Tensor Norms and Operator Ideals

TENSOR NORMS AND OPERATOR IDEALS NORTH-HOLLAND MATHEMATICS STUDIES 176 (Continuation of t h e Notas de Matematica) E...

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TENSOR NORMS AND OPERATOR IDEALS

NORTH-HOLLAND MATHEMATICS STUDIES 176 (Continuation of t h e Notas de Matematica)

Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester N e w York, U.S.A.

NORTH-HOLLAND -AMSTERDAM

LONDON

NEW YORK

TOKYO

TENSOR NORMS AND OPERATOR IDEALS

Andreas DEFANT Fachbereich Mathematik Universitat Oldenburg Oldenburg, Germany

Klaus FLORET Fachbereich Mathematik Universitat Oldenburg Oldenburg, Germany and I ME CC/Un icamp Campinads.F, Brasil

1993 NORTH-HOLLAND -AMSTERDAM

LONDON

NEW YORK

TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211,1000 AE Amsterdam, The Netherlands

ISBN: 0 444 89091 2

0 1993 ELSEVIER SCIENCE PUBLISHERS B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U S A . - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands

Contents

V

Contents

Introduction

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

Chapter I: Basic Concepts . . . . . . . . . . . . . . . . . . . . . .

1

7

1. Bilinear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Continuous bilinear mappings, 1.5. non-validity of Hahn-Banach theorem, 1.7. non-validity of open-mapping theorem, 1.8. canonical extension to the bidual.

7

2. The Algebraic Theory of Tensor Products . . . . . . . . . . . . . . 2.2. Universal property and construction of tensor products, 2.4. examples, 2.5. trace, 2.6. trace duality, 2.7. tensor product of operators.

15

3. The Projective Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Minkowski gauge functional, 3.2. basic properties, 3.3. Bochner i n t e grable functions, theorem of Dunford-Pettis, 3.4. compact sets, 3.6. nuclear operators, 3.7. trace, 3.9. A does not respect subspaces, 3.10. extension property, 3.11. Grothendieck’s characterization of L1, 3.12. lifting problems, 3.13. t;-spaces, Ex 3.24. Radon-Nikodfm theorem for operator valued measures.

26

4. The Injective Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Basic properties, 4.2. examples, 4.3. E does not respect quotients, 4.5. lifting of vector-valued continuous functions, compact extension property, 4.6. integral bilinear forms, Ex 4.3. Fourier matrices.

46

5. The Approximation Property . . . . . . . . . . . . . . . . . . . . . . 5.2. Survey about counterexamples, 5.3. compact operators, 5.4. characterization with nuclear operators and the trace, 5.5. injectivity of completions of tensor products of operators, 5.6. operators with nuclear dual, Ex 5.17. compactb approximable operators.

58

6. Duality of the Projective and Injective Norm . . . . . . . . . . . . 6.3. Dense sequences of finite dimensional Banach spaces, Johnson spaces C,,6.4. embedding theorem, 6.5. weak principle of local reflexivity, 6.6. principle of local reflexivity, 6.7. extension lemma for integral bilinear forms, Ex 6.4. Lindenstrauss’ compactness argument.

70

vi

Contents

7. The Natural Norm on the p-Integrable Functions . . . . . . . . . 7.1. Bochner pintegrable functions, A,, 7.2. continuous triangle inequality, 7.3. positive operators, density lemma, 7.4. A, respects subspaces and quotients, quotient lemma, 7.5. Fourier transform, 7.6. Hilbert transform, 7.7. type and cotype, 7.9. a Beckner-like result, Ex 7.1. averaging operator in L,.

77

8. Absolutely and Weakly p-Summable Series and Averaging Tech-

90

9. Operator Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Quasinorms, 9.4. criterion, 9.6. examples, 9.7. injective ideals and the injective hull, 9.8. surjective ideals and the surjective hull, 9.9. dual ideals, 9.10. composition ideals, Ex 9.8. space ideals, Ex 9.13. quasinuclear operators, Ex 9.16. K-convex operators.

108

10. Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Characterization with the trace, 10.4. examples, 10.5. factorization, 10.7. characterization with T @ id^, 10.8. and 10.9. summability of the diagonal of infinite matrices, Ex 10.4. extension and lifting properties of integral operators.

118

11. Absolutely p-Summing Operators . . . . . . . . . . . . . . . . . . . 11.1. Basic characterizations, 11.2. positive operators, 11.3. GrothendieckPietsch domination theorem and factorization, 11.4. Dvoretzky-Rogers theorem, 11.5. composition, 11.6. Hilbert-Schmidt operators, 11.7. Pietsch lemma, 11.8. Kwapieii’s test, 11.9. absolutely 2-summing norm of i d E , 11.10. absolutely p-summing norm of the identity of finite dimensional Hilbert spaces, 11.11. little Grothendieck theorem, 11.12. operators with absolutely 2-summing duals and a characterization of Hilbert spaces, Ex 11.10. extension property of absolutely 2-summing operators, Ex 11.13. the ideal of Hilbert-Schmidt operators, Ex 11.16. Banach-Mazur distances between finite dimensional Banach spaces and the Kadec-Snobar result about projections, Ex 11.18. factorization of Hilbert-Schmidt operators.

127

niques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Absolutely p u m m a b l e and weakly p-summable sequences, 8.2. representations of operators on or into l,, 8.3. unconditional summability, 8.4. general scheme of averaging, 8.5. Rademacher functions, Khintchine inequality, 8.6. type and cotype of L,, 8.7. Gauss functions, 8.9. Orlicz property, Ex 8.9. Rademacher versus Gauss averaging, Ex 8.12. absolutely ( T , s)summing operators.

vii

Contents

Chapter 11: Tensor Norms. . . . . . . . . . . . . . . . . . . . . . . .

146

12. Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . 12.1. Reasonable norms and the metric mapping property, 12.2. criterion, 12.4. finite and cofinite hull, 12.5. Laprestb’s tensor norms 12.6. com12.8. the diagram of Laprestk’s tensor norms, pletion with respect to 12.9. tensor product representation of weakly psummable sequences, Ex 12.7. tensors of finite rank.

146

13. The Five Basic Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 13.1. Approximation lemma, 13.2. extension lemma, 13.3. embedding lemma, 13.4. density lemma, 13.5. 2,p-local technique lemma.

159

14. Grothendieck’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . 14.1. Idea of proof, 14.4. proof of Grothendieck’s inequality in tensor form, 14.5. matrix form, 14.7. estimates for the Grothendieck constant K G , Ex 14.1. the original proof - more or less.

166

15. Dual Tensor Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1. Trace duality, 15.2. dual norms, 15.5. duality theorem, 15.6. accessibility of tensor norms, 15.7. conditions for the good behaviour of duality, 15.9. tensor norms and their duals on “symmetric” finite dimensional spaces, 15.10. duality of A,, the Chevet-Persson-Saphar inequalities, 15.11. tensor norms closest to Ap, 15.12. another proof of the Beckner result, Ex 15.10. weakly conditionally compact subsets of tensor products.

177

16. The Bounded Approximation Property . . . . . . . . . . . . . . . 16.1. Topologies on 2 ( E , F), 16.2. characterization with the cofinite hull f of the projective norm, 16.4. results involving the Radon-Nikodjm property, 16.5. and 16.6. duality of 6 and T , 16.7. duality of the operator ideals ff, %, and 2, 16.8. non-nuclear operators with nuclear dual, Ex 16.4. - Ex 16.8. reflexivity of 2 ( E , F) for special spaces.

190

17. The Representation Theorem for Maximal Operator Ideals

...

17.2. Maximal operator ideals, 17.3. tensor norms and operator ideals which are associated with each other, 17.4. right-tensor norms and a general way of constructing maximal operator ideals, 17.5. the representation theorem, 17.6. the embedding theorem, 17.7. the transfer argument, 17.8. the dual ideal, 17.9. the adjoint ideal, 17.10. - 17.13. examples, 17.14. Grothendieck’s theorem, 17.15. and 17.16. characterization with tensor product operators T 8 id^, 17.19. ideal norms of identity operators in symmetric finite dimensional spaces, 17.20. injective embedding of E& F into the space of operators, 17.21. unit ball of % ( E ,F), Ex 17.16. continuity of S 8 2 ” ‘ , Ex 17.17. density lemma for maximal normed operator ideals.

200

viii

Con tents

18. ( p , q)-Factorable Operators . . . . . . . . . . . . . . . . . . . . . . . 223 18.2. The norm of the integrating functional, 18.4. ultraproducts, 18.5. factorization through positive functionals, 18.6. p-factorable operators, 18.7. p-integral operators, 18.9. Maurey’s factorization theorem, 18.11. the factorization theorem, Ex 18.4. - Ex 18.11. properties of ultraproducts. 19. ( p ,q)-Dominated Operators . . . . . . . . . . . . . . . . . . . . . . . 241 19.2. The basic estimates, 19.3. Kwapieli’s factorization theorem, 19.4. composition of factorable and dominated operators, 19.6. Grothendieck inequality for C*-algebras. 20. Projective and Injective Tensor Norms . . . . . . . . . . . . . . . . 20.3. Duality relations, 20.4. examples, 20.6. the projective associate, 20.7. the injective associate, 20.9. finite dimensional characterization, 20.10. general rules for associates, 20.11. - 20.13. relations with operator ideals, 20.14, results about gp, 20.15. table for w ~ , d 2 , g 2and their adjoints, 20.17. Grothendieck’s inequality in its original formulation (operator version), finite dimensional Grothendieck constants, 20.18. Banach spaces satisfying Grothendieck’s theorem, 20.19. a result of Saphar and the best constants in the little Grothendieck theorem.

250

21. Accessible Tensor Norms and Operator Ideals . . . . . . . . . . . 275 21.2. Accessible operator ideals, 21.4. total accessibility of certain composition ideals, 21.5. accessibility of Laprestk’s tensor norms, 21.6. a result about the bounded approximation property, 2 1.7. the a-approximation properties, Reinov’s results in the case Q = gp, 21.11. some results of Kisljakov and Bourgain-Reinov, H w , Ex 21.3. principle of local reflexivity for operator ideals. 22. Minimal Operator Ideals . . . . . . . . . . . . . . . . . . . . . . . . 22.1. The minimal kernel of an operator ideal, 22.2. the representation theorem for minimal operator ideals, 22.3. examples, 22.6. the dual of amin(E, F), 22.7. weak-*-continuous operators in Qmin(F‘, E), 22.8. the dual ideal of the minimal kernel, 22.9. a counterexample, Ex 22.7. extension and lifting properties of minimal operator ideals.

287

23. l!;-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1. Local techniques, 23.2. various characterizations, 23.3. relations with the Cp-spaces of Lindenstrauss-Peiczyriski, 23.4. dual characterization, 23.5. 25-and Ci-spaces, 23.6. the projection constant, 23.7. quasinuclear operators, 23.8. characterization of space ideals with integral operators, 23.9. coincidence of absolutely l-summing and nuclear operators, Hardy spaces, 23.10. Grothendieck’s theorem for l!;-spaces, a characterization of Hilbert spaces, Ex 23.8. the extension norm of an operator.

300

ix

Contents

24. Stable Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 24.1. The linear dimension of e, and L,, 24.2. positive definite functions and Bochner’s theorem, 24.3. moments of stable measures, 24.4. LCvy’s theorem about the embedding of e, into L, (LCvy embeddings), 24.5. embeddings into L,, 24.6. - 24.7. results due to Saphar, Kwapied and Maurey about absolutely q-summing operators with values in t, or spaces with cotype, 24.8. stable type and Rademacher type, Ex 24.1. Schur product, Ex 24.5. stable type. 25. Composition of Accessible Operator Ideals . . . . . . . . . . . . . 327 25.1. Representation of the minimal kernel of accessible operator ideals, 25.4. cyclic composition theorem, 25.5. Persson-Pietsch multiplication table, 25.6. quotient ideals, 25.7. quotient formula, 25.8. the adjoint of composition ideals, 25.9. the regular hull and characterizations of the associates of ‘cp,q and 2,,in particular, of the associated space ideals - results of Kwapieri, 25.10. isomorphic characterization of subspaces, quotients, etc. of L,, 25.11. the minimal kernel of the injective resp. surjective hull of an operator ideal. 26. More About L, and Hilbert Spaces . . . . . . . . . . . . . . . . . . 26.1. Inequalities about a,,( coming from the Khintchine and Grothendieck inequalities, 26.2. factorization through Hilbert spaces of the identity mapping + 26.3. continuity of operators between spaces of Bochner pintegrable functions, complexification of operators, 26.5. Kwapied’s result about the factorization of operators L, + L, through L p , 26.6. tensor norms and ideals on Hilbert spaces, 26.7. the Hilbert-Schmidt tensor norm u,26.8. Schatten’s result about self-adjoint, symmetric extensions of u to Banach spaces, 26.10. limit orders of tensor norms, Puhl’s result, 26.11. unEx 26.6. unconditionally summable sequences conditional bases in e&,&, in L p ( p 8 v).

344

27. Grothendieck’s Fourteen Natural Norms . . . . . . . 27.2. Grothendieck’s diagram, 27.3. the original notations.

361

C,

........

Chapter 111: Special Topics . . . . . . . . . . . . . . . . . . . . . . .

365

28. More Tensor Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 28.1. Three new classes of tensor norms, 28.3. description of the projective associate of 28.4. characterization of operators in ‘ ; 2 28.5. a characterization of operators factoring through a Hilbert space, 28.6. - 28.8. description of the composition ideals 2, o Zq and its adjoints, 28.9. table of results, Ex 28.14. complexification of operators.

X

Contents

29. The Calculus of Traced Tensor Norms . . . . . . . . . . . . . . . . 378 29.1. The tensor contraction, 29.4. the associated operator ideal of a traced tensor norm, 29.5. characterization of p-dominated operators, Tp-spaces, 29.6. the calculus of traced tensor norms, 29.7. the tensor product of two tensor norms and of maximal normed operator ideals, 29.8. properties of 4 %, 29.9. - 29.11. tensor products of special tensor norms, Ex 29.8. ultrastable ideals. 30. The Vector Valued Fourier Transform . . . . . . . . . . . . . . . . 30.1. Fourier operators, 30.3. their characterization, 30.5. Rademacher and Gauss type and cotype, Kwapieh’s type/cotype theorem, characterization of Hilbert spaces, 30.6. the main theorem, 30.8. type and cotype with respect to orthonormal bases.

394

31. Pisier’s Factorization Theorem . . . . . . . . . . . . . . . . . . . . . 407 31.1. K-convex operators, 31.2. duality of type and cotype, 31.4. Pisier’s factorization theorem, 31.5. factorization of compactly approximable operators through Hilbert spaces, 31.6. non-accessible tensor norms/operator ideals, 31.7. “abstract” proof of Grothendieck’s inequality.

32. Mixing Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 32.2. Reformulation of former results, 32.3. tensor product characterization, continuity of tensor product operators between spaces of Bochner p-integrable functions, 32.4. a domination theorem, 32.5. Maurey-Pisier extrapolation theorem, 32.6. a characterization of 32.7. Maurey’s splitting theorem, 32.10. and 32.11. relation with absolutely ( r ,s)-summing operators, 32.12. a finite dimensional result.

ME’,

33. The Radon-Nikodfm Property for Tensor Norms and Reflexivity 33.1. Duality of E and ?r revisited, 33.3. Lewis’ theorem, 33.4. permanence properties, 33.5. LaprestC’s tensor norms, 33.6. coincidence of pnuclear and p-integral operators, 33.7. p-strong operators, 33.8. - 33.11. reflexivity of tensor products and components of operator ideals.

430

34. Tensorstable Operator Ideals . . . . . . . . . . . . . . . . . . . . . . 445 34.1. P-tensorstable and metrically P-tensorstable operator ideals, 34.2. strongly P-tensorstable operator ideais, 34.3. examples, 34.4. permanence properties, 34.5. factorization arguments, 34.6. projection constant of the injective tensor product, 34.7.stability of space ideals, 34.8. double Khintchine inequality, stability of tensor products of Hilbert spaces, 34.9. 6- and s-stability of (p, q)-factorable operators and their relatives, 34.10. distribution of eigenvalues, Pietsch’s tensor product trick, 34.11. a result of Kwapieli unifying Orlicz’s, Littlewood’s and Grothendieck’s inequalities, 34.12. improving weak inequalities with tensor products, mixing operators tl -+ tp.

Contents

xi

35. Tensor Norm Techniques for Locally Convex Spaces . . . . . . . 469 35.2. Tensor norm topologies, 35.3. traced tensor norms, 35.4. locally convex space ideals, 35.6. injective and projective tensor norms on locally convex spaces, 35.7. tensor product of direct sums, 35.8. lifting of bounded sets, property (BB), 35.9. probkme des topologies, Taskinen’s counterexample, 35.10. injective tensor product of (DF)-spaces.

Appendices: A. Some Structural Properties of Banach Spaces . . . . . . . . . . . A l . Subspaces and quotients, A2. dual systems, A3. lemma of Ky Fan, A4. bases, A5. Banach algebras, A6. lattices, A7. abstract L,-spaces and their representation.

489

B. Integration Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .

493

Extension procedure and the basic theorems: B1. Daniel1 functionals, B2. the convergence theorems, B3. measurable functions, the fundamental theorem of the Daniell-Stone integration theory, B4. product measures; The Lp-spaces: B5. Holder and continuous triangle inequality, B6. duality, Radon-Nikodim theorem, Segal’s localization theorem, Lebesgue decomposition, B7. strictly localizable measures; Borel-Radon measures and Riesz representation theorem: B8. 7-continuity and Bourbaki’s extension procedure, Kolzow ’s theorem, B9. representation of Borel-Radon measures, B10. L1 is complemented in its bidual; Bochner integration: B11. measurability of Banach space valued functions, B12. .C,(p, E), B14. Pettis integrability, B15. variation lemma. C. Representable Operators . . . . . . . . . . . . . . . . . . . . . . . . 508 Grothendieck’s characterization: C1. Riesz-densities, C3. nuclear operators, C4. factorization through e,; The Dunford-Pettis theorem: C6. a general result about representability, C7. strong version of the DunfordPettis theorem. D. The Radon-NikodQm Property . . . . . . . . . . . . . . . . . . . . 517 Basic properties and examples: D1. reduction to the Lebesgue measure, D3. examples, D4. dual of Lp(p, E); Pietsch integral operators: D5. and D6. relation with other operator ideals; The Radon-Nikod9m property and operator ideals: D7. and D8. characterizations in terms of (Pietsch) integraknuclear, D9. integral operators which are not Pietsch integral. Bibliography

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527

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

545

List of Symbols Index

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555

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Introduction

1

Introduction

1. Grothendieck’s “Rhum6 de la thkorie mktrique des produits tensoriels topologiques”, submitted in 1954 and published in 1956 in the Bulletin of the Mathematical Society of SZo Paulo, is one of very few papers which have deeply influenced the course of Functional Analysis: It not only demonstrated the enormous possibilities for the using of tensor products in Banach space theory, but it anticipated the study of Banach spaces in terms of finite dimensional subspaces - the secalled “local” theory, which has 80 much enriched our understanding of Banach spaces. The “R&umk” (as this paper is nowadays called) contained no proofs (with the exception of the main theorem) and apparently few people at the time understood it or even noted its importance. This, in spite of the fact that Grothendieck’s theory of locally convex tensor products and nuclear spaces (also poorly understood when it first appeared) had already won great appreciation. There was simply some reluctance in the Functional Analysis community to accepting the idea of thinking in terms of tensor products. It was only in 1968 that the “R6sumk” received more attention through the famous Studia Mathematica paper of Lindenstrauss and Pelczyliski [173] “Absolutely summing operators in &,-spaces and applications” , which presented Grothendieck’s deep “thkor8me fondamental de la thbrie mktrique des produits tensoriels” (the main result of the “ f i u m k ” ) as an inequality concerning n x n matrices and Hilbert spaces. Without using tensor products, various fascinating applications were given, mainly dealing with the class of absolutely psumming operators. Banach space theory (which in the mid-sixties had been considered to be nearly complete by some people) was reactivated in an incredible way - and many of its important results nowadays trace their roots back to the “Risum6”. It is astonishing to see that many (certainly not all) of the ideas of Banach space theory in the last 20 years are already contained in Grothendieck’s paper, although sometimes in a hidden way. The phrase “this result is implicitly contained in the Risumk” became almost a clichC for a certain time, but was nevertheless quite often true. 2. Tensor products apparently appeared in Functional Analysis for the first time during the late thirties in the work of Murray and John von Neumann on Hilbert spaces. The first systematic study of classes of norms on tensor products of Banach spaces is due to Schatten who in 1943 continued his work in a series of papers (some together with von Neumann). Schatten’s influential monograph “A Theory of CrossSpaces” contains what was known in 1950: the most beautiful applications of the theory dealt with operator ideals on Hilbert spaces [249], the Hilbert-Schmidt operators, the traceclass or more generally the Schatten-von Neumann-classes 6,. Many of the more elementary aspects of Grothendieck’s theory were known to Schatten, but he was

2

Introduction

not aware of the important role of the finite dimensional behaviour of tensor norms. Therefore, he did not succeed in developing - for general Banach spaces - a useful theory of duality; the reader will get a more precise idea of why this happened in section 15. On the other hand, the idea of ideals of operators was always apparent in the study of tensor products - in Schatten’s as well as in Grothendieck’s work. When writing the “R&umB”, Grothendieck was not aware of Schatten’s work. In 1968 Pietsch and his school began a systematic investigation of operator ideals on the class of Banach spaces and, ignoring tensor products, developed a method of thinking in a “categorical” manner which is just as powerful as thinking in terms of tensor products - but has the advantage that it is certainly much easier to learn the basics of operator ideal theory than the basics of the theory of tensor products. This development culminated in 1978 in the publication of Pietsch’s book “Operator Ideals” which contains in a nearly encyclopaedic way everything that was known about operator ideals at the time. Although many of the ideas and results clearly came from the “R&um6”, tensor products were never used in his book.

3. There was a bias against tensor products in Banach space theory, but the projective tensor norm K and the injective norm E proved their usefulness rather quickly through simple examples involving vector valued function spaces. New results about general tensor norms were only obtained around 1970; may be the most important of these “early” papers are those of Saphar and Kwapien’s Bordeaux-paper on operators factoring through L,-spaces. This latter paper, however, was written in the language of operator ideals. Later on (in 1973) Lotz based his investigation of maximal normed operator ideals (he had called them Grothendieck ideals) on tensor products. The rich paper of Gordon-Lewis-Retherford on operator ideals was stimulated by Schatten’s and Grothendieck’s work and uses tensor product methods explicitly. It was Pisier’s work in Banach space theory (starting around 1975) which finally drew new attention to the tensor product approach. A highlight is certainly his solution of the last problem stated in the “Rbumk”: There exist infinite dimensional Banach spaces P such that P @e P = P @ x P holds isomorphically. Pisier’s book “Factorization of Linear Operators and Geometry of Banach Spaces” is centered around the problem of determining conditions under which an operator between Banach spaces factors through a Hilbert space. These investigations finally led to solutions of all six of the problems stated at the end of the “RCsumk” except that the exact constant in the Grothendieck inequality (as the “thCorkme fondamental” is nowadays called) is not yet known (the approximation problem was solved in the negative by Enflo in 1972). Reading Pisier’s book, it becomes strikingly clear that thinking in terms of operator ideals and in terms of tensor products is quite useful. Further support for this way of dual thinking is given by a trick due to Pietsch from 1983 where he uses tensor products of operators in order to give a simple proof of the famous result concerning the distribution of eigenvalues of absolutely psumming operators due to Johnson, Konig, Maurey, and Retherford (see 34.10. for some details).

3

Introduction

4. Unfortunately, the beauty and power of “tensorial” thinking, becomes clear only after one is really accustomed to using it. The “Rksumk” is very difficult to read and for this reason various attempts have been made to present the theory of tensor norms in a clearer fashion (Amemiya-Shiga [3], Lotz [180],Losert-Michor [179], Michor [191], Gilbert-Leih [81] are known to us), but there seems to exist none which is easily accessible and, at the same time, incorporates the wonderful theory of operator ideals as it exists nowadays. We have already expressed our point of view in a sort of prebook in [43]. Our central aim with the present book is to convince the reader that the theory of tensor norms is much less difficult than it sometimes seems and that it gives one a deep and beautiful insight into various questions in Banach space theory - in particular into the theory of operator ideals, which is our main concern apart from the forementioned objective. We hope that we can convince the reader (and the historical development gives clear evidence for this) that both theories, the theory of tensor norms and of normed operator ideals (if we consider them for a moment to be really different), are more easily understood and also richer if one works with both simultaneously. It should become obvious that certain phenomena have their natural framework in tensor products and others in operator ideals. 5. We assume that the reader of this book knows the basics of Banach space theory as they are usually presented in an introductory course on Functional Analysis. Some additional information is collected in the four appendices - in particular, our point of view for the theory of measure and integration. Appendices C and D about the Radon-Nikodim property will not be needed before section 16. The book is divided into three chapters: Basic Concepts, Tensor Norms, and Special Topics. The first chapter may serve as part of an introductory course in Functional Analysis since it already shows the powerful use of the projective and injective tensor norms T and E , as well as the basics of the theory of operator ideals. The second chapter is the main part of the book: it presents the theory of tensor norms as designed by Grothendieck in the “Rhumk” and deals with the relation between tensor norms and operator ideals. This relation is dominated by the following definition (17.3.) which constitutes a one-to-one correspondence between finitely generated tensor norms a and maximal normed operator ideals (%, A) : They are called associated (in symbols a (a,A)) if %(MIN ) = M‘&N

-

holds isometrically for all finite dimensional Banach spaces M and N.We believe that the following theorems are fundamental for the understanding of the interplay between operator ideals !2l and their associated tensor norms a : The representation theorem for maximal operator ideals (17.5.)

!2l(E,F’) A ( E g a 8 F)’

( is o m e t y )

and the representation theorem for minimal operator ideals (22.2.)

~ ‘F 2, 6 a m~i n (El F )

( m e t r i c surjection),

Introduction

4

where E and F are arbitrary Banach spaces and Qmin is the minimal kernel of Q. In view of applications it is natural to first study tensor norms on finite dimensional normed spaces and later extend them to arbitrary normed or Banach spaces. There are two ways to do this - an inductive procedure + a ( z ;E, F) := inf{a(z;

M ,N) I z E M 8 N; M ,N finite dim.}

and a projective procedure %(z; E , F ) := sup{a(Qf 8 Q E ( z ) ;E / L , F / K ) I E / L , F / K finite dim.}

.

Both coincide if (and in some sense only if, see 16.2.) both spaces have the metric approximation property. Grothendieck chose the first approach and this is justified when one looks at examples. However, we have found it very useful for our investigations to also have the “cofinite hull” % at hand and we hope that we can convince the reader that it is helpful for structuring one’s thoughts and is often quite useful for finding and working out proper statements and proofs. For operator ideals the cofinite hull gains importance through the embedding theorem, which states that

E I ~ + FL Q ( E ,F ) a

(metric injection)

if a and Q are associated (see 17.6.). As has already been mentioned, these three theorems describe the relationship between tensor norms and operator ideals. Chapter I11 deals with special questions; for this, as well as a detailed account of the contents of the other two chapters, the reader may consult the table of contents. 6. Each section is accompanied by a series of exercises and the reader is strongly advised to solve them or at least give them a close look. They are of varying degrees of difficulty: if the result is not immediate from what was said in the same section, then there will be hints for the solution, which - we hope - make the solution accessible. Clearly, after a while (more or less after section 23) the reader should be continually mindful of the three fundamental theorems just mentioned as well as the local techniques. We do not claim that a result presented as an exercise is easy (“just an exercise”) - we only say that at this stage of the development of the theory this result can be obtained (together with the hint) without too much difficulty.

7. The book treats only tensor products of two “variables” - and not norms on of more than two Banach spaces. Though there is certainly tensor products El@-. -@En a good part of the theory which easily extends to the case of several variables, there exist new interesting phenomena which we do not study. Moreover, we deal mostly with normed, not quasinormed operator ideals (with a few exceptions: for example the cyclic composition theorem in section 25): this is a consequence of the fact that tensor

Introduction

5

norms and their duality theory cannot be used to study general quasinormed ideals. One may consider this to be a disadvantage, but one should not forget that most of the interesting operator ideals are actually normed and that the theory of operator ideals (as it is presented in Pietsch’s book for example) also deals mainly with normed (maximal or minimal) ideals. The following list contains some topics which we did not include in the book - partly because of lack of space (we never thought that the book would grow so large), partly because we did not feel competent: - Pisier’s example of a space P with P aCP = P P - Geometry of tensor products, stability properties - Tensor products of Banach lattices - Tensor products of C’-algebras - Applications to Harmonic Analysis. 8. A short word about citations: We are very concerned that we state the origin of those results and methods which we consider to be important for the ideas which are presented. Nevertheless, we find it boring to constantly repeat “due to Grothendieck” during most of the first two chapters. Primarily in the first chapter we label many results as now being folklore - a judgement which may be rather subjective; we apologize for any omissions which may occur in this way - although we hope that there are none.

During the preparation of this book we received great support from many colleagues and friends - and we learned very much from them: In particular, for our understanding of the basic ideas of tensor norms and the “Rksumk” we owe much to the work of Harksen, Lotz and Saphar. Our knowledge about operator ideals stems for the most part from Pietsch and his school - his book on operator ideals serves as a permanent reference for us. We profited much from the study of the papers of Gilbert-Leih, Gordon-Lewis-Retherford, Kwapieli , Lewis, Lindenstrauss-Pelczyliski, Maurey and Pelczyfiski. And our vision of the topic was widened when we read in Pisier’s work. We are also grateful to those colleagues who helped us in solving special problems during the preparation of the book - in particular, R. Alencar, B. Carl, M. Defant, P. Harmand, H. Jarchow, A. Pelczyliski ,N. Tomczak-Jaegermann and J . Voigt; we are happy that G. Pisier provided us (shortly before finishing the book) with an example of a non-accessible tensor norm. We thank the universities of Ann Arbor, Campinas, Lecce, Niteroi, Oldenburg, Pretoria, Rio de Janeiro and Zurich for having given us the opportunity to present a series of lectures or seminars on our ideas; we acknowledge the critical assistance of colleagues and students at these universities. Various students from the University of Oldenburg wrote diploma theses related to our work: those of L. Behrens (section 27) and M. Meile (section 33) directly influenced our presentation.

6

Introduction

We gratefully acknowledge the support of the Land Niedersachsen, the Volkswagen foundation, the Gesellschaft f i r Mathematik und Datenverarbeitung GMD, the Brazilian CNPq and the DAAD during the preparation of this book. There were many colleagues and friends who spent a great deal of time and energy helping us in the difficult task of proof-reading: Klaus Bierstedt, Bernd Carl, Jeslis Castillo, Martin Defant, Ute Defant, Peter Harmand, Hans Jarchow, Mario Matos, Jorge Mujica, Peter Stollmann, J o k Prolla and Jurgen Voigt. Bruce Hanson had a close look at our English. We thank all of them wholeheartedly - not only for the errors which they found, but also for their many observations which helped improve the readability of the text; in this respect we are particularly grateful to Hans Jarchow. Nowadays when the publisher’s task is often reduced to copying and selling (we do not say that the latter were easy) the secretarial work of producing a manuscript has gained special importance. We feel that Mrs. 1. Matziwitzki from the University of Oldenburg has done an excellent job - with dedication and a good sense of aesthetics. We are deeply grateful for this - it is always a pleasure to work with her. Our feelings of gratitude go to the first author’s wife Ute for her understanding and patience. It is a particular pleasure for us to have this book published in the series Notas de Matembtica edited by our friend and colleague Leopoldo Nachbin; according to the introduction of the %sum6 Grothendieck and he intended at the time to write a monograph about locally convex spaces including many aspects of the metric theory of tensor products. Unfortunately, this never happened. We thank Leopoldo Nachbin for always having encouraged us to write our book and the North Holland/Elsevier Science Publishers for accepting it for publication.

Campinas and Oldenburg, September 1991 Andreas Defant Klaus Floret

1. Bilinear Mapping8

7

Chapter I Basic Concepts After some remarks on bilinear mappings and the introduction of algebraic tensor products, the projective and injective tensor norms and some of their most important properties are studied in sections 3 - 6. Vector valued pintegrable functions, summability of sequences and averaging methods are treated in section 7 and section 8 - always from the perspective of tensor products. The remaining sections are devoted to the basics of the theory of operator ideals and integral and absolutely psumming operators between Banach spaces, culminating in the so-called little Grothendieck theorem. The reader should be familiar with some fundamental tools from Banach space theory, such as the Hahn-Banach theorem, the Mackey theorem / uniform boundedness principle, the weak- and weak-*-topologies, continuous linear operators and the open mapping and closed graph theorems. This clearly includes some simple knowledge about the classical Banach spaces C ( K ) and L p , as well as the sequence spaces co and l,. In Appendix A some additional information about the structure theory of Banach spaces is collected.

1. Bilinear Mappings This first section treats bilinear mappings and puts special emphasis on the fact that in many respects they behave differently than linear mappings, although they are intimately related with them.

1.1. Let E , F, G be vector spaces over the same scalar field K = R or (Cof the real or complex numbers. A mapping i9: E x F - G

is called bilinear if the mappings

8

1 . Bilinear Mappings

are linear for each z E E and y E F , in symbols : 9 E Bil(E, F; G) if E L(F,G)

and

eYE L(E,G)

for each 2 E E and y E F. For simplicity: Bd(E, F) := Bil(E, F ; K ) . If E , F,G are normed spaces (or more generally: topological vector spaces), the set of continuous bilinearmappings E x F -+ G will be denoted by %il(E, F;G) and %i[(E, F) if G = K. From 9(z, y) - 9(zoryo) = O(z - z o , y

- Yo) + q z -

2 0 ,

yo)

+

@(so,

Y - Yo)

the following result is easily deduced:

PROPOSITION: For 9 E Bil(E, F; G) the following assertions are equivaleni: (a) 9 is continuous. (b) 9 is continuous ut (0,O). ( c ) There is a constant c 2 0 with ll@(z,y)llG 5 cllzllEllyllF for all (z,y) E E x F

.

It is easy to see that

11@11 := min{c 2 0 I c as in (c) } = sup{((@(z,y)ll, I z E B,,y E B,} defines a norm on %i[(E, F; G) which is even a complete norm if 11 ' IIG is. Note that continuous bilinear mappings are not uniformly continuous since, e.g., the restriction R , (2, y) 4zy to the diagonal is the function R 3 z u* z2 E R. of Ra

-

1.2. A bilinear mapping 9 E Bil(E, F;G) is separately continuous if all and 9, : E .--+ C are continuous.

:F

-, G

THEOREM: Let E, F, G be normed spaces and E complete. Every separately continuous bilinear mapping 9 E Bil(E, F;G) is continuous.

PROOF: The set D := { Z ' O @ ~J Z ' E B G I E, ~BF} c E' is n(E', E)-bounded since for each z E E

Mackey's theorem / the uniform boundedness principle shows that D is uniformly bounded, i.e. there is a c 2 0 such that for all t' E BGIand y E BF

9

1. Bilinear Mappings

It follows that llOll 5 c . 0

1.3. Some examples of bilinear mappings: (1) For t’ E E’ and g‘ E F’

L Z’@Y’l(Z,

Y) := (t’,4 ( d ,Y)

defines a continuous bilinear form and unit balls and (A,) E e l , then

llz’&dll= 11z‘11 115/11. If t k

c

and y’, are in the

8

cp(Z,

Y) :=

An[&aY;l(zI

Y)

n=l

is well-defined and IlppIl 5 C,”=llAnl; bilinear forms of this kind are called nuclear bilinear forms. (2) The evaluation map on the space C(E,F) of continuous linear operators

&(E,F)X E (T,z)

--

F Tx

has norm 1 (if E and F are not trivial). (3) If E and F are finite dimensional, then all bilinear mappings E x F continuous (use bases). (4) The convolution mapping

Ll(R) x Ll(R)

-

--*

G are

Ll(R)

(!,! -I f) *s

is bilinear. ( 5 ) Take the continuous functions on a compact space K and E a normed space. Then

C ( K )x E (!I4

--

C ( K ,E ) f(.)Z

is bilinear. 1.4. The mappings

give isomorphisms of vector spaces and are inverse to each other. Since

10


this isomorphism reduces for continuous bilinear forms to an isometry of normed spaces

% s i [ ( E , ~ A) c(E,F')

1 1 ~ ~ =I lIIPII

and

llhll = IlTll.

This relationship is basic for the understanding of the ideas which will be presented in this book: the continuous bilinear forms on E x F are exactly the continuous operators E 3 F'. 1.5. Since there is no Hahn-Banach theorem for operators, there is none for bilinear continuous forms in the following sense: Let G c E be a subspace and cp E %iI(G, F ) ; does there exist an extension $ E %3i((E,F ) of cp ? This would mean, by the identification of bilinear forms and operators, that every T E C(G,F') would have an extension F E C(E,F'):

E

To see some examples of operators which are not extendable take the special case in which G = F' and T = idG the identity-mapping: the extension ? would be a projection of E onto G.

(1) Due to a famous result of Lindenstrauss-Tzafriri [175] every infinite dimensional Banach space which is not isomorphic to a Hilbert space has a non-complemented closed subspace. (2) To see a more concrete example take the Rademacher functions defined on [ 0,1] rn(t) :=(-I)~ if

(they form an orthonormal system in L2[0,1] injection f2

-

tE

;[-,

,,l[

, Lebesgue measure) and

consider the

L1[0,1]

n=l

The Khintchine inequality (8.5. and 8.6.) will show that L1 induces an equivalent norm on &. But f 2 cannot be complemented in L1 : a projection P : L1 -,4 Q L1 would be weakly compact so P = P2 would be compact by the Dunford-Pettis property of L1 (see Appendix C7.), which is not possible since e2 is infinite dimensional. (Another argument for the non-complementation follows from the fact - which will be proved in section 11 - that every operator L1 -,f 2 is absolutely 2-summing.)

11

1 . Bilinear Mappings

1.6. Extension to the completions, fortunately, is no problem. Recall that there is no uniform continuity!

PROPOSITION: Let E , F, G be normed spaces and G complete. Every d E B i I ( E , F ; G ) bas a unique edension 6 E %i[(,@, P;G ) . Moreover, Il0ll = 11611. This follows easily from the isometric relation

B i [ ( E ,F ; G ) = 2 ( E , C ( F ,G ) ) and the extension of linear continuous operators.

1.7. There is also no open mapping theorem for bilinear continuous surjective mappings.

REMARK: If E is a normed space of dimension at least 2, then the scalar multiplication K x E + E is bilinear, continuous, surjective but not open. PROOF: Take an open non-void set V C E and a functional 3/ E E’ with inf l(y‘, V ) (> 0 . If U is the open unit ball in K, then 0 E U . V, but 0 is not an interior point of U . V since U . V n ker y’ = (0). 0 It is also possible to give examples d E B i I ( E , F ; G ) which are surjective and not open in zero, i.e. zero is not an interior point of +(BE,BF):In 1973 Cohen [33] constructed a relatively complicated bilinear mapping 41 x --* 41 of this kind, but two years later Horowitz [118] found the following simple example @ : K3 x K3 + K4

(see also Ex 1.8.). There is, however, a closed graph theorem (Ex 1.11.). 1.8. Another negative property of continuous bilinear mappings is that they do not remain continuous for the weak topologies: the unit vectors en in 4, (real) converge weakly to zero but (enlen)cl = 1 does not (see also Ex 1.2.).

1.9. For normed spaces E and F the following isometries hold: 0 : %iI(E, F ) A 2 ( E ,F‘)

T

-

c ‘ .2 ( F “ , E’) A B i I ( F ” , E ) T’

%it(E,F ” )

,

12


where the last equality is the obvious "transposition" V'(z,y) := V(y,z). For each E %i[(E,F ) define 'p" := @(p)E Bil(E, F"); it satisfies II'p'II = II'pII = llL,ll and

'p

~"(z, Y") = (Lb(Y"),Z ) E ~ , E= (Y", L , ( Z ) ) F ~ ~=, F(Y", ~ ~ ( z. ), ) F ~ ~ , F I for all (2, y") E E x F". Since, by definition ~ ( zy), = ( y ,L,(z))F,F, for all z E E and y E F , the map 'p" extends 'p from E x F to E x F" with equal norm. 'p" is called the canonical (right-) extension of 'p.

-

PROPOSITION: Let E and F be normed spaces and 'p E %i[(E,F ) . Then 'p" is the K unique bilinear separately u(E,E')-u(F", F')-continuous mapping 1c, : E x F" which extends 'p. PROOF: That 'p" is such an extension is clear from the equation following the definition; uniqueness follows from the u(F", F')-density of F in F". 0 Clearly, there is also a canonical left-extension it follows that "'p(Z'Ii

y)

How are the functionals the following is true:

= (zll, (Lb ("'p)"

and

"'p

on E"x F defined by "'p := (('pz)")z;

= (2'1,

IE~)Y),qif,,qi

"('p")

V ( * lY ) ) E i i , E f *

on E" x F" related? Surprisingly enough,

COROLLARY: For 'p E %i[(E,F ) the following three statements are equivalent : (a) The two "canonical" extensions ("'p)" and "(9") of cp to El' x F" coincide. (b) There is a E Bil(E", F") which is separately c(E", E')-u(F'', F')-continuous and extends 'p. (c) L, : E -+ F' is weakly compact. I n this case the II, in (b) is equal to ("'p)" = "('p").

+

PROOF: The proposition implies easily that (a) is equivalent to (b). To see the equivalence of (a) and (c) first note that, by Ex 1.12. and Ex 1.14.

Now recall that L, is weakly compact if and only if L:(E") c F'

.0


13

Exercises:

Ex 1.1. If E and F are finite dimensional, then every p E Bil(E,F ) has the form for some xk E E' and

4 E F' .

k=l

Ex 1.2. (a) If E and F are normed spaces, then every (o E !&[(ElF ) which is continuous for the weak topologies has the form as in Ex 1.1. with continuous x i E E' and 4 E F' . (b) If E and F are vector spaces, then every u(E' , E)-a(F* ,F)-continuous bilinear form (o on E' x F' has the form n (p

=

for some X k E E and

Xk&k

yk

EF

.

k=l

Ex 1.3. Let E be a Banach space and F be normed. If S C %iI(E,F) is a set of separately continuous forms such that {=pI p E S} C F' is equicontinuous for each x E E, then sup{ll(oll I (o E D} is finite. EX 1.4. Using the space cf of "finite" sequences ( a ) (i.e. en E K and o with

+ E)M E ~ F E r ( i E @ b ~ )and , ~ ( zE;, F) 5 (1 +

E ) Z ~ &E

+ &)'z

63

and (1

0

E ) - ~

< 1.

0

Now x has to be calculated more explicitly: For z E B E and y EBF property (2) implies that x ( z @ y; E, F) 5 1, hence for all (z,y) E E x F

But there are

t'

E BE' and y' E B p with (z',z) = llzll and

11~11llyll =I

(2' @

(y',y)

= llyll, hence

Y', 2 @ ar) 15 112' @ y ' l l ~ ( z8 Y; E , F ) 5 x ( c @ y; E , F ) .

This is (3) - and everything is ready for proving the initial formula for triangle inequality (and an obvious notation)

r ( z ;E, F) 5 infoo(z)5 infj(z) On the other hand, take r ( z ;E, F) < A, i.e.

k=l

which implies

A:

By the

for z E E 8 F.

9. The Projective Norm

29

and therefore inff(z) 5 n(z;El F ) . Clearly, the inf-formula implies ( 5 ) , and it will also be used for the proof of (4) since it is handier than the original definition: Take E C(Ei,Fi) and n

z=

C

X: QD Z:

E El

QD E2

.

k=1

Then

For brevity, the notation TI &, T2, and Tl&T2 for its unique extension to the completions E1&E2 4 Fl&F2, will be used. By considering elementary tensors, it even follows that llT1 8% T211 = 11T111 llT211 holds.

3.3. Let ( 0 , p ) be a measure space and E a Banach space. Then

Cl(f4 8 E c C l h E ) - the

space of Bochner integrable functions and J u P )8 E

c Ll(P7E)

for the equivalence classes. It is clear from the definition of Bochner integrable functions (see Appendix B12.) that the vector space e ( p ) 8 E of E-valued step functions n

EX,,@xk

Ak p-integrable,

Xk

EE

k=l

is dense in & ( p , E ) and hence the space S ( p ) €3 E of classes is dense in & ( p , E). It follows that

are dense inclusions. PROPOSITION: Let p be an arbitrary measure and E a Banach space. Then


30

is an isometric dense embedding; in other words,

f o r each f E L1(p) 8 E , and L1(p)6&E = L1(p,E ) holds isometrically.

PROOF: Obviously L1 x E

L1(E) has norm

5 1, hence

Since it is easy to see that S(p) 8 E is ?r-dense in L1(p) 8 E (see 3.9. or Ex 3.3.), it is enough to check, by Ex 3.4., that

holds for step functions N

n=l

Clearly, the An can be assumed to be pairwise disjoint, therefore

This example is quite important; for 1 < p see sections 7 and 12 for a discussion if p

5 00 it is not true that L,&E = L,(E), < 00 and 7.1. and Ex 6.8. for p = co .

What is the dual of L1aT F ? In other words, is there a good description of (L1BTF)’ = (Li6LF)’ = & ( p , F)’ ?

THEOREM OF DUNFORD-PETTIS (weak version): If p i s a strictly localizable measure F’ a continuous linear operator into the dual of a Banach on R and T : Ll(R,p) space F, then there is a function g :0 4 F’ such that --+

(1) g is weak-*-measurable (see Appendiz B11.) and g(R) C T B L , ~ ular, Ilg(w)llFI 5 IlTll f o r all w E R . (2) g is a weak*-densiiy of T , i.e.: for each z E F and f E L1

(F’lF)

s

I n other words, T f = f g d p as a a(F‘, F)-Pettis integral.

. In partic-


31

PROOF:Call 'p E ?ZW(L1, F ) = (-L1 ~3~F)' 3 ~ ( L F') I , the bilinear form associated with T and define ('pz,f) := 'p(f,z) = ( T f , z )for each 2 E F and f E L1. Then pt E L: = L , . Since p is strictly localizable, there is a "lifting" p : L , + 2 , (see Appendix B7.), i.e. p is linear, positive, p ( i ) = 1 and p(f) f j.It follows that Ip(f)(w)l

I IIflL~.

For each w E R the functional g(w) E F' is defined by ( g ( 4

for all z E F ,

4 := P(cPZ)(W)

hence ( g ( - ) ,2 ) = p(p,) is measurable for all z E F and I(S(414I

= IP('pz)(4l 5 Il'pzllt, = sup{l(Tj, 41 I f E

BLJ

which means that g(w) E ( T B L ~ ) " = T B L I 0( F ' r F ) by the bipolar theorem. Clearly, the following representation holds:

( T f ,4 = ('pZ1 f) =

Jnp(cp,)(w)f(w)p(dw)= J,f ( w ) ( s ( w ) M, d w )

0

I f T E 2(L1, F ) is weakly compact, then the set T B L ~c F is weak-*-closed in F" and therefore:

COROLLARY: If p is strictly localizable, F a Banach space and T : L l ( R , p ) --* F weakly compact, then there is a weakly measurable function g : R + F with values in TBL~ such that

(Zt,

for all ( x ' , j ) E F' x L1

Tf)=

J,

f(W)(ZI,

s(w))p(dw)

. In other words, T f = S fgdp

as a Pettis integral.

In Appendix C it will be shown that the function g in this corollary can be chosen to be Bochner integrable if p is a finite measure. Appendix D4.demonstrates conditions under which the dual of L l ( E ) can be described by bounded measurable El-valued functions. 3.4. The following theorem - as most of them in this part of the theory: due to Grothendieck - describes the compact subsets of E& F .

THEOREM: Let E and F be nonned spaces and K c E& F relatively compact. Then there are zero sequences (2,) and (yn) in E and F, respectively, and a compact subset H C L1 such that for each z E K there i s a (A,) f H with

n=l


32

PROOF:The idea (following Pietsch) is to expand each z E E&F convergent series z = C zn and to choose a representation

in an absolutely

- and all of this somehow simultaneously in K, with the aid of its precompactness. For this, take eo := 1 and En > 0 with Cr., en = 1+ e and define

Assume K C B, ; then there is a finite Do C B, with

K C Do

+ 81

and, by induction, finite sets Dn c Bn with

There are (pairwise disjoint) finite indexing sets A, and BT with 0, := {zY I j E An} and representations zj” = a:kxy,,t 8 $,k kEB?

with llzjlkll

= ll$,kll

=En

and

c

5 En

+

k€B?

The set IIYzkll

-+

r

:= { ( n , j , k ) I n E N U { O } , j E An,k E BY} is countable and I l z y k l l = 0 under some enumeration of I’. Take as the desired compact set in Ll(I’)

For z E K there are jnE An with z

- zj”, E ( K - 0,) n B1= F~

- z ; ~- ~ j ’ ,E ( z - z~”O - ... - z;,,

z

F~ ol)n B2 E (Fn- D,) n Bn+l

...


33

It follows that

which is the desired representation. 0 3.5. Bounded subsets C c E&F A c E and B C F such that

are “liftable” in the sense that there are bounded

ELF

C c r ( A @B )

(why?) - the theorem implies that the same holds for relatively compact sets:

COROLLARY 1: Let E and F be normed and C c E&F relatively compact. Then there are compact sets A C E and B C F such that E&F

C C r ( A8 B ) The fact that

E = E&#

gives another important special case:

COROLLARY 2: Every relatively compact subset K in the completion of a normed space E is contained in the closed absolutely convex hull of a zero sequence in E . Moreover, for every E > 0 the z, can be chosen to be in [(1+ &)supzEK11z11]B~. Since for every z,, -, 0 there is A, 00 such that Anz, -+ 0 , it follows that for every compact subset K of a Banach space (K c I?{ z,}) there is a compact Banach disc L (take L := I”{Anzn})such that K is compact even in [t]. Another consequence of the theorem is that every element z E E&* F has an absolutely convergent series expansion of the form z = Cr.l z, €9 y, with 2, E E and y, E F .

COROLLARY 3: For z E E& F

PROOF:By the triangle inequality for ?r the right hand side defines a norm r m 3 ?r on E&F which , by 3.2., coincides with ?r on the ?r,-dense subspace E €9 F of E&F; this implies the result - e.g. by Ex 3.4.. An application: Every f E L l ( p , E ) has an expansion m

f = CL.zfl fl=l

34


00

Clearly, T is continuous in this case and

defines a norm on the vector space %(E,F) of all nuclear operators; it is easy to see that %(E, F) becomes a Banach space with this norm. Recall from 2.4.(5) the injective map E' @ F S ( E ,F ) C % ( E ,F ) C 2 ( E , F ) *'@!I zQy

--

with (z'sy)(x) := (x', z)y. With this notation nuclear operators have an expansion m

n=l

being absolutely convergent in %(E,F) and hence in 2 ( E , F), which implies that they are compact. The definition of %(E,F), its norm and corollary 3 give the

PROPOSITION: The map

is a metric sujection.

Note that the shorthand definition for J waa used: first on elementary tensors, then finite sums, and finally extension to the completion by continuity; it is therefore clear that for x E E [J(Z)I

(XI

=

00

00

n=l

n=l

[CX L B Y ~ ] (x)= C ( x k z)Yn

for each representation z = C,"==, zh @ yn. Unfortunately, it will turn out that J is not in general injective - and hence is not an isometry on E' @= F. This has to do with

35


the approximation property and will be investigated carefully in section 5 (for N on $(E, F ) = G @ F see Ex 5.16.). Clearly, if E or F is finite dimensional, then

E'&F = El@=F = % ( E ,F )

(via J)

and J is an isometry.

-

3.7. Since I(d,z)I 5 Ilz'l11z11, the trace defined in 2.5. trE : E' @= E

K

has norm 1 (if E # (0)) and hence extends to the completion t r E E (El& E)'

, lltr~ll= 1

with (by continuity) n=1

-

for any convergent representation z = C,"==l z', 8 zn.If the mapping

J : E'&E

% ( E ,E )

is injective, then the trace is defined for nuclear operators - but it will be seen in section 5 that this is not always possible. The trace is a very useful tool for many calculations. To see an example:

REMARK: N ( i d E ) = dim E for all finite dimensional Banach spaces E PROOF: Take an Auerbach basis (el,

a * ,

.

en) of E with coefficient functionals e i

Ilekll= I l e i l l = 1 and

(see Appendix A4.). Then idE = CrZle&eb = and

, i.e.

(eS,et) = 6k,f

= J(C;=le i @ e b ) , hence N ( i d ~ )5

C;=Ille611 llekll

since J is an isometry. 0 For many calculations the following result will be helpful:


36

PROOF: By linearity and continuity it is enough to check this for elementary tensors z=z@y:

In particular,

3.8. For % E 2 ( E j , Fi) the operator T I ~ & T: ~ E1G3,E2 -+ Fl&Fz is well-defined; is it a homomorphism if the are?

PROPOSITION: If Qi : Ei

1

+ Fi

tare two metric surjections, then

is a metric surjection. In particular,

is a metric surjection as well.

0

0

PROOF: Qj(BEi) =BF, and the description 3.2.(2) of the open unit ball in the projective tensor product imply

which is the claim. 0 This property of

?r

is the reason for calling it projective.

3.9. Let G c E be a subspace. If G@., F were isomorphically a subspace of E@* F, the Hahn-Banach theorem would imply that every linear operator in E(G, F’) = ( G a rF)’ could be extended to E -+ F’. There have been given various examples in 1.5. showing that this is not true; therefore: T does not respect subspaces, not even isomorphically ! Later on (5.8.) it will even be shown that Tl&&T2 may fail to be injective if the T i are. But A respects complemented subspaces, dense subspaces and the embedding into the bidual:

PROPOSITION: Let E and F be normed. (1) If G C E i s complemented with a projection P : E

4

G , then

37


for all z E G @ F and P @ idF i s a projection of E @, F onto G @= F . (2) If G c E is dense, then G @= F is an isometric dense subspace of E @= F and, consequently, of B B~P. ( 3 ) I respects the natural embedding into the bidual isometrically:

PROOF:(1) follows from the mapping property and (2) from the unique extension of a continuous bilinear form to its completion (1.6.). To see (3), observe first that IlidE @ ~ ~ 5 11, hence 1 n(z;E , F") 5 I(%;E, F) for each z E E m F , Take a p E ( E @ =F)' of norm

5 1 with

and its canonicaI right-extension p" E (E @= F")' (see 1.9.). Then

The result (3) also follows from Ex 3.19. on weak retractions. The following example (taken from Jameson's book [119]) shows in a simple way that x does not respect subspaces isometrically: Take for n 2 2

G = %(G,G) Q %(G, ti') = GI & li' and zo corresponding to i d c . and consider G' Then, by remark 3.7., x(z,;G',G)=n-l .

c,

hence zo = corresponds to the embedding G + canonical basis. It is easily checked that IleilJGt = 1/2 and therefore

zo E GI @=

x ( z o ;GI,

since q

@=

e i 8 ek for the

q )= n / 2 ,

GI = G(G') holds isometrically by 3.3..

3.10. The following characterization holds:

PROPOSITION: For every Banach space E the following assertions are equivalent: (a) E @= respects subspaces isomorphically .


38

(b) There is a A 2 1 such that: Whenever G c F is a subspace, then ~ ( zE; , G)5 AT(%; E , F ) for all z

E E 8G .

(c) E’ has the A-extension property for some A 2 1. (d) E’ is an injective Banach space. Moreover, the constants A in (b) and (c) can be taken to be the same.

PROOF: (a) w (d) and (b)

-

(c) follow from the Hahn-Banach theorem applied to

C(F,E’) = ( F @r E)’ (d)

TC

restriction

(G @, E)’ = E(G, E’)

.

(c) was shown by the reader in Ex 1.7..

If E = LI(p), then for every

f E L1(p) 8 G

hence L l ( p ) @ ~respects ~* subspaces isometrically. For a localizable measure (i.e. L,(p) is order complete, see Appendix B6.) Lm(p) = (Ll(p))’ holds, hence:

COROLLARY: If p is localizable, then L,(p)

has the metric extension property.

A theorem of Nachbin-Goodner-Hasumi-Kelley states that the spaces with the metric extension property are exactly the C ( K ) where K is compact and extremally disconnected (=Stonean), see Day [38], p.123 or Lacey [164], p.92 . This fits with the Gelfand representation of the C*-algebra L, . Moreover, due to a theorem of Kakutani (see Appendix A7.), abstract L-spaces L are isometric to some L1(p) (where p is a Borel-Radon measure on a locally compact space and, in particular, is strictly localizable). This implies that:

(1) L respects subspaces isometrically. (2) L‘ has the metric extension propertg. The duality between abstract M - and L-spaces has various consequences; first observe that every C ( K ) is an abstract M-space, hence C(K)” is the dual of an abstract L : (3) C(K)” has the metric extension property for each compact space K ; in particular, For G C F every operator T E C ( G , C ( K ) ) can be extended t o some operator T E C ( F ,C ( K ) ” )with the same norm Il5?((= llTll - but there is in general no ettension T :F 4 C ( K ) .

e)

39


The last statement follows from the fact that co = C(N U (00)) is not complemented in 1,. So, in general, C ( K ) is not an injective Banach space - but it will be seen in 4.5. that compact operators with values in C ( K ) can be extended.

3.11. The following characterization is due to Grothendieck [91]

THEOREM: Let E be a Banach space. Then E @, and only if E is isometric to some L l ( p ) .

respects subspaces isometrically if

PROOF: If E @=I respects subspaces isometrically (the other direction was shown before), then, by the last proposition, El has the metric extension property and therefore El = C ( K ) by the theorem of Nachbin et al.. Grothendieck showed (see Appendix A7.) that this forces E to be isometric to some L 1 ( p ) . 0 3.12. Dual to the extension problem is the lifting problem: Given a surjective mapping + G between Banach spaces and T E C ( E ,G). Is there a ? E C ( E ,F) with

Q :F

T=QOF?

P

It is easy to see (lift the T e , ) that for E = !1(r) and E > 0 there is always a lifting with IjFll (1 e)llTII - but, in general, the same norm cannot be achieved (Ex 3.21.). Conversely, a result of Kothe [151] (with the aid of Ex 3.22.) implies that each Banach space E such that all T's are liftable (with arbitrary norm) is isomorphic to some el (r).

+

-

PROPOSITION: Let E be a Banach space such that E ~3~ respects subspaces with a constant X (as in proposition 3.10.(b)). If Q : F --Y G is a metric surjection between normed spaces, then for every T E C(E,G ) there is an operator E C ( E ,Fii) such that ~ C G o T = Q" o and 11?[1 5 AIITII:

PROOF:Since

40


the Hahn-Banach theorem shows that there is a p E C ( E ,F”) with 11p1 ;15 AllTll and id^ @ Q’)’(P).It is easy to check that the latter means that ICGoT = Q”oT . 0

T=

COROLLARY 1: Let E be a Banach space such that E @T respects subspaces with a constant A and F , G normed spaces. If for Q E C ( F , G ) and T E C ( E , G ) there is a bounded and absolutely convex subset C c F with Q(C)3 T ( B E ) ,then there is a “lifling” E C ( E ,F”) with ICG o T = Q” o p and BE) c ACoo.

PROOF: Obviously, the mapping [q-+ [&(C)]is a metric surjection; moreover, with I : [q Q F, it is clear that I”(B[c])c Cooc F“. So it is enough to apply the proposition. 0

(See also Ex 3.23..) Clearly, any space L1(p) satisfies the assumption with A = 1 : COROLLARY 2: Let Q : F -W G be a metric surjection between Banach spaces. (1) For every measure p and every T E 2 ( L 1 ( p ) , G )there is a p E C ( L l ( p ) ,F”) with llTll= IlTll and K G o T = Q” of’ . (2) For every measure p , every compact operator T E C ( L l ( p ) , G ) and every E > 0 there as a compact lifting p E C ( L l ( p ) ,F ) with Ilrl’ll 5 (1 &)llTll.

+

PROOF:(1) is clear. For (2) observe that T ( B L ~C) I’{yn} with

(y,,) a zero sequence

+ &)IITIIBc(see corollary 2 in 3.5.). For z,,E F with llznll 5 (1 + ~)IIy,,ll and Qz, = y,, take the compact set C := I’{zn}, which clearly satisfies Q(C)3 T(BL,). Since Coo = C C (1 + ~ ) ~ 1 1 T l holds, l B ~ corollary 1 implies that im(T) c F and in (1

1 1 ~ 1 15 (1 + 4211Tll.0

See also Ex

3.31..

3.13. The metric results on extensions and liftings connected with the spaces C ( K ) ,L , and L1 have natural isomorphic counterparts for spaces which “look locally like” Pw or 6.

DEFINITION: For 1 5 p 5 00 and 1 5 A < 00 a normed space E i s called an C:px-space, if for every finite dimensional subspace M c E and E > 0 there are R E C ( M ,q )and S E C ( q ,E ) for some rn E N such that SRz = 2 for all z E M and IlS((IlRll 5 A+& :

41

3. The Projectiue Norm

E is called an 2;-space if it is an 2;,A for some A. In section 4 (Ex 4.7.) it will be shown that each Lp(c() is an A!:,l-space. The 2;-spaces are very closely related to the $-spaces of Lindenstrauss-Pelczyriski [173] ; they will be studied to some extent in Chapter 11, section 23 (where the exact relationship between 2,- and Zg-spaces will also be clarified). A starter for the “local techniques” (i.e. the method of investigating a Banach space in terms of its finite dimensional subspaces) is the following PROPOSITION: If E is A!T,A-space and G C F is

a subspace, ihen

?r(z;El F ) 5 ~ ( zE ;, G)5 An(%;El F )

for all z E E 8 G ; in other words, E

@x

-

respects subspaces up to the constant A.

PROOF:Take z E E 8 G and E > 0. Since ?r is finitely generated (proposition 3.2.(5) and Ex 3.30.), there is a finite dimensional subspace M c E with z E M 8 G and ~ ( zM, ; F ) 5 (1

+ e ) r ( z ;E , F ) .

For a factorization I E = SR of the embedding I E : M llSll llRll5 A E the mapping property gives

+

4

E through some

with

It follows that if E is a 2{,A-space its dual E’ has the &extension property (3.10.) and E has the lifting property expressed in 3.12. . Grothendieck’s result 3.11. (and EX4.7.) says that the 2:,l-spaces are exactly the Ll(p)-spaces. It will be shown in section 23 that the C:-spaces E are those such that E @ x respects subspaces isomorphically.

Exercises: Ex 3.1. Show that ?r(.; E, F) is the Minkowski gauge functional of I’(BE 63 BF). Ex 3.2. Show that 112’1 Bx 2’211 = 11T111 llT”ll . Ex 3.3. Prove directly, by using the infimum-description of the ?r-norm, that

42


is a dense isometric injection if E, c E and F, C F are dense. Ex 3.4. Let G be a vector space with two norms 11 111 and II.112 which coincide on a 11 Ill-dense subspace Go c G. (a) The norms are equal on G under each of the following three conditions:

-

-

-

(1) II * 112 I All 111 for some A. (2) There is a norm 11 . 113 on G with 11 Ils-dense.

-

11 - Ili 5

Ail1 . 113 for some A i and Go is

(3) (G, 11 112) is a Banach space and there is a separated topological space X and an injective 0 : G + X which is continuous for both norms. (b) Give an example of two non-equivalent Banach space norms which coincide on a subspace which is dense for both norms. Hint: 11~112:= llSzll1 defines a norm for each injective S E L(G, G) - not necessarily continuous. Ex 3.5. A subset K C E g 3 , F is compact if and only if there are a compact H C el, zero sequences (2,) in E and (yn) in F such that

If K C Eg,, F is absolutely convex or convex, then the set H C of the same type.

el

can be chosen to be

E x 3.6. Let T E C(Ll(p),Ll(v)) and S E C ( E ,F ) . Then there is a unique continuous operator U E C(Ll(p, E),Ll(v, F)) with U ( f 8 z)= Tf@sz

.

E x 3.7. Let T E C(Ll(p),Ll(v)) and (fn) a sequence in L l ( p ) . Then

for all 1 5 p < 00

. Hint: Consider T 8 idt, .

Ex 3.8. If j E L1(p, F) and f E L w ( p ) , then fj is Bochner integrable. Show that T :L , ( p )

-

F

is nuclear and N(T) 5 SIlglldp. Actually, there is equality of norms; see Appendix c3.. Ex 3.9. Show that there is a metric surjection t 2 G 3 , t 2 ---+ el; in particular, ep$,t, is not reflexive.

43


Ex 3.10. Closed graph theorem: Let E , F and G be Banach spaces. Show that every linear operator T : E @= F + G with closed graph is continuous. Hint: Ex 1.11.. Ex 3.11. Does ti 8% ti = hold isometrically? Ex 3.12. If G C E has codimension n, then

for all z E G 8 F . Can this inequality be improved in general? Ex 3.13. (a) Show that t1&F -+ 2(co, F) is injective (F a Banach space). (b) If I : G 4F is a metric injection, then: T E %(co, G ) if and only if I o T E %(co, F ) ; in this case, N ( T ) = N ( I o T ) = C,"=lI(Te,II. Ex 3.14. N ( T ) 5 min{dim E , dim F } IlTll for all T E 2 ( E , F ) . Ex 3.15. Let E c and Fc be complex Banach spaces and ER and FR the same spaces considered as real spaces. (a) For every @linear T E C ( E ,F )

holds. Is it true that N d T ) = N R ( T ) ? Hint: 3.7.. (b) Denote by QDK the tensor product over K. Show that the natural map

from Ex 2.11. is a metric surjection. Ex 3.16. For U E C(Eo,E ) and S E C(F,F,,) (all spaces are complete) the diagram

W'6-S

I

.

1

z

commutes. Deduce N ( S T U ) I IlS((N(T)[lUll.Clearly, this also follows directly from the definition of nuclear operators. Ex 3.17. If G is a closed subspace of a Banach space E with %(F,G) = F'&G for some F and F' @% G L-, F' @, E is not an isomorphic embedding, then there is a non-nuclear T E C ( t P ( F ) , t p ( G )which ) is nuclear as an operator t p ( F )--+ t P ( E ) .Hint: The operator F + G associated with z E F' @ G has a norm I T(Z;F', E ) . Ex 3.18. Let E and F be Banach spaces such that F' c E isometrically. (a) Then there is a projection P : E -W F' with norm 5 X if and only if ~ ( zF', ; F) 5 AT(%; E , F ) for all z E F' @ F .

44


(b) If p(F‘, E ) := inf{ llPll I P : E + F‘ projection } < 00, then there is a projection with norm p(F’, E) . Ex 3.19. For a metric injection I : G 4 E of normed spaces the following statements are equivalent: (a) I @ idF : G Br F + E @r F is an isomorphic [resp. metric] injection for every normed space F. (b) I @ idGn : G @ , ,G’+ E a,, G‘ is an isomorphic [resp. metric] injection. ( c ) There is U E C(E,G”) with U o I = ICG [and llUll 5 11 . (d) There is S E C ( C ,El) with I’ o S = id^ [and llSll 5 11 . Hint: Ex 1.6.. In this case, properties (c) and (d) justify calling I a weak retraction. Ex 3.20. Let M be an abstract M-space and G C F. Then every operator T : G -+ M can be extended to an operator f : E -+ M” with 115?11 = llTll . Ex 3.21. Let H be a closed subspace of a Banach space F and Q : E -+ F/H the metric surjection. (a) If T E C(tl(I’2, F / H ) and E > 0, then there is a “lifting” 5? E C(t,(I’), F ) of T (this means: T = Q o T ) with I l f l l 5 (1 ~)llTll. (b) Every T E i!(tl,F / H ) is liftable with the same norm if and only if H is proximinal in F (i.e. every element in F has a best-approximation in H ) . (c) If F is a dual space and H weak-*-closed, then every T E C(tl(I’), F/H) is liftable to F with the same norm. Ex 3.22. Assume that the canonical surjection t l ( B ~ -W) E has a right inverse S E C ( E , ~ ~ ~ BThen E ) ) for . every metric surjection Q : F -W G and T E C ( E ,G) there is a lifting T E C ( E ,F) with I l f l l 5 (1 ~)llSllIITll. Ex 3.23. Show that in 3.12., corollary 1, the assumption Q(C) 3 T ( B E )may be replaced by Q(C)3 T ( B E ) .Hint: Use E g r (F‘/ker mace). Ex 3.24. Radon-Nikodim theorem for operator valued measures: (a) Let p be a strictly localizable measure, E and F normed spaces and T : Ll(R,p) -+ C ( E ,F ) continuous and linear. Use the weak version of the Dunford-Pettis-theorem 3.3. to show that there is a a(C(E,F”), E&F’)-density D : fl-+ C ( E ,F”) such that

+

+

(as a weak-*-Pettis integral in (E&F‘)’ = C(E,F”)). (b) Let A be a a-algebra of subsets of R and p a (non-negative) measure on A. If E and F are Banach spaces and M :A --t C ( E , F) is a-additive, then there is an operator T E C ( L l ( p ) ,C ( E ,F ) ) with T ~ =AM ( A ) if and only if for each z E E and y‘ E F’ the signed measure (M(.)z, y’) is p-absolutely continuous with density in C m ( p ) . Ex 3.25. A special case of the foregoing Radon-Nikodfm theorem is E = K. Show for the Fourier transform

45


that the values of the density cannot be chosen in F = co (see also Appendix C). Ex 3.26. Every space L m ( p ) has the compact extension property: If F C E and T E f f ( F , L m ( p )(the ) space of compact operators), then there is, for every E > 0, an , with 11?;11 5 (1 ~)llTll. Hint: Dualize 3.12., corollary 2. extension ? E f f ( E L,(p)) Ex 3.27. Show that for arbitrary measure spaces (nilpi)

+

Ll(Pl@P2)

A ~ 1 ( P l I L 1 ( P 2 ) A) Ll(Pl)&Ll(P2)

*

1

c

@ p2) by Fubini-Tonelli. Hint: Ll(p1) @r Ll(p2) Ex 3.28. (a) Let ( e n ) be an orthonormal system in 12 and (An) E 11 with non-negative An. Using the trace, show that r(C,"==, Anen @ en;12,12) = CZzIAn . (b) If (en) and (fn) are two orthonormal systems and (A,) E 1 1 , then m

m

*(CAnen@fn;P;r,12) = n=l

C1Anr.

*

n=l

Hint: Use U e , := fn. Recall the spectral decomposition theorem: Every compact operator 12 -+ 12 has the form CAnen&fn with (A,) E c, and (en) and (fn) orthonormal systems . (c) Take E = 1, (for 1 5 p 5 00) or c, and T := C:=l A n e n e n E 2 ( E ) for (An) E co. Then T is nuclear if and only if (An) E 11. Hint: Use l t r ( q r-, lP%lp -W C)I 5 N(T) and adjust the signs. Ex 3.29. If G C F is a subspace and E @* F does not induce an equivalent norm on E @, G, then there are MnE FIN(E),Nn E FIN(F) and zn E Mn@ (N,, fl G) with r(zn;Mnr Nnfl G)> nn(zn;Mn, Nn).

Hint: Use the fact that r is finitely generated . EX 3.30. r ( z ;E l F) = inf{r(z; M ,F) I z E M @ F, M c E finite dimensional}. Ex 3.31. If E is an 2:,A-space and Q : F -W G a metric surjection, then there is for every compact operator T E 2 ( E ,G ) a compact operator ? E 2 ( E ,F ) with Q? = T and Ili'll 5 (A ~)llTll. Hint: As in corollary 2 in 3.12. use a lifting of compact sets. EX 3.32. If G C E is a subspace and T E %(G,F), then for every E > 0 there is an extension ? E %(ElF ) of T with N(F) 5 (1 & ) N ( T ). EX 3.33. If F C E is a complemented subspace (with projection P) and E an 2:,A-space, then F is an .i2;,A,,p,,-space. EX 3.34. Let E and F be Banach spaces, E complemented in its bidual E" with projection P . If T E 'X(E',F ) is u(E', E)-u(F, F')-continuous, then for every E > 0 there are xn E E and yn E F such that

+

+

c m

T=

m

2n&Yn

and

n=l

Hint: Look at T' : F'

4

C llxnll

n=l

E

L,

E"

P

+ E.

IlYnII

5 IPI")

+

&

*

46

4 . The Injective Norm

4. The Injective Norm The tensor product E @ F of two normed spaces is canonically embedded into the normed space %il(E‘, F‘) = ( E ’ @ r F‘)’of continuous bilinear forms on E’ x F‘. This space induces another natural norm on E @ F, the injective norm e, which is the norm of uniform convergence on the Cartesian product BE’ x BF’ of dual unit balls.

4.1. Let E and F be normed spaces. Then for x’ E BE’ and y‘ E B p the linear form x’@y‘ : E @ F -P K @ K = K is defined. For z E E @ F

gives exactly the norm which is induced on E @ F by the embedding into %i[(E’,F‘) =

(E’ @r F’)’since ~ E ’ o - F ’= r ( h E ’ @ E& F.

ip).Notation: E aeF and for the completion

PROPOSITION: Let E and F be n o m e d spaces. Then

- and even E Be F A (E‘ @= F’)‘ if both spaces are finite dimensional. ( 2 ) For norming subsets A c BE’ and B C B p

(3) E ( X @ y; E , F ) = 1 1 ~ 1 1llyll for all x E E and y E F

.

( 4 ) e S n o n E@F.

( 5 ) e saiisfies the metric mapping property: If ?;. E Z(Ej, Fi), then

Recall that a subset A C BE’ is called norming if l l z l l ~= sup (I(x‘,x)J I x’ E A} for all x E E . The set e x t B p of the extreme points is norming since every linear weak-*-continuous functional attains its supremum on the weak-*-compact unit ball BE! at an extreme point.

4. The Injective Norm

47

PROOF:(1) was the definition; for (2) take any representation of z and calculate: m

E(Z;

b,C

E,F ) =~~p{l(z’@

2 ,

I

@ Yn)l

2’

E BE',^ E B F I }=

n=l rn

(3) is clear from the definition, (3) implies (4) and ( 5 ) follows from (2: @

z;, (Tl @ TZ)(Zl @ z2)) = (T:z: @ T;z;, 2 1 8 z2)

.

0

Since BE c BE&* is norming, it follows that

E ( Z ; E ’ , F ’ ) = S U P { I ( Z @ Y , ZE) IB~EZ , Y € BF}= = SUP{ [ ( w ,z)IIw E r(BE 8 BF)}=

= SUP{l(W, .)11.<w,

E , F ) I 1)

1

in other words,

E‘ BCF’

c ‘ . (E

@w

F)’

A C ( E ,F’) .

This shows that even though T and E coincide for elementary tensors they are far from being equal : E is connected with the operator norm, T with the nuclear norm of operators! For explicit examples see Ex 4.2. and Ex 4.3.. 4.2. Examples:

(1) Since E @ F 4 $(El, F ) L C(E’, F )

E @c F

(E’ @r F’)’, it follows that C(E’, F )

- and similarly F = $ ( E , F)

C ( E ,F ) ,

and therefore (for Banach spaces E and F)

~ ‘F 6Z(E, ~ F) , the space of all approximable operaiors T E C(E,F), i.e. those for which there exists a sequence (Tn) of finite rank operators with [IT- T’ll + 0 ; the space Z ( E , F) is equipped with the operator norm.

4. The Injectiue Norm

48

An element z E E @ F gives a u(E’, E)-a(F, F’)-continuous operator in e(E’, E). It follows that

{TE e(E’, E ) I a(E’,,!?)-u(p,F’)-continuous}

E&F

since the latter space is closed in the Banach space g(E‘,E);see also EX 22.2.. Note that E‘& F operators

-

is injective - a fact which was not true for r! (2) C ( K )@ E is the space of all continuous functions K 4 E the image of which is contained in a finite dimensional subspace. If K is compact, then the Dirac functionals

& (for i E K) form a norming subset for C ( K ) ,hence

&(f;C(K),E)=sup{l(61~b,Cf, @.,)I I t E K , d E BE‘} =

I

= s u P { l ( d , ~ f n ( t ) ~ n ) lt E K , d E B E ’ }

= sup{llf(t)llE

I

E K } = IlfllC(K,E)

>

where C ( K ,E) is the space of continuous E-valued functions. Hence

C ( K )BCE

C ( K ,E)

.

A simple argument using uniform continuity and a partition of unity shows that this space is even dense, hence

C(K ) & E A C(K , E ) for each Banach space E.

(3) If K and L are compact topological spaces, the same reasoning with norming subsets shows that C ( K ) C& C ( L ) c C ( K x L ) holds isometrically. The Stone-Weierstrass theorem gives density, hence C (K )& C (L )

C(K x L)

C(K,C(L))

- where

the latter follows from example (2) (check that the identifications are the “natural” ones!) or by direct verification of C(K x L ) = C ( K ,C(L ) ) .

(4) It is impressive to see how the natural mappings usually fit together and give new information. This is one reason for the power of tensor product methods. Consider the following mappings for the space R(E, C ( K ) )of compact operators:

Z ( E , C ( K ) )c R ( E , C ( K ) )

T

rw+

C ( K ,E’) = C(K)&E’ [t * T’&]

4. The Injective Norm - note

(a

49

that 0 is defined on ff. Since

is isometric. For T = xl&f E E’ @ C ( K ) C R ( E , C ( K ) )it follows that

(there was a dualisation /transposition involved!) and hence, by example (l), @ is surjective on T ( E ,C ( K ) )and therefore

Every compact operator with values in C ( K ) is approzimable. 4.3. Concerning subspaces and quotients, E behaves “dually” to n. This is in some sense clear from the definition, but will be made much more precise in the sequel.

PROPOSITION: (1) If Gi c Ei are subspaces, then

i.e.

(2)

E

respects subspaces isometrically. In particular, E is finitely generated, i.e.

If Ti E C(Ej,Fi) are injective, El and E2 Banach spaces, then Ti@CTz: Ei&E2

-

FigCF2

is injective. 1

( 3 ) E does not respect quotients - not even isomorphically: If Q : F -U G i s a metric surjection, then, in general, idE QC Q need not be open and its completion idE& Q need not be sujective.

PROOF:(1) The mapping property implies that for z E G1 @ G2

and the Hahn-Banach theorem gives equality. For (2) consider the commutative diagram

4. T h e Injective Norm

50

Since all operators from E1&Ez are u(E{, El)-u(Ez, E4)-continuous (see Example 4.2.(1) - the completeness of El is used here) and Ti has a u(E{,El)-dense range, it is clear that T1&T2 is injective . (3) By the open mapping theorem it is enough to find a metric surjection Q : F G and an E such that

id,rp&Q :

F = Z ( E ,F) S

+.

--

$(E, G) = E‘geG QS

is not surjective - in other words: an approximable operator T : E -+ G which is not of the form Q S An “advanced” example runs as follows: Take E = G = L2 and Q : l l --y l z (every separable Banach space is a quotient of L1). It will be seen later on (Ex 11.18.) that every operator LZ -, LZ which factors through l l is Hilbert-Schmidt. But there are Lz which are not Hilbert-Schmidt. compact=approximable operators l z A more “elementary” example uses “local techniques” and the fact that .n does not respect subspaces: By Ex 3.29. there are finite dimensional spaces En,Gn c Fn and zn E En @ Gn such that

.

r(zn; En, Fn) For the natural quotient map Qn : FA norm of

1

< -T(zn; E n , Gn) n

-, FA/Gt

-

= Gk this implies that the quotient

cannot be surjective, because of the open mapping theorem and the fact that all spaces EL @e FA are l-complemented in lz(EL)&l2(FA) . 4.4. However,

-

respects metric surjections, i e . : If Q : F PROPOSITION: C ( K )Be surjection between normed spaces, then id @c Q : C ( K )@e F

-

C ( K )@e G

1 --Y

G is a metric


51

is a metric surjection and therefore also

“Philosophically” this is a consequence of the facts that L1 @r * respects subspaces, is dual to T and C ( K ) is in some sense dual to L1 by the Kakutani theorem (at least C(K)” = L l ( p ) for some p ) . Using the “local techniques” it is possible to make this precise and prove the statement of the proposition: First assume K to be finite, then clearly E

e”,@cE=%(E)

-

9

i.e. Enwith the maximum norm; individual lifting of i d @ Q : e”, @c F

(21,

...,z,,)E G” gives that

e”, @e G

is a metric surjection. [ If F and G are Banach spaces, this follows also from id @ Q’: (&(GI)’ = q(G’) r-+ q ( F ’ ) = (e“,(F))’

1. The strategy of the proof is to next show that C(K)looks

being a metric injection! locally like .

LEMMA: For every e

>0

and every finite dimensional subspace M C C ( K ) there are

S E 2 ( M ,em,) and R E 2(e, C ( K ) )for some rn E Bv with llRll llSllI 1 + E and RSf = f

for all f E M ; in other words, C ( K ) is a 2k,1-space (see 3.13. for the definition).

PROOF: (a) Take ( f 1 , ...,fn; zi,...,xi) an Auerbach basis of M (i.e. l f’II = llzill = 1 and (zi, fi) = S i j ) and E < 1/2 . Then there is a partition of unity in C ( K ) such that there are ti E K with (of(ti)= 1 and gk E M, := span { p i ) with

((oi)zl

llfk - g k l l w

5 E/n

k: = 1, -..,12

For S1 := C x f , @ k : M -+ M , it follows that for each f E M

c n

IIf - S l f l l w I

l(xk,f)l

llfk

- stllw

I ~11f11w

1

k=l

hence IlS1ll 5 1 + E . Since M, is isometric to (with the mapping g * ( g ( t i ) ) i ) , the desired factorization is nearly obtained: Only the embedding M, + C(K)needs a small pertubation. (b) To do this, first observe that

llflloo I Ilf - Slfllw + llslfllw I E l l f l l o o + Ilslfllw

I


52

11s;'

hence : S 1 M + Mil _< ( l - ~ ) - '. Now extend the functionals (SF')'zi E (S'M)' to 3/, E C(K)' with the same norm, note

and define

n

dB(S1fk - fk) : C ( K )

R1 :=

+

C(K)

k=l

which has norm operator

5 ~ l sz ~k IIziII l ~ llgk ~ - f k l l m 5 &(I - E)-'

< 1 . Therefore the

Rz := i d c ( ~-) R1 :C ( K )+ C ( K )

+

satisfies llRzll 5 1 e(1 - &)-' = (1 - &)-' and R & f k = f k , hence R 2 S 1 is the embedding M cr C ( K ) . Since S 1 ( M ) c Mo and Mo is isometric to em,, the desired factorization is found. 0 Note that the operator Rz : C ( K ) + C ( K ) which was constructed in the proof is even I ( l - ~ ~ R 1 ~ ~5) -(1-2e)-'. ' This means that the subspace an isomorphismand 11&-'11 M is contained in a subspace (namely R2M0) which is nearly isometric to em, .

PROOFof the proposition: Take a metric surjection Q : F -W1 G and

zo E C ( K )@ G. Then there is a finite dimensional subspace M c C ( K ) with zo E M @ G. Since E respects subspaces, E(zo;M , G ) = E ( 2 , ; C ( K ) ,C ) .

Now take 6 > 0 and a factorization R o S of M IlRll IlSll I 1+ 6 :

Choose yo E

@c

C ( K ) through some

F which lifts S @ idc(to)and satisfies

Then it is clear that yl := R @ id&,)

which was to be shown. 0

lifts to= RS @ id^(^,) and

em, with


53

-

It is clear, from the proof, that E QDC respects metric surjections for each Z&,,-space E, and also: that 2&,A-spacesrespect metric surjections up to the constant A. It will be shown in 23.5. that this property charactizes the 2L-spaces. 4.5. The proposition has interesting analytic consequences.

COROLLARY 1: For every metric sujection Q : F --y G between Banach spaces, every continuous function f : K --+ G on a compact space K and every E > 0 there is a continuous function g : K -+ F with Q o g = f and llglloo5 (1 E)llflloo .

+

PROOF: This follows from

The argument can be applied to function spaces other than C ( K ) , see the work of Kaballo [134], [135] and 23.5..

COROLLARY 2: If G c E i s a subspace, T E R(G,C(K)) and edension E R(E, C ( K ) ) of T with 115?11 (1 ~)llTll.

+

E

> 0, then there is an

PROOF:In 4.2.(4) the relation R(F,C(K))= F’&C(K) was shown, hence

For L,(p) this compact extension property was already shown in Ex 3.26.. Lindenstrauss ([171], p.92) has given examples showing that corollary 2 does not hold for e = 0 - not even for finite dimensional E. Consequently (look at the proofs above) also corollary 1 is false for E = 0 and even for finite dimensional F; see also Ex 3.21.(b). 4.6. The dual of ( E @3r F)’ is the space of continuous biIinear forms. Which bilinear forms cp are continuous even with respect to E , i.e. cp E (E C& F)’ ?

THEOREM: Let cp E ( E @ F ) * .Then ‘p E (E&F)’ if and only if there ezists a (positive) Borel-Radon measure p on BEIx B p (equipped with the weaklc-topologies) such that for all z E E €31 F (cp,

.>J =

B E /x BFt

(2’

€3I y‘, .)Cl(d(z’, y‘))

*

The measure p can be chosen such that 11pl1 = ~ ( B EX IB F I )= I / ’ ~ ~ ~ ( E B . F ) )

.

This is why bilinear forms in ( E B CF)’c (EmDr F)‘ = %il(E,F ) = Z ( E , F’) are called integral. The operators E -+ F’ associated with integral forms are called “integral” as well; they will be studied in detail in section 10.

54

4. The Injectiue Norm

PROOF: (a) By the very definition of the injective norm

is an isometric injection. Then if cp has such a representation, it follows that

F)’ a signed and, vice-versa, the Hahn-Banach theorem gives for every cp E (E measure p E C ( B J x~ B p ) ’ =: M ( B p x B p ) which represents cp as indicated. (b) But p may not be positive. To achieve this, the metric surjection I’ : M ( B p x B p ) ++ ( E @c F)’ has to be examined; denote by M;’ the convex, weak-*-compact set of probability measures on BE’ x B p and D C M t the set of Dirac measures. By the definition of the injective norm

I’(o)O = B E ~ . F holds; moreover M ( D ) c I‘(D) for all 1x1 5 1 and hence the convex hull and the absolutely convex hull of I’(D) coincide. The bipolar theorem gives (closure with respect to the weak-*-topology in (E @c F)’):

It follows that for every cp E (E @, F)‘ there is a p E M;’ with cp = IIcpllI’(p) - this was to be shown. 0 4.7. If p is a positive Borel-Radon measure on a compact set K, then

pI : C ( K ) & C ( K ) = C ( K x K )

-

-

K

Jh(w,w)p(dw)

is clearly continuous, i.e. integral on C ( K ) x C ( K ) ,and of norm p ( K ) . The theorem says that this is somehow the typical integral bilinear form (the details will be given in a moment). REMARK:If p i s a finite measure, then the “integrating functional”


is continuous and

55

= p(Q)

PROOF: Choose a lifting p : L,(Q, p ) -, 2,(Q, p ) (Maharam’s theorem, see Appendix B7.). Hence, a f ) := P ( f K 4 defines an element in Lb, of norm 1 for every w E Q. The claim follows from

Note that for these two bilinear forms the norm in ( E @< F)’ and ( E @r F)’ coincide.

COROLLARY: For ‘p E (El @I E2)* the following statements are equivalent: (a) (o i s integral. (b) There exists a compact set Q and there exist a positive Borel-Radon measure p and

PROOF:Clearly, if /?I is the “integrating” form on C(S2)8 C ( Q )or L,(p) @ L,(p), then, as noted before, CP = PI 0 (R1 @c Rz) is integral. It remains to show (a) f i (b) : The representation theorem gives a p on the compact set BE‘ x B p - and the operators [ R l z l l ( 4 , 4 ) :=

[R24(4 4 ) := f

give the factorization. 0

,

(2’121)

(4.2)


56

the norms T and E are not equivalent on E’ 8 F whenever there is an approximable operator which is not nuclear. For a long time it was an open problem whether or not there are infinite dimensional Banach spaces E such that

holds isomorphically! Pisier [224] succeeded in 1981 to construct such Banach spaces PI thus solving one of the toughest problems Grothendieck left behind for the functional analysts; P can be chosen to be separable and even to satisfy

P‘& P

-+.

%(PI P ) L $(PIP ) = P’& P

and this mapping is not injective, hence ?r and E are not equivalent on P‘ 8 P. (For the very first step of Pisier’s construction see Ex 4.15..) John [128] showed that every compact operator T : P -, P’ ist nuclear.

Exercises: Ex 4.1. Does t?j8ct?j= q hold isometrically? Check that ?r # E on 4: 8 tg Ex 4.2. Show that E and T are not equivalent on t 2 8 22 . Hint: Ex 3.28.. Ex 4.3. Consider the “Fourier matrix”

.

Show that (a) T ( w n ; G , G ) = n3I2 . (b) 1 S E ( w n ; q , G ) I n (c) llid : Q 4 c i6 ~1 12 n1I2 . (d) E and ?r are not equivalent on t1 8 t1 . Hint: For (b) use the fact that wn represents a unitary operator 4!j --..* 4!j . A result essentially due to Hardy and Littlewood [loo], [178] (see also Dineen-Timoney [57]) says that there is a constant c > 0 such that for all n E N c n ~ ( pI)

e ( w n ; q ,$1 5 nT@)

) l/p - 1/2 if 2 5 p 5 00 and y(p) = 2/p - 1 if 1 _< p I 2 (for p where ~ ( p = c = 1 works) . For (c) and (d) see also proposition 11.1..

2 2 even


57

Ex 4.4. For Q = E or a = A show that

Ex 4.5. Let E be a real normed space and define a complex multiplication on the real tensor product Ec := E fT by a o z := id^ @ Eiidc)(z) . Show that for a continuous R-linear E + F the operator

.

is @linear and llTc11 = llTl/ Ex 4.6. Show that every z E & & l 2

has a representation

with orthonormal systems (en) and (fn) and (An) E co . In this case the norm of z can be calculated as follows: E ( z ; & , & ) = max IAn[ . Hint: Expansion of compact operators. Ex 4.7. (a) Use the arguments from the proof of lemma 4.4. to show: If E is a normed 2:,A-spacel then the completion @ is an 2 ;,p p a c e . For the converse see Ex 6.6.. (b) For 1 5 p 5 00 every L p ( p ) is an 2:,,-space. Hint: Step functions, and (a). (c) For 1 < p < 00 the Hardy spaces HP are 2z-spaces. Hint: By the M. Riesz theorem they are complemented in Lp[O,27r] , see Duren [60], p.54 and 67. (d) If E is a Banach space which has an increasing family (E,) of subspaces which are X:,,-spaces and the union of which is dense in E, then E is a 2;,A-space. (e) Just for fun: If a (real or complex) vector space E is the union of a (not necessarily increasing) sequence of subspaces En,then every finite dimensional M c E is contained in some En . Ex 4.8. (a) Show that too @c E is isometrically contained in

{(z~) E l,(E)

1 {Zn} c E relatively compact}

and deduce from this that l,&l, c l , ( N x N ) is a proper subspace. (b) If PBv denotes the Stone-Cech compactification of N equipped with the discrete topology, then C(PBv) = t, (note that PnV is the Gelfand space of the C*-algebra 4,). Show that the natural mapping P(Bv x Bv) + PN x PN is not bijective. Ex 4.9. Show that for every measure p and Banach space E

holds isometrically, but the spaces are different in general. Hint: B L is ~ norming for L , ; both norms are dominated by T , hence it is enough to take step functions.

58

5. The Approzimation Property

E x 4.10. Show that A C E& F is bounded if and only if sup l(x'@ 3 / , A ) [ < 00 for all

x' E E' and d E F' . E x 4.11. A sequence (z,,) in E& F is a weak Cauchy sequence if and only if for all x' E E' and d E F' the sequence ((z'@d, zn))converges. Hint: Lebesgue's dominated convergence theorem. Ex 4.12. EBCF = E a r F holds isomorphically if and only if every continuous bilinear form on E x F is integral. Ex 4.13. Let G c E a subspace and (o an integral bilinear form on G x F . Then there is an integral bilinear form on E x F which extends (o . Ex 4.14. Every nuclear bilinear form (see 1.3.) is integral; the natural mappings E ~ &F' E'&F E& F

---

( E @c F)' ( E @e F')' (El F')'

all have norm 1 whenever E and F are different from (0). Ex 4.15. Let 1, : En L-* En+l be a sequence of isometric injections of normed spaces En such that sup IlI,, @ I,, : En BEEn 4 En+l Br E,,+lll =: C < 00 . Then

E S T L C E on E @ E for the space E :=

u,"==l E,, with its natural norm.

5. The Approximation Property Is every compact operator approximable? Is E'&F = % ( E ,F ) ? Is the trace defined for nuclear operators? Is an operator nuclear if its dual is? Is Tl&Tz injective if TI and Tz are? These questions find an answer with the help of Grothendieck's approximation property.

5.1. Let E and F be normed spaces. C,,(E, F) denotes C ( E ,F) equipped with the topology re, of uniform convergence on all absolutely convex compact subsets of E; it is a separated locally convex space with seminorms

PK(T) := SUP IlTKll

for all T E 2 ( E ,F )

where K C E is absolutely convex and compact; EL, := ZC,,(E,K ) .


59

DEFINITION: A normed space E has the (1) approximation property if idE E S ( E , E).=O ; in other words, for every absolutely conuex compact K and E > 0 fhere is a T E 5 ( E ,E ) with llTx - xi1 5 E for all t E K . (2) A-bounded approximation property for X E [1,00[ if idE E AB~(E,EI‘ co ;in other words, fhere is a net (Tv) of finite rank operators with llTv;ll 5 A and Tv + idE in &o(E,E). (3) metric approximation property := 1-bounded approximaiion property. (4) bounded approximation property if it has the X-bounded approximation property for some X 2 1

.

Some consequences are immediately clear: (1) E has the approximation property if and only if for all normed spaces F (or only F = E) the space S(E, F) is r,,-dense in Z ( E , F) . (2) E has the A-bounded approximation property if and only if there is a net (Tv)in S ( E , E) with llTqll 5 A and Tvz -+ t for all t E E . This follows from the fact that on equicontinuous subsets of t ( E ,F ) the pointwise

topology and rcocoincide. (3) If a Banach space E has a basis, then it has the bounded approximation property and an equivalent norm Il.llo such that (E, II.llo) has the metric approximation property. This norm is given by 1 1 ~ 1 1:= ~ sup llPnx[[,where the operators P,,are the expansion operators of the basis. 5.2. For many years it was an open problem whether or not every separable Banach space has a basis - or at least the approximation property. This was solved in the negative by Enflo [63] in 1972; the present “state of the art” is the following:

(1) All .fp (for 1 5 p 5 0 0 , p # 2 ) and co have a closed subspace without the approximation property ( 2 < p 5 00 and co : Enflo; 1 5 p < 2: Szankowski [262] ); Pisier’s spaces P mentioned in 4.8. cannot have the approximation property (by theorem 5.6. below). (2) Szankowski [261] constructed a separable Banach lattice without the approximation property.

(3) The space .C(&,&) does not have the approximation property (Szankowski [263], see also Pisier [221]). That this space does not have the approximation property is scandalous! All the “usual” spaces (which means: not artificially constructed) have the metric approximation property. For C ( K ) this follows from arguments with partition of unity, for L p ( p ) with averaging operators. Exception :

(4) It is not known whether the (also non-separable) space H m ( D ) of all bounded holomorphic functions on the open unit ball of Chas the approximation property (but it has the gp-metric approximation property for all 1 < p < 00, see Bourgain-Reinov

60

5. The Approximation Property

[17] and section 21)' while the disc algebra A ( D ) (the subspace of H w ( D ) of functions with continuous boundary values) has the metric approximation property. All the implications between the various approximation properties and the property of having a basis are either false or trivially true: ( 5 ) There is a Banach space with separable dual and with the approximation property but without the bounded approximation property (Figiel-Johnson [66]). (6) There is a Banach space with separable dual and the bounded approximation property but without the metric one (this follows from theorem 1 of Figiel-Johnson [66] with the aid of a construction due to Lindenstrauss, mentioned in this paper; see also Lindenstrauss-Tzafriri [177], I, l.e.20.). It is not known whether a Banach space with the bounded approximation property can be renormed in order to have the metric one. (7) There is a reflexive separable Banach space with the metric approximation property and without a basis; Szarek [264]. Szarek's method was recently refined by Mankiewicz and Nielsen [181]. A positive result will be proven in section 16 with the Radon-Nikodim property: (8) A reflexive Banach space (or a separable dual space) with the approximation property even has the metric approximation property. 5.3. The first question to be attacked is the approximation of compact operators:

E'&F = T ( E ,F ) C ff(E,F ) . When does equality hold?

PROPOSITION: (1) A Banach space E has the approximation property if and only if F'& E = ff(F,E ) f o r all Banach spaces F . (2) The dual E' of a Banach space E has the approximation property if and only if E'& F = R(E, F ) f o r all Banach spaces F . It is enough to check the conditions for reflexive F (see Ex 5.8.).

PROOF:(1) Since TBF is relatively compact for T E ff(F,E), the condition is easily seen to be necessary. Conversely, take K C E compact. Then there is a compact Banach disc L such that K is a compact subset of [L] (see 3.5., corollary 2). The embedding J : [L]+ E is compact = approximable by assumption, hence there is an operator T = xi 8 y k E EL]' 8 E with "

for all x E K

.

k=l

The dual J' : E' + [L]' has a weak-*-dense = [Ll:,-dense range; therefore the xi 's can be approximated on K by $k E El and 9 := dk8 y k gives the result.

ct=l

61

5. The Approzirnation Property

(2) If T E R(E, F ) , then by the approximation property of E‘ there is an S E E“ @ Et such that

llT‘ - ST’II 5 E and hence also llT” - T”S‘ll 5 E . If S =

x:

@J

a$, then

n

TI’S’ =

E E’

xi @I T”a$

@J

F

k=l

since T”(F”) C F ; it follows that [IT - Czi @ T“zrll 5 E . To see the converse it , is approximable. Since, by assumption must be shown, by ( l ) ,that every T E f f ( F E’) T‘ o ICE E R(E,F’) is approximable, it follows that T=(T’o/c~)’o)cF

is also approximable. 0

COROLLARY: If E’ or F has the approximation property, then

E’&F = R ( E ,F ) ( E ,F Banach spaces): each compact operator E

+F

is approximable.

5.4. Is E’&F = % ( E ,F ) ? Is the trace defined for nuclear operators (see 3.7.) ? To solve these types of questions, observe first that there are natural mappings

E’&F E‘&F E’&F

T ( E ,F ) L-, f f ( E ,F ) 4 Z(E, F ) c* %i[(E,F’) E’&F c* f f ( E F , ) c* E ( E , F )c* %[(E,F’) F’)’ ( E @J=F’)’ = %il(E,F‘) . -+ ( E

-

--y

-+

The last mapping comes from the fact that nuclear bilinear forms are integral, Ex 4.14.; all these mappings have norm 1 (provided E and F are # { O } , of course) and they are all the same in the following sense : just E’&F -+ %iI(E,F’) naturally. So, if one i s injective, all three are injective . The same holds for the three mappings

E@*F -+ T ( E ‘ , F ) R(E‘, F ) c* Z(E’, F ) c-,%i[(E‘,F‘) E&F + E&F L+ R(E’, F ) c* %i[(E‘,F’) E& F (E’ @Jc F’)’ 4 (E’ @ J F’)’ ~ = %i[(E‘,F‘) Q

-+

(all are injective if one is); also the same is true for the natural mappings

El& F’

+

... c* %ir(E, F ) .

The question is under which circumstances

E& F -+

operators

/

bilinear forms

62


is injective. The main result of this section, due to Grothendieck, answers this question in a very satisfactory manner - a result which is basic for the study of tensor products and operators. 5.5. For this, a description of the dual of &,(E, F) will be needed; the Banach space

E will be fixed. Denote by @F the natural mapping

LEMMA:*F : Ec,(E, F) -+ [(F'&E)', weak-*-topology ] is continuous and therefore

i s defined; if z = C;=,dn@ xn E F'gZ E i s an arbitrary expansion, then

f o r all T E E ( E , F )

PROOF: It has to be shown that for z = C,"==, y,' @ xn E F'& E the mapping

is continuous: Choose C,"==, 11&/,11

< 00 and 2,

+ 0;

then

and the latter is a continuous seminorm. Thus DF is defined and the formula follows from m

C (An,Tzn) =

(trF, idF'&'r~(z)). 0

n=l

If F is reflexive, @F is an isomorphism (onto) and therefore also D F .

PROPOSITION:

DF : F'&.,E

-

(&,(El F))'

i s surjective. If F is reflexive, then DF is even an isomorphism (of vector spaces).


63

PROOF:Given p E (C,,(E, F))‘ , take a compact set K C I’{zn} C E (where (zn) is a zero sequence, see 3.5.) such that

Consider J! : L ( E , F )

T

--

G(F) (Tzn) .

Then the inequality for p and the Hahn-Banach theorem give a @ = (&) E t , ( F ’ ) = bo (c,(F))’ such that @ o J! = cp . For z := Cn=l y,‘ 8 2, E F’&E it follows that

5.6. For every F there are natural mappings

En

&&zn (it is sometimes good to maintain the notation where Jp(C, & 8 2), = y ‘ z for the operator associated with y‘ 8 z).It follows that

for all (z’,y) E E’ x F and z E F’& E

. In particular,

Now E has the approximation property if and only if

since DE : E’&E + (&,,(El E)‘ is surjective, the Hahn-Banach theorem and (**) imply that this is equivalent to : If z E ker J E ~then ,

in other words, E has the approximation property if and only if ker J E I C ker t r E This means that the trace : E‘& E -, K factors through the nuclear operators.

.

64


THEOREM: For a Banach space E the following statements are equivalent: (a) E has the approximation property. (b) ker J E ~C ker trE , 1.e. there as a continuous linear functional i f E on %(E, E ) with = it-E o JE, (such a functional is called a trace on the space of nuclear operators).

tPE

(c) The natural mapping

J E ~: E'&E

-

%(E,E )

is injective.

(d) For all Banach spaces F (or only F = El) the natural map

is injective.

Usually the trace i r E on %(E,E ) will also be denoted by t f E - if it exists.

PROOF: The equivalence of the first two statements was already shown, clearly (d)& (c) TC (b) and it remains to show that (b) implies (d) : For this take z E F&E with JF(z) = 0 and 'p E ( F @,I E)' with the associated operator L, E E(F, El); note that

(see 3.7.). A look at the commutative diagram

shows (p, z ) = 0 , hence z = 0 . 0 It will be seen later on (21.9.) that it is enough to check (d) for separable reflexive spaces F only. It follows from 5.4. that E has the approximation property if and only if for all Banach spaces F (or only F = E')

is injective.

65

5. The Approzirnation Property

5.7. Other consequences are immediate.

COROLLARY 1: If E' or F has the approtimation property, then E'g3,F = % ( E , F ) holds isometrically (E and F Banach spaces).

-

If E' has the approximation property, then, by the theorem,

E

E'&E

is injective, hence - again by the theorem - E has the approximation property:

COROLLARY 2: A Banach space E has the approximation property whenever E' has it. The converse is false : It will be seen in Ex 12.9. that the space .&&& has a basis but its dual (ty&&)' = l!(.f2, &) does not even have the approximation property. Since it will be shown in 16.3. that coroilary 2 has an analogue for the metric approximation property, 5.2.(8) implies that the examples in 5.2.(5) and (6) also have duals without the approximation property.

COROLLARY 3: Let E and F be Banach spaces. If F is reflexive or: if F' or E has the approximat ion proper2 y, then

F'&E = ( Z c 0 ( E F))' ,

(via

DF)

.

We do not know whether or not DF is always injective.

PROOF: The reflexive case was already treated in 5.5.. If F' or E has the approximation property, then J p .FI& E -+ F'& F

-

is injective. Since, by the formula (*) in 5.6., ker DF C ker J p the result follows. 0 This is a duality result between E and ?r : if one defines the injective tensor product for locally convex spaces, then Ei0gt F = C,,(E, F ) (if E or F has the approximation property), hence

E&,,F' = (E:,&F)' whenever E or F' has the approximation property. The duality between be studied in sections 6 and 16.

E

and

?r

will

66

5.

The Approximation Property

5.8. If E A.!(Ei,fi) are injective operators and Ei are Banach spaces, then the operator T1&T2 : E1&E2 -+ Fl&F2 is injective (see 4.3.). What about Tl&Tz ? The commutative diagram

and the theorem lead to

COROLLARY 4: (1) If the Banach space El or E2 has the approzimation property, then Tl&Tz is injective whenever both E C(Et,Fi) are injective. (2) Conversely, i f for a Banach space E and its natural metric injection IE into ~ , ( B E I )the mapping idEn&IE : E'&E

---+

E'&e,(Bp)

is injective, then E has the approzimation property.

For (2) use the fact that eoo(BEn) has the approximation property.

n=l

and this shows at the same time that N ( T ' ) 5 N ( T ) .

PROPOSITION: Let E and F be Banach spaces, E' with the approzimation property and T E C ( E ,F ) . IfT' is nuclear, then T is nuclear and N ( T ) = N ( T ' ) holds. PROOF:First recall that F&rE' Q F"&E' is a metric injection (3.9.). For a functional 'p E (F"&E')' the following diagram holds (recall that transposition for tensors = dualization of operators): E t 6 r F L C ( E ,F ) 3 T

I

I. 3


67

The three horizontal maps in the middle are injective since E’ has the approximation property. If T’ is nuclear, there is a t E F”& E’ with J ( z ) = T’ . It must be shown that I E E’& F : for this, take, by the Hahn-Banach theorem, a cp E (F”& E’)’ which is zero on E‘&F ; this means precisely that L, : F“ -t EN is zero on F. Since T‘ is nuclear, T is compact and hence T“(E”) c F, which implies that L, o T” = 0 and the diagram shows that (cp, t) = 0 . 0 This result will be improved in 22.8.; but in general there are non-nuclear operators the duals of which are nuclear (see 16.8. and 22.9.). For the nuclear norm it may even happen that N(T’) < N(T) for finite rank operators (see Ex 16.10.)!

Exercises: Ex 5.1. ( 2 ( E ,F ) , r6)‘ = E @ F’, where rs is the topology of pointwise convergence in 2 ( E , F ) . Hint: Use the Hahn-Banach theorem as in proposition 5.5.. Ex 5.2. (a) The completion ?J! of a normed space E has the X-bounded approximation property if E has it. (b) For every normed space E there is a net (Tq)of finite rank operators which converges pointwise to the idE. Hint: Finite dimensional subspaces are complemented. (c) Every normed space of countable dimension has the approximation property. Hint: Banach discs are finite dimensional. (d) The statement (a) is not true for the approximation property.

Ex 5.3. A separable Banach space E has the bounded approximation property if and only if there is a sequence (Tn) in $(El E ) converging pointwise to i d E . Ex 5.4. If E and F are Banach spaces with the approximation property (resp. bounded a.p.; metric a.p.), then E& F has the approximation property (resp. bounded a.p., metric a.p.). Hint: Lift compact sets.

Ex 5.5. The property of Banach spaces having a basis, the approximation property, the bounded or the metric approximation property is not stable under forming subspaces or quotients. The bounded approximation property and the approximation property are inherited by complemented subspaces. The latter is not true for having a basis this follows from a result of Pelczyriski’s [200]together with Szarek’s counterexample mentioned in 5.2.(7). Ex 5.6. (a) Every abstract L-space has the metric approximation property. (b) Using the fact that E has the metric approximation property if E’ has it (this will be proven in 16.3.), show that abstract M-spaces have the metric approximation property.

68


Ex 5.7. If E has the A-bounded approximation property for all X the A,-bounded approximation property.

> A,,

then it has

Ex 5.8. Use the Davis-Figiel-Johnson-Pekzyhski factorization theorem (see 9.6.) to show that in proposition 5.3. it is enough to test for reflexive spaces F .

Ex

5.9. Let I : E2

1

1

G be a metric injection, Q : E --y El a metric surjection and T E ff( El ,E2) such that I o T and T o Q are approximable. Is T approximable? Ex 5.10. (a) If F C F” is complemented and T E C ( E ,F ) with TI E %(F’, El), then T is nuclear as well. What about the nuclear norms ? (b) If F is reflexive, then T E C ( E ,F ) is nuclear if and only if its dual is; moreover, N(T) = N(T’). (c) If T’ is nuclear and S weakly compact, then S o T is nuclear. Ex 5.11. The natural map J : E’&F -+ %(El F ) is a metric surjection. Is it possible to find a finite rank operator T E %(El F) such that for each n there exists zn E E‘& F (or El @, F ) with “(Zn; E‘, F ) 2 n and J ( Z n ) = T ? Ex 5.12. Take z E E&F such that for all ‘p E E’ L,

Is it true that z = 0 ? Under which conditions ?

Ex 5.13. Let E and F be Banach spaces, G C E a closed subspace with canonical injection I and assume that F‘ has the approximation property. Then the following two statements are equivalent:

(a) Each T E C(F,G ) is nuclear if I o T is nuclear. (b) F’ @, E induces an equivalent norm on F’

@r

G

.

J

Hint: F’&G L, F‘&E 4 C(F, E ) are injective and M := J-’(C(F,G)) is closed, hence (a)n(b) is simple. For the converse try to use an argument as in the proof of 5.9. and observe that for S E C ( E , F ” ) = (F‘&E)’ with SIC = 0 and z E M the operator U E C(F”, F”) coming from i d p & S ( z ) is just S o J(z)” = 0 .

Ex 5.14. If E is a Banach space with the approximation property, S E C ( E ,F) invertible, then ( t r E , T ) = ( t r F , STS-’) for all T E % ( E , E )

Ex

5.15.

.

If (e,) is an orthonormal basis of a Hilbert space, then

for each T E % ( H , H )

.

69


Ex 5.16. Denote by J : Et 8 F

4

5 ( E ,F) the canonical isomorphism and define a

new norm on $ ( E , F) by

N"(J(z)) := n(z;El, F )

.

(a) ($(E, F),No)'= C(E', F') holds isometrically with the duality bracket (trace duality)

(S,T)g,r::= t r p ( T o S') = trE'(s' o T ) (the trace is defined for finite rank operators).

(b) For T E 2 ( E , F )

(c) If F is reflexive, S E S ( F , E), then there is a T E C ( E , F ) with IlTll = 1 and

N"(S) = trE(S o T ) . (d) If E' or F has the approximation property, then No= N on $(E, F) (e) N = No on S ( E ,E) if and only if E has the approximation property

.

Ex 5.17. The operators T E $(E, F)"O C C(E,F ) are called compactly approtamable. With the notation of 5.5. and 5.6. observe that ( t r F , T o J p ( z ) ) = ( D ~ ( z ) , T ) f o r a l l T E C ( E , F ) a n d z EF'C3Eand

for all T E C ( E ,F) . Show that for T E C ( E ,F) the following statements are equivalent: (a) There is a linear functional 7 on % ( F ,E) such that for all z E F'6&E

(c) T is compactly approximable. Note that 17 is necessariIy continuous if it exists.

70

6. Duality of the Projective and Injective Norm

6. Duality of the Projective and Injective Norm The injective norm E was defined to be the “dual” of the projective norm 1 in the sense of trace duality; surprisingly enough, z is not dual to E in this sense - but nearly. A consequence of this restricted duality is the (weak) principle of local reflexivity which says that the bidual E“ of a normed space looks “locally” like E - a fact which is fundamental for the understanding of modern Banach space theory. 0

0

0

6.1. The fact that BE’~,,F’= r(BEl €3 B p ) implied, by the very definition of the injective norm E , that 1

c (E’ €3.,

E&F

F’)‘

.

Using the fact that the e-norm on E’ @ F can be calculated on norming subsets of the unit balls instead of BEII€3 B p (e.g. BE c B E I I ) ,it follows in the same way that 1

E’

F

( E @,. F’)‘ 1

F‘

.

(E€3., F)’

C--+

In this sense E is dual to a on all pairs (E, F) of normed spaces. 6.2. Again by the definition of E, each x’ €3 g‘ E BE’ €3 B p defines a functional on E @< F of “integral” norm I 1 (and x‘ @ y on E QDr F‘, ...). This implies that

11 E’@.,F’ 11 El@., F 11 EB., F

c

(E€3cF)’ 1151 ( E @ c F’)’ 11 I 1 ( E t a F’)‘ 11 I1

.

Note that this also follows from dualizing the natural embeddings in 6.1.. Since the space (E’I& F’)‘ is complete and continuously embedded into (El@.,F’)’ = C(E’,F”), the study of the approximation property showed that the extension to the completion

E& F

-

(E’

F’)’

may fail to be injective - in particular, E@.,F (E’BCF‘)‘ may not be an isomorphic embedding. However, it will be shown that these embeddings are even isometries if one of the spaces is finite dimensional. The proof of this needs some preparation. 6.3. The Banach-Mazur distance of two normed spaces is defined by

I

d ( E , F ) := inf {llTll l ~ T - l ~T~E C (E ,F ) bijective 1 E [I, 001

LEMMA:For each finite dimensional normed space M and of some such that d ( M , G ) I 1 + ~.

E

.

> 0 there is a subspace G


Clearly, by dualization, there is also a metric quotient of

71

with this property.

PROOF: Choose an €-net (z;,...,zL) in the dual unit ball B M ~ (i.e. min,1126 for all x' E B M I )and define

-z'II 5 E

Every finite dimensional space M (for short M E FIN) is isometric to K" with some norm. Choosing the functionals z i E (K")'= K" in the preceding proof with rational components, it is clear that there is a sequence (Gn) of normed spaces, each a subspace of some such that for every finite dimensional normed space M and E > 0 there is an n with d ( M , Gn) 5 1 E .

+

If (Gn)nEN is such a sequence, Cp := ep(Gn) @p ep(GL)

+

is said to be a Johnson space (1 5 p 5 m). Every M E FIN is (1 &)-isometric to a 1-complemented subspace of Cp. All the C, are reflexive for 1 < p < 00 and it is easy to show that for 1 5 p < 00 the Cp have the metric approximation property (Ex 6.3.). However, Cm does not even have the approximation property since (by Ex 6.5.) every reflexive separable Banach space is 1-complemented in Cm,hence also those which do not have the approximation property. 6.4. With the lemma the announced duality result can be demonstrated.

(canonical embeddings).

If F is finite dimensional, then the mappings in (2) and (3) are surjective.

72


PROOF: The third isometry follows from (2) and 3.9.(3). For the proof of (1) and (2) assume E to be finite dimensional. Since by 6.1. E'

aCF' = ( E @= F)'

holds isometrically, the isometry (1)follows by dualization. The statement (2) is more difficult to prove - it will be shown in three steps: (a) Assume first E = P ,, then by the examples 3.3. and 4.2.(2)

F' = q ( F ' ) = ( C ( F ) ) ' = (Pm (b) If I : G

F)' isometrically

.

P , is a subspace, then the diagram

commutes. Since ?r respects quotients and E subspaces metrically, the vertical mappings are metric surjections and therefore J is an isometry (clearly, onto as well). (c) Finally, s u m e that E is an arbitrary finite dimensional normed space. Choose, ! and a bijection T : E 4G with IlTll ~ ~5 1T E 1 . The~ by 6.3.,a subspace G c & diagram

+

implies 11J-'11 follows. 0

5 llT-'llllTll 5 1 + by~ the metric mapping property and the conclusion

Since q ( E , F ) = E'&F if E' or F has the approximation property, the following is a simple consequence of (3).

COROLLARY: If E and F are Banach spaces, one of them finite dimensional, then

(V, E))'

%(E, F ) 2+

holds isometrically, the duality given by the trace. Clearly (%(El F))' = (El F)' = C(E', F') in this case. We shall come back to the question of the duality between C,ff, and 9l in section 16.

~


73

6.5. This kind of duality between E and r produces an important tool for the local

techniques in Banach space theory: the "local reflexivity" of all normed spaces; more precisely:

WEAK PRINCIPLE OF LOCAL REFLEXIVITY: If M is a finite dimensional n o m e d space, F an arbitrary normed space, and S E C ( M ,F"), then for every E > 0 and every finite dimensional subspace N of F' there is an R E C ( M , F ) with IlRll 5 (1 ~)llSll such that for all x E M and y' E N

+

PROOF: It follows from the duality relations that

A check on elementary tensors shows that T

*r,

KF o

T

C(M ,F ) 4 C(M , F") = (C(M , F))" is exactly the canonical injection of a normed space into its bidual. The space

L :=

N

c M @F'=

(~(M,F))'

is finite dimensional, hence the result follows from

HELLY'SLEMMA:If E is a normed space, L c El finite dimensional and xt E El', E > 0 there is an z, E E with 1 1 ~ ~ 5 1 1 (1 +~)IIzflland

then for every

(XI,2,)

= (zff,2')

for all x' E L

.

PROOF:If Lo is the polar of L in E" and OL := Lo n E the polar of L in E l then it follows that dim E/OL = dim L < 00 and hence the natural mapping

is a surjective isometry. This implies the result. 0 6.6. Although in most cases the weak principle is sufficient, it is interesting to know

that the operator R can be chosen such that Sx = Rx whenever Sx E F, and injective if S is - with control of the norm of the inverse:


74

PRINCIPLE OF LOCAL REFLEXIVITY:Let M and F be normed, M finite dimensional, S E 2 ( M , F“) and N a finite dimensional subspace of F‘. Then for every E > 0 there is an R E 2 ( M , F ) such that (1) IlRll 5 (1 + 4llSll * (2) (z’,Ry) = (Sy,z’) for all x’ E N and y E M . (3) Ry = S y whenever Sy E F . If S is injective, then R can be chosen to be injective such that additionally (4) llR-lll

I (1 + 4llS-’ll

-

*

-

PROOF:Denote by J the embedding Mo := {y E M I Sy E F} M and define So : M , F by Soy := Sy. Now choose, according .to the weak principle, for every finite dimensional N C L C F’ an operator RL E C ( M , F) with 11R~l15 (l+e)llSll and ( d ,R L ~=) (Sy,z’) for all 2’ E L , y E M ; in particular, ( z ’ , R L J ~=) (.’,Soy) for all 2’ E L, y E Mo . Since (C(Mo,F))’= (ML F)’ = Mo@ F‘, this means that the net ( R L J )converges in 2 ( M o ,F) to So with respect to the weak topology on C ( M o ,F); it follows that So is in the weak=norm closure of the convex set

hence there is for every 6 > 0 a UoJ E C with

-

clearly, (z’,(So U o J ) y ) = 0 for all Q : M + M, and define

2‘

E N and y

E

M,

.

Now take any projection

Then (2) and (3) are obvious and (1) follows from

IlRll 5 l l s o - UJll

IlQll + lluoll I 6llQll+

(1 + ~)llSll

*

(b) Now suppose that S is injective and put A := llS-lll-l . Then there is an el-net (2i)!=i in the unit sphere of M and in the unit sphere of E’ such that


hence U, is invertible and llU;lll the perturbation W := R

75

5 (1+s)llS-'ll if €1 is chosen sufficiently small. Now

- Uo = (So- V0J)Q E 2 ( M , F )

which was found in (a) has to be controlled. The operator

is invertible if ]]U;lW]l

5 (1+ E ) 6 l]S1l1llQll is small and

by the Neumann series. It follows that

if 6 was taken sufficiently small. 0 An immediate consequence is the

COROLLARY: For every finite dimensional subspace M there is a subspace N c E with d ( M , N ) 5 1+ E .

t: > 0

c E"

of a normed space E and

This property is called: E" is 1-represented in E - and this notion serves the purpose that if F is represented in E, then F shares those properties with E which are determined by the finite dimensional subspaces. Clearly, the bidual E" is much more than l-represented in E : The subspace N and the bijection R : M + N can be chosen such that M n E c N and R t = t for all t E M n E . 6.7. An application: Every 'p E % i [ ( E ,F ) = (E @, F)' has, by 1.9., a canonical extension 'p" E %i[(E,FII) = (E @, F")' with llcpll = ~ ~ c p "What ~ ~ . if cp is continuous with respect to the injective norm?

EXTENSION LEMMA: Let E and F Be normed and cp E (E@,,F)'. Then 'p E ( E a t F)' if and only if

'p"

E ( E at F'I)'; in this case,

1

PROOF: Since 'p is the restriction of 'p" and E @c F E BCF" holds isometrically, it follows that 11'p11... 5 llcp"ll ... . Take, conversely, z E E a t F" and finite dimensional subspaces N c E and M c F" with 2 E N @ M . Then, by the weak principle of local reflexivity, there is (for every E > 0) an R E 2 ( M , F) with IlRll 5 1 + B such that for all y" E M and t E N

(d',&t)F",F' = (&Z,

R#)F',F

76

(L, :E


4

F' the linear operator associated with cp); this means

and therefore (y^,z ) = (cp, ( i d E C3 R ) ( z ) )for all z E E 8 F". It follows that

This lemma will be used in section 10 for the investigation of integral operators.

Exercises: Ex 6.1. Show that ( C ( E ,F ) ) ' & % ( F ,E ) under trace duality if the Banach spaces E and F are reflexive, and one of them is finite dimensional. In particular,

N(T : F

-+

I

E ) = sup((tr(S0 T)I S E C ( E , F ) , llsll I 1) .

Ex 6.2. Let T E $(F', E') , A E C(E', M I ) and suppose that M is finite dimensional. Then, given E > 0, there is an operator R E C ( M ,E ) such that IlRll _< (1 ~)llAlland R' o T = A o T . Hint: Apply the weak principle of local reflexivity to S := A' and N :=im(T). Ex 6.3. Check that the Johnson spaces C, have the metric approximation property if 1sp 1 (one says: do not have proper coiype/type), and infinite dimensional L 1 ( p ) do not have proper type. In the next section type and cotype of

86

7. The Natural Norm on the p-Integrable Functions

L,-spaces will be studied and some results will be presented which begin to show the importance of these notions. For p > 2 (resp. q < 2) there are no non-zero operators satisfying the type p (resp. cotype q ) estimates; see Ex 7.15.. 7.8. For the Hilbert transform 3 : Lz(R) + L2(R) operators (and spaces) such that A @ A ~T is continuous were studied by Burkholder [19], Bourgain [15] and M. Defant [50], [51]; for the Paley projection La[O, 2x1 4 La[O, 2 4 by Pisier [223], see also the surveys of Figiel [64] and Pelczyriski [204] during the IMC 1983. For the projection R of La[O, 11 onto the Fbdemacher functions see Ex 9.16. and section 31.; the operators T such that R @ T is continuous will be called K-convex. Neither the el-valued Fourier transform (7.5.) nor and the el-valued Hilbert transform (7.6.) was continuous - and el did not have proper type. That the counterexamples were always in is not by chance: PROPOSITION: Let T E &(L,(p), L,(v)) such that

T 8 idE : Lp(p) @A? E

-

L,(v) @ A ~E

is continuous for E = el. Then it is continuous f o r all Banach spaces E. PROOF:T 8 idll being continuous implies that there is a constant c 3 0 such that

for all f1, ...,fn E Lp(p). The density lemma 7.3. gives that

T 8 idt,(r) : Lp(p) @A? e i ( r )

-

Lq(v) € 3 e i~( r )~

has norm 5 c for all r. Since every Banach space is a quotient of some tl(I'), the fact that L, 8~~ respects quotients (theorem 7.4.) yields the result. 0 Together with proposition 7.3. this implies for each T E &(L,(p), L,(v)) (where v is supposed to be localizable if q = co)the following: If T @ idE is continuous for E = el o r E = too, then T is regular and T @ idE is continuous for all Banach spaces E.

-

-

7.9. In particular, T @ S : L, @A, E L, 8~~F is continuous for all S E &(E,F) if T @ idll is continuous. For operators T 8 S : L, 8 L, L, 8 L, there is another quite useful positive result in this direction - which even holds for subspaces of L,. For this denote for a subspace El C L,(p) and a normed space E2 by

El @Aq

E2

E2 (see Ex 7.17. for more the space El @ E2 with the norm inherited from Lq(p) information about this definition). This notation was already used in 7.7. for Rad.

87


THEOREM: Let p and v be measures, El C L q ( p ) a subspace, E2 and F normed spaces, T E C(Ei, F ) and S E C(E2,L p ( v ) ) .If 1 5 q 5 p 5 00, then

For Ei = L,(pi) this result is due to Beckner [8]; the present form is due to FigielIwaniec-Pelczyhski [65]. For another proof of Beckner’s result see 15.12.. Note the special case [IT @ : Lq @Aq Lq L~ @A, L ~ l l = llTll llsll

s

-

for q 5 p. This is false for q > p : see 26.3. for an example; some positive results will be presented in 18.9.’ 26.3. and 32.3..

PROOF: Recall that idL, @Sand T @ i d ~are , continuous. Now the continuous triangle inequality in the form of proposition 7.2. and the diagram

-

The general question about the continuity of operators S @ T: L, @ A ~E L, @A, F is rather involved; the reader will find at various places of this book more information about this problem - in particular, in the sections 18, 26 and 28 - 32. For a survey on this topic see [46].

Exercises: Ex 7.1. Let L p ( p ) be infinite dimensional (1 5 p 5 w). For a sequence (An) of pairwise disjoint integrable sets with positive measure the averaging operator P : L p ( p ) + L p ( p ) is defined by

7. The Notuml Norm on the p-Integrable Functions

88

Show that for any normed space E

.

E is a norm-one projection such that im(P @ i d E ) is isometric to eP E x 7.2. Use the fact that L,&L, = A(L1, L,) c (L1G3,L1)'to show that in general L,(Y) is not dense in L,(p @ Y) for finite measures (see also Ex 4.8.). L,(p) Ex 7.3. Let p be a measure, l/p = 1/r l / q and g E Lr(p). Then for every normed space E the multiplication operator

+

L,(P,E)

f

--

J%(P,E)

s*f

is continuous and has norm llgllr (if E # (0)). If Mg : L&) multiplication operator in the scalar case, then

llMB@ T : Lq @A* E

L p @ A ~Fll

+ L p ( p ) denotes

this

= IIgIIr IITII

€or all T E C ( E ,F). Hint: Holder inequality. Ex 7.4. If S E C ( L , ( p ) , G) and T E C ( E ,F), then

for each 1 5 p 5 00 . Ex 7.5. If T E C ( E ,F) factors through a positive operator To E C ( L P ( p l )Lp(p2)), , , the operator then for all S E 2 ( L p ( p )L,(v))

s @ T :Lp(p)

E

-

Lq(v) @ A ~F

is continuous. Ex 7.6. If E and F are Banach lattices, then every positive operator E F is continuous. Hint: Assume not, then there exist tn 2 0 such that lltnll 5 2-" and IlTZnII 2 2" . Ex 7.7. Use the metric surjection E'& F + %(E,F) to show that the kernel condition in the quotient lemma 7.4. is indispensable ! Ex 7.8. (a) If E C(Lp(pj), Lp(vj)), then -+

llT1@T2 : Lp(p1) @Ap Lp(P2)

(b) If S : L2(p) space, then

-+

-

Lp(v1) @Ap Lp(Y2)11

= 1IT1'111

llT2ll

.

L ~ ( Yis) a (not necessarily surjective) isometry and H a Hilbert

s 63 idH : L2(p) @ A ~H

-

L2(v) @ A ~H

is an isometry. Hint: (SflSg) = (flg) or look at A;

.


Ex 7.9. If H is a Hilbert space, then T z ( H )= C z ( H ) = 1 . Ex 7.10- T p , ( T )I T p , ( T )and c , , ( T ) I C , , ( T ) if 1 I P I IP Z I 2 I Ex 7.11. If S o To U is defined for three operators, then

89

q1

i qz S 00 .

Cq(SoToU)I Ils~lc,(T)llUll

- and the same holds for the type constants T, .

Ex 7.12. (a) If F is a subspace of E , then C , ( F ) 5 C , ( E ) and T , ( F ) I T,(E) . (b) If (E,) is a family of subspaces of a normed space E with dense union such that for all a,P there is a y with E, U Ep C E,, then

C , ( E ) = SUP C q ( l a: E , + E) = sup C , ( E , ) Q

.

Q

The same holds for the type constants T , . Ex 7.13. Use the unit vectors e k to show that T p ( G )2 n1-l/p . Deduce that no space E which contains the uniformly (i.e. there are subspaces M, c E with sup d ( M n , q )< oo) does have proper type (Pisier [217]proved the converse; see also Beauzamy [7]). In particular, all infinite dimensional L1(p) and abstract L-spaces do not have proper type; by 23.3. this is also true for &:-spaces. Ex 7.14. Show that: (a) sup, Tp(Pw)= 00 for 1 < p 5 2. Hint: Ex 7.13.and lemma 6.3.. (b) C , ( c ) 2 nl/q for 2 5 q < 00. (c) A space which contains the uniformly has neither proper type nor proper cotype. In particular, this is true for all infinite dimensional C ( K ) ,L,(p) and abstract M spaces (use local reflexivity); by 23.3. this is also the case for arbitrary Z&,-spaces. Ex 7.15. Show with x k = 1 that the space R has neither type p for 2 0 there is a finite codimensional, closed L c E such that (1 - E)llxI} _< } / Q f ( x ) }I } 1 1 ~ 1 1for all x E M . L can be chosen to be weak-*-closed if E is a dual space. Hint: Proof of 6.3.. (b) Show that for f E L, @ E

Ap(f; L,, E) = inf{A,(f; L,, M) I f E L, 8 M , M c E finite dimensional} = = sup{A,(id~,. @ @(f); L,, E / L ) I L c E closed, finite codimensional} .

Ex 7.17. (a) Let E i j c L,(pi,j) be subspaces such that Ei,l is isometric to Ei,a (for i = 1,2).Use the generalization 7.9. of Beckner’s result to show that El,1 @A, E2,1 is . shows that, for fixed p , the definition of El @A? E2 is isometric to E1,2 @A? E ~ JThis independent of the L p ( p i )in which the Ei are embedded. However, Rad@A,E is, in general, not even isomorphic to t 2 @A, E , hence El @A, E (where E is an arbitrary 1 Banach space) depends on the embedding El + L p ( p ) .

90

8. Absolutely and Weakly p-Summable Series and Averaging Techniques

(b) Use the fact that L1[0,1] has a non-complemented reflexive subspace E (e.g. isomorphic to &f via Rademacher functions) to verify that A, is not equivalent to ?i on E €4 Ll[O, 11. Ex 7.18. Consider the Fourier transform T : Lz(Z) + L2[0,2?i] Tek

for all k E Z

:= (2?i)-’l2exp(ik-)

.

Show that

T €4 i b , : t 2 ( Z ) € 4 t i~ ~L 2 [ 0 , 2 ~€ ] 4 t i~ ~ is not continuous. Hint: C e k €4 e k . Ex 7.19. Recall from 7.3.that an operator between complex Banach lattices is regular if it is of the form S1 - S2 + i(S1 - 5’2) with positive operators Sj. (a) Show that the Fourier transform on L 2 ( R ) and the discrete Hilbert transform on .fp(Z)(for 1 < p < m) are not regular. (b) Let v be a localizable measure. Then all operators in 2 ( C ( K ) ) L,(Y)) , and all operators in C(L,(p),L,(v)) (where p is an arbitrary measure and 1 5 p 5 m) are regular. Hint: 7.3. and the mapping property of E . -+

8. Absolutely and Weakly p-Summable Series and

Averaging Techniques

This section treats various types of series in Banach spaces which are related to E and Ap. Rademacher and Gauss averaging techniques will be introduced, in particular the Khintchine inequality will be proved. As first applications a theorem of Orlicz’ on the convergence of series in L p ( p ) and the calculation of certain nuclear norms are given.

8.1. A sequence (zn) in a normed space E is called absolutely p-summable (1 5 p if 00 tp(zn; E )

:= t p ( 2 n ) :=

(C

0, but Talagrand [266] recently constructed a Banach lattice with the Orlicz property which does not have cotype 2. Note that, by Ex 8.l.(c) the Orlicz property of E means that there is a constant c satisfying

+

for all 2 1 , ...,z, E E . In Ex 8.11. the reader finds a nice application of Orlicz’ theorem to pointwise convergence of series.

COROLLARY: If E has cotype 2 and 1/r 5 l / q - 1/2, then

i.e. every weakly q-summable sequence in E i s absolutely r-summable.

In particular] this applies to all L, with 1 5 p

5 2.

PROOF:For A = (An) E Biq, the diagonal operator D,+: t, therefore] by the proposition,

+ t1

has norm

5

which implies

for all z i , ...,znE E

for all r

2 ro . 0

. For

l/ro := l / q - 1/2 the left supremum is t r 0 ( Z k ; E), hence

1;

104

8. Absolutely and Weakly p-Summable Series and Averaging Technique8

The idea of Orlicz in 1933 to use the Rademacher functions and Khintchine’s inequality for the investigation of Banach spaces may be considered to be the beginning of the “modern” averaging techniques.

Exercises: E x 8.1. (a) Show that for all 2 1 , ...,2, E E

E x 8.2. If (cn)is weakly psummable and ( a n ) E c,, then (CrnZ,) E tp&E = q f o ( E ) . E x 8.3. Give an example of a zero sequence in a Banach space such that

is bounded but not relatively compact. Hint: 8.2.. E x 8.4. (a) Use the fact that every continuous operator c, -, tl is compact to show that (co & c,)’ = ll&t1. (b) Pitt’s theorem states that 2(tp,tq) = ff(tp,tq) whenever 1 5 q q. Hint: 8.2.(1). to show for p, q E]1,00[that $(!,) (c) Show that e ( E ) C c,(E) if and only if e ( E ) = c e o ( E ) Hint: . Use the BessagaPekzyliski result mentioned at the end of 8.3.. (d) Every block basis sequence (yn) with respect to the unit vector basis ( e n ) in t,, (this means ark E span { e n k + i ,. . . , G , ~ + ~for } some 0 = nl < n2 < ... and llykll = 1) is weakly p’-summable. (e) Bessaga-Pelczyfa’ski selection principle: Let ( e n ) be a basis of a Banach space E with coeficient functionals ( e ; ) . If (yn) is a sequence such that JJy,,Il 2 c for some c > 0 and lim (eh,yn) = 0

cfo(t,)

n-m

8. Absolutely and Weakly p-Summable Series and Averaging Techniques

105

for all m E N ,then (yn) has a subsequence (yW(,,)) which is equivalent to a block basis sequence (z,) with respect to (en)' i.e. C(Qntn) converges if and only if C ( a n y q ( n ) ) converges. For a proof see Diestel [54], p.46. Use this to show the following result, which is due to Castillo [25]: Take 1 < p < 00. Then $(E) C c,(E) if and only if q ( E ) = q + ' ( E ) . Hint: An operator T : .fpt E is compact if and only if ~~2'yn~~+ 0 whenever yn converges weakly to zero. E x 8.5. Give examples of unconditionally summable but not absolutely summable sequences. --+

E x 8.7. Use Khintchine's inequality to prove that for all 21,...,2, E el C 4 .fl(zk;ez)

Ih%(zk;el)

*

Ex 8.8. Let 0 : 0,-* Dn be bijective; then for all f : Dn

a'

E x 8.9. (a) Show that for all normed spaces E and elements

21,

..., 2, E E

Hint: Integrate 11 x E k ( W ) g k ( V ) Z k I ( P = 11 c & k ( w ) signgk(v)lgk(u)lzkllp with respect to dyndpn and dpndyn; Ex 8.8. and 11 J 11% i J 11 11% . (b) Show by using the unit vectors in P , that there is no converse inequality for any 1 I p < 00. Hint: Use the fact that max{lgk( k I n} converges to infinity y-a.e. in

I

RN. Maurey and Pisier 11871 showed that for a Banach space E the converse inequality , uniformly. holds (with an additional constant) if and only if E does not contain the P Note that, by Kahane's inequalities for the Gauss- and Rademacher functions, this need only be checked for one p. E x 8.10. Let H be a Hilbert space and (zn) a sequence in H . Use the parallelogram identity to construct by induction a sequence (tun) in {-l,l} such that 11 qzkl12 2 11Zk112 for all TI. Deduce that every unconditionally summable sequence in H is absolutely P-summable. E x 8.11. (Orlicz) Show that for every unconditionally convergent series in L l ( Q , p ) the series C(lfn(z)i2) converges for p-almost all 1: E St . Ex 8.12. (a) For 1 5 s 5 r _< 00 call T E C ( E ,F ) absolutely (rl s)-summing if there is a constant c 2 0 such that

c;=,

c(fn)

.fr(Tzk;F ) I CWa(Zk; E )

106

8. Absolutely and Weakly p-Sumrnoble Series and Avemging Techniques

for all finite sequences (xk) in E; the smallest such c is denoted by Pr,#(T).By using the idea of 8.9. show that Pr,g(T)5 C2(T) if l / r _< 1/s - 1/2 .

(b) Why does the definition of absolutely ( r ,s)-summing operators only make sense for s 5 r ? (c) Show that T E C ( E , F ) is absolutely (r,s)-summing if (Txk)E tr(F) for all sequences (zk)E Gt"(E). Ex 8.13. (a) Show with the unit vectors e k that

(b) Deduce that infinite dimensional L p ( p ) do not have type for any r have cotype for any s p and do not

Ex 8.14. (a) If 1 5 p 5 2 and T E C ( E ,F), then

= J ( C E ~ & . , C Q or T dualize Z ( ) ~the ~ ~tensor product Hint: Use that C(T'&.,xs) characterization. (b) Show that the converse inequalities are false - even with an additional constant. = .', . See Pisier [223] and also section 31 for positive results if T = idE . Hint: (c) With the weak principle of local reflexivity show that Cp(T")= C , ( T ) and Tp(T")= Tp(T)

for all T E C ( E , F ) . Hint: Use 4 E B p with (T"zg,y',)M ~ ~ T "forz the ~ ~type ~ ;use discrete Rademacher functions and the same idea.

Ex 8.15. For each finite set A c N define the Walsh function

if A # 8,and E ) := 1. Show that { E A I A c W finite 1 is an orthonormal basis of Lz(D,p). Hint: Do this first for D,. Ex 8.16. For 1 5 p 5 00 and elements 21,...,x, in a normed space E use the convexity of the function

(and Ex 8.8.) to show that (with CR = 1 and cc = 2)

8. Absolutely and Weakly p-Summable Seriee and Averaging Techniques

107

for all a1, ...,a, E P( . This is a very special case of Kahane's contraction principle (see e.g. Marcus-Pisier [183], p.45).

Ex 8.17. For 1 2; for 1 < p < 2 define a projection on L, n L2 by Pf = C(flrn)rn, estimate IIPf(lp5 $(C J(fJrn)12)1/2and apply Holder's and again Khintchine's inequality to

to aee that P is continuous on L, fl La C L,. The Rademacher functions are not complemented in L1[0,1], see Appendix C7.. l a k l for real 0). In the complex Ex 8.18. (a) Show that 11 Crkrkll&~o,l]= case these two norms are equivalent. Note that (in the real case) this result provides an isometric embedding Q C , l!g .

(b) Modify the second part of its proof to show that the Khintchine inequality holds also for 0 < p < 1 .

Ex 8.19. Take zi,...,zk E E'. Show that

where the latter minimum is taken over all factorizations

with diagonal operators DX . It is enough to take Dx with non-negative entries.

108

9. Operator Ideals

9. Operator Ideals The notion of a Banach operator ideal is simple but nevertheless important since it structures the way of thinking and hence also the proofs. It was Pietsch who in 1968 noticed the relevance of this notion, and in the following years he and his school investigated all aspects of the abstract theory of operator ideals - a research which culminated in the incredibly rich monograph “Operator Ideals” in 1978. In this section we collect some (not all) of the fundamental notions of the theory which we will need later -on.

9.1. An operator ideal 31 is a subclass of the class C of all continuous linear operators between Banach spaces such that for all Banach spaces E and F its components

%(E,F ) := C ( E ,F ) n 21 satisfy (1) %(ElF) is a linear subspace of C(E,F ) which contains the finite rank operators. (2) The ideal property: I f S E 21(Eo,F0),R E C(E,E,) and T E C(F,, F ) , then the composition TSR is in %(E,F ) .

The ideal property of certain classes of operators has already been used quite often. The ideal 5 of finite rank operators is obviously the smallest operator ideal and C the largest one. Note that for the theory of operator ideals we are only considering operators between Banach spaces. 9.2. For a good analytic theory of operator ideals one needs convergence in 2l. Recall that a function 11 : E -+ [0, m[ on a K-vector space E is said to be a quasinonn if there is a constant c 2 1 such that

If E is complete with respect to the uniform structure coming from the quasinorm, it is called a quasi-Banach space. For 0 < p 5 1 a p n o n n is a quasinorm satisfying

(3’)

IIZ + YllP 5 ll~llp+ IIYIIP

for all Z,Y E E ;

in this case c := 2’IP-l may serve as a constant in the quasi-triangle inequality (3). If (E, 11 11) is quasinormed with constant c and p ~ ] 0 , 1such ] that c = 2l/p-’, then

9. Operator Ideals

109

defines a p n o r m on E satisfying

llzllo I 1141I 2”’11~110

for all z E E,

hence for every quasinorm there is an equivalent p n o r m . We omit the tricky proof since it is not important for our purposes; see e.g. Pietsch [214], p.92.

9.3. A quasinormed operator ideal (%,A)is an operator ideal % together with a function A : ?2l+ [0, CQ[ such that

(1) AI~x(E,F) is a quasinorm f o r all Banach spaces E and F. (2) A ( i d ~ = ) 1. (3) If S E %(E,, F,), T E 2(F,, F ) and R E 2(E, E,), then the composition satisfies A(TSR) I IITIIA(S)llRII. If all the components %(E,F) are complete (with respect to A) ,then (a,A) is called a quasi-Banach operator ideal. A simple argument (with lp-sums) shows that the quasitriangle constant may be chosen independently from (E, F);therefore the construction in 9.2. shows that there is always a p E]O,1] and an equivalent A, such that (a,A,) is a quasinormed operator ideal and A, a p n o r m on each component. The advantage of a p n o r m is that it is continuous (for the topology it generates) while a quasinorm need not be! See Ex 9.2. for an example of this “quasinorm catastrophe”. If A is a norm (resp. p-norm) on each component %(E,F), then (2, A) is called a normed (resp. p-normed) operator ideal. In the case that the components are complete one speaks of Banach and p B a n a c h operator ideals.

The following statements hold f o r each quasinormed operator ideal PROPOSITION: (%,A):

llTll 5 A(T) for all T E Q . A(z’&y) = 11x/11 llyll f o r all x‘ E E’,y E F.

(1)

(2)

PROOF:(1) follows from the fact that for all z E BE and y’ E B F ~

and (2) with the same idea of factorization: 1121 ‘1

IlYll = llZt&Yl1

I A(Z‘&Y) = A((idKBY) 0 idK 0 4 I

I l l ~ ’ l l ~ ~ ~ ~ K=~1 l1 4 l1 Y IlYll ll . 9.4. It is rather simple to check the following very useful

CRITERION: Let be given a subclass Q of 2,a function A : 31 + [0,CQ[ and 0 < p 5 1. Then (a,A) is a p-Banach operator ideal if (and only i f )

110

9. Operator Ideals

n=l

Observe that (3) accumulates linearity of the components, the triangle inequality and completeness. 9.5. A norm on a vector space E is good for analytic investigations of this space if it is unique, up to equivalence, under all “reasonable” norms; if E is complete under these norms, the closed graph theorem usually helps. The ideal norm on an operator ideal is always “reasonable” in this sense since it dominates the operator norm:

PROPOSITION: If (a,A) and (%, B) are two quasi-Banach operator ideals such that TZL C 23, then there is a constant p > 0 such that for all T E Q

PROOF:Since the embeddings Q(E,F) and B(E,F) into 2 ( E ,F) are continuous, the closed graph theorem (it holds for quasi-Banach spaces) shows that there is a constant ~ E , F such that B ~ E , F A on %(ElF). An obvious indirect argument with tp-sums shows that the P E , F can be chosen to be uniformly bounded. 0 The reader who is not used to the “obvious” argument with ep-sums (metric injection of the En’s into tp(Em),metric projection, ideal property) should check the details. 9.6. Examples:

(1) The approximable operators a Banach operator ideal

(3,II 11);

3 (see 4.2.(1))

together with the operator norm form

recall

and that z ( E , F) = ff(E,F) if E’ or F has the approximation property (see 5.3.). The ideal ff of compact operators and the ideal ?IDof weakly compact operators form (again with the operator norm) a Banach operator ideal; ideals of this type are called closed operator ideals (=: their components are closed in X ( E , F)). A famous theorem of Calkin states that the only proper (i.e. different from 2 and (0)) closed ideal in the algebra .C(t2,e2) is the ideal of compact operators; this is also true for c, and ep (if 1 5 p < w), but not for Lp[O,I] (if 1< p < 00 and p # 2, see Pietsch [214], p.84). Weakly compact operators satisfy an important factorization theorem due to Davis, Figiel, Johnson and Pelczyriski 1371: If T E m(E,F), then there is a reflexive Banach

9. Operator Ideals

111

space G and operators R E C(E,G) and S E 2(G,F) such that T = SR and IlTll =

IlSll IlRll : T

E-F G For a proof, including the relation about the norm, see Pietsch [214], p.55. (2) The class % of nuclear operators (see 3.6.) with the norm N is a Banach operator ideal: Completeness is clear since N is the quotient norm of

the ideal property is easy to verify (and corresponds to the mapping property of the projective norm) and N ( i d ~ )= 1 (- it is just a joke to argue for this with K = K = % ( K ) ) By . the very definition of the nuclear norm and the fact that K

for all operator ideals, it follows that (%,N)is the smallest Banach operator ideal. Recall that the nuclear norm of a Glinear operator may increase if it is considered as an BZ-linear operator; this follows from N ( i d ~= ) dimK E (see 3.7.).

(3) Let (Oi,pi) be two measure spaces, p , q E [ l , w [ , G a subspace of &(PI) and S an operator in & ( G , L , ( p 2 ) ) of norm 1. Take Cs to be the class of all operators T E C ( E , F) such that @ T : G @A, E

+

Lq(pLz)@ A ~F

is continuous. Together with Ls(T) := IlS@T : @ A + ~ @A,II the class 2 , is a Banach operator ideal (this is an exercise). As a special case one gets: The ideal ('Xp,Tp) of type p operators and the ideal (C,, C,) of cotype q operators are Banach operator ideals. Other ideals of this type and results about 2s are treated in Ex 9.15., Ex 9.16. and 17.4..

-

9.7. An operator ideal Q is said to be injective if for each metric injection I : F G an operator T E C ( E ,F) is in Q if I o T E 2l. A quasinormed ideal (a,A ) is called injective if moreover, A(T) = A(I o T ) . Clearly, the Banach ideals of compact and of weakly compact operators are injective, but the ideals of approximable and of nuclear operators are not, see Ex 3.17., Ex 5.13. and Ex 9.13. for the nuclear operators and Ex 9.12. for the approximable ones.

9. Operator Ideals

112

PROPOSITION: Let (24, A) be a quasinormed operator ideal. Then there is a (unique) smallest injective operator ideal 24'"j which contains 24. If IF denotes the canonical injection F 4 e , ( B F ~ ) , then T E C ( E ,F ) belongs to mini if and only i f IF o T E 24. With Ainj(T) := A(IFT) (2l'"J,A'"]) is a quasinormed operator ideal which is quasi-Banach if (24, A) is quasiBanach.

PROOF:Define ai"j to be the class of all T E ' C ( E , F) such that IF o T E 24 and A'"j as in the statement. Then it is easily seen that (24'"j,Ai"j) is a quasinormed

operator ideal (for the ideal property use the metric extension property of l,(r)) with complete components if % has. If I : F 4 G is a metric injection, the diagram

shows

A'"~(T)= A ( I F T ) = A ( P I G I T ) 5 I ~ P ~ ~ A ' " ~5( IA'"~(T) T) , hence (24inJ,Ai"j)is injective. If 23 3 24 is any injective operator ideal, then clearly 93 3 24'"j and the proposition is proven. 0

(Dinj,A'"j) is called the injective hull of (24, A). In Ex 9.13. it will be shown that 92'"' is the class of quasinuclear operators.

COROLLARY: If the Banach space F has the X-ettension property and (%,A) is a quasinonned operator ideal, then B(E, F ) = %'"j(E,F ) for each Banach space E and A(T)5 AA'"j(T) for all T E 24'"J(E,F)

.

This is immediate. 9.8. The dual concept is the following: An operator ideal 24 is called surjective if for each metric surjection Q : G --y E an operator T E C ( E ,F) is in 24 if T o Q E 24. A

quasinormed operator ideal (%,A)is called surjective if moreover, A(T) = A(TQ). The Banach ideals R . and !2D are surjective, 5 is not (Ex 9.12.) - and also the nuclear operators do not form a surjective ideal; this may be seen as in Ex 3.17. or by the following argument: Take a metric surjection Q : G 4 E and F with the approximation property such that id^ is not an isomorphic injection, for example (see 1.5.)

113

9. Operator Ideals

I : tzL, L1[0,1] the embedding of the Rademacher functions, Q := I' : L , F = ez. NOW

+t 2

and

commutes; since the subspace %(G,F ) t l imO is closed in %(G,F), it cannot coincide with O ( n ( E ,F ) ) by the closed graph theorem; in other words, there is a non-nuclear T E C ( E ,F ) such that TQ E %(G,F ) : The ideal % of nuclear operators is not surjective.

PROPOSITION: Let (%, A) be a quasinormed operator ideal. Then there is a (unique) smallest surjective operator ideal which contains %. If QE is the canonical surjection el(BE) --y E , then T E L!(E,F ) belongs t o %"' if and only if T o QE E a. With A'"'(,) := A(TQE) (Be"',A'"') is a quasinomed operator ideal which is quasi-Banach if Banach.

(a,A) is quasi-

PROOF:The argument follows the same pattern as that in the case of the injective hull - the metric extension property is replaced by the lifting property of tl(I'), see 3.12.; the central point is to show that %'"', defined as indicated in the statement of the proposition, is surjective: For every E > 0 there is a lifting S, of QE

hence

+ E ) IAdU'(T)(l+

A"'(T) = A(TQE) = A(TQQGS,)IAsU'(TQ)(l

E)

and this gives AsU'(T)= AdU'(TQ).0 (%(""', A'"') is called the surjective hull of

(a,A) . The same diagram gives the

COROLLARY: If (%,A) is a quasinorrned operator ideal, then for all sets Banach spaces F %(el(r),F ) = n y e l ( r ) ,F )

r

and all

114

9. Operator Ideals

holds isometrically.

A description of Wurwill be given in Ex 9.14.. 9.9. If

(a,A) is a quasinormed operator ideal, then %dual

._ .- { T E 1: I T' E a}

Adual(T):= A(T') defines a quasinormed operator ideal, the dual ideal of (a,A). If (a,A) is quasiBanach, then (21dua1,Adua1) is as well. Well-known results state: R = Rdual and ID = IDdua1; ideals Q with a = adua' are called completely symmetric. !Yl c !TIdua1 (see 5.9.), but % # ndua1 as will be seen in 16.7.. The approximable operators 3 are completely symmetric, see Ex 9.12.. 9.10. Finally, the notion of the product (or the composition) of two ideals 2l and % will be needed: T E &(E,F ) belongs to the product 2l o (23 of Q and '13 if there is a Banach space G and R E %(ElG) and S E a ( G , F) such that

E-F

T

G If % and % are quasinormed, then

A o B(T) := inf A(S)B(R) , where the infimum is taken over all such factorizations. It is not difficult to see that (a o '13, A o B) is a quasinormed operator ideal. The completeness (if IZI and 23 are) follows from the

PROPOSITION: If (a,A) is a q-Banach and ((23, B) a p-Banach operator ideal, then (Q o % , Ao B) is an r-Banach operator ideal where 1/r = l/p + l / q .

PROOF:Only property (3) of criterion 9.4. needs to be checked: For a sequence (T,) in rzL o %(El F ) with 00

n=l

choose factorizations T, = S, o R,, through G, with

B(R,,)'I (l+&)[AoB(Tn)]' A(&)' I(1 E ) [A 0 B(Tn)Ir

+

115

9. Opemtor Ideals

and put G := lz(Gn) (denote by I,,, and Qm the canonical injections and projections). Again by the criterion

n=l

with

n=l

C,"=lTn = SR and

n=l

hence ( A o B)(CZ==,Tn) 5 [(1+

E)

C:==,[A

0

B(Tn)]"]"' - for all

E

>0 .

Unfortunately, the product of two normed operator ideals need not be normed. For example, if 3 is the ideal of integral operators (to be defined in the next section), the composition ideal 3 o 3 is not normed (see 29.8. for an argument showing this). Moreover, %o% is not normed; this followsfrom the facts that operators in %o%(E, E) have summable eigenvalues, but nuclear operators have, in general, only square-summable eigenvalues (see e.g. Konig [147], 2.b.14. and 4.a.5.) and 'YI is the smallest Banach operator ideal.

Exercises: Ex 9.1. If

11 11 is a quasinorm with triangle inequality constant

Ex 9.2. (a) If (E,11 11) is p-normed, then (b) For S E 1:define

11 11 : E

-*

c,

then

R is continuous.

Then (2, A ) is a quasi-Banach operator ideal, but A is not continuous on 2 ( E ,E) (with respect to the A-convergence) if E is infinite dimensional. Ex 9.3. Check that the ideals 'XI of completely continuous operators ((Tzn)is norm convergent if (zn) converges weakly) and X of operators with separable range are closed Banach operator ideals.

116

9. Operator Ideala

E x 9.4. If 'I has a sufficiently large cardinality, then the algebra C(!Z(I'),&(I')) has infinitely many closed ideals. Ex 9.5. If T E ff(E,F ) and n E IV,then there are reflexive Banach spaces E l , ...,En and operators To E ff(E,El), T k E f f ( & , E k + 1 ) and T n E R(En, F ) such that T = Tn o ... o To and = IlTnIl ...I1Toll. Hint: 3.5., corollary 2. E x 9.6. Let 21, ...,z,, y1, ...,yn E K and M the matrix ( z k $ J ( ) ; , f = 1 considered as an operator K" 4 K". If (a,A) is an operator ideal, calculate A(M : + Ex 9.7. If (%,A)and (B,B) are quasi-Banach operator ideals, R E C(Eo,E) and S E C(F, F,) such that % ( E ,F ) WEO,Fo) T SoToR

C).

--

is defined, then this map is continuous. E x 9.8. If % is a quasinormed operator ideal, the class space(%) of all Banach spaces E with i d E E % is called the space ideal of a. Show that it is stable under isomorphic bijections, complemented subspaces and finite Cartesian products. If % is injective (resp. surjective), then space(%) is stable under forming subspaces (resp. quotients). Ex 9.9. For T E C ( E ,F ) put where the infimum is taken over all factorizations T = SR through any Hilbert space. Show that the class 2 2 for all operators with Lz(T) < 00 is (with Lz) an injective and surjective Banach operator ideal. Ex 9.10. Let 24 be an operator ideal, T E C ( E ,F ) and G a Banach space. (a) Let S E % ( G , F )such that im T c im S . If S is injective or % surjective, then T E %. (b) Let S E %(E,G)such that (lTc11 5 cIlSeII for some c 2 0 and all z E E. If S has dense range or % is injective, then T E %. Ex 9.11. Let (%,A)be a quasi-Banach operator ideal such that for every metric injection I the operator T is in % if and only if I o T E 8 .Then 8 = ainjand A and A'"' are equivalent quasinorms. E x 9.12. (a) Use the fact that there exist Banach spaces without the approximation property to show that the ideal 3 of all approximable operators is neither injective nor -8uc surjective. Moreover, $nJ = 3 = ff. -dual

-

(b) Show that f is completely symmetric: 5 = 5. Hint: If T E ff(E,F) and S E $(E,F") such that llT - sll I E , choose an e-net T x k of TBE and apply the principle of local reflexivity to span{imS U { T e a } } . ( c ) Deduce from (b) that f is regular, i.e.: if ICF o T E z ( E , FN),then T E $ ( E , F). Ex 9.13. An operator T E C ( E ,F ) is called quasinuclear if there exist ( z i ) E t!,(P) such that for all z E E n=l

117

9. Operator Ideals

(a) With Q N ( T ) := inf{C,"=l IIziII 1 (2;) as above } the class Q9l of quasinuclear operators is an injective Banach operator ideal. (b) If T or T' is nuclear, then T is quasinuclear and Q N ( T ) 5 N ( T ' ) . (c) Give examples of quasinuclear, non-nuclear operators. Hint: Ex 3.17. and Ex 5.13.. (d) If F has the X-extension property, then Q%(E, F) = %(E, F ) and N ( T ) X Q N ( T ) . Hint: Factor T through a subspace of L1 and extend. QN). (e) ( ( n i n j , N i n j ) = (f) Quasinuclear operators are compact. (g) Use the fact that there is a metric surjection Q : G --* F such that Q' @, idt, is not an isomorphic injection (since L, is no Ci-space by 8.6. and Ex 7.14., this will follow from 23.5.) to show that Q9l is not a surjective operator ideal.

(a%,

(h) Q%(co, F) = 91(co, F) holds isometrically. Hint: Ex 3.13.. Ex 9.14. Show that T E C ( E ,F ) is in the surjective hull ndur of the nuclear operators if and only if there is a sequence (yn) E t l ( F ) such that

Hint: Ex 9.10..

Ex 9.15. (a) Show that the operator ideal (Cs,Ls) defined in 9.6.(3) is always injective. If G = L p ( p l ) , then (Cs,Ls) is even surjective. Hint: 7.4.. (b) What does (a) imply for the ideal 2, of type p and the ideal Cq of cotype q operators? Show that Cq is not surjective; in particular, if G C L p ( p ) , then G @ A ~ does not in general respect metric surjections. (c) Let 1 < p < 00. If T is an operator which factors through a subspace of a quotient of some Lp(v)(or: through a quotient of a subspace of some Lp(v)-this is the same, as will be seen later on), then for all S E C ( L P ( p 1 ) L , p ( p 2 ) ) the operator

s @ T : Lp(P1) @ A p E

-

Lp(P2) @Ap F

is continuous. Hint: Fix S, consider first T = i d L p and use (a). It is a celebrated result of Kwapieli that these T are the only ones which make S @ T continuous for all S as above; we shall prove this in Chapter 111, 28.4.. Ex 9.16. Denote by R the orthogonal projection of L z ( D , p ) onto the closed linear span Rad of the discrete Rademacher functions (see 8.5.). (a) If En := L1(Dn, pn), then

llR @ j d E , : L2(D,p ) @A2 Hint: Use the function f := inequality.

En

-

nF=,((l+ Q )

L2(D, p ) @Aa En11 2

8

*

E L z ( D , p ) @ En and Khintchine's

@J ~ 6 )

118

10. Integral Opemtors

(b) Deduce that

R €3 idt, : La(D,P) @

A ~ti

-

L2(D1P) @ A ~t i

is not continuous. (c) Following Maurey and Pisier [187], an operator T E S ( E , F) is called K-conwez if

K(T) := llR @ T : Lz(D,P) € 3 E~ -* ~ L2(D,P) @ A ~Fll< 00

-

Show that the class of K-convex operators together with the norm K is an injective and surjective Banach operator ideal. It follows that a space which contains the uniformly is not K-convex; it is a deep result of Pisier [223]that the converse is true. See section 31 for more information about K-convex operators.

Ex 9.17. Show that Q o (!I3 o C) A (Q o 23)o C for quasinormed operator ideals Q, '13 and

e.

Ex 9.18. Let (a,A) be a quasinormed operator ideal. Then ('zr(E,F),A) is complete if E or F is finite dimensional. Hint: llTll 5 A(T) 5 IITIIA(~~F).

10. Integral Operators Grothendieck's integral operators are related to the injective tensor norm in the same way as the continuous operators are related to the projective tensor norm: the associated (bi-)linear form on E €3 F' is continuous with respect to E (instead of r). This section gives the basic factorization theorem for integral operators and some applications. Many of the arguments will be the same as those given in Chapter I1 where the study of maximal operator ideals using tensor norms will be taken up. So the present section is partly a training program for later on.

10.1. Recall that the continuous bilinear forms on E x G (i.e. the continuous linear forms on E & G) are exactly the continuous linear operators E + G':

119

10. Integral Operators

Every operator T E L!(E, F) defines a continuous linear functional

'p

on E & F' by

and hence 'p = P K F oE~(E @= F')' is the functional associated with K F o T : E .-,F". Recall that (E @t G)' C (E @% G)'. An operator T E C ( E ,F) between Banach spaces is called integral if its associated linear functional P n F o on ~ E 8 F' is continuous with respect to the injective norm E (i.e. is integral in the sense of 4.6.). Define the integral norm by

Then the class 3 of integral operators with I as a norm is a Banach operator ideal; the ideal property of J comes from the mapping property of E , obviously I ( i d ~ = ) 1 and it remains to show the completeness. Since (E @c F')' + ( E a,, 3'')' = C ( E ,F") is continuous and C ( E ,F) c C ( E ,F") is closed, the space

3 ( E ,F ) = ( E ~3~F')'

f l

C ( E ,F )

(the intersection taken in C ( E ,F")) is closed in the Banach space (E acF')', hence complete (with respect to the induced norm).

PROPOSITION: J(E,F') = ( E Qc F)' (via T u* pT) holds isometrically for all Banach spaces E and F . PROOF:T E C ( E ,F') defines a functional 'p = pT E (E @= F)'. The canonical right-extension 'p" of 'p to E @ F" (see 1.9.) satisfies

: extension lemma 6.7. for the injective norm e says that hence 9" = P K F r o ~The cp E (E F)' if and only if 'p" E (E @c F")' (with equal norms). This is exactly the statement of the proposition. 0 10.2. One should always keep in mind the tensor product representations of 3 ( E ,F) and J(E, F'). These representations imply the

COROLLARY 1: T E 3 ( E , F ) if and only if coincide: I(T) = I(KFo T).

KF

o T E J ( E ,F"); moreover, the norms

Ideals with this property are called regular.

COROLLARY 2: T E J ( E ,F ) if and only T' E 3(F', El); moreover, I ( T ) = I(T').

120

10. Integral Operatore

In other words, (3dua',Idu0') = (J,I) by the definitions in 9.9..

PROOF: If T E C ( E ,F) and

hence

'p'

'p

is its associated functional on E @ F', then

= TI . Since, clearly,

I I ~ I ( F J ~ , E ) # =IIPII(E~~F~Y the result follows from the proposition. 0

10.3. The definition 3 ( E , F) = (E@.,F / ) / n C ( E F) , of integral operators, the isometric equality F'@., E = 5(FlE) and the trace duality 2.6.imply that T E C ( E ,F) is integral if and only if there is a constant c 2 0 such that

clearly, I(T) is the minimum of these constants c . 10.4. Some examples: If E or F is finite dimensional, the duality of E and 6.4.) gives that %(E, F) = El @r F c* ( E a t F')'

T

(theorem

holds isometrically. Since C ( E ,F) = Z(E, F ) = E' @ F in this case, one obtains the

PROPOSITION: If E and F are Banach spaces one of which is finite dimensional, then for all T E C ( E ,F ) %(ElF ) = J ( E , F ) and N(T) = I ( T ) .

The results in 8.8. imply

EXAMPLE: The canonical injection I : -!I PROOF: It has to be shown that

'p

L,

: 11 @8 -!I

co is integral with integral norm 1

-

K defined by

.

121

10. Integral Opemtors

is continuous with norm 1 . Since p is clearly continuous on l 1 18C1 % with norm 1, it is enough (by the density lemma 7.3.) to check the &-continuity of p on the ?r-dense Q I8 : Since it was just seen that I(id : &) = 1, the result subspace follows. 0

U;=,

-

e

The ideal property implies that

which later in this section will be seen to have interesting consequences. Which operators are not integral? Integral operators are weakly compact (see Ex 10.1.) - this provides a good number of examples; the product of two integral operators is nuclear (this will follow from the corresponding property for absolutely 2-summing operators) which gives further examples. To see a more concrete example of an operator, recall the Fourier matrices from Ex 4.3.: They provide operators

A, : P , with llA,II

-

5 n and I(A,) = N(A,) = n3I2. Take 1 < d < f i and define T, := (2d)-,Ap

:

Then llTrnll 5 d - , and I(Tm) = (4. d-l)m c, el

lz

-+

-*

00;

2m

.

since

= co(t,2m ;m = 1,2, ...) = tl(e:"; m = 1,2, ...I ,

the operator T := @Tm: co -+ can be defined: it has norm 5 (d - l)-', but is not integral since the integral norms of T, = &,TI, (canonical injection and projection onto the m-th component) tend to infinity.

10.5. The integrating form /?I : ( j ,i )-+

f g d p on L , (Q, p ) C%

JL (Q, p ) has norm

p(Q) by 4.7., hence, by definition, the canonical embedding

I :L o ( P )

&(PI

is integral and I(I) = p(Q) for all finite measures p. If p is a Borel-Radon measure on a compact space Q, this also implies that

I : C(Q)

-

Ll(Q,P)

is integral and I(I) 5 p ( Q ) = 11111 5 I(I), hence this canonical map also has i n t e gral norm p(Q). These are the typical integral mappings - this will follow from the description of the elements of (E BCF)' as integrals.

122


THEOREM: For a n operator T E C ( E ,F) between Banach spaces the following statements are equivalent: (a) T is integral. (b) There are a Borel-Radon measure p on a compact set R, operators R E C ( E ,C(R)) a n d S E Z ( L l ( p ) , F") such that

is commutative. (c) There are a finite measure space ( a , p ) , a n operator R E C ( E , L , ( p ) ) and a n operator S E C ( L l ( p ) ,F") such that

is commutative. Moreover, in both cases (b) a n d (c) the measure p and the operators can be chosen i n such a way that llRll= llSll = 1 (if E and F # (0)) and p(R) = I(T). In particular, it follows from the ideal property and llIll = p(R) = I(I)that

I(T) = minllRll

IlSll llIll

with factorizations as in (b) or as in (c).

PROOF:Only (a) c-+ (b) is not obvious: If 'p E (E BCF')' is the linear functional associated with T E 3 ( E ,F), then corollary 4.7. of the representation theorem for integral bilinear forms gives a Borel-Radon measure p on a compact Q with p(R) = I(T), and operators R1 E C(E,C(R)) and R2 E C(F',C(R)) with llRlll = llR2ll = 1 such that (with I. : C(Q) L,(R,p) canonical)

-

10.

Integral Operators

123

hence KFT= S I R which is the desired factorization. 0 This factorization theorem is very important for the understanding of integral operators.

10.6. The canonical maps L , ( Q , p ) Even more can be said:

-

L l ( Q , p ) and C(Q)

--+

Li (Q , p) are integral.

PROPOSITION: Let K be compact, p and v arbitrary measures. If E = C ( K ) o r L M ( p ) , then every positive operator T E 2 ( E , Ll(v)) is integral and I ( T ) = IlTll . This is also true for sublattices of El see Ex 10.5.. For the proof note first that theorem 7.3. implies that for all Banach spaces G

Therefore the conclusion follows from the following extremely useful tensor product characterization of integral operators:

10.7. THEOREM: A n operator T E 2 ( E ,F ) between Banach spaces is integral if and only if for all Banach spaces G (or only G = F')

is continuous. In this case,

PROOF:If T E J ( E ,F ) , then for each

'p

E ( F @r G)' = 2 ( F , GI) and

z

E E

@

G the

formula (PIT @ i d G ( z ) ) = (PL,oT, z )

holds (check for elementary tensors) and hence, since L, o T is integral,

which shows: x(T @ i d G ( z ) ;F I G ) I I(T)&(z; E , G ) - this is one direction of the statement of the theorem. Assume, conversely, that T @ i d p is continuous as indicated and let ' p E~ ( E @ r F')' be the functional associated with T : For z E E @ F'

(VT,2) = (h, T @i d ~ ~ ( . t ) ) (again a check for elementary tensors), hence

124

10. Integml Operators

with c := llT@i d p : E BEF' -, F was missing from the proof. 0

F'll. It follows that I(T) 5 c and this is all what

10.8. Another nice application of this characterization is the following

PROPOSITION: Let E be a Banach space, (zn) E q ( E ) and summable. Then

(2:)

E e ( E ' ) weakly

00

C I(zk,zn)I I

~ ( 2 ; E')wl(zn; ;

E)

*

n=l

PROOF: First recall that by 8.1. n

€(C.k@ p r ; G , G ) = wl((Yk);=l;G) k=l

for

91, ...,yn

in a Banach space G. Since I(q C?JC E

by the theorem, it follows that for

21,

E

and a shows that

It follows that for all zi,...,: z E E'

which implies the result. 0

COROLLARY: Ifp E %ii(co, co), then 00

n=l

PROOF: First note that

Pw) = 1 by 10.4. and hence

e", @ r Ell 5 1

...,2, E E

k=l

The duality of

4

C?Jr E

-

(G

El)' holds isometrically, hence


If L , E E(co,ll) is the operator associated with

'p,

125

then

and hence the proposition implies that 00

00

n=l

n=l

These results (also due to Grothendieck [93], p.95; see also Ex 10.7.) are connected with an inequality of Littlewood's [178]from 1930:

for all 'p E !B3i[(co,co)where c is a universal constant - and the exponent 4/3 best possible. This will be shown in 34.11. as well as an estimate for the constant c.

10.9. The last corollary was a consequence of the fact that N(C c, Pm)= 1 holds; this was obtained by Rademacher averaging. Using this method directly, one obtains the more general

PROPOSITION: For l / p

+ l/q + 1/r

= 1 and 'p E %i[(lP,k',) the following inequality

holds: Il(v(eklek))k€NlIfr Ill'pll

*

One may replace &, by co. The result and the following proof are due to Aron (unpublished), see also a recent paper of Zalduendo [278]. PROOF: For

Ck

:= p(ek, ek) choose (Yk and

,fjk

E

(if p, q, r E [l,m[). Rademacher averaging gives n

n

k=l

k=l

with

126

10. Integral Opetutors

If some of the p, q, r are 00, some obvious modifications have to be made. 0

10.10. The ideal of integral operators is neither injective nor surjective: This can be deduced from the fact that E does not respect quotients (4.3.) - but the details are omitted since in Chapter I1 the relationship between E and 3 will be studied systematically in a more general context. On the other hand the fact that E respects subspaces implies extension and lifting properties of the ideal 3, see Ex 10.4..

Exercises: Ex 10.1. Every integral operator is weakly compact, but not necessarily compact (and hence also not necessarily nuclear). Ex 10.2. For which 1 5 p Hint: 10.4..

I q 5 00

is the canonical embedding

e,

c,

l,, integral?

Ex 10.3. If F is complemented in F”, then an operator T E S ( E , F ) is integral if and only if it admits a factorization E -+ L, 4 L1 -, F ; what about the integral norm I ( T ) ? Examples: F reflexive or a dual space, F = L1(p).

Ex 10.4. (a) Eztension property of integral operators: If G is a closed subspace of a Banach space E and T E 3(G,F ) , then there is a ? E 3 ( E , F”) extending KFTwith I @ ) = I(T).Hint: E respects subspaces or: the factorization theorem. (b) Lifling property of integral operators: If Q : G --y F is a metric surjection and T E 3 ( E , F ) , then there is a F E 3 ( E , G”)satisfying Q ’ o p = K F O T and I @ ) = I(T).

Ex 10.5. Let M be a norm-closed sublattice of an L,(p) or C ( K ) (with K compact) and T E S ( M , L l ( v ) )positive, then T is integral and I(T) = llTll. Hint: Kakutani’s theorem shows that T“ satisfies the assumption of 10.6.. Ex 10.6. Let p be a finite measure. Then T E SC(E,Ll(p))is integral if and only if TBE is lattice bounded. Hint: For one direction use an h 2 ITtI for all t E BE and the measure dv := hdp to obtain a good factorization; for the other direction recall L1(p)) is regular. from 7.4. that every operator S E S(LI(V), Ex 10.7. Use 10.8. to prove the following result: If ( E n ) and (F,) are two sequences of Banach spaces and A E %ir(e,(E,),C,(F,,)), then

Ex 10.8. Let Ti E C(Ei, Fi) be two operators one of which is integral. Then the operator TI 8 Tz : El QDrE2 4 F1 8-FZ is continuous.

11. Absolutely p-Summing Operators

127

11. Absolutely p-Summing Operators The Banach ideal of absolutely p u m m i n g operators is in some sense at the heart of Banach space theory. For p = 1 these operators are due to Grothendieck (“applications prkintkgrale droite” in the Rksumd or “semi-intkgrale droite” in his thesis); they were generalized in the mid sixties (Mityagin, Pekzyriski, Pietsch) to p > 1; the basic results on factorization and composition are due to Pietsch. The great advantage of absolutely p-summing operators is that - with the aid of various handy characterizations - it is often relatively simple to check whether or not an operator is of this type. Outside Banach space theory these operators had an important impact on the theory of nuclear spaces and the theory of distributions.

11.1. An operator T E S ( E , F) between Banach spaces is called absolutely p-summing (for 1 5 p < co)if there is a constant c 2 0 such that for all finite sequences ( 2 1 , ...,z n ) in E

Define P,(T) to be the infimum (=minimum) of these constants and write ppfor the class of all absolutely psumming operators; for obvious reasons it is sometimes P m ) := (C, 11 11). Clearly, the defining inequality for absolutely useful to define p-summing operators also holds for infinite sequences. Recall from section 8 that for the absolutely p-surnmable sequences

(vw,

and for the weakly p-summable sequences

Now using the definition and an obvious application of the closed graph theorem, gives the

PROPOSITION: Let E and F be Banach spaces. For each T E C ( E ,F ) and 1 5 p the following staternenis are equivalent:

( 4 7-E

V A E ,F ) .

< 00

128


(b) T maps weakly p-summable sequence sequences into absolutely p-summable sequences: T N ( G W )c tP(F) . (b') T N :$ ( E )

-,tp(F)is

continuous.

(c) T N ( q o ( E )c) t P ( F ) . (c') TN : c i o ( E )-,t p ( F )is continuous. (d) i d @ T : tpaCE --* tp@A? F is continuous. (e) There is a c 2 0 such that f o r all n E IV

In this case,

Recall that A1 = r;Ex 4.3.implies that P1 ( i d l ; ) 2 consequence of the definition:

fi.The following is an immediate

REMARK: (vp, Pp) is an injective Banach operator ideai. However, it is not surjective, see 11.12.. For p = 1 part (c) of the proposition means, by 8.3., that T E t ( E , F ) is absolutely summing (:= absolutely l-summing) if and only if it maps unconditionally convergent series into absolutely convergent series. The characterization 10.7. of integral operators by operators of the form idG @T shows (property (a)) that each integral operator is absolutely summing and

11.2. The following example is basic: If K is compact and /J a (positive) Borel-Radon measure on K , then the canonical map

is absolutely p-summing and P p ( I p )= llIpll= p(K)'/P. This follows from

Much more is true, namely the


129

COROLLARY: Let E = C ( K ) for a compact K or E = L,(p) for some measure p. If u is any other measure, then every positive operator T E C ( E ,Lp(u)) i s absolutely p-summing and Pp(T)= IlTll.

PROOF: It was shown in theorem 7.3. that

hence part (d) of the proposition gives the result.

11.3. To a large extend the theory of absolutely p-summing operators is governed by the following result:

GROTHENDIECK-PIETSCH DOMINATION THEOREM: Let 1 p < 00 and K C BE‘ a a(E’,E)-compact norming subset. Then T E C ( E ,F) is absolutely p-summing if and only if there is a (positive) Borel-Radon measure p on K with

for all t E E. In this case, Pp(T) = minp(K)l/p, where the minimum is taken over all such p .

PROOF: If T satisfies such an estimate, then n

. n

hence PP(T) 5 p(K)’/P. Conversely, assume Pp(T)= 1, hence n

n

It is easily seen that on the Banach space C R ( K )of reatvalued continuous functions on K the functional w defined by

is sublinear and satisfies

130

11. Absolutely p-Summing Operetors

The Hahn-Banach theorem gives a p E CB(K)’with p 5 w . Since p ( f ) 5 w ( f ) 5 0 for f 5 0, it follows that p is positive, hence a positive Borel-Radon memure by the Riesz representation theorem. For z E E consider -I(-, z ) l P E C*(K):

.>IP)

5 w(-l(.1 41P) 5 -llT4IP which is exactly the desired estimate since p ( K ) = p ( 1 ) 5 1. 0 P(-I(-1

?

The theorem can be rewritten in terms of a factorization - this is why it is sometimes called the Grothendieck-Pietsch factorization theorem.

COROLLARY 1: Let 1 5 p < 00 and T E C(E,F ) . Then the following are equivalent: (a) T E F) . (b) For each norming weak--compact subset K C BE, there are a Borel-Radon measure p on K , a closed subspace G C L p ( p ) , operators R E C ( E ,C ( K ) )and S E C(G,F ) such that S I p R x = Tx for all x E E : E - F

T

(c) There are a finite measure space ( Q , p ) , a closed subspace G c Lp(p)l operators

R E C ( E , L,(p)) and S E C(G,F ) such that E - F

T

I n this case, Pp(T) = minllRll llSll l[plll/P, where the minimum is taken over the factorizations in (b) or over those i n (c).

PROOF: Take T E Yp(E,F) and choose p from the theorem such that for all x E E and Pp(T)= p ( K ) l / p . Define RX := (*,z) E C(K)

G := Ipo R ( E ) S ( I p o R(x)) := T z ;

131

11. AbsoIuteIy p-Summing Opemtors

then S is well-defined by the inequality and both operators R and S have norm hence (b) is proved. The implication (c) A (a) follows from the fact that

5

1,

vP.

The equalities for the norm come by the corollary in 11.2. and the injectivity of Pp)is an injective normed operator ideal. 0 from the fact that

(vp,

In special situations the factorization can be improved.

COROLLARY 2: Let 1 p < 00 and T E Z ( E ,F ) . (1) If p = 2 or F has the metric extension property, then T E TP,(E,F ) if and only if at admits a factorization

E-F

T

(where, as before, K is any weablc-compact nonning subset of BE,). In this case,

= C ( K ) , then T E V p ( E F, ) if and only if there are a Borel-Radon measune K and S E C ( L p ( p ) , F )with

(2) If E p on

C(K)

T

F

In this case, P,(T) = min llsllP ( K ) ~ / P .

It followsthat in general a T E Z ( E , F) is absolutely psumming if and only if it admits a

factorization

132

and PP(T)= minllRll


llSll IIIpll.The space C ( K ) can be replaced by L m ( p ) .

For p = 1 the factorizations show that the “typical” absolutely l-summing and the “typical” integral map is the same (see 10.5.), only the way of factoring through this mapping is slightly different: The definition of the injective hull of an operator ideal (9.7.) and Ex 10.3. give the first statement of the following result.

COROLLARY 3: (1) ( W ,PJ) = (?I

,P I )

- the

absolutely summing operators form

the injective hull of the integral operators.

(2) If K is compact, then J ( C ( K ) , F )= ? P l ( C ( K ) , F )holds isometrically for all Banach spaces F. Clearly, (2) follows from 10.5. and corollary 2(2). Another immediate consequence of the Grothendieck-Pietsch domination theorem is that:

COROLLARY 4: If 1 5 p 5 q < 00, then

VP C Pqand P q ( T )5 Pp(T) .

The inclusion ppC !Jlq isstrict, see Ex 11.23.(a). One can also show that the canonical mapping C[O,11 4 L,[O, 11 is in no !JJp for p < q (see Pietsch [214]).

11.4. A first application is the THEOREM: A Banach space E is finite dimensional if (and only DVORETZKY-ROGERS if) every Prnconditionally Convergent series is absolutely convergent.

PROOF:Since, by 8.3., q l ” ( E )is the space of unconditionally convergent series, it follows from ll.l.(c) that the condition implies i d E E !& C p2, hence i d E factors through a Hilbert space by corollary 2:

H Since S must be surjective, E is isomorphic to a Hilbert space. But this implies that E is finite dimensional because ( t e n ) E q l o ( t 2 ) \ t , ( t 2 ) . 0 See

Ex 11.9. for the Dvoretzky-Rogers theorem for p-summable sequences.

11.5. For many applications Pietsch’s multiplication theorem is important:


THEOREM: If 1 S r, p , q C

00

with 1/r = l / p

+ l/q,

then gpo gqC

PROOF:One may aasume Pq(T : E + F ) = Pp(S: F that

Wr(t1

133

, ...,2,; E) 5 1 implies that

+ G)

Vr and

= 1. It remains to show

If p is a probability measure on BE' with

define C k := (lB,, I ( z ' , ~ k ) ] ~ p ( d z ' )and ) ~ / note ~ that assume T x k # 0, all C k are different from zero. Now

c;)=,5 1. Since one may c:

The Holder inequality for l/r'+ l / q + l / p = 1 implies for these ( L Y E ) and z' E BEIthat

It follows from this that

(*) and this inequality conclude the proof. 0 This shows in particular that V z o '& c and, consequently, that the composition of 2n absolutely psumming mappings is absolutely lsumming if 2n 1 p . Clearly, the

134

11. Absolutely p-Summing Opemtors

vp

inclusion o p qc Ex 11.23.(b).

is true if 1/s

5 l/p + l/q but it is false if 1/s > l/p + l / q , see

11.6. In order to study the relation between absolutely psumming operators and other operator ideals more presicely recall that an operator T E S ( H 1 , H 2 ) between Hilbert spaces is called a Hilbert-Schmidt operator if for some orthonormal basis (e,) of H i a

notation: T E A 6 ( H 1 , H 2 ) . It is easy to see that the definition of the Hilbert-Schmidt norm HS is independent of the specific orthonormal basis:

HS(T)~=

c

W e uI

=

C I(ea I ~

* f pt2 )= ~HS(T*)~ ~

a,B

a*B

where T* : H2 -+ H I is the Hilbert space adjoint; in particular, T E 4 6 if and only if ir E 46. The basic facts about Hilbert-Schmidt operators are treated in Ex 11.12., Ex 11.13. and Ex 11.18..

PROPOSITION: T E C(H1, H 2 ) is Hilbert-Schmidt if and only if it i s absolutely 2summing. Moreover, HS(T) = P2(T) .

PROOF: If T E v z ( H 1 , H 2 ) , then for all orthonormal ( e l , ...,e n )

(2

IITekll2) 1'2

5 ~ 2 ( T ) w ( e 1...,e,;H1) = P2(T)

k=1

and hence HS(T) 5 P2(T). If, conversely, T E A 6 ( H 1 , H 2 ) and an orthonormal basis ( e , ) of H2 -

21,

...,z,

E H I , then

- with

n

k

k=l

a

a

k

Khintchine's inequality actually implies that

Vp(H1,Hz) = A G ( H 1 , H z ) for all 1 5 p < 00 (with equivalent norms) - this will be done later in a slightly more general setting (see 26.6.).

11. Absolutely p-Summing Operotors

135

11.7. For testing whether or not an operator is Hilbert-Schmidt the following lemma is of great practical importance.

PIETSCH LEMMA: If H is a Hilbert spacel p a Borel-Radon measure on a compact set K and S E C ( H , C ( K ) ) ,then T := 12 o S : H 5 C ( K ) A L z ( K ,p ) ISHilbertSchmidt and H S ( T ) 5 ~ ~ S ~ ~ p (. K ) 1 ~ 2 This follows from the fact that Pz(12) = p(K)'I2, the ideal property and the last proposition. But, interestingly enough, this can be proved directly: If ( e l l ...,en) is orthonormal, then

5

J llS% llLP(d4 5 IlS'll"(K)

by the Bessel inequality (and H = H').

COROLLARY: The product of three absolutely 2-summing operators is nuclear. PROOF: Consider the factorization

and use the fact that the product of two Hilbert-Schmidt operators is a nuclear operator (see Ex 11.13.). Later on this result will be improved to the composition of two Operators, thus generalizing the case of Hilbert-Schmidt operators (see also Ex 11.14.). It follows that the composition of 3n absolutely psumming operators is nuclear if n 2 p / 2 . In many concrete cases this is an appropriate tool to show that a given operator is nuclear! In particular, absolutely p-summing operators in a Banach space are power compact and hence have a discrete spectrum; the sequence of eigenvalues of a T E v P ( E E , ) is in tP if p 2 2 . This famous result of Johnson-Konig-Maurey-ktherford I1311 is the basis for the recent theory on eigenvalues of operators (see the monographs of Konig [147] and Pietsch [216]); in 34.10. one step of the proof will be presented. 11.8. A nice consequence of !&(H, H) = 4B(H, H ) is

Pz(idp)= HS(idq) = f i

.

136


In order to have this for all n-dimensional spaces the following test, seemingly due to Kwapied, will be useful.

PROPOSITION: For T E E ( E ,F ) and 1 I p < 00 the following are equivalent: (4 T E 9 p m F ) * (b) There is a c 2 0 such that Pp(TS)I cllSll for all S E A!($#, E) and all n E N . (c) TS E Pp(tp#, F ) for any S E Z(lp#,E ) . Moreover, Pp(T):= sup(Pp(TS) n E N , 1 1s: -+ Ell 5 1) .

I

5,

PROOF:(a) implies (c) and (c) implies (b) by a closed graph argument. For the remaining implication recall from 8.2. that for 21,...,2, E E

and therefore k

k

- C“llS(lP . 1 = ePwp(z,;
IlSoll IdZ I J;; . Conversely, the Grothendieck-Pietsch factorization theorem gives a factorization idE

E - E

H

5 n,


through a Hilbert space H with S is bijective, hence

137

llSll = 1 and Pz(R) I P z ( i d ~ ) .One may assume that

f i = PZ(idH) = P2(RS) 5 P2(R)IISII 5 PZ(idE) . 0 This result (due to Garling-Gordon [79]) has interesting consequences for the BanachMazur distance between Banach spaces of the same finite dimension and the existence of certain projections on finite dimensional subspaces of Banach spaces, see Ex 11.16..

11.10. Using Gauss averaging it is possible to calculate the p-summing norm of the identity map on 4; the following result is due to Gordon [82],[83] (see also Pietsch [214], 22.1.3.), who also calculated various other P p ( i d ~for ) finite dimensional E.

PROOF: The Khintchine-type equality for the Gauss measure says (8.7.) that for all

with vn(dw) := Ilglll&llwll;?i,(dw) (which is obviously a finite measure). This equality is the key to the proof (see also Ex 11.19.). The easy part of the GrothendieckPietsch domination theorem gives Pp(id~;)5 v,,(q)l/P . To obtain the converse, choose - again by the domination theorem - a measure p on the unit ball B of such that

It follows that

by the KhintchineGauss equality. This implies that

which is the claim. 0 The explicit calculation for p = 1 (for other p see Ex 11.24.) gives the following: If un-l is the size of the surface of the euclidean unit ball of En,then

138

11. Absolutely p-Summing Operntors

from 8.7.. Then (with an obvious notation for 6 )

and

What is the behaviour of these constants like as n + oo? Using again

and Wallis' formula a 2

- = lim n-m

one easily gets

since clearly Pl(idR;)

5 Pl(idRn+I)>this is also true for the odd n. It follows that:

COROLLARY:

11.11. Gauss averages will also give the

139


THEOREM: If H is a Hilbed space and T E 2(c, H ) , then

where KfG = @ an the real case and KEG = 2/+

in the complex case.

It is not by chance that the constants are the same as in the preceding corollary: In 20.19. it will be shown with the aid of this corollary that these constants are actually the best possible ones. The theorem (and the following corollary) is sometimes called the little Grothendiect theorem since it already has the flavour of Grothendieck’s fundamental theorem of the metric theory of tensor products which will be treated in Chapter 11.

PROOF:One may, by the ideal property of !l32, assume that H = (proposition 11.1.) that for all T E 2(c, 5)and all n E EV

c.It is to be shown

Since

by the duality of E and

A

(see section 6), this is the same as showing

for all S E 2(q,c). Fix S =

zcld e i and take

z

=

xk@ee, E

and hence

By the Khintchine-type equality 8.7. for the Gauss measure it follows that

@q.Then

140

11. Absolutely p-Summing Operutors

by the biorthogonality of the

gk

(see 8.4.). An appeal to 8.7. gives that

KLG

=~

b

1

~

~

~

,

~

~

b

1

~

~

2

,

~

is m indicated. 0

COROLLARY (“little Grothendieck theorem”): If E

is an C&,A-

or an C:,A-space and

H a Hilbert space, then

C ( E ,H ) = % ( E , H )

and

Pz(T) 5 KLG~ITII.

In particular, all operators from C ( K ) ,L , or L1 into a Hilbert space are absolutely a-summing. The “full” Grothendieck theorem will show that all operators L1 -+ H are even l-summing!

PROOF:(a) Assume first that E is CL,, and T E 2 ( E , H ) . To show the relevant !&-inequality take 2 1 , ...,x,, E E; then there is a factorization

em, and hence by the theorem

P2(TSR) 5 IIRIIP2(TS) 5 IIRIIKLGllTsll 5 KLGllTIIA(1 -k E )

.

(b) If E is an C : A-space, then the same reasoning shows that it is enough to demonstrate that for ali T E C ( c , H )

P2(T) 5 KLGllTll To do this take, by Kwapierl’s test 11.8., some S E C ( L 5 , q ) .Then P2(TS) = H S ( T S ) = HS(S’T’) = P,(S‘T‘) 5

5 p2(s’ :

+

e!>llT’ll 5 KLGlls’ll

11T‘11

by the theorem, hence Pz(T) 5 K L G ~ ~ T . IJ Note the simple dualization argument for ??32(., H) with the aid of Hilbert-Schmidt operators in part (b) of the proof. More systematic dualization arguments for operator ideals will be developped in Chapter 11.

141


One can also prove the corollary for Ci-spaces F instead of the Hilbert space H (with additional constant) just by adding a factorization of TS : -+ F through an 15. But Ex 11.17. implies that then F is isomorphic to a Hilbert space: nothing is gained but the 8n

REMARK:The 2i-spaces are exactly the Banach spaces isomorphic t o Hilbert spaces. This result will also be a trivial consequence of the general characterization of 2;spacee given in section 23.

vp

11.12. The operator ideal is injective - but not surjective: The little Grothendieck theorem implies that the canonical quotient map l,+& is in 9 2 but clearly idt2 is not since absolutely 2-summing maps are power compact.

-

The canonical embedding 11 4 1 2 is in ez

VZ (even in !?3,;

see Ex 11.5.) but its dual

co 4 1,

is not in pz : This follows from the fact that the sequence ( e n ) of unit vectors is weakly2-summable, aIthough C Ilenllzo = 00. This shows that the statements “T E !&” and “T‘E &” do not imply each other. Between Hilbert spaces, however, these statements are equivalent since v z ( H 1 , H z ) = AG(H1, H z ) . More generally, if T E f&(H, F),then T’E ?&(F’, H‘) (where H is a Hilbert and F a Banach space) since, by the factorization theorem, T factors through a Hilbert-Schmidt operator. Cohen [30] and Kwapieli [160] proved the following:

PROPOSITION: A Banach space E is isomorphic t o a Hilbert space if and only if the dual T’E f& whenever T E ‘&?(E,ez).

PROOF:One direction was just shown. Conversely, embed

I :~ - t , ( r ) . The idea of the proof is to show that the metric surjection I’ is in Qz, hence factors through a Hilbert space and consequently E’ is isomorphic to a Hilbert space. To see that Z’ E ‘J.32 take

vz(~w(r),~2)

(A) E q(t,(r)l)= .c(tw(r),~z) =

by 8.2. and the little Grothendieck theorem, and consider its associated operator S := CS/n&en. Since the composition SI E !J?z(E,~z), the assumption on E implies that Z‘S‘ E !$?z as well, therefore

C llI’Y‘n112 = C III‘S’enl125 P z ( l ’ S ’ ) 2 w z ( e n ;

12)’

n

n

< 00 ,

142


which shows that I' E V2

.

0

We shall come back to the relation between T and TI being in

VPin 25.9..

Exercises: Ex 11.1. Show that 3 c Vp and P,(T) 5 I(T) for all 1 5 p < 00. Ex 11.2. Prove that for all Banach spaces F

holds isometrically. Hint: T = C engTen.This result is true in many other cases: this can be shown with the Radon-Nikodfm property, see Appendix D7. and D8.. Ex 11.3. If a Glinear operator between complex Banach spaces is considered as &linear, does the p-summing norm change? Ex 11.4. For g E Lp(p)the multiplication operator

-

4 7 : ~w(c1)

LP(cL)

defined by M,(f):= fg is absolutely p-summing and P p ( M B )= llgllp. Hint: 11.2.. Ex 11.5. Use Khintchine's inequality to show that

p l ( q 443 I fi Pz(q+g)=l.

-

Deduce that the canonical embedding I : .tl .t2 is in C V Zwith Pz(I) = 1 5 PI(]),< 4.Recall from 10.4. that I is not integral! Ex 11.6. An operator T E C(E,F) is absolutely p-summing if there is a finite measure space (Q,p ) and a a(E', E)-measurable ip : Q -+ B p such that

In this case, Pp(T)I p(Q)l/P. Ex 11.7. For E = L w ( v ) show that T E v p ( E F) , if and only if it admits a factorization E 4 C ( K ) + L p ( p ) -+ F. Ex 11.8. (a) Every absolutely p-summing operator T is weakly compact and completely continuous (this means: if 2, 40 weakly, then llTznII + 0), but in general not compact. Hint: Lebesgue's dominated convergence theorem. (b) VpoVpC Ppo?2DCff, but I D o V p ff; moreover, V P= ?2DoVPif 1 < p < 0 0 .


143

(c) Every quasinuclear operator is absolutely l-summing, but not conversely. Ex 11.9. A Banach space E is finite dimensional if and only if $i"(E) = P,(E) (as sets) for some (and then for all) 1 5 p < 00. This is the Dvoretzky-Rogers theorem for p-summable sequences. Ex 11.10. Extension property for absolutely 2-summing operators: If G C E is a closed subspace of a Banach space E and T E T2(G,F), then there is an extension r?l E !432(E1F) of T with P z ( F ) = P2(T). Hint: Factorization through L , and 3.10.. Ex 11.11. If T E '&(El E ) , then there are a Hilbert space H, operators R E C ( E ,H) and S E C ( H , E) such that SR = T and T := RS is a Hilbert-Schmidt operator and such that HS@) 5 P2(T).One says that T and are related operators, a concept due to Pietsch which is extremely useful since related operators T and T have the same non-zero eigenvalues (with the same multiplicity). E x 11.12. (a) An operator T E C ( & , & ) is Hilbert-Schmidt if and only if H S ( T ) 2 = ck,( I(Teklef)12 < O0 * (b) An operator T E C ( L 2 ( Q 1 , p l ) ,L z ( Q 2 , p 2 ) ) is Hilbert-Schmidt if and only if there is a k E L 2 ( p 1 @ p2) with

In this case, H S ( T ) = ( l k ( (.~ ~ E x 11.13. (a) (46,HS) is a Banach operator ideal in the class of Hilbert spaces: (1) AB(H1, H2) C C(H1, Hz) is a linear subspace containing the finite rank operators. ( 2 ) HS(idK) = 1 . (3) H S ( T R S ) 5 ((TI(HS(R)l(S(( if the composition TRS is defined. (4) (AB(HI,H~),HS) is a Banach space. (b) If (e,) is an orthonormal basis of H , then

defines a scalar product on f i B ( H ,H) with associated norm H S : It follows that the space (AB(H, H), HS) is a Hilbert space. (c) The finite rank operators are HS-dense in AB(H1, Hz). (d) Hilbert-Schmidt operators are compact. (e) If T = C,"=lA n e n & f nwith two orthonormal sequences (en) and (f,) and a bounded sequence (A,) of scalars, then T is Hilbert-Schmidt if and only if (A,) E &. In this case, H S ( T ) 2 = C,"=l(A,('. (f) Use the spectral decomposition theorem for compact operators (see Ex 3.28.) to show that the composition of two Hilbert-Schmidt operators is nuclear and

N(ST) 5 HS(S)HS(T) .

144


(g) Show that (SIT) = trH(T*S)if S, T E A6;(H, H ) . Hint: Ex 5.15.. Ex 11.14. p z o ! & ( H ~ , H z ) C % ( H l , H z ) for Hilbert spaces H I and H z . Hint: FactorHilbert-Schmidt operators. This result also holds for arbitrary ization theorem for Banach spaces, see 19.4.. Ex 11.15. Pz(T) 5 I I T I I J a for all T E &(ElF ) . Ex 11.16. Prove the following corollaries of theorem 11.9.: (a) The Banach-Mazur distance from an n-dimensional Banach space E to is I f i . Hint: Use the fact that absolutely 2aumming maps factor through Hilbert spaces. (b)d(E,F) 0 and choose for j = 1,2 representations

such that

12. Definition and Examples

151

It follows that

(2) There is nothing to prove for l / p l + l / q l = 1, hence assume r1 < 00 and define

and (by Holder’s inequality)

12.6. To describe the completion of the normed space E @‘a,,q F recal1 from section 8 the notation $ ( E ) = ((211) E E~ I wp(zn) < 00)

$*‘(E) = ((zn) E C(f-9I wp ( ( Z n ) ? h )

-+

01

-

PROPOSITION: (1) For (An) E tr (in c, i f r = oo), ( t n ) E q ( E ) and (gn) E C , ( F ) the series

C ( X n t n 8 un) converges unconditionally in E&L,~,F .

152

12. Definition and Examples

( 2 ) For every z E EgUp,,F there are (An) E & (in co i f r = oo), (yn) E t$"(F) with

(2")

E elfl,"(E) and

where the infimum as taken over all such (finite or infinite) representations.

PROOF: (1) follows easily from the fact that (An) E tr (or co) forces the series to be an ap,q-Cauchy series - independent of the ordering. To prove (2) take for z E E~L+,F and e > 0 elements z, E E @ F with z = C,"=l zn and C,"==, ap,f(zn) 5 ( l + ~ ) a ~ , .~ ( z ) Choose (AT), (zr)and (yr) (finite) with zn = CiATz? @ gr and er

((Ar)i)

I (ap,q(Zn)(l+ &))'Ir

wql ((zr)i) I(ap,q(zn)(l+ wp'

((Yi")i) L

(ap,q(zn)(l+

&))"ql

E))"~'

.

Then

&I2

4, ( ( A T ) i , n ) wq' ((zT)i,n)Wpj (br)i,n) I a p , q ( z ) ( l + and (Ar)i,n E t, , (tr)i,n E Ct'"(E)and ($)i,,,E T J B o ( F. )Now (1) implies that m

m

n=l

n=l

n ,i

i

Moreover, if P denotes the seminorm defined by the infimum in the statement of (2), then for all z E E&,,, F . P(z) I aP,f(4 Conversely, if z = C,"='Anzn@ gn and z N :=

N

Anzn@ y,, then

4(An)Wq'(zn)wpl( ~ )n2 2 er

( < ~ n ) n N , iwp' ) ((2n)L)

wpl

( KLG - E ; this establishes the existence of 21,...,zm E P , with m

C l l T ~ ~ 12 1( K~ L Gk=l

E ) and ~

wa(z1,

...,zm;e",)= 1 .

174

14. Grothendieck's Inequality

(c c)'

The functional cp E ar defined by ( c p , ~8 y) := (Tz,Ty) has norm 1 and therefore (with some flk of modulus 1 in the complex case) m

m

m

It will be shown in 20.19 that the best (real and complex) KLG are actually the ones given in 11.11. : K f G = and K f G = 2/+, hence A

1.57079... = - < K E 24

1.27323... = - 5 K c

5
1 only if p is a finite measure!):

-

PROPOSITION: Let p , q, r E [l,001 with l/p + l / q = 1 + 1/r and p u finite measure on a set s1. Then the integrating functional P I : Lq’(P)

B LP4.4 fBg

--

IK Jfgdp

is ak,q-conlinuous. In other words, the embedding I : L q j ( p ) c* L p ( p ) is ( p , q ) -

factorable; momover,

Lp,q(I)

= IlIll = P ( W r = l l ~ I l l ( L q i @ ~ ; , q L , l ) ~

Clearly, p(R)l/OOis understood as being 1 if p(s1) > 0

*

.

PROOF: Since q’ 1 p , the embedding I is defined and continuous (the norm is p(s1)’jr) and 80 is the integrating functional /?I. Therefore, by the density lemma 13.4., it is enough to check the a;,,-continuity of PI on the dense subspace of step functions. Since a:,q is finitely generated, it is enough to take pairwise disjoint integrable sets Ci, ...,Cm C 0,the subspace

and M , := (M, 11 ll~,).Then it has to be shown that

18. (p,q)-Foctomble Opemtors

has norm

225

5 p(Q)'I'. For this define the functionals ' p k on M by

it follows that m

m

for all f i g E M and therefore

corresponds to , B , I M ~ M under the isometry (M,, it is immediate that

Mp/)' = Mi,

M i , . Since

it follows from the definition of ap,fthat

Thus, the first step towards the factorization theorem has been completed: the integrating functional (or the embedding), which will turn out to be the typical a;,,-continuous functional, actually is a;,,-continuous.

18.3. For finite dimensional M and N the typical factorization of p E ( M G q q N)' = MI

N'

goes as follows: the functional p has a; f-norm < c if and only if there exist operators R E C ( M , q t ) , S E C ( N , $ / ) and X E [0, oo[" such that

where 6~ is the bilinear form A k e k 8 e k and llRll llSll llbxll < c; this was already mentioned in 17.10.. The functional 6~ is positive (i.e. (ax f 8 g) 2 0 whenever f 2 0 and g 2 0). Since a;,( is finitely generated, cp E (E @ F)' is a;rq -continuous if and ~ uniformly bounded a;,,-norm (this is another only if the restrictions c p I ~ , g ,have

226

18. (p,q)-Factomble Opemtors

formulation of the maximality of Sp,().The ultraproduct technique (which will be explained now) allows one to study cp in terms of its finite dimensional parts. 18.4. Let I be a non-empty set. If 'XI is any filter on I, Zorn's lemma implies that there is an ultrafilter LL finer than 'XI. Recall that a topological Hausdorff space X is compact if and only if every ultrafilter on X converges; in particular, i f f : I -+ R is a bounded map, then the family (f(I)),€r converges along every ultrafilter U;the limit is usually denoted by limu f(6). Now fix an ultrafilter LL in 1. If (EL),Eris a family of Banach spaces, consider in the Banach space

the closed subspace c t ( E , ) of all (2,) with limu 11z411= 0

.

The ultraproduct of (E,),€r along LI is defined to be

(&)u

:=

foo(E')/C,U(EL)*

With the quotient norm this is a Banach space, the elements of which are denoted by ( 2 , ) ~(whenever ( % , ) , € I E &,(EL)), and after a moment's reflection one sees that

Il(2hll = 1p

112L11

In the case that each E, = E the ultraproduct (E,)uis called the ultrapower of E along U. Perhaps the reader is tempted to picture the ultraproduct as being something like tW/co - this is erroneous: The right analogue is c/co (where c is the space of convergent sequences). This indicates that ultraproducts are much smaller than one might at first guess; see in particular Ex 18.5. and Ex 18.10.. If (E,),E~ and (FL)&€r are two families of Banach spaces and T, E C ( E , , F L )with llT,II 5 c < 00 for all I E I, then

-

T' : &o(Ei) (2')

&o(J")

(T,d

is defined and llTrll 5 c. Since llZzL~~ 5 cllzcll holds, T I maps c f ( E , ) into c:(F,), hence there is a unique map T E f!((E,)u,(FL)u)such that

227

18. (p,q)-Factorable Opemtortr

commutes; T is called the ultraproduct (T,)u of the operators T, . It is clear from the description of the norm in the ultraproduct that

defines a continuous linear functional on iionr $ = limu 4'.

(E,)u

(F,)u with Il$ll = limu 11&11

. Nota-

PROOF: The mapping

is bilinear. With c := sup 11$,11,

I($', that

2' QD YJ

it follows from

- ($'YE' QD Y')I I 4 1 2 '

- 51 ' 1 IlV'li

+ 112;111 llsk - l7'11)

4 factors through the map $ of the statement. Since

one obtains

II$II 5 limu 11$,11

and even equality by choosing appropriate x' and y,

.0

If each E, is a Banach lattice, then .f!,(E,) is also a Banach lattice under componentwise ordering. Since c ! ( E , ) is a closed, solid subspace (since the norms are lattice norms), it follows (see Appendix A6.) that (E,)u is a Banach lattice under its natural quotient ordering. If each ELis an abstract Lp-spaces (1 I p < m), i.e.

1. + YllP = 1141"+ llYll"

if x A y = 0 ,

then their ultraproduct is also an abstract Lp-space: take z = ( c , ) ~and y = ( y r ) u with z A y = 0 ; then (2' A y') E c 2 ( E t ) and (2' - 2, A y') A (y' - 2' A y L ) = 0, hence

The same holds for abstract M-spaces

(112

+ yll = max{llzll, llyll} if c A y = 0).

228

18. (p,q)-Factorable Operators

The ultraproduct techniques in Banach space theory were developed around 1970 by various authors; recent references are Heinrich [lo41 and Sims [257].

18.5. To apply the ultraproduct technique for the description of a 4 E (E @a;,q F)’, with the help of its finite dimensional restrictions, define on the index set

I := FIN(E) x FIN(F)x]O, 11 an ordering by

(MI,NI,EI) I (M2,N2,&2) whenever MI C Mz, N1 C NZ and E I 5 ~2 , and take ?D the order filter which has as basis the sets { I E Ilk 2 I ~ and } LL any ultrafilter finer than ?D; note that L( is not unique. For z E E and I = (M, N , E ) define

x, := and E, := M. It follows that

{

z 0

--

ifzEM ifxeM

JE : E (E,)u z (ZJU is a linear map -which is an isometry by the definition of the norm in the ultraproduct. With F, := N for L = (M, N , e ) there is also an isometry JF : F + (F,)u Everything is now prepared for proving the

.

THEOREM: Lei l/p+ l/q 2 1 and p E ( E

F)’. Then there are strictly localizable measures p and v, operators R E 2(E,Lqt&>) , S E 2 ( F , and a positive functional 6 E ( L q I ( p )@* Lpl(v))’with (p, t @ y) = (6, Rx 8 Sy), i.e. Lpj(v))

-

It will follow from corollary 18.10. below that llvll.., = IIRll llSll llSll actually holds. (A linear functional p : E @ F K on the tensor product of two Banach lattices is called positive if (p,z @ y) 2 0 whenever z 2 0 and y 3 0.)

229

18. (p,q)-Factorable Opemtors

Now consider (see lemma 18.4.)

It follows, by the very definitions (recall the definition of 2, for t E E and similarly y,) that for z 8 y E E 8 F

and therefore

Since all 6, are positive, it is clear from the definition of the ordering in the ultraproducts and (.$:)u that 6 = limu6, is positive . By what was said above G, := (ce)u is an abstract L,-space if s < 00 and an abstract M-space if s = 00. The representation theorems (see Appendix A7.) show that G, = L , ( p ) (with norm and order) for some strictly localizable p if s < 00 and that for s = 00 the bidual GC is an L , ( p ) . Therefore, if p', q' E [1,00[, the theorem is completely proven. If, for example, q' = 00 then 6 factors

(q;)~

1

6 : G, BT Gpl

GL BrGpi = L,(p) 8rLp'(v)

A6

K

through its left canonical extension "6 which, clearly, is also positive and has the same norm as 6 . 0 The structure theorems imply that p and v can be chosen to be Borel-Radon measures on locally compact spaces. In terms of operators the theorem implies that for each T E f!p,q(E, F) there are strictly localizable measures p and v, operators R E C ( E ,L q i ( p ) ) ,S E C(Lp(v), F") and a positive operator U E f ! ( L q , ( p )Lp(v)) , such that

230

18. (p,q)-Factorable Operotors

18.6. Lp(id~,)= 1 by what was said at the beginning of 18.2.. Therefore this factorization and Lp(T) 2 IlRll IlUll llSll 2 IlRll IlU o Sll immediately imply:

COROLLARY 1: An operator T E C ( E ,F) is p-factorable if and only if IEF o T factors through some Lp(c() :

Moreover, Lp(T)= minllRll

llSll, the minimum taken over all such factorizations.

The measure can clearly be chosen to be a Borel-Radon measure. For p = 2 the fact that every subspace of a Hilbert space is l-complemented gives the following:

COROLLARY 2: T E C ( E ,F) is 2-factorable if and only if ii factors through a Hilbert space. Moreover, Lz(T) = min llRll IlSll, where the minimum is taken over all operators R E C ( E ,H) and S E C ( H ,F) with T = S o R (and H an arbitrary Htlbert space). In general, the factorization cannot be chosen to be through an L p ( p ) with a finite measure! Take T = idl,(q and a factorization

(t,(l?)is complemented in its bidual), so t,(I') is isomorphic to a subspace of L 1 ( p ) . If I' is uncountable,

c(

cannot be u-finite by a result of Pelczyliski [199], p.244.

18.7. For the ideal Jp N gp = ap,lof p-integral operators (1 5 p tion 18.5. reads

< 00) the factoriza-

231


Since, by 17.17., the positive operator U is even p-integral and absolutely p-summing with Pp(U) = I J U ) = IlUll, it follows from the Grothendieck-Pietsch factorization theorem (recall that L,(p) A C ( K ) for some compact K and use 11.3., corollary 2) that U factors

COROLLARY 3: For an operator T

E

E(E, F ) and 1 5 p
1 every

+

233


( p , 0)-factorable operator factors (into F“)through a positive operator L q , ( p )+ Lp(v) for some measures p and v (this is a consequence of theorem 18.5.), we shall proceed as follows: (1) Every positive L q t ( p ) + Lp(v)factors through a multiplication operator

-

Mg : Lq+)

-

Lp(v); Mg(f) := f g .

(2) Multiplication operators MB : Lq8(p) -+ L p ( p ) factor through an embedding L p ( p o ) for some finite measure pol hence are ( p , q)-factorable by what was said in 18.2.;moreover, Lp,q(MB) = llMBll= llgllr . Note that q’ 2 p . The fact (2) will be shown in 18.10. and (1) follows from Maurey’s factorization therem, which will be proven now. Lq8(po)

18.9. Every positive operator for all S E C ( E ,F )

ItT@

T : L q ( p i ) + Lp(p2) has the property (see 7.3.) that

-

: Lq(pl) @Aq

( p , q E [l,m] arbitrary). In particular, @

s : L q ( p l ) @A:

Ldv)

Lp(p2) @Ap

-

FII = llTll llsll

Lp(P2)@Ap Fll = IlTll llsll

for each measure v and each S E C ( L q ( v )F). , So T satisfies condition (a) of

MAUREY’SFACTORIZATION THEOREM: Let 1 5 p 5 q 5 00 and define r E [1,m] by l / p = l / q 1/r . For T E C ( E , L p ( p ) ) the following two statements are equivalent: (a) For every measure v and every S E C ( L q ( u ) F , ) (or only: S := idt,) the operator

+

T @ s : E @A:

Lq ( u )

-

Lp ( p ) @ A p F

i s continuous.

(b) There are g E L , ( p ) and R E C ( E ,L q ( p ) )with T = Mgo R. In this case, idt, : E @A; l q

-

Lp(p) @ A ~eqII = min IlRll IIgllr

1

where the minimum is taken over all factorizations as in (b).

PROOF: Assume first (b). Then it was proven in Ex 7.3. with the help of Holder’s inequality, that llMg

@

s : L q ( p ) @A:

Lq(’)

-

Lp(p) @A,,

Fll = 11~11rllSll

*

234

18. (p,g)-Factomble Opemtors

It follows that

has norm 5 llR 8 id : . * * 11 llMg 8 S : - 11 = llRll IlglirllS((. Conversely, assume (a). If p = q, the function g = 1 gives the desired factorization. So assume (a) with S = idt, for r < 00; this implies that c := llT 8 idd, : ..-I1 is a constant satisfying

...,Zn) in E . for all finite families (21, For q = 00 (and hence p = r < 00) the sum has to be replaced by a maximum, which means, in particular, that the increasing net ( f ~ ) fA := max{lTzl

12

E A } C &(p)

(where A C BE is finite) is norm-bounded by c. The type of order completeness which 5 c and

L , ( p ) has (see Appendix B5.) gives a non-negative g E L r ( p ) with llgllr lTzl 5 gllzll for all z E E. This shows that R E Z ( E , Lo3(p))defined by T Z

RX := 9

(5

:= 0)

has norm 1 and that T = Mgo R is a factorization as in (b) with IlRll llMsll 5 c. For 1 5 p < q < 00 (this is the remaining case) one has to find a g E L r ( p ) with 11gIlr I 1 and

Then Rx := 9-l . Tx and Mg again give the desired factorization T = Mg o R with llRll llMgll 5 cllgllr 5 c . For this, define s := q / p so s' = r/p . To apply Ky Fan's lemma (see Appendix A3.) observe that

K := {f E L&)

I f 1 0, 11fIld I 11

is weakly compact and convex. For every p-measurable h 2 0 the function [0, m] defined by if this integral exists if not

iph

:K

-,

235

18. (p,q)-Factomble Opemtora

is convex. The convex set {f E K I @h(f) 5 a} is norm-closed by Fatou's lemma from integration theory and hence weakly closed; this means that @ h is lower semiEE continuous. Therefore the set of functions @ z l , . . . , z n : K -+ [-m, 001 for 2 1 ,...,zn given by @=I,

...,sn(f):= cq

2

11Xk11'

-

12

ITzklqf-'@

k=l

k=1

is a convex set of concave, upper semi-continuous functions. For z1,...,X n E E the function f defined by

is in K and satisfies, by inequality (*),

(the case a = 0 is trivial). Ky Fan's lemma gives a function ...,z n E E the inequality a21,...,z n ( f o ) 2 0

fo

E

K such that for all

21,

holds. In particular,

aI(f0)2 0, which means that

J I T z l q f 7 d P 5 cql141q

1

for all x E E. The function g := f;lp satisfies llgllr 5 1 and (w). 0 The proof shows that T factors through a multiplication operator (as in (b)) if and only if (*) holds and this is how Maurey's factorization theorem was originally stated [186]. Recall from 7.9. that T C3 S is always continuous if q 5 p . Other factorization results in the spirit of this theorem will be given in Ex 18.14. and Ex 18.15..

18.10. For positive operators L,,

-

L, this implies:

COROLLARY: Let l/p+l/q 2 1. I f T : L q j ( p ) -+ Lp(v)i s a positive operator, then there are a finite measure po and operators R E 2 ( L q j ( p ) Lqj(po)) , and S E C(Lp(po), Lp(v)) such t h a t T = S o I o R

Ts

236

18. (p, q)- Factorable Operators

and IlTll

= llSll llIll IlRll . In particular, T

is ( p , q)-faciorable and L,,,(T)

=

PROOF: Since q' 2 p and T satisfies condition (a) of Maurey's factorization theorem (this was already noted at the beginning of 18.9.), T decomposes into T = M, o R, with llTll = IlgllrllRoll . It remains to factor the multiplication operator Mg : L,I (v) 4 Lp(v); recall that l/p = l/q' + 1/r . For this, take g1 := sign(g)lgl-'/q'

and

and the probability measure dp, := Igl'llgll;'dv liM9l Lq'W

-

l l K 2

and MB = Mgao [ I : Lq'(p,)

--

g2

:= Igl'/P

(recall 0/0 := 0). Then

L,'(clo>ll = l l ~ l l ~ r ~ q '

: LP(P0)

L P W I I = 11g11?"

Lp(p,)] o Mgl . It follows that

T=M,,oIoM,,oR,. Since llIll

= Lp,q(I)= 1 by 18.2., this factorization satisfies

LPAm 2 llTll = Il~ll'llRoll= l l ~ 9 a I IllW, I1 IlRoll 1 L IlMgaII I141 llM9i 0 R o l l = IIMg,ll LP,q(I) llM9i

0

Roll 2 Lp,q(T)

and everything has been proven. 0

18.11. Summarizing these results, one obtains the general factorization theorem for (p, 9)-factorable operators.

THEOREM: Let p , q E [l,oo[such that l / p + l/q the following two statements are equivalent:

> 1. For each operator T E E ( E , F )

(a>T E EPAK F ) * (b) There are u finite measure p , operators R E C(E,L q / ( p ) )and S E C(LP(p), F") such that K;F0 T = S OI 0 R E

T

F

F"

. In this case, LP,,JT)= minllSll 11111 IlRll over all such factorizations. Recall that the case l/p 18.6..

+ l/q

= 1 of pfactorable operators was already treated in

18. (p,q)-Factomble Operators

237

PROOF:If T has such a factorization, it follows from proposition 18.2. (and the regularity of maximal operator ideals, 17.8.) that

Conversely, if T E gP,,(E,F ) the ultraproduct theorem 18.5. gives a factorization

with some measures p and v, a positive operator U and

By corollary 18.10. the operator U can be decomposed into U = So o I o R, with LP,,(U)= llUll = IlS,ll 11111 llRoll and I : L q t ( p o )+ L p ( p o )for some finite measure p o . It follows that KFOT = S o S o 0 l o RooR and

LP,Q(T)L llsrl IlUll IlRIl

I tlS0 S o l 1 11I11 t w o 0 Rll 2 LP,,(T)

- the last inequality by the first part of the proof. 0 By examining the way in which the measure p is found it becomes clear that p can be chosen to be a finite Borel-Radon measure on a locally compact space R - and hence, even on a compact space (by passing, e.g., to the onepoint compactification of R).

If the range space F is 1-complemented in F” (or p = 2), then the factorization can be made into F. Simple manipulations show that:

COROLLARY: Take l / p + l / q > 1. For each

cp E ( E @r

F)’ the following statements

we equivalent:

(4 cp E ( E

@pa;,,

F)’

(b) There are a finite measure p , operators R1 E &(E,L , , ( p ) ) and Rz E f!(F,L,t(p)) such that (9,z @ Y) = L ( R i + ) . (RzY)~P .

In this case, l l c p l l ( ~ ~ F~ ), I = minllRlll l[R~Ilp(fl)’/~, the minimum taken over all such Psq factorizations.

238

18. (p,g)-Factomble Operotors

Exercises: Ex 18.1. Use the factorization 18.1. of T E C,,,(M,G> and the little Grothendieck theorem to show that for all 1 5 q 5 00

whenever H is a Hilbert space and E an arbitrary normed space. Ex 18.2. (a) An operator T E C(E, F) is in the injective hull f!? of the pfactorable operators if and only if T factors through a subspace of some L p ( p ) . Moreover,

where the minimum is taken over all R E C ( E ,G)and S E C(G,F) with T = S R and G C L p ( p ) for some measure p. (b) T E C ( E ,F) is in the surjective hull Crrof 2, if and only if ICFo T factors through a quotient of some L p ( p ) . Moreover,

the minimum taken over all such factorizations. Ex 18.3. If E = C(K) or L,(p), then P p ( E ,F) = 3,(E, F) holds isometrically and the operator S in the factorization 18.7. has necessarily values in F (instead of the bidual F"). Ex 18.4. (a) Let II be the ultrafilter of all sets containing a fixed point L* E I. What is the ultraproduct along II of a family (E,)u of Banach spaces? (b) Describe the relationship between ultrafilters on I and points of the Stone-Cechcompactification P I of I (discrete topology). Ex 18.5. If II is an ultrafilter on I and E, := K for all I E I, then (E,)u = K . Hint: limuf(L) defines a linear functional on &,(I). See also Ex 18.10.. Ex 18.6. (a) If T, E C(E,,F,) are metric injections, then (T,)u is a metric injection. (b) Show that the construction of (Z)u establishes an isometry

Ex 18.7. (a) The ultraproduct of C'-algebras is a C'-algebra. (b) The ultraproduct of C(K)-spaces is a C(K)-space. (c) For every Hilbert space H and 1 5 p < 00 there exist a strictly localizable measure p and an isometry I : H 4 LP(c().For 1 < p < 00 the Hilbert space is isomorphic to 1 a complemented subspace of some Lp(c().Hint: 6 4 L p ( p n ) for some pn via Gauss functions, see 8.7.; for the second part use Rademacher functions, see Ex 8.17..

18. (p,q)-Factomble Opemtors

239

E E: is a uniformly bounded family of functionals, then (cp, ( 2 , ) ~ := ) limu((p,, 2,) defines a linear functional cp on (E,)u with llcpll = limu 11cp,11. It follows that this defines an isometric injection

Ex 18.8. (a) If cp,

(b) Take an operator T E C ( E ,G) and a special ultrafilter U defined on the index set I := FIN(E) x FIN(G’) x ]0,1] as in 18.5.. For L = (M, N, 6) E I define E, := M, G, := GINo and T, := Q$ O T O : E, = M C , E -, G + GINO = G,

IZ

- the finite parts of T. Then there is a factorization as follows:

In the index set one can drop the interval ]0,1] for this construction, but (as one saw in the proof of 18.5.) it is sometimes quite useful to have it. The decomposition KG o

T = QG

0

(Z)u 0 J E

of T is sometimes called the canonical decomposition of T into its finite parts (with respect to U). Note that QG and JE are independent of T. (c) Use (b) to prove the factorization result 18.5. for operators T E $,,(I?, F). Ex 18.9. (a) Let F be a normed space and U an ultrafilter on FIN(F) finer than the order filter. Define Fu to be the ultraproduct of all N E FIN(F) along U. Then there is an isometry JF : F 4 Fu. (b) Let T E C(F’, E’) and (ON E (N BT El)’ satisfy

@T El)’ with and supN I I c p ~ l l< 00. Then there is a cp E (FL(

and IIcpII = limu I I c p ~ l l. If S E C(E’, Fh) is the operator associated with the functional then the relation Jk o S o T = i d p holds. In particular, im(T) is complemented in F’ and T : F’ 4 im(T) invertible. Hint: Lemma 18.4.. This result is just “Lindenstrauss’ compactness argument” treated in Ex 6.4. in its ultraproduct version. tp,

240


Ex 18.10. (a) If dim E, 5 n for a family (E,) of Banach spaces, then its ultraproduct (E,)u has dimension In as well. Hint: For dim E, = n take Auerbach bases (Z,,k,(p&,k)k1, ...,n and define (pk := limucp,,k (Ex 18.8.). Then ((pk,(z,)~)= 0 for all k implies ( t , )=~ 0. (b) If T,E C ( E , ,F,) all have rank 5 n, then ( q ) u has rank 5 n. Ex 18.11. Let T E C ( E , F ) be such that for all 6 > 0 there is an n E N such that for all M E FIN(E) there is an S E C ( M , F ) of rank 5 n with IlTlw - Sll 5 E . Use the factorization of T into its finite parts (Ex 18.8.(b)Land Ex 18.10.(b) to show that KF o T : E -+ F" is approximable. Since the ideal 3 of approximable operators is regular (Ex 9.12.(c)), it follows that T is approximable. Ex 18.12. Use the previous exercise to prove the following result of Heinrich [104]: Let T E C ( E , F ) be such that for all separable subspaces Eo c E the restriction TIE. E C(Eo,F) is approximable, then T is approximable. Hint: Negation and Ex 18.11.. Ex 18.13. For 1 5 p 5 q < 00 and T E C(E,t,) the following statements are equivalent (l/p = l / q + l/r): (a) For all tl,..., z nE E

-

(b) There are an R E C ( E , Q , )with llRll5 1 and A E Bf, such that T = Dx o R, where Dx : t4 Q, is the diagonal operator associated with A. Ex 18.14. Let 1 5 q 5 p 5 00 and l / q = l/p+ l/r. For T E 2(Lp(p),F) the following statements are equivalent: (a) For each S E C ( E ,L4(v))(or only S = idto)

T @ s : Lp(c()

@ A ~E

F @A; L4(v)

is continuous. (b) There are a g E Lr(p) and R E 2 ( L 4 ( p ) ,F") such that K F o T = R o Mp. Hint: Dualize Maurey's factorization theorem. Ex 18.15. The following variation of Maurey's factorization theorem 18.9. was given to us by Pisier; according to him, it "has been known in the neighbourhood of Maurey for a long time". The idea of proof is exactly as in 18.9.. Take 1 5 p < r < q < 00 and T E i ! ( L 4 ( p ) L,(v)) , and define s , t by 1/s + l / q = 1/r and l/t 1/r = l / p . (a) If there are non-negative functions go E L , ( p ) and h, E Lt(v)such that

+

for all f E L 4 ( p ) ,then there are multiplication operators M,, and Mh, and an operator E C ( L r ( p ) ,L,(v))such that T = Mh, o o Mgo and 11p 1 l 5 c . Hint: The closure of {go! I f E L 4 } is l-complemented in Lr .

? !

241

19. (p,q)-Dominated Opemtore

(b) Assume that T satisfies

for all

fi,

...,fn E L , ( p ) and define u := q / r and v := r / p . Use the function Ifkl'gu'dp -

@fl,...,jrn(9, h ) := c' k=l

12

ITfkI'h-"dv

k=l

(for non-negative g and h in the unit balls of L ( u ~ ) 2 ( pand ) L v # ( v ) respectively) , to deduce from the lemma of Ky Fan that there are functions go and h, as in (a). Hint: The first term is norm-continuous since

(c) Verify from these results the following theorem for 1 5 p

llT 8 idtV : Lq(p) 8~~t r

-

< r < q < 00

:

&(v) C ~ G AI1 ~5 c

if and only if there are functions g E L,(p) and h E Lt(v) and !i!E .C(Lr(p),Lr(V)) such that T = h!fh o !i!o Mg and Ilgll.ll!i!ll llhllt 5 c . Note that the case r = q is a special case of Maurey's factorization theorem and r = p a special case of Ex 18.14.. Ex 18.16. The ideal .CP,,of (p, q)-factorable operators is contained in the ideal 'LD of weakly compact operators if and only if q' < 00 or p < 00.

19. (p,q)-Dominated Operators

In this section the Grothendieck-Pietsch domination theorem for absolutely psumming operators will be appropriately generalized to the ideal ap,,= Ei8,q,of ( p , q)-dominated operators. The interpretation in terms of factorizations will give Kwapieri's factorization theorem: ap,q = g ~ a r o-~a result p which shows the central role of the absolutely p-summing operators in the theory. At the end of this section some comments about the so-called "non-commutative" Grothendieck inequality will be made.

242

19. (p,g)-Dominated Operators

+

+

19.1. For p , q E [l,001, with l/p l/q I 1 (or equivalently: l/p' l/q' 2 l ) , the of (p,q)-dominated operators was defined to be the ideal associated with ideal 9p,q the tensor norm = In other words (see 17.11.), T E 9 P , q ( EF) , if and only if

+

lr((d,%)) I c ~ p ( z k E; ) w q ( d ; F ' )

where l / p + l / q 11.' = 1, the norm D,,,(T) being the minimum of these c. This estimate w u a consequence of the obvious characterization of the aqt,pt-continuous bilinear forms: Take l/p l / q = 1 l/r, then p E (E @ F)' is aplq-continuous if and only if &f((P,zk @ Yk)) 5 C wq'(zk;E)wp'(Yk; F )

+

-

19.2. Since Qp g;f = dLf = theorem implies that I(cp9

z €9 Y)I I IIL9.ll

+

(see 17.12.), the Grothendieck-Pietsch domination

IlYll I PP(L9)

(J

I(%',4 I P P ( W ) l/pllPll

BE#

for each p E (E @d,f F)' = V p ( E F') , and vice-versa. For arbitrary ap,q-continuous forms (or, equivalently, for ( p , q)-dominated operators, see the corollary below) there is also a powerful characterization of this type - a result which is essentially due to Kwapieli [160]:

+

THEOREM: Let p, q E [I, m] with l/p l/q 2 1 and K C BE' and L C B p w e a k compact norming sets for E and F , respectively. For p E ( E @ F)' the following two statements are equivalent:

( 4 9 E ( E @a,,* F)'. (b) There are a constant c 2 0 and normalized Borel-Radon measures p on K and on L such that for all z E E and y E F

Y

(where the integral must be replaced by 11 11 if the ezponent is m). In $his case, ~ ~ p ~ I ( ~ g =,min{c ~ , , ~I -c )as~ in (b)}.

PROOF:For p' = m (or, by symmetry, for q' = m) this is just the GrothendieckPietsch domination theorem 11.3.; so it may be assumed that p and q are > 1, hence r' < 00. If (b) is satisfied, take 21,...,z,, E E and yl, ...,y,, E F. Since r'/q'+ r'/p' = 1, it follows from Holder's inequality that

243


The converse will be proven using Ky Fan’s lemma (see Appendix A3.). For this assume ~ ~ y ~ ~ ~= ~ 1 ,8take ~ the p , convex q ~ yweak-*-compact set Mf(K)c C(K)‘ of probability measures on K and define

c := M f ( K ) x M f ( L ) c C(K)’ x C ( t ) ‘ Consider the set 5 of all functions yl, ...,gnE F such that

I

f : C 4R for which there exist 2 1 , ...,zn E E and

for a11 ( p , v) E C. It is clear that all such f are continuous and aEne and that the set 5 is a convex cone. Since K and L are weak-*-compact and norming, there exist for f E 5 (coming from ( z k ) in E and (yk) in F) elements zb E K and y‘, E L such that n

The inequality a / s measures

n

+ b/s’ 2 al/abl/d’(for s €11,co[and a, b 2 0), implies for the Dirac

since ~ ~ y ~ = ~ 1 ~: All ~ the ~ assumptions ~ p , qfor ~Ky )Fan’s ~ lemma are satisfied! It follows that there is a ( p , v) E C with f(p, v) 3 0 for all f E 5;in particular,

:= (Jq 1(~’,z)1q’p(dz’))’/Q‘and p := (J, 1(y’,y)lP’~(d3/))~/P’ (if a = 0, then I((o, Xz @ y)l‘ 5 r’/p’sL l(y’, y)IP’v(dy’) for all X and hence (9,z @ y) = 0). This is the

for

(Y

inequality in (b). 0

-

Since 9p,qair,pland

244

19. (p,q)-Dominated Opemtors

(isometrically) by the representation theorem for maximal operator ideals, the theorem has the following consequence:

COROLLARY: Take p , q E [l,001 with l/p+ l / q 5 1 and K C BE' and L c Bpi weak*-compact norming sets. For T € C ( E ,F ) the following statements are equivalent: ( 4 T E3P,9WlF) * (b) There are a c 2 0 and probability measures p and Y such that

holds for all z E E and y' E F', (replace the integral by In this case, D p , q ( T )= min{c I c as in (b)}.

11 11 if the ezponent

is

00).

19.3. The proof will show that the following result is just a reformulation of the characterization of the ( p , q)-dominated operators which was just obtained.

K W A P I E ~FACTORIZATION ~S THEOREM: Let p , q E [l,001 such that l/p Then Dp,q

+ l/q

5 1.

= %ya' 0 P JP

holds isometrically. In other words, T E C ( E ,F ) is ( p , 9)-dominated if and only if T admits a factorization

E-F

T

G such that R E V P and S' E Pq . Moreover, D p , q ( T )= min{Ppa'(S)Pp(R) I T = S o R } Recall that

I

qm:= C,so the result is nothing new if p or q is 00.

PROOF: If T has such a factorization and 1/r = l/p W Y i l

T 4 ) = [r((S'Yi,

RZk))

+ l/q,

then

5 tp(RZk)tq(S'ylk) L

L pP(R)wP(.k)p~(s').lq(Yi) and therefore, D P , q ( T )5 P,d""'(S)P,(R) (by what was repeated in 19.1.). Conversely, take T E 3 p , q ( EF, ) . Then, by the corollary above, there are probability measures p B p such that

on BE' and Y on

245

19. (p,q)-Dominated Operotors

(the casa p or p = 00 need not be treated). Define R,x := (., x) E C(Bp)and consider the diagram T

E

where Rz := I(R,z). Since Pp(Z) = 1 and Vp is injective (see 11.1. and 11.2.), it follows that Pp(R)5 1. The operator S is defined on R(E) by

SRz := Tx

- and this definition makes sense because

(1 l(d',d)l'"&''))

l(d,SRz)I IDp,q(T)IIRx11G

l"

.

Bprr

It follows that S is continuous on R(E) and has a unique extension to R ( E ) = G; moreover, the inequality implies that

which means that S' is absolutely q-summing and

Pdua'(S) I = P,(S') 5 Dp,,(T). This ends the proof. 0

-

-

19.4. Since .CP,' aP,' and !Di,rq'a;,,,it follows that the ideals .Cp,, of (p, 9)factorable operators and Dp',q~ of (p', q')-dominated operators are adjoint to each other (see 17.9.). Anticipating the fact that ap,' and a;,' are accessible tensor norms (to be proven in section 21), it follows from the inclusion a* o Q c J (see 17.19.) that

CP,' 0 BP',,' c J and I(TS) I L P , ' ( ~ P P ' , & ) 9p"q' 0 Cp,, c J and I(TS) IDP',Q'(TNP,Q(S) (for l/p

+ l/q

PROPOSITION

2 1). The composition is even nuclear in most cases: if 1/p + l / q 2 1 and (p, q ) 4 ((1, I), (1, oo),(00, I)}, then B)p',q' 0 .Cp,e

c 3 and

N(TS) IDP',q'(T)LP,q(S) .

246

19. (pfq)-Dominated Opemtors

This relation i s false for the excluded pairs ( p , q ) .

For the nuclearity of operators in

cp,qo

9pf,qi

see Ex 19.3..

PROOF:The best way to check that an integral operator is nuclear, is to use the Radon-Nikodjmproperty. Take S E 2 p , q ( EF) , and T E 9p~,qt(Fl G ) and assume that 1 < q < 00; then

9pf,q' c a,;1,

cm

(weakly compact operators)

by Kwapieri's factorization theorem (or more simply: since di 5 a;,q). Therefore T " ( P ) C G which implies that the astriction T" : F" -+ G of T" is also (p',q')dominated and D P ~ , 4 ~ (= T "DPt,qf(T) ) (since TI' is, and maximal ideals are regular, mix 17.8.). The factorization theorem 18.11. for ( p , q)-factorable operators gives

with Lp,q(S)= Lp,q(Sf')= llUl1 llIll llVll . Since I E C p , q ,it follows that T" o V o I is integral and hence nuclear because L,I is reflexive and has the approximation property (see Appendix D8., corollary); moreover,

N(T* o V o I ) = I(T" o V 0 I ) 5 D p ~ , q ~ ( TllIll ) ~.~ V ~ ~ It follows that T o S is nuclear with the desired estimate. If q = 1 and 1 < p < 00, observe first that the Grothendieck-Pietsch factorization theorem (11.3., corollary 1, (c)) implies that PP# = ?~DO?J?~~ holds isometrically; hence,

Dpl,oo0 cp,l= 9pf,m 0 9;,?= Pp10 Pi,c m 0 3 c n and N 5 11 11 o I, where the latter is one of Grothendieck's results related to the Radon-Nikodjm property (see Appendix C8.). To see the negative results, first take (PId = (131): a,,, 0 c1,1= c 0 3 = 3 @ n . For the remaining two cases ( p , q ) = ( 1 , ~ or ) (00,l) recall that co does not have the Radon-Nikodjm property (Appendix D3.), hence there is a T : C[O,11 -, co which is absolutely summing but not nuclear (Appendix D7.). Since C[O,11' has the approximation property (it is an abstract L-space), TI is also not nuclear by 5.9.; since, clearly, idcpl] E 2 , and idt, E 21 (by 18.6., for example), it follows that

T E PIO 2 , = 91,aO 0 em,1 T' E PPa1 o2 1 = !Doo,l o 2 1 , ~

247

19. (p,q)-Dominated Operutors

- and this completes the proof. 0 The following important result of Grothendieck is a special case:

COROLLARY: The composition of two absolutely 2-summing operators is nuclear; moreover, N(TS) 5 P2(T)P2(S). Since 32 = rlp2 (see 18.7.), this follows from

19.5. If K and L are two compact topological spaces, then T

_< K C W ~ on C ( K )c3 C ( L )

by Grothendieck’s inequality (see 14.5.). Since ( C ( K ) theorem 19.2. implies:

C(L))’= % i [ ( C ( K )C, ( L ) ) ,

PROPOSITION: For each (p E Bif(C(K),C ( L ) )there are normalized Borel-Radon meaon K and Y on L such that for all ( f , g ) E C(K ) x C ( L )

sures p

Obviously, the smallest constant having this property (for all K and L) is the Grothendieck constant (recall proposition 14.6.).

19.6. A state X on a C*-algebra A is a linear functional A E A’ with 11X11 = 1 which is positive, i.e. X(zz*) 2 0 for all x E A. Since every commutative C+-algebra is isometric (as a C’-algebra) to some C ( K ) and the probability measures on K are exactly the states of C ( K ) , it follows: Let A and B be commutative C*-algebras and cp !Bil(A, B). Then there are states XI on A and A2 on B such that I d t t

Y)l

5 K:llVII

2 112

M142)112~2(lYl 1

*

This is another formulation of Grothendieck’s inequality which (with another constant) also holds for arbitrary C*-algebras A and B if (different from the usual definition of the modulus) /.I2 := 1/2(22’ z*z);this deep result was conjectured by Grothendieck (R6sum6, problkme 4), first proven by Pisier [220], assuming the approximation property, and then by Haagerup [97] in the general case. The result is sometimes called the Haagerup-Pisier-Grothendieck inequality or the non-commutative Grothendieck inequality. If K , denotes the best possible constant, then a simple example in A = B = R(t?2,&) shows that K , 2 2 (see Pisier [225], p.119), which implies, in particular, that K,$ < K, . Actually, Haagerup [97] proved the following stronger version, which easily implies that K , 5 4, and in which the constant 1 cannot be improved:

+

248


If A and B are C*-algebras, I,O E %it(A, B ) , then there are states A',

A2

on A and

p1,pz on B such that lcP(z1

Y)l

I l l ~ l l ( ~ l ( = *+) ~ 2 ( z * ~ ) ) ' / 2 ( ~ 1 ( Y+v *P2(Y*Y))'/2 )

*

Unfortunately, it is not true (as it is in the commutative case) that the tensor norms x and w2 are equivalent on the tensor product A @ B of two C*-algebras: Since 1 2 is l-complemented in the C*-algebra E(&,!2), this would imply that x and w2 were equivalent on 42 @ f a , but, clearly, idt, is not 2-dominated, i.e. tr $$ ( t 2 BW,4)'; this observation is due to Pisier. Since w2 5 w: (see Ex 19.5.), the following result is weaker; recall that w2 22, the ideal of operators factoring through a Hilbert space (18.6.).

-

COROLLARY (Haagerup [97]): If A and B are C*-algebras, then every continuous operator T : A + B' factors through a Hilbert space; moreover, L2(T) other words, s52w: onAc3B.

I 211Tll.

In

PROOF: The non-commutative Grothendieck inequality implies that

+ A2(z*z))11~2IIIYY*II + llY*Yll11~2 5 5 2llTll[2-'(~1(+x*) + .x2(z*41]1/2 IlYll

I(TZ1Y)lS IlTllJZ [2"(Al(ZZ*)

*

The definition (.I%) := 2-1(A'(z%*)

+ A2(%*.))

gives a non-negative sesquilinear form on A x A such that the associated seminorm p satisfiea p ( z ) := ( z ~ z ) ' 5 / ~11~11. It follows that A / ker p (with its natural norm coming from p) is a preHi1bert space and hence its completion H is a Hilbert space. The canonical map R : A -+ H has norm I 1. The inequality at the beginning implies that llTZ11 I 2lITlI l1WH *

If So : R(A) -+ B' is defined by So(Rz) := Tx, then So is well-defined, has norm 5 211Tll and extends to an operator S : H -+ B' with llSll = llSoll. Clearly, T = S o R and therefore,

LdT) I IlSll IPll I 211Tll Since

( A @r B)'

a

A f!(A,B') = &(A, B') A ( A

B)'

by the representation theorem for maximal operator ideals, the result is proven. 0 For a nice application of this see Ex 19.9. - Ex 19.11.. The reader will find more information about the Haagerup-Pisier-Grothendieck inequality in Pisier's book [225]. Using the ideas of Pisier and Haagerup, Grothendieck's inequality for C*-algebras was


249

extended by Barton-F'riedman [6] to continuous bilinear forms on the larger class of so-called JB*-triples (see also Chu-Iochum-Loupias 1291).

Exercises: (where p or q are different from oo),then E is finite dimensional. Ex 19.1. If i d E E gp,q Ex 19.2. Using the Grothendieck-Pietsch domination theorem, show directly that every operator T E o ppsatisfies an integral estimate a8 in corollary 19.2.. Ex 19.3. If (p,q ) {(I, l), (1, oo),(oo,l)} and T E ( f ! p , q o DPtIqt)(E, F), then T' is whenever T = R o S is such a factorization. nuclear and N(T') 5 Lp,q(R)Dpt,qt(S) In particular, K;Fo T is nuclear and, if E' has the approximation property, then T is nuclear as well. If ( p , q ) E {(l,l),(l,m),oo,l)}, then it may happen that none of the operators T, T' or K F o T is nuclear. Hint for the negative part: T at the end of 19.4. has a bidual which is not nuclear. Ex 19.4. Deduce from 12.5. and 12.8. the following results: (a) If PI 5 PZ and q1 i q z , then ( 9 p l ,919 Dp1,ql) C (Bpa,q:, Dpa,ga)(b) If p,q €11,a[, then go,,, C PZ. (c) If p , q E 12, oo[,then 9p,q = DZ. Ex 19.5. Show that w2 5 gz _< w; . Hint: Operator ideals. Ex 19.6. If p or q is < 00, then 9p,q C !Zl . Ex 19.7. (a) Every operator C ( K )+ C(L)' is in 9 2 . Hint: Grothendieck's inequality. (b) 9 2 (t ff . Hint: Consider the operator T : L,[O, 11 3 f-fdA E L,[O, 11' for the Lebesgue measure A. Ex 19.8. (a) Show with the little Grothendieck theorem that P$'"'(E, F) C %(E, F ) whenever E is an 2!-space. Hint: Prove this first for l , and then use the CP-local technique lemma. (b) There is a set I' and a Banach space F such that

!J?ral

!Z are injective and '432 $t ff, but '732 o Z !D C R . Hint: (.p2 and P2 o ? (c) Deduce from (a) and (b) that 9 2 !$ ! Z l o D z . Ex 19.9. Use corollary 19.6. and the fact that w; and w2 are totally accessible (to be proven in section 21) to show the following results for arbitrary C*-algebras A and B. (a) The canonical map A&B --+ A&B is injective. (b) w z 5 2e on A' 43, B'. (c) The natural embedding A & B (A' & B')' is an isomorphic embedding; more precisely, n 5 2% on A 8 B .

-

250

20. Projective and Injective Tensor Norma

Clearly, (c) improves (a). Recall that the C'-algebra 2(&,&) does not have the approximation property. E x 19.10. Let H be a Hilbert space. Then every operator C ( H , H ) --+ 'YZ(H,H) factors through a Hilbert space. E x 19.11. If A is a C'-algebra such that A' is isomorphic to a C'-algebra, then A is finite dimensional. Hint: Use the fact that infinite dimensional C'-algebras are not reflexive (this follows, e.g., from Exercise 4.6.12 in Kadison-Ringrose [139]).

20. Projective and Injective Tensor Norms

The projective norm 7r respects metric surjections and the injective norm E respects metric injections. In this section these properties are studied systematically for arbitrary tensor norms. In particular, injective and projective norms (i.e. those respecting metric injections or surjections, respectively) closest to a given tensor norm are constructed. The associated operator ideals will be described. Finally, some more aspects of Grothendieck's inequality will be treated.

20.1. A tensor norm Q on NORM (or on FIN, or other classes) is called right-injective on NORM (or on FIN, ...) if for all metric injections I : F L* G the operator

is a metric injection for E , F, G E NORM (or E FIN, ...) and right-projective on NORM (or on FIN, ...) if for all metric surjections Q : F --y G the operator

is a metric surjection for E, F, G E NORM (or E FIN, ...). If a*is right-injective (resp. right-projective), then the tensor norm Q is called left-injective (resp. left-projeciive) on NORM/FIN/ ...; if Q is right- and left-injective (resp. right- and left-projective), it is called injective (resp. projective) on NORM/FIN/ ... . Clearly, E is injective and I is projective. Observe that i d E 6 ) I : E&,F Q EgaG is a metric injection whenever i d E @ I is and it follows from the quotient lemma 7.4. that : E g a F --y E&,G is a metric surjection whenever i d E €4 Q is.

251


The duality implies the

REMARK:a is right-injective on FIN if and only if a' is right-projective on FIN. 20.2. This remark will be extended to tensor norms on NORM. The right-injective

tensor norms are relatively easy to handle, but when treating the right-projective ones some problems occur when passing from Banach to arbitrary normed spaces. So their study will be prepared by a precise investigation of their behaviour with respect to dense subspaces. For this, let ,D be a tensor norm on NORM x C (where C is either the class of all Banach spaces or the class of all normed spaces), and recall from Ex 12.3. the definition of the right-finite hull 'p

p'(z; E l F ) = inf{p(z; El N) 1 N /3+ is a tensor norm and ,f?I/I'

E FIN(F), z

E E 8N }

.

+

I/I .

LEMMA: (1) Ifp is right-projective on NORM x C , then p = 'p on NORM x C . (2) I f p is a tensor norm on NORM such that p = /I' on NORM x BAN, then /3 = 'p on NORM x NORM and

E ~ ~ F C - L E ~ ~ F all ( E lF ) E NORM x NORM. (3) I f p is a tensor norm on NORM, right-projective on NORM x BAN, then it is right-projective on NORM x NORM.

for

PROOF:(1) If G E C, then there is a metric surjection 1

Q:F-G such that F has the metric approximation property (if G is complete, take F := l l ( B ~ ) and in the general case an appropriate dense subspace of t , ( B & )which necessarily has the metric approximation property by 16.3., for example); then, for each normed space E the approximation lemma 13.1. implies that

It follows that for each element z E E 8 G there is an N E FIN(F) and a 2 E E 8 N with idE €3 Q ( i = ) L and P(2; E , N ) I(1

+ c)P(t.; El G )


252

and therefore ,

P(z;El G ) I

P-'(z;E, G )I

P ( t ; E , Q N )I P ( i ;

El

N )5 (1 + E)P(z;E , G )

*

(2) Take z E E 63 F. Then the metric mapping property gives r ( z ; E , P) = P ( z ; E l P)

5 P(z; El F ) 5 p-'(t;

El F )

.

For N E FIN(F), with t E E 63 N and P ( z ; El N ) 5 (1

+ &)P-(z; El E )

1

use the principle of local reflexivity 6.6. to find an operator R : N 4 F with IlRll 5 1 + ~ and Ry = y for all y E N n F. The operator T : E' 4 F associated with t has values in N n F, which implies that

Consequently,

P-*(t; El F ) I P ( i d ~63 R(z);E l R N ) I IIRIIP(t;E , N ) I I (1 + &)'P-'(z; El P) , which proves (2). To see (3), recall again the quotient lemma in 7.4. and take Q : F --y G a metric surjection between normed spaces; then the completion Q : -+ G is a metric surjection with ker Q = and, by assumption,

idE @ Q : E @p

P

-

E @p 6

is also a metric surjection. Since by (1) and (2)

E@pF-E@pF

and E @ p G - E @ p e

are metric injections and, clearly (idE@Q)(EQDF) = E @ G ,it follows from the quotient lemma that idE @ Q : E @ p F E QDp G

-

is a metric surjection if

id^ €3 0) = E €3 ker Q But this is obvious since ker 0 = ker

(see 2.7.).

EC3d

ker( idE @ Q)

.0

20.3. Now the announced duality between right-injective and right-projective tensor norms can be proved. At the same time, and this is in some sense natural, a preliminary

253


observation concerning the accessibility of these tensor norms will be made (a more careful investigation will be given in section 21).

PROPOSITION: Let a be a tensor norm on NORM. (1) If a is right-injective on FIN, then % and (2) If a is right-projective on FIN, then

d

d

are right-injective on NORM.

is rightprojective on NORM.

( 3 ) If a is finitely or cofinitely generated, then: a is right-injective on NORM if and only if a‘ is right-projective on NORM. (4) If a is right-injective or right-projective on FIN, then a i s right-accessible. 1

PROOF:(1) and (4): If a is right-injective on FIN, then for F cr G and z E E 8 F d ( z ; E, G) 5 Z ( z ; E, F ) = = inf{a(r; M, N n F)lM E FIN(E), N E FIN(G), z E M @ N} =

= inf{a(z; M, N ) [...} = d ( z ; E , G),

d is right-injective. To treat the cofinite hull, first (4) will be shown: For this, take (N, F) E FIN x NORM and o E N 8 F and assume a is right-injective on FIN. Then, by what was already shown and by the approximation lemma, it follows that

so

hence a is right-accessible. Now recall that a is right-accessible if a’ is (see 15.6.). Hence, if Q is right-projective on FIN, the dual a’ is right-injective on FIN, therefore a’ is right-accessible and so is a. Now it is possible to show that % is right-injective on NORM if a is right-injective 1 on FIN: For F L,G and z E E 8 F the following holds by the two results which were already shown:

254


1

(2) Using lemma 20.2.(3), it is enough to consider a metric surjection Q : F 4 G between Banach spaces. By (4) the tensor norm a' is right-accessible, hence for every N E FIN result (1) implies that

and therefore idN 8 Q : N 8F a

-

N 8; G

is a metric surjection. Now take E an arbitrary normed space: ~ ( zE ;, G)

{ N , G ) I N E FIN(E), z E N B G } = = inf {Z'(w; N , F) I N E FIN(E), i d N €3 Q(w) = z } = = inf

z(%;

= inf { s ( w ; E, F ) I i d E 8 Q(w) = z }

.

Finally, statement (3) follows from (l),(2) and the remark at the end of 20.1.. 0 It is not true that the cofinite hull % is right-projective on BAN if a is right-projective 1 on FIN; to see an example take a = A and . t l ( B ~--y) F for a Banach space F without the metric approximation property. Then 1

F'@+~~(BF = )F ' ~ , ~ ~ ( B F ) - - Y F ' € ~ ~ FF .# F ' ~ ~ A

Since there is no Hahn-Banach theorem for general operators between Banach spaces (1.5.), A is neither right- nor left-injective. Dually (this means using (3) of the last proposition), E is neither right- nor left-projective; see also 20.20.. 20.4. For the a,,,-tensor

norms the following result holds:

PROPOSITION: Let 1 5 p 5 00. Then ( 1 ) d, is right-projective; consequently, gp is left-projective and g; = d; is rightinjective. ( 2 ) az,, is right-injective, C Y , , ~left-injective and right-projective. In particular, w2 is injective and w; = wi i s projective.

PROOF:Since


255

result (1) is a direct consequence of the following observation: If Q : F --y G is a metric surjection, E > 0 and y1, ...,y, E G, then there are & E F with Q($i) = yi and

To ax that z E E @ F and

E

is right-injective, take a metric injection I : F of z in E @ G with

> 0. Choose a representation

c, G,

Then (by what was explained in 17.10.) the associated operator T, : E' factorization

an element

+F

admits a

z ; G). The subspace S - l ( I ( F ) ) is 1-complemented with llRll110~11IlS((5 ( l + ~ ) a z , ~ (E, in 6 ,hence there is an So : 6 -+ F with llSoll5 llSll and T, = Soo D Ao R. It follows that n

z

=

CX~CL @ S O e h E E @F k=l

and

which shows that 0 2 , ~is right-injective. The other statements in (1) and (2) follow easily by dualization and transposition. 20.5. To see an application of this, recall from 15.10. the Chevet-Persson-Saphar

inequalities dp

5 Ap Ig;*

on L P W @ E

and also that dp and g;.' are the closest tensor norms to Ap (see 15.11.). I t was shown in 15.10. that dp = Ap = g i l if E = L p ( v ) .More generally,

PROPOSITION: If E i s a Banach space which can be obtained by starting with some Lp(v) and then taking metric quotients and metric subspaces finitely many times, then

for every measure p .

256

20. Projective and

Injective Tensor Norms

PROOF: Fix p and consider the maximal normed operator ideal ,!A such that L p ( T ) := I I i d ~ , ( p )@ T : Lp(cI) @d, E-Lp(p)

@A,

of all T E &(E,F )

F)ll < 00

(see 17.4.). Since d, is right-projective and Ap right-injective (7.4.), this operator ideal is injective and surjective. This impLes that the class of spaces E such that L,(idE) = 1 is stable under forming metric subspaces and metric quotients (both # {0}, clearly). Since L , , ( i d q v l ) = 1 by what was said earlier, it follows that dp = Ap for those spaces satisfying the assumption. For gp, consider the ideal of those T such that

IlidL,(p) @ T : Lp(p) @A, E-Lpb)

@g;f

Fll< 00

.

Observe that it is again injective and surjective since g;, is right-injective and A, is right-projective (7.4.), and argue in exactly the same way. 0

In Ex 20.8. and EX 20.9. it will be shown that the spaces in the assumption of the proposition are precisely the subspaces of quotients of some L, (equivalently: quotients of subspaces of some L p ) . It will turn out in 25.10. that actually these are exactly the spaces E with dp = Ap = g;’ on L, @ E . For Ap = d;f and Ap = g , see also 25.10.. 20.6. Every tensor norm a is less than or equal to 7r and 7r is projective. Hence it is reasonable to search for a smallest tensor norm /3 2 a which is projective.

THEOREM: Let a be a tensor norm on NORM. Then there is a unique right-projective tensor norm a/ 2 a on NORM with the following property: If /3 2 a is right-projective, then /3 2 a/.Moreover, a/ is finitely generated from the right, i.e. a/ = (a/)+. The right-projective associate

of a can be calculated using the following property:

(Y/

If E is normed and F a Banach space, then

i s a metric surjection. If E and F are arbitrary normed spaces and z E E @I F, then

(*I

I

a/(%; E , F ) = inf{a/(z; E l N) N E FIN(F), z E E @ N }

.

1

The fact that a/ respects quotient mappings F-FIG - is of course responsible for the use of the symbol a/.

PROOF:Uniqueness presents no problem. Existence: a/ will be constructed first on NORM x BAN and then extended, using the introductory lemma 20.2..

257


(a) If (E, F ) E NORM x BAN, define a/ to be the quotient seminorm on E @ F given by the mapping E @a ti (BF)-E @F . Since E a 5 rn and x is right-projective, it follows that E 5 a/ 5 rn on the class NORM x BAN. To see the metric mapping property on NORM x BAN, take operators Z E E(Ei, Fi) and use the lifting property of the spaces t,(I')(see 3.12.) to obtain a T 2 with

5 11Z;11Il!&i;l It follows that llT1@af";II of a/)that llTl@ T2 : El

@a/

E2-F1

5 ~ ~ T 1 ~ ~ ( l +and ~ )hence ~ ~ T(by2 the ~ ~definition , @a/

F2ll

5 111;11(1+

w 2 1 1

for all E > 0. Thus, a/ is a tensor norm on NORM x BAN. (b) To show that a/ is right-projective on NORM x BAN take a metric surjection Q : F --y G between Banach spaces and again use the lifting property:

The diagram

implies that i d s @ Q is a metric surjection. (c) Lemma 20.2.(1) now implies that a/ = (a/)+on NORM x BAN. This means that the definition on NORM x NORM a/ := (a/)+ gives a tensor norm which extends a/ from NORM x BAN to NORM x NORM. Lemma 20.2.(3) shows that a/ is right-projective on NORM; moreover, by the definition a 5 a/ on NORM x FIN, and therefore, a 5 a+ 5 (a/)- = a/.


258

0. Then there are R E a ( E ,G) and S E %(G,F ) such that T = S o R and B(S)A(R) _ < e)(B ( 1 o A)(T). +Since is injective, one can choose this factorization such that R(E) = G, hence S(G) c T ( E ) and S is also a finite rank operator. Since % is totally accessible and left-accessible, T factors as follows: T

E

F

F

Consequently,

B o A(So 0 R,)

I B(S,)A(R,) I( 1 + c)B(S)(l+ E)A(R)I (1 + E ) ~ 0BA(T) ,

which proves the result. 17

21. Accessible Tensor Norms and Opemtor Ideals

279

For more results on the accessibility of composition ideals see Ex 21.2. and 28.6.. 21.5. Everything is prepared for giving an easy proof of the following important result, apparently due to Gilbert-Leih [Sl]:

THEOREM: Let p, q E [l,001 satisfy l / p + l / q 2 1. (1) ap,qand (.CPlq,Lp,g) are accessible. ( 2 ) a;,q and ( 9 p 8 , q t , DP,,(t)are totally accessible. PROOF:Since the tensor norms and operator ideals in question are associated and a is accessible if a* is (by 15.6.), it is enough to show that 9p,,q~ is totally accessible. Kwapieli's factorization theorem 19.3. states that

-

-

9pr,ql

=

!py!ppl. 0

Since '&, g; and pdu'" g;' are totally accessible (by corollary 21.1. and 21.3.) and the operator ideal &,I is injective, the preceding proposition shows that is totally accessible. 0 21.6. A structurally interesting application of the total accessibility of certain tensor norms is the

PROPOSITION: Let (a,A) be a maximal normed operator ideal with associated tensor norm a. If a* is totally accessible, then every Banach space E with idE E 24 has the bounded approzimation property with constant A ( i d E ) . PROOF: In order to use the characterization theorem 16.2. (concerning ?r 5 X f ) for * E'@ E by 17.15.. the bounded approximation property, observe that ?r 5 A ( i d ~ ) aon Consequently, ?r

5 A ( i d ~ ) a=* A ( i d E ) z 5 A ( i d E ) f

on E' 8 E

,

which gives the result by 16.2.. 0

COROLLARY 1: For 1 5 p 5 00 and X 2 1 every .C:,x-space E has the bounded approzimation property with constant A.

-

PROOF: Since .Cp wp is accessible and E 5 wp 5 X E on G@ E for every Banach space G by the local technique lemma (see 13.5.), it follows from the characterization 17.16. that L p ( i d E ) 5 A. The fact that wp' is totally accessible implies the result. 0 This was a "positive" application of the proposition. But the proposition also has a negative flavour: if there is an E E space()#) (this means, by definition i d E E Q)

280

21.

Accessible Tensor Norms and Operator Ideals

without the bounded approximation property, then a* is not totally accessible! An example: Each subspace E c 4, is contained in space(2y); since there are subspaces of tpwithout the approximation property (if p # 2, see 5.2.), the adjoint of wp\ il? cannot be totally accessible:

-

COROLLARY 2: For 1 5 p 5 00 and p

# 2 the tensor norm wL/

is not totally accessible.

21.7. Let a be a finitely generated tensor norm. A Banach space E is said to have the a-approximation property if for all Banach spaces F the natural mapping

is injective. E has the bounded a-approximation property with constant X 2 1 if for all Banach spaces F the natural mapping

) ~A ~= 1 one speaks of the metric a-approximation satisfies “ ( 2 ; F, E) 5 X ~ ~ I (...;z if property. For a = T the usual approximation properties are obtained. For a = gp this notation was introduced by Saphar [243] under the name “approximation property of order p” (and generalized to ap,pby Diaz, L6pez Molina and Rivera Ortun [52]).

PROPOSITION: (1) A Banach space with the approximation property has .the a-approximation property for all tensor norms a. (2) E has the bounded a-approximation property with constant X i f and only if Q 5 A% on F €3E for all Banach spaces F. I n particular, i f a is totally accessible, then each Banach space has the metric a-approximation property. (3) If a is accessible, then E has the bounded a-approximation property i f it has the bounded approximation property (same constant). (4) It is enough t o check the definitions for the (bounded) a-approximation property for dual spaces F. (5) If El’ has the (bounded/metric) a-approximation property, then E has it. PROOF: (1) is a reformulation of 17.20, and (2) follows from the duality theorem 15.5.. The approximation lemma and (2) imply (3) and the rest is a consequence of the embedding lemma. 0 Thus, every Banach space has the metric g2-approximation property since 92 = g; is totally accessible. It will be shown in 31.7. that Banach spaces E of cotype 2 satisfy

F

E = F &ig,E

for all p E [2,m]

,

21. Accessible Tensor Norms and Operator Ideals

281

hence they have the gp-approximation property for these p. There are spaces of cotype 2 without the approximation property: Szankowski’ssubspaces oft, (where 1 5 q < 2, see 8.6. and 5.2.(1)). In 1982 Reinov [228] showed for 1 5 p < 00 and p # 2 that E -, El& E is not injective. In particular, (1) There is a Banach E such that E1hgv E does not have the gp-approximation property and the tensor norm gp is not totally accessible. (2) There is a Banach space with the gp-approximation property but failing the bounded gp-approximation property. (3) There is a Banach space E with the approximation property which does not have the bounded gp-approximation property for any p as above.

21.8. That the (bounded) a-approximation property is really a property of “approximation’’ of certain operators can be seen as follows. If !2l is the maximal Banach operator ideal associated with a,the adjoint operator ideal satisfies

?Z*(E,F”) = ( E @ a t F’)’ = ( E hat F’)‘ isometrically. For the weak topologies u := u(%*(E,F”), E @ F’) and

8 := u(?Z*(E,F “ ) , E hatF’)

lemma 16.2. reads as foIIows:

PROPOSITION: Let E and F be Banach spaces, a a tensor norm and A 2 1. Then on F ‘ @ E is equivalent to each of the following two statements about the unit

a 5 A= balls:

(4

B n * ( E , F J Jc ) ABEI&.F*

.

(b) B ~ * ( E , F Jc J) . In particular, E has the bounded a-approzimation property (with constant A) if and only if (a) or (b) is satisfied f o r all Banach spaces F. The latter follows from proposition 21.7.(4).

21.9. For the bounded gp-approximation property (1 < p < 00) it is enough to check the definition for reflexive spaces. This will now be shown.

LEMMA: Let 1 5 p 5 00 and E , F arbitrary Banach spaces. For every z E F6bg,E there are a separable refletive Banach space GI an injective operator S E 2(G,F ) and 20 E G 6 g v Ewith S @I idE(W) = z .

PROOF: For p = 00 read c, instead of looin this proof. By 12.7. the element z E FGgrE has a representation m

282

21. Accessible Tensor Norms and Operator Ideals

with (An) E t,, a zero sequence (2,) in F and wpr(yn;E) 5 1. Define K to be the absolutely convex, closed hull of {zn}.Then [ K ] is a Banach space such that the embedding [ K ] ~1F is compact, and hence factors injectively through a reflexive space G1 by the Davis-Figiel-Johnson-Pelczyliski factorization theorem (see 9.6.); take G to be the closed linear span of {z,} in GI. Since {z,} is bounded in [ K ] ,it is bounded in G, and w := C,"==l Anzn@ yn E G&, E gives the result. 0

In Ex 21.5. there is a metric version of this lemma.

PROPOSITION 1: For 1 5 p 5 00 a Banach space E has the g,-approzimation property if (and only i f ) IF : F 6 g ,E - F ~ L E is injective for all separable, reflexive spaces F. Recall that the gl-approximation property is the usual approximation property.

PROOF: Take F to be an arbitrary Banach space, z E FGgVE and choose G, S and E G&,, E as in the lemma. Then the diagram

w

F & ~E, S@g,idE

IF

F& E

-T

S@,.idE

IQ

Ggg, E

G& E

implies the result since IG and S & i d ~ are injective. 0 For the bounded gp-approximation property the analogous result holds at least for l 0, every finite rank operator T : E --+ &(I?) has the form T = C:=, &Byrn , with tP(z&)5 (1 &)Pp(T) and wpj(ym) I 1. Hint: gp+, = gp\ and accessibility. (b) For each T E 5 ( E ,F) with Pp(T) < 1 there are z;,...,z; E E' with l,(zA) 5 1 and

+

for all z E E.

Ex 21.5. Check that in lemma 21.9. for every E > 0 the operator S E C(G,F) and w E G&,, E can be chosen 80 that llSll 5 1 and gp(w; G, E) 5 (1 + E)gp(z; F, E ) .

287

22. Minimal Operator Ideals

E % ( E , F ) and E > 0, then there are reflexive spaces G I and G2, operators R E 2 ( E , GI),To E %(GI,G2) and S E .C(G2,F ) such that T = S o To o R and IISllN(To)llRII5 (1 s)N(T).Hint: Ex 21.5.. Ex 21.7. (a) Take l / p + l / q 2 1. Then for each operator T E Pqt(F,G)and each Banach space E

Ex 21.6. If T

+

5 Pq#(T).

llT@ id^ : F6a,,qE-G6g,Ell

(b) Use Kwapie6's factorization theorem to show that a Banach space E has the Q

~ , ~ -

approximation property if it has the gp-approximation property. Hint: For each functional cp E ( F E)' and L , = U o S one has (cp, z) = tr,@ 8 i d ~ ) ( @ S idE)(z). This result is due to Dim, L6pez Molina and Rivera Ortun [52].


The smallest Banach operator ideal which coincides with a given Banach operator The main result ideal for finite dimensional spaces is called its minimal kernel amin. of this section is the representation theorem of minimal Banach operator ideals in terms of the tensor norm which is associated with the maximal hull am'" of a.

22.1. If (a,A) is a quasi-Banach operator ideal, then its minimal kernel is defined to be the quasi-Banach operator ideal

(a,A)min:= (amin, Amin):= (5,II 11) o (a,A) o

11) ,

where f is the ideal of all approximable operators; in other words (see 9.10. for the definition of the composition of operator ideals),

I

Amifl(T) = inf{ IISllA(To)llRIIT = S o Too R with S,R E 3 and To E

a} .

(a,A) is called minimal if (a,A) = (a,A)min. - REMARK:5 o 5 = 3 holds isomeln'cally. PROOF:Take T E E((E,F) and choose T,

C,"=lT,

(with respect to the operator norm

E 5 ( E , F ) and 1

11 11)

and

5 A, t

00

with T =

c,"=l LllTnll 5 llTll(1+ e). If

288

22. Minimal Opemtor Ideals

Nn := im(Tn), then

This implies that 5 = 505 = 50C o 5, hence 5 is the minimal kernel of all operators: Cmin = 8. Moreover,

(isometrically) - in other words,

amin is a minimal operator ideal.

Since, obviously,

amin c Bminif a c %, these facts show that 8 is the largest minimal operator ideal.

If is pBanach, then aminis q-Banach for some q by 9.10.. To avoid the quasinorm catastrophe the following result will be formulated only for p-Banach operator ideals; but recall from 9.3. that every quasi-Banach ideal has an equivalent p-norm.

PROPOSITION: Let (a,A) be a p-Eunuch operator ideal. (1) amin(M,N ) = a"""(M, N ) = % ( M ,N ) holds isometrically for all MIN E FIN. (2) For each T E amin there are T, E 5 o 9l o 5 with Amin(Tn- T) 40. min = (a,A)min . (3) = (a,A)""" . (4) (a,A)min (5) If (%, B) is a q-Banach operator ideal with B 5 cA on FIN for some c 2 1, then a m i n c B m i n and Bmin5 cAmin. In particular, ( 5 ) shows that (a,A)min is the smallest q-Banach operator ideal (where q is arbitrary) which induces the norm A on FIN.

PROOF: The relation for amax in (1) is immediate from the definition 17.2. of maximal operator ideals. A 5 Amin is always true and Amin 5 A on C ( M , N ) follows from T = idN o T o id,+,. Clearly, (1) implies (4). To 8ee (2), take T E amin(E, F ) , an E > 0 and afactorization T = SoTooR E f o a o g with IlSllA(T')llRII 5 A m i n ( T ) ( l + ~ )For . Sn,R, E 5 with IIR-&l( -+ 0 and IIS-Snll + 0 define the operators Tn := SnoTooR,. Since Amin is an r-norm for some r, (2) follows from

To see (3), take T E am'" gives the factorization

. Then this construction (applied to a"""instead of 9l)


where To,nE ama"(Mn,Nn)

289

A %(MnlNn). It follows that

AGn) I llsnIIAmat(~o)li~ll

A(Tn

- Tm) I((IIsn - Sm(IAmaZ(To)I(R,IOP + (IISmIIAma"(To)IIh - RnII)p]l'p

1

hence (T,) is an A-Cauchy sequence. It converges in g ( E ,F) and its limit is clearly T. Since A is a p n o r m , it follows that

A(T) = n-oo lim A(Tn) 5 n-m lim IJSnllAmaC(To)llhll 5 Am"" This means that

%ma' min

min (T)(1+ E )

*

c !2l and A 5 AmaZ amin

g m a z min

which implies that and Amin 5 Amaz min

The reverse, however, is obvious. The remaining statement ( 5 ) follows from (1) and (3). 0 The proposition implies that for the study of the maximal and minimal operator ideals QmaZ and Jllmin one may always assume that % is maximal. 22.2. Let (%,A) be a maximal n o m e d operator ideal and (Y its associated finitely generated tensor norm, i.e. % ( M ,N) = M' @a N for all finite dimensional M, N (see section 17), and denote by

J : E'

@F

-

Qmin(E, F)

the natural embedding J ( z ) := T, . For M E FIN(E') and N E FIN(F) the diagram

El @a F

I

Win(E,F ) 3 I:

o T o Q&,

obviously commutes. If z E E'@ F and u E M @ N with Is' @ I;(.)

= z, then

Amin(J(t))= Amin(I{ o T, o QZ0)I A(T,) = a ( u ; M , N ) ,

290


which implies that Am'"(J(z)) 5 a(%;E', F ) . Even more is true:

THEOREM FORMINIMAL OPERATOR IDEALS:Let (%,A) be a BaREPRESENTATION nach operator ideal and a the tensor norm associated with (%,A)mas. Then the canonical map J : E'@,F 2 P i n ( E ,F )

-

is a metric surjection for all Banach spaces E and F. In particular, the minimal kernel of a Banach operator ideal i s normed.

PROOF:One may assume that % is maximal. By what was said above J is defined and has norm 5 1. By definition, M' @, N = %(MIN ) . holds isometrically whenever M ,N E FIN. Take operators R E S ( E , G 1 ) , T o E O ( G 1 , G z ) and S E S ( G z , F ) ; if Z S ~ T , ~:= R J-'(S o To o R), then a ( z S o T o o R ; E',

F ) 5 IlsllA(To)llRII :

Indeed, take (as in the proof of 22.1.) the factorization SoT,oR

E - F

Then a(zSoT.oR;

CZI 1z(zT1);E', F)IIIKIIIII~II~(~T,;M', N )=

E', F ) =

= Il~llA(T1)IllKll Il~lllA(~0)ll~ll = II~IIA(T0)llSll *

This inequality is the key to the following reasoning. Take T E W'"'"(E, F ) , factor T = S o To o R with ((S((A(T,)((RII5 (1 + &)Am'"(T) and choose &, S, E 5 with llR -, 0,IlS - Snll -+ 0. The proof of 22.1.(2) showed that T,, := S, o Too & converges to T in Qm'"(E, F ) . Denote r, := J-'(S,, o Too &) E E' @I F ; then Z,

- zm = J-'((Sn

- S m ) o T o o Rn) + J-'(Sm

and therefore by the inequality,

- &))

o To o (R,

291


It follows that (zn) is a Cauchy sequence in E'&,F. Its limit z E E'&F j ( z ) = lim ~ ( z , ,= ) lim T, = T n+oo

n+W

in

satisfies

amin

since J" is continuous. Moreover,

a ( z ;E', F ) = lim

n+w

This, together with

cy(zn;E',

F)

+E ) .

lim llSnJJA(To)ll%ll5 Amin(T)(l

f8-w

ll.fll 5 1, shows that .f is a metric surjection.

0

A special case: The projective norm 1 is associated with the ideal 3 of integral operators, hence the description 3.6. of nuclear operators gives ( 3 ,I)min= (%, N) .

This is the reason why the operators in 21mi"(E,F ) are called a-nuclear if 21ma"N a Moreover, it follows that the ideal % is the smallest normed minimal operator ideal.

.

COROLLARY 1: Let a be associated with the mazimal hull of the Banach operator ideal

a. If E'

or F has the approcimation property or if a is totally accessible, then

E'&QF = 21min(E,F ) holds isometrically.

This follows from corollary 17.20.; clearly, it would be enough to assume that F had the a-approximation property or E' the a'-approximation property (see 21.7.).

22.3. Examples: KP4) := (.cP,P, f.'p,4)min Np) := V p , (Hp, Kp) := (Cp, Lp)min (fiP,41 ( q 7 ,

the ideal of ( p , q ) - compact operators. the ideal of p - nuclear operators. the ideal of p - compact operators.

The associated tensor norms are a p , 4gp , = ap,l and wp = aPs~. The representation theorems 12.6. and 12.7. for E'ga,,,F and the representation theorem for minimal operator ideals give

292


The formula for Kp,q also shows that the (p,q)-compact operators are those which factor

E - F

T

.1eql -Ts

DA a diagonal operator (e, can be replaced by c,) ,

D X

eP

.

-

and KP,Ji?) = inf llSl1 ~IDAIIlIRl1 Since Qmi" = 5 o !2lmi" o 5,the operators R and S can be taken to be compact. Note the special cases %, = Rp,l and fip = ffp,pr; it follows that the p-compact operators are just those which factor compactly through the sequence space 4,. 22.4. The following consequence of the representation theorem for minimal operator ideals is trivial, but useful:

COROLLARY 2: If E and F are Banach spaces, a on E' @ F, then (Ijmin(E,F ) C )2Lmi"(E,F ) and

-

2l and p

-

% such that a

5

cp

Ami"(T)5 c Bmifl(T) for all T E %mi"(E, F). By the embedding theorem (and 15.7.) the next result (again for maximal normed ideals and fixed Banach spaces E and F ) appears as a special case of this corollary: If %(E, F ) C %(E,F ) and !2l is totally accessible (or: 2l accessible and E' or F has the bounded approximation property or: El and F have the bounded approximation property), then

!Bmin(E,F ) c !21min(E,F ) .

Since I 5 &ga0

on

63

(see 14.8.), the local technique lemma implies that

5 AKGg, 5 AKGgp for every Hilbert space H,every 2!,A-pace rJmin - %,, it follows that:

,

on H @ F

F and all 1 5 p 5

00.

Since g p

-

3, and

PROPOSITION: %(H, F ) = %,(H, F ) and N ( T ) 5 A K G N ~ ( T for ) every Hilbert space H , 2f,A-space F and p E [l,001. This is in some sense a "nuclear" form of Grothendieck's inequality. 22.5. The representation theorem for minimal operator ideals represents the third of the three basic links between the metric theory of tensor products and the theory of


Banach operator ideals: If the maximal Banach operator ideal generated tensor norm (Y are associated, i.e. if

M'

293

(a,A) and the finitely

N = %(MlN )

@a

holds isometrically for all M, N E FIN, then for all Banach spaces E and F the following theorems are valid (17.5., 17.6. and 22.2.): (1) The representation theorem for maximal Banach operator ideals:

%(E,F')

A (E

and

F)'

F ~ )nI q

% ( E ,F ) A ( E

~F ) , .

(2) The embedding theorem:

6q ~ F ) .,

E I ~a - F

( 3 ) The representation theorem for minimal Banach operator ideals: E1gaF

1

21min(E,F ) .

This troika of results is a powerful tool for investigating maximal and minimal n o me d operator ideals. A first example is the

COROLLARY: Let (a,A) be an accessible maximal Banach operator ideal. If E' or F has the metric approximation property or i f

(a,A) Is totally accessible, then

amin(E, F ) = 5(E,F)*

% ( E ,F )

The proof follows from the fact that, in this case, (Y = 'Z on E' @ F (see 15.7. and 21.3.). As before, it would be enough to assume that, for example, F has the metric a-approximation property. A consequence of this corollary is:

since g,;

- Vp

P F n ( E ,F ) =

mP' d+P p ( E ,F )

is totally accessible.

22.6. Another nice example of the interplay between the three basic theorems is the following alight extension of a result due to Schwarz [254].

Let (%,A)be a maximal Banaeh operator ideal. If (%,A)is totally PROPOSITION: accessible or i f E or F' has the approximation property, then

(Bmin(F,E))' = %*(El F") holds isometrically.

294

22. Minimal Opemtor Ideala

-

PROOF: If a is the tensor norm associated with Q, then 3L* Q*(E,F I I )

(E

FI)'

' a and therefore

A ( F I ~ ~ E. ) '

Since F'g3,E = amin(F,E ) by 22.2., corollary 1, the result follows. 0 The duality bracket can be calculated by using the trace: Use 17.15. to see (first for elementary tensors) that for T E 2l*(E,F") Qmin ( F ,E ) = F'&

S-

E

- 1 idFr&T

-

F'& F"-

%(F', F')

S'

O

T'

0

KFI

trFl

(T,S) E K

and

where S" : F"

4

E is the astriction of S"; it follows that P,S) =

{

trp(S' 0 T' 0 K trE(S" o T)

p )

if F' has a.p. if E has a.p. .

In the case of a being totally accessible, the duality bracket cannot always be calculated with the trace on operators. This can be seen as follows: The diagram shows that S' o T' o KFI and S" o T are nuclear operators - and it may happen that all nuclear operators in %(F', F') [or % ( E ,E)] appear this way; therefore, if F' [ or E ] does not have the approximation property, the trace is not defined. For an example, take a = E , hence Q* = 3 and amin = 3, and G a reflexive space without-the approximation property. Then %(G, G) = %(co,G ) o T(G, c,) and %(G, G) = $(el, G ) o %(GI41). Since 41 and G' have the Radon-Nikodfm property, it follows from Appendix D8. that

%(G,G)= ~ ( C O , G ) ~ ~ ( G= ,~% ( ~)i , G ) o J ( G , 4.i ) Therefore ( E , F ) = (c,,G') and ( E ,F ) = (GI!,) show that for a = E none of the two operator trace formulas holds for all pairs ( E ,F ) of Banach spaces. Even more: Taking the direct sum of these two counterexamples one can verify that there is no quasi-Banach operator ideal a with a continuous trace such that for each ( E ,F ) either all S' o T' o KFI E %(F', F') or all S" o T E % ( E ,E ) ; see Ex 22.9. for the definition of such ideals and the main argument for this.


295

22.7. If T is nuclear, then T’ is - but the converse is not true by the example given in 16.8.. For other nonned minimal operator ideals the question of duality reads as follows: If (a,A ) is a maximal normed operator ideal with associated tensor norm a, then the transposed tensor norm at is associated to 2Idua1,the ideal of all T such that TI E !2l (see 17.8.). The diagram

shows that Ti is a-nuclear if T is &‘-nuclear and Amin(T’)5 Adua’min(T)holds; see also Ex 22.5.. Under which circumstances is the converse true? In other words, when does a m i n dual ( E , F ) %dual min ( E lF ) c !2lmin ( E ,F )

t

hold? If F is a dual space, then it is easy to see that this is true (Ex 22.5.). As for further situations where the answer is positive an appropriate description of the a(F‘, F ) - a ( E , El)-continuous operators in amin( F‘, E) will be helpful:

PROPOSITION: Let % mapping

F6aE

-

a be associated and El F Banach spaces. Then the canonical

-

I

{T E Qmin(F’, E ) T u ( F i ,F ) - a ( E , El)-continuous}

is defined and has norm

5

1. It is a metric surjection under each of the following

conditions:

( 1 ) F is 1-complemented in F“. (2) F”/F has the approzimation property.

( 3 ) E has the approximation property.

PROOF:The first statement is clear since, by 4.2., all operators coming from F&E are a(F’, F ) - .(El El) continuous. The representation theorem also gives that, for a given a(F‘, F ) - a ( E , E’) continuous To E amin(Fi,E ) , there is an zo E F”@QEwith j ( z o ) = To and F”, E ) 5 ( 1

&(to;

It remains to prove that ( 1 ) If P : F”

---+

tois

+ &)Amin(To) .

(or can be taken to be) in F&E.

F is a projection with IlPll = 1, then the diagram

296

22. Minimal Operutor Ideal8

commutes. Since ICF o P o Ti = Ti by assumption, it follows that

To= J’ ( ( I C F 8 ~ d E ) ( P ~ i d E ) ( Z o ) ) and hence P&dE(Zo) E FGaE represents To. (3) If the space E has the approximation property, it will be shown that z, is already an element of F&E. For this, take cp E (F’16aE)twith c p p 8 ~= 0; it has to be shown that (cp,~,) = 0. If L, E C(F”, E’) is the operator associated with cp, then, by 17.15., L,&dE

: F”&E

is continuous and (9,z,) = trE(L,&dE(z,))

E‘& E

-

E’&E

:

-

C(El, El) 3 L ,

3 S’ . 0

Since L,IF = 0 and Ti(E’) C F, it follows that L, o Ti = 0. The space E has the approximation property, hence the lower map I is injective; therefore L , o Ti = 0 implies that L,@lidE(Z,) = 0 which gives (cp, z,) = 0. (2) Finally, assume that F“/F has the approximation property. As before, take a functional p E (F’t&aE)’ with c p l =~ 0.~ Denote ~ by U E $ ( E l FI’I) the operator associated with cpt E (E&tF”)’ = %*(E,F’“). Since im(U) C Fo

F’”

(the polar taken in F”’) and F o is complemented in F”‘ (by Z ” ’ - ~ ’ ’ ’ - I C F I ( ~ ’ ’ ‘ I,)), it follows that the astriction U, : E + F o of U is in a* as well. Note that F o = (F”/F)’. With the canonical surjection Q : F” --* F”/F, it follows from 17.15. that Q&Jo

F”8aE

-

F’t&(F”/F)t -F”/F&(F”/F)’

is continuous. A check on elementary tensors shows that the diagram

297


commutes. Since Ti(E') c F, it follows that Q o Ti o 17:= 0 and the same argument as before gives the result. 0 The space (F"/F)' is complemented in F"' (this was just shown in the proof), hence F"/F has the approximation property if F"' has it. It will be shown in a moment (22.9.) that the statement of the proposition does not hold if only F" is required to have the approximation property. 22.8. Coming back to the question under which conditions

amin c

%dual min

holds, the proposition gives the

(a,A) be a Banach operator ideal and El F Banach spaces. If F is l-complemented in F" or if E' or F"/F has the approximation property, then

COROLLARY 1: Let

g m i n duel ( E ,F

) = %dual

min

-

(El F )

holds isometrically. If a (Qmazl Amax)this means that: T E f ! ( E ,F ) is at-nuclear if and only if T' is a-nuclear; the respective norms coincide.

= ( Q l A)dua1 , one may assume (a,A) to be PROOF: Since clearly, (a,A),,= maximal. A weakly compact operator S E f!(F', E') is u(F', F ) - u(E', E") continuous are compact, the if and only if it is a dual operator. Since all operators in amin proposition implies that the canonical map

-

~ 6 3 ~ ~ (TI ' E lZLmin(F',E') IT E 2 ( E l F ) } is a metric surjection under each of the hypotheses of the corollary. Since

FgaE'

l

-

commutes, this (and El&+ F a-nuclear; moreover,

-t+

C(F', E') 3 T'

T

21dua1

C ( E lF ) ) shows that T is a'-nuclear if T' is

The converse is always true. 0 The fact that S' is p-nuclear if S is Pt-nuclear easily implies the following: The dual T' of T E f ! ( E ,F ) is at-nuclear if and only if t c ~ o T= T"otc~: E + F" i s a-nuclear. Therefore, corollary 1 (applied to a t ) gives the

298

22. Minimal Opemtor Ideal8

COROLLARY 2: Let (a,A) be a Banach operator ideal and El F Banach spaces. If F is I-complemented in F" or if El or F " / F has the approzimation propedp, then T E amin(E, F ) if and only if KF o T E Bmin(E,F"); moreover, Ami"(T) = A m i " (o~T~)

.

In other words, T is a-nuclear if and only if K F o T is a-nuclear (where a is the tensor norm associated with am,.).Clearly, the result is trivial under the first assumption that F is complemented in F"; for more information about these questions, see 25.11., Ex 22.6. and Ex 25.12.. 22.9. For a =

these results were already proven if F is complemented in F" (see Ex 3.34.) and if El has the approximation property (see 5.9.) - but not yet under the hypotheses on F"/F. The example in 16.8. showed that it is not enough to assume that E has the approximation property. It is also not enough that F" has the approximation property; the key to a counterexample is the following result due to Lindenstrauss [172]: If G is a separable Banach space, then there is a Banach space F such that F" has a basis and F"/F is isomorphic to G. If one takes G to be a space without the approximation property, one obtains a space F" with a basis such that F"/F does not have the approximation property.

PROPOSITION: There are Banach spaces E and F such that F" has a basis and there is a non-nuclear T E G ( E , F ) with nuclear dual. PROOF: For E := F"/F, as before, it is enough to find a non-nuclear T E C ( E ,F )such that KFOT is nuclear. For the quotient map Q : Fit -+ F"/F consider the commutative diagram

(F"/ F)'& F"/ F

A %(F"/ F, F"/ F ) 3 Q o S

T

The construction of F implies that 52 and 53 are injective, but J1 is not! Therefore there is a z, E (F"/F)'& F" such that w, := id&Q(z,) # 0, but its associated operator JI(w,) is zero. If To := Ja(z,), then Q o To = 0, hence im(T,) C F and To = K F o T for some T E f!(F"/F, F ) . If T would be nuclear, then the diagram would imply that To = 0. 0

299


The results of 22.7.-22.9. are taken from Losert-Michor's exposition [179] of Grothendieck's %sum& A counterexample similar to the above one was also recently constructed by Oya-Reinov [196]. The investigation of 24."' min and m i n i min ill be for accespostponed to 25.11. since it requires some more flexible descriptions of amin sible operator ideals 24.

Exercises: Ex 22.1. Use the Davis-Figiel-Johnson-Pelczyliski-factorization theorem (see 9.6.) to show that for every T E 21min(E, F) Amin(,) = inf IISllA(To)llRII ,

%(GI, G2)for reflexive GI and Gz and, obviously, R, S E 5 and T = S O T ,o R. Ex 22.2. Let E and F be Banach spaces, one of which has the approximation property. where To E

Then

I

E& F = {T E ff(E',F ) T a(E', E)-a(F, F')-continuous}

.

Hint: 22.7. if F has the approximation property; dualize and approximate T' in the other case. Ex 22.3. Give conditions (like the approximation property, accessibility) which imply that Amin(,) = AmaZ(T)on $ ( E , F). Check this for the ideals: ff,%,gp,q,2p,Dp, q - l , q , and v p . Ex 22.4. If ('2, A) is a maximal Banach operator ideal, F an CL-space, then a i n j min (E, F) = )2Lmin(E, F)

for all Banach spaces E. What about the norm? Hint: 20.8.. For nuclear operators the converse is true, see 23.7.. Ex 22.5. Use factorization to show that %dual min

min dual

C%

and

5 Adual

~min(Tt)

min

(T)

for all quasi-Banach operator ideals. Moreover, %dual min (E, F')

=amin

dual

(E,FO

with equal norms. Hint: Ex 22.1.. Ex 22.6. Let (a,A) be a quasi-Banach operator ideal. The regular hull (a,A)reg of (a,A) is defined to be the class of all T E t ( E ,F) such that K F o T E and A r e 9 ( T ) := A ( K Fo T). is a quasi-Banach operator ideal containing a. (a) Show that (a,

23. 2;-Spaces

300

(b) (a,A)dua'reg = (Q, A)dua';in other words, (Q, A)dua'is regular. (c) (Q, A)dua'min reg = (a,A)mindual, Hint: Ex 22.5.. Ex 22.7. Eztension and lifting properties of minimal operator ideals: Let a be a tensor norm with associated maximal normed operator ideal (Q, A). 1

(a) a is left-projective if and only if: Whenever I : G-E and T E Qmin(G,F), then for every E > 0 there is an ectension f' E 21min(E, F) with Amin@) 5 (1 +&)Amin(T) and T = ? o I. (b) a is right-projective if and only if: Whenever Q : G A F and T E Qmin(E, F),then for every E > 0 there is a lifting f' E IZLmi"(E,G) with Amin@) 5 (1 &)Amin(T)and T = Q o F. Hint: Representation theorem and 20.3.(1) and (2). Ex 22.8. Let (Q, A) be a maximal normed operator ideal with associated tensor norm a. Use the characterization 17.15. to show that

+

- in

other words, T o S is nuclear if S is a-nuclear and T is a*-integral. Conversely,

T E Q*(E,F ) if for all S E !21min(C2,E) the operator T o S is nuclear. Hint: Lemma 17.16. and 22.2., corollary 1. Ex 22.9. A continuous trace T on a quasi-Banach operator ideal (%,A)assigns to each T E Q a number T(T) E K such that (1) T restricted to each Q(E,E) is linear and continuous. (2) T ( z ' @ ) = (z',z) for t' E E' and z E E. As for the existence of ideals with a continuous trace see the books of Konig [147] and Pietsch [216]. Show the following: If the Banach space G does not have the approximation property, then 'JI(G, G) 6 Q(G,G) for each quasi-Banach operator ideal which admits a continuous trace. Hint: Closed graph theorem, 5.6. and t r ( z ) = T ( J Q ( z ) ) for all z E G'&G.

23. 2;-Spaces

This section gives a systematic presentation of the class of all 2;apaces. These spaces were introduced in Chapter I and have already been used at various times. Locally they look like and hence share many properties oft,. They are very closely related to the class of 2,apaces investigated by Lindenstrauss and Pelczyliski in their famous 1968-paper; in order to make them easier to use we modified their definition

23. i!; -Space8

301

slightly and as a result they behave better under dualization and complementation. The main results about .C,- and hence E:-spaces presented in this section are, at least in spirit, due to Lindenstrauss, Pekzyriski and Rosenthal.

23.1. Recall from 3.13. that for 1 5 p _< 00 and 1 5 A < 00 a normed space E is called an Ci,.+-space, if for each M E FIN(E) and E > 0 there are R E C ( M , q ) and S E C ( q ,E) for some rn E N factoring the embedding 15 such that llSll llRll 5 A+& :

E is called an t:-space if it is an Zip,.+ for some A 2 1. It has already been shown that all C ( K ) and L,(p) are C&,,-spaces (see lemma 4.4.) and that all L , ( p ) are .C;,l-spacea (Ex 4.7.). It is easy to see (Ex 4.7. and Ex 6.6.) that E is an 2;,A-space if and only if its completion?!, is an ,Ci,.+-space. The local techniques for .2;-spaces, i.e. passing from statements about the el: to all t;-spaces, are summed-up in - the 2p-local technique lemma about inequalities between tensor norms (see 13.5.). - its generalization to the continuity of operators ide; @ T (see Ex 13.7.). - the &-local technique for minimal operator ideals (Ex23.13.). - and the following lemma for maximal operator ideals (which by corollary 2 of 22.3. also has a consequence for minimal ideals):

.Cp--Loc~~ TECHNIQUE LEMMAFOROPERATOR IDEALS:Lei (%,A)and (%, B) lie quasi-Banach operator ideals, (23, B) maximal, c 2 0 and E a Banach space. (1) If B 5 cA on C($, E ) for all n , then %(F,E ) C B(F,E ) and

B(T)5 cXA(T) for all Z;,.+-spaces F and T E %(F, E ) . (2) If B 5 CA on C(E,%) for all n, then %(E,F ) C %(ElF ) and

B(T)5 cAA(T) for all 2;,,-spaces F and T E %(ElF ) . (3) If%($,I?,) C %(tP,lq) and B 5 cA on %(I?,,t?,),then %(ElF ) C %(ElF ) and

B(T)5 c b A ( T )

302

23. .f!;-Sjmces

for all Ci,x-spaces El C:,,-spaces F and T E %(ElF).

PROOF: Since % is maximal, consider for M E FIN(F) T

M C ' . F - E

A/.

IlRlI

IlSll i A + &

9

c

which gives ( 1 ) . For (2) anticipate the dual characterization in corollary 4 below (23.4.):

E-F

T

(2'" c

IlRll IlSll 5 A + E

for L E COFIN(F) - and again maximality gives the result. (3) is a consequence of (1) and (2). 0 The ideal (Cp, Lp) of p-factorable operators turned out to be the class of all operators T E C(E, F ) such that K F o T factors through some L p ( p ) ;moreover,

L p ( T )= min{llRII

IlS(( I

KF

R S o T : E-Lp-F")

.

This was obtained in 18.6. by ultraproduct techniques.

COROLLARY: The assumptions of (3) imply the formulas C , O % O ~ ~ C and %

B~c(L,oAoLp).

PROOF:If U E C, , then U" factors through some L,(p). Take T o S o R E C, o 2l o Cp; then

Since B(TlS"R2) 5 cA(TlS"R2) by ( 3 ) of the proposition, it follows that (TSR)", and hence T S R is in '13 - with the desired norm estimate. In the spirit of this corollary, statement ( 1 ) of the proposition reads E) and (2) 2, o %(El.) c % ( E , .) with the norm estimates. %(a,

0

ep(-, E) C

23.

2;-Spaces

303

23.2. The ideal (XP,Lp)of p-factorable operators is associated with the accessible tensor norm w,, the adjoint w; of which is even totally accessible (21.5.). This implied that every 2:,Aapaces has the X-bounded approximation property (see 21.6. and also Ex 23.1.). Recall from Ex 9.8. that the space ideal space($) is the class of all Banach spaces such that idE E 2,.

THEOREM: Let 1 5 p 5 co and 1 5 X statements are equivalent: (a) E i s an 2i,A-space.

< 00. For each normed space E the following

(b) idg is p-factorable, i.e. 6 , E space(2,), and L,(idE) 5 A. (c) For all Banach spaces G (or only G = E’ or G some predual of E )

w; 5 a 5 Xw;

on

G @ E.

on

G@ E.

(d) For all Banach spaces G (or only G some Johnson space C,) E

_< wp _< X E

(e) There as a factorization idE” = S o R through some L p ( p ) (where p is strictly localizable) with llSll IlRll 5 A.

PROOF: By what was already said in 23.1. and 13.4. one may assume E to be complete. . , in section 17 (see 17.15, and The characterization of the accessible operator ideal C 17.16.) applied to T = idE gives the equivalences between (b), (c) and (d). The last statement means that L,(idEv) _< A (note that Lp(T)= min ...), so (e) is equivalent to (b) since T E I# if and only if T” E (21 for each maximal normed operator ideal I# (see 17.8.). Since G @c A$ = G (see 12.9.), the tp-local technique lemma 13.5. shows that (a) implies (d). To see the converse take M E FIN(E) and denote by ZM E M’ 61E the element associated with :1 E S ( M , E). Then

which, by the definition of w, (see 17.10.), gives the desired factorization 1; = S o R through some with IlS(l llRllL X + E . 0 The last calculation was in some sense the crucial point of the proof of the theorem. 1: (1) A normed space E is an 2:,A-space if and only i f i t s dual E’ is an COROLLARY .Ci, -space. (2) Complemented subspaces of 2;-spaces are L!;-spaces; more precisely, i fG c E i s p-complemented and E an 2iaA-space,then G is an L!i,pA-space.

304

29. 2F-Spaces

(3) E is an 2;-space if and only if its bidual E" is isomorphic to a complemented subspace of some L p ( p ) . The bidual of an i!;,l-space is isometric to a l-complemenied subspoce of some L p ( p ) .

-

-

PROOF: The relation 2pdY' w;, = w, 2, means: idE E i!, if and only if id^)' = id~~pl E 2,t (with the same norms, see 17.8.). Statement (2) is obvious from (c) or (d) of the theorem and the last statement follows from this and (e). 0 In particular, the class of 2:-spaces is exactly the class of spaces isomorphic to a Hilbert space (this is already known from 11.11. where this result was shown using the little Grothendieck theorem). Since wP 5 cPwz for all 1 < p < 00 for some cp 2 1 (see 12.8.), it follows that

COROLLARY 2: Halbert spaces are C;-spaces

for all 1 < p

< 00.

However, they are neither 2&- nor Ef-spaces (if they are infinite dimensional) by the little Grothendieck theorem (11.11., corollary).

23.3. Lindenstrauss-Pelczyliski [173] called a Banach space E an Zp,x-space if for each M E FIN(E) there is an N E FIN(E) with M c N and the Banach-Mazur distance

Obviously, g,,x-spaces are i!i,A-spaces. Infinite dimensional Hilbert spaces fail to be l$,,x-apaces (for p # 2 ) since d ( 4 , q )= n11/2-1/PI + 00 for n 4 00 (see Ex 26.4. for a proof of this well-known fact); but this is the only difference: Lindenstrauss-Rosenthal [174], theorem 4.3., showed exactly the non-trivial implications of the following two statements: 1 < p < 00 : A Banach space is an 2i-space if and only if it is an CP-space or isomorphic to a Hilbert space. p = 1 or 00 : A Banach space i s an 2!-space if and only if ii is an 2,-space. However, the constants may vary! Cg-spaces are much handier than 2,p-spaces; for example, it is not true that the 2,Cpl-constant of the dual of an Cp,A-space is 5 A since there are infinite dimensional Cm,l-spaces, but no infinite dimensional 21,1-spaces! See Lindenstrauss-Tzafriri [176], p.199. It seems to be unknown whether or not the is an 2,~pl,~+~ for every E > 0. dual of an 2 , ~ The Lindenstrauss-Rosenthal characterization of 2;-spaces implies that 2;-spaces E which are not isomorphic to a Hilbert space contain the $ uniformly: There are M,,E FIN(E) with sup d ( M n , $ ) < 00. For p = 1,00 this can be seen as follows: By local reflexivity (see explicitly corollary 6.6.) and theorem 23.2.(e) one may assume that E is complemented in some L p ( p ) .If p = 00, it follows from a result of Pelczyliski

23.

2;-Spaces

305

[197] that E contains a subspace isomorphic to c,, which implies the result. But this observation is helpful also when p = 1 : In this case it follows that the Ck-space E‘ contains colhence there is a surjection E” -W el; this easily implies (by lifting the unit vectors) that E” has a subspace isomorphic to L1.

COROLLARY 3: (1) Infinite dimensional 2& -spaces have neither proper type n o r proper cotype.

( 2 ) Infinite dimensional 2f-spaces have no proper type, but do have cotype 2.

( 3 ) C;-spaces have type min{p, 2 ) and cotype max{p, 2 ) for 1 5 p < 00. (4) space(q3) # space(&) i f P

#q

and PI q E [I, 4.

PROOF: By what was said above (1) follows from Ex 7.14. and (2) from Ex 7.13. and 8.6. - this latter reference also gives (3). These statements and Ex 8.13. give (4). 0 In particular, no infinite dimensional space can be simultaneously an CS,- and an Cf-space. 23.4. A dual characterization of C;-spaces is the

COROLLARY 4: A Banach space E is an C;,x-space if and only if for every subspace L E COFIN(E) and E > 0 there i s a factorization Qf = S o R through some with IiSIl IlRll I + E E

9,”

EIL

PROOF: If E is an 2;,x, then its dual is an C;,,x. Thus, if L E COFIN(E), there is a factorization

hence Qf = Rb o Sb o K E . Conversely, and in essentially the same way, the condition implies that E‘ is an 2pB,,x+pace. 0

23. 2 ;-Spaces

306

23.5. In 20.14. it was shown that r\ /T

= w;

= w& and E / = wl; transposition yields and

\E=

wco

.

Thus, theorem 23.2. gives a characterization of 2 :- and .C&-spaces in terms of respecting subspaces and quotients; for 2 : the result is due to Stegall-Retherford [259].

COROLLARY 5: Let E be a Banach space. (1) E is an 2:-space if and only i f E & respects subspaces isomorphically. More 1 precisely, E is an 2;,A-space if and only i f for each G- F and z E E 8 G ~ ( zEl ; G) 5 A r ( z ; El F )

.

(2) E is an Sg-space i f and only i f E C& respects quotients isomorphically. More 1 precisely, E is an 2k,A-space if and only i f for each Q : F + G

PROOF: That the conditions are necessary follows from the properties of r\ and E / . Conversely, observe that if E respects quotients, then, using !l(Bc) --y G (recall theorem 20.6.), there is a AG with E / 5 AGE on E@G- and a simple lp-sum-argument shows that the constant can be chosen independently from G. The inclusion about the unit balls gives as well E / 5 AE on E 8 G for all G. Since E / = w k , this implies (2). Similar arguments give the result for (1). 0

-

Grothendieck’s theorem 3.11. says that the 2f,l-spaces are exactly the spaces of the form L1(p) for some strictly localizable measure p. Moreover, it follows that Lf-spaces share the interesting lifting property expressed in 3.12., corollary 1 - in particular, the lifting of compact operators, see Ex 3.31.. The result in 3.10. now reads that E i s an 2:-space (resp. an 2:,A-space) if and only i f E’ is an injective Banach space (resp. has the A-extension properly). For 2f,,,-spaces this means (by the duality result in corollary 1) that a dual space is A!&, if and only if it has the A-extension property. Moreover, as in 4 . 5 , the fact that for an 2&,A-spaceE ff (G,E ) = G’& E = G’& E (L&+paces have the approximation property) shows that these spaces E have the compact extension properly with constant X : For every isometric embedding G L+ F, every operator T E R(G, E ) and E > 0 there i s an extension ? E R(F, E ) of T with 11p1 l 5 (A e)llTII. This, as well as its converse (see Ex 23.4.), is - in its spirit - due to Lindenstrauss [171]. Going to the completion, the corollary gives the following result due to Kaballo [134], which can be used for the lifting of vector valued functions, see also 4.5., corollary 1.

+

23.

.Ci-Spaces

COROLLARY 6: A Banach space E is an C",space

307

if and only if the completion E&*

respects quotients isomorphically.

PROOF:The quotient lemma 7.4. implies that E&,* respects quotients; 80 the condition is necessary. Conversely, take a metric surjection !l(r) --y C2 onto a Johnson space Cz. Then the condition implies that E&,,,C2 = E&C2 holds isomorphically and hence E / 5 X E for all E 8 G by the very properties of C2, see lemma 17.16.. 0 23.6. It was shown that dual C&-spaces are injective. This is not true for arbitrary Lsapaces; take the Cs-space co which is not complemented in tm.Every injective space E has the X-extension property with X = IlPll, where P : !,(BE') 4 E is a projection (Ex 1.7.). Define the projection constant of a Banach space E to be

I

X(E) := inf{llPII P E c(&(BEf), E) projection onto E} E [0, m] . PROPOSITION:

Lm(idEf) = >(El)for each Bonach space E .

'

PROOF: If P is such a projection, then i d E f = f ' o IET ( B E f f ) is a factorization through an Lm(p), hence L , ( i d p ) 5 llf'll. Conversely, if X = L,(idEf), then, by what was said above, EJ has the X-extension property and therefore X(E') 2 A. 0

Note that it was shown that El has the X(E')-extension property, in particular, X(E') =

*in{ llPll I ...}.

-

Since the adjoint operator ideal C& w& = g& is the ideal 'J31 of all absolutely l-summing operators, it follows from 17.19. that for each M E FIN

X ( M ) P l ( i d u )3 dim M and there is even equality if the norm on M satisfies certain invariance conditions (e.g. M = 5).This gives a possibility of calculating projection constants; for the euclidean space 4 see 11.10,. See also Ex 23.8. for the study of the extension of operators.

-

23.7. In order to obtain some operator characterizations of C&-spaces recall J x for the integral operators, Pin = % and the representation theorem for minimal operator ideals. The diagram

308

29. C;-Spacea

commutes, and also the left vertical map is injective since r\ is totally accessible by 21.1.(2). Arguing as in 5.9. or Ex 5.13. with the Hahn-Banach theorem, one obtains

(.3inj)min(E, F) = %(E,L,(Bp)) n 2 ( E , F ) = %'"j(E, F ) if E' has the approximation property. In 25.11. a much more general result will be obtained, showing that this also holds if F has the approximation property. By the characterization of quasinuclear operators (see Ex 9.13.) this means exactly that:

LEMMA:If E' or F has the approximation property, then E'&\ F = U%(E, F ) holds isometrically.

The following characterization is due to Stegall-Retherford [259].

PROPOSITION: For each Banach space E the following statements are equivalent: (a) E is an 2%-space. (b) QZ'JI(E,F) = %(ElF ) f o r all Banach spaces F (or only C2). (c) u f R ( F ,E) = %(F, E ) f o r all Banach spaces F (or only Cz).

PROOF:If E is an C&, its dual E' is an Cf-space; since x\ = wit, theorem 23.2. implies that r\ and 17 are equivalent on E' 8 F, which implies (b) by the lemma. Conversely, (b) gives ?r\ M x on E' 8 C2, which shows (17.16.) that ?r 5 Ax\ for some X on all E' 8 F. This implies that E' is an Cf-space (by the theorem) and E is an C& -space. That (a) implies (c) is a consequence of the fact that a NN a\ on F' 8 E for all tensor norms a if E is an C&-space (see 20.8.). For the converse, it follows that (again by the lemma) ?r M x\ on E' 8 E. The theorem implies that E' is an Zf-space. 0 In Ex 23.9. the reader will find a characterization of C$-spaces in terms of pcompact operators Rp = CPn.

23.8. The preceding characterizations were in terms of minimal operator ideals. To obtain a characterization with p'-dominated operators (recall 2; = !Dp/) the following lemma, which is a special case of the much more general quotient formula to be proven in section 25, will help. LEMMA:Let 2 be an accessible maximal normed operator ideal. Then f o r each Banach space E the following statements are equivalent: (a) E E space(%).

23. 52;-Spaces

309

(b) W ( E ,F ) = 3(E,F ) f o r all Banach spaces F . ( c ) %'(F, E ) = 3(F,E ) f o r all Banach spaces F. Q of 2l is accessible. Since a* o !2i C 3 and 17.19., statement (a) implies the others. Assume (b), then

PROOF:The associated tensor norm Q o 'ZL'

c J by

is continuous and hence G @c E + G BU E, which implies that idE E %, see 17.16.. If (c) is satisfied, use the fact that a* is accessible and CZhas the metric approximation property, to see that the diagram

I proves the continuity of C2 and theorem 17.15..

E

1 +

CZ

E, which implies idE E 2l by lemma 17.16.

It is clear from 17.19. that, for example, I(T) 5 A ( i d E ) A ' ( T ) in (b). Since 2; = Pp,, the p'-dominated operators, it follows that:

PROPOSITION: For a Banach space E and 1 5 p 5 00 the following

are equivalent:

(a) E is an 52;-space.

(b) !D,t(E, F ) = 3 ( E ,F ) f o r all Banach spaces F. (c) PDp#(F, E) = J(F,E ) f o r all Banach spaces F. Recall that 91 = !$JI and Pm= Vdua' 1 . 23.9. A!;-spaces

enjoying the Radon-Nikodjrm property are rare:

PROPOSITION: Let the 52:-space E be complemented an EJJ.Then E has the RadonNihodim property if and only if id is isomorphic to some ll(r). PROOF:Observe first that ll(r) has the Radon-Nikodim property (see Appendix D3.). Conversely, theorem 23.2.(e) implies that E is isomorphic to a complemented subspace G of some L1(p). By proposition 2 of Appendix D2.(Lewis-Stegall theorem), the projection P : L 1 ( p ) 4 G factors through some l1(r0):

310

This shows that G (and hence E) is isomorphic to a complemented subspace of tl(I',) and, as a consequence, has the lifting property (see 3.12.). It follows that E is isomorby Kothe's result mentioned in 3.12.. Cl phic to some This result and the following interesting consequence are due to Lewis and Stegall[170].

COROLLARY 1: The dual E' of a Banach space E is isomorphic t o some

t,(r) if and

only if

Pl(E,F ) =

F)

for all Banach spaces F.

PROOF: It follows from 23.8. and 23.2. that P l ( E , .) = J ( E , if and only if E' is an Ciapace. Appendix D8. gives that J(E, F') = % ( E ,F') for all F if and only if E' has the Radon-Nikodjm property. This, together with the proposition and 5.9., easily implies the result. 0 0

)

The Hardy spaces HP are $;-spaces for 1 < p < 00 since they are complemented in Lp[O,274 (see Ex 4.7.). Pelczyliski [201] showed that H" is not a quotient of any C ( K ) . Since H" is a dual space ([202], p.ll), it cannot be 20, by theorem 23.2.(e).

COROLLARY 2: The Hardy space H w is not an ZCg,-space, the Hardy space H' i s not an 2:-space.

PROOF:It is known that H' is a dual space ([202], p.12) and separable, and hence has the Radon-Nikodjm property. If it were an Ciapace, it would be isomorphic to 41 (by the proposition), hence weak and strong convergence of sequences would coincide: The sequence (exp(in.)) converges weakly to xero by the Riemann-Lebesgue lemma, but each function has norm 1, a contradiction. 0

23.10. Recall from 17.14. Grothendieck's theorem

2(tiIt2)= Pi(el,tz) and P l ( T ) 5 KGllTll . By the Cp-local technique lemma 23.1. for operator ideals this can easily be extended: GROTHENDIECK'S THEOREM: If E is an C:,A-space and H a Hilbert space, then every T E C ( E , H ) as absolutely I-summing and

Pl(T) 5 KGAllTII

23. 'cp"-Spaces

311

This improves the little Grothendieck theorem, which stated that T E V 2 ( E ,H ) . The same argument as before shows that the first theorem in 17.14. reads as follows for C+paces:

THEOREM: Let E be an Ck,A- and F an Cf,l,-space. Then every T E C ( E ,F ) is 2-dominated and (hence) absolutely 2-summing; moreover,

In the next section it will be shown that for 1 I p 5 2 the sequence space lp is isometrically contained in some L t ( p ) - the Lkvy embedding. Since V2 is injective,

T : &, implies that C(L,, .fp) = !&(4',

-

l p L L i ( p )

,tp)and hence, by the Cp-local technique for operators:

1: If E is an C&,A- and F an C;,,-space for 1 I p COROLLARY

5 2, then

-

Clearly, the result also has a tensor norm formulation, see Ex 23.15.. Note that = 9 2 (with equivalent norms) since w$\ 92 = g ; (see the table in 20.15.). See Ex 25.5. and Ex 25.6. for more information about this result. Proposition 20.19. gives the "adjoint" form:

COROLLARY 2: If E and F are Banach spaces, E an C;,A-space for 1 5 p 'Y2(E,F ) = V i ( E ,F )

and

5 2, then

P i ( T ) I KcAP2(T) .

I

which shows that p(u)

P(.) = suP{I(uI .)I

5

I

a(u)a'(z) and

If ( M 4

5 a'(u)a(r) (and

so

on)

1

f(u); therefore,

I f(4 I 1) Isup{I(u, 41 I @(.) I 11 = B ( 4 = P * ( 4

*

The statement concerning the Riesz-condition follows from the fact that f satisfies it whenever a (and hence a',a* and at)does. I7

26.10. For u(") := C;=lek 8 ek E K" 8 K" and p, q E [I,001 the inequality

a ( u ( " ) ; q , q )< 7r(u'"';q,q) 5 n holds. The limit order X(a;p,q)of a tensor norm a on FIN is the infimum of a11 X E [0,1] such that there is a CA 2 0 with

ff(u(");q,q)I cxnx

for all n E I?.

356

26. More About

Lp

and Hilbert Spaces

The limit order A(%; p , q ) of a Banach operator ideal (24, A)is defined to be the infimum of all A E [0,1] with a cx 2 0 such that A(id :

C

-c)

5 cxnX

for all n E N.

This notion is due to Pietsch; see [214], sections 14 and 22, for the explicit calculation of various limit orders. Obviously,

if a and 24 are associated and clearly, X(a;p, q ) = X(a'; q , p ) . Since

It may happen that the sum is even 2 (see Konig [144]), but in many cases it is 1. For example (Pietsch [214], 22.4.12.),

if 2 5 p 5 00 and (r,s) E [l,21 x [l,001. If a is a tensor norm and Pff the one constructed in proposition 26.9.

then, clearly,

LEMMA: If A(a;p,q)+A(a;q',p') = 1, then

f o r all tensor norms 7 with Pa 5 7

PROOF: Inequality (**) gives

pff.

357

26. More About Lp and Hilbert Spaced

so the first terms in the two sums coincide. Again by (**) it follows that

This implies the statement. 0 Take a = g; for 2 5 p _
0 such that

basis of a Banach space E. Then there is a

PROOF:e1&E is the space of unconditionally convergent series in E and

by the results of section 8. Since

m

m

n=l

n=l

is continuous (closed graph theorem), it follows that there is a c > 0 such that

which is the result. 0

PROOF (ofthe theorem): It is clear that it is enough to show that a and u are equivalent on 484 with constants independent of n . Define for z := aL:,tek8 e t f 484

ci,t=l

26. More About L, and Hilbert Spaces

359

(canonical embedding). Then, by the lemma, there is a constant c > 0 such that

Now fix z E t?!j QD l!j and choose two isometries Ul,lJ2 : l!j

+ l!j

such that

2 0 - this follows, e.g., from the polar decomposition UA of the operator t?!j (where U is an isometry and A positive) and the diagonalization A = DV, hence U-'T,V-' = D. with

Xk

T, :l!j

Isometries (onto) preserve tensor norms, therefore, n

.

u(z) = k=l

Now take the Fourier matrix (see Ex 4.3.)

and

Since all operators involved are isometries, it follows that a ( w ) = a(%).But

which implies a(z) = Cr(lw1) 5 c a ( w ) = c a ( 2 ) u(z) = a(lw1) 2 c-'a(w)

- and this was to be shown. 0

= c-la(t)

360

26. More About

Lp and Hilbert Spaces

Exercices: E x 26.1. Let p, q €11, m[ with l/p+ l / q 2 1 and 1 5 s 5 2 5 r 5 00.

(a) G OCr

c &,q

(b)

9p',p' 0

2, C 9,'

(c)

.CroSp',q'

c

*

.

Hint: 26.1., 23.1., cyclic composition theorem.

Ex 26.2. Let p,q E]1,00[ and

l/p+ l / q

5 1 and

1 58

5 2 5 r 5 00. Then

Hint: 26.1..

Ex 26.3. For p , q €11, cm[and l/p+l/q 2 1the projective associate is equivalent to wz and /a;,q\ is equivalent to w2. Hint: Grothendieck's inequality, 20.9. and 26.1.. EX 26.4. Show that L z ( i 4 ; ) = d(q:,G). Deduce from this and 26.2. that

for all p E [ l , 001.

Ex 26.5. Take g:, ...,gA E LP(pl) and g:,

...,g$

E Lp(p2).Then

if 1 5 q' 5 p 5 00 (with constant c = 1) or if 1 5 p 5 2 5 q' 5 00 (with constant = Ki). Hint: Use 11 Cea&n : --* Lpll = w , ( g t ; L p ) from 8.1. and 26.3..

c

q,

Ex 26.6. Use the foregoing exercise to show: If ( g : ) a E N are weakly q-summable sequences in Lp(pi) and 1 5 q' 5 p 5 00 or 1 5 p 5 2 5 q' 5 00, then (9: 8 gz),,tEN is weakly qaummable in Lp(pl 8 pa). In particular (see 8.3.), if p 2 and (9;) are unconditionally summable in L p ( p i ) , then (9: 8 g&,tEN is unconditionally summable in Lp(pi 8 p2). Using the counterexamples in 26.3., check that the result is false if 1 5 p < q' < 2 or 2 < p < q' 5 00. These results are due to Rosenthal-Szarek [233].

s

27. Grothendieck's Fourteen Notuml Tensor Norms

Ex

361

26.7. Deduce from 26.6. that on Hilbert spaces : %*,=A : A1 = A,

P,PEILoo[ p €11, oo[

-

= 9Ip = 36;

compact operators Hilbert-Schmidt operators.

1

Hint: Hi & H2 %(HI, H2) for maximal normed operator ideals. Ex 26.8. Show that the Schatten-construction in 26.8. gives an accessible tensor norm y whenever a is acceasible. Ex 26.9. (a) Take 1 q 5 p 5 00. If E C L F ( p ) and T E Z(E,L:(v)), then llT"ll = llTll. Hint: Use 7.9. as in 26.3.. (b) The best constants in the Khintchine inequality are the same in the real and complex case.

27. Grothendieck's Fourteen Natural Tensor Norms

The second chapter ends with a reference to Grothendieck's Rhumk: It will be shown that if one starts with the injective norm e and takes the adjoint, transposed, right or left injective or projective associate finitely many times, then one obtains, up to equivalence, only 14 different tensor norms. This Grothendieck result is, in a certain sense, a capsule summary of the results in the Rkumk.

27.1. Let (9 be the smallest set of (finitely generated) tensor norms which contains the injective norm E and is stable under the operations a-a', a f ,a\, and hence stable under a-a*, a/,\a,\a/,/a and /a\. A finitely generated tensor norm a is called natural if it is equivalent to a tensor norm in (9. Grothendieck's inequality states that /z\-

w2

and

\E/-

w;

,

hence the set Q 0 , consisting of the tensor norms Wl\ , \ E = % o , /w, , T\ = w&, , wL/, / T = W ; , \wi , wa/ [- d2 = 41 , \w2 [- ~2 = dl , w2 ,

E , e/=w1

1

T

(see 20.14. and the table in 20.15.) contains only natural norms.

362

27. Grothendieck's Fourteen Notuml Tensor Norms

THEOREM (Grothendieck): Every natural tensor norm is equivalent t o one of the fourfeen tensor nomas in @., Actually a more precise statement can be made, which (due to an idea of Grothendieck's) may be summarized in the following table of natural norms (and their associated maximal Banach operator ideals):

27.2.

J

I

I

1

c

I

J

1

1

J

I

c

1

1

I

I

J

1

1

J

I

I

1

I I

I

J

I

&

c

I

(See 20.12. for the factorization properties of the ideals VI/ and \'J33Sa'.) This table of the fourteen norms in 0, has the following properties, the first of which proves theorem 27.1. and the third of which shows (in particular) that all the fourteen norms in 0, are mutually non-equivalent :

(1) If a E a,, then a',at and a\ are equivalent t o a noma in 0,.

27.

Grothendieck's Fourteen Natural Tensor Norms

363

(2) A n arrow a -+ p means that a dominates p, i.e. there is a constant c with /3 5 ca. In other words, Q c 2?for ihe associated operaior ideals. ( 3) There are no other dominations in the table than those indicated. (4) If the locations of Q and p are symmetric with respeci to the point * in the middle, then a' and /3 are equivalent. (5) If the locations of Q and /3 are symmetric with respect t o the horizontal dotted line, then a' and p are equivalent. (6) If the locations of Q and /3 are symmetric with respect to the vertical dotted line, then at = p.

-

PROOF: The symmetries (4) - (6) are obvious - the only necessary work was already done in section 20 when showing that \w2 g2 = 92.. To see the stability property (1) - by the symmetries - it is only necessary to show that a\ is equivalent to a norm in 'Po for each a E 'Po : For a = \ ( / x ) , observe first that \w2 = (\w2)\ (by Ex 20.11.); now use the injectivity of w2, Ex 20.5.(b) and Grothendieck's inequality to see that \w2

= (\w2)\ 5 ($/XI)\

5 \(/r$ 5 KG\w2

-

-- -

The remaining non-obvious casea are simpler: (/x)\ w2 by Grothendieck's inequality (wz/)\ ~2 by 20.15. (\E/)\ w;\ g2 by 20.15. ((x\)/)\ = T\ by EX 20.5. ( \ ~ 2 ) \ N \ ~ by 2 20.15. (\E)\ = (\di)\ = ( ( / d l ) / ) ' = d: = E by 20.14. (/(\&))\ = /woo\ = /gw\ = / E = E by 20.14.. The domination arrows on the right hand side are either trivial or can be deduced from the following (the others are a consequence of transposition; recall apz,ga 5 aPlrq1 for p1 5 p2 and q1 5 42 from 12.5.):

5 K L G ~=~KLc&/ / 5 K.cGSk/ by Ex 20.10. \w2 s; 5 \w2 Q; 5 w; w2 5 Q2 woo = goo 5 Q2 /woo = (g&/)* 5 KLGW;*= K L G W ~ by the first case. W;

--

It remains to show that there are no other dominations in the table than those indicated; the arguments will mainly use operator ideals: It will be shown that each ideal on the

364

27. Grothendieck's Fourteen Natuml Tensor Norms

right side of the diagram is only contained in those ideals indicated by the arrows. The rest then follows by transposing. , (see e.g. 23.3.), it is (by the definition of the surjective Since the space L1 is not in C hull) also not in Cr, hence 2 C r : Using the symmetry with respect to adjoints, this implies PI/ $ 3. By corollary 3 in 23.3. the ideals 21, C2, and 2 , are mutually uncomparable: It follows that 2; = ,pa' , 2 : = 9 2 , and 2: = g 1

c

are incomparable as well. In particular, y1 $ C 9 2 . By again using the symmetry , . Clearly, 2, $ !& $ !& - and hence with respect to adjoints, one gets 22' $ 2 all the vertical arrows on the right side are not reversable. As for the righthand side it remains to show that %31/

c %3y c 1

%31

21

1

%32

$21 1 2 , $tC y '1:2

$q

j

*

It is a result of Gordon-Lewis [84] that $t21, and hence V 2 $ 2 1 as well,. Moreover, the fact that t 1 4 space(C2') implies that 21 $ C g r ; this gives 2 , $ C;"j and also Cg' .?!A Using the symmetry with respect to the point *, one obtains !Q1/ from C , C? .

c~ ~ "

c

The deep result of Gordon and Lewis which was used solved one of the open problems in the Wum6. But note that $'2 $ 21 is much easier to show: If the non-compact embedding C[O,11 4 L 2 [ 0 , 1 ] actually factored through some L 1

then it would be in

o9 2

C

%32 o !fS2

C 3 by Grothendieck's theorem.

27.3. For the convenience of the reader - here is a list of Grothendieck's notation:

365

28. More Tensor Norms

Chapter III Special Topics In the first two chapters we presented the theory of tensor norms, the theory of operator ideals and their intimate relationship with each other. With this knowledge at hand, the final chapter touches some questions of a more specific character. Section 28 discusses examples of more tensor norms which are related to Laprestk’s 29 presents the powerful technique of traced tensor norms, which mimics for tensor norms the composition of operator ideals. Though at first glance a little bit formal, it also provides a very general setting for understanding the various tensor product characterizations of operator ideals given in Chapter 11. L Y ~ , , Section .

Sections 30 and 31 deal with the factorization of operators through Hilbert spaces in terms of type and cotype. In particular, this leads to the construction of a nonaccessible tensor norm/operator ideal due to Pisier. Maurey’s mixing operators are studied in some detail in section 32.

-

An important motivation for the sections 28 to 32 is to answer the question: Under what conditions are operators of the form S 8 T : L, @A, E L , 8~~F continuous? The results given include various characterizations and the study of two special operators S : the Fourier transform and the projection onto the Rademacher functions. Sections 33 and 34 treat two interesting topics from the theory of operator ideals: Lewis’ Radon-Nikodfm property for tensor norms, which serves to decide when 21min(E,F) = 2Pac(E,F) holds, and the tensor stability of operator ideals. The last section introduces the so-called tensor norm topologies on tensor products of locally convex spaces. It demonstrates that tensor norm techniques can be successfully applied to questions from the theory of locally convex spaces; in other words, there is also some type of a “local theory” (i.e. in terms of finite dimensional spaces) for locally convex spaces.

28. More Tensor Norms In this section some additional tensor norms will be studied. They are closely related to LaprestC’s tensor norms which were so important. The associated operator ideals have interesting descriptions in terms of factorization, inequalities and the continuity of operators A 8 T : e,,t 8 E -+ .!,8 F.

366


28.1. For p, q E [l,w] with l/p

+ l/q

2 1 (equivalently: q1 2 p ) and

Pp,q(z; E , F) := inf Il(ak,t) : GI

rp,q(z; E , F ) := infll(ak,f) : 6p,q(z; E l F ) := infll(ak,t) :

G t

qt

+

+

+

z E E @ F define

$llwqt(xt; E)wpi(yk; F ) $)&i(Zt;

E)epi(yk;F )

$lleqi(Zt;

E)wpi(yk;F )

where the infima are taken over all finite representations n

ak,tzi@yk E E 8 . F ;

Z= k ,t=1

note that Aet := a k , t e k for the mapping A := ( a k , t ) : GI + $ . With exactly the same proof as for ( Y ~ (see , ~ 12.5. - or Ex 12.8.) one can show that these are finitely generated tensor norms satisfying Ppa,q3

IPP1,Ql

(analoguously for yp,qand lip,q) and PP,O

if

P1

L PZ and

41

5 q2

Pi,q = Pq,p(and this also for Y ~ , ~ Moreover, ).

Ia P , P and

PP4 I 6 P d

IYP,q .

Note that for p or q = 1 some of these norms coincide. Recall (Ex 20.2.) that rP,( is projective and hence rL,qis injective. It follows from 21.1. that yp,qis accessible and $,q totally accessible. It will be seen (28.6. and Ex 28.7.) that the tensor norms ,8p,q and are accessible as well. 28.2. For the study of the associated maximal (and minimal) operator ideals note that z = x k , t a k , ( t : @ y k E E1@F= Z ( E , F ) means that the associated operator Tzfactors

Recall from section 8 (in particular Ex 8.19.) that

L , ( x i ; El) = minIIS : E

-+

llDx : e", -+

CIl

where DAo S = x:Bek : E + is a factorization with a diagonal operator DA with non-negative entries. Since Dx is a positive operator, the s-integral norm satisfies

I,(Dx :

-

-

c c )= llDx :c c((

367


by 17.17.. In particular, lpl(yk;

F) = minilS : F‘ = min IIR : q

= minIIR : q

--

5-

P , l llD, : P ,

F”ll /IDp: ~ “ 1 1 I,~Y’(D,:

411=

-

el(= C q ),

ci=l

with D, a positive diagonal operator such that Ro D, = ek@c&k) : * F” . It is clear now that the maximal operator ideals associated with pp,,,rPlq and bp,q are related with composition ideals of &,JP and J;fual.We shall treat Pp,, and 7p,qin order to demonstrate the methods and leave bP,, as an exercise.

28.3. First -yP,, : Take for finite dimensional spaces M, N n ak,LZ:

8 ?./k E M’

@yp,p

N

-

k,kl

Then for the associated operator T, : M

+N

and any factorization

TR the inequality (IFa’o I,I)(T,) 5 llSllI,~(Dx)llAII I$’a’(D,) IlRll holds. Passing to the infimum over all representations, it follows that (I$ o Iqi)(T, “ :M

-*

N ) 5 yP,,(z; M’, N )

-

by what was said before. To see that this is actually an equality, take for z E M’ @ N a factorization T, = R o 5’ through some Banach space G. Since J,i q , l and Jdual N O ~ , are ~ I accessible (section 21), the operators can be factored (see 18.1.) as P’ follows:

M-G

P ,

s

-q DA

R

G-N

368


It follows that M' @7p,q N = (3$',' o 3 , ) ) ( M ,N) holds isometrically. This is in some sense the core of the following

PROPOSITION: Let l/p+ l/q 2 1. Then

and the associated maximal nonned operator ideal is \9ql,p,/

= (3;v

0 Jq,)reg

.

Recall from 25.9.that the regular hull %reg of a quasi-Banach operator ideal (a,A) is the class of all T E Z ( E , F ) such that t c o~ T E 2l, and Areg(T) := A(KFo T).The ideal \9q,,p,/ is, by 20.12.,the ideal of all operators T which factor

for some compact K and measure p.

vq,

PROOF: 9 q ~=, q$"" p~ o holds isometrically by Kwapieli's factorization theorem 19.3..Moreover, the formula \g; = gel from 20.14. implies that \QqJ

! @ a ' /

--

-

= gq1 3,) g;t/ = (\gi)' = g;,

\g;

-

5pv

by the relationship between tensor norms and operator ideals. 25.9. now gives that

\9q/,p// = \(!Q$O'

0

holds isometrically. Since remains to verify that

M'

!Qql)/

-

@7p*p

= ((W$O'/)

0

(\yqt))reg

= (3;Y'

0

Tq,)reg

is maximal, it and (3$'*' o DqOre9 = \9q,,p,/

N = 'a:$(

0

J q ' ) ( M ,N )

- and this was done before. 0 In particular, it follows that (3;Yoro 3 q ~ ) r e gis a n o m e d operator ideal - a fact which is not a priori clear. 28.4. The "adjoint" version of this result describes the operator ideal

[(3$0'

0 Sq/)+eg]*

-

= (3;Y 0 q)* y;,, = 7;,p = (\&J)r

= /aq,p\

-

c qi n , pj m r

28. Mom Tensor Norms

369

- and this implies a description of the operators in Cinj sur q,p

-

0

Jq4*

-

9

which for p = Q' is an operator form of a result of Kwapieri's [l60] and completes the characterization of subspaces of quotients of Lq-spaces given in 25.9. and 25.10..

THEOREM: Let l / p + l / q 2 1. For each operator T E $(El F ) the following statements are equivalent: (a) T E 'Ur(E,F). (b) For all A E C(tPl,1,) the operator

.CE/

A @ T :tpl

E

-

eq

@Aq

F

is continuous. ( c ) There is a constant c 2 0 such that for all natural numbers n E N ,all matrices (ak,f), all 21,...,zn E E and all yi, ...,y', E F' n

In this case,

PROOF: An obvious application of the closed graph theorem shows that (b) implies the existence of a constant c with llA @ TI1 _< cllAll. Since

and, for A = (ak,t),

it is easy to see that (b) implies (c). The converse is true by the density lemma 7.3.for p > 1 ; for p = 1 one has Am = E , and hence this implication follows by &,-local technique. By what was said before, the operator ideal ':2 l U ris associated with 7i,p.The representation theorem for maximal operator ideals shows that T'E:2 aur if and only if its associated bilinear form satisfies pT E (E @-,q,9 F')' . Therefore the definition of 74rP gives the equivalence of (a) and (c). 0

370


In statement (b) the sequence spaces tp appear: they can be replaced by arbitrary infinite dimensional L p ( p ) ; this could be proved directly using ; : 2 Our = 3;' o (see 25.9.) - a fact which is related to the statement anyway. But there will be a general result in 29.12. showing this.

'J3Fa'

w2 5 ap,q5 a for p , q E [l,2], it follows that 3 = &, I C C 22. Moreover, 2 2 is surjective and injective and 3"'j IuP = 2 2 by Grothendieck's inequality in the operator form 20.17., hence

28.5. Since

(1) For p, q E [l,21 the relation ;:2

= 2 2 holds isomorphically.

Our

In particular, w2 and 7L,p(for 1 5 p, q 5 2) are equivalent tensor norms. For p E [l,m] the relation gil = gp\ (see 20.14.) implies that

and therefore,

(2) 2yj *P

=

'J3Fa'

IUc

and 22'

= Pychold isometrically (1 I p 5 m).

lUr

Clearly, the theorem is related with proposition 1in 26.3.: Since C(L,, L B )= C2(l+! L , ) if s 5 2 5 r (by 26.1.), it follows from (1) that every operator T : L, + L, is in ; : 2 IUr and hence, by (b) of the theorem (and 29.12.),

A @ T LpI(P1) @A,#

Lc(P2)

+

Lq(v1) @Aq L d ( y 2 )

-

is continuous. Note that in 26.3. this was shown for all p , q. Certainly, one of the most interesting cases is w2 = ~y;,~ 2 2 , the operators factoring through a Hilbert space.

COROLLARY: T E C ( E ,F ) factors through a Hilbed space if and only if there is a constant c 2 0 such that f o r all isometric n x n matrices A = (ak,t) and X I , ...,zn E E n

n

n

n

PROOF: By condition (b) or (c) of the theorem the condition is necessary. Conversely, fix t t ; then the convex function g(A) on the left side attains its maximum on the extreme points of the unit ball of C(4,G) - which are the isometries (see Halmos [99], p.263 ). It follows that the inequality holds for all matrices A = ( a k , L ) of norm 5 1 and this easily implies (b) or (c) of the theorem . 0 This result, due to Lindenstrauss-Pelczyliski [173], is in some sense central to the theory of operators factoring through a Hilbert space - e.g. in the proof of Kwapieri's

371


important type/cotype theorem 30.5.. Note also that w; is projective, hence w; = 72,2 (by 28.3.) is an explicit description of the tensor norm associated with the adjoint ideal 2; = 9 2 of 2-dominated operators. See Ex 28.l.(d) for a 3ther description of 22. 28.6. Next, the tensor norm pp,q(z; E , F ) = inf{

ll(ak,L) :

*qllwqJ(tt)Wp’(vk)

and M’@ P , , ~N’ = (M @pi N)’, gives that for cp f (M @ N)’ OQ

lt’pll(M88;,qN)’

= inf Il4l IlRll IlSIl

7

where the infimum is taken over all factorizations ’p = 6 o (R @ S) with an operator R :M + an operator S : N -+ P,and 6 E @= $,)’ for some n E N.Following the ultraproduct-technique-proof oftheorem 18.5. word by word (with the exception that the forms 6 are positive there) gives the implication (a) r). (b) of the

c,,

THEOREM: Let l/p + l / q 2 1 and

(q,

’p

E ( E @ F ) * . Then the following statements are

equivalent:

(a) ’p E ( E @P;,q F)’ * (b) There are measures p and v, operators R E .C(E,L,,(p)),S E E ( F , L , ~ ( v ) and ) 6 E ( L * ’ ( p )@r LP’(V))‘ with

The measures can be taken to be Borel-Radon measures on locally compact spaces and

372


PROOF:It remains to show that (b) implies (a): since ($# the 2p-local technique easily implies that

A ($)

@r $#)I

$#)I,

holds isometrically, which gives the result. Simple factorization arguments show the

COROLLARY 1: For l/p+ l/q 2 1 the maximal nomaed operator ideal associated with the tensor norm

Ppl9as isometrically

equal to the ideal

(XP0 f q r e 9

*

Again: (EPO C ~ ' )is~normed '~ - a fact which is not obvious. If % and '13 are accessible operator ideals, straightforward factorization and the strong principle of local reflexivity give that (% o %)reg is accessible as well. The ideals 2* wd are accessible, hence:

-

COROLLARY 2: The tensor n o m s pp,9are accessible, the tensor n o m s , i 3 ~ are , ~ even iot a1ly accessible.

The latter follows from the fact that 2 2 o Sq,= ( 2 2 o 1 3 9 ~ ) re9is an injective operator ideal, hence &,9 is accessible and right-injective, hence totally accessible by 21.1.(2). are maximal Note that .CPo22 = ( 2 , 0 C ~ ) ~ (by ~ g Ex 28.5.) and C202,t = normed operator ideals (1 5 p I 2).

28.7. This explicit description of the operator ideal associated with information about pp,9for special ( p , q ) :

pp,9gives further

PROOF: The description of \% in proposition 20.12. shows that

-

-

pp,l (2,0 C,)re9 = (\.Cp)'e9 = \2, \wp .

-

This gives (1) - and (3) follows from this using 20.14.and Grothendieck's inequality (tensor form 20.17.). (CPo C 'J32 g2 for 1 5 p 5 2 is corollary 1 in 23.10. (again Grothendieck's inequality), whereas 9 2

c .cz 0 c,

c (SP 2 o o y 0

373


for 1 < p < 00 follows from the Grothendieck-Pietsch factorization theorem and the inclusion 2 2 c 2,. Statement (4) is a consequence of (2,o 2q’)reg C 22 for p 5 2 5 q’ (see 26.1.) and 22 c 2*o 2,’for 1 < p, q’ < 00. The last claim is obvious by corollary 1. 0 28.8. The adjoint operator ideal (2,o 2,,1)*= [(2,o 2q’)reg]* description in terms of inequalities:

-

p;,, has also a nice

COROLLARY 4: For l/p + l/q 2 1 and T E 2 ( E , F ) the following statements are equivalent: (a) T E (2,0 C , ’ ) + ( E F). , (b) For all A E 2(e,t,tq) the operator

A @ T : e,i

E

@lE--*

eq@ x F

is continuous.

(c) There is a constant c 2 0 such that for all n X n matrices and all Y;, ...,3/, E F’

I

(ak,t), at/ 2 1 ,

...,2, E E

n

a k , t ( T Z t , d ) l5 c ll(ak,t) : $’ - ~ I l w p ’ ( c t ) w q ‘ ( ? h )

*

k,t=l

In this case,

(L, o L,’)*(T) = sup IIA @ T ll4lI1

: @c

-

@xll

= inf c

.

1

PROOF:Since l, axF LI (e,! Bt F‘)’ and @en @ un;er,G) = Wr(Un;G) by 6.4. and 12.9.,it is clear that (b) and (c) are equivalent by the closed graph theorem and 2,-local techniques. The representation theorem for maximal operator ideals and

p’,sP = p*P 4

N

(2,0 &‘)*

1

show that T E (2,o Zqt)*(E,F ) + (E @pp,pF’)’ if and only if n

a k , t z f ‘8 k,k1

&)I 5

c Il(ak,t) :

-* ~

flwp’(zt)wq’(dk)

- and this is the equivalence of (a) and (c). 0 The relations in corollary 3 (and 17.12.)give:

(2, 0 2 , ) .

9y (2,0 Go)* = Y2 (210 Cm)*

(2,o

= 22

2,‘).= I D 2

for 1 I p s 00 for 1 < p s 2 for 1 < p , q I 2.

374


One of the main objectives of this section was to establish conditions under which for a given T E 2 ( E l F ) the operator

28.9.

A @ T : tp#@ E

+t, @ F

is continuous (for certain norms) for all A E 2(tpt,l,). The results of 28.4., 28.8., Ex 28.8. and Ex 28.11. can be summarized in the following table:

r tensor norm

I

operator ideal

I

A@T:tp#@E-tq@F

Recall that ($Yo1 o 3,'). = 2ej * W . It will be shown in 29.12. that for these four characterizations A E 2(tpl,A!,) can be replaced by A E 2(Lpj(p)lL , ( v ) ) - with fixed measures p and v such that Lp'(c() and L,(v) are infinite dimensional. For related characterizations of this type for ; : 2 and XJ"' see Ex 29.4. and Ex 29.5.. PJ

Exercises:

Ex 28.1. Let 1 I p < 00. For z1,...,zmly 1, ...,y, m

m

k=1

k=l

E E write (zk)
_ 1 such that Wpl

5 Wpl @G

5 AWpl .

- -C s,

R,

Hint: Definition of up# and id :$ (b) If a Banach space G satisfies E @G 29.5.). Hint: Consider

(5@c

Wpl

E

G with IlRnll llSnII 5 A. 5 AwPi, then it is a Tp-space (in the sense of

G') 8 s (G@c

$1)

By the result from 1421 mentioned in 29.5. and (a) is true if p = 1,2 or 00.

+$ E @G E

@c

5t

< w,'

*

@G

wpl,

the converse of

30. The Vector Valued Fourier Transform

394


The Fourier transform 5 : &(R) + L2(aZ) is a powerful tool in Analysis so it is worthwhile to determine the conditions under which the vector valued Fourier transform is continuous: Unfortunately, the Fourier transform for E-valued functions is continuous only if E is a Hilbert space. This is the main result of this section - a result which can also be formulated in terms of vector valued Fourier series (i.e. in terms of the Fourier transform L2[-7r, r] -+ & ( Z ) )For . the proof one first uses Gaussian averaging to get Kwapied's important type/cotype factorization theorem C2 o 2 2 C 2 2 : The composition of a type 2 with a cotype 2 operator factors through a Hilbert space. All the main results of this section are due to Kwapieri (1611.

30.1. The Fourier transform

5 :L2(R)

-+

L z ( R ) , defined by

[5(f)l(t) := (2x)-1/2 J

m

f(s) exp(-ist)ds

i f f E L 2 ( R ) vanishes outside of a compact set, is an isometry onto L 2 ( R ) . Its inverse is, up to a sign, of the same form:

where

3 is the isometry of &(R)defined by

The maximal normed operator ideal ( 2 3 , L3) of all operators T E 2 ( E , F) (between complet Banach spaces) such that

(see 17.4.) is injective and surjective: this is a direct consequence of proposition 7.4.. The operators in 2 3 are sometimes called Fourier operators. REMARK: (23,Lg)

= (28-1,Lg-1)

.

PROOF: This follows immediately from the fact that for f

E L 2 ( R ) C9 G


for all Banach spaces G and formula for 5 8 T . 0

395

5-' 8 T = (8 8 T)o (3 8 i d E ) - and the analoguous

Since (8fIg)La

= (fIS-lg)La

for f , g E L2(R), it follows that 5' = 8 : L 2 ( R ) + L 2 ( R ) (watch the complex conjugate in the scalar product!). It is easy to see (use 15.10.) that gdual

3

= 23'

holds isometrically and hence:

COROLLARY: 2 3 = Z$'O'

holds isometrically, i.e. the ideal 23 of Fourier operators is

completely symmetric.

It was shown in 7.5. that the identity map on el is not a Fourier operator - and in Ex 7.8. that L 3 ( i d ~= ) 1for Hilbert spaces H. The ideal property implies for the ideal 2 2 of operators factoring through a Hilbert space that 2, c 2 3 and L3(T) 5 L2(T) for all T E 2 ( E , F ) .

30.2. Obviously,

holds for all h E &(R) 8 E . The following simple result is basic:

LEMMA:For a > 0,

a complex Banach space

E and x - ~ x-n+l, ,

..., xn E E

k=-n

PROOF: The first formula is obvious. For the second one, observe first that a a 2exp(-i-)sin(2) = i[exp(-ia) - 11 2

define

80. T h e Vector Valued Fourier Transform

396

and hence, for all k E Z and t E R,

nr+:

exp(-ist)ds = -i [exp(-i(k

+ 1)at) - exp(-ikat)]

=

t

a

2

at

at

= - exp(-ikat) [exp(-iat) - 11 = - exp(-ikat) exp(-i-) sin( T t t 2

With the substitution t

).

= 21ru/a this gives

But for all u E RZ \Z

c

mEZ

sin2(nu) a2(u+m)2 =

(consider Parseval's identity for the Fourier coefficients of the function exp(iu.)) and therefore.

Another obvious subsitution gives the result. 0 30.3. For T := [-r,r[ and f E L ~ ( T ) ~ A = , EL 2 ( T , E ) C L l ( T , E ) the Fourier coefficientsare defined by

f(n) := (2n)-'/2

/1

f(s)exp(-ins)ds E E

J-1

for n E Z; here E is again a complex Banach space. Recall that ((2n)-'I2 exp(in*))nEz is an orthonormal basis in L2(T) and that for f E L2(T) Parseval's identity


397

holds. It is now quite interesting to see that T being a Fourier operator means that T satisfies certain Parseval/Beasel-like inequalities.

THEOREM: Lei E and F be complez Banach spaces, T E 2 ( E , F ) and c 2 0. Then ihe following jive siaiements are equivaleni: (a) T E C s ( E ,F ) and L3(T) 5 c . (b) For all n E N and x-n, ...,Xn E E

(b') For all f E &(T, E )

(c) For all n E N and x-n,

...,t n E E

(c') For all f E &(T, E )

PROOF: T is a Fourier operator if and only if A2(3 8 T ( h ) )5 c A z ( h ) for all h E &(R)8 E. Since the functions h of the form treated in lemma 30.2. are dense in L z ( R ) 8~~E, the equivalence of (a) and (c) is obvious. Moreover, for h as in the lemma

( 2 IITzkll2)'lz = Az((3-l 8 T) (8 8 i d E ) ( h ) ;~52(R), F) 0

h=-n

1

holds. Since 2 3 = 28-1,this shows that (a) implies (b) - and the converse is also ) dense in L 2 ( R ) 8~~E (recall true because the functions of the form ($8i d ~ ) ( hare that 5 is an onto isometry). The Fourier coefficients of the function n

(2~)-'/~exp(ik.)xtE L2(T, E)

f(.) = k=-n

398

SO. The Vector Valued Fourier livlnsform

are f(k) = xk for lkl 5 n and zero otherwise; moreover, these vector valued exponential polynomials are dense in Lp(T)&E and hence in L2(T)&3a2E : these two observations prove the equivalences (b) c, (b’) and (c) c+ (c‘). 0 30.4. For T = idE the theorem implia that if in Parseval’s identity

one inequality holds for all f E L2(T,E), then there is equality - and this is equivalent to L s ( i d ~ )= 1. A neat trick due to Kandil [141] (see Ex 30.2.) shows that then E satisfies the parallelogram identity and hence is (isometrically) a Hilbert space:

COROLLARY 1: A complex Banach space E satisfies L s ( i d ~ )= 1 if and only if it i s isometric to a Hilberl space:

The isomorphic version of this result is more complicated and will follow in 30.6. from Kwapieli’s type/cotype theorem 30.5.. Before entering this topic, an example:

COROLLARY 2: For 1 5 p 5 00 and n E N Ls(idi;) = n11/p-1/21 .

PROOF: Since By 26.2.

.C3

is completely symmetric, it is enough to check this for 1 5 p 5 2.

La(idp)

5 L z ( i d 4 = n1/p-1/2

holds. For the converse note first that

In particular, 1, 4 space(&) for 1 5 p 5 general result is true.

00

and p

#

2 - but as announced, a more

30.5. A rather general theorem about factorization of operators through Hilbert spaces will be proved next. It is based on Gauss averaging. The following lemma shows that sometimes Rademacher averages can be replaced by Gauss averages.


399

LEMMA:Let E , F be Banach spaces (real or complex), T E %(El F ) and define dR := 1

and d c := &. (1) If T i s of type 2 , then for a l l z l , ...,zn E E

(2) If T is of cotype 2, then for all 2 1 , ...,z, E E

This result can be expressed as follows: Rademacher type 2 (resp. cotype 2) implies Gauss type 2 (resp. cotype 2). The additional constant 4 in the complex case is natural since the complex Gauss functions Q k have norm

a.

PROOF:The proof is - as in Ex 8.9. - a simple application of F’ubini’s theorem and the invariance of the Gauss measure ‘yn under isometries: Assume T to be of type 2.

for all w E K”. Integration with respect to 7n gives

where

Ilgkll2,K

= d~ by 8.7.. Statement (2) follows in the same way. 0

For T = idE Gauss cotype 2 implies Rademacher cotype 2, but for arbitrary operators this is false (see Pisier [225], p.36/37). Ex 8.9. implies that Gauss type 2 operators are of Rademacher type 2. For Gauss type/cotype p see Ex 30.3.. Note that with Gn := span{gk I 1 5 k 5 n} C Lz(m) and

Gauss type 2 means that

400

90.

The Vector Valued Fourier Tmnsform

and Gauss cotype 2

where, as in section 7, the norm A2 on Gn 8 E is the one induced from L2(yn) Q D A ~ E. The following type/cotype theorem is quite important.

THEOREM: The composition S o R of a type 2 operator R and a cotype 2 operator S factors through a Hilbert space, i.e. CZ o 3 2 c $2. Moreover,

In particular, a Banach space i s isomorphic to a Hilbert space if and only if it has type 2 and cotype 2 .

This characterization of Hilbert spaces is due to Kwapieri [161] - and it was observed by Maurey (see Kwapieli [162]) that the result also holds for operators.

PROOF:Let T = S o R E E(E, F) with R E %2(E,X) and S E C, ( X, F). Using the characterization 28.5. of 2-factorable operators, it suffices to check that for all n E N and all isometries A E .C(q,G)

Fix such an operator A and let A :L2(Kn,7") 4 L2(Kn,7") be the map fIt is obvious that A(Gn) c Gn and

f o A'.

A = I - ~ o A ~ , , ~ I : ~ - ~ . Moreover, A QD idx is an isometry on L2('yn) Q D A ~ X (by the invariance of 'yn under isometries). It follows that the diagram

commutes - and the remarks made before the theorem show that

which ends the proof. 0


401

The natural quotient map Q : 11 + 1, (for 2 < p < 00) does not factor through a Hilbert space. Since 11 has cotype 2 and 1, type 2, it follows that in general neither the operators in 2 2 o C2 nor those in 2 2 n C2 factor through a Hilbert space. Note also that Kwapieli’s type/cotype theorem improves one of the results in 26.1. stating that every T : 1, -+ 1, factors through 1 2 if 1 5 s 5 2 5 r < 00. On the other hand, Pisier’s factorization theorem (to be proven in the next section) extends Kwapied’s result in certain situations since E’ has cotype 2 whenever E has type 2 (by Ex 8.14.).

30.6. Fourier operators are of type 2 and cotype 2 - this is the content of the next

PROPOSITION: max{Ts(T), Cz(T)} L 2 Lg(T) for all T E C ( E ,F). PROOF:In order to show that T E C3(E, F) has type 2 fix 30.3.(c) shows that for all w E 0,

z1,

...,t n E E. Theorem

Integrating with respect to p n gives

Kahane’s contraction principle (see Ex 8.16.) implies that

so that Tz(T) 5 2 Lz(T). The proof that the operator T has cotype 2 follows along the same lines. The constant 2 is actually superfluous: at the end of this section a more general result (with constant 1) will be proven - but we preferred for the sake of simplicity and elegance also to give the proof above. Now the main result of this section follows easily:

THEOREM (Kwapieri): A complex Banach space E is isomorphic t o and only if the E-valued Fourier transform

is continuous.

a

Hilbert space if

402

SO. The Vector Valued Fourier Transform

PROOF: If this is so, i d E E 23 and hence E has type 2 and cotype 2 by the previous propmition. Therefore, idE = i d E o i d E is 2-factorable by Kwapied's type/cotype theorem, which concludes the proof. 0 Recall that the isometric version of this result was already given in 30.4.: If L 3 ( i d ~ )= 1, then E is isometric to a Hilbert space. It is not known whether the operator version of this result holds: Does every Fourier operator factor through a Hilbert space? In other words, is 2 3 c CZ?A partial answer is the

REMARK: 2 2 = 23 o 23

.

PROOF: Clearly 2 2 C 2 2 o 2 2 C 23 o 2 3 and conversely 2 3 o 2 3 Kwapieri's type/cotype theorem. 0

c &2 o 2 2 c 2 2

by

30.7. It is worthwhile to have a look at the structure of the proof of the results above: If one says that an operator T E C(E,F ) has Fourier type 2 if it satisfies for some c 2 0 and all t k E E

and - only for the moment (since this notion is not the usual one) - Fourier coiype

then the following has been shown in this section: (a) An operator is a Fourier operator if and only if it is of Fourier type 2 and if and only if it is of Fourier cotype 2 (theorem 30.3.). (b) Fourier type 2 implies Rademacher type 2, the same for cotype (proof of proposition

30.6.). Rademacher type 2 implies Gauss type 2, the same for cotype (lemma 30.5.). (d) The composition S o R of a Gauss type 2 operator R and a Gauss cotype 2 operator S factors through a Hilbert space (proof of Kwapied's type/cotype theorem). (c)

30.8. In light of this structure the following type/cotype result with respect to an arbitrary orthonormal basis of an L2(p) is of some interest (the result is also due to Kwapieri [lSl]). For this, take an orthonormal basis ( f n ) n E ~of some separable La(R, p ) and consider the isometry

I : L2(p)

-

42

30. The Vector Valued Fourier Tran~form

403

defined by g-((glfn))nEN. For T E $(El F) continuity of I @ T : L ~ @ A -+~ E &@A~F means a sort of cotype 2 inequality:

- and the continuity of I-'

(g, T means a type 2 inequality (with respect to (fn)). It is obvious that every isometry I from L2(Q, p ) onto l 2 fixes (via the unit vector basis of 4 2 ) an orthonormal basis of L 2 ( n , p ) .

PROPOSITION: Let ( Q ,p ) be a non-atomic, separable measure space, I : L2(R, p ) -, 42 a surjective isometry and T E 2 ( E ,F ) . (1) If I @ T : Lz(Sl,p) ( g , E ~ -+ ~ C2 € 3 F~ i s ~continuous, then T is of (Rademacher) cotype 2 and C2(T) 5 111 (g, T : . .II. ( 2 ) If T : e 2 @aa E -+ L2(R, p ) € 3 F ~i s ~continuous, then T i s of (Rademacher) fype 2 and Tz(T)5 11I-l €3 T : . .)I.

PROOF:Clearly, one may assume that p is normalized. Then, by the Halmos-von Neumann theorem (see Brown-Pearcy [MI, p.181) the measure spaces ( Q , p ) and ([0, l ] , A ) (Lebesgue measure) are isomorphic, i.e. there is a measure preserving, monotone, additive and bijective map Qi from the classes of p-measurable sets onto the classes of A-measurable sets. Clearly, Qi induces a bijective isometry

between the classes of E-valued step-functions, hence a surjective isometry L z ( p , E) + L2(A, E ) . This means one may restrict one's attention to the Lebesgue measure on [0, 11. Thus, in case (1) there is an orthonormal basis (fn) of Lz[O,11 such that

for all 2 1 , ...,en E E. One has to show that the f k can be replaced by the Rademacher functions r k ; this will be done with a gliding hump approximation of the functions r k by certain fm. Fix x1 , ...,Zn E E. Since for the discrete Rademacher functions

holds for every strictly monotone 9 : N -+ N,the same is true for the sequence ( r k ) and 80 it is enough to find a subsequence of the (rk).

90. T h e Vector Valued Fourier Transform

404

Take c

> 0 and $( 1) E 4v such that for do) 9

:= E(rlI.fk).fk k=l

the function hl := Ilgll;'g there are

satisfies

11.1

-

5

2 - l ~ .Assume,

p(1) := 1 < ~ ( 2 < ) ... < P(n) and $(O) := 0 < $(1)

hk E S p a

{fm

1 $'(h - 1) < m Id'(k)I

such that llr9(k)- h k l l 2 5 2 - k ~: Since (rjlf,,,) p(n 1) > p(n) such that

+

lIhkIl2 4

< ... < $ ( n )

=1

0 for j

4 00,

+(n)

E

l(f$Y(n+1)1fk)I2

is small ;

k=l

+ 1) > +(n) and some normalized hn+l E span {fm I $(a) < m 5 4(n + 1))

this easily leads to some $(n

such that [lr9(n+~l - hn+l[12 5 2-("+')~. The construction showed that h n l h , whenever n

# m.

by induction, that

Now

- for all c > 0. The second claim follows in the same way. 0

there is an index

405


The result does not hold for arbitrary orthogonal sequences: see Ex 30.7.. One application: If G is an infinite abelian compact group with normalized Haar measure, then the dual group G is a discrete topological group and the Fourier transform 5 : Lz(G) -+ &(G) is an isometry. If Lz(G) is separable, then C is countable.

COROLLARY: Let G be a compact, abelian infinite group such that Lz(G) is separable. If for a Banach space E the E-valued Fourier transform

5 8 idE : Lz(G)

E

6 9 ~ 2

as continuous, then

-

P Z ( ~8)A a E

E is isomorphic to a Hilbert space.

PROOF:Since G has no discrete points and each atom of a Borel-Radon measure contains a one-point atom, the Haar measure has no atoms in this case. Therefore the proposition shows that idE is of (Rademacher) type 2 and cotype 2, hence the claim follows from Kwapieli’s characterization 30.5. of Hilbert spaces. El The assumption that Lz(G) is separable is actually superfluous: this was shown by Rubio de Francia and Torrea [234].

30.9. Peetre defined a Banach space E to be of Fourier type p (for 1 Fourier transform satisfies the Hausdorff-Young inequality

p

5 2) if the

for all f E L , ( R ) 8 E. It is obvious that for p = 2 this means i d E E 28, which is the same as having Fourier type 2 in the sense of 30.7.;it follows that these are the spaces isomorphic to a Hilbert space. A deep result of Bourgain statys that each Banach space of type p > 1 has Fourier type r for r with r‘ = 18Tp(id~)P. See Konig [149] for references and the behaviour of Fourier coefficients of certain differentiable E-valued functions.

Exercises:

Ex 30.1. Recall that a norm (on a real or complex vector space E) comes from a scalar product if and only if for all t,y E E it satisfies the parallelogram identity IIt

+ Y1I2 + 1.

- Y1I2 = 2(11412 + llYll2)

-

(&)If II~+y11~+llz-y11~ for a l l z , y E E), then the norm 11 11 satisfies the parallelogram identity.

406


(b) If for a Banach space E the type 2 constant T z ( i d E ) or the cotype 2 constant C z ( i d ~ is ) 1, then E is isometric to a Hilbert space. Ex 30.2. Let E be a complex Banach space. (a) For z , y E E define f ( t ) := z for - A 5 t < 0 and f ( t ) = y for 0 5 t < A. Show that f E &(T, E) and

c Ili(.)l12(1. =

+ Y1I2 + 112 - Y1I2)A/2

*

nEZ

(b) Deduce from (a) and Ex 30.1.: If

for all f E L2(T,E) (or 2 for all such f), then E is isometric to a Hilbert space. This argument is due to Kandil [141]. Ex 30.3. Define Gauss type p and Gauss cotype q for operators. Use Kahane's inequality for Gauss averaging (8.7.) to show that for T E 2 ( E , F) and 1 5 p 5 2 5 q<m:

(a) T is of Gauss type p if and only if it is of Rademacher type p. (b) If T is of Rademacher cotype q, then it is of Gauss cotype q. Hint: 30.5. and Ex 8.9.. Ex 30.4. Use Kwapieri's type/cotype theorem to reprove the result from 26.1. that for 1 5 q 5 2 5 p < 00 every operator from an X;-space into an Xi-space factors through a Hilbert space. Ex 30.5. If 1 5 q 5 p 5 00, then every Fourier operator tp -+ t, factors through a Hilbert space. Hint: First q 2 and 30.5.. Ex 30.6. Show that for 1 5 q 5 2 5 p 5 00,every operator T E X ( t , , t , ) is a Fourier operator. Hint: 26.1.. Ex 30.7. If fn := 2 - " / 2 ~ p - 1 , 2 - n + 1 ~ , then (f,) is an orthonormal system in L2([0,11) and

holds for all Banach spaces E and 21, ...x, 6 E. This example (showing that proposition 30.8. does not hold for arbitrary orthonormal systems) is also due to Kwapieli.

Ex

30.8. Let

E be a complex Banach space.

(a) Show that 11f(n)11B 5 11f112 for all f E Lz(T,E) and n E N . (b) The E-valued exponential polynomials are norm-dense in L2(T, E). Hint: Use C(T, E) = C(T)&E. (c) Riemann-Lebesgue lemma: If f E L z ( T , E ) , then the sequences ( f ( n ) ) n E N and (f(-n))nEN converge (in E ) to zero.

31. Pisier’s Factorization Theorem

407

Ex 30.9. Show that for each Banach space the following statements are equivalent: (a) E is isomorphic to a Hilbert space. (b) If C= :o 11Zn112 < 00,then C2=oexp(in.)Zn E L2(T, E). (c) The sequence of Fourier coefficients of every f E &!(TIE)is in & ( E ) . Ex 30.10. If H is a Hilbert space and f E L2(T, H ) , then the Fourier series

converges in L 2 ( T , H ) to f.

3 1. Pisier’s Factorization Theorem

Kwapieli’s type/cotype theorem states that every operator T from a space of type 2 into a space of cotype 2 factors through a Hilbert space. Pisier’s factorization theorem - in some sense an “abstract” version of Grothendieck’s theorem - extends this to compactly approximable operators T E C(E,F) if E’ and F have cotype 2; note that E having type 2 is more than E’ having cotype 2. The proof of the theorem is taken from Pisier’s book [225]. Moreover, a non-accessible tensor norm, also due to Pisier, will be presented.

31.1. Let R : &(D,p ) + L2(D,p ) be the orthogonal projection onto the closed space Rad of the discrete Rademacher functions (see 8.5. and Ex 9.16.). As an orthogonal projection R is self-adjoint in the Hilbert space L2(D, p ) . Identifying L2(D, p ) with its dual through the usual duality bracket J fgdp, it can easily be seen that

- -

R’f= R‘f= Rf . Since the Rademacher functions en are real-valued, it is obvious that RT = R f and hence R’ = R. An operator T E C ( E , F) is called K-convez if

K(T) := llR @ T : L z ( D ,P ) @Aa E

-*

L2(Di P ) @Aa FII < 00 .

It follows (see 17.4. and 7.4.) that the class f f C of K-convex operators together with K forms a maximal, injective and surjective Banach operator ideal. Since

408

91. Pisier's Factorization Theorem

commutes, ffC C ffCd""' holds and hence, ffC = ffCd""' isometrically (17.8.): The operator ideal (ffC, K)of K-convex operators is completely symmetric. It is easy to see (using A:) that the identity operator on a Hilbert space and hence all 2-factorable are K-convex; moreover, K 5 L2. Recall from Ex 9.16. that idl, is not K-convex (proposition 7.8. stated that if there is a non-K-convex operator, then idc, is not K-convex) and Pisier's result that E is not K-convex (if and) only if E contains the 6 uniformly. By another result of Pisier's (mentioned in Ex 7.13.) it follows that E is K-convex if and only if it has proper type.

31.2. The dual of a type p operator has cotype p' (see Ex 8.14.) - but the converse is false: co does not have type 2 (Ex 7.14.), but el has cotype 2 (see 8.6.). It is clear from the definitions of type p and cotype q operators that this gap in the duality relation may be filled by K-convexity; more precisely,

PROPOSITION: For 1 < p 5 2 the following two relations hold: (1) RCo CTa'C 2 , and T , 5 K o C$"' . (2)

%, o ffC c %Par and TP"'5 C,) o K .

PROOF: Since ffCduo'= ffC, it is clear that L2

L~

R

* Rad J

(2) implies (1). For (2) use the operators JP'

-

+ l,# IP

Rad

e,

9

which satisfy (J,' o A)' = J o 1,. If S E ffC(E, F) and T E Cp#(F, G), the definitions show that the operator

has norm

5 C,#(T)K(S). The diagram

yields that T,((To S)') satisfies the same norm estimate - which is the claim. 0

409


In particular, if E is K-convex (i.e. - by the result of Pisier - does not contain the uniformly), then E has type p if (and only if) E' has cotype p'. 31.3. The following two lemmas prepare the way for the proof of Pisier's factorization theorem. Recall from Ex 8.15. that the Walsh functions E A (where A runs through the finite subsets of N)form an orthonormal basis of &(D,p ) .

LEMMA1: For every 6E]O, 11 there is a signed Bore1 measure D = { - l , l } N such that

I k & ~ d v 651 6 if A C N and IAl

(3)

V6

on the compact group

>1.

PROOF: For 0 E [-1,1] and n E N define n

It follows that

If"eld/l = fi(1-k k=l

and, for A

(kdpl)

=1

Di

c { 1, ...,n},

For each n E N put

1 Fa := 2 4

m).

Then the sequence F f d p in C(D)' is norm bounded (by Since C(D)is separable, it has a weak-*-convergent subsequence with limit V6 : It is obvious that this measure satisfies all the requirements.

K 5 Lz - but this estimate can be improved in a certain sense: This will be a crucial point in the proof of Pisier's factorization theorem. LEMMA2: Take T E &(E, F ) and 6 E]O,l]. Then

410


PROOF: Fix

n f = x & A k @Xk k=1

ELz(DiP)@E

with distinct A1 ,...,An. The idea of the proof is to construct a decomposition

these estimates imply that

as desired.

Let

v6

be the measure constructed in lemma 1 and define n

.

since SD&,,,dv6 = 1. Note that the Rademacher functions E,,, (and hence all Walsh functions) are multiplicative on the group D - and p is actually the normalized Haar measure on D,i.e. invariant under multiplication. It follows that


411

m,

since llvall I and hence A 2 ( i d €3 T(f i ) ) 5 llTllJ17'8a~( f). The operator T is 2-factorable - so Ex 28.2. can be used to obtain the estimate for f 2 : Indeed, take 2' E E'; then

by the biorthonorrnality of the Walsh functions and lemma 1. It follows from Ex 28.2. that A 2 ( i d € 3 T ( f 2 ) L21 ; F ) I 6 L 2 ( T ) A 2 ( f L2, ; E) 1 which is the remaining estimate. 0

31.4. Now the main result of this section can be proven. It was obtained by Pisier [222] in 1980. PISIER'S FACTORIZATION THEOREM: Let E and F be Banach spaces such that E' and F have cotype 2. Then every operator T E & ( E l F ) (in particular, every finite rank operator) satisfies L2(T) I (2c2(E')c2(F))3/211Tll. In terms of tensor norms: E

5 ~2 5 ( 2 C 2 ( E ) C 2 ( F ) ) 3 / 2 ~

onE@F

whenever E and F have cotype 2.

PROOF:Since w2 and E are totally accessible, C z ( E ) = Cz(E") and 22 and w2 are associated, it is clear that the claim about the tensor norms follows from the one about the operators. Given T E & ( E l F ) Kwapiefi's type/cotype theorem 30.5. yields L2(T) I T 2 ( T ) C z ( F ).

Proposition 31.2. and lemma 2 imply that

+

1

T2(T)L CZ(E')K(T) I C 2 ( W [6Lz(T) zllTll] for every 6 E ] O , l ] ,and hence

412

91. Pisier’a Factorization Theorem

Choosing 6 := (2Cz(E‘)Cz(F))-’,gives the conclusion. 0

31.5. An operator T E C ( E ,F ) is called compactly approzimable whenever there is a net (T,) of finite rank operators converging to T uniformly on all compact subsets of E (see Ex 5.17. for a characterization of these operators). It is called A-compactly approzimable if the T, can be chosen of norm 5 X(lTI1.

COROLLARY 1: Let E’ and F have cotype 2 . ( 1 ) Then every A-compactly approzimable operator T E 2 ( E ,F ) i s 2-factorable and

( 2 ) In particular, if one of the spaces E or F has the bounded approzimation properly, then &(ElF ) = & ( E , F). Pisier [225] actually proved that this result holds for all compactly approximable operatom - and therefore ( 2 ) is also valid if E or F only has the approximation property.

PROOF: The fact that the unit ball of &(E, F) is closed in 2 ( E ,F) with respect to the weak operator topology (see 17.21.) together with the theorem shows immediately part (1) of the claim. 0 Using the fact that 2 ( E , E ) = C z ( E , E ) implies that E is isomorphic to a Hilbert space, gives the

COROLLARY 2: A Banach space E with the (bounded) approzimation property such that E and E’ have cotype 2 , is isomorphic t o a Hilbert space. As has aIready been mentioned at various times (see 4.8.) Pisier has constructed infinite dimensional Banach spaces P such that P P = P @c P; his construction even gave that P and P’ have cotype 2 . Obviously, such a space cannot be isomorphic to a Hilbert space and hence i d p shows that in corollary 1 (and 2 ) an approximability hypothesis is indispensable. 31.6. Pisier’s space P and his factorization theorem serve also to find a non-accessible tensor norm, or equivalently, a non-accessible maximal normed operator ideal. The following construction was given to us by Pisier. For M, N 6 FIN define

it is easy to see that this defines a normed operator ideal on FIN. Denote by ( % p , Ap) its maximal hull and by a p the associated (finitely generated) tensor norm.


THEOREM (Pisier):

( a p ,A p )

and

cup

413

are neither lef2- nor right-accessible.

PROOF:By 21.3. it is enough to treat the operator ideal. Assume that Q p is rightaccessible; it will be shown that this implies that P is isomorphic to a Hilbert space. By the maximality of the ideal of 2-factorable operators it is enough to check that La(15 : M c,P) is uniformly bounded for M E FIN(P). Obviously, Ap(Z&) = 1 and 80 accessibility gives M&P

It is straightforward from the defining inequality of cotype 2 that with P' also &(P') = (&(P))' has cotype 2. Since 1; o S has finite rank, Pisier's factorization theorem gives 0 S) I ~ ( ~ C ~ ( ~ ~ ( P ' ) ) C Z . (P))~'~ Lz(15) 5 JJRJJL2(Ig

: P -+

To see that % p is not left-accessible one considers in the same way Lz(Q! for L E COFIN(P).

P/L)

Qp

The injective hull of 2lp (which is associated with the tensor norm ap\) is rightaccessible (by 21.1,); however, the same argument as before, the metric approximation property of .&, and the fact that the ideal 2 2 of 2-factorable operators is injective imply the

COROLLARY: (a?, A?) and

cup\

are right- but not left-accessible.

These were non-accessible maximal ideals; see Ex 31.4. for non-accessible ideals consisting of approximable operators.

31.7. Since tk, and 4, are C:-spaces and hence have cotype 2 (see 8.6.), Pisier's factorization theorem (more precisely, its corollary 1) implies that

(*I

2(&J14) = .c2(&0,4)*

This is nothing other than Grothendieck's inequality, which can be seen as follows: The little Grothendieck theorem ( 2(!, ,.&) = !?32(.f,, &)) and Kwapieri's factorization theorem give

qt, ,4 ) = .c2(&0,e l ) = Y32du41 PZ(k3,e l ) = 9 2 ( & , 4 ) Since w i = wz

-

0

'

the representation theorem for maximal operator ideals yields is r 5 ew2 on t,@c, for some c :This is Grothendieck's inequality in tensor form. See Ex 31.3. for the constant c. 92,

( . f o o @ l r ~ o ) '= ( t , ~ w 2 c o ) ' which

414


Thus, Pisier’s factorization theorem provides a new proof of Grothendieck’s inequality - and this justifies labelling this factorization theorem as an “abstract” form of Grothendieck’s inequality (in the form (*)). Note that toodoes not have type 2 - and hence Kwapiexi’s typelcotype theorem does not give (*). The use of the little Grothendieck theorem also implies part (1) of

COROLLARY 3 (Maurey): Let E have coiype 2. Then (1) .C(lm E ) = vz(tw E ) (2) !J3z(E,F) = v l ( E ,F ) for each Banach space F. 1

1

The second part follows from the first through Saphar’s result 20.19.. The content of the corollary was obtained in 1974 by Maurey [186]. Other proofs of Maurey’s result will be given in 32.9. and Ex 32.7.. 31.8. The last result and proposition 26.4. read:

-

!&,(ElF ) = PI(E,F ) i f p E [I,21 and E has cotype 2 P q ( E ,F ) = !&(El F ) if q E [2, co[and F has cotype 2

.

Since ipp gi, is totally accessible, the embedding lemma implies the following result concerning the equivalence of tensor norms:

COROLLARY 4: Lei p €]1,2] and q E [2, m] and El F Banach spaces. Then an E @ F g& gi i f E’ has cotype 2 dw d, i f E has cotype 2 g: g; i f F has cotype 2 d2 dp i f F’ has cotype 2 .

---

Exercises:

If E’ and F have cotype 2, then Z ( E ,F) C & ( E , F). Ex 31.2. (a) If E and F have cotype 2 and p , q €11, oo[satisfy l / p + l / q 2 1, then E F = E BeF holds isomorphically. (b) For the dual result let E‘ and F‘ have cotype 2: Then f and w; are equivalent on E @ F, but x and wz are in general not; a sufficient condition for x and w; being equivalent on E @ F is that E or F has the (bounded) approximation property. If E and F‘ have cotype 2, then it follows in the same way that x and wz are, in general, not equivalent on E’@ F, but Pisier ([225], p.47) showed that the nuclear norm N and Ex 31.1.

the 2-dominated norm Dz (associated with w:) are equivalent on %(El F).

32. Mizing Operatore

415

Ex 31.3. Show that the “abstract” proof 31.7. of Grothendieck’s inequality gives the estimate KG 5 8 K i G for the Grothendieck constant. Hint: 8.6.. Ex 31.4. (a) Show for the factorization ideal 2lp from 31.6. that %p o 3 is left- but not right-accessible. Hint: Ex 25.3. and proof of 31.6.. Recall that the minimal hull 2 p l = 5 0Qp o 3 is accessible by 25.3.. (b) Define the quasi-Banach operator ideal (Cp, C p ) by Cp(E, F) := {T E E(E, F) IT = Ti CP(T) := inf{(Ap

0

+ T2, Ti E ap 05,Tz E 5 0ap}

II ll)(Tl) + (I1 II 0 AP)(T2)1 -

Show with Ex 21.2. (b) that (Cp, C p ) is an ideal of approximable operators which is neither right- nor left-accessible.

32. Mixing Operators

The following situation has appeared already at various times: An operator T (or idE of a Banach space) has the property that S o T is absolutely p-summing whenever S is absolutely q-summing. This section is devoted to a systematic study of these

“mixing” operators/spaces. Mixing operators are characterized by interesting integral inequalities and by a certain splitting property (which explains the name “mixing”); moreover, they are almost equal to absolutely (r,s)-summing operators for appropriate r and 8 . Many of the results to be presented are at least implicitly due to Maurey [186]; the ideal of mixing operators was studied systematically by Pietsch in his monograph on operator ideals and by Puhl [226].

32.1. Let p,q E [ l , ~ An ] . operator T E E(E, F) is said to be (q,p)-mizing if for all Banach spaces G and S E V q ( F G) , the composition S o ‘ I is‘ absolutely p-summing. With Mq,p(T) := sup{Pp(SoT) Pq(S) I 11 it is clear that, in the terminology of quotient ideals introduced in section 25, the ideal 9JIq+, of ( q , p)-mixing operators is isometrically g;’ o gp:

I

(9JIq,p,Mq,p)= (V;l

O

Vp,

pi’ O PP) .

The quotient formula 25.7. implies that M,,,) is a maximal Banach operator ideal - and later on it will be easy to see that it is injective but not surjective (Ex 32.1.). The name “mixing”, due to Pietsch, will become clear in 32.7..

416

$2. Mixing Opemtors

Since VP is injective and 3y = ‘Q, it is also easy to see that mq,p

= 3;’

0

Vp

holds isometrically. The ideals Vp and DP are accessible and 31, = 3pmin is the ideal of pnuclear operators, hence the quotient formulas 25.7. give a result due to Puhl [226]:

PROPOSITION: The following relations hold isometrically:

mf,p:= ! p i l o

q.tp = 3;’

0

qp= p,t 0 3;’ = 3,, 0 371 = 31,)

0

3 ’1 ;

=

= pP’ 0 V,)*= pP’ 0 3J . Note that this result implies various composition formulas, as for example

32.2. Some observations - and translations of former results into the language of mixing operators deserve to be stated explicitly: (1) It is obvious that 9Jl,,p = C whenever q I p and M,,p = yp.So only the case 1 5 p < q < 00 gives something new. (2) Pietsch’s composition formula !Jlqo Pr C VPfrom 11.5. gives:

V r C M,,p and

M,,p 5 P,

if p , q , r E [1,00] with l / q

+ 1/r = l/p ;

this will be extended to absolutely (r, s)-summing operators at the end of this section. Since in Pietsch’s composition formula the index p is best possible (see Ex 11.23.), it follows that if 1 I P - 6 < P I q ; m,,p $t m,,p-a see also Ex 32.14.. (3) Saphar’s result 20.19. means that:

E E space(Mf,l)

if and only if C ( l , , E ) = ‘13q~(~wr E) ;

I

moreover, M , , l ( i d ~ = ) sup{P,t(T) 112’ : loo+ Ell 5 1). This also follows from the quotient formula % ,1 = I , 3,t o 3k1= !+oIf 2L1. ! F’urthermore, this formula also implies that 1F ) = V &J 1 F )

m,&a

holds isomorphically for all F. (4) The little Grothendieck theorem says that a(&,H) = V z ( l , , H) for every Hilbert space H , hence idH E space(9Jl2,l) by (3). Looking at the constants in Saphar’s result, one gets

M z , i ( i d ~5 ) KLG

417

$2. Mizing Operators

- even = KLG,see Ex 32.4.. Much more generally: E E space(9Jl2,l)

f o r all colype 2 spaces E

- this is just a reformulation of Maurey’s result in 31.7.. Conversely, every Banach lattice in space(m2,l) has cotype 2 (see Pisier [225], p.108) - but it seems to be unknown whether or not this holds for arbitrary Banach spaces. In 32.9. and Ex 32.7. alternative proofs of Maurey’s result are indicated. ( 5 ) For cotype q > 2 less is true. Theorem 24.7. gives:

E E space(%lq’--r,l)

if E i s of colype q > 2 .

Since L , is of cotype max{q, 21,it follows for the r-factorable operators that

2,.c mP,l

if 2 < r

< p’ < 00

and

2,

c M2,l

if 1

Ir 5 2 .

(All this was already stated in 26.4.(1)and (2)- including the constants involved.) (6) In particular, the sequence space el is in space(9Jl2,~).One can show, however, that el is not in space(mq,l)for q > 2;this will follow - as will some other counterexamples -from considerations about the ideal of absolutely ( r ,s)-summing operators: m,,~c !J?,I,l by 32.10. and t1 4 space(Vq,,l) by proposition 34.11.. Since C[O,l] c, L,(A) (Lebesgue measure) is in Vqbut not in VP for p < q (this was mentioned in 11.3.),

C(P, 11) 4 space(M*,p) and (by local techniques) also 1, !,I

if 1 I P

0. For all operators S E 2(L,+,(p1),Lp(v)) and

i s continuous.

PROOF: Take s = q . 0 32.4. There are two characterizations of mixing operators - one in terms of integral inequalities and one in terms of a splitting property. These descriptions of the quotient = pi1o ppof (q,p)-mixing operators make them important in many ideal 9Jtq,p situations. First the integral description, which has the flavour of the GrothendieckPietsch domination theorem:

PROPOSTION: Let 1 < p

1, hence the result does not hold for operators. Moreover only the case q < 2 is interesting by 32.10..

PROOF: Fix a probability measure p on B E ’ . By the previous proposition one has to find a probability measure Y such that for all z E E

- with a constant c independent of p. Let vo := p. By assumption and induction there are probability measures un such that

where co := Mq,p(idE). Define v := Then Holder’s inequality gives 00.

Since p = vo,this implies that

c,”==, 2 - ” ~ n and 0 ~ ] 0 , 1by [ l/p = Q+(l-O)/q. M

.

422

32. Mizing Operators

which gives M,,l(idE)

5 (2Mq,,(id~))'ie .

For an application to spaces which satisfy Grothendieck's theorem (the G.T. spaces from 20.18.) see Ex 32.9.. 32.6. Maurey's factorization theorem 18.9. gives a very useful characterization of operators in me', which - for spaces - is also due to Maurey [186], th6orhme 23.

<
max(2,p)

of!&,p (and hence the space ideal of ?Blq,p) consists of finite dimensional spaces only.

PROOF: If E E is infinite dimensional, it contains all uniformly by Dvoretzky's spherical sections theorem (see [Sl]). Since 'Qr,pis injective, this implies that Pr.p(i4;)

for some c independent of n. But

Ic


427

which is a contradiction.

32.11. The inclusion Mq,pc !$.?r,p (if l/q+l/r = l/p) is in general strict: For 1 < q < 2 it is known from 32.8. that L,, fZ space(Mq,l),but L,’ E space(?13qi,l)according to Ex 11.22. (or remark 24.7.). The inclusion %Iq,,c ?J3,p,1 is even strict for all 1 < q < 00; see 32.12.. Talagrand’s Banach lattice (mentioned in 8.9.) which has the Orlicz property (hence is in space(!&,l)) but does not have cotype 2, shows that space(M2,l) # space(v2,l) since a Banach lattice is of cotype 2 if and only if it is in space(M2,l) (see 32.2.(4)). Nevertheless, the ideals Mq,pand ?J3r,p almost coincide: This result is contained implicitly in Maurey’s thesis [186] and was stated and proved explicitly by Puhl [226] :

THEOREM: %,p C Onq-c,p zf l / q

+ 1/r = l/p

and

E

> 0.

PROOF:Recall first from 32.1. that

and from proposition 24.7. Pq‘J

0

2 , c Pq’+q

+

(since ! J J q t , 1 c ?J3r,,,, if 1 < s1 < q’ < r1 < q‘ q with appropriate r1,s1 - see Ex 11.21.). Take T E ‘J3r,P(E,F) and S E JPz(G,E). By the factorization theorem 18.7. for p’integral operators and the fact that U E % if and only if U” E 2l for maximal normed ideals one may assume that G = C ( K ) ,which is an XL-space. It follows that

and hence by Ex 32.2.(c)

For appropriately chosen q this gives the result. 0

32.12. The “border case” E = 0 in the last statement was investigated by Carl-D. [21] : If T E C ( E ,F) has rank n, then

-

where l / q + l / r = l/p. For p = 1 and 1 < q < 00 this “growth” is sharp in the following sense: There exist a constant cq > 0 and operators T, : P, P, with cq(l

+ log n)’/q 5 Mq,~(Tn) and Pq,,l(Tn)= 1 .

428


(For (q,p) = ( 2 , l ) these results were first proven by Jameson [120]). In particular, it follows that

MgJ c ! P g # , 1 #

for 1 < q < 00. For q < 2 this result was already obtained in 32.11. - but with totally different methods (32.8. and 26.3.).

Exercises: Ex 32.1. Show that the operator ideal Hint: Splitting theorem, L1. Ex 32.2. Show that

is injective but in general not surjective.

(4 Me,,,, c M,,,P,

if Pl I P2 and 92 I Q1 * (b) m4,r 0 M‘,P c M9,P if P, q, s E [I, 4. (c) !Pp,6 0 m6,r c fPg,r if r, s,p,q E [I, m] such that 1/s l / q = I/p l/r. Hint: Maurey’s splitting theorem for s > r and Ex 11.21, for s 5 r. In all three statements there are no additional constants in the norm estimate. Ex 32.3. If E is an 2&-pace, then fPg#(E, F) = Mg,l(E,F) for every Banach space F. Hint: 3, = 2,. Ex 32.4. Show that

+

+

if 1 I p I 2. Hint: 32.2.(3), 11.11. and 23.10.. Ex 32.5. Show for 1 I p 5 q I 00 :

Hint: Quotient formula, Ex 29.1., 32.1., 32.6. and 18.10.. Ex 32.6. Show that T E 2 ( E , F’) is (q,p)-mixing (for 1 there is a c 2 0 such that

In this case, Mg,P(T)= rninc.

I p I q < co) if and onIy if

429

92. Mizing Opemtors

Ex 32.7. Use 26.4.(3') and the Maurey-Pisier extrapolation theorem to prove Maurey's result from 31.7.: E E space(m2,l) if E has cotype 2. Note that this again shows that t1 E space(m2,l) which, together with the little Grothendieck theorem, implies c(t1, t2) = 'JIl(t1, t2). This is another proof of Grothendieck's inequality. Ex 32.8. Show that the following statements are equivalent for a Banach space E : (a) E E space(m2,i) . (b) For one (and then for all) q E 12, m] 32(Edq) = 3(E,tq) (c) For one (and then for all) p E [l,2[: Every compact operator E' Hint: 32.9. and representation theorems for ideals. Ex 32.9. (a) If E is a Banach space such that

-+

t, is 2-factorable.

for some 1 5 p < 2, then L(E,&) = 'JIl(E,t,), i.e.: E is a G.T.space (Grothendieck theorem space) in the sense of 20.18.. Hint: E E space(9l?z,,). (b) G.T. spaces are in space(m2,1), in particular, they have the Orlicz property. Ex 32.10. Let 1 5 p < q 5 00 and q > 2. If a Banach space E is such that for all T E Z ( E , t p ) the operator

T 8 i4,: E BA; eq

-

tp@A,, eq

is continuous, then it is finite dimensional. Hint: 32.6. and 32.10.. Ex 32.11. space(m,,,) = FIN if and only if q > max(2,p). Ex 32.12. Show that Mq,,(tl, F) = mq,1(t1,F) if p 5 2. Ex 32.13. (a) Show that T E 2 ( E , F) is (q,p)-mixing if and only if @T : pq(F,&o)

-

@T(S) := S o y

'JIpfE'ec~)

is defined (and hence continuous). In this case,

ll@Tll= MQ,P(T) *

(b) Show that in (a) the space t , can be replaced by L,[O, 11. Ex 32.14. Use Ex 11.23. to show that

whenever 1 < r 5 q

< 00.

430

99. The Radon-Nikodh Property for Tensor Norms and Refleziuity

33. The Radon-Nikodfm Property for

Tensor Norms and Reflexivity

The Radon-Nikodfm property led to an understanding of the full duality of the projective and injective tensor norms, namely, describing the conditions under which E’&,,F‘ = (E & F)’ holds. Noticing the central role of the sequence space t!, for this question, Lewis [169] succeeded in obtaining many results of the form E‘&,,F‘ = (E&, F)’ or, in other words, results about 21min(E,F’) = % ( E lF’) if 2l is the maximal normed operator ideal associated with a. This section presents Lewis’ ideas including applications to the reflexivity of tensor products and operator ideals. The main results are due to Lewis, but the proofs are sometimes taken from Meile [189].

33.1. Let a be a (finitely generated) tensor norm and (a,A) its associated maximal normed operator ideal. The objective of this section is to find conditions under which the natural map

is a metric surjection (the reader may already be accustomed to recognize the use of the representation theorem for minimal and maximal ideals). Note that J is injective if E’ or F‘ has the approximation property (see 17.20.) or if F’ has the a-approximation property (see section 21) or if a is totally accessible. For a = T this question was investigated in section 16: It turned out that in order for

t o be a metric surjection a sufficient condition is that F’ (or E‘) has the RadonNikodfm property (16.5.). An alternative proof to the one given in section 16 (or Appendix D7.) uses two ingredients: (a) The Lewis-Stegall theorem: I f t h e Banach space G has the Radon-Nikody’m property, then every T E 2 ( L 1 ( p ) ,G) factors through e l ( & ) (or through el zfp is finite), and IlTll = inf llRll IlSll, where the infimum is taken over all these factorizations ( A p pendix D2., proposition 2; one may take R the natural quotient map e l ( & ) + G). (b) 3(E,L1) = ( E & co)’ = (cO(E))’= e l ( E t )= e,&,,E‘ = %(E,e1) isometrically. Now take T E 3 ( E , F’) with F‘ having the Radon-Nikodjmproperty. The factorization theorem 10.5. for integral operators and (a) give

99. The R a d o n - N i k o d h Property for Tensor Norms and Reflexivity

431

m

and it follows from (b) that T factors through

A %(E,tl) , which implies that T E %(ElF') and N(T) II(T)(l + E ) . To = S1IR E 3 ( E , t l )

33.2. This simply structured proof leads to the following definition, due to Lewis:

DEFINITION: A finitely generated t e n s o r n o r m a has the Radon-Nikodim property if el

=(E

c,)'

holds isometrically for all Banach spaces E .

It was just shown that a has the Radon-Nikodfm property - but follows from 1 E'&tl A & ( E , t l ) C(E,ti) = (E@X ~ 0 ) ' .

E

does not: This

The fact that l 1 has the metric approximation property implies that the mapping J : E'&,tl --+ ( E c,)' is a metric injection whenever a is left-accessible. Since the usual tensor norms are accessible, the important part of the definition is the surjectivity of J - in other words, it will be important to know when

21mi"(E,t1)= Q ( E , L l ) holds (where 2l is the associated operator ideal). Since

E'

el =1 E'

@m !I

and

E'

@af\ c ,

1

= E'

@af c,

(see 20.8.), it follows that

PROPOSITION: a has the Radon-Nikody'm property if and only if a/ has i t , does not have the Radon-Nikodfm property, but In particular, E / = d , = wl = it will soon be seen that all the other ap,gdo!

33.3. The following is simple:

LEMMA: If(%,A) i s a quasinormed operator ideal and E a Banach space such that Qmin(E,!l)= %(El4 ) holds isometrically, then

amin(~,e,(r)) = a(~,e,(r))

432

33. The Radon-Nikodb Property for Tensor Norma and Reflexivity

holds isometrically for a11 sets I'. Since 6 is 1-complemented in e l , the finite case is obvious. If r is infinite PROOF: and T E 2l(E,f!l(l?)), then for each bounded sequence (zn)in E the sequence (Tz")is contained in some 1-complemented subspace el(I',) with countable r0 C .'I It follows from the assumption that (Tz")has a convergent subsequence, hence T is compact. Therefore, T has separable range, which - again - is in some 1-complemented l1(r0), as before. This implies that T E 21mi"(E,!1(r)) (with the same norm). 0 This observation and the fact that operators in 2l/ factor through some L1(p) (see 20.12.) give a result which can be seen as a justification of Lewis' definition:

THEOREM (Lewis): Let a be a tensor norm with the Radon-Nikody'm property and (%, A ) its associated maximal normed operator ideal. If F' has ihe Radon-Nikody'm property, then E'&,/F' + ( E F)' is a metric surjection and (21/)mi"(E,F') = % / ( E ,F') holds isometrically for all Banach spaces E.

PROOF: The definition of T E 2l/(E,F') and the Lewis-Stegall factorization theorem give E

T

F'

.1 7 f. Q Since clearly, A/(R) = A(R) and since a/ has the Radon-Nikodim property (by proposition 33.2.), it follows from the lemma that

and hence T E (21/)m'"(E,F') and (A/)mi"(T)

-

5 A/(So o R)llQll 5 A/(T)(l

+E). 0

This result does not hold when a/ is replaced by a : It will be seen in a moment that a := w2 2 2 has the Radon-Nikodim property, but

33. The R a d o n - N i k o d ~Property for Tensor Norms and Reflexivity

433

33.4. It is interesting to note that whenever E'6pF' -W (E @pi F)' holds for some tensor norm p, then this holds also for p replaced by some injective or projective associates of p. This will be an easy consequence of the

LEMMA:Let P be a finitely generated tensor norm, E and F Banach spaces. (1) I f L w ( E ~ j ) ' 6 p F '--w ( ~ ~ ( B@pi E IF)' ) is a metric surjection, then

~ 'F' 6( E ~ ~F)' -W

is a metric surjection as well. (2) I f l W ( B ~ ) & p F-W' BE) @pi F)' is a nietric surjection and F' has the approximation property or: p i s right-accessible and E' has the approximation property, then

E

~

--o

(~ E qSlp)l ~ F)' ~

F

~

is a metric surjection.

PROOF:With I : E + L C O ( B ~part i ) , (1) follows from the diagram

I1&dFi

To see (2) take B ately that

-

3

1

-

p and observe that 23""' /p. The assumption implies immedi(Bmin)sur(E, F')

A BsUr(E,F') . 1

Proposition 25.11.(1) implies that (%mi")8ur(E, F') = (Bdur)min(E, F') under the conditions stated, which means that

~ ' 6f, (~~ a u~r ) m~i n ( F') t~ , A B ~ EF'), = ( E B

( ~F)'~ .) ~

Taking F := co, and recalling that \a and /a are always left-accessible, this shows the

COROLLARY 1: If a has the Radon-Nikodgm property, then \a and /a also have it. Together with proposition 33.2. this implies that along with a a/, \a,/a and \a/ 1 /(a/) (a/)/ 9

(and so on) also have the Radon-Nikodim property. But note that a\ need not have it: This follows from e = w,\ (see 20.14.) and from 33.5. below.

434

33. The Radon-Nikoddm Property for Tensor Norma and Rejlezivity

Combining this with the lemma and theorem 33.3., leads to another result of Lewis':

COROLLARY 2: If the tensor norm a has the Radon-Nihody'm property, then so do a/,\a/,/(a/) and (/a)/.If E and F are Banach spaces (E' or F' with the approximation property i f /3 = /(a/))such that F' has the Radon-Nikoddm property, then E'&p F' --Y ( E @pi F)'

p =

a metric surjection and Brnin(E, F') = % ( E lF') holds isometrically for the maximal operator ideal (%, B ) associated with /?.

is

33.5. LaprestB's tensor norms and their adjoints (this means: the ideals CP,,of (p, q)factorable and !Dp,q of (p, q)-dominated operators) nearly all have the Radon-Nikodjm property. The result is due to Lewis:

= E and 0 1 , ~= w l = E / do not have the Radon-Nikodfm The tensor norms = g& has it or not; since property. Unfortunately, we do not know whether 9; !J31 = Jinj, this is the same as asking whether or not

-

(by proposition 25.11.(2)) holds: Is every absolutely summing operator E nuclear (see Ex 9.13.)?

-

tl

quasi-

PROOF: Since all these tensor norms (and their associated ideals 2f) are accessible, it remains only to show that 5 0 Q(E,ti) = Q ( E ,ti) (see the initial remarks in 33.2. and 25.2.). (1) Each operator T E X,,,(E,tl) factors as follows

Ts

33. The Radon-Nikodh Property for Tensor Norms and Refleziuity

435

Thus, by the very properties of I,, it is enough to show that S is weakly compact (=compact=approximable): For 1 < p < 00 this follows from the reflexivity of L,, for p = 00 this is a consequence of the fact that every operator L , -,I1 factors through a Hilbert space (e.g. by Grothendieck’s inequality, see 17.14.). For p = 1 this is not true - a more complicated argument is necessary.

So assume that p = 1 and 1 < q < 00 (the case a 1 , 1 = 7r is already clear). It is enough to show that +(p)

C

L L(P)

.Crp.

s

I1

( p a finite measure) is in Since I, has the Radon-Nikodim property, the operator S has a Ftiesz-density g = (gn) with gn E L1(R,p) and

for all w E R. The sequence A = (An) defined by

is in Iq.It suffices to prove that the operator S o l factors through the diagonal operator Dx : 1,) + It and that m

The latter follows from

For the factorization define U : I, makes sense since for (a,) E Iq

-, L , ( p )

by Uen := A;’gn

- and

this definition

It is easy to verify that S o I = Dx o U‘ - and this ends the proof of (1). (2) For 1 < p < co take T E Zg/(E,Ll) and factor

33. The Radon-Nikodh Property for Tensor Norms and Reflexivity

436

Then it is obvious that R, E ZEj and So E Z((M,tl) since M is reflexive. It follows that T = So o R, E 3 0X:i(E,ei) = (2Ei)min(E,.t~) by the accessibility of ;:2

-

-

.

CY~,~\

(3) First recall that 3 p ~=, q ?pal ~ o q p t by Kwapieh's factorization theorem 19.3..Sinced;/ = d,, (by 20.14.), it followsfrompart (1) of the proof that for 1 < q 5 00 the tensor norm d;/ and hence di has the Radon-Nikodim property (33.2.). Since d; p$"', this means that

-

YfYo'(E,4'1)

A (!.@''')min(E,lI)

Ex 25.15.(a) (about (B o 21)min(E, G ) = B o B ( E , G)) gives the result. In the remaining case q = 1 one has has the = g;l = gpt\, and therefore - and

Radon-Nikodim property by (2) whenever 1 < p < 00

.0

33.6. The fact that the (right-projective) tensor norm dp has the Radon-Nikodim property (for 1 5 p < ca) implies that

-

is a metric surjection if E' has the Radon-Nikodim property (theorem 33.3.). Since $ = g p gP (the ideal of p-integral operators) and V, = grin (the ideal of pnuclear operators), it follows that

COROLLARY 1: Let 1 5 p property, t h e n

< 00

and E , F Banach spaces. If E' has the Radon-Nikodgm

J p ( E ,F') = %,(ElF ' ) holds isometrically.

For p = 1 this was shown in Appendix D. Recall from 21.10. the Pietsch pintegral operators, which factor as follows (1 5 p < m) :

33. The Radon-Nikodh Property for Tensor Norms and Reflezivity

437

T

E-F

Since Lp is 1-complemented in L : for 1 5 p

< 00, it follows from corollary 1 that

P&(Io R ) = Ip(Io R ) = Np(Io R ) whenever E' has the Radon-Nikodjrm property, and hence,

COROLLARY 2: For 1 5 p < 00 and E' with the Radon-Nikodgm property W p ( E ,F ) = %(El F ) holds isometrically.

33.7. One application of the coincidence of p-nuclear and p-integral operators: In Ex 28.1. an operator T E C ( E , F ) was called p-strong (for 1 5 p < co> if there is a constant c 2 0 such that

whenever

21,

m

m

k=l

k=l

..., zm,y1, ...,y, E E are such that m

m

k=l

k=l

for all x' E E'. It was shown that every operator T E p-strong operators are in C y I"'. Even more is true:

THEOREM: A n operator is in

Cy is p-strong - and that all

Cy if and only if it i s p-strong

(1 5 p < 00).

For T = idE this is a result of Lindenstrauss-Pelczyriski [173] (see Ex 33.5.) - and for arbitrary T the following proof was given to us by Tomczak-Jaegermann.

Cy

PROOF:That each T E is p-strong with constant c = L y ( T ) was shown in Ex 28.1.. For the converse take a p-strong T E &(ElF ) with constant c . It will be shown that L y ( T ) 5 c. Recall from 25.9. that = ppo (pFa')-l.Therefore it is sufficient to show that for S E !#Fa'(G, E ) with PpduO'(S)= 1 the composition TS satisfies P (TS) 5 c. By the definition of T being p-strong and the fact that Ci"j are maximal operator ideals, one may assume that E and G are finite P and $IFaP dimensional. Since 3y = QPholds isometrically, it follows that

Cy

IoS' :El

s'

I

G' k!,(Bc)

438

33. The Radon-Nikodh Property for Tensor Norms and Reflezivity

is p-integral and

4 ( Z o S’) = Pp(S’)= 1, hence

with Np(ZoS’)= 1 by corollary 1 before (or by the representation theorems 22.5. since E is finite dimensional and gp accessible). Therefore it is a consequence of the definition of gp (see e.g. 17.10.) that I o S‘ factors as follows

- or

dually

Now take z 1 , ...,zm E G with and define xk

:= S r k

Wp(%k;G)

5 1, assume without loss of generality

E E and y k := R i D A e k = A k R ’ , e k

These two m-tuples of elements in E satisfy for each xF E E’ m

m

k=l m

m

k=l

k=l

Since T is p-strong, it follows that m

m

m

k=l

k=l

k=1

which shows that P p ( T S )5 c . 0

EE

.

rn = n

$3. The Radon-Nikodh Property for Tensor Norms and Reflezivity

439

33.8. At the end of this section conditions under which the tensor product

and/or the component

% ( E ,F ) of the associated operator ideal is reflexive will be investigated. If E and F are # (0) (which will be assumed from now on) it is a necessary condition that the complemented subspaces E (or E') and F of E 6 a F (or %(ElF)) are reflexive. Clearly, this is not sufficient since, for example,

(see section 16) is not reflexive. Note that

(see 26.1. and Grothendieck's inequality, operator form) - so the spaces Cp,q(E, F) and qyr((E, F) are also not necessarily reflexive if E and F are. Recall that a reflexive space E has the approximation property if and only if its dual has - and this implies that E and El both have even the metric approximation property.

THEOREM: Let a be a finitely generated accessible tensor n o r m , E and F Banach spaces one of which has the approximation property. T h e t e n s o r product

hold isometrically.

PROOF: Clearly, the condition is sufficient. Assume now that E g a F is reflexive. Then E and F are reflexive and, e.g., E and El have the metric approximation property. It follows that the natural injections

are isometries. The commutativity of the diagram

440

33. The Radon-Nikalghn Property for Tensor Norms and Rejlezivity

shows that

11

is surjective and 1; injective: This is the claim. 0

33.9. If 24 is the associated maximal operator ideal, then (a')' = a* N 24. and the representation theorems for maximal and minimal operator ideals give

COROLLARY 1: Let (a,A) be an accessible, maximal normed operator ideal and E , F Banach spaces one of which has the approximation property. Then % ( E ,F ) is reflexive i j and only i j E and F are reflexive and 2lmi"(E,F ) = % ( E ,F ) (2i*)min(F, E ) = a * ( F ,E ) hold isometrically. Using again E @a F = F @ a t E , another consequence is immediate as well:

-

COROLLARY 2: Let a (a,A) be accessible and E , F Banach spaces one of which has the approximation property. (1) 1f E 6 a F is reflexive, then E'&IF' , F&tE

and

F'&-E'

are reflexive as well. If % ( E ,F ) is reflexive, then

(2)

Q*(F,E) and !21dua'(F', E') are reflexive. (3) %(E,F ) is reflexive if and only if E'G,,F is reflexive. 33.10. Theorem 33.8. shows that Lewis' theorem 33.3. and its corollary 2 in 33.4. provide conditions for reflexivity. Recall again that reflexive spaces have the RadonNikodjm property.

COROLLARY 3 (Lewis): Let a be a tensor norm such that a and a* have the RadonNikodgm property. For p := /(a/)or p := (/a)/and E , F reflexive Banach spaces, one with the approximation property, the spaces E & F , E ~ @ , FE,& F ,

E&=F

33. The Radon-Nikodbirn Property for Tensor Norms and Refleziuity

441

are reflezive.

The analogous statements for operator ideals are obvious.

PROOF:By the foregoing corollary and (/(a/))* = (/(a*))/ it is enough to prove that E&pF is reflexive for ,8 = (/a)/. The tensor norms /3 = (/a)/and p' = /(a*/)are tensor norms as in 33.4., corollary 2, hence (using the approximation property),

Since p' is (totally) accessible (by proposition 21.1.(3)), and therefore /3 accessible, theorem 33.8. gives the result.

33.11. Some examples: It has already been seen that tP,P(W2)

is not reflexive whenever p , q E]l,co[ and hence reflexive as well. What about gp and dp = gi?

L2&3a,,,12,

by corollary 2(3), is not

PROPOSITION 1: Let E and F be reflexive Banach spaces, one of which having the approximat ion property. (1) E@g,F is reflexive f o r 1 < p 5 00, but not i n general f o r p = 1 . (2) 3 , ( E , F ) , 3FaI(E,F ) are reflexive f o r 1 < p 5 00, but not i n general f o r p = 1 . (3) !&tp(E, F),!-@'"(E, F ) are reflexive f o r 1 5 p

< 00, but not in general f o r p = 00 .

For p = 1 the statement (3) is due to Gordon-Lewis-Retherford [85], the other results seem to be due to Lewis [169].

-

-

PROOF:Since gp DP and gpl pplit is enough (by corollary 2) to deal with gp or gi = dp = ( / d p ) / (see 20.14.). For 1 < p < 00 the tensor norms dp and d; have the Radon-Nikodfm property by 33.5., therefore the result follows from corollary 3. The space &&& = &$$jslf?2 is certainly not reflexive, so it remains to prove that E&d, F is reflexive: It follows from Ex 33.6. that

-

hold isometrically - this is one condition of the reflexivity theorem 33.8.. For the other one it remains to check (recall d, = w l 21)that

442

33. The Radon-Nikodim Property for Tensor Norms and Reflexivity

is surjective ( d , = w1 is accessible, one of the spaces has the metric approximation property). Take T E 21(E', F ) , use the Radon-Nikodjm property of F and the LewisStegall factorization theorem (33.1.) to see that T factors as follows:

E-F

T

Ll(P)

-

el(&)

so

where Q E 21 and So o R is weakly compact=approximable since E is reflexive. It follows that T E 21 o 3 = 2yinby 25.2.. 0 More results in this direction follow by considering the tensor norm 33.11.).

(see also Ex

~ t ~ , ~ /

+

PROPOSITION 2: Let p , q E [l,oo] such that l/p l / q 2 1 and El F reflexive Banach spaces with E having the approximation property. (1) For q > 1 the spaces

E 6 a p ~Fq /and E&a;,,\ F 2 P > 9 / W F ) and \ 2 9 , P W E ) !D$$,(E, F ) and !D;&(F, E)

are reflexive. (2) For q = 1 each of these statements fails for E = F = &.

PROOF: It is clear from 33.9., corollary 2, that it is enough to show the reflexivity of E 6 a p , q / F. The case al,oo/= d,/ = d, was settled in proposition 1. For 1 < q < 00 the tensor norm ap,qhas the Radon-Nikodjm property (33.5.), hence Lewis' theorem 33.3. gives

E 6 a v , , / F = (E'

F')' .

Applying theorem 33.8. again, it remains to show that

(!D$,$)min(E, F') = E'ga;,p\ F' (note that

A (E

@I(~;,~\)S

F)' = !D:,i,(E, F') ;

is totally accessible by 21.1.). Since

(!D7$,)min(E, F') = (!D7:$)inJ(E, F')

99. The Radon-Nikodbim Property for Tensor Norms and Reflexivity

443

by 25.11.(2), it is enough to show that

Since 'Pqj(E,G) = !PYin(E,G ) for all G (see Ex 33.6., here the approximation property of E - and not of E or F - is needed), this follows from Kwapieli's factorization theorem

Dqt,p,= y3pdF 0 PqI and Ex 25.15.(b) (concerning (23 o %)mi" = % o a). For q = 1 it is a consequence of Grothendieck's theorem (this was already mentioned in 33.8.) that

is not reflexive. 0

Exercises:

Ex 33.1. An operator T E 2 ( E ,F) is called a

Radon-Nikody'm operator if for every

S E C(L1[0,1], E ) the operator T o S is representable. Denote by r2) the ideal of all 11 11) is an Radon-Nikodfm operators and give it the operator norm JJ 11. Show that (9, injective Banach operator ideal which is not surjective and not maximal. Hint: Identity of k'l and L1[0,I]. Ex 33.2. If p is a finite measure, S E ~ ( L I ( , u )E) , and T E r2)(E,F), then T o S is representable. Hint: Proof of theorem Dl.. Ex 33.3. The ideal of operators T E C ( E ,F) which factor

with the norm inftlRll llSll is not maximal. through some Ex 33.4. Use corollary 2 from 33.4. in order to give conditions under which

Give an example for which %(ElF) = Qmin 22.9. for the counterexample.

reg (ElF

) # %""n(E, F). Hint: 31 = 3 and

444

33. The Radon-Nikod&n Property for Tensor Norms and Reflezivity

EX 33.5. For 1 5 p < 00 a Banach space E is isomorphic (resp. isometric) to a subspace of some L p ( p ) if and only if there is a c 2 1 (resp. c = 1) such that for all m E N and 25, gk E E the condition m

m

for all x’ E E’ k=1

k=l

Ex 33.6. If E’ has the Radon-Nikodh property and: E’ or F has the approximation p < 00 property, then for all 1 I

holds isometrically. Hint: 33.6. and 25.11.(2). Ex 33.7. Let a be a finitely generated left-projective tensor norm with associated maximal operator ideal Q. If at and E‘ have the Radon-Nikodjm property, then

Q(E,F’) = Qmin(E,F’) holds isometrically for all Banach spaces F. E x 33.8. Let Q be an accessible maximal normed operator ideal. Show that Q ( E ,F ) is reflexive for all reflexive E , F such that E has the approximation property if and only if Q*(E,F ) is reflexive for all reflexive E , F such that F has the approximation property. Ex 33.9. Prove again the result from 16.7. that for reflexive Banach spaces E and F (one with the approximation property) X(E, F ) is reflexive if and only if X(E,F ) = ff(EF , ) . Hint: 33.9., corollary 1. E x 33.10. Let E , F be separable Banach spaces, one of which has the approximation property, and Q an accessible maximal normed operator ideal. If Q ( E ,F ) is reflexive, then it is separable. Ex 33.11. Use the fact that the tensor norm a p , 2 is left-injective (for 1 5 p < 00) to show that is reflexive if E and F are reflexive Banach spaces and E or F has the approximation property. Hint: (/a)/. Note that proposition 2 in 33.11. did not cover the case that F has the approximation property. E x 33.12. Use corollary 26.1. to show that for p, q E ]1,00[ the spaces

are not reflexive.

94. Tensorstable Operator Ideals

445


This section is devoted t o the following natural questions: Given an operator ideal ?2l and a tensor norm p; is S 6 p T E % for all S , T E a? Is E 6 p F E space(%) if E and F are? Although this happens rarely for general % and p, there are some positive situations which have interesting applications. On the other hand, the number of results which handle these questions in the special case where 1,3 = E or p = ir is too large to cover all of them here. We only present a systematic approach for the more general setting. At the end of this section a method will be presented which improves inequalities with the help of tensor stability.

34.1. Let p be a tensor norm. A quasi-Banach operator ideal (%,A)is called ptensorstable if for all Ti E %(Ei,Fi) the continuous tensor product operator

is in the ideal %. It can easily be seen using an ll-sum argument that in this case there is a constant c 2 1 satisfying

say: % is P-tensorstable with constant c. For c = 1 the term metrically ptensorstable will be used. See Ex 34.2.(c) for an example showing that not every P-tensorstable ideal is metrically p-tensorstable. . 11) is metrically pThe metric mapping property of tensor norms says that (2," tensorstable for all tensor norms. The relation - we

5 of approximable operators is metrically p-tensorstable for all p. If Ti E C(Ei,Fi) is compact, then BE,) is relatively compact and hence

also shows that the ideal

is also relatively compact, which shows that the ideal ff of compact operators is metrically n-tensorstable. The ideal (%, N)of nuclear operators is easily seen to be metrically p-tensorstable for all p : just use expansions. Since e & 4 and l 2 & & are not reflexive, the ideal !ID of weakly compact operators is neither ir- nor E-tensorstable.

446

34- Tensorstable Operator Ideals

34.2. The metric mapping property implies that even llT16pT~ll= 11T111 llT211. Under what conditions does a /3-tensorstable operator ideal (a,A ) satisfy an estimate from below: A(Tl)A(Tz) 5 cA(Tl&pT2)? Using an argument of Pelczyriski ([203], lecture 14), the following can be shown for the adjoint operator ideal %* :

PROPOSITION: Let (Q, A ) be a quasi-Banach operator ideal which is P-tensorstable with constant c 2 1. Then

-

PROOF: Using the definition 17.9. of A*, take N, E FIN(Ei) and L; E COFIN(Fi) as well as Ri : Fi/Li Nj with A(R;) 5 1. Then A(R18pRz) 5 C, and hence

-

since tr(S1 8 S 2 ) = trS1 trS2 (check with rank-one operators). The definition of A* now yields the claim. 0 Since Q** = 2l for maximal normed ideals, this result (applied to %*) implies the

COROLLARY: Let (2l,A) be a maximal Banach ideal. If ( % , A ) and (%*,A*) are metrically P-tensorstable, then

for all TI, T2 E %. Operator ideals satisfying this relation will be called strongly P-tensorstable.

REMARK:The ideal (3inJaur,pnj aur) is g2-tensorstable, but not strongly gz-tensorstable.

PROOF: By Grothendieck's inequality (in the operator form 20.17.) 3'"J bur = 2 2 holds with equivalent norms. Since H&H = H&H (Hilbert-Schmidt norm; 26.7.) is a Hilbert space, simple factorization shows that Cz and hence 3'"j is g2-tensorstable. Assume now that I,, := 1'"j satisfies

447


Since Io(idt2)= KG,the Grothendieck constant, this would imply

which is not possible because KG > 1. 0 We do not know whether every P-tensorstable ideal Q necessarily admits a lower estimate A(Tl)A(T,) 5 cA(T16pT2).

34.3. More examples: Since llTl6pT211 = 11T1;11llT211, the ideals 2 and P-tensorstable and ff is strongly n-tensorstable. Using

5 are strongly

and the tensor stability of 5, it follows that the ideal ff of compact operators is strongly P-tensorstable whenever P is injective, e.g. /3 = E or w2. A similar argument with L1(BEl)6p11(EEa)shows that ff is strongly P-tensorstable if P is projective, for example, P = w i . It seems to be unknown whether or not ff is tensorstable with respect to arbitrary tensor norms.

LEMMA: For p = 1,2 the mapping e ,

-

e , @ e , extends t o a n isometry

el c ' . ep&.*ep. PROOF:T(C:=~ Anen 8 e,;eP,ep) 5 C,"==, IA,l gives one inequality. The other one follows from e163*e1= el(N x nV) and, for p = 2, from (see Ex 3.28.) m

If I : P1

L,

8

La, then obviously

commutes and this implies the

-

COROLLARY 1: I f t h e operator ideal Q i s injective and L1 !$ space(%), then the operator 16% I : el&.*el e26.*e2i s not in Q.


448

I n particular, for 1 n-tensorstable.

5 p < 00

the ideal

!Jlp of

absolutely p-summing operators is not

The latter follows from I E !J31 C Pp(see Ex 11.5.). Since the ideal Q% of quasinuclear operators is the injective hull ninjof the nuclear operators (Ex 9.13.),the same argument which was used for ff = gives that Q% is metrically P-tensorstable whenever P is an injective tensor norm. But Holub [113] observed that:

?J

COROLLARY 2: The ideal Q% of quasinuclear operators is not ?r-tensorstable. PROOF: Take

(An)

E co and T := C,”==l Anengen : el

-

P2,

Then the operator

is nuclear by Ex 11.5. and Appendix C8., corollary (recall that 3‘“j = PI),and therefore T is quasinuclear. Now look a t the diagram

To

lJ fl

,

‘J

Toen := Aien

.

fl

I f T&T were quasinuclear, then To would be quasinuclear and hence m

n=l

nuclear (T: E Q9Z2c J!J: in f4 (Ex 3.28.). 0

c !J3; c %). But this is only the case if the sequence (An)

is

For a positive result in this direction see Ex 34.3.. 34.4. Many of the arguments already used are valid in a more general setting. The following statements (taken von Carl-D.-Ramanujan [23]) allow a more systematic treatment of tensor stability.

PROPOSITION: Let (a,A) and (23, B) be Banach operator ideals and P a tensor norm. (1) If(%, A ) and (%, B) are P-tensorstable (with constants a and b, respectively), then (Sa o %, A o B) is P-tensorstable (with constant a . b).

449


(2) If (a,A) as P-tensorstable, then its minimal kernel (amin, Amin)is P-tensorstable (with the same constant). ( 3 ) If ,O is injective, (21max,Amax)right-accessible and A(Ti @p T2) I c .A(Ti)* A(T2) for operators Ti and T2 between finite dimensional Banach spaces, then the maximal hull ( ~ m a zAmax) , of is P-tensorstable (with constant c). (4) Let (a,A) be P-tensorstable (with constant c). If P is injective, then (IzLinj,Ainj)is /3-tensorstable; if ,8 is projective, then (Zl'"', A*"') is P-tensorstable (same constants). (5) If /3 is projective, (a,A) is P-tensorstable (with constant c) and (Qmar, Am,=) is

left-accessible, then

Amaz

( a m a x dual 1

dual

1

is P'-tensorstable (with constant c).

PROOF:(1) and (2) are obvious since 2 P n = 5 o a o 5 and 5 is P-tensorstable. Statement (4) follows as in the case of ff = $n' = 5""'at the beginning of 34.3.. TOprove (3) take Zl E a m a r ( E i , F i )and put E := El @p E2. Then, by the density lemma for maximal operator ideals (Ex 17.17.), it is enough to show that A ((T16pT2) o I i )

5 c . Amax(Tl). Ama2(T2)

for cofinally many M E FIN(E). Since for each such M there are Mi E FIN(&) with M C Mi @ M2 and P is injective, it follows that

M

Mi @p Mz

El @p E2

this implies that M of the form MI @p M2 suffice. Bmazis supposed to be right-accessible, hence there are Nj E FIN(Fi) and Sj E L!(Mj, N j ) with

- and

. I;: o I s i = 12 o Si and A(Si) I (1+ &)Ama"(Z) Now the desired inequality is a consequence of A ((Ti6pT2)o I & ~ ~ M =A ~ ((Ti ) 0 Izl)6p(Tz 0 12)) =

=A

((12 sl)6p(~zs2))= A (1NlqN2 0

F,&&

0

0

(slB~s2))I

< c * A(S1) - A(S2) 5 ~ (+lE)~A'"''(T~)* Amax(T2). To see ( 5 ) first observe that for operators TI and T2 on finite dimensional spaces, one has (Ti @ p t Tz)' = Ti @p Ti,and therefore,

-

Adua'(Tiapt Tz)= A(T{ Bp Ti)5 c A(T:) A(Ti) 5 c * Adual(Ti). Adua1(T2) .

450


Now (3) applied to !21dua' and

p' gives the result. 0

Note that ( l ) , (2) and (4) also hold for quasi-Banach operator ideals. 34.5. The most important positive examples come from factorization techniques (apparently first used by Holub [113] in this context). Factorization of an operator in 2 p , q ( EF , ) usually goes into F" (rather than F ) , and therefore the following simple observation is useful: If /3 is finitely generated, then any of the two maps 0, from the extension lemma 13.2. make the diagram

commutative; recall that we do not know whether or not the

0j

are isometries.

If Ti E 3(Ei,Fi), then the factorization 10.5. for integral operators yields for each finitely generated tensor norm @

1 C(K1 x Kz)= C ( K l ) & C ( K 2 ) - and

-

/

Ll(Pl)@rLl(PZ)= Ll(P1 63 P 2 )

the observation above implies the

REMARK:T h e ideal (3,I) of integral operators is metrically @-tensorstable for all finitely generated tensor n o m s p. Exploring this factorization idea for T E 2 p , q ( EF , ) , the factorization theorem 18.11. for (p, q)-factorable operators leads to the diagram

1 and hence the

/


451

PROPOSITION: Let p and 7 be finitely generated tensor norms such that

A,) 5 7 on L,t @ L,) P 5 A, on Lp @ L p Then f o r

E

2p,q(Ejl

Fi) the operator

-

is defined, ( p ,q)-factorable and LP,,(T16Tz)5 Lp,q(T1)Lp,q(Tz). It follows from 29.11. and 29.4. that in the hypotheses L,I and L, can be replaced by and ,! respectively. The result is interesting because of its special cases 3, = f!, and J?"' = 2 1 , , : the fact that dil = dp = A, = g;, = g p on L, 8 L, (see 15.10.) implies the !,I

COROLLARY 1: (1) For 1 5 p 5 00 the ideal 3, ofp-integral operators is metrically P-tensorstable f o r all finitely generated tensor norms satisfying P 5 A, on L, @ L,. Examples: E , d;, ,d,, g;l ,g,, wp,wpj and all smaller tensor norms. (2) For 1 5 q 5 00 the ideal 3F"' is metrically P-tensorstable for all finitely generated tensor norms satisfying P 2 A,, on L,I @ LqI. Examples: T , d ; , dqtlg; , 9,' , w;, ,w; and all larger tensor norms. It follows that for 1 5 p 5 2 the ideal 7, is metrically w,-tensorstable for all r E [l,001. Proposition 34.4. allows one to verify tensor stability for many other operator ideals (see Ex 34.2.). Restricting to E and T , it gives the related to 3, and 3":'

COROLLARY 2: Take 1 5 p 5 00. (1) The ideals 3, of p-integral, $' 3, of absolutely p-summing, and

'JlY

of quasi-pnuclear operators are strongly E-tensorstable, the ideal %, of p-nuclear operators i s metrically e-tensorstable. (2) The ideals 3Fa',q?"' and A:; are strongly ?r-tensorstable, the ideal Al,, is metrically x -1 ens o rsl a ble. Because of its application to the theory of eigenvalue distribution (see 34.10.) perhaps the most important statement here is that q, is E-tensorstable; this result is due to Holub [113].

PROOF:Corollary 1 and proposition 34.4. give that all these ideals are metrically tensorstable with respect t o E or A. Since 3; = $'3,' and V; = J , I , corollary 34.2. gives that the maximal ideals in (1) are even strongly E-tensorstable. Since g p = gp\ = g;l

452


on E' 60 looand g;, is totally accessible, the representation theorems for maximal and minimal ideals imply that

for all E , F - and this shows that with relations

!&, also T$'j

is strongly c-tensorstable. The

(ppal)*= 'Jpdyal give that the maximal ideals in (2) are strongly 7-tensorstable and

ffiY;(E, F ) that this is also true for

pra'(E, F)

ffb,y.0

34.6. Two applications of these results. First, recall from 23.6. that the projection constant A(E) of a finite dimensional Banach space is L , ( i d ~ )= I ,(i d~). Hence the relation I p ( m 3 c T 2 ) = I p ( W p ( ~ 2 )7

which was just proved, implies that

for all finite dimensional Banach spaces E and F. This is a well-known result. However, there is no such relation for infinite dimensional spaces! Even more:

PROPOSITION: Let E and F be infinite dimensional spaces. Then the Banach space E& F is not injective - i n other words: A(E& F ) = 00. This result was given to us by Pelczyhski.

PROOF: First take E = F = l, : By a result of Kalton [140] the space t , @ ~ ~=t , ff(41,l,) is not complemented in C ( t ,,t,) - and is therefore not an injective Banach space. Now, take arbitrary E and F and assume EI& F to be injective. Then, as complemented subspaces, E and F are injective as well. A result of Rosenthal's states (see [177], I, p.105) that t, is isomorphic to subspaces E, c E and F,, c F ; they are complemented (since toois injective), and hence t,@,,t, is isomorphic to the complemented subspace Eo&F,, of the injective space E&F. It follows that t,&t, is injective as well which is a contradiction. 0 Another application for bases (see Ex 12.9.) is also due to Holub [113] :


REMARK:The s n i t vector basis (en 8 em) in . t l & t ,

453

is not unconditional.

-

PROOF:The embedding I : l 1 L, l 2 is absolutely l-summing. By corollary 2 the l2&& is also in and hence maps unconditionally operator I&I : l l & l l convergent series to absolutely convergent series. If (en @ em) were an unconditional basis of 1, &el, then every expansion

would converge absolutely in r?2&4 which means that C lAn,ml c 00 would hold. In other words, every operator in f f ( c o ,e l ) = .f1&11 would be nuclear; but this is not true (see e.g. Ex 4.3.). 0 34.7. What about tensor stability of space(%)? Negative results will follow from the

LEMMA: Let spaces. Then

p

be an injective tensor norm and E l , E2 infinite dimensional Banach contains the uniformly.

El&@E2

PROOF:By Dvoretzky's spherical section theorem, which was already mentioned in 32.10., the spaces Ei contain subspaces Mi" with d ( M r , e ) 5 2. Since /3 is injective, all it4r @p M," are isometric subspaces of E163pE2; therefore, it is enough to show that Pa is isomorphic to a subspace of Ej @ p 6 with control of the norm. For /3 = E

is obviously an isometry. But t = 202 on 6 @ 4, hence Grothendieck's inequality (and again the injectivity of p) give

- this ends the proof. 0 Note that the lemma is false if /3 is not injective: Take the Hilbert space 12&)ga.t2 as an example. The following theorem and its corollaries are due to Carl-D.-Ramanujan 1231 :

THEOREM: Let (%,A) be a Banach operator ideal and /3 a tensor norm such that (a) /? i s injective and supA(idt&) = 00, or: (b) /? is projective and sup A(idt;) = 00. Then E1@3E2

4 space(%) whenever El

and E2 are infinite dimensional Banach spaces.


454

PROOF:Assume E163gEz E space(%) and the first condition (a): The lemma implies that there is a c 2 1 such that

which is not possible. Under condition (b) the dual follows that

P' of /3 is injective, and therefore totally accessible. It

6( E ~ ~ D E E~ space(0max )' dual 1

E I ~ ~ E ;

(recall 17.8., corollary 4) and hence,

Since sup Arnax

dual i n j

( i d f k ) = SUP A(idf;) = 00

,

the tensor norm P' and the Banach ideal Smar injsatisfy condition (a), therefore E;&E; 4 space (amax 'nj ) - a contradiction. 0 Since sup A(&) yields the

1 (since 22' = is the positive case). For the ideal i$"i = E-tensorstability would imply (see 34.7., corollary 1) that Loo E space(Cil"f) but this is not true since, for example, Loo does not have cotype 2. If p, q €11, w[, then 2E; c !?JJ, the weakly compact operators. Now L 2 E space(ilp,q)C space(CE4) by 17.13., but L ~ & I ~isLnot ~ reflexive.

vP

2y

It remains to check the case ::2 quotient mapping e l --o e 2 is in

c !?JJ for 1 < q
k. Using Weyl's inequality in Hilbert spaces, one can show that Hilbert-Schmidt operators have eigenvalues (An(T))in Lz. Since every absolutely 2-summing operator T : E -+ E admits a factorization T = R o S through a Hilbert space such that S o R is HilbertSchmidt, it follows from the principle of related operators (due to Pietsch) that T has eigenvalues in 12 as well. In particular, nuclear operators have eigenvalues in 1 2 ; in Hilbert spaces they have eigenvalues in and it can be shown that a Banach space E is isomorphic to a Hilbert space if all T E %(E,E) have summable eigenvalues (Johnson-Konig-Maurey-Retherford [1311). In order to prove eigenvalue estimates the following result, due to Pietsch [215], is sometimes helpful:

PROPOSITION: Let (a, A ) be a quasi-Banach operator ideal of Riesz operators such that for all Banach spaces E and all T E B(E, E ) the eigenvalue sequence (A,(T)) is in Lp (for 0 < p < w). If there is a tensor norm p such that (24, A ) is P-tensorstable with constant c, then

n=l

for all T E % ( E ,E ) .

PROOF:First observe that there is a c1 2 1 such that the inequality (*) holds with c1 : If not there would be Tm E %(Em, Em)with A(Tm)5 (2d)-" (where d is the constant 2 m. Considering EmTm in in the quasi-triangle inequality for A ) and II(An(Tm))llp &(Em), gives a contradiction. Now take c1 t o be the best constant in the inequality (*). Since for eigenvalues A and p of T E % ( E ,E) one has N m ( A i d ~- T ) @ N , ( p i d ~ (8ee Pietsch

- T ) c i V , ( A p i d ~ @ -~ T 8 T )

[216], 3.4.7., for a proof), it follows that

94. Tensorstable Operator Ideals and hence that c1 5 (c1c)1/2 since follows that c1 5 c. 0

c1

461

was chosen to be the best possible constant. It

Note that for this “trick” only T&T E !2l was used and that the constant c does not depend on p : This will be important in a moment. By the way, the trick was already used in remark 34.2.. Using Konig’s Weyl inequality for Banach spaces (see [145]), one can show that in the case 2 5 p < 00 there are constants cp with SUP

IAn(T)ln”P 5 ~

p p p ( ~ )

for all T E yp(E,E ) (see [145], [131]). In particular, this “weak inequality” implies that (An(T)) E .t‘p+rfor all E > 0. Since !Jlpis metrically E-tensorstable (see 34.5.), it follows from the proposition that

n=l

for all E > 0 and T E V p ( E E) , and hence,

THEOREM: If T E 2 ( E ,E ) is absolutely p-summing for 2 5 p < 00, then

n=l

This important theorem is due to Johnson-Konig-Maurey-Retherford [131], the idea of proving it with the “tensor product trick” is due to Pietsch [215]. For more details, see the books of Konig [147] and Pietsch [216]. The proposition about the tensor product trick also has a negative flavour. For example, the inequality sup IXn(T)ln’IP < 00 n

actually holds for all T E P p , 2 ( E , E )with 2 5 p < 00, see [131] - but for 2 < p the eigenvalues are not in (see Konig-Retherford-Tomczak-Jaegermann [1501). The above reasoning implies the

!,

REMARK: For 2 < p < 00 the ideal Vp,2of absolutely

( p , 2)-summing operators i s not

tensorstable for any tensor norm.

See also Ex 34.8. and Ex 34.9. for the tensor stability of the ideal pv,*. 34.11. The ideals VP,land 9 X p j , ~ nearly coincide (see 32.10.-32.12.). A variant of the tensor product trick will give a nice improvement of the following theorem, due to Kwapieri [157].

462


THEOREM: For p E [l,001

1/r = 1 - Il/p

and

- 1/21

f o r T E 2(t,,t,) and 8 := 11 - 2/pl .

Note, the special cases p = 1 : 1 1 has the Orlicz property (see 8.9.). p = 2 : Grothendieck's theorem/inequality. p = 4/3 : an extension of Littlewood's result mentioned in 10.8. - this will be clarified in a moment.

PROOF (taken from Tomczak-Jaegermann [272]) : Using I!,-local to find a constant cp such that

technique, it sufices

pr,l(T) I CplITIl

for all T E Wl(zi;q)

2(e,$); equivalently:

= 1. Fix such

(21,

satisfies IPP

Since

:W

l

$

e

.tr(Tzi;$) I cpllTll for all 2 1 , ...,zm E with Then one has to show that CP,(T) := ( T z i )

...,zm).

-

t

T($)lI3 c p .

)

= P,(q), this means

I!(e,q) =P ,

-

It@, :e",(qJ)

-

Y(qJ>II 1 CP .

0, as a map KnX" Kmx"does not depend on p, so this is a typical situation for using interpolation. By Grothendieck's inequality and P z , l ( i d t , ) 5 C z ( P 1 ) 5 a 1 = f i (Orlicz's theorem 8.9., 8.6.) one gets

-

11%

:e",(e;)

11%

: c(q)

w3IlI q ( q ) l l sK G We).,)Il I Jz

II'P.0 : G(e).,)

*

First take 1 5 p 5 2 and 8 defined by

-1 = -1 - 0 P 2

+ -01.

Then 1/r = (1 - Q ) / l + 8 / 2 . Complex interpolation of P,(.) (n-tuples!) and of vector valued L, (see Bergh-Lofstrom [111, p. 107) gives

lppllI 2012KA-Q . For 2 5 p 5 00 start with l/p = (1 - 0 ) / 2

+ 0/00- and do the same. 0

463


.Before proving that the index r in this result is best possible the connection with Littlewood's result will be clarified: For p E %i[(co,co) = 2(co,t1) = c ( t 1 ) one has llL,+,ll = llpll = wl(L,e,;tl) (see 8.2.(1)) and hence Littlewood's inequality implies

E %il(c,,co), which means exactly that the embedding Z : tl 413 is in So for p = r = 4/3 the theorem generalizes Littlewood's result from Z to all operators T E f!(tl,t4/3). The theorem gives c 5 2'/4KAJ2 for the best constant in Littlewood's inequality. for all

'p

Q

?4/3,1.

PROPOSITION: L e t p , r E [1,00].lf2(k'l,tp) = Pr,1(t1,tp), then 1/r 5 1 - Il/p- 1/21. For p 5 2 this conclusion holds if only P , 1 ( Z : t, ~ -t,) r < 00. In particular, this means that the index r in the theorem cannot be improved (a result also due to Kwapieli [157]) and that in the Littlewood inequality the exponent p = 4/3 is best possible as well.

PROOF: Walsh matrices will be used. They are defined by induction:

-

A1 :=

2-"I2A, and A,

: t:"

o A,

(i

-:>

(2 -2,)

----5") and An+l :=

.

t:" is orthogonal and self-adjoint, hence llAn : tg" k':"ll = 2"12 = 2"id. Moreover, llAn : t:" too 2" 11 = 1 and the factorization

give llAn : 1% c 2 1 satisfying

-

,,

A, : t& 1:"ll

id

t;"

A n

e;"

id

t:"

5 (2n)3/2. To prove the result, observe first that there is P J ( T : t;-

a

5 CllTll t;" if 1 5 p 5 2 (and all n E N ) . Take

for all T E 2(t:", $") or only T = I, : t:" t!:" one has the latter case first: For A, : tz,"

k=l

hence, ((2n)r/p2n)1/r 5 ~ ( 2 " ) ~which / ~ , is only possible if the claim about r is satisfied. If 2 5 p 5 00, consider

464

- and


again r satisfies the desired inequality. 0

34.12. Next the announced improvement (due to Carl-D. [21])of Kwapieri's theorem 34.11. will be given. Recall that !Dlq,l C PSp,,1.

THEOREM: Let p , q E [l, m] with l/q = 11/2 - l/pl. Then

In particular, this space coincides with Pqt,l(tl,tp) and therefore the index 4 cannot be improved by proposition 34.11..See Ex 34.11. for two interesting reformulations. As was already mentioned, the proof will use a trick involving tensor stability - this time with respect to the right-tensor norms A,. The idea is condensed into the following lemma which shows how to improve weak inequalities.

u:,.

LEMMA 1: Let Zl c .C(K",K " ) be a set of operators and A , B : % two functions such that the following three conditions are satisfied:

(1) For every e > 0 there is

a c(e)

-

[0,oo[ be

> 0 such that

A ( T )_< c(€)n'B(T)

-

for all T E .C(Zi", K " ) n2l .

(2) [T@T : Kn@ K" = Kna K"'] E ZI whenever T E %. (3) There are constants a and b such that for all T E ZI A(T)25 a A(T 18 T ) and B(T €3T ) Then A(T)5 a b B ( T ) for all T E ZI

PROOF: Fix E

 0. Then for each T E C(K",K") in 2l

A(T)25 a A ( T @ T ) 6 ac(c)n2'B(T €0 T )

< abc(e)n2'B(T)I ,

hence A(T) 5 d m n ' B ( T ) . Replacing C ( E ) by d m and iterating the argument, leads to A(T)5 (ab)c:=i 2-' ( c ( E ) ) ~ - '. 'n . B ( T ) . Taking first k -+

00

and then

E -+

0 shows the result.

The reader may have noticed the similarity between the proofs of lemma 1 and proposition 34.10..

466


- and this is the desired weak estimate. Now apply lemma 1. 0 Littlewood's result was that I E v 4 / 3 , 1 ( t 1 , t 4 / 3 ) (see 34.11.) -the theorem shows that this operator is even (4,1)-mixiig which means: Every absolutely 4-summing operator T : 1413 F is absolutely l-summing when restricted t o e l . The 2,--local technique lemma for operator ideals 23.1. shows that the theorem implies 2, o 21 C !2Jlq,l - or, by the definition of (q, 1)-mixing operators

-

Cyclic composition (see Ex 25.16.) gives the

COROLLARY: Let p, q E [l,m] and l / q = 11/2 - l/pl. Then O V q02 , c ??3y ~ 1 o ~ c , o V q c ~ o , , XP 0 21 0 2 , c J,f , 2 , o ff102 , c n,, ffp 0 210 EWc n,/ , ep0 21 0 ff, c nqf.

c,

?

For an application of this to the question of which absolutely q-summing operators factor through a Hilbert space see Ex 34.12.. &) and recently Bennett [9] and Carl [20] have investigated the norms Pr,#(I:t,, Carl-D. [21] the norms Mq,,(I : 1, t")for other indices u, u. The latter paper also gives further applications of the tensor product trick presented in lemma 1.

-

-

Exercises: Ex 34.1. Let (%,A)be a quasi-Banach operator ideal. If T163pT2 E rz( and T2 # 0, then TIE % and A(Tl)IIT'11 I A(T163pT2). Ex 34.2. (a) Show that 2, is metrically P-tensorstable for @ = gp,gp+,,dp, d;/, and that for these @ the @-tensor product of two 2zspaces is an 2;-space. See also Ex 23.17.. Hint: 34.5.. (b) 9 2 is strongly @-tensorstable for @ = dz and 9 2 . Hint: 34.5., corollary 1. (c) (21o is P-tensorstable, but not metrically P-tensorstable for the tensor norms p = d2 and g2. Hint: (b), D2 IK G ( L o~Lw)'eg and the tensor product trick 34.12..

$4. Tensorstable Opemtor Ideals

--

467

Ex 34.3. (a) Use factorization and the fact that L1 @, * respects subspaces to show F16pF2 is absolutely l-summing if TI E 3 and T2 E Sp1. that T16pT2 : E16pE2 D

-

(b) Use the factorization El co 2 t1 T16pT2 is quasinuclear whenever TI E 3 and

T2

F1 of nuclear operators to show that E 0%.

These results are due to Holub [117].

Ex 34.4. (a) For 1 5 p 5

00 the ideal C , is neither w2 nor wi-tensorstable. Hint: , $ C, and for p = 00 Grothendieck's inequality, For 1 5 p < 00 use the fact that 2 tensor form, and 34.7..

(b) For 1 < p 5 2 the ideal 2 , of type p operators is neither w2 nor wi-tensorstable.

(c) For 2 5 q < 00 the ideal Cq of cotype q operators is not w2-tensorstable. However, it is not known whether or not C2 is r-tensorstable; recall from 34.7. the open question of whether or not t 2 6 3 n t 2 6 3 r t 2has cotype 2.

Ex 34.5. (a) Prove the following extension of Schauder's theorem due to Vala [273] : If T E C(E1, E2) and S E 2(E3, E4) are compact, then

is compact as well. Hint: Use the compactness of E16,,EI, that Hom(T, S") is compact.

TQPS'

E2&E4 to show

(b) Use the same type of argument to show that for 1 5 p < 00

if S E ?J$, and T E SppdUO'.

Ex 34.6. (a) Let 1 5 p 5 00. If a Banach space E contains all 5 uniformly compleG = E @, G holds topologically, then G is finite dimensional. Hint: mented and E Use 29.5. to show that idG E 9,. (b) There are no infinite dimensional Banach spaces E, F and G such that EBCF&G = E @, F @= G holds topologically. Hint: & r-* E F = E @, F.

Ex 34.7. Let (%,A)and ( % , B ) be two quasi-Banach ideals such that 2l

c % and

3 B(.) _< cA(.). If both ideals are /3-tensorstable and for all T E 2 B(T)2 5 B(T6bpT) and A(T6pT) then

c

5 A(T)2 ,

can be chosen to be 1.

Ex 34.8. Show that V r , is~ not E-tensorstable for 2 5 r < 00. Hint: This result even holds for all r €11, 00[

E space(!&,l). which will be shown in the next exercise: 12

468


Ex 34.9. (a) Show that

Hint: Use 4.6. or Ex 8.l.(a) and $, @A,, (b) Use this and 32.4.(c) to prove that

= qjm.

= ?&,p. Hint: (c) Assume that l / q + 1/r = l/p. If !&,p is 6-tensorstable, then Use the first estimate in 32.12. and lemma 1 of 34.12.. (d) Use (c) and 32.12. to show that '&,I is not c-tensorstable for all 1 < r < 00. Ex 34.10. (a) Deduce from Kwapieri's result 34.11. and Ex 11.21. that

for p, r, s E [l,001 such that 1/r 5 1/s - Il/p - 1/21. (b) Show that this equality also holds for 1 5 p 5 2 5 s I00 and r := p's/2 by interpolation between the case p = 1 (trivial) and p = 2 (little Grothendieck theorem). Hint: Proceed as in 34.11.. The second result is taken from Bennett [lo]. Ex 34.11. Let p,q E [l,001, l / q = 11/2 - l/pl and T E 1 2 ( e 1 , e p ) . (a) For every probability measure p on Bf;there is a probability measure v on Bf,

for all x E e l . (b) For each weakly summable sequence (2,) in L1 there is a sequence (An) E l,i and a weakly q-summable sequence (yn) in lpsuch that Tx, = Any, for all n E N. In particular, for every unconditionally summable sequence (zn)in .f!1 there is a sequence (An) E t4/3 and a weakly 4summable sequence in e d / ~such that Z n = Any, for all n. Hint: 34.12. and section 32. Ex 34.12. Show that for p, q E [l,m]

(every absolutely q-summing operator on Lp factors through a Hilbert space) if and only if l / q 2 Il/p - 1/21 . Hint: 2 2 = '.= I( ) 21$ o 2-)*.

95. Tensor Norm Techniquesfor Locally Convez Spaces

469

35. Tensor Norm Techniques for Locally Convex Spaces

The final section shows that tensor norm techniques can be successfully used for the study of problems in the theory of locally convex spaces. As in the case of Banach spacea, tensor norms (different from E and T) and local Cp-techniques help one to better understand various phenomena concerning the projective and injective tensor product of locally convex spaces - such as the investigation of topological-geometric properties, Grothendieck’s (DF)-problem and the negative solution of the “problhme des topologies” given by Taskinen. Moreover, the so-called tensor norm topologies serve to classify locally convex spaces. Tensor norm topologies were introduced by Harksen in his dissertation 1979 (see also [102]). We assume that the reader is familiar with the basics of the theory of locally convex spaces (see Jarchow [123], Kothe [152]).

35.1. The topology and the uniform structure of a locally convex space are given by the family cs(E) of all continuous seminorms on E - or by the family U ( E ) of all absolutely convex closed (or open) neighbourhoods of zero. For p E CS( E ) denote I C ~:

E

-

E/kerp ; Ep := (E/kerp,p) ; Ij(kp(x)) : = p ( z ) .

Ep is a normed space which sometimes is also denoted by Eu if U := {x I p ( z ) 5 1) and one wants to point at the neighbourhoods instead of the seminorms. It is clear that it is enough to consider bases &(E) of U ( E ) or bases cs,(E) of cs(E) (this means: for every p E cs(E) there are a q E cs,(E) and c 2 0 such that p 5 cq ). If p 5 cq, then there is a natural map I C ~ :, Eq ~ Ep, and the completion of a separated locally convex space E can be represented as the projective limit of the net ( & p ) p E e s , ( ~ ) :

-

-

E = proj Ep . PE~J.(E)

The strong topologies on the dual E‘ and the bidual E” will be indicated by EL and E f ; the space E“ endowed with its natural topology of uniform convergence on equicontinuous subsets of El will be denoted by EF. 35.2. Given a tensor norm a (on NORM) and locally convex spaces E and F, define for p E cs,(E) and q E cs,(F) the seminorm p @a q on E 8 F by

P @a q(z; El F ) := a ( . p 8 ‘ E q ( ” ) ; Ep, Fq) for

I

E E 8 F. The family

I

{P @a 4 P E cso(E),Q E c s o ( F ) )

470


defines a locally convex topology on E @ F (notation: E F and E G a F for the completion), which is certainly independent of the special bases cs,(E) and cs,(F) which were taken. This so-called tensor norm topology was introduced by Harksen. For cr = B or T it coincides with the usual injective or projective topology on the tensor product of locally convex spaces. In many respects the tensor norm topologies behave very similarly to the tensor norms on normed spaces.

PROPOSITION: Let a be a finitely generated tensor norm. (1) If E and F are separated, then so is E @a F. (2) Mapping property: If 9i C C(Ei,Fi) are equicontinuous, then { T I 8 7 ' 2 :El @a

E2

-

FI @a F2

IZE g i }

is equicontinuous. ( 3 ) Embedding lemma: The natural map

E @ F~CE @ F,"~ is a topological embedding. (4) E @ ,, F, + Eaa%' is a dense topological embedding which in particular means that E&F = E&f' holds topologically. ( 5 ) Extension lemma: If D c ( E @,, F)' is equicontinuous, then the set

D" := {'p^l'p € D ) of

right-canonical extensions is equicontinuous in ( E

F,")'

The assumption of a being finitely generated is not needed for (1) and (2); the statements (3) and (4) also hold for cofinitely generated tensor norms.

PROOF: For 0 # z E E @ F it is easy to see with corollary 2.3. that there are seminorms

c ', @ tcq(z) # 0; this implies (1). (2) is obvious from the definition of the seminorms p @a q. For (3) observe that for U E U ( F ) the dual of Fu is isometric to the Banach space [Uo] c F' (see 3.1. for this notation) and that

p , q with

(F")uoo

[U']'

A (Fu)"

is an isometric embedding. The embedding lemma 13.3. for normed spaces shows that

hence the conclusion.

35. Tensor Norm Techniquesfor Locally Convex Spaces

471

It follows from 13.4. (with an obvious notation) that the mapping

is dense and isometric - and this implies (4). (5) also follows directly from the normed case since every equicontinuous D C ( E B aF)' factors through an equicontinuous subset of some ( E u QDa Fv)'. 0

-

It is easy to see that the trace t r E : Ei @= E K is continuous if and only if the locally convex space E is normable. Therefore using the trace requires more precautions than in the Banach space case. Nevertheless, under certain circumstances the trace can be defined on the space %(E, E) of all nuclear operators (i.e., for complete El those operators which factor through a nuclear operator between Banach spaces):

where the inductive limit runs over all p E c s ( E ) and Banach discs B c E; see [44] for details concerning this topic.

35.3. For the study of right-projective tensor norms (to be done in 35.6.) and traced tensor norms a little lemma is helpful. For a given surjective map Q : F + G and a seminorm p on F, define the quotient seminorm pQ on G to be

-

LEMMA: Let Q : F G be continuous and surjective. Then Q i s a topological surjection if and only if for every p E c s , ( F ) there is a q E cs(G) such that the canonical map FP

-

is defined, a metric surjection and ker ti.,

G,

c kerpQ C G.

PROOF:If Q is topological, then q := pQ is continuous on G - and this gives the conclusion. Conversely, it is enough to show that pQ 5 q. Note that (p")". = q on G,. Take q(z) < 1. Then (PQ)"q(.q(4> = C(ti.,(.)) < 1 1 hence there is y E ker K , c kerpq such that pQ(z PO(, Y) PQ(Y) < 1. 0

+ +

+ y) < 1. It follows that p Q ( z ) 5

Recall the definition of a traced tensor norm a @G p from 29.1.. Its fundamental property also holds for locally convex spaces; the following result is taken from [41]:

472

35. Tensor Norm Techniques for Locally Convez Spaces

THEOREM: Let CY and p be two tensor norms (on NORM), G a normed space and 6 := (Y @G p. Then for all locally convez spaces E and F the tensor contraction

is a topological surjection.

PROOF:For pi E c s ( E ) and pz

E cs(F) consider the following commutative diagram:

For p := (pi @,9 11 I l ~ i8% ) (11 IIG @a pz) and q := p1 86 p2 the lower tensor contraction C,$' is a metric surjection - and, by the lemma, it remains t o show that pcQ(z) = 0 for all

+

= (kerpl @ F) ( E @ kerpz)

z E ker(rcp, @ rcpa)

Decompose z = (u', u ) E G' x

G with

.

+ Cy='=m+l Z j 8 yj according to this equation and choose

zj @ yj (u', u) = 1. Then

i=l

and

n

pCQ(z)5 P ( ~ u5)C P l ( z i )

II~'IIII~IIP Z ( Y ~ ) = 0

*

i=l

35.4. Let a and p be finitely generated tensor norms with Q accessible. For a given locally convex space E, under which conditions is the operator

continuous for all locally convex spaces G 7 It is enough to check this only for Banach spaces G - or even just G = Cz a Johnson space: Apply lemma 17.16. t o

However, using traced tensor norms, continuity of this latter mapping means that

Z q , p E 9' where 1)-p@CY*


473

(see theorem 29.4.). Since a*= (% o %*)* by 29.8., one obtains

PROPOSITION: Let a and p be finitely generated tensor norms, a accessible, with associated maximal normed operator ideals % and 23, and E a locally convex space. Then

-

is continuous for all locally convex spaces G _(Or only C2) if and only i f for every p E c s ( E ) there is a q E c s ( E ) such that Zq,p : E, Zp is in (% o a*)*,the operator ideal associated with (p @J a*)*. If Ctl is a quasi-Banach operator ideal, call a locally convex space E to be in space(%) if for all p E cs,(E) there is a q E c s ( E ) such that iE,,p E %. For Banach spaces this coincides with the notation which was used 90 far. For locally convex spaces the books of Jarchow [123], Junek [133] and Pietsch [214] give much information about space(%). Assume % to be maximal and normed and take a to be its associated tensor norm. Then !2 a = (a*)* = ( E @ a*)*= (a*@ T * ) *

-

(see 29.8.), hence the proposition implies the

COROLLARY: Let Sf be a maximal normed accessible operator ideal with associated tensor norm a. For each locally convex space E the following statements are equivalent: (a) E is in space(%). (b) G@Ja. E = GI& E (topologically) for all locally convex spaces G (or only G = (22). (c) G

E = G c&,

E (topologically) for a11 locally convex spaces G (or only G = Cz).

For the equivalence (a)

H

(b) accessibility of % is not needed.

35.5. Some examples: The spaces in space(%) are usually called nuclear spaces. Since for all 1 5 p < 00 there is an n E N with qpo ..o !JlP c % (n-times composition, see 11.7.), it follows that

space(%) = space(qp) = space(JP) = space(3)

-

€or all 15 p < 00. To be in apace(3) means that E = * BXE (topologically) since 3 r, which is Grothendieck’s original definition of nuclear spaces. See also EX 35.2.Ex 35.4.. The locally convex spaces E in space(EP) have the property that for all p E cs,(E) there is a q E c s ( E ) such that N

474

95.

Tensor Norm Techniques for Locally Convez Spaces

factors through some L p ( p ) . We do not know whether or not each complete locally convex space E E space(Cp) is the reduced projective limit of a net of C;-spaces (the Banach spaces in space(Cp)) - except in the case p = 2 : The spaces in space(&) are exactly the Hilbertizable locally convex spaces. The Cp-local technique reads as follows:

REMARK:Let (1) If f o r T

a and /3 be tensor norms, /3 finitely generated.

E C(F1, F2) (locally convez spaces)

is continuous, then

idEgIT: E @ p F1

-

E B a FZ

i s continuous f o r each locally convex space E E space(CP).

(2) If id : tp@ ~ p tq-+

tq is continuous, then

tp

id:E@DpF - E a a F is continuous f o r all locally convex spaces E E space(Cp) and F E space(&).

PROOF:(2) is a consequence of (1). For (1) take for U2 E U(F2) a U1 E U(F1) such that i 4 , @ Tul,u, : tp€ 3 (F1)u1 ~ 4, @ a (F2)u,

-

-

is continuous. Then Ex 13.7. gives that e, can be replaced by Lp(p). Thus, if the operator iiv,,~,:bv1 &, is pfactorable, factorization into the bidual gives

1 and hence the result. 0 In [44], 4.4., it was shown that if a’ i s totally accessible, then all locally convex spaces E E space(%) have the approzimation property. Banach spaces in space(%) even have the bounded approximation property (see 21.6.) - but for general locally convex spaces (and a = T ) this is already false for Frtkhet spaces since there are nuclear Frechet spaces without the bounded approximation property. Since w; is totally accessible, locally convex spaces in space(2,) have the approximation property.


475

35.6. Corollary 35.4. showed that E E space(Cp) if and only if

GmCE = G B W ,E or G@"; E = G @ wE for all locally convex G or only G = Ca (see also section 23). Since woo = w; = / T (see 20.14.), it follows, as in the case of Banach spaces, that:

E E space(&) i f and only i f E 8- = E @*\ E E space(Ze,) i f and only i f E mC* = E @c/

\E

and

.

Thus it will be important to know that injective norms respect subspaces topologically and that projective norms respect quotients topologically also for locally convex spaces. The following result is due to Harksen [loll, I1021 :

-

THEOREM: Let a be a tensor norm (on NORM) and E , F, G locally convez spaces. (1) If a is right-injective, then idE @ I : E

@a G

is a topological embedding whenever I : G

L,

E ma F

F is.

( 2 ) If a is right-projective, then

is a topological surjection whenever Q : F

+G

is.

PROOF:(1) follows from the fact that the sets V n G with V E U ( F ) are the typical neighbourhoods of G and

for all U E LL(E). For ( 2 ) lemma 35.3.will be applied. Take a typical seminorm p = p1 @a pz on E @a F. 1 Then, by the lemma, there is a q E cs(G) such that ker K , C kerpf and FPa + G , is a metric surjection. Since

is a metric surjection, it remains to show (again by the lemma) that each element z E ker(Kpl @ Ic,) satisfies pid@Q(z)5 E for all E > 0. Write z

=

m

n

i=l

i=m+l

Czi@ Yi + C

Zi

@ yi E ker(npl @ K , ) = kerKp, @ G

+ E @ ker

K,

476

35. Tensor Norm Techniques for Locally Convez Spaces

and choose elements Ji E F such that Q(gi)= yi; since ker tcq C ker p28 one can achieve p2(gi) 5 C E for i = m 1, ...,n with a constant c 2 0 to be chosen in a moment (1).It follows that

+

i=l m

n

i=l

i=m+l

The theorem generalizes the well-known (and simple) facts that E respects topological embeddings and A topological surjections. The characterizations of space( C,) and space(&,) given above allow one to extend the characterization 23.5. of f!! and .C&spaces from Banach to locally convex spaces; this result is due to Hollstein, [lo71 and [lo91 :

COROLLARY: Let E be a locally convex space. (1) E E space(&) if and only if E @= respects topological embeddtngs. ( 2 ) E E space(2,) if and only if E @c respects topological s u j e c t i o n s .

-

PROOF:It follows from the theorem that both conditions are necessary. For the sufficiency take G = Cz a Johnson space and look at the diagrams

These results have many applications to the extension and lifting of operators and to the theory of vector valued functions. 35.7. For the duality theory of E@ it is not difficult to see that E h w the Radon-Nikodfm property if and only if (for every finite p ) E-valued measures of a certain type (p-absolutely continuous and of bounded variation) have a density with respect to p; this is shown in Diestel-Uhl [55], p.63 - we do not need this result. In other words, the above definition is equivalent to the one given in Diestel-Uhl's book. The situation is the same as in the scalar case (see Appendix B6.) where the scalar Radon-Nikodfm theorem (for p ) is equivalent to the statement that Ll(p)' = fJhJ(p): this operator theoretic aspect of the RadonNikodim theorem will be iniportarit for us in the vector valued case; see D4. for more information about this point. It is not necessary to check for all finite measures p in order to know that a given Banach space has the Radon Nikocljm property.

THEOREM: A Banach space E has the Radon-Nikody'm properly if and only if e v e r y operator T : L1[0,1] 4 E (Lebesgue measure) i s representable. The following proof of this well-known result is essentially taken from Botelho [13].

PROOF:Clearly, the condition is necessary. Conversely, take an arbitrary finite measure p on Q and decompose it into its continuous and purely atomic parts ( Q c , p c )and (Q5,pa) : L l ( Q , p) = Ll(Qc,C(C) @ Ll(flZaip4) .

By proposition C1.(4) each operator T E .C(L,(p), E) has a density on the atomic part one may assume p to have no atoms (and to be normalized). If L , ( p ) is separable, then t l ( p ) is isomorphic to L1[0,1] due t o a theorem of Carat h h d o r y (and Halmos-von Neurnann, see Brown-Pearcy [18], p.181) and therefore each T is representable by thc t!--factorization theorem of Lewis and Stegall C4.. If L l ( p ) is arbitrary, then corollary C1. will be applied. One has to show that for T : L l ( p ) -* E the net (9:) is Cauchy: this is the case if (gZm)converges for every increasing sequence (C,) of p a r t i h n s . Given such a (Cn),take the p-complete calgebra C generated by UC,,;then L l ( E , p ) is separable and, by what wasjust proven, - hence

Appendiz l3: The Radon-Nikody’m Property

the restriction of T to &(C, p ) is representable. Since the algebra C, corollary C1. gives that ( g z J n is Cauchy. El

519

U C,

is p-dense in

D2. The following results will be used frequently when dealing with the RadonNikodfm property:

PROPOSITION 1: Let p be a finite measure and E a Banach space. ( 1 ) If E has the Radon-Nikodfim property, then every operator T : L1(p) -+ E has a separable range. (2) Every T : L1(p) -+ E which faclors through a space with the Radon-Nikody’m property is representable. (3) Let I : L m ( p ) c, L1(p) be the canonical embedding for also I : C ( K ) L t ( p ) if p is a Borel-Radon measure on a compact K ) and led E have the Radon-Nikody’m property. Then T o I is nuclcar f u r all T E Z ( L l ( p ) ,E ) and N(T o I ) 5 llTll 11111. -+

This is immediate from what was shown in Appendix C about representable operators particular Grothendieck’s characterization C3.. The Lewis-Stegall theorem C4. reads (with a slight extension) as follows:

- in

PROPOSITION 2: A Banach space E has the Radon-Nikody’m property if and only if for each measure p (finile or nol - or: only p = A the Lebesgue measure on [0,1]) each operator T : L l ( p ) -+ E faclors fhrough some l l ( r ) .In this case,

- the infimum taken over all such factorizations. One may take natural metric surjection .fr(D,y) --y E .

r = BE and R

the

PROOF: For finite measures this is just C4. (and the result in D1. about the Lebesgue measure; note that L1[0,1]ie separable). For arbitrary p recall from B7. that

for some index set I and finile mewiires p,. This easily gives the desired factorization through . f l ( I ; l l )= !,(I x N).Tho fact that every operator l l ( r )-+ E is “liftable” (see 3.12.) to some operator CI(I‘) l l ( B ~* ) E implies the result. 0 -+

Another characterization is alno quitc useful:

PROPOSITION 3: For each Banach space E the following statements are equivalent: (a) E has the Radon-Nikodim properly. (b) All closed subspaces of R have fhe Radon-Nikody’m property.

520

Appendiz D: The Radon-Nikody’m Property

(c) All separable closed subspaces of E have the Radon-Nikodim property.

PROOF:That (a) implies (b) follows from C1.(6) and that (c) implies (a) is a consequence of proposition C6.for IZL := 2.0 D3. The following Banach spaces E have the Radon-Nikodim property: (1) E = .tl (see C4.) or more generally: Every Banach space with a boundedly complete amemI n E aV} bounded implies the convergence of anen; inspect basis (i.e. the proof of proposition C4. with the equivalent norm supn 11 C:=l cr,e,ll).. (2) E = Ll(I’), where r is infinite (proposition 2 in D2.). (3) E is a separable dual space (thin follows from the weak version of the Dunford-Pettis theorem C5.). (4) E reflexive (this is the Dunford-Pettis theorem, strong version, C7.).

{x:

x?

The following spaces E do not have the Radon-Nikodim property: (1) E = c, (see C2.). (2) E = &,(r),where r is infinite (by D2. since c, c &(I?)). (3) C(K)whenever K is an infinite compact space (same argument). (4) E = L1(p) whenever p is not purely atomic (see C3. if p is finite - but the general case is easily reduced to this). (5) M ( K ) := C(K)‘,the space of signed measures on a compact space K for which there exists a measure p which is not purely atomic (since L1(p) C M ( K ) isometrically). It follows, in particular, that neither the Radon-Nikodim property nor its negation is inherited by quotients (take el (r)+ c, in the first case and L1[0,1] +I!. in the latter) and that E having the property iR not related to E’ having the property (consider C,

,el, ew 1.

In Diestel’s and Uhl’s book [55], p.217-219, the reader can find a list of almost 30 equivalent formulations of the Ratloti-Nikodim property and a list of spaces with or without this property.

D4. It was observed in B6. that the Radon-Nikodim theorem for a (finite) measure p simply means Ll(p)’

= L,(p). For the vector valued case recall from €312. that

I

Lm(p, F ) := { g :R -+ F p-essentially bounded, p-measurable }/

-

(equivalence classes with respect to llgllw = 0). Now 3.3, the universal property of the projective tensor norm, and Appendix B12. imply that

Obviously, equality holds if and only if every operator L 1 ( p ) -, E is representable.

Appendiz D: The Radon-Nikodkm Property

PROPOSITION: Let E be a Banach space, p E [l,oo[ and

l/p

521

+ l / p = 1. Then

holds isometrically for all finite measures p (or only the Lebesgue measure on [0,1]) if and only if E’ has the Radon-Nikody’m property.

PROOF: If this equality holds, take T : L 1 ( p ) + E’ and let pT E ( L , ( p ) @= E)’be its associated functional. Since

is continuous, it follows that ,&o I E L , ( p , E’) : There is a g E L q ( p ,E‘) C L1(pl El) such that, in particular, ( T X A , z)= (h’0 11X A @ ) = L ( d w ) i z ) k ( d w )=

(L

gdP1 z) I

sA

hence T X A= g d p and it foflows, by C1.(3), that g is 8 Riesz-density for T. Conversely, take ‘p E L p ( p ,E)’ and define (as in the scalar case) an E’-valued measure M by ( M ( n ) z) , := (PIX A @ ) This measure (we omit the details Nince the result is not needed; see Diestel-Uhl [55], p.61) is of the type which has a density g E L l ( p , E ’ ) (see Dl.). For the sets Bn := { w E C4 I Ilg(w)IIE) 6 n) it follows that cpn defined by

coincides with ‘p for step functions f which are zero outside B,. This implies that IbnII I IId (in L p ( p , E ) ‘ ) and hence llgIIqdp)”q = IlV’nll I IId; Beppo Levi’s monotone convergence theorem and the theorem of Banach-Steinhaus show that the function g belongs t o Lq(plE‘) and that g represents cp . 0

(sB,

Pietsch Integral Operators D5. Integral operators T : E

-+

F are characterized by their factorization

Appendiz D: The Radon-Nikodim Property

522

(see 10.5.). In general, there is no such factorization with S taking values in F (see

D9.below for an example). If this can be achieved, then T is called Pietsch integral (or strongly integral). The class

PI(T) := inf{llRllll~llllsll

p3 of all Pietsch

integral operators together with

I T :E A L , ( ~ ) L

I

S

~1(p)

~ , finite p

1

is easily seen t o be a Banach operator ideal.

PROPOSITION : (1) 'J1 C y3 C 3 c

and all these inclusions are strict; moreover,

Pi 5 1 5 P I 5 N . ( 2 ) If F is A-complemented in F", then

p3J(E, F') = 3( E , F ) and PI(T) 5 AI(T) . In particular, P3(E,GI) = 3( E , C') holds isometrically. ( 3 ) If F has the A-extension property, then

P 3 ( E,F ) = 3( E , F ) = P l ( E ,F ) and PI(T) 5 APl(T) . In particular, T3(E,F) = 3( E , F ) = y l ( E ,F ) hold isometrically if F = too or L , ( p ) , (4) If T E 3, then the dual T' E y3 and PI(T') 5 I(T). (5) If T E 3(E,F ) , then ICF o T E y3 and PI(ICF o T ) = I(T).

PROOF:The only non-obvious statement in ( l ) , (2) and (3) is 'JJ3 # 3 # : this follows from Ex 11.5. and an example which will be given in D9..If T E 3, then T' f 3 (by 10.2.), hence (4) follows from (2). Since I(KFo T) = I(T) (the regularity of the operator ideal 3, see 10.2.), (5)is also a consequence of (2). 0 In general, T 4 '!433 if T' E '$3 : this would imply that P 3 = 3. For the same reason, the ideal 'J33is not regular.

D6. The Grothendieck-Pirtnch factorization theorem (see 11.3., corollary 2) shows that:

PROPOSITION: If K is compact and F a Banach space, then P l ( C ( W ,F ) = J(C(K),F) = W(C(K), F) holds isometrically.

In particular, the same relation holtlti for E = L , ( p ) (where p is an arbitrary measure).

Appendix D: The Radon-Nikody’m Property

523

The Radon-Nikodfm Property and Operator Ideals D7. The key to understanding the role of the Radon-Nikodym property in the theory C3.,which states that an operator

of Banach operator ideals is Crothendieck’s theorem T : L l ( p ) + F is representable if and only if

ToI:

C(I0

or Lmb)

is nuclear; moreover, N(T o I) = Riesa-density of T.

- I

J llglldp 5

T

L1(p)

Ilglloop(S2) =

F

IlTll IlIll whenever g is a

THEOREM: F has the Radon-Nikndkm p r o p e d y if and only if p 3 ( E , F ) = % ( E , F ) E.

for all Banach spaces

This is a consequence of t h n following proposition, which contains some interesting details concerning this result.

PROPOSITION: (1)If F has the Radon-Nikodim property, then the relations

Pi(C(K),F ) = 3((31

Tensor Norms and Operator Ideals

Tensor norms and operator ideals

Tensor norms and operator ideals