f2 ) is a Hilbert space. We want to show that it is not 2-absolutely summing. Since WJ (1r2 ) c Wf (lf2 ) it suffices to show that id: WJ (1r2 ) -+ wJ - 1 (1f2 ) is not 2-absolutely summing. Let us put fn,m (O, r.p) = (m + n) - 1 ei nll ei m'P, n, m > 0. We have 11.

L

O < m +n 2kc,0 (2k)l·

ll cp(Tc ) ll ll e ll ll"'ll � l (cp(Tc)e, .,.,} l = =

(28)

=

Observe that Theorem III.E.9 gives polynomials cp with ll cp lloo � 1 but as big as we please.

I Lk C2kc,0 (2k) I

Proof of the proposition.

{ cno m b(n, m) = { cno m

a(n, m) = and

Let us put +

if 0 m � 0, if m � n � 0. One easily checks that a(n, m) is a matrix of a bounded operator A on £1 while b(n, m) is a matrix of a bounded operator B on £00 with norms bounded by a constant independent of On sequences (en )�= O we formally define operators

c.

Sk ((�n) �= o ) = (�k , �k+b · · · ) and SA; ((�n )�=o) = (O, . . . , O , �o. � l , · · · ) · ........_.. k t i mes

213

III.F Absolutely Summing And Related Operators § 1 6.

One checks that for finite sequences (en)�= O one has ASZ - Sk A = Sk B - BS;'�fVk . From these relations we infer that Vk acts on both £ 1 and €00 with norms uniformly bounded in k. Thus by interpolation (see I.B.6) Vk acts also on f2 and II Vk ll :5 C. We now define H as a completion of £ 1 EB £ 1 with respect to the scalar product

where ( ·, · ) is the usual scalar product. We define Tc: H - H by Tc(f, g) = (Sif, S1g ). Clearly r: (f, g) = (Sk f, Sk g ). Since

li T: ( !, g) II � = II Sk f ll� + II ASk f + Sk g ll� :5 II/II � + ( II Vk/ 112 + Il Sk A! + SkYII 2 ) 2 :5 11/11� + (C II /11 2 + II A J + Yll2 ) 2 :5 C ll (f, g) ll � we get ( 26 ) With e = (eo, 0) and TJ = (0, eo) we get for n = 0, 1 , 2, . . . .

((T0 e , TJ)) = (( (en, O), (O, eo) )) = ( A(en), eo ) = a(n, O ) = cn . a

So (27) also holds.

Since the estimate (25) does not hold for every power-bounded operator one has to seek other estimates for ll cp(T) II · For a polynomial cp(z) = L n 0 we get from II.E. 13 an isomorphic embedding i: X ---+ e;;, with II i ll . ll i- 1 11 � 1 + E . Since e;;, = C(S) where s = { 1, 2, . . . ' n } we apply Theorem 23 to get a measure v. Since id: C(S) ---+ Lp(S, v ) is the same as �: e;;, ---+ e: with � = ( 6n ) ;;' 1 where 6n = v ( { n }) � we get the p-nuclear factorization. a Proof:

=

25 Corollary.

If T: X ---+ Y

with

X and Y finite dimensional,

then

220

III.F Absolutely Summing And Related Operators §26.

From remarks made after Proposition 22 we know that 11'2 ( · ) =

h ( · ) (always) so the claim follows from Corollary 24. Proof:

a

The reader should consult the exercises to find examples showing that the above classes are different in general. One of the reasons that � nuclear and �integral operators are important is that they are connected with �absolutely summing operators via duality. Before we proceed we have to discuss duality as applied to Banach operator ideals. 26. Given a Banach operator ideal I(X, Y) we want to describe the dual space I(X, Y) * . In general this subject is quite involved; the theory of general tensor products and the approximation property play impor­ tant roles. We will discuss it only for finite dimensional Banach spaces X and Y. Despite this restriction, the results can be applied in the study of �absolutely summing operators on infinitive dimensional spaces. This follows from Proposition 5. If T: X --> X (X finite dimensional) is a linear operator then T has a representation (non-unique of course) in the form T(x) = Lj xj (x)xi . The trace of T is defined as tr(T) = Lj xj (xi ) · As is well known and easily checked this definition is correct, i.e. it does not depend on the particular representation of T. Obviously the trace is a linear functional on L(X, X). Given T: X --> Y and S: Y --> X we see that tr(ST) = tr(TS) . For every operator S: Y --> X the formula r.ps (T) = tr(TS) , for T E L(X, Y ) , defines a linear functional on L ( X, Y). Counting dimensions we realize that we can identify L(X, Y)* with L(Y, X) if the duality is given by the trace, i.e. (S, T ) = tr(ST) . All this is elementary linear algebra and can be found in most textbooks on the subject. If we have a norm i on L(X, Y ) then the norm i* on L(Y, X ) is dual to i (with respect to trace duality) if for every T E L(X, Y ) we have

i(T) = sup{tr(ST) : i * (S) � 1}. 27.

(37)

The following important theorem identifies 71'; .

Theorem. Let X and Y be finite dimensional Banach spaces. Then ( IIp ( X, Y ) , 11'p )* = (Np' (Y, X), np' ) , 1 � p � oo , with the trace duality.

Let us fix T E L(X, Y ) . For S: Y --> X with the representation S(y) = Lj yj (y) · xi we have

Proof:

j

j

221

III.F Absolutely Summing And Related Operators §28.

(38) ( � i!Yj ll p' ) ? ( � I! Txi ii P) ; :5 ( � IIYj li p' ) 11'p (T) sup { ( � l x * (xi W ) "P : x * E X* , ll x * ll :5 1 } · :5

J

J

1

pr

J

1

J

Since (38) holds for every representation of S we get l tr(TS) I :5 np' (S) ·

11'p(T).

On the other hand let us fix X1 , , Xn in X such that Ej l x* (xj) I P 1 :5 1 for every x* E X* with ll x* I :5 1 and such that ( Ei II Txi l i P ) "P � (1 - c:)11'p(T) . Let us choose yj E Y* such that IIYj ll = 1 and yj (Txi ) = li Txi II for j = 1, 2, . . . , n. Let us also fix numbers ai � 0, j1 = 1, 2, . . . , n such that Ei a� = 1 and Ei ai II Txi I = ( E i II Txi li P ) "P . We define S: Y -+ X by the formula S( y) = E;=l ai yj (y)xi . Since •

•

•

I

np' (S) :5 and

(� ) J

a�'

1

pr

·sup

{ ( � l x* (xi ) I P) J

-;;1

}

: x * E X* , l! x * ll :5 1 :5 1

tr(TS) = L ai yj (Txi ) � (1 - c:)11'p(T) j

we get the claim.

a

28. Our aim now is to establish the dual version of Grothendieck's theorem. Because of later applications in III.I we give a more abstract presentation than is really necessary.

Proposition. Let X be a finite dimensional Banach space. The fol­ lowing conditions are equivalent:

(a) for every T: X -+ £1 we have 11'2 (T) :5 C II T II ; (b) for every T: X* -+ £2 we have 11' 1 ( T) :5 CI !I T II · More precisely if (a) holds then C1 :5 Ka · C. If (b) holds then C :5 C1 . (a) =?(b) . Clearly we can restrict our attention to T: X* -+ i!f . From Theorem 27 we see that we have to estimate tr(T S) for every

Proof:

222

III.F Absolutely Summing And Related Operators §28.

00

i!f -+ X* with n (S) = 1. Using the standard approximation (see Proposition II.E. 12) we get the diagram

S:

s

�iM00 /fi

i!f

X*

T

X

T*

i!f

with II a ll · 11 .8 11 ::; 1 + c. Dualizing this diagram we get

iN2 a

s•

\iM/

iN2

�

1

From Corollary 25 and Theorem 7 we get using ( a )

l tr(TS) I = l tr(S*T* ) I ::; 1r2 ( a * )1r2 (,8 *T* ) ::; 7r2 (a * )7r2 (,8 * ) 11 T* II ::; Ka ll a * II C II .B* II · II T* I I ::; Ka C(1 + c) II T II ·

Since c was arbitrary we get ( b ) . ( b ) => ( a) . As before we use Theorem 27 so we have to look for tr(TS) where we have the following diagram:

with ll a ll · 11 .8 11 · I A ll ::; 1 + c. Dualizing this diagram we get

III.F Absolutely Summing And Related Operators §29.

223

Using ( b ) we get 1r1 (S*T*) :5 ll a* II II� * II II T* II 7r i ( .B * ) :5 C1 (1 + c) II T II · But S*T* is an operator on .e� so by remarks made after Proposition 22 we have 1r1 (S*T* ) = i 1 (S*T* ). Corollary 24 gives i 1 (S*T*) = n 1 (S*T*) so we have

l tr(TS) I = l tr(S*T* ) I :5 n 1 (S*T*) :5 C1 ( 1 + c) II T II · This gives ( a) .

•

29 Theorem. For any C(K)-space (in particular for L oo (!-£) ) and 1 :5

p

:5 2 we have

L(C(K) , Lp) = Ih (C(K) , Lp) with 1r2 (·) :5 Ka ll · II · From Proposition 5 and II.E.5 we infer that it is enough to show 1r2 (T) :5 Ka li T II for every T: .e� -+ lif , 1 :5 N, M < oo . The Grothendieck theorem ( Theorem 7) says that X = if satisfies ( b ) of Proposition 28 so we get

Proof:

1r2 (T) :5 Ka i i T II for every T: .e� -+ .efl.

(39)

If 1 < p :5 2 then lif is isometric to a subspace of LI [O, 1] ( see III.A. 16) . From II.E.12 and an easy approximation we see that we can assume a lif c .efl' so (39) and Proposition 5 give the theorem. 30. This dual version of the Grothendieck Theorem also has some nice applications. We will discuss some of these, connected with harmonic analysis. Let G be a compact, abelian group with dual group r. The following is a classical result of Orlicz, Paley and Sidon.

Let A = (>. -r ) -r er be a function on r. The map A(!) = b)) (>.-r f -r er maps C (G) into £1 (r) if and only if A E £2 (r) .

Theorem.

If A E £2 (r) then A acts not only from C (G) into £ 1 (r) but also from L 2 (G) into £1 (r) , simply because characters form a complete orthonormal system in L 2 (G). Conversely if A: C(G) -+ £ 1 (r) , then by Theorem 29 1r2 ( A ) :5 Ka ll A l . The Pietsch factorization theorem ( Theorem 8) and (13) give Proof:

!

II A /11 :5 Ka ll A l! ·

( i if(g) l2dm(g) )

1

2•

224

III.F Absolutely Summing And Related Operators §31 .

But this means that

(

L 1 -X'Y I I ib ) l :5 Ka llAl! L 1] ( 7 ) 12 'YEr "Y Er

!

)2

a

31. Let us recall that a subset S C r is called a Sidon set if { 'Y h es C C (G) is equivalent to the unit vector basis in i 1 ( S ) . For S c r the symbol Cs (G) will denote span {'Y : 'Y E S} C C(G). The following

theorem shows that the Banach space structure of Cs (G) determines if S is Sidon or not. Theorem. If Cs(G)* is isomorphic to some Sidon set.

C( K )

space than S is a

Proof: Let us fix f E Cs (G) and let us consider the operator T1 : M(G) --+ C(G) given by the formula Tt (J.t) = f * J.t · We can factorize

this operator as follows:

M(G)

Tr

_.

Cs (G) � C(G)

•\ jr ___

M(G)/Is

where q is the quotient map and Is = {J.L E M(G): [l,('y) = 0 for 'Y E S} . Clearly M(G)/Is � Cs(G) * rv C( K ) and also the space Cs(G) can be considered as a subspace of Cs (G) ** rv M (K) so by Theorem 29 we get '11'2 (T) < oo. This implies '11'2 (Tt) < oo. But Grothendieck's theorem implies that also '11' 1 (Tt) < oo. The proof is completed by an application of the following. If an operator T: M(G) --+ C(G) given by T(J.t) = L: 'Y e r a'Y[l,('y)'Y is 1-absolutely summing then L: 'Y e r l a'Y I < oo, and con­

32 Lemma.

versely.

Let L c r be a finite set and let E c C(G. ) be such that d( E, i�) :5 1 + c and 'Y E E for every 'Y E L. Let P be a projection from C(G) onto E such that II P II :5 1 + c . Let Lf = span{'Y : 'Y E L} C L1 (G) c M(G) . For the operator S: Lf --+ E defined by S(J.t) = PT ( J.t ) Proof:

225

III.F Absolutely Summing And Related Operators §33.

we have 11" 1 (8) :5 (1 + c)11"1 (T) . For every sequence TJ = ( TJ he L with ,. I TJ,. I = 1 we define a map u71: E --> L f by u11 (!) = ( 'L. ,. e L TJ,. f ('y ) · r} Since I 'E e L TJ,. J('y ) · 'Y II 1 :5 I 'E e L TJ,. J('y ) · 'Y II2 :5 ll/ll2 :5 11 /ll oo we get ,. ,. ll u11 ll :5 1 and thus also noo (u 11 ) :5 1 + c. From Theorem 27 we get

I I:>,.a,. l = 1 tr(u11 8) 1 :5 11"1 (8) · n00(u11 ) :5 (1

+

')'E L

Since this holds for every c > 0, every finite set L get 'E ,. e r l a,. l :5 11" 1 (T). The converse is obvious.

C

c) 2 7r1 (T).

r and every 17 we a

Now we return to the study of p-absolutely summing operators. One lesson to be learnt from our previous discussions is that it is very useful to have equalities of the form Ilp ( X, Y) = L ( X, Y). This is only rarely true so we try to look for equalities of the form Ilp( X, Y) = II q (X, Y). These are also useful ( see III.H. 12). Before we proceed let us state some easy but useful facts. 33.

Proposition.

( a) Let 1 :5 p :5

oo

and let T: X -->

Y.

Then

11"p(T) = sup{7rp(T8): 8: I!� --> X, 11 811 :5 1 , m = 1 , 2, . . . } .

(40)

( b ) If T: X --> Y is an operator and 0 < p :5 oo and (0, J.L) is any measure space and f(w) is an X-valued Bochner integrable function (see III.B.28) then

( J II Tf(w ) II PdJ.L(w) )

1

'P

:5 11"p(T)

sup

x• e x • , Ux* l l 9

( j l x* (f(w )) I PdJ.L(w) )

1

'P .

(41)

Proof: ( a) Clearly the quantity on the right hand side of (40) does not exceed 11"p(T) ( see (4) and Theorem 3). On the other hand let us take 1 XI . , Xm C X such that sup { ( 'L-:;'= 1 lx* (xJ ) I P) 'P : llx* II :5 1 } = 1 and 1 ( 'L-:;'= 1 II T(xj ) li P) 'P 2: 11"p(T) - c. Let us define S: I!� --+ X by S(e1 ) = x1 .

•

.

226

III.F Absolutely Summing And Related Operators §34.

for j = 1, 2, . . . , m. We have

li S I = sup = sup = sup

{I � I { t, { (�

ai xi :

t, l ai lp' 1 } �

ai x * (xi ) : ll x * ll

m

l x * (xi ) I P

)

.!

P :

� 1 and

ll x * ll

� 1

�

l ai l p'

�1

} = 1.

}

Also

(b) By a standard approximation argument it is enough to check ( 41) for step functions of the form :E;'!: 1 Xj �A; for disjoint sets Aj , j = 1 , 2, . . . , m. If for some j we have JL(Ai ) = oo then both sides of (34) are 1 infinite. If for all j we have JL( Ai ) < oo then we put Yi = JL( Ai ) "P xi for j = 1, 2, . . . , m and we see that (41) takes the form n

( j=l L II TYi ii P)

l. P

�

trp(T )

sup

llx II ::::;I ,x . E X •

n

•

(L: l x * (xi ) I P ) j=l

!. P

which is clearly true. 34.

II

We now return to the investigation of the identity Ilp (X , Y) =

IIq (X, Y). We will prove an important extrapolation type result. It is

analogous to Exercise III.A.2. We will use it in III.H and it also permits some improvements of our previous results. It is also the first result, besides the Pietsch factorization theorem, which deals with p-absolutely summing operators for p < 1. Theorem. Let X b e a Banach space and let 1 � for some number p, with 0 < p < q we have IIq (X, Y) =

Ilp (X, Y)

q

� 2. Suppose that

for all Banach spaces Y.

Then for all Banach spaces Y and for all numbers p such that 0 have IIp (X, Y) = Ilq (X, Y ) .

we

(42) < p < q

227

III.F Absolutely Summing And Related Operators §34.

(42)

First note that implies that there exists a constant C such that for every and for every operator we have Proof:

Y

T: X -+ Y

(43) (43) does not hold then there are Tn: X -+ Yn with 7rq(Tn) = 1 but 1rp(Tn) > 4n , n = 1,2,3,n . . . . Then the operator T: X -+ ( E� 1 Yn) 2 defined by T(x) = (2 - Tn(x))� 1 is q-absolutely summing but not absolutely summing. Let IC denote ( Bx· , ( X * , X)) and let P denote the set of all prob­ ability measures on We will identify X with its canonical image in C(IC). Using Theorem 8 and (43) we see that (42) is equivalent to If

rr

a

/C.

for every .>. E P there exists .>.1 E P such that

( l l x (x* Wd>.(x*))

1

9

5 C

1

( l l x (x*) I Pd>.1 (x*)) ;; for x E X. (44)

Analogously we see that in order to show the theorem it is enough to show that for every r < r < p and for every .>. E P there exists E P and Cr > 0 such that

1,

( l l x (x* Wd>.(x* ))

1

9

5

J.L

1

Cr

( l l x (x* )r dJ.L (x * )) ;: for all x E X. (45)

(44)

Note that if .>. = .>.1 in then the Holder inequality ( see Exercise for all r, O < r < q with = .>.. In general however III.A.2 ) gives .>. =I= >.1 . We put .>. = and inductively applying we get a sequence .>. 2 , . . . in P such that for all E

(45)

>.o, >.I.

We fix

8,

0
.o

x X,

J.L

(44)

such that � = � + (l � O) and put a n = 2- n - l . For

228

III.F Absolutely Summing And Related Operators §35.

x E X we have n 2': 0

< C L an ll x ii L ( >,n +I ) · ll x ll t1>- n +tl n 2': 0 l - (J (J < L an ll x ii Lr ( A n +t l L an ll xii L .(>. n+ t l n 2': 0 n 2': 0

c(

)(

)

Thus Since

r

:::;

1 we get

From (46) and (47) we obtain

where /-l (45) .

L n ;::: o 2 - r (n+ l ) An. Normalizing

/-l appropriately we get a

35 Corollary. {Grothendieck-Maurey) .

Every operator from L1 (!-l) into a Hilbert space H is p-absolutely summing for every p > 0. Proof: From Theorem 7 and Corollary 9 we get IT 2 (L1 (/-l) , H) = IT1 (L1 (!-l) , H) = L(L1 (/-l) , H). Theorem 34 gives the claim. II

Note that this is a strengthening of Theorem 7 which does not follow from our proof of it, simply because the operator P: A --+ £2 ( see

229

III.F Absolutely Summing And Related Operators §36.

Proposition 6 ( a)) is p-absolutely summing for p < 1, ( see Exercise III.I.2 ) . Note also that for the above argument to work we do not need the full power of Theorem 7. It is enough to know that (J.L) , H) = (J.L) , H) for some p < 2. This fact can be derived from Proposition 6 ( b) exactly like Theorem 7 was derived from Proposition 6 ( a) . If we avoid the use of Proposition 6 we can obtain some equalities of the form = without knowing that = The following theorem is a useful example of such situation.

not

L(L 1

llv(L 1

Tip( X, Y) llq (X, Y)

llv(X, Y) L(X, Y).

X is a Banach space of cotype 2 then for any Banach llv(X, Y) = TI2 (X, Y) for all :5 2.

36 Theorem. If space Y

p

Let us start with the following. 37 Lemma. If Y is a Banach space of cotype 2, then for any Banach

X and any C2 (Y) · Cv · 1rv(T).

p

space

� 2 we have

Tip(X, Y) = TI2 (X, Y), and 1r2 (T) :5

T Tip(X, Y)

,

X.

Let us take E and X I . Xn E From the definition of cotype ( see III.A. 17 ) and Kahane's inequality ( see III.A.20 ) we get

Proof:

•

•

•

p (48) ( � 1 Txi 1 2 ) 2 ::; c( J I � ri (t)T(xi ) l dt) p " ,; C (/ l r ( t, r; (t ) x; ) I dt) l , with C = C2 (Y) · Cv · Applying Proposition 33 ( b ) and Khintchine's inequality to ( 48) we n

l

n

.!.

get

:5 C1rp (T) , and the constant is of the right form. The remaining inclusion is always true. a

so 1r2 (T)

230

III.F Absolutely Summing And Related Operators §Notes.

We know from Theorem 34 that it is enough to show the theorem for a fixed p, 1 < p < 2. Also it is enough to consider X, Y finite dimensional and to keep track of the constants (see Proposition 5) . Under these assumptions, for T: X --+ Y Theorem 27 and Lemma 37 give

Proof of Theorem 36.

1rp (T)

sup{tr(ST) : S: Y --+ X, np' (S) ::5 1 } ::5 sup{tr(ST) : S: Y --+ X, 1rp' (S) ::5 1 } ::5 sup{ tr(ST) : S: Y --+ X, 1r2 (S) ::5 C2 (Y) · Cp} = C2 (Y) · Cp · sup{tr(ST) : S: Y --+ X, 1r2 (S) ::5 1 } . =

Applying Corollary 25 we get a

Notes and Remarks.

Much of this chapter, as well as much of modern Banach space theory, is the outgrowth of Grothendieck [1956] . The work of Grothendieck was phrased, however, in the language of tensor products and bilinear forms. This language, although still used by some and known by many, seems to have been generally replaced by the language of operators. In our book we adhere to this usage and avoid tensor products almost entirely. The notion of Banach operator ideal emerged in the late 60's, mainly as a result of many attempts to understand Grothendieck [1956] . From this time on, A. Pietsch and his students and collaborators have stud­ ied many aspects of the abstract concept, contributing greatly to the creation of the theory of operator ideals as presented in Pietsch [1978] . The p-absolutely summing operators (probably the most important op­ erator ideal) were introduced by Pietsch [1967] as a generalization of Grothendieck's 'application semiintegrale a droite' which are now called !-absolutely summing operators. In this paper A. Pietsch proved the basic properties of p-absolutely summing operators, in particular the fundamental Theorem 8 (the idea of using the separation argument in the proof is due to S. Kwapien) and Corollary 9. The Grothendieck theorem ( Theorem 7) was proved by Grothendieck [1956] who called it the fundamental theorem of the metric theory of tensor products. The proof was understood and presented in the language of 1-summing op­ erators by Lindenstrauss-Pelczynski [1968] . These authors proved the Grothendieck inequality (our Theorem 1 4) directly and derived The­ orem 7 from it. The prmtf presented here was found by A. Pelczynski with some help from the present author and was published in Pelczynski

III.F Absolutely Summing And Related Operators §Notes.

231

[1977] . Numerous other proofs have been published for various versions of the Grothendieck theorem. We refer to Pisier [1986] , Haagerup [1987] and Jameson [1987] for references. Considerable effort has gone into eval­ uating the Grothendieck constant Ka. It is known that this constant is different for real and complex scalars. The most precise estimates for the complex case are 1.338 :5 Kg :5 1.40491 (see Haagerup [1987] for the proof and a discussion of the known results) . There exist also C* -algebra versions of the Grothendieck theorem. They are quite involved, but useful in the theory of C* -algebras. As an example let us quote the following Theorem A. If A is a C* -algebra and Y is a Banach space of cotype 2, then every linear operator T: A -+ Y factors through a Hilbert space.

This was proved in Pisier [1986a] . This theorem is rather in the spirit of Chapter III.H but it is also the most Banach space theoretical statement. A more detailed presentation of even the most important results in this area requires, quite naturally, some familiarity with the theory of C* -algebras. We refer the interested reader to Pisier [1986a] for the proof of this result and for a detailed description of and references to the earlier works. Theorem 1 0 was proved in Lindenstrauss-Pelczynski [1968] . The­ orem 1 1 is a special case of a result proved by Kislyakov [1976] . This theorem should be compared with Proposition III.A.3. It is a Banach space manifestation of a phenomenon common in harmonic analysis that certain continuity results for important operators which hold for 1 < p < oo fail for p = 1 or p = oo . Incidentally one gets a rather crazy proof that the multipliers considered in Lemma III.A.4 are not contin­ uous in L1 (T2 ). The fact (Exercise III.G. 13) that C 1 (11'2 ) is not iso­ morphic to any C(K)-space was stated by Grothendieck [1956a] and the first proof was published by Henkin [1967] . Actually Henkin proved the much stronger result that C k (11'e) for k � 1 and i > 1 is not homeomor­ phic to any C(K)-space with the homeomorphism and its inverse being uniformly continuous. The basic idea of the proof of Theorem 1 1 has been applied in a similar but much more general context in Pelczynski­ Senator [1986] . Theorem 13 is taken from Bourgain [1987] . Its main point is the estimate (20) . Apart from this estimate facts of this type are well known and much used in harmonic analysis (see Rudin [1962a] 2.6) . Problems centred around the von Neumann inequality (25) are among the most interesting in operator theory on Hilbert space. Even

232

III.F Absolutely Summing And Related Operators §Notes.

our small sample shows the remarkable variety of methods used. The first example of a power-bounded but not polynomially bounded opera­ tor was given by Lebow [1968] . He showed that an example constructed by Foguel [1964] of a power bounded operator which is not similar to any contraction has this property. We present here some results taken from Peller [1982] and Bozejko [1987] . More precisely the proof of Proposi­ tion 15 is taken from Bozejko [1987] while Theorem 16 and Lemma 1 7 and Corollary 1 8 are due to Peller. The direct proof of Corollary 18 was communicated to the author by G. Pisier. Observe that estimates of the type 11 (T) II � (3(4>) where (3 is some norm on polynomials lead to a functional calculus for T on some class of functions. This is a very important subject in operator theory. It has many connections with other branches of analysis. The reader may consult Nikolski [1980] for a more complete picture. This subject has also a branch in the theory of Banach algebras. A nice result (once more relying on the Grothendieck theorem) is a theorem of Varopoulos [1975] that any Banach algebra X isomorphic as a Banach space to a C(K)-space is algebraically and topo­ logically isomorphic to some subalgebra of the algebra of all operators on a Hilbert space. It is an open problem asked by P. Halmos if every polynomially bounded operator T is similar to a contraction, i.e. is of the form T = VT1 v - 1 with II T1 I I � 1 and V an isomorphism of an underlying Hilbert space. The best partial result in this direction seems to be contained in Bourgain [1986a] . He proved Theorem B. If T: H -+ H satisfies ll p(T) II � M II P II oo for every polynomial p and if dim H = N < oo then there exists S: H -+ H such that II STS - 1 11 � 1 and II S II II S - 1 11 � M log N.

4

The proof is quite complicated and uses, among other things, Grothendieck's inequality and Theorem 111.1. 10. For applications of the Grothendieck theorem in the theory of stochastic processes the reader may consult Rao [1982] . For applications to interpolation theory the pa­ pers of V.I. Ovchinnikov should be consulted (e.g. Ovchinnikov [1976] and [1985] ) . The notion of 1-nuclear operator goes back to Grothendieck. It is a generalization of a1 operators on a Hilbert space (see Remarks af­ ter III.G. 18). The theory of p-nuclear operators has been developed in Chevet [1969] and Persson-Pietsch [1969] . The important and useful Theorem 27 can be found in Persson-Pietsch [1969] . It is only a small sample of various duality results for other operator ideals. There is also

III.F Absolutely Summing And Related Operators §Exercises

233

a (more complicated) duality theory for operators on infinite dimen­ sional spaces. Proposition 28 is folklore. It is a formalization of the connection between Theorem 7 and its dual form, Theorem 29. This connection was already known to Grothendieck [1956] and was quite explicit in Lindenstrauss-Pelczynski [1968] . The notion of p-integral op­ erator and all our results about them can be traced to Persson-Pietsch [1969] . Actually our p-integral operators are quite often called in the lit­ erature strictly p-integral, with the name 'p-integral operator' reserved for operators T: X -+ Y such that iT is strictly p-integral (i.e. integral according to the definition given in 21) where i is the canonical em­ bedding of Y into Y** . Such operators appear naturally in the duality theory for operator ideals when the spaces are infinite dimensional. Theorem 30 is a classical result of the theory of Fourier series. It is very similar in spirit to Theorem III.A.25. The theory of Sidon sets is an interesting part of commutative harmonic analysis. The standard, but a bit outdated, reference is Lopez-Ross [1975] . More recent advances in this area are connected with the use of Banach space methods. Our Theorem 31 is one such example. It is a variant of a result of Varopoulos [1976] . A stronger result proved in Bourgain-Milman [1985] is Theorem C. If G is a compact abelian group with dual group r and if for S C r the space Cs ( G ) has ootype p for some finite p, then S is a Sidon set.

Theorem 34 and 36 are due to Maurey [1974] . Our proof of Theorem 34 follows Pisier [1986] and is due to Maurey and Pisier. These theorems

will be useful later on in Chapters III.H and 111.1. There are many places where the theory of p-absolutely summing, p-integral and p-nuclear operators is presented. We conclude these re­ marks by listing some of them: Pietsch [1978] , Pisier [1986] , Tomczak­ Jaegermann [1989] , Kislyakov [1977] , Jameson [1987] . Exercises

1. 2.

Let p , q, � 1 be such that � + � = � and let T E IIp (X, Y) and S E IIq (Y, Z) . Show that ST E IIr(X, Z) and 7rr(ST) :5 7rp(T)7rq (S). r

Show that id: £1 -+ £2 is not 1-integral. Is it p-integral for some > 1? Show that id: £1 -+ £00 is 1-integral.

p

3.

(a) Let T: L1 [0, 1] -+ C [O, 1] be given by Tf (x) = J; f ( t)dt. Show that T is not !-absolutely summing.

234

III.F Absolutely Summing And Related Operators §Exercises

(b) Let a: if" � i� be defined as a((ei }.f= 1 ) = (}::: j= 1 ei ) := 1 · Show that 1r 1 (a) "' C log(N + 1 ) . n 4. For I = .E� 0 anz E H1 (D) we define T(f) = ( v'�� 1 ) := o · Show that T: H1 (D) � £2 is bounded but not !-absolutely summing. 5. Let K(x, y) be a measurable function on [0, 1] x [0, 1] . Let T l (x ) = J; K(x, y)l(y)dy. Find necessary and sufficient conditions for K(x, y ) so that the operator T maps C[O, 1] into itself and is !­ absolutely summing. 6. Show that id: C[O, 1] � Lp [O, 1] is not q-absolutely summing for any q < p. 7. (a) Show that every p-nuclear operator, 1 $ p < oo, is compact. (b) Show that if 1 $ p < oo and X is reflexive then Ip(X, Y) = Np (X, Y). 8. Show that if id: X � X is p-absolutely summing for some p < oo, then X is finite dimensional. In particular in every infinite di­ mensional Banach space there exists an unconditionally convergent series which is not absolutely convergent. 9. (a) Let G be an infinite, compact, metrisable abelian group and let f..L E M(G) be such that jJ,('Y) � 0 as 'Y � oo, f..L � 0 and f..L is singular with respect to the Haar measure m. (Note that III.C.6 shows that such a f..L exists.) Let T�-' : C(G) � L 1 (G, m ) be defined by T�-'(f) = I * f..L · Show that T is a compact, 1integral but not 1-nuclear operator. (b) Let r.p E L00 (T)\C(T) . Show that Tl = I * r.p considered as an operator on C(T) is compact and 1-integral but not 1-nuclear. 10. Show Theorem 7 assuming Theorem 14. 11. Suppose that I E L 1 (T) is such that / ( 2 k ) = 1 for k = 1, 2, . . . , N and f-r I l l $ 1 + c. Show that for every a, 0 < a < 1 , there exists a constant C = C ( a , c ) > 0 such that 1 {£: /(£) =/:- O} � CNo. tn N .

l

12. (a) Suppose that X is a Banach space with unconditional ba­ sis ( xn) ;;::'= l · Show that every !-absolutely summing operator T: X � Y factors through £ 1 . (b) Show that C[O, 1] does not have an unconditional basis. This is a special case of 11.0. 12 but try to prove it using (a) .

III.F Absolutely Summing And Related Operators §Exercises

235

(c) Suppose that F C N is a A(2) set (see I.B. 14) and suppose that L: = span{ ein9 } n EF C Lp(T), p > 2, has an unconditional basis. Show that the characters are unconditional in L:. The same holds for Cp = span { ein 9 } n EF C C (T) .

13. Show that for every p =f. 2, there exists a subset F C N such that idp : Cp -+ L: (for notation compare Exercise 12 (c)) is p­ absolutely summing but not p-integral.

III. G . Schatten-Von Neumann Classes

In this chapter we consider Schatten-von Neumann classes of operators on a Hilbert space and their applications in the theory of Banach spaces. We start with the notion of an approximation number of an operator between Banach spaces. We prove that the approximation numbers of an operator and its adjoint are the same. Then we study operators on Hilbert space. We prove the Weyl inequality and basic facts connecting eigenvalues, s-numbers and approximation numbers. Various character­ izations of Hilbert-Schmidt operators are presented. We also show the classical Fredholm-Bernstein-Szasz theorem about Fourier coefficients of Holder continuous functions. Next we give results about summability of eigenvalues of p-absolutely summing operators on a general Banach space and apply them to eigenvalues of Hille-Tamarkin integral opera­ tors. 1.

Given an operator T: X --+ Y we define its approximation numbers

an(T) = inf{ II T - Tn ll : Tn : X --+ Y, rank Tn < n } ,

n

= 1 , 2, . . . .

Clearly a1 {T) = II T II and the sequence { an (T))�= l is decreasing. If an (T) --+ 0 then T is a norm limit of finite dimensional operators, thus compact. One proves routinely that if Y has b.a.p. (see II.E.2) and T: X --+ Y is compact then an(T) --+ 0. 2 Proposition. The following inequalities hold for every n , m � all operators T, S:

1 and

an + m - l (T + S) $ an (T) + am(S); an + m - l (T S) $ an(T) · a m (S).

{1) {2)

o

us

Proof: The argument for {1) is obvious. To prove {2) let take any Tn with rank Tn < n and any Sm with rank Sm < m. Then

Since

rank {Tn S + TSm - TnSm ) � rank{Tn(S - Sm )) + rank(TSm ) < n+m- 1

238

Ill. G. Schatten-Von Neumann Classes §3.

we get

For 0 < p < oo we define AP(X, Y) to be the set of all operators T: X - Y such that 2:: :::'=1 an(T) P < oo. We denote (2:: :::'=1 an (T)P) ! as ap(T) . 3.

The quantity ap (T) is a quasi-norm on AP(X, Y) for with this quasi-norm is a quasi-Banach operator

Proposition.

0 0 let us take V, a finite c-net in T ( Bx ) . Since T is compact T ( Bx ) is norm-dense in T** ( Bx·• ) so V is also an c-net in T** ( Bx·· ) . Fix also an operator Tn : X** --+ Y** with rank Tn < n and li T** - Tn ll � an (T**) + c. Put F = span { V U Tn (X**) } . From the principle of local reflexivity II.E. 14 we get an operator cp: F --+ X with ll cp ll � 1 + c and cp I F n X = i d. In particular cp j V = i d. For x E X with ll x ll � 1 let us fix v E V such that II Tx - vii � c. Then we have Proof:

an (T), n =

II Tx - cpTnxll � c + ll v - cpTnx ll � c + ( 1 + c) ll v - Tnxll � c + (1 + c) ( ll v - Tx ll + ( II T** x - Tnxll ) � c + ( 1 + c ) ( 2c + an (T**)). Since cpTn has rank less than n and c was arbitrary we obtain an(T) �

an(T**).

a

Let T: X --+ X be an operator such that Tk is compact for some k . Such operators are called power-compact. For a power-compact operator T we define the sequence (A n (T))�= l which consists of all eigenvalues of T counted with multiplicities ( cf. I. A. 18.) ordered in such a way that I A 1 (T) I � I A2 (T) I � I A 3 (T) I � · · · . Let us recall that A E ([ is an eigenvalue of an operator T if Tx = Ax for some x E X, x =/= 0. Since T is power compact, the Riesz theory holds for T ( see I.A. 16-19.) so the multiplicity of each eigenvalue is finite. Let H and L be Hilbert spaces and let T: H --+ L be a compact oper­ ator. Then I T I = -/T*T is a positive compact operator and there exists an isometry U: I T I (H) --+ L such that T = U I T I . The spectral theorem for compact, positive operators shows that there exists an orthonor­ mal system ( vn)�= l such that I T I (x) = E�= l An ( I T I ) (x, vn)Vn · Since T = U I T I we have that for an arbitrary compact operator T: H --+ L there exist orthonormal systems (vn )�= l and ( un )�= l such that 6.

00

(4) T(x) = L An( I T I ) (x , Vn }Un · n= l Clearly Un = Uvn, n = 1 , 2, . . This is called the Schmidt decomposi­ tion and the numbers A n( I T I ) are called the singular numbers of T and are denoted sn (T). .

.

III. G. Schatten-Von Neumann Classes § 7.

240

If H and L are Hilbert spaces then AP (H, L) is called the p-th Schatten-von Neumann class and denoted ap( H , L) and ap(T) is then denoted ap(T). This may seem confusing but the hilbertian theory is so rich and special that it has its own traditional language. 7 Theorem. Let H, L be Hilbert spaces and let T: H pact operator. Then s n (T) = an(T), n = 1 , 2, . . . .

Proof:

----t

L be a com­

Using (4) we get immediately

Conversely, given any operator Tk : H ----t L with rank Tk < k let us take x = E7= l /3j Vj (where (vj ) �1 is given by (4) ) such that ll x ll = 1 and Tk x = 0. From (4) we get

I! Tx !l = so

(� k

A n ( I T I ) 2 1 /3i l 2

)

.!

2

?

Bk(T) I! x ll •

8. Our basic tool for the study of Schatten-von Neumann classes is the following.

Theorem. (Weyl's inequality) . Let T: H ----t H be a compact oper­ ator on a Hilbert space H. Then for evezy n = 1 , 2, . . .

n

n

k= l

k l

II ! A k (T) I :5 II= ak (T).

(5)

Proof: Clearly we can assume A n (T) =/= 0 since otherwise there is nothing to prove. Using the spectral theorem for compact operators and the Jordan decomposition in finite dimensional spaces we infer that for every n there exists a subspace Hn C H such that

= n, T(Hn) Hn, T I Hn : Hn Hn dim Hn

c

----t

has eigenvalues

At (T) , . . . , An (T).

(6) (7) (8)

III. G. Schatten-Von Neumann Classes §9.

241

Let Tn denote T I Hn : Hn ---+ Hn. Let us fix an orthonormal basis (xj ) j= 1 in Hn , and for any operator S: Hn ---+ Hn let det S = det [ (Sxk , Xj) ] k,j =t · From well known properties of determinants of finite matrices we get det S = Il �= l .Ak (S). Thus (8) gives

n

n

k= l

k= l

II I.Ak (T) I = II I.Ak (Tn ) l = I det Tn l ·

(9)

Using the polar decomposition we get Tn = U I Tn l where U is a unitary operator. Obviously o:k (Tn ) :::; o:k (T) , k = 1 , . . . , n and Theorem 7 gives o:k (Tn ) = o:k ( I Tn l ) = A k ( I Tn l ), k = 1, 2, . . . , n. Thus we get

I det Tn l = det I Tn l =

n

n

II= l I.Ak ( I Tn l ) l ::=; II= l o:k (T) k

k

and comparing (9) and (10) we get the claim.

( 10)

•

9. To use all the information contained in (5) we need a lemma about sequences of positive numbers.

Lemma. Let ( o:k )f= 1 and ( f3k )f= 1 be decreasing sequences of positive numbers such that L:�= l O:k :::; L:�= l f3k for n = 1 , 2, . . . , N and let r.p: 1R ---+ 1R be a convex function such that r.p(x) :::; r.p( l x l ) . Then

N

r.p( o:k ) 2::: k= l Proof:

N

r.p(f3k ) · :::; 2::: =l k

In 1RN we define a convex set V by

V = conv{ (c- k f3u(k) )f= 1 : E" k = ± 1 and a is a permutation of the set { 1 , 2, . . . , N}}. If ( o:k )f= 1 f/. V then by the Hahn-Banach theorem I.A. lO there exists a functional on :JRN such that 4>/V :::; 1 and ¢((o:k )f= 1 ) > 1 . Since V

is invariant under permutations of coordinates and changes of signs and (o:k )� 1 is a positive, decreasing sequence we can assume that (( x1 Jf= 1 )

=

N

:�:::>1 x1

j=l

with

c

1 � c2 > · · · � eN � 0.

242

Ill. G. Schatten-Von Neumann Classes § 1 0.

But 1
.j 's with E Aj = .>.je{.Bu; (k)

)

we have ( a k )k'= 1 = 1 and Aj � 0. This

8

N

j=1

k= 1

$ L L Aj c,o (e{ ,Bu; (k) ) $ L Aj L c,o ( .Bu; (k) )

k= 1 j = 1 N = L: c,o ( ,ak ). k= 1

10 Theorem. For every compact operator T: H < oo and every N = 1 , 2, . . . we have

p

• --+

H and every p, 0
.n (T)) II P $ ap(T) .

Without loss of generality we can assume .>. N (T) � 1 and O! N (T) � 1 . Applying Lemma 9 for O!n = p log .>.n (T) and .Bn = p log an (T) and c,o (t) = exp t we get the claim.

Proof:

II

11. Weyl's inequality also allows us to show that ap ( H ) , 1 $ p < oo , is actually a Banach space. As we know ( see 3) , this is not true for general Banach spaces. We have

Proposition. Let T and S be compact operators on a Hilbert space H. For every n = 1 , 2, . . . and every 1 $ p < oo we have

Ill. G. Schatten-Von Neumann Classes §12.

243

Using the Schmidt decomposition (4) and Theorem 7 we can

Proof: write

(S + T)x = L a k (S + T) (x, vk )uk . k Let U be a partial isometry defined by U( vk ) = U k and orthogonal projection onto span{ v k }k= l · We have n

let

P

be the

n

L a k (S + T) = L( U* (S + T)vk , vk) = trPU* (S + T)P k= l k= l ::; i tr(PU*SP) I + i tr(PU*TP) I . Since for finite rank operators the trace equals the sum of eigenvalues the last expression is majorized by

Thus from ( 1 1 ) we get for n

n = 1 , 2, . . . n

n

L ak (S + T) ::; L ak (PV* SP) + L a k (PV*TP) k= l k= l k= l n

n

k= l

k= l

(12)

$ L a k (S) + L a k (T) so our proposition is proved for p = 1 . Applying Lemma 9 for a k = ak (S + T) and f3k = ak (S) + ak (T) and cp(t) = tP we get (use (12)) E�= l a k (S + T)P ::; E�= l (a k (S) + ak (T)) P . Now the Holder inequality gives the claim. a 12. The operators of class a2 are called Hilbert-Schmidt operators. Here are some equivalent characterizations.

Proposition. Let H and L be Hilbert spaces and let T: H ---> L. The following conditions are equivalent: (a)

T E a2 (H, L) ;

(b) for every orthonormal basis

LjEJ II Th1 ll 2 < oo ;

(h1 ) 1 0

(c) there exists an orthonormal basis

l:j EJ II Th1 112 < oo ;

in the space H we have

( h1 ) 1 EJ

in the space H such that

Ill. G. Schatten-Von Neumann Classes § 1 2.

244

(d)

(e)

T admits a factorization

for some Lt ( JJ, ) -space;

T admits a factorization

T

H

L

� � C(K)

for some C ( K ) -space; (f)

(g)

T E Tip(H, L) for every p, T E Tip(H, L) for some p,

1�p< 1 �p
( b ) . Let ( hj ) jE J be any orthonormal basis in H. Using the Schmidt decomposition (4) we get

I

L II ThJ II 2 = L L ak (T) (hj , vk )u k jEJ jEJ k = L L a k (T) 2 i (hj , Vk) l 2 jEJ k

l2

( 13)

jEJ k 2 = L a k (T) = a2 (T) 2 . k

( b ) => ( c ) . This is obvious. (c ) => ( d ) . We define v : H -+ ft (J) by and �: £1 (J) -+ L by

v( x ) = ((x , hJ ) II ThJ II ) JEJ with the convention

� = 0.

245

Ill. G. Schatten-Von Neumann Classes §13.

Since T = E o v we get the desired factorization. ( d ) =? ( f ) . By Grothendieck's Theorem III.F.7 and Corollary III.F.9. ( f ) =? ( g ) . Obvious. ( g) =? ( a) . Since Hilbert space has cotype 2 ( see III.A.23 ) we get from Lemma III.F.37 that T E II2 (H, L) . From Corollary III.F.9 ( b ) we get that T is compact thus the Schmidt decomposition gives T ( x ) = 1:�1 A k ( x, vk) u k . The definition of 2-absolutely summing map gives

L I.Xk l 2 = L 11 Tvk ll 2 $ 1r2 ( T ) 2 sup { L l (x, vk) l 2 : ll x ll $ 1} = 1r2 ( T ) 2 k k k

(14) so a2 (T) $ 1r2 (T). Since ( e ) is a dual condition to ( d ) and ( a) is a self dual condition ( see Proposition 5) we infer that also ( e ) is equivalent to all the others.• If an operator T: H - L satisfies any of the conditions of the Proposition then

Remark.

(

1r2 (T) = a2 (T) = � 11 ThJ II 2 J

)

1

2

( 15)

for any orthonormal basis ( hJ )je J is H. We see from ( 13) and (14) that only 1r2 (T) $ a2 (T) remains to be proved. If (lJ ) JeJ is any orthonormal basis in L and h E H then

11 Th ll 2 = L I (Th , lj) l 2 = L l (h, T*lj) l 2 . jEJ jE J

When we view this inequality as a special case of III.F. ( 9 ) we see that

1r2 (T) 2 $ LjEJ II T*lj ll 2 . From ( 13) we get 1r2 (T) $ a2 (T* ) = a2 (T) .

13. One of the reasons why the Hilbert-Schmidt operators are impor­ tant is that they admit a nice integral representation. Proposition. An operator T: L 2 (0, J.L) - L 2 ( E , v ) is Hilbert-Schmidt if and only if there exists a function K E £ 2 (0 x E, J.L x v ) such that

Tf( a) =

k K (w, a) f(w)dJ.L(w)

(16)

III. G. Schatten-Von Neumann Classes § 14.

246 Proof:

From the Schmidt decomposition we get

Tf(a) = L >.. k r f(w)vk (w )df1 (w)uk(a) k ln = >.. k vk (w ) · u k (a) f (w )df1 (w ).

In [ �

]

and vk (w ) · u k (a) is an orthonormal system in £2 (!1 x I:, 11 x v ) we get the desired function. Conversely let ( fj (w ))j EJ and ( h 8 (a)) 8 E S be orthonormal bases in L2 (!1, 11) and L2 (I:, v) respectively. Then ( fj (w ) · h8 (a)) (j, 8 )EJ x S is an orthonormal basis in the space £ 2 (!1 x I:, 11 x v ) . We have Since I: l>.. n l 2
2 and there is no function K (x, y ) on [0, 1] x [0, 1]

such that

Tf(x) =

1 1 K(x, y)f(y)dy

a.e.

where the integral is understood as a Lebesgue integral. Observe that if such a representation exists then the function n(x) = J I K (x, Y ) l dy is finite almost everywhere, so

I Tf(x) l S 11/l loo · D (x).

(17)

We define our example as n = 1, 2, . . . , n = 0, - 1, -2, . . . ,

where hn is the Haar system ( see II.B.9) and An = n- ! · log n. Obviously T E ap(L 2 [0, 1]) for p > 2. Since sup n I T(e 2,.in y) (x) l = sup n l >..n hn (x) l = oo a.e. we see that (17) does not hold. 14. Now we would like to apply these general notions to investigate the connection between the smoothness of functions and the size of Fourier coefficients.

247

III. G. Schatten-Von Neumann Classes §15.

Theorem. The Fourier coefficients of every function I a $ 1 belong to f.p for every p > (2 a� l ) .

E Lipa (T), 0
. s. Suppose that for some E with =I= 0 we have

v V

is impossible, so for W = A(V) we have dim W = k. >.w==0.AvButE this W we have

so For

(>. - T)8w = jt= l ( -1 )j (�) >.s -i (AB)i w = jt= l ( -1)j (�) >.s -i (AB )i Av = A(t( - 1 )i (;) >.s -i (BA)i v) = A( >. - 8)8v = 0. J

J

This shows that W is contained in the eigenspace of T corresponding to >.. Repeating the same argument with and T interchanged we get the a lemma.

8

16.

Directly from Lemma

15 we get

Theorem. IfT: X -+ X is a p-absolutely summing operator with p :::; 2 then (>.n (T) )�= l E i2 .

It is enough to assume 1r2 (T) Then we have a factorization

Proof:

X

C(K)

8:

T

= 1 (see Corollary III.F.9(a)). X

id

8 = o o B is

The operator L 2 (JL) -+ L2 (JL) defined as id i similar to T and from Proposition 12 we get a2 < oo. Theorem 10 completes the a proof. 17.

(8)

Surprisingly this is an optimal result. Namely we have

Proposition. There exists a nuclear operator T such that (>.n (T))�=l � lp for any p < 2.

249

III. G. Schatten-Von Neumann Classes §18.

This follows directly from III.A.25 and the following. 18 Proposition. Let G be a compact abelian group with Haar measure *

m, and let T: C(G) - C(G) be given by Tf = f h for some h E C(G) . Then T is nuclear.

It is clearly sufficient to show that n1 (T) :5 ll h lloo for h a finite combination of characters. Such T being finite dimensional is nuclear. It follows from II.E.5(e) and the definition of the nuclear norm that T can be approximated in the nuclear norm by operators of the form P1 TP2 where P1 and P2 are projections onto finite dimensional subspaces in C(G) with d(ImP1 . £�) :5 1 + c and d(ImP2 , £::!,) :5 1 + c. From III.F.12 we infer that 1r1 (T) :5 ll hlloo so it is enough to show that for 8: £� - £::1, we have n1 (8) :5 1r1 (8). This follows directly from Corollary III.F.24 since 1r1 (8) = i1 (8) (see remarks after III.F.22) . a

Proof:

The above Proposition 17 contrasts with the situation in Hilbert spaces. For T: H - H we have n1 (T) = a1 ( T ) . If T has a nuclear representation T(x) = Lj ( x, Xj) Yi then from Proposition 1 1 we get a1 (T) :5 Lj a1 ( ( ·, xi ) Yi ) = Lj llxi ll II Yi ll so a1 (T) :5 n1 ( T ) . The converse follows directly from the Schmidt decomposition. Thus Theorem 10 shows that any nuclear operator on a Hilbert space has absolutely summable eigenvalues. Remark.

19.

The behaviour of eigenvalues of p-absolutely summing maps for

p > 2 is given in Theorem. then

If T: X - X is a p-absolutely summing operator, p � 2, (18)

The proof of this theorem follows from the following two facts. 20 Lemma. There exists a constant Cp such that for every operator T: X - X we have

21 Proposition. Let P be given. Suppose that for some q there exists a constant Cq such that for every operator T: X - X and any Banach

250 space

III. G. Schatten-Von Neumann Classes §21.

X

Cq = 1 .

we have

(2:�= 1 1-Xn (T) I q ) � ::; Cq rrp(T) .

Proof of Theorem 19.

1

Then we can take

It follows from Lemma 2 0 that for every

q > p we have (2: I .Xn (T) I q F ::; Cq rrp(T). Applying Proposition 21 and a

passing to the limit as q --+ p we get (18) .

From finite dimensional linear algebra we infer that for every n = 1 , 2, 3, . . . there exists a subspace Xn C X, dim Xn = n such that T(Xn ) C Xn and Aj (T I Xn ) = Aj (T), j = 1 , 2 , . . . , n. Clearly 1rp(T I Xn ) ::; 7rp(T) so applying the Pietsch factorization Theorem III.F.8 we have the factorization Proof of Lemma 20.

T IXn

Xn i

l

x=

Xn

I

id

a

XPn n where rrp(id) ::; 1 , ll a ll ::; 7rp(T) and X� is an n-dimensional subspace of some Lp(/-L) · From Corollary III.B.9 we get operators A and B such that X��f'2�X� and such that BA = i dx:;, and II A II · II B II ::; n ! - � . The operator T I Xn : Xn --+ Xn is related to the following composition which we will call S: on B XnP "' Xn i xn= id XnP A {.on2 · {. 2 -----+

-----+

-----+

-----+

-----+ 1

1

Since rrp(S) ::; rrp(id) II A II · l l i l l · ll a ll · II B II ::; n 2 - "P rrp(T) we get from Proposition 12 that there exists a constant Cp such that (20) From Theorem 10 and (20) we get

so (19) follows.

a

Proof of Proposition 21. Let us denote the smallest possible Cq by K. If K > 1 then there exists an operator T: X --+ X such that

251

Ill. G. Schatten-Von Neumann Classes §21 .

1rp(T) = 1 and (�=:"= 1 1-Xn (T) I q ) � > .[K. Without loss of generality we can treat X as a subspace of C(O) for some compact spac� n. From III.F .8 we get a probability measure f..L on n and an operator T: Xp -+ X with II T II = 1 where Xp is the closure of X in Lp(O, f..L) . Let Y c C(O ® O) be the closure of the set of functions of the form -r;;=l x; (wl ) · z; (w2 ) where x; , z; E X, j = 1 , . . . , n and let Yp be the closure in Lp(O X n, f..L X f..L) of Y. We define an operator T ® T: y - y by the formula T ® T("[; x; (w l )y; (w2 )) = 'E T(x;)(w l ) · T(y; )(w2 ) · One easily

checks that T is continuous and that 00

00

00

L 1-Xn (T ® TW � L 1-Xn (TW · L 1-Xn (T W . n =l n= l n= l

(21)

Let T ® T: Yp -+ Y be defined by

Since the formal identity from Y into Yp has norm at most 1 we get 11'p(T ® T) � li T ® T il· But for F = 'E;=l f; (wl ) g; (w2 ) E Yp we have

I ( t,

) l l ) (fn i � n p = II T 11 (fn i T ( � f; (wi ) g; ) (w2 ) 1 df..L (wi ) ) n p 2 T � II II ( i l � f; (w1 ) 9; L df..L (w l) ) n p 2 T II I1 (1 l ?: /; (w1 )9; (w2 ) 1 d(f..L X f..L ) (w b w2 ) ) J =l

T (g;)(w2 ) /; (wl ) I (T ® T)(F) (w1 , w2 ) l = T n T (g;)(w2 ) /; (wl ) df..L (wl ) P � II T 11 �

P

�

P

�

P

�

P

�

nxn

� II T II 2 II F II Yp · Thus 11'p(T ® T) � 1, so (21) together with the choice of T gives

This contradicts the definition of K.

a

252

III. G. Schatten-Von Neumann Classes §22.

Remark. The reader familiar with tensor products will easily see that the above argument gives that an c:-tensor product of p-summing maps is p-summing. 22. As an example of the applicability of previous results let us consider Hille-Tamarkin integral operators. Let (0, p.) be a probability measure space and let K(w1 , w2 ) be a function on n X n such that

2�p
Lp( O, p.). For such an operator we have 2:�= 1 IAn (TK ) I P < oo. This follows from Theorem 19 and the fact that TK is p-absolutely summing. To see this put cp(w1 ) = (J0 I K(w1 , w2 ) 1 P dp.(w2 )) _!_v' . One checks that S(f) = TK (f) · cp - 1 is a linear map from Lp (O, p.) into £00 (0, p.) so we have to check that for cp E Lp the map f �----+ f · cp is p-absolutely summing from £00 (0) into Lp (n, p.) . This was observed in III.F.4. Note that Hille-Tamarkin integral operators are direct generaliza­ tions of Hilbert-Schmidt operators (see Proposition 13) . Let G be an abelian compact group with normalized Haar measure m and dual group For f E Lp' (G, m) , p � 2 we define a kernel K(g1 , 92 ) = !(91 - 92 ) · This kernel clearly satisfies (22). Since TK (9) = f * g we see that eigenvalues of TK coincide with j('y ) so we get the Hausdorff-Young inequality I

f.

(

L l f('y ) I P

I'Er

)

1

p

� IIJII P1 > p � 2.

(23)

Notes and Remarks.

There are two excellent books which treat the matters explained in this chapter, and much more. They are Pietsch [1987] and Konig [1986] . The concept of approximation number is so natural that we have been unable to trace proper historical references. Nowadays it is an example of the general notion of s-numbers (see Pietsch [1987] ) . Proposition 5 is due to Hutton [1974] . It is a quantitative version of the classical Schauder Theorem asserting that an operator is compact if and only if its adjoint is compact. Our material on Schatten-von Neumann classes is classical and can be found in many places. The above mentioned books contain nice

III. G. Schatten-Von Neumann Classes §Exercises

253

presentations but also Gohberg-Krein [1969] and Simon [1979] should be mentioned. Proposition 12 showing the connection between Hilbert-Schmidt and p-absolutely summing operators on a Hilbert space was proved by Pelczyiiski [1967] . Theorem 14 is the classical result of Fredholm [1903] but in the theory of Fourier series it is usually associated with Bernstein [1914] and Szasz [1922] . Our proof is taken from Wojtaszczyk [1988] . Theorem1 6 and Proposition 1 7 and 18 are basically due to Grothen­ dieck [1955] . In the present generality Theorem 16 was proved by Pietsch [1963] . The fact that the eigenvalues of a nuclear operator on Hilbert space are absolutely summable actually characterizes spaces isomorphic to Hilbert space among all Banach spaces (see Johnson-Konig-Maurey­ Retherford [1979] ) . This paper contains also the first proof of Theorem 19. Our proof of Theorem 19 is a modification of a proof given in Pietsch [1986] . The application of Corollary III.B.9 allows us to avoid the use of the general theory of Weyl's numbers. One should be aware that the subject of eigenvalue estimates of operators on X is related to best projections on finite dimensional subspaces of X. This is made clear in Konig [1986] 4.b where the estimates for eigenvalues are used to prove Corollary III.B.9. Our discussion of Hille-Tamar kin integral operators is taken from Johnson-Konig-Maurey-Retherford [1979] . Exercises 1.

Show that up (£2 )* = uq (£2 ) for 1 < p < oo and � + � = 1 , and also O'oo (£2 )* = 0'1 (£2 ) and 0'1 (£2 )* = L(£2 ). The duality is given by

(T, S)

2. 3.

=

trTS.

Show that the space u1 (£2 ) has cotype 2.

For A C N let PA : £2 ----t £2 denote the natural coordinate projection defined by PA (E� 1 a;e;) = L; E A a;e; . (a) Show that maps T �--+ TPA and T �--+ PAT are contractions on O'p(£2), 1 � p � 00 . (b) Show that if p =1- 2 then the operators Ti; , i, j = 1, 2, . . . defined by Ti; (E;;: 1 a k e k ) = a i ei do not form an unconditional basis in O'p ( £2 ) ·

(c) Show that O'p, P =1- 2 and p =1subspace of Lp(J-L) .

oo,

is not isomorphic to any

III. G. Schatten-Von Neumann Classes §Exercises

254 4.

If X is an infinite dimensional subspace of a00 (i2 ) , then X contains an infinite dimensional subspace x1 complemented in O'oo (i2 ) such that either X1 rv Co Or X1 rv £2 .

5.

Show that

6.

Show that there is no norm on A 1 (i00, il ) which is equivalent to the quasi-norm a1 (·).

7.

Show that for every (.X n )�= 1 E i2 there exists a nuclear operator i 1 EB 00 --+ i1 EB 00 such that the eigenvalues of T are precisely

O'p

rv

CL:

O'p)p for 1 :5 p
. -P / 2 = C(>.) . One checks that this function C(>.) satisfies the desired conditions so we have (b) . (b)=>(c) . Let us fix a function C(>.) as in (b) and for a given c > 0 let us fix a number R such that C(R) < c. Let us consider the following condition on a subset F c !l 3x E X, ll x ll $ 1 such that J.t(F) · I Tx(w) I P > RP for all w E F. If no subset F have

c

n satisfies (*) then for every x E X with ll x ll t-t{w e n: I Tx(w) l > >.} $

( �Y ·

=

(*) 1 we

If there are sets in n satisfying ( *) let us fix a maximal family of disjoint sets (Fj ) satisfying (*) with corresponding Xj E X, ll xi ll $ 1. For ci = J.t(Fj ) l !P we have E ll cixi iiP $ 1 and supi I T(cjXj ) (w) l > R a.e. on F = Ui = l Fi . Condition (b) yields t-t(F) $ C(R) $ c. We will show that E = 0\F satisfies (c) with Ce = RP. If not, there exist x E X with ll x ll $ 1 and a number >. > 0 such that J.t{w E E: I Tx(w) l � >.} > (Rj>.)P. Thus {w E E: I Tx(w) l � >.} satisfies (*) and is disjoint with F. Since ( Fj ) was a maximal family we get (c) . 1/n, n 1 , 2, . . . let En and Cn CE n be (c)=>(a) . For c n given by (c) . Let us fix a sequence of positive numbers O n such that I: Cna ;;- P 1 and for Fn = En\ Ui0 tt{w : s�p I Txk (w) l A} $ 2. I i {t E [0, 1] : I Tgt (w) l A}i dtt (w ) :::; 2 · l [ l {t E [o , 1] : l ut l ;:::: v'X}I (1) + l {t E [0, 1] : I Tut (w) l v'XI I ut l }l ] dtt(w) = 2 · l {t E [0, 1] : I Yt l ;:::: v'X}I + 2 . I JL{w E n: I Tgt (w) l ;:::: v'XI I ut l } dt. Since X has type p, the Markov inequality, sometimes also called Cheby­

Thus for every

2::

2::

2::

shev's inequality, gives

T

l {t E [0, 1] : I Yt l ;:::: v'X}I :::; T, (X)P . A -p/2 • A -+

Since is continuous at zero, Proposition 2 shows that the last integral in (1) tends to zero as oo. This verifies condition ( b ) of Proposition a 5 and proves the theorem.

X g 19

7 Corollary. Let G be a compact abelian group with Haar measure m. Assume that is a Banach space of type p and that we have a represen­

X. T9f(h) = f(h + g) T: X -+ T9TI9-1 = T g

tation �--+ of G into the isomorphisms of Let for E L0 (G, m) . Assume that L0 (G, m) is a continuous sub­ linear operator such that for all E G . Then T is of weak type q with q min (p, 2 ) .

f

=

Using the Nikishin theorem we see that condition (c ) of Propo­ sition 5 gives a set E C G with � such that

m(E) > m( {g E G : I Tx(g) l > A} n E) :::; c ( I � I r for all E X. Since T9TI9-1 = T we get {g E G: I Th (x)(g) l > A} = h + {g E G: I Tx(g) l > A}. Comparing (2) and (3) we see that for every h E G

Proof:

X

(2)

(3)

263

III.H. Factorization Theorems §8.

Integrating (4) with respect to h and using the fact that h E G we get

l h l � C for

m {g E G: I Tx(g) l > .X } � m(E) - 1 c ( " � " r·

a

8 Comment. The Nikishin theorem and Corollary 7 formalize several well known equivalences between existence almost everywhere and weak type (1-1) for certain operators. In particular we have

Let f E £1 (T) and let f(rei 9 ) denote its harmonic extension. Let f(rei9) denote the harmonic function conjugate to f(rei9 ). For f � 0 the function G(rei9 ) = (1 + f(rei9 ) + if(rei9 )) - 1 is bounded and analytic. The Fatou theorem shows that lilllr --+ l G(re i 9 ) exists a.e. . This implies that limr--+1 j(re i9 ) exists a.e. . This implies that such a limit exists a.e. for arbitrary f. For the background on this see I.B.19 and the references given there. So (see Proposition 3) Mf(O) = maxo< r 0 and all j's. But then for all N we have

2:::

(�

i3 I T(xi ) IPdf.L) 1 /P

2:::

(cN) 1 /P

which is impossible for large N. Thus we can assume (see 9) that {f!, f.L) is a probability measure space. (a)=?{b) . This is easy. We have

(fo ( t I T(xi W) pfq df.L) 1 1p p/ q /p q (In gP ( t j u(Xj ) i ) df.Lr =

265

III.H. Factorization Theorems § 1 0.

j

j

g

J..L)

The implication (b):::} ( a) is more difficult. Let us observe that ac­ tually we are looking for a positive function in Lr(O, such that The case q = oo is rather simple. In this case (b) � reads

l g- 1 T(x) l q K l x l ·

This implies that

g = sup{ I T(x)l:x E X, l x l � 1 } exists in Lp ( O, J..L ) and I 9 I � 1 (see Exercise III.A.1). This is the desired g (note that for q = weP get p = ) In the case q
C I .Xn 1 2 · so {18) holds. Conversely, assuming {18) we have 00

0

=

00

=

Now we want to show that the integrand is a bounded function. For 1- 2- N r 1- 2- N - 1 we have for some q independent of N, 0 < q < 1 k N 22sk {1 - 2- N - 1 ) 2· 2k k 2 2· 2 s 2sk {1 - 2- N - 1 ) 2· 2k 2 2 r L L L k=O kN=O k=N+ 1 k - +l) L 22s k L 22s k q2 (N k =O k = N + 1 C22sN C22sN kL= 1 22sk q2k C22sN C{1 - r) - 28 so the integrand is really bounded and so {16) holds. 29 Proposition. The sequence = {.Xn)�= O is a multiplier from i2 into Bp(D), 0 < p 2 if and only if 1 1 1 r 2 p {20) �

oo

0

�

�

+

�

+

�

+

�

oo

oo

00

�

�

a

A

- + -

=

- .

281

Ill.H. Factorization Theorems §29.

nk

(nk )�0

be any sequence such that 2 k :5 < 2k+ 1 . Let Passing to polar coordinates and using Khintchine's inequality I.B.8 (see also Exercise III.A.9) we get that there exists a constant such that for any such sequence and any sequence of scalars Proof:

C (ak )�0

(nk )�0

(21) From (21) we see that for a multiplier A: £2 - Bp (D) we have

(an)�0

(nk )�0

Since (22) holds with the same constant for all scalars and all sequences as above we get (20) . In order to prove the other implication we use the following inequality

I t. -.z· l . , ( � 1 ··�· -.z· ID

*

n, n L:::'=o l an l 2

·

(23)

For p :5 1 this is just the p-convexity of the space Lp (D) (see I.B.2) . For p = 2 it follows directly from orthogonality of z = 0, 1, 2, . . . , in L 2 (D). The remaining cases follow by standard interpolation. = 1 and such that For a sequence satisfying (20) we obtain from (23) and the HOlder inequality

(an)�=O

(.Xn)�=O

282

III.H. Factorization Theorems §30.

a

If A satisfies (20) this shows that A is a multiplier.

Now we are able to prove the description of multipliers from X8 into Bp(ID) . From Propositions 27 and 28 and 29 and Holder's inequality we get 30 Theorem.

The sequence A =

(.An)�=O is a multiplier from X8 into

Bp(ID) , 0 < s, 0 < p � 2 if and only if

n< 2• + 1 ns- � I .An i )P < k=O 2•:s:;sup 00

L(

oo .

Notes and Remarks.

Various sublinear operators are of paramount importance in modern harmonic analysis; they include a variety of maximal operators, square functions or area functions etc. Texts such as de Guzman [1981] , Garcia­ Cuerva-Rubio de Francia [1985] , Folland-Stein [1982] , Torchinsky [1986] etc. make their importance absolutely clear. In our presentation we give only the most general results which fall naturally into the scope of Banach space theory. Kolmogorov [1925] has shown the weak type (1,1) of the trigonometric conjugation operator ( see 8 Example 1). This was probably the first paper where finiteness almost everywhere was shown to imply weak type (1,1). The principle was generalized in Stein [1961] where a version of our Corollary 7 was proved. E.M. Nikishin was led to consider his general theorems by problems connected with the structure of systems of convergence in measure for £2 . His main results in this area are published in Nikishin [1970] . This paper basically contains Theorem 6. Later Maurey [1974] gave a more abstract presentation. It is his approach that we follow in this book. A sequence of functions C Lo [O, 1] is called a system of convergence in measure for £2 if every series with E converges in measure. Clearly there is a correspondence between £2

(¢n)�= l

L::=l an¢n

(an)

283

III.H. Factorization Theorems §Notes.

systems of convergence in measure for £2 and continuous linear operators T: £2 --+ L 0 • Some other similar notions have been investigated (see Exercises 13, 14 and 16). 8 Example 2 is an old theorem of Calderon (see Zygmund [1968] XIII. 1.22) . The fundamental theorem due to Carleson [1966] (for p = 2) and extended by Hunt [1968] to 1 < p < oo asserts that for every f E Lp(T) its Fourier series converges almost everywhere. An example of an L 1 -function whose Fourier series diverges a.e. was given by Kolmogorov [1923] . The example was improved in Kolmogorov [1926] to yield an L 1 -function with everywhere divergent Fourier series. Very recently the Armenian mathematician Kheladze gave a remarkbly simple construction of an L 1 -function for which condition (c) of 8.Example 2 fails. Inspired by Nikishin [1970] and Rosenthal [1973] B. Maurey under­ took his study of operators from X into Lp(O, which factors strongly through Lq (O, His results are presented in Maurey [1974] . Our presentation of 10-12 and 15 follows that monograph, with the excep­ tion that the proof of (b)=>(a) in Proposition 1 0 is taken from Pisier [1986a] . We recommend the reader to consult this paper. It contains many additional results, also in the setting of C* -algebras. Its main interest is to present necessary and sufficient conditions for the operator T: X --+ Lp (n, to factorize strongly through Lq,oo (n, that is the topic between Nikishin's and Maurey's theorems. Corollary 13 is one of the main results of Rosenthal [1973] . This paper played a very important role in the development of the theory. We would like to mention that the notion of type used in Maurey [1974] is different from the one used in this book. We use the type and cotype which is sometimes called in the literature 'Rademacher type' and 'Rademacher cotype', while Maurey uses the so-called stable type. Let 1 � p � 2 and let ei be a sequence of independent, identically distributed standard p-stable random variables. A Banach space X is called of stable type p if there exists a constant C such that for any finite sequence (xi ) in X we have

J.L) .

J.L)

J.L)

J.L) ,

Actually the use of the exponent 1/2 in the left hand side integral is irrelevant. It can be replaced by any number q < p. With this, one checks that Lp, 1 � p < 2, is not of type p but is of type s for any s < p. The following fact analogous to Theorem 6 holds.

284

III.H. Factorization Theorems §Notes.

If X is a Banach space of stable type p, 1 :S p :S 2, then every linear operator from X into Lr(O., JL ) , r < p factors strongly through Lp (n, JL ) .

Theorem A.

We have practically proved this theorem during the proof of The­ orem 12. The usefulness of the notion of stable type can be seen from

this proof. There is an obvious analogy between Definitions 4 and 9 and be­ tween Proposition 5 and Proposition 1 0. Let us note that conditions (b) of those propositions can be interpreted as vector valued inequalities (see the remark after Proposition 5) . This point of view is explained in detail in Garcia-Cuerva-Rubio de Francia [1985] , as is the equivalence between factorization and weighted norm inequalities. We do not discuss this important subject here. Our informal discussion in 14 is more or less folklore. It can be found in full detail in Maurey [1974] . Proposition 14 is a special case of the following result due to Maurey [1974] . Theorem B.

The following conditions on the Banach space X are

equivalent:

(a) co is not finitely representable in X; (b) there exists a q < oo such that ll q (C(K), X )

= L(C(K), X ) .

This result and its consequences for Banach space theory are dis­ cussed in great detail in Rosenthal [1976] . The Menchoff-Rademacher Theorem 22 was proved in Menchoff [1923] and Rademacher [1922] improving many earlier results. This is the best result. Menchoff [1923] constructed an orthonormal system on [0,1] such that for every sequence with 1 :S :S . . 2 and (log divergent al­ n there exists a series o most everywhere and such that < oo . The connec­ tion between the theory of p-summing (or radonifying) operators and the Menchoff-Rademacher theorem was noted in Schwartz [1970] and Kwapien-Pelczynski [1970] . This last paper also contains some general­ izations of the Menchoff-Rademacher theorem in the spirit of Corollary 25 which was proved in Maurey [1974] . Later Bennett [1976] gave an­ other, more elementary, but in fact closely related, treatment of such generalizations. Theorem 1 7, proved in 0rno [1976] , shows that se­ ries unconditionally convergent in measure (in particular unconditionally convergent in Lp ) are closely related to orthogonal series. Our Lemma

Wn =

)

(wn)�=l (wn)�=l = w 1 w2 . L::=l an Wn 2 L::=l l an l wn

Ill.H. Factorization Theorems §Notes.

285

18 is a classical result of I. Schur, published first in Rademacher [1922] (see also Kashin-Saakian [1984] ) . The dilation Theorem 19 is classical by now and is a basis of a large part of the theory of operators on Hilbert spaces (see Nagy-Foias [1967] ) . Our proof of the Menchoff-Rademacher theorem is a mixture of various published proofs like K wapieti-Pelczytiski [1970] , Bennett [1976] , Nahoum [1973] , Schwartz [1970] . Our Corollary 20 was proved in 0rno [1976] . It improves an earlier result of Kashin [1974] . Theorem 26 for a = 1/2 was proved by Mitiagin [1964] and the general case was shown by Bockariov [1978] . Our proof follows Woj­ taszczyk [1988] . Actually it is possible to obtain analogous results for systems more general than orthonormal and for more general moduli of smoothness. We refer the interested reader to Wojtaszczyk [1988] for formulations, proofs and the history of the subject. Theorem 30 and its proof are taken from Wojtaszczyk [P] . We would like to mention also the paper Bichteler [1981] where factorization theorems are applied to the theory of stochastic integration. The factorization theorems are basically a type of Tauberian the­ orem; they assert that the operator is actually better that it seems to be. This is useful both ways; we get stronger information once we prove something weaker or conversely we show the 'very' bad behaviour once we show a 'moderately' bad one.

286

1.

2.

3.

4. 5.

6.

7.

8.

Ill.H. Factorization Theorems §Exercises

G f) x)

Exercises

Let ( ( = [M( I / 1 2 )] 1 1 2 where M is the Hardy-Littlewood max­ imal operator. Show that is a sublinear operator on £ 2 [0, 1] which is of weak type (2,2) but not continuous on £ 2 [0, 1] .

G

T: X* Lp (O, J.L) factors strongly Lq (O, J.L) X* p < 1) then IIp (X, lq ) = IIq (X,lq ),p Let (cpj ) �1 be a sequence of independent, p-stable random vari­ ables in LI(J.L ) and let X = span(cpj ) �1 � lp . Show that i*: L00(J.L)�X*, where i is the identity embedding of X into L 1 (J.L), is not p'-absolutely summing, � + � = 1 . Show, without using Proposition 15, that every operator from L00 into Lq , 1 � q � 2, factors strongly through L 2 . Let T: Lp Lo, 1 � p < be a continuous sublinear operator. Assume moreover that T is monotone (i.e. if 1 / 1 � I Y I then also I T/ I � I Tg l ) . Then T factors strongly through Lp, oo · Let T: Lp Lo(O, J.L) be a positive (i.e. if f � g then Tf � Tg) linear operator and assume J.L(f2) = 1 and p � 1. Then T factors strongly through Lp (O, J.L ) . Show that there exists a positive linear operator (see Exercise 6) T: lp -+ Lq (O, J.L) , with 1 > p > q > 0, and (0, J.L) a probability measure, which does not factor strongly through Lp (O, J.L ) . Let X be a Banach space and 0 < < p < Show that the fol­ lowing properties of the bounded linear operator T: X L r ( O, J.L) are equivalent: (a) there exists a constant C such that for all finite sequences (xi ) X we have I s�p I T(xi ) ll l p � c ( � l xi l p) (b) there exists a constant C' such that there exists a function I E L1(f2, J.L), f � 0, J0 fdJ.L = 1 such that for all x E X and all measurable subsets E f2 have I T(x) · XEI I r � C'l l x ll ( l fdJ.L) � - � ; Show that if every operator -+ through (and has b.a.p. if < q, q � 1 .

-+

oo ,

-+

r

oo .

-+

C

1

•

c

we

p;

287

III.H. Factorization Theorems §Exercises

(c

9.

10. 11. 12.

13.

)

the operator T admits a factorization of the form T

M

X ---+Lp,cxo (O, fdJ.L) ---+Lr (J.L) with f bounded, f E L 1 (0, J.L) , J � 0, In fdJ.L = 1 where M is an operator of multiplication by f�. is Every operator from Lv [O , 1] into iq , 1 < q < 2 < p :::; compact. Let (fn)':'= 1 be an orthonormal system in L 2 [0 , 1]. Show that N 1 2:::= 1 fn -+ 0 almost everywhere. For a given number x, 0 < x < 1 let (cj (x))j�1 be its dyadic expansion (cj = 0 or C"j = 1). Show that for almost all x E [ 0 , 1] we have N- 1 2:: := 1 €j (x) -+ 1/ 2 , i.e. almost every number has asymptotically equal number of O's and 1 's in its dyadic expansion. Let Un)':'= 1 be a complete orthonormal system in L 2 [0 , 1]. Show that 2:: := 1 I fn i = on a set of positive measure. Let Aa , 0 < a < 1, be the space of all functions in the disc algebra such that l f (e i 9 ) - f(e i ( 9- h ) ) i :::; Cj h j . Show that there exists a f E Aa such that f' (j. N ( N denotes the Nevanlinna class) . A system of functions Un)n>1 L0[0 , 1] is called a system of con­ vergence in measure for £2 if every series E n>1 anfn with (an) E £2 converges in measure. Show that Un)n�1 iS a system of conver­ gence in measure for £2 if and only if for every c > 0 there exist a set EE [0, 1] with l Ee I > 1 - c, a constant CE and an or­ thonormal system (cpn)n�1 on [0 ,1] such that fni EE = CEcpn i EE for n = 1, 2 , . . . Show that the following conditions on a system of functions Un)n>1 L0[0 , 1] are equivalent. a.e. ( a) For every (an) E £ 1, we have E i anfn l < ( b) For every c > 0 there exist EE [ 0 , 1] with l Ee I > 1 - c and a constant CE such that supn IE, I fni :::; CE. Show that there exists a function f E L 1 (Y2 ) such that n+m j(n,m)ein9 eim

1 d� = oo where q = 2p( 2 - p) and 0 < p < 2, then there exists � function f E C[O , 1] such that En�1 l dn (!, I{Jn } I P = oo. Show that the map f = E �oo J(n)einB �--+ ( .Xn J(n))';= oo maps C(1r) into lp, 10 < p < 2, if and only if E�oo 1 -Xn l q < -oo where q = 2p( 2 - p) - .

17. If

18.

·

19. Show the von Neumann inequality using Theorem 1 9. Let us recall (see Exercise III.B.8) that the von Neumann inequality says that for a contraction on a Hilbert space (i.e. II T II � 1 and any polynomial = we have II � IIPIIA · 20.

T ) n p(z) En=O anzn anT E n=O l (a) Suppose Y :::> X and assume that both X and YIX have some type p > 1. Show that Y has some type q > 1 . (b) Let X, Y, Z be Banach spaces with Y :::> X and let T: X --+ Z. Show that there exist a space V, an isometric embedding j: Z --+ V and an operator T1:Y --+ V such that jT = T1 I X and the spaces YI X and VIZ are isometric. (c) Suppose that X C(S) is a subspace such that C(S)IX is reflexive. Show that every operator from X into ip, 1 � p � 2, is 2-absolutely summing. (d) Suppose A is a Ap-set for some p > 1 (see I.B. 14) and sup­ pose that (r.pn );:'= 1 is a complete orthonormal system in L2 (Y) . Show that there exists f E C(Y) such that j (n) = 0 for n E A and 2::= 1 I ( !, I{Jn ) I s = for all < 2. c

c

7l

oo

s

289

III.H. Factorization Theorems §Exercises

21. ( a) Describe the coefficient multipliers from and 0 < p :::; 2.

Xs

into f.p, for s > 0,

(b) Describe the coefficient multipliers from Bp (D) into Bq (D) for 0 < q :::; 2 :::; p < 00 .

111.1. Absolutely Summing Operators On

The Disc Algebra

We start this chapter with the construction of a non-compact, !-absolutely summing operator from any proper uniform algebra into £2 . This shows in particular that such an algebra is never comple­ mented in C(K) . Then we study p-absolutely summing operators on the disc algebra A. We construct an 'analytic projection' which maps some weighted Lp('l', �d.X) spaces onto the closure of analytic polyno­ mials and has properties analogous to the properties of the classical Riesz projection n. Then we show that every p-absolutely summing operator from the disc algebra is p-integral, p > 1 . We also show that A* has cotype 2 and derive some corollaries of these results. Next we study reflexive subspaces Y C Ld H1 and show that any linear operator T: Y --+ H00 extends to an operator T: L I /H1 --+ H00 • This is applied to some interpolation problems on D x D. In this chapter we present the detailed study of p-summing and re­ lated operators defined on the disc algebra A. Such a study is motivated both by the intrinsic beauty of the problems and by important appli­ cations. Actually we have already seen one application. In Proposition III.F.6(a) we have exhibited an absolutely summing operator P: A --+ £2 which was later used to prove the Grothendieck theorem III.F.7. This example suggests that absolutely summing operators on A may have some rather unexpected properties. Actually this phenomenon is not restricted to the disc algebra, but is shared (to a certain degree) by all proper uniform algebras. Let B C C(S) be a proper, point-separating subalgebra of C(S) , with 1 E B. 1.

2 Proposition.

operator T: B

There exists a non-compact, 1-absolutely summing

--+ £2 .

Proof: Take J.L

E M(S) such that J.L E B l_ and J.L ¢. B l_ where B =

E

C(S) : f E B}. It follows from the Stone-Weierstrass theorem that B =I B so such a J.L exists. Let v denote the Hahn-Banach extension of J.L I B to C(S) . We assume that I v ii = 1 . Thus we can take a sequence ( /n)':=l C B with 11/n ll oo $ 1 and such that J fn dv --+ 1 for n --+ oo . Considering {!

292

III.I. Absolutely Summing Operators On The Disc Algebra §2.

this sequence in L 2 (ivi + i JL i ) we can pass to a subsequence and convex combinations to get a new sequence (still denoted by Un)�= 1 ) such that In E B for n 1, 2, . . . and ll lloo � 1 and limn _, oo I 1 and In (ivi + I JL I )-almost everywhere for some F (see III.A.29 but in fact easier) . Clearly we get I Fdv = 1 = ll v ll so = 1, l vl-a.e. Now let us consider V Bl_ n L1 ( l v l + i JL i) and define T1 : V - by T1 In order to show that T1 is continuous it is (J enough to show that

ln

F, =

fndv = IFI i

= 2 k (a) = F da)k:: 1 •

2

Rn: C(Y) - C(S) R�:

for every n with the constant independent of n. Let be defined by = H(cp) o I where H(cp) is the harmonic extension of C B we get that Bl_ - Al_ = HP cp. Clearly � 1. Since so by the Paley theorem (see I.B.24)

Rn(cp) I Rn l

n Rn(A)

2 = � i (R� (a) , z:2k W z2 J;. r 1 l k k a Rn( a )d J d = J � �I � CI I R� ( a) l i 2 � C l i a ii 2 . Now we look for the operator i : B - V given by i( f ) = I · p for some p E V, such that T = T1 i will be non-compact. Clearly) such T is 1-absolutely summing (see III.F.4) . We take p = JL (v - p F2 • Since for i E B J ldp = J ldJL J F2 ldv - J F2 ldP, = 0 J F2 ld(v - JL) = 0 we see that p E V. Moreover lim sup i ( TI, e k ) i = lim sup I JF2 k ldp l k k 11/11 :::::: 1 11!11 9 ;::: lizn s�p I J F2k 1�k - 1 dp I ;::: lizn J I F2k p2k - l dp I - I f{ I FI =l } Fdp l = I J{ IFI = l } FdJL J{ I F I = l } Fd;; - J{ I FI =l } Fdp,l F dJL F dJL - [ = 11 ( [ ) I· J{ I F I =l } J{ I FI = l l } o

+

+

+

+

+

·

293

III.I. Absolutely Summing Operators On The Disc Algebra §3.

Since the bracket above is purely imaginary we see that sup lim k

11 ! 11 9

I (Tf,

ek

) l ?: 1

a

so T is not compact. 3.

From this proposition we get

Corollary. Let B be a proper, closed subalgebra of C(S) separating points and with 1 . Then B is not isomorphic (as a Banach space) to any quotient of any C(K)-space. In particular B is not complemented in C(S) . Proof:

onto

T: To

Suppose that there exists a map q: C(K) ---+ B . Let B --+ f.2 be a 1-absolutely summing, non-compact operator. Then q: C(K) --+ f.2 is a 1-absolutely summing, non-compact operator. From III.F.8 we infer that we have the factorization

Since L1 (K, 1-L) has the Dunford-Pettis property ( see III.D.33,34 ) and both S and id are obviously weakly compact we get that q is a compact. This contradiction finishes the proof.

To

Remark. The above corollary shows that some algebraic properties of a uniform algebra are determined by its Banach space structure. In particular there does not exist a multiplication on the disc algebra A which makes it into a commutative C* -algebra.

There are some very natural limitations to what can be proved about the Banach space structure of a general uniform algebra. We have the following.

4.

Proposition. For every complex separable Banach space X there exists a uniform algebra Ux such that X is isomorphic to a 1-complemented subspace of Ux .

294

III.I. Absolutely Summing Operators On The Disc Algebra §5.

Proof: Let K be the unit ball in X* equipped with the a ( X* , X ­ topology. We define Ux to be the smallest closed subalgebra of K containing all functions for E X and constants. For a = function E Ux we define = 2 rr Since the i elements E�= l ( · , Xr ) r for Xr E X and ir E N are dense in Ux we easily see that for every E Ux , Pf ( k ) = for some E X. Since clearly a 1 we get the claim ( see II.A. 10 ) . �

f

I PI

'Px (k) (k,x) -x1 i -i Pf ( k) ( ) J e 8 f(e 8 k)d0. f (k,x) x T: A --+

A.

) C( )

5. Now we return to the disc algebra Suppose we have a p-absolutely summing operator X, 1 � p < oo. The Pietsch theorem III.F.8 gives that there exists a measure on the circle such that � which is the In other words extends to the space closure of the polynomials in Thus it is natural that we start the detailed investigation of p-absolutely summing operators on with some observations about spaces

C(f l f !PdJ.L) ; .

J.L

J.L T Lp(1I', J.L) . Hp(J.L) .

1I'

Hp(J.L)

l i T/I I

A

J.L J.Ls fd

be a probability measure on 11', with the Let 6 Proposition. Lebesgue decomposition = >. where >. is the normalised + Lebesgue measure. Then

91 Lp (J.Ls ) 92 Hp( fd ) h A (1I') l 9t 9 h i Lp (tt. ) l 92 -h i Hp ( fd>.) 1I'� 1 92 A. J.Ls l_). � (JT\t�. l 9t! PdJ.Ls ) to ( J1f\L:I. I 92 I PdJLs ) to · () cp A cp l � 9tl � (JT l cp i P fd>.) � J'lf\ L:I. I cp ! PdJ.Ls '¢ A 1 '¢ 1 '¢ 1 ( J I '¢92 I P fd>.) � h cp '¢)92 · h.

What we really have to show is the following: given E such that and E >. we have to find E � c. We can obviously assume that is � c and continuous and E Since there exists a closed set � C with and < < >. � = 0 such that Using Proposition III.E.2 we find a E such that = and � e / 10. Let us use Proposition � 1E0 and III.E.2 once more to get a function E such that = 1, � = 1 and < e / 10. We put = + ( 1 One checks that a we have imposed enough conditions to make it the right Proof:

T: A

Note that Proposition 6 in particular implies that a p-summing operator --+ X whose Pietsch measure is singular with respect to X. extends to a p-summing operator T: measure the Lebesgue

C(1I') --+

7. Now we want to give a heuristic indication of what will be done in the subsequent sections 8, 9 and 10. We would like to reduce the inves­ tigation of p-summing operators on A to the study of certain operators

295

III.I. Absolutely Summing Operators On The Disc Algebra §8.

6

on C(1r) or Lp (1r) . From Proposition we see that this requires inves­ tigating a projection from Lp (fd>.. ) onto Hp(fd>.. ) , where f is a positive function and we impose the normalization J fd>.. = 1 . In general we cannot assume any properties of f. The only freedom which we retain is that we can replace f by any It � f (because if f d>.. was a Pietsch measure for an operator T, then ft d>.. also is) . However in order to keep things under control we have to control J ft d>.. . The classical case when f 1 , i.e. Hp(fd>.. ) = Hp (1I') , correspond, to operators which are rotation invariant (see Exercise 2) , and in this case there is the Riesz projection whose properties are well known (see I.B.20) . The main result of this chapter, Theorem 10, will give a very useful analogue of the Riesz projection in our general case. Technically, our projection will be built by patching up pieces of the Riesz projection More precisely (but not really precisely) we will construct a sequence of Hcxo-functions (depending on f) such that 1 and we put = Formally is a projection onto an­ alytic functions. In Theorem 9 we will construct such a sequence of functions having very elaborate properties, and the projection will be constructed in Theorem 10. =

'R

'R.

P

('Pi )� Pg 2::� 1 'Pi1'R(cpi g).

1

2::� 1 'PJ

P

=

P

G £1

There exist constants C and M such that for every f E E (1r) = 1 we have a function and a sequence of Hcxo-functions such that

8 Lemma.

('Pi )�0

L (1r) such that f � 0 and J f d>.. CX)

""" rn - 1' � rJ -

i =O

I � I 'PiiP I =
> < oo and put 'If:

0

·

0

0 0

°

0

· . . . ·

0

We define G to be

G=

{1 - e) - 3M j=O L Mi l 1 - 'I/Ji l 3 + 1.

With the above definitions we have to check conditions mostly routine. Ad Follows directly from Ad {2) . Observe that implies �

{1 ).

{12) {1)- (6). This is

{9)-{11). {8) 1 '1/Ji l {8 - 8 1 1 - '1/Ji +s l ) -1 so I 'I/Ji l l 1 - '1/Ji +s l � 88 for j = 0, 1, . . , J - 1 and s = 1, 2, . . . , J - j. {13) For a given t E '][' let n be the smallest integer such that 1 '1/Jn ( t )l > £. We see from {10) and {11) that I 'Pi ( t )l � 2£n - i for j = 0, 1, . . . , n, so n P { 14) L j =O I 'Pi (t)I � C(e) . .

III.I. Absolutely Summing Operators On The Disc Algebra §8.

297

(I ) I I - ,Pn+s (t) l � 6; so (9) and (IO) yield J+l P j=Ln+l l .. :::; CI I 'R I � f I J I P t:::.d>.. .

by (22) by (I.B.20)

3

3

P f3 L00(Y, >..)

by (19)

Lp (!:::.d>.. ) I 'R I I J3 I oo f3

This shows that is continuous in and the estimate for the norm follows from the well known estimates for ( see I.B.20) . = 1. Then for Let us take E with ;:::: 0 and we have ('][', C

f L1 t:::.d>.. )

00

j I P/I ! f3!:::.d>.. = j I � Oi rj'R(rf f) 1 2 f3!:::.d>.. by (18) :::; / if=O hi 2 I 'R (rj! ) I ! J3t:::.d>.. :::; by (22) :::; c 'f,=O ci j !riii 'R (rj! ) I ! J3d>.. i ! by (25) � C t, c; (! l rfil d>. ) (! l r; l /3 d)y ! � C ( t, c; j l rf f l d>. ) ( t, c; j l r; I /3d>. ) ! ! � C (! 1 1 1 ( t, c; l r; I ' ) d>. ) (! /3 ( t, c; l r; 1 ) d>. ) :::; c ( j l f l t:::.d>..) ( j f3t:::.d>..) Since this holds for arbitrary f3 we infer from Lemma 1 1 that P is of weak type (1-1) with respect to the measure t:::. d>.. . 1

l

1

2

1

2•

III.I. Absolutely Summing Operators On The Disc Algebra §12.

Given

303

f E Lp(l:!.d>..) and h E Lp'(!:!.d>..) we have j hP(f)!:!.d>.. = Jf=O j hOi rjR(rj f)!:!.d>.. = f J rj JR (hOjrj f:!. ) d>.. j=O 4 = j f L 7i R ( Bjrj · !:!. · h) i:!. d>.. j=O f:!. (X)

so

(X)

4 P*(h) = j=O L 7 R(Bjrjl:!. h) . f:!. 1

P* is of weak type (1-1) let us take f3 E L=(1f, .X) with f3 1 /3 1 = = 1. Then we have J 1 P*h l ! f31:!.d.X :S C J Jf=O 1 ri i 2 I R(Bjrji:!.h) l ! f3�d>.. by (22) :S C f .jCj J l ri 1 3/ 2 f31 R (Bjrj f!.h) l ! d>.. j=O ::; c f; vs ( / I Bjrjl:!. h l d>..) 2 ( / hl 3 1 2 f3d.X) 2 by (25) ::; c f; ( / l rf h l i:!. d>.. ) 2 ( / l rjl 3 1 2 cjf3d.X) 2 ! ( / ( t, l r; l 'i' c; ) f3dA) ( h ( ) C ) d I / ! i t, A t. ' lr; ,; ! ! by (19) . ::; c ( J l h l !:!.d>.. ) ( J f31:!.d.X) From Lemma 1 1 we infer that P* is of weak type (1-1) with respect to the measure f:!. d.X. To check that � 0 and

1

1

00

!

00

1

a

12. Now let us see how we can apply our considerations to estimate integral norms of absolutely summing operators on the disc algebra.

Theorem. Let T: A p < oo. Then

-->

X be a p-absolutely summing operator, 1




q

>

p, and every (), 0 < () < ¢ where ¢ =

we have

1-� (26)

!:id>..

Let J.L = + J.L8 , be a Pietsch measure for T (see Theorem III.F.8) . Applying Theorem 10 to the weight we get a measure jl = + J.Ls which is also a dominating measure for T with � C. Proposition 6 and Theorem 10 show that there is a projection from and also of weak type Lp (jl) onto of norm � C max(p, (1-1) . Let us consider the commutative diagram Proof:

!:id>..

!:i

(p � l ) )

Hp (jl)

id i

l il lP

where and with various subscripts denote the formal identity acting between spaces indicated, and is the natural extension of T which exists because T is p-absolutely summing. This diagram clearly shows that T is p-integral with

T

1 ip (T) � I P I P · I T- I � C max(p, p)7rp (T) -1 so we have (a) . In order to show (b) let us note that

iq(T) � l i T o iq,p o P o ioo ,q o id l � l i T iq,p o P I = l i P* i�,p T* I · Given x* E X * with l x * l = 1 we define = P*i � , p T*(x*) E Lq' (jl) , so 0

0

0

1. Since £i is !-complemented in it suffices to consider the operator T: A ---. £i given by = (j( O ) , � ]( 1 )) . We have

T (f)

1 II T IIA �R� = sup { l f ( O) I + 2 1/ (1) 1 : 11/ll oo S 1 } '

'

{ 2� j f(O) (a + �/Je - i0)de : lal 1 , 1/JI 1 , 11 /lloo = 1 } = I l + � ei O I 1 / 1 I G e - i O + 1 + � ei O ) t 1 = 1 . l LH = sup

S

S

S

O n the other hand if f : C (1I') ---. £i is an extension of T then, by a standard averaging argument we get IITII � II T1 II where T1 = ]( 1 (J ( o), � ) ) for E C(1I') . But

(f)

f

20. We would now like to discuss the reflexive subspaces of LI/ H1 . We start with the following proposition which is analogous to Corollary III.C. l8.

X

c LI/ H1 be a closed subspace which does not Proposition. Let have type p for any p > 1. Then contains £ 1 .

Proof:

X

This follows directly from Theorem III.C.16 and III.D.31.

XC

a

LI / H1 be a reflexive subspace and let q denote the natural quotient map from L 1 onto LI/H1 • There exists a reflexive subspace L 1 such that q i is an isomorphism from onto 21 Theorem. Let

XC

X

X

X

X.

It follows immediately from Proposition 20 that has type for some p > 1. We see from Proposition III.H.l4 and Corollary 13 that L(A, = IIq(A, for some q < oo. In particular the operator A ---. defined by = for is q-absolutely summing, so by Theorem 12 q-integral. Thus extends to an operator ---. M (1I'). For E A and Let us take C (1I') L 1 (1I') and so = = = we have is an isomorphic embedding and we This shows that qo = 1:1 can put Proof:

p

X* ) X*) r.p: X * r.p(f)(x) x (f) x E X r.p f : Ll /H1 --+ H00 by T4> (u) (z) = (u, ¢(z, ·) ) . The correspondence ¢ +-+ T4> establishes an isometry between L(Ll /HI . Hoo ) and H00 (D x D) . Proposition.

Clearly T4> is a well defined linear operator into H00 • Since (Ll / H1 ) * = H00 we have

Proof:

IIT ll = sup sup l (u, ¢(z , ·) ) l = sup ll ¢(z, · ) ll oo = ll¢ll oo ,

z EID

z E D llull9

so the map ¢ �---+ T4> is an isometry. If we are given T: Ll/ H1 --+ H00 then we define ¢(z1 , z2 ) = T(Oz1 ) (z2) where for z E D the symbol Oz denotes the functional on A 'value at z ' . The Poisson formula shows that Oz E Ll/H1 . Obviously ll ¢ll oo � IITII · Since Oz1 E Ll/H1 for Z1 E D we get that ¢(z1 , z2 ) is analytic in z2 . But T(6z1 ) (z2 ) = T* (Oz2 ) (zl ) where T* : H� --+ H00 so ¢ is also analytic in z 1 . This shows that ¢ E Hoo (D

x

•

D) .

27. Let us recall that a set of integers A is called a Ap set, 1 < p < oo , if there exists a constant C such that for all sequences of scalars ( aj ) j E A we have

Corollary. If A is a A2 subset of positive integers and sequence of H00 functions such that

sup L lcpk (z) l 2
k x k ll � M · ( L l ak l 2 ) ! for all sequences of scalars.

k =l

Then for some z E

2 (see Kislyakov [1981b] ) . The concrete applications of our main results which we present are mostly routine. Some more are to be found in the Exercises. Corollary 27 is a two-dimensional analogue of III.E.9. Let us also recall that Theo­ rem 1 0 has applications to operators on Hilbert space which we discussed in Notes and remarks to Chapter III.F. It was also instrumental in con­ structing counterexamples to some old conjectures of Grothendieck (see Pisier [1983] ) . The very important technical aspect of our work in this chapter is the extrapolation argument based on interpolation inequalities. The first time this appeared in this book was in the proof of Theorem III.F.27.

III.I. Absolutely Summing Operators On The Disc Algebra §Notes 319

In this chapter we have two interpolation inequalities, the abstract one namely Theorem 12(b) and a very concrete one namely Lemma 23. It was recognized by Kislyakov [1987a] that inequalities like in Lemma 23 hold for other natural operators and can sometimes serve as a substitute for the weak type (1-1) .inequality. This has some nice consequences. It is also interesting to note that our work is essentially restricted to spaces of analytic functions of one variable. Bourgain [1985] and [1986] constructed for every p > 2 an operator on A(IBa), d > 1, and on A([)n ) , n > 1 , which is p-absolutely summing but not q-integral for any q. This shows that most of the results presented in this chapter about the disc algebra are false both for A(IBa) and for A([)n ) with d > 1 and n > 1 . It seems to be a very interesting problem to figure out what goes on in several variables. Added in proof (May 31, 1 990} After this book was submitted I received a very interesting work of Kislyakov [P1 ] . It contains a new, simple proof of Theorem 12 b) . The main truncation lemma (refered to in the title) asserts the following: If b. 2:: 8 > 0 is such that b- 1 1 2 ::; cb- 1 1 2 for some c then every

H1 (b.d>.. ) can be represented (i) /I E Hoo and II /I lloo :=; 24c2

fE

as

f

= /I + h

:=; s < oo if f E Hs (b.d>.. ) ;s 1 . (f l h I s bo d>.. ) /s ::; 1 06c4 �� � �1 I ll s b.d>..

(ii) for every s , 1

(

r

where

then /2 E

Hs (b.d>.. )

and

Some further extensions are contained in [P2 ] . Exercises

1. Show (without appeal to Theorem 1 0 ) that every multiplier T: £2 ---+ A (i.e. any operator of the form T(�n) = 2:: ;:'= 0 �ntnz n ) has !-absolutely summing adjoint. Let T: A ---+ H2 be a multiplier (i.e. T(2:: ;:'= 0 anz n ) = 2:: ;:'= 0 antnz n ) . Show that the following conditions are equivalent: 2.

(a) T is p-absolutely summing for some p < 1 . ; (b) T is 1-nuclear;

(c) T factors through an L 1 (/-L) space;

320 III.I. Absolutely Summing Operators On The Disc Algebra §Notes Note that it follows from this exercise and Paley's inequality I.B.24 ( or Hardy's inequality I.B.25 ) that the map i d : A (1r) --+ H1 (Y) does not factor through any £1-space. A second such example is in Ex­ ercise III.G. lO ( d ) . 3.

Let X be a finite dimensional Banach space and let us fix numbers K and p, 1 < p < oo. Show that the following conditions are equivalent: ( a)

ip(T) ::; K1rp (T ) for every operator T: X --+ Z, where Z is an arbitrary Banach space;

( b ) for every subspace

Y C Lp' (JL) , � + � = 1 , and for every linear operator T: Y --+ X there exists an extension f: Lp' (JL) --+ X with II T II ::; K II T II ·

4.

Let X C C(S) be a closed subspace. Show that the following con­ ditions are equivalent: ( a)

X@C(S) is closed in C(S)@C(S); ( b ) every operator T: X --+ £1 (JL) extends to an operator T : C(S) --+ Ll (JL); ( c ) II 2 (X, £1 ) = L(X , i l ) and X* has cotype 2. 5. Show that every reflexive subspace of C 1 (1r2 )* embeds into some Lp space for some p > 1. 6. ( a) Define an operator T: H1 (Y) --+ H00 (1r) by T(L:�...., 0 anz n ) = L::'= o � zn . Show that T is continuous and does not have an extension to f: L1 (Y) --+ Hoo (1r) .

( b ) Show that, if

X

c

L 1 (T) is a non-reflexive subspace, then there exists an operator T: X --+ H00 (1r) which does not have an extension to f: £1 (1r) --+ Hco (1r) .

( c ) Show that, if

X c Ld H1 is a non-reflexive subspace, then there exists an operator T: X --+ H00 (1r) which does not have an extension to f: Ld H1 --+ Hoo (1r) .

7.

8.

Show that Theorem 25 holds with the space Ld H1 replaced by the space £1 (0, JL, Ld HI ) of Bochner integrable (Ld H1 ) -valued functions ( see III.B.28 for definitions ) . Let R be a reflexive subspace of £1 (0, JL) and let T: R --+ Hoo (T) . Show that T extends to an operator T: L1 ( f2 , JL) --+ H00 (1r) .

III.I. Absolutely Summing Operators On The Disc Algebra §Notes 321

9.

(a) Suppose that ( /j ) f=- 0 H00 (T2 ) into £1 by

c

Lt (T) and define an operator T from 00

T(g) = ( (gj , fi} ) 'f= 0 where g(z, w ) = L Yi (z) wi . j =O Show that T is bounded if and only if

1 2 ) 2 : hi E Hr for j = o, 1 , 2, ... } oo . { h ( f; Show that the matrix (an, m ) n, m ;::: 0 is a coefficient multplier from A(D2 ) into £1 (N x N) if and only if:L": n, m ;::: o l an, m l 2 oo . Suppose that M (mk 1 , k N )k 1 , ,kN ;::: o is a coefficient mul­ N tiplier from A(D ) into £1 (N N ) . Show that there exists a inf

(b) (c)

00

l.

I Ii + hi


1 is not isomorphic to the disc algebra A(ffit ) ·

A(D2 ).

Hints For The Exercises

1. If there is a metric, the balls have to be unbounded in norm. This would give a weakly null sequence which is not norm-bounded. 2. Each basic neighbourhood U(O; c, xi , . . . , x�) restricts only countably many coordinates from r. This implies that if U1 ::) U2 ::) · · · are weakly open sets in B£2 ( r ) and 0 E U; for j = 1, 2, . . . then n;: 1 U; has contin­ uum cardinality. 3. Use the Riesz representation theorem I.B.ll and the dominated convergence theorem. 4. Take X = eo and ( x �)�= 1 the unit vectors in £ 1 = c0 . X = £ 1 and x� E f00 , x� = L:;:n e; also works. 5. Both topologies are metrizable so it is enough to check the convergence of sequences. Work with Taylor coefficients. 6. Use Exercise 5. Note also that T is 1-1 so r - 1 (BA (D) ) does not contain a line, so it is not u(H00 , Ld Hl )-open. 7. Show that the products are positive and have integral 1 . Use the Fourier coefficients to show that the cluster point is unique. 8. This is basically the same as Exercise 7. 9. First show that for 'Y E r we have T7 = a"'f'Y · Take as f..L the w*-limit of T(gn) where 9n = II XA,. II 1 1 XA,. where An are neighbourhoods of the neutral element in and I An l -+ 0. 10. Put fr (e i6 ) = f(r ei6 ) and note that functions fr are uniformly bounded in L 1 (T). Take the u{M(T) , 0{11'))-limit. II.A.

G

II.B. 1 . See Proposition 3. 2. (a) Think what it means in terms of the unit balls. The proof is in Pelczynski [1960] . {b) Note that if ll x + Y ll2 = ll x ll2 + I IY II 2 then x = >.y for >. � 0. (c) Show that I l l · I l l of (b) is strictly convex. (d) Consider the subspace £0 = { x E foo (r): card{"': x ('Y ) =f. 0 } � No } . For x, y E fD such that ll x ll oo = IIYIIoo = 1 write x < y if y ('Y ) = x('Y) for all 'Y E r such that x('Y) =f. 0. Put Fx = {y E £0 : x < y}. Let I l l · I l l be any equivalent norm on £0 . Put mx = inf{ I I IYI I I : x < y } and Mx = sup{ I I IYI I I : x < y}. Use transfinite induction to get z E £0 , ll z ll oo = 1 such that for all y > z we have my = mz and My = Mz . Then mz = Mz . This is only a glimpse of renorming theory. For detailed exposition the reader can consult Diestel [1975] . 3. Look at the partial sum projections. Consider fn = :L:;= 1 e; in CQ . 4. Examine how in the proof of Corollary 18 we used the assumption that (xn) is weakly null. 5. If the sum X + Y is not closed the map ( x + y) �---+ x is unbounded so there are x E X and y E Y with ll x ll = I I Y I I = 1 and ll x - Y ll arbitrarily small. 6. (a) Approximate the Faber-Schauder system. (b) Approximate the Haar functions, the n-th function in Lp,. where Pn /' oo. Examine

324

Hints For The Exercises

what happens if 8 � 1 in Proposition 15. 7. (a) The Haar system is basic in L00 [0, 1] . (b) Interpret Haar functions as functions on 6. . 8. Find the coefficient functionals explicitly. This shows l an l < en- "' for [O, 1] . Conversely write f = E :'= l an'Pn = 2::, 1 !k where /k = f E�Lipa 1 E n = 2 k + l an'Pn and for t, s such that I t - s l 2 - N estimate separately E �=l !k and E'; fk . Compare with 111.0.27. 9. (a) Permute the trigonometric system and use the Riesz projection. This was shown by Boas [1955] . (b) The derivative maps Lip1 [0, 1] onto L00 [0, 1] and is almost an isomorphism. (c) , (d) Look at (b) . 10. (a) Show that contains a closed subset homeomorphic to 6.. This is done in Kuratowski [1968] 111§36.V and Lacey [1974] . (b) Compare the dual spaces. 11. Use the decompositoin method. For (a) represent (EC[O, 1])o as a subspace of C[O, 1] . 12. Note that for 1 � p < oo, p =f. 2 two functions J, g such 1 that II / + ag ii P = ( II / II � + IIYII � ) :P for all scalars a with lal = 1 have to be disjointly supported. 13. Use Theorem 4 and Exercise II.A.3. 14. Start first with finite E1 . For existence in the general ca.Se use the Radon-Nikodym theorem. Such projections are called conditional expectations, and are of fundamental importance in probability theory. They are studied in almost every introductory book on probability. 15. (a) Take any countable dense set in Bx and map the unit vectors onto this set. (b) f1 (r) is a subspace of C[0, 1] * . Use (a) . (c) On f00 there exists a sequence of functionals (x�);;"= 1 such that if x� (x) = 0 for n = 1 , 2, . . . then x = 0. 16. Try to repeat the proof of Proposition 6. "'

S

1. Define P(x*** ) = x*** l i(X) E X* . 2. Use the domi­ nated convergence theorem. Construct Rademacher-like functions. 3. Find a sequence (xn);;"= l C X with ll xn ll = 1 for n = 1, 2, . . . such that (Txn);;"= 1 is a basic sequence in Y and there is an y* E Y* such that y* (T�n) � 8 > 0 for n = 1 , 2, . . . . This was proved in Lindenstrauss-Pelczynski [1968] . 4. (c)::::} (b) follows from the Fejer the­ orem I.B.16 and for (a)::::} (c) consider fe = (2e:) - 1 X ( - e , e ) and show that w*-lime o Tp. (fe) E L1 (Y) . 5. Note that TK : L2 [0, 1] -+ Loo [O, 1] . To see that TKo is not weakly compact look at the images of the Haar functions. 6. Suppose lp, (ni ) l > E: for ni -+ oo. Let f..Loo be the w* -cluster point of { e in;ll f..L }�1 and let v be the w* -cluster point of {e - in; 9 Vn; * f..L }�1 . The F.-M. Riesz theorem yields v absolutely con­ tinuous. Also [1,00 (n) = D(n) for n � 0, so f..L oo is also absolutely con­ tinuous. On the other hand writing f..L = fdt + f..Ls one checks that f..L oo is singular. This is a result of Helson [1954] . Compare with Exercise 7. 7. If not, take n(p) and m(p) in N tending to oo with p so that p, (n(p) - j ) = p, (m(p) - j) for j = 1 , 2, . . . , p but p, (n(p)) =f. p, (m(p) ) . II.C.

.....

325

Hints For The Exercises

Write fL = fdt + /L s and look at { (e - i n (p}l1 - e - im ( p ) li )/Ls } �1 C L1 ( l l"s l) . The weak cluster point /Loo exists, belongs to £1 ( 1 /Ls i ) and is not zero. On the other hand F.-M. Riesz Theorem gives that /Leo is absolutely continuous. This is from Helson [1955] . Compare with Exercise 6. 8. (a) The very definition of a shrinking basis gives that (x�)�= 1 is a basis in X* . It is boundedly complete by the Alaoglu theorem. (b) Boundedly completeness give the *weak compactness of the unit ball. (c) Put together (a) and (b) . (d) By (a) (x�)�= 1 is a basis in X* so every x** E X** can be identified with a sequence of scalars. This is old and well known to specialists (see Lindenstrauss-Tzafriri [1977] ) . Some parts are already in Karlin [1948] . 9. (a) Cauchy sequences in I · I I J are coordinatewise Cauchy. (b) Consider vectors (1, . . . , 1, 0, 0, . . . ) . (c) Show that if n1 < m1 < n2 < m2 < · · · and Xk = '£';�n k O.jej then ll '£ := 1 xk ii J :s; ( '£ := 1 llxk ll ) ) ! . (d) Use Exercise 8 (d) . (e) Look at the isomorphism between Co and c given in II.B.2(a) . (f) The number dim ( J** f J) is an isomorphic invariant. All this except (f) can be found in James [1950] . (f) is due to Bessaga-Pelczynski [1960a] . 10. This and much more can be found in Davis, Figiel, Johnson, PelczyD.ski [1974] . 1 . Use the form of the partial sum projection as given in the 2 k+l proof of II.B. 10. 2. Look at L 00 k =O '£ 2 k + l an'Pn · 3. Note that the series is we;tkly unconditionally convergent and use Proposition 5. 4. Use � £1 is not compact then Theorem 13 and Theorem 6. 5. If T: '£:'= 1 T* (en) is a weakly unconditionally convergent but not uncondi­ Proposition 5 and Theorem 6 lead tionally convergent series in to a contradiction. 6. Consider the Orlicz property. 7. (a) It is enough to consider Hilbert space. It is possible to prove it by induction on n. (b) Show that if x E X is a limit of some subsequence of partial sums of the series '£:'= 1 Xn then x E U(xn) · Both (a) and (b) are due to Steinitz [1913] . (c) Take functions {± 2-� l hn i } �=O · Show that they all can be ordered into a series whose sum is 0 and into another series whose sum is 1 . Since each function takes only values 0 and 1 every sum will take integer values. (d) The example is in Kadec-Wozniakowski [P] . It is a bit too complicated to repeat it here. 8. (a) This is almost obvious. (b) Show that for every N there is a measure preserving transformation of [0,1] which transforms { gn,k} , n 1, . . . , N, k = 1, . . . , 2 n onto the first Haar functions. (c) Like in the proof of Theorem 10 produce a block-basic sequence as in (b) . (d) Find blocks of (cpn)�= 1 behaving like those considered (gn,k) in (b) . This basically reduces the problem to the Haar system. This exercise shows the fundamental role played II.D.

C(K)

M(K).

=

326

Hints For The Exercises

by the Haar system in the study of Lp [O, 1]-spaces and in theory of or­ thonormal series. (d) is a result of Olevskii (see Olevskii [1975] p. 75). A Banach space theoretical version of these phenomena is presented in Lindenstrauss-PelczyD.ski [ 1 97 1 ] . 9. (a) Look at the formulas and com­ pute carefully. (b) Apply to the dk 's the procedure applied to the Haar functions in order to get the 9n,k 's of Exercise 8(b) . Note that the con­ clusions of (a) hold. Represent it as blocks of the Haar system. (c) This follows directly from Exercise 8(c) . All this is due to Burkholder [1982] and [1984] . II.E. 1 . (a) This is just reformulation of the definition. (b) Take x** E X** , llx** ll = 1 and a net (x'Yh E r C Bx tending to x** in the a ( X** , X*)-topology. Use (a) to show that this net converges in

norm. This is a classical result of D.P. Milman. This proof is due to Ringrose [1959] . (c) One has to show the Clarkson [1936] in­ equalities I I (u�v) 11: + II (u;v) 11: :::; ! ( llull � + llv ll � ) for 2 :::; p < oo ' l and II (u� v) 11:' + II (u;v) 11:' :::; ( ! llull � + ! llv ll � y - for 1 < P :::; 2. The first one is the integration of the corresponding numerical inequal­ ity while the second follows from the appropriate numerical inequal­ ity and the inequality ll lul + l v l ll q 2:: llull q + llvll q valid for q :::; 1 . We apply it t o q = p - 1. (d) Observe that if liz + vii :::; 1 and liz - v ii :::; 1 then ll z ll :::; 1 - cp(l l v ll ) . Apply this observation induc­ tively to the finite sums. This is due to Kadec [1956] . (e) This is quite obvious. Use (b) to show the existence of the best approxima­ tion. 2. (a) Replace max by the average in the definition and estimate 2 from below J0 '��" 1 1 + be i9 ld0. (b) Use the ideas of the proof of Exer­ cise l.d. The notion of complex uniform convexity was first studied by Globevnik [1975] . 3. Take (Pn );:"= 1 and ( qn );:"= 1 two disjoint se­ quences, dense in [1, 1.5] such that P1 = 1. Take X = ( :L: :'= 1 .e�J 2 and Y = ( :L::'= 1 .e�J 2 • Show that Y does not contain .e� isometrically. 4. (a) Adapt the proof of Theorem 9. This is a correct estimate (see III.B.22) . (b) Consider everything on [ -1r, 1r] . Look at the translation of the square of the Dirichlet kernel. If Vr (t) = sin(r + 2)t/2 sin !t then Vr (O) = r + ! and Vr (08) = 0 for 08 = ( 2;s_;l ) , 8 = ± 1, . . . , ±r. Put /r,s (t) = [Vr (t - 08 ) ] 2 , 8 = 0, ± 1 , . . . , ±r. Then :L: := - r /r,s = (r + ! ) 2 and this helps to show that { (r + � ) - 2 fr, s } �= - r is isometrically equiv­ alent to the unit vector basis in .e� + 1 . This is a classical interpolation problem (see Natanson [1949] ). 5. Apply Theorem 9 twice. 6. Every open ball contains infinitely many disjoint balls of equal radii. 7. This is a compactness argument. For each o: E r and x* E X* define a function

327

Hints For The Exercises

on Y by the formula IP

{

a (x * ) (y ) = x * (Sa0 (Y )) ifif Yy ¢E Ya, Ya.

Taking a pointwise cluster point we find cp: X* --+ Y* which is bounded and linear and T*cp is a projection onto T* (Y* ). 8. (a) Note that ( E :'= 1 �) 00 = ( E :'= 1 �) ; and use Exercise 7. (b) If d(X, Y) is small one can represent X and Y as norms on Rn (or ccn ) such that the unit balls are close, so the norms are close as functions on Rn (or ccn ) . Now we see that the limit exists, so we have completeness. For total boundedness use the Auerbach Lemma. (c) Use (b) and Exercise 7. This and Exercise 7 can be found in Johnson [1972] . Exercise 7 is an improvement of an earlier result of C. Stegall. 9. (a) On each finite dimensional subspace of X we have a uniformly convex norm, uniformly close to the original. Use a compactness argument. (b) Similar to (a) . 10. Show that if E C Lp [O, 1] is finite dimensional, 1 � p � oo, then there exists F C Lp [O, 1] , F � t;: such that E is close to a subspace of F, where both n and 'closeness' are controlled. The case p = oo is relatively easy (use Lemma 1 1 or Proposition 10) . For the case p < oo show that f(t) = sup { J x (t) J : x E E, ll xJJ � 1} is in Lp [O, 1] and consider g(t) max(f(t) , 1). Take the isometry J: Lp [O, 1] --+ Lp( [O, 1] , gPdt) defined by Ih = h · g - 1 . Note that IBE C BLoo ([o , 1j , gP dt) · Now we can follow the case p = oo. This is taken from Pelczynski-Rosenthal [1975] . 11. Take the quotient map from £1 onto £� (see Exercise II.B.15 (a) and dualize. Or use II.B.4. Show that a finite dimensional subspace of co is also a subspace of £� (for some so has a finite number of extreme points. =

N),

J

1. Note that if f � g and f =1- g in Lp (J.L) then I f - g J P dJ.L > 0. Since everything is below g we can reach the max in a countable number of steps. 2. Estimate J x l q using the Holder inequality. 3. Assume Lp (J.L) = Lp [O, 1] and find in Y a block basis (Yn )�= 1 of the Haar system equivalent to the unit vector basis in fw Take (y�)�= 1 , the sequence of biorthogonal functionals such that y�( Yn ) � epJJ y� J I IIYn ll and such that y� are in the same block of the Haar system as Yn for n = 1, 2, . . . . The projection P(f) = E :'= 1 y� (f)yn works. 4. Consider sets Me = {! E Lp [O, 1] : I f l i P � c-l l fl l 2 } for c > 0. If X C Me for some c > 0 then X rv £2 and is complemented . If f E Me then there exists a set A c [0, 1] with I AI < c and II ! · XA l i P � (1 - c-P) � II f l i P · From this, if X is not in any Me we can find a sequence in X close to the sequence of disjointly supported III.A.

J

328

Hints For The Exercises

Lv (J.t)

functions. This is a result of Kadec-Pelczyiiski [1962] . 5. Each norm-1 projection in is a conditional expectation projection {for definition see Exercise II.B.14) . To see this is a rather tedious process. We check that if E ImP and supp g C supp then supp Pg C supp We also check that P ( h sgn = I P ( h sgn !) I sgn for E ImP and h � 0. The details and references are in Lacey [1974] . 6. Use the finite dimensional version of Proposition 7 and Theorem 6. 7. Use Exercise 6. 8. If P1 , . . . , Pn E P, Q1 , . . . , Qn E Q with PkPj = 0 and QkQj = 0 for k -# then the norm of Ej ,k PkQj can be estimated by twice applying the Khintchine inequality. Note that Ej= 1 ± Pj is uniformly bounded. This is due to McCarthy [1967] . 9. From the Khintchine inequality 1 we get II E := 1 f; . To estimate it ( f0 ( E:= 1 from above we use the � convexity of the norm {for p $ 2) or the Holder inequality {for p � 2). To estimate from below we replace 1 by = { t: 1 $ t $ 1 - 2-n- } . 10. X where Follow the proof of Theorem 8 (P = Po ) . For (a) note that P is a selfadjoint (and so orthogonal) projection. This is classical, due to Bergman. A similar exposition on 1Bd can be found in Rudin [1980] . 11. The operator T9 : Bp (D) -+ Bp (D) is compact (see Exercise 16) . Consider the spectrum of T9 . This is due to Axler [1985] . 12. Start with n = 2 and write P explicitly, then use the multiplier theorem I.B.32. 13. Apply Proposition 9. 14. Apply Proposition 9c with y(x) = X01 for right a. 15. Use Proposition II.B. 17 (or see Exercise II.B.4) or its modifications for p < 1 to show that the existence of a non-compact operator T: lp -+ lq implies that lp -+ lq is bounded. 16. All except (e) are variants of Theorem 25 and can be found in Wojtaszczyk [1988] . For (a) , {b) , {d) repeat the proof. For (c) apply (a) inductively. For (e) take a system such that E := 1 J < oo (e.g. a subsequence of the Haar system. 17. Apply definitions. 18. Use Remark 20. 19. (a) is an example of Schreier [1930] . The original construction requires some familiarity with ordinal numbers. The other way is to invent any Banach space with the sequence violating (a) and use Theorem II.B.4. One such example is to define ll (xj ) �1 11 = sup{ E ;= 1 I x : n = 1 < < For {b) use (a) and apply Theorem II.C.5 to the h < operator T: £ 1 -+ C[O, 1] given by = n = 1, ... .

f

f

f)

f.

f f

j

l an l 2r2k+ 1 )! rdr) Lv ; 2 2-n

anfn l v "'

r2n+l r2n+l · En

En

id:

l fPn l

· · · jn} · 2

T(en) fn,

2,

ik l

j

1. Show that L00 [0, 1] embeds into £00 (see Exercise II.B. 15(a) , use Theorem and the decomposition method. This is due to Pelczynski [1958] . 2. span { z k w - k : k = 0, 1, . . . , n } is such a subspace. 3. Identify t k with cos k (} E T:O , k = 0, 1, . . . , n. This is a classi­ cal device; see Natanson [1949] . 4. Compute the relative projecIII.B.

n

Hints For The Exercises

329

tion constants of subspaces of polynomials of degree at most n. 5. (a) Use estimates for d(.e; , .e�) and d(.e;, � ) . (b) Dualize. 6. Let �n = { - 1 , 1 } n . Note that li is isometric to the span{ri }J= l C C(� n ) , where rj (ct . . . . , en) = cj · Observe that this can b e identified with the span of the first n Rademacher functions in C(�) . Apply Theo­ rem 13. Projection constants of £; spaces can be found in Theorem VII. 1.9 of Tomczak-Jaegermann [1989] . 7. Identify Y with eo and put (x�)�= l the norm-preserving extension of coordinate functionals. Find z� E X* n y.L such that x� - z� ---+ 0 in a(X* , X)-topology. Put P(x) = (x� (x) - z� (x))�= l · This fact is due to Sobczyk [1941] and the proof indicated here to Veech [1971] . 8. (a) Use the following set­ theoretical result due to W. Sierpinski. If N is a countable set then there exists a family { A'Y } 'YE [O , l ] of infinite subsets of N such that A'Y, n A'Y2 is finite for all 'Yl -# ')'2 • For the proof of this identify N with the set of rationals in [0,1] and put A'Y any sequence of rationals tending to 'Y · (b) Use Exercise II.B.15c. This is a classical result of Phillips [1940] . The argument indicated here is taken from Whitley [1966] . (c) Suppose P is a projection onto X and i: eo � X is an isomorphism. Extend i to j: £00 ---+ £00 and show that i- 1 Pj is a projection onto CQ . 9. Start the induction in the proof of Theorem 21 with the polynomial p. 10. Show Proposition 19 for cp being a lower semi-continuous function on Bd and with the inequality in (a) holding on Bd. This can be found in Rudin [1986] . 11. Take f E H1 ( Bd ) , f = E':= o fn where fn is a homogeneous polynomial of degree n. First note that Rf = E':=o nfn and next show that 1:. ';:'= 1 n - d ll fn lloo :::; ll flh · For this use the Hardy inequality on one-dimensional complex subspaces of ccd and estimate the ratio between ll fn lh and ll fn l l oo like in the proof of Proposition 18. This is taken from Ahern-Bruna [1988] . 12. Dualise and use the weak type (1-1) of the Cauchy projection. For details see Wojtaszczyk [1982] . 13. (a) This is a direct calculation. (b) Use functions from (a) . This can be verified by the direct calculation or by appeal to Corollary III.H.16. 14. Use the ideas from Exercise 13 and the polynomials constructed in Proposition 18. Better results can be found in Ullrich [1988a] . 1. Reduce to the case T: eo ---+ X, I Te n I :::; n- 2 • This is not a semi-embedding because 1:. ';:'= 1 Ten E X. This can be found in Bourgain-Rosenthal [1983] . 2. For each t consider ( 2c) - 1 X [t - e , t+e ] and let x; E X* be a a( X*, X)-cluster point. Show that there are uncount­ ably many t's so that xi are far apart. This is a classical result of Gelfand. Modern generalizations can be found in Diestel-Uhl [1977] . 3. Take two sequences convergent to different limits. The desired sequence Ill. C.

330

Hints For The Exercises

consists of long stretches of one sequence separated by long stretches of the other. 4. If it is not so, build in H the unit vector basis in £ 1 . 5. Ob­ serve that spanT{Ll {fl, JL)) = spanT{L2 {fl, JL)) . 6. Simply a uniformly integrable sequence convergent in measure converges in norm. 7. One 2n .!. possible candidate is { ! E LI [O, 1] : L n ( J n 1 l l (t) i l + ;;1 dt) ( n + 1 ) $ 1 } . (n + 1 ) 8. Factor In = BnFn and show that a subsequence of Bn and Fn converge weakly. Consider ffn in H2 {Y) . This is due to Newman [1963] . 9. Modify the proof of Lemma 15. This is a result of James [1964] . 10. If £1 were finitely representable in X* then the � 's would be uniform quotients of X. But X has type p, p > 1 so by Exercise III.A. 17 � would also have type p. 11. Instead of characteristic functions use their smooth approximations. 12. Find closed, disjoint sets Fn C 1] and functions hn E H such that infn JFn l hn (t) ldt > 0. Find In E C O , 1] such that JFn l hn (t) ldt rv JFn hn (t) ln(t)dt, ll ln l l oo $ 1 and In I Fk = 0 for k < n. Put 'Pn = IJ�;:: (1 - I Ik l ) ln · This is due to Pelczyll.ski More general results are in III.D. 13. Use Lemma 10 and Proposition III.A.5.

[0,

[ [1962] .

III.D. 1. Write T E L(lp ) as T(x) = E : 1 l'{ (x)e i . For F E L(lp ) * define G E L(lp )* by G{T) = E: 1 F( Ti ) where Ti (x) = f'{ (x)e i . Show II G II = II F II and on compact operators G agrees with F. Show that II F II = II G II + II F - Gil · This is due to Hennefeld [1973] . For p = 2 see Alfsen-Effros [1972] . 2. For h* E H * define Eh* = JL I S where JL is any measure on T which extends h* to C(T). Use the def­ initions to check that it makes sense. 3. For K = [0, 1] the Faber­ Schauder system shows this. For the general case use the same ideas. 4. Show that dist(f, Irp(C[O, 1] )) = ! sup{ l l(s') - l{s") l : cp(s') = cp(s")} so C(�)/Irp(C[O, 1] ) � eo , with unit vectors corresponding to points s' -# s": cp(s') = cp(s"). If there is a projection we can lift these unit vectors to In E C { � ) . On the other hand since those points are dense in � we can find a subsequence such that I I E;= l In; I I 2: en . This was proved by M.l. Kadec. The proof is in Pelczynski §9. 5. One example is: K1 is a disjoint union of the interval [0,1] and the inter­ val [0,1] with circle attached at each end. K2 is the disjoint union of two intervals each with one circle attached at one of the ends. Check that it works. The details are in Cohen [1975] . 6. For (a) and (b) reduce to the case of selfadjoint operators and write (Ax, x ) explicitly. (c) follows from {b) . More details can be found in Kwapien-Pelczynski [1970] and Bennett [1977] . 7. If nk is very lacunary then you can analyse sgn ei n k B and conclude that (e i nk 9)�= l is in sup-norm equiv­ alent to the unit vector basis. As a model think about Rademacher

[1968]

331

Hints For The Exercises

functions. A more efficient way is to use Riesz products (see Exercise II.A.7) . 8. Assume II Tn ll = 1 and take p(x) such that p(xo) = IIPI I · Then T� (8x0 ) (p) --+ 1 so T� (8x0 ) --> 8x0 in w*-topology. Note also that the mass of T� ( 8x0 ) has to concentrate around xo . From this get the con­ vergence of Tn (f) for smooth f's. This is an improvement of the original Korovkin theorem (see Korovkin [1959] or Wulbert [1968] ) . 9. (a) Note that I;�= O (� ) (1 - x) n - k xk = 1 . (b) Use the Korovkin theorem (Exerx x2 cise 8) and compute that Bn (1) = 1 , Bn (x) = x, Bn (x 2 ) = x 2 + ( � ) . This is a modern version of S.N. Bernstein's proof of the Weierstrass approximation theorem. 10. Use the remark after III.A.12 to show that X8 rv £00 and that (X2)** = X8 • The fact that X2 rv Co is more involved. Analogously as in the proof of Theorem III.A. l l show that X2 is isomorphic to a complemented subspace of co. 11. If not then R: A(JBd ) --+ B 1 (JDd ). Like in III.A. l l we show that B1 (1Bd ) is iso­ morphic to a subspace of £1 . Thus (use DP and Pelczynski property) R is compact. But for the polynomials Pn (z) constructed in III.B.18 we have II RPn ll 2: c > 0 for all n. This contradicts the compact­ ness. 12. (a) The desired embedding of T::O into £� + 1 is given by p t--> (p(exp(27rk8/(4n+ 1))k� o (see II.E.9) . This is due to Marcinkiewicz [1937a] (see also Zygmund [1968] chapter X §7) . (b) Take small 8 and a maximal Jn separated subset of the unit sphere §d C <ed considered with the quasi-metric p((, ry) = 1 - l \ ( , ry ) l . The embedding into £00 is given via the point evaluations (see Wojtaszczyk [1986] ) . (c) Identify f� with L00 ( { 1 , . . . , N}, J.t ) where J.t ( {k}) = 11 for k = 1, 2, . . . , N. Take a maximal set (x1 ) j= 1 C E such that

k

II L i xj l ll oo :::; 1 + 8. j=l

Then span(xj) j= 1 = G. To estimate k show that if I { i: I:;= l l xj (i) l > 8} I :::; �n then III.B.9 implies that (xj ) j= 1 is not maximal. This is Corollary 6.2 of Figiel-Johnson [1980] . 13. Instead of Aj 's consider the basic splines bj , i.e. functions such that b1 is continuous and b1 l (s k , Sk + l ) is a quadratic polynomial for all k and b1 ( s1 ) = 1 and b1 is non-zero only in three of the intervals (s k , S k+ d · (The numbers sk are those defined in 20. ) Show that such b1 's exist and check their properties. Follow the proof of Proposition 21. 14. (a) Note that B c £1 [0, 1] . (b) Show that ll fn ii B :::; C(n + 1)- ! . To do this expand fn into the Haar series. The antiderivative of 2:::7= 1 Un, hi )hi is a piecewise linear function. Write it in terms of A1 's as defined in 20 and differentiate back. Conversely it is

332

Hints For The Exercises

enough to estimate the Franklin coefficients of a special atom. Estimate separately small coefficients ( n � - log 2 I IJ) and big ones. For details see Wojtaszczyk [1986a] . 15. The general strategy is similar to the proof of Theorem 27. The details and generalizations can be found in Ciesielski [1975] . 16. As a simple model show directly that the Haar system is not an unconditional basic sequence in L00 [0, 1] . Next use the same idea to show that derivatives of the Franklin system are not an unconditional basic sequence in L00 [0, 1] . 17. Use Proposition 21b) to show that N

f �--+ sup I L ( !, fn ) fn l N

n= O

is a weak type 1-1 map. This can be found in Ciesielski [1966] . 18. (a) We can follow the proof for the Franklin function or perform a direct and rather explicit calculation. (b) The orthonormality follows easily from the definitions. The completeness follows from the fact that continuous functions are dense in L 2 (R) . This and much more can be found in Stromberg [1983] . 19. Suppose Xn�O and Yn are bounded and such that ll xn ll = Xn ( Yn) · Find a weakly Cauchy subsequence Ynk and consider T: X --t c defined as x �--+ x(ynk ) . 20. Use the Dunford­ Pettis property. 21. One example is ( L:: := l .ey) 1 ; use Exercise II.E.8. 22. Modify implications (e)=?(d) and (d)=?(b) of Theorem 31. This requires the use of nets. The argument is in Bourgain [1984b] . 23. Use the Pelczyiiski property of £00 and Exercise III.B.7 to show that the existence of an operator that is not weakly compact would imply that £00 has a complemented subspace Y isomorphic to eo . This is impossible; see Exercise III.B.8(c) . 24. Use the Ascoli theorem. 25. Identify C 1 (11'2 ) with a subspace X of C(T2 , £�) by f �--+ (j, {h f, lhf) . Consider the annihilator X .L of X, X .L c M(T2 , £�) , where M(T2 , £�) is the space of measures with values in £� . Consider the space G C M(T2 , £�) of all measures J.L such that limn -+oo J fn dJ.L = 0 for all sequences fn E X such that if fn = (gn , 01 9n, fhgn) then 01 Yn and fhgn tend to zero pointwise on T2 • Show that G is complemented in M(T2 , £�) (use that it is a C(T2 ) module) . Show that G/X.L separable. Also show that the kernel of a projection onto G is isomorphic to M(Y) . The details can be found in Pelczynski [1989] . 26. The closedness follows from Lemma 6 like in the proof of Corollary 7. To show that it is an algebra, use III.B.20 to show that for a Lipschitz function cp E C(S) and h E Hoo (S) the function cp · h E H00 + C. This is from Rudin [1975] . 27. The proof of closedness is similar to the case n = 1 (see Corollary 7) . To show that H00 (11'n ) + C(r) is not an algebra for n > 1 take f E H00 (11')\A(11')

Hints For The Exercises

333

and show that zn f ( z 1 ) E L00 {T"" ) but is not in H00 {T"" ) + C{T"" ) . This is from Rudin [1975] . that one can assume they are both in £ 1 (11') . This implies that there is cp E H00 {11') such that both cpp, 1 � 0 and 'Pf.-£ 2 ;:::: 0. Since for every 'ljJ E H00 {T), J 'ljJcpdp, 1 = J 'ljJcpdp,2 we infer f.-£ 1 = f.-£2 · 3. If you have a set of extensions that is not relatively weakly compact {it is enough to assume these extensions are in £1 (11')) then use Theorem III.C.12 and Lemma III.C.20 (or Exercise III.C. l l ) to produce a eo-sequence showing that the original set of functionals was not relatively weakly compact. Use the methods of Theorem III.D.31 or Exercise III.C.l2. 4. (a) Use the Hardy inequality. {b) Easily follows from the fact that diagonal multiplication by 7.: in £2 is not 2-absolutely summing (see III.G. 12) . To find an elementary example look at lacunary series. 5. The isometry in both cases is given as [/ ] t--t { G.ij )i,j � O with G.ij = j( - (i +j)) . For f E L00 (T) consider the operator HJ (g) = P(f ·g) where g E H2 (T) and P is an orthogonal projection from £2 (11') onto H2 (11') . To evaluate II HJ II use the canonical factorization I.B.23. These are classical results of Nehari and Hartman (see Nikolskii[1980] ) . 6. (a) Use condition {b) of Theorem 4. These examples are due to Hayman and Newman and can be found in Hoffmann [1962] . For {b) take { A n );:-'= 1 such that {An } c D is such that {An } n 11' has positive measure but is not T. 7. (a) For F( z ) = � (z + � ) we have F{1Pr) = Er is an ellipse, so A( E) � A(D) . The function F induces an isometric embedding of A( E) into A(1Pr) and the image is !-complemented. {b) Consider the map z t--t e z from an appropriate strip onto 1Pr. Consider A{1Pr) as a space of functions on this strip. Use the ideas of the proof of Theorem 12. This still requires some effort. For details of (a) and {b) see Wolniewicz [1980] . (c) The exact computation of this norm is in Voskanjan [1973] . Consider a very thin annulus. 8. (a) Use the canonical factorization, Theorem I.B.23. {b) Show that J::_'lr (limr --+1 J::_'lr log l w'l'( r ei t) l dt)dcp = 0 and use (a) . (c) Use {b) to approximate inner functions by Blaschke products. Use Exercise III.B.9 to approximate an arbitrary function by the inner functions. {d) From (c) follows that functions f ( z ) = B ( r z ) with B a finite Blaschke product and r < 1 are dense in BA . Represent explicitly Ba ( r z ) = s�;;J) as a convex combination of Blaschke products. (e) Show that for a Mobius trans format ion p (z) = (z - A) (l - Xz) - 1 we have ll p (T) II :::; 1. Use ( d ) to show that it extends to any f E A. (b) is a classical result of Frostman (see Koosis [ 1980] IV.9 or Garnett [1981] ). (c) is even older, it goes back to Nevanlinna. ( d ) is a result of Fisher [ 1968] . ( e ) is a classical and important result of von Neumann [1951] . The proof we indicate here is from Drury [ 1983] and is close to the original. For more about this inequality see III.F . 15. A different

334

Hints For The Exercises

{t

{!

proof is indicated in Exercise III.H. 19. 9. (a) Put Max{!) = E T: 1/1 = 11/11 } . Show that E ImP: I Max{f) l = 0} is dense in E. Find a sequence (et , . . . en) C E, ( e i , . . . , e�) C E* , n = dim E such that Max(ej ) = O, j = 1, . . . , n, ll ei ll = ll ej ll = ll ej (ei ) ll = 1 and the matrix (ei {ej)) f.i = 1 is non-singular. Then ft = ei P, i = 1, . . . , n, o

A

span ImP* . (b) If the disc algebra is a 1r1-space then (a) and the F.­ M. Riesz theorem show that locally looks like l� but this is not the case. This is taken from Wojtaszczyk [1979a] (see also Exercise III.I. 12.). 10. (a) Use the u(L00 , L1 )-compactness of the closed ball in H00 • (b) Regularize using the Poisson kernel. (c) Use duality to find F E HP ('I') such that II F II = 1 and (21r) - 1 Jy F / = di s {f , H00 ) . This gives that for any best approximation g E Hoo to f we have f - g = 1�1 • (d) Take f = L:: :'= 1 Un'Pn where II 'Pn lloo = 1 and supp 'Pn C ( ( n� 1 ) ' �) C (-11", 11"] = T and an ---+ 0 slowly enough. All this is quite old. A nice presentation, references and much more can be found in Garnett [1981] .

t

1 . Work with the definitions and the Holder inequality. 2. Think of l1 as a span of Rademacher functions in L00 [0, 1] and l2 as a span of Rademacher functions in Lp [O, 1] , 1 ::; p < oo. 3. Consider the Haar system. This gives (a) and can be used to get the lower es­ timate in (b) . To get the upper estimate in (b) you can follow the proof of III.H.24. 4. Use the fact that ( v';+ 1 )�= 1 ¢ l2 . 5. Pietsch's theorem shows that T must map some L1 ( [0, 1] , JL) into C[0, 1] . What can be said about JL? 6. Use the Pietsch theorem and the fact that Lp [O, 1] is not equal to any L q [O, 1] . 7. (a) Use the factorization. (b) First note that every T E Ip (X, Y) is compact (use the Dunford­ Pettis property) . Next use the Pietsch theorem and arguments like in Lemma III.A.12 to show that T is a sum of absolutely convergent se­ ries in Np (X, Y). This is due to Persson [1969] . 8. Use Corollary 9. 9. (a) Since TIL (L1 (m) ) C L1 (m) , we see that TIL is 1-integral. Since fl('y) ---+ 0, TIL is compact (look at L2 (G)). If TIL is 1-nuclear, then the definition yields (look also at T; ) that TIL (f) (x) = fa K(x, y)f(y)dm(y) for some K E L1 ( G x G, m x m) , but this is impossible for singular JL · (b) Use Proposition 12. Show also that translation invariant, nuclear operator on C('l') is a limit in the nuclear norm of operators of convolu­ tion with a polynomial. Compare with III.G. 18. 10. Take (xj)j= 1 c lf such that L::;= 1 1 x* (xi ) l ::; C ll x* ll and apply Theorem 14 to the matrix k (xj ( i )) i,i = 1 • 1 1. Estimate ( L:: := 1 e - i 2 9 ) f(O)d0 from above using the Holder inequality. The Khintchine inequality will yield the esti­ mate from below for 11/ll oo· The details are in Bourgain [1987] . 12. III.F.

J1r

335

Hints For The Exercises

(a) Note that if x = I:�= l anXn then I:�= 1 Jan J JJTxn J I � CJJx J J . (b) id: C[O, 1] --+ £1 [0 , 1] does not factor through £1 . (c) Let X be ei­ ther L: or CF . Fix x E X, � E X* and (cn)n E F , En = ±1. Define A: Lf --+ X by A ( I: n E F aneinO) = I: n E F ani:(n)en einO and B: X --+ Lf by B(x) = 'L: n E F �(ein°)i:(n)ei nO . Show that 1r1 (A* ) � CJJxJJ and 1r1 (B) � CJJ�JJ and next JtrBAJ � 1r1 (A* )7r1 (B) . This is a weak form of a result of Pisier [1978] . 13. Show that if idF is p-integral then the orthogonal projection from Lp(lr) onto L: is bounded. Then use II.D.9.

III.G. 1 . Use Proposition 4 in one direction. For the other use the Schmidt decomposition. 2. For selfadjoint A1 , A2 , . . . , An E a 1 (f2 ) such that I:7= 1 a1 (Aj ) 2 = 1 take selfadjoint Bb · · · , Bn E L (£2 ) such that I:7= 1 JJBi J I 2 = 1 = 2::7= 1 tr(Ai Bi ) · For k = 2k 1 +2 k2 + · . · +2 k; with o � k1 < k2 < · · · < kj � n we define (t) = I: k Bk 1 Bk; rk1 (t) . . . rk; (t) . 1 Estimate JJ ( t ) JJ and evaluate J0 tr ( (t) I:7= l rj (t)Aj )dt. This is taken from Tomczak-Jaegermann [1974] . 3. (a) is obvious. For (b) compare the ap-norm of an n x n unitary matrix all of whose entries have absolute value Jn with the ap-norm of the matrix all of whose entries are Jn · (c) follows from Exercise III.A.8 and (b) . 4. Use 3(a) and gliding hump arguments. Note that if Uj E a00 (£2 ) are such that PA; UjPA; = Uj =/= 0 for some disjoint sets Ai C N, j = 1, 2, . . . then span(uj ) �1 "' c0 and is complemented. On the other hand if PB uj PA; = Uj =/= 0 for some finite set B C N and disjoint Aj C N then span( Uj ) �1 "' £2 and is complemented. This is due to Holub [1973] . More results of this type are in Arazy-Lindenstrauss [1975] . 5. Use the projections PA of Exercise 3 and the decomposition method II.B.23. 6. Consider the diagonal operators. If A 1 (£00 , £ I ) had an equivalent norm, then every diagonal operator would be in A 1 (£00 , f I ) . To see that this is not so, show that for idn: � --+ if we have ak (idn) 2: (n - k + 1) - 1 for k = 1 , 2, . . . , n. 7. Write An = O:n · f3n with (o:n) E £2 and (f3n) E co . Define T(x, y) = (o:(y) , f3(x)) where o: (y) = ( o:; yn );;:-:'= 1 and f3(x) = (f3;xn);;:"= I · This example is due to Kaiser-Retherford [1984] . 8. If K(x, y) is square integrable this follows from Proposition 13 or Theo­ rem 19. The general operator differs from this case by one dimension. 9. Show that TK admits a factorization Lq ' �Loo .!Lq ' where f3 is an oper­ ator of multiplication by a function. Use Theorem 19. 10. (a) For the lower estimate take ej ®e i and see that I: i , j Jx* (ej ® e i ) J = I: i ,j Jx* (i, j) J where x* is really an operator on f2 with n1 ( x* ) � 1. Use Exer­ cise 12. For the upper estimate note that a2 (u) � cn ; J (u(ut), vt) Jdt where Ut = n ! (r i (t) , . . . , rn (t)) and Vt = n ! (rn + I (t) , . . . , r2n(t)) where •

•

•

J

336

Hints For The Exercises

(rj (t)) J!! 1 are Rademacher functions. (b) Let U be an n x n unitary matrix with lu (i, j) l = Jn and let It have diagonal r1 (t) , r2 (t) , . . . , rn (t) , zeros otherwise and let Jt have diagonal rn + l ( t) , . . . , r2n ( t) , zeros oth­ erwise. Show that a2 (u) :5 3yn J; ltr(ultUJt ) l dt. c) Passing to the adjoint note that /'l (Jn) = n00 (id: a2 (£�) ---+ a00 (£�)). Use the trace duality III.F.24. (d) Glue together the finite dimensional operators from (c) . (e) This is almost the same as Exercise III.F.12a. (f) Use (e) and (c) . All this is taken from Gordon-Lewis [1974] . 11. Use III.G (13) . 12. Consider one-dimensional operators. 13. (a) Fac­ torize T as C(K) � L2 (J.L) � Ll (J.L) �.e2 and write a id: L2 (J.L) ---+ £2 as L:: :'= l an with an of finite rank and a2 (an) :5 2- n . Show that n 1 (an id) = i1 (an id) :5 2- n . (b) Consider the factorization C(S)�C 1 ('1'2 ) �Wf (T2 )...i.. L 2 ('1'2 ) where i is the identity and j is also the identity (see I.B.31) . Show that it is not nuclear and use (a) or show that it fails (c) . (c) Given V9 C L 1 (J.L) and T: L1 (J.L) ---+ L2 (v) consider M9 : L00 (J.L) ---+ Ll (J.L) , M9 (!) = g · f and use the fact that TM9 is nuclear by (a) . 14. (a) First do it for .e� . For general E factor id: E ---+ E as E�£� L E with 1r2 (a) = 1 and 1 1.811 = 1r2 (id) . But then ,Ba = idt� . This is a result of Garling-Gordan [1971] but the proof indicated is due to Kwapien. It can be found in Pisier [1986] . (b) Use (a) , (c) and (d) . Use also III.F.8. 15. Assume that one norm is given by the usual scalar product on Rn (or CC") and show that the other can be chosen to be (x, y ) 2 = L::j= l ajxj'jjj with aj > 0. Take X1 spanned by an appropriate block basis. 16. (a) Take x to be an extreme point in BE . (b) Identify £� with L 2 (N, J.L) where N = { 1 , 2, . . . , N} and J.L is a probability counting measure. On E we have two Hilbertian norms, from L 2 (N, J.L) and the one given by the distance. Use (a) and Exercise 15. 17. (a) , (b) Use Exercise 16b. 1. Use the well known fact (see Katznelson [1968] , Zygmund [1968] etc.) that M f is of weak type 1-1 but not continuous on Ll [O, 1] . 2. It is enough to work with finite dimensional spaces. Apply III.F .33 and dualize. This works fine for q > 1 . The case q = 1 requires more care. This shows that Proposition 15 is actually an equivalence. Like Proposition 15 this is from Maurey [1974] . 3. If i* is p' summing use Pietsch theorem and dualize. This shows that i factors through Lp . Use Proposition 10 to show that this is impossible. This was observed by Kwapien [1970] . 4. Use III.F.29 and Corollary 1 1 . 5. and 6. Use Propo­ sition 5. These can be found in Maurey [1974] . 7. For p < 1 there exist positive stable random variables (where stable is understood in a more III.H.

Hints For The Exercises

337

general sense than in III.A. 14) for which III.A.16 holds. The construc­ tion can be found in Feller [1971) or Lukacs [1970] or in other books on probability theory. 8. Modify the proof of Proposition 10. The details are in Pisier [1986a) . 9. Use Proposition 16 and Exercise III.A.15. 10. Show that if (ai )� 1 are numbers such that Cn = E �= l ai converges then E�= l (*) ai - 0 as n - oo. Apply this and the Menchoff-Rademacher theorem to an = n - 1 fn (w) . This is due to Banach [1919] . The argu­ ment indicated here was shown to me by Mr Wojciechowski. 11. If E;:"= 1 I fn i < oo a.e. then the map (�n)�= l 1---+ E;:"= 1 �nfn is a contin­ uous operator from £00 into Lo [O, 1] . Use Corollary 16. 12. If every f' E N then the map f �---+ f' (ei8) would be well defined into L0 (T) , so would admit a factorization through Hp (T) , p < 1 (use Corollary 7) . This is impossible. 13. Note that a system Un)n �l is a system of convergence in measure if and only if there exists T: £2 - £0 [0, 1] , a continuous, linear operator such that fn = T ( en ) · Use Proposition 5, Corollary 1 1 and the dilation theorem 19. This is due to Nikishin [1970] . 14. Use Proposition 5. 15. If so then the double Riesz projec­ oo n n n tion " L.. n , m � O anm ei B eimcp would be of weak L.. n+, m= - oo anm ei B ei cp 1---+ " type (1-1) (use Corollary 7) . The Marcinkiewicz theorem (see I.B.7) gives a bound on the norm in Lp (ll'2 ) , p > 1 , which is false. 16. Use Corollary 1 1 and show using the structure of the ,system (more precisely the ergodicity of measure preserving maps t �---+ nt) that the multipli­ cation operator equals the identity. Compute the Fourier coefficients of sgn sin x. This yields the coefficients of sgn sin nx. Use it to estimate the L2 norm of E := l sgn sin nx. 17. Use Corollary 16. 18. Use Corollary 16. 19. Check the von Neumann inequality for unitary maps. 20. (a) Use Theorem III.C.16 and Lemma III.C.15. (b) Put V = (Y E9 Z) t /H where H = { (x, -Tx) : x E X } . (c) If T: X --t £2 use (b) to extend - V and use (a) to show that V* has some type > 1 . T to T1 : Use Proposition 1 4 and Lemma III.F.37. (d) Use similar arguments to III.A.25. An analogue of Lemma III.A.26 follows from (c) and the Pietsch factorization theorem. 21. This is like Theorem 30. For (b) ob­ serve that Bp (D)* = Br (D) where � + � = 1 , 2 :5 p < oo (this follows from the boundedness of the Bergman projection in Lp (D) , 1 < p < oo ; see Exercise III.A. 10) . We use Proposition 29 twice.

C(S)

1. By III.F.36 it is enough to show 1r2 (T* ) < oo , so by dual­ 111.1. ity (see III.F.27 and III.F.25) we have to show that aT* is nuclear for summing multiplier. Factorize a and use a: £2 --t Ltf Ht . a 2-absolutely 1 2 the fact that ( E ::'= o !tn l ) 2 :5 !I T ! ! . This is due to Kwapien-Pelczyiiski [1978] . 2. (d) => (c) and (b) => (c) are obvious while (c)=> (a) follows from

338

Hints For The Exercises

A

III.F.35. For (a)=> (d) take a �---+ fa. E with fa. (t) = E;= O eiit eii a. , apply III.F.33 and use the Kolmogorov theorem I.B.20. For (d)=>(b) note that T extends to L1 (T) , so use Exercise III.G. 13(a) . This is due to Kwapien-Pelczynski [1978] . 3. The proof is based on duality the­ ory and clever diagram chasing. This is due to Maurey [1972] (see also Pelczynski [1977] ) . 4. (c)=> (b) follows from elementary properties of 2-absolutely summing maps. (b)(c) follow from the Grothendieck theorem III.F.29. 5. Use Exercise III.A.12 and Theorem III.D.31 to show that such a subspace embeds into some Lp for p < 1. 6. (a) Show by averaging that if there is an extension, then the invariant extension is bounded. This is false. (b) By III.C.l8 and III.C.l6 the space X contains l1 almost isometrically, so comple­ mented, even in L1 (T) . Use the finite dimensional version of (a) . (c) Is almost the same as (b) but one has to use different theorems. This is noted in Bourgain [1986] to show that the reflexivity assumption in 111.1.25 and III.I.Ex.8 is needed. 7. Use Remark III.E.13 and observe that all arguments in the proof of Theorem 25 are local. 8. Follow the proof of Theorem 25. One does not need Proposition 22. This is due to Bourgain [1986] . 9. (a) the 'if' part is easy. For the 'only if' part use Exercise 7 and follow the ideas of sections 26 and 27 with L1 / H1 replaced by L1 (T, LI/H1 ) · For (b) use (a) . The details are in Bourgain [1986] . (c) Like in Exercise III.D.l2 show that span{z � 1 z�_,.N } k 1 :5K is uniformly a subspace of f� with s = {lOOK N) N . Next show that if X c f� and T: X --+ £1 then 1r2 (T) ::; CJlOgS II T II · This uses III.F.37 and the estimate Cp ::; cy'P for the constant cp appearing in III.F.37. This estimate follows from the estimate given without proof in Remark III.A.20. This is due to Kislyakov [1981] . 10. (a) Dualize and use the F.-M. Riesz theorem and Paley's projection. (b) Take Xp , the closure of X in Lp (T) , p < 1 , and show that Xp = span{ e in 9 } n E A E9 Hp (T) . (c) Average and note that X/ "' £2 but span{ e in 9 } n E A C C(T) is isomor­ phic to £1 . This can be found in Kislyakov [1981a] . 11. Use III.H.13 and Exercise III.A.2 to show that R is isomorphic to a subspace Y of Hp (T) , p > 1 , such that for y E Y we have II Y II v ::; c ll y l h · 12. If such a sequence exists then sup d(Jm Pn , f�) < oo. This would imply that !-absolutely summing operators on behave like !-absolutely sum­ operators on C(T). (see also Exercise III.E.9b) . 13. Show that ming There are many other Theorem 12(a) fails for id: --+ H1 ways. The result was first proved in Henkin [1967a] . 14. The idea is to observe that Proposition 6 implies that if T: --+ £2 is a !-absolutely summing operator onto then T* ( £ ) is 'essentially contained' in LI/H1 . On the other hand in we have a continuum of complex lines La. such • • •

A

A(D2 )

([2

2

A(D)

(D2 ). A(D)

Hints For The Exercises

339

that � n La are disjoint, so we have a continuum of !-absolutely sum­ ming operators A(JB2 ) -+ £2 defined by Ta(f) = P( ! I S2 n La) where P is a Paley operator. Also 82 n L a are disjoint peak sets so T� (£2 ) tend to behave like an £ 1 sum. This is too much to fit into Ld H1 . The details (quite complicated) are in Mitiagin-Pelczyiiski [1975] .

List of symbols General symbols

8x 8n, m

IAI

] (n) [x]

XA

u

v

rn ( t)

_

the Dirac measure at the point x the Kronecker symbol; 1 if n = m, 0 otherwise the absolute value of a number; the cardinality of a finite set; otherwise the Lebesgue measure the n-th Fourier coefficient of a function f integer part of a real number x indicator function of a set A the normalized Lebesgue measure on Sd when it denotes a measure on JBd it is the normalized Lebesgue measure Rademacher functions, I.B.8. Sets

open unit (euclidean) ball in (Cd complex plane unit disc in