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
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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
1. Bilinear Mappings
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
1. Bilinear Mappings
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
1. Bilinear Mappings
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
E-G'.
4
T 1
Ex 1.6. Let G c E be a subspace and I : G L* E the isometric injection. Every (o E %iI(G,GI)is extendable to a E Qi[(E,G') if and only if there is a U E C ( E ,G") with U o I = IEG . Ex 1.7. A Banach space F is called injective if for every Banach space E l every subspace G C E and every T E C(G,F) there is an extension E & ( E , F )of T . If there is a constant A 1 1 such that 11fl1 5 AllTll can be achieved, then F is said to have the A-extension properly; F has the metric extension properly if A = 1. (a) tm(I')has the metric extension property for every set l'. (b) F is injective if and only if F is complemented in some &,(I?). If P : &(I?) -+ F 1 is a projection, then F has the IIPII-extension property. Hint: F ~ * t , ( B p ) .
14
I . Bilinear Mappings
Ex 1.8. If @ : K 3 x K 3 + K4 is Horowitz' surjective bilinear mapping which is not open in zero (see 1.7.) and E and F are normed, then
6 :( K 3x K
-
x E ) x ( K 3x F ) ( X I r zzr 23;Y1,Y2)
K4 x F
,I.(@
Yl), 22Y2)
is also continuous, surjective, and not open in zero. In particular, there are such mappings t, x 1, -+ t, for each p E [l1m]. Ex 1.9. Let 3 ( X ) be the space of all scalar-valued functions f : X + K with finite support [f # 01 := {z E X I f(z) # 0). Then
[fa # 01 = [f # 01 x [g # 01 in order to construct a bilinear mapping R2x R2+ R4 the range of which is not a linear subspace; other examples of this
is bilinear, Use
type are easily deduced from 1.3.(5) . Ex 1.10. Under what circumstances is a bilinear mapping injective? Ex 1.11. Let E, F and G be Banach spaces and a, 3 E B d ( E ,F;G). (a) Closed graph theorem: If 0 has a closed graph, then 0 is continuous. (b) Banach-Steinhaus theorem: If 3,(z, y) 4 3(z, y) for all (z,y) E E x F and all 3, are continuous, then 0 is continuous. Hint: In both cases show the separate continuity of @ . Ex 1.12. Show that for the canonical embedding K E : E c* E" of a Banach space and the canonical projection PE' : El" ---+ E' the following relations hold: (a) KL = P E ~ .
(b) P p 0 K$ = idEli. (c) P& = ICE" if and only if E is reflexive. (d) 6% = K E ~ Jif and only if E is reflexive. Ex 1.13. If 9 E %i[(E,E') is defined by ~ ( zz',) := (z',z ) then ~ L, = K E Ex 1.14. Let p E %i[(E,F).Show that L(,A) =
KFI
o
L, : E
-
L(EL,+,) = Pp o L :~ E l 1 -
.
F"' F' .
Hint: Use proposition 1.9. for ' 9 . Ex 1.15. Let @ E %iI(E1,E2; F) and let Ki C E; be compact convex sets; then for the set of extreme points the following holds: ext (conv(a(K1, K z ) ) )
c
QP(ext Kl,ext K z ) .
2. The Algebraic Theory of Tensor Products
15
Hint: Use the fact that the extreme points of conv(L) with L compact are in L and the concept of an extremal set D of a convex set C (i.e. art (1 - a ) y E D with a ~ ] 0 , 1 [ and 2, E C implies 2, E D ;see e.g. f701).
+
2. The Algebraic Theory of Tensor Products With the aid of a certain universal object the study of bilinear mappings can be reduced to the study of linear mappings. The construction of this new vector space E @ F is, at least for an analyst, very simple.
2.1. For an arbitrary set A define T ( A ) to be the set of all functions f : A + K with finite support, i.e. f ( a r ) = 0 except on a finite subset of A. For fixed cr E A the “&-unit vector” is the function e , E T ( A ) defined by e,(P) :=6ap
,
the Kronecker delta. It is clear that each f E F ( A ) has the unique representation
0EA
in other words: (e,),EA is an algebraic basis of F (A ) . Now take two sets A and B and consider, as in Ex 1.9.,the bilinear map q0 : F ( A ) x T ( B )
(fld
--
T ( Ax B)
f(-)s(-*)
-
Since e(,,p) = \ko(ea, e p ) , it is clear that span im8, = F ( A x B )
.
(Note that e , has a double meaning if A n B #jI- but this clearly does not cause any G) define T E L ( F ( A x B), G) by problems.) For ip E B d ( F ( A ) ,F(B); T ( e ( a , p ) ):= we,, e p )
(and linear extension). It is obvious that T is the unique linear map T ( A x B) with To\Eo=@ .
G
16
2. T h e Algebraic Theory of Tensor Products
It follows that
-
L(F(A x B ) ,G ) = BiI(F(A),F(B); G) T To*, is a linear isomorphism of vector spaces. I t is worthwhile to look at G := K :
F ( A x B)* = Bil(F(A),F(B))= L ( F ( A ) ,F(B)*) linear functionals on T ( A x B), the bilinear forms on F(A) x F(B)and the linear mappings F( A) + F(B)*coincide ! This fundamental algebraic relation allows one to consider certain operators T as linear functionals on a new space depending on the domain and range of T - and this idea is essential to the theory presented in this book.
- the
2.2. The universal property of the pair ( F ( Ax B), \ E d ) is expressed in the
DEFINITION: Let E and F be K-vector spaces. A pair ( H Iq,) of a K-vector space H and a bilinear map qo E Bil(E, F ; H ) is called a tensor product of the pair ( E lF ) i f f o r each K-vector space G and each 0 E BiI(E,F ; G ) there is a unique T E L ( H , G ) withO=To*,:
ExF-G
In other words,
L(H,G) T
--
Bil(E,F;G)
To*,
is injective, surjective (and linear) for all G. This easily implies the following properties:
PROPOSITION: (1) If ( H I8,) is a tensor product of (ElF ) , then span im 9,= H . (2) If ( H I ,*:) and ( H 2 , qk,2) are two tensor products of ( E lF ) , then there i s a unique linear isomorphism S : H1 -+ H2 with S o Q: = f&: . ( 3 ) If Ti : Ei -+ Fi are linear isomorphisms (onto) and (HI*,,) is a tensor product of (F1,Fz), then (HIQ,, o (TI x Tz)) is a tensor product of ( E l ,E2) . If E is a vector apace and
(x,),~A
an algebraic basis of El then
2. The Algebraic Theory of Tensor Products
17
is obviously an isomorphism (onto). The construction in 2.1. and the proposition give the Every pair (E, F) of vector spaces has (up to a linear isomorphism as in THEOREM: (2)) a unique tensor product ( H , Q,,).
It will be shown in 2.4.(4) that the space
E @ F : = 93i(((E’,a(E’,E)),(F*,a(F*,F))) with the bilinear mapping
ExF Y)
(2,
--
-
%i(((E’,a(E’,E)),(F*,a(F’, F))) [(C*l
Y’)
(z’,4(Y’, YN
is a tensor product of (E,F). This space has the advantage that it is constructed “naturally”, not with “coordinates” (i.e. with the help of a basis) - let us call it the tensor product of (E,F).Clearly, much of the power of the theory of tensor products stems from the fact that in specific situations other spaces H can serve as a tensor product - it is customary to speak of other “realizations” of the tensor product and to write
H=E@F - the bilinear mapping !Po : E x F + H being either obvious from the specific type of spaces or being given explicitly. In this sense
F ( A x B ) = F(A) @ F(B)
.
Recall other instances in mathematics where we are used to working with a universal object - without always thinking of a specific realization: the completion of a metric or normed space, the inductive or projective limits of spaces - or even the real numbers! The same will be done with E @ F . Before going on to various examples and methods of checking whether or not a space is a / the tensor product, note again the fundamental properties of tensor products:
BiI(E, F ;G)= L(E @ F,G)
(E@ F)’ = BQ(E,F) = L(E,F’) .
-
The tensor product mapping
ExF will be denoted by has the form
(2,y) v*
E@F
z @ y . By (1) of the proposition every element z E E n k=l
@F
18
2. The Algebraic Theory of Tensor Products
elements of the form z €3 y are called elementary tensors. If 0 E Bil(E, F; G) and T E L(E €3 F, G) with q z ,Y) = T(" €3 Y) 1 the then T is called the linearizafion of 0 (sometimes T is denoted by (PL) and associated bilinear mapping of T . From this one-to-one correspondence it is clear that defining a linear T : E 8 F ---+ G is the same as defining a bilinear 0 : E x F ---+ G. So definitions of T very often will look like
-
T:EQF-G c €3 y
and one must check whether
( 2 , y) u+
...
... is bilinear.
2.3. It is clear from the construction in 2.1. and (3) of the last proposition that for ~ B of E@F. bases (z,),€A of E and ( y p ) p E ~of F the family ( Z ~ € ~ Y ~ ) ( , , ~ )is~ aA basis This observation leads to the following test: CRITERION: Let E , F and H be vector spaces and Qo : E x F space (spanQo(E x F ) , Qo) is a tensor product of (ElF ) span im(\k,) = E @ F
(via
H bilinear. Then the
\ko)
i f and only if (QO(Zi,gj))i=1,..., n;i=l,..., m is linearly independent whenever and (yl, ...,ym) are linearly independent in E and F , respectively.
(21,
...,2,)
PROOF:The condition is clearly necessary. Conversely, let (z,),EA and ( y p ) p E ~be bases of E and F. Then, by the condition, (Qo(z,, Y ~ ) ) ( , , ~ ) ~ A ~isBa basis of the linear hull of \k,(E, F ) , and hence this space is isomorphic to F ( A x B) = E @ F and \ko serves as the tensor product map. 0 An easy calculation with the criterion gives
REMARK:If ( e a ) a € A is a basis of F with coeficient functionals e: : F E(A) := e a E ~ isEa tensor product of (El F )
+
lK,then
E @ F = E(*) via E x F 3 (z,y) u+ ((e:,y)2),E~ E E ( A ) .In particular, for every z E E @ F there exists a unique family ( 2 , ) E E(A)with
2. The Algebraic Theory of Tensor Products
Now assume z = ~ ~ =@ eaj l and ~ order j ( t l , ...,zn) such that of span(t1, ...,t n } . Then
19
(ti,
...,t k ) is a basis
hence the
COROLLARY: For every 0 (21, . . . , t n )
and
#z
E E @ F there are an n E
N and linearly independent
( ~ 1 ,...,gn) with n j=1
2.4. Some examples of tensor products (use the criterion or remark 2.3. - readers who are not familiar with tensor products should check the details; in order t o become acquainted with a mathematical object one also has to get used to it ...):
(2) If K and L are compact, then
C ( K )@ C ( L )=
This is why the notation g@h
:= g( .)h( ..)
is used for the product of two functions in different variables. The Stone-Weierstrass theorem shows that C ( K )@ C ( L ) c C(K x L ) is dense with respect to uniform convergence; is there a natural norm on E @ F which, in this case, is the sup-norm? For the polynomials
P ( F )@ P ( P ) = P(CF+") holds and for the entire holomorphic functions
20
2. The Algebmic Theory of Tensor Products
where, by Taylor-expansions, it is easily seen that 'H(P)@'H(P) is dense in 'H(P+") with respect to the topology of uniform convergence on compact sets.
(3) If S ( X ) is any vector space of functions on X of a special type (continuous, differentiable, integrable - or sequences with special properties) and E a vector space, then
Clearly, the notation g @ z for g(.)z will be used. (4) Let E and F be vector spaces. Then, using the criterion and Ex 1.2.(b) it is clear that 'L3i(((E', 4E'l E ) ) ,( F ' , 4 F ' l F ) ) )
is a tensor product of (ElF ) via the map
xBy:
E ' x F'
-
K
This was formally "the" tensor product of (E, F ) , see 2.2.. This means that z&y is t @ y and the notation 3 is not necessary. Nevertheless we shall from time to time use the notation xBy for the bilinear form on E* x F' since it has some advantages in the future; for example, if the mapping bilinear forms/operators/ functions tensor product were extended to larger spaces by density, the injectivity might be lost! This could happen easily in the following dxample as will be seen in section 5 .
-
(5)
E* @ I F= {T E L ( E ,F ) I rank(T)
< m} =: F ( E ,F )
via (x*,y)~.,[z'&y: z ~ * ( x ' x)y] , (still easy with the criterion). If E and F a r e normed spaces, then
E' @ F = {T E 2 ( E , F ) I rank(T) < m} =: $(El F )
,
the space of continuous finite rank operators. This identification will be often written as k=l
k=l
hence T ( x = ) C;=l(zi,t)yk; note, that by the very definition of the tensor product, this is independent from the special representation of z. If E or F is finite dimensional,
E'@F=C(E,F),
2. The Algebraic Theory of Tensor Products
21
in particular, @ Krn= q K " , Krn) = K""
K"
,
the space of matrices as was already seen in example 1. In summary, the basic examples of tensor products, also from the point of view of applications, are vector valued functions functions of two variables finite rank operators
.
-
2.5. If E is a vector space, then the evaluation map
E' x E
K ;
(z*,t)-(z*,~)
has a unique linearization in (E' @ E)' : it is called the trace t r E on E evaluation
E'xE-K F ( E , E ) = E' 8 E
To check that this is the usual trace of matrices for dim E = n, take a basis ( e l , ...,en) of E with coefficient functionals e ; . For T E L ( E , E) with
and representing matrix ((e; ,T e k ) )
n
rn
n
k=l
k l
k=1
hence the trace of T is the sum of the diagonal elements of the representing matrix of the mapping T. 2.6. Take E and F vector spaces and 'p E (E @ F)* = BiI(E, F ) = L ( E , F') with associated operator T = L,. Using the transposition (see Ex 2.1.), the identification
E@F z
--
F @ E=~~~I((F*,u(F',F)),(E',u(E*,E))) -F(F*,E) Zt
2. The Algebraic Theory of Tensor Products
22
is as follows: For z =
xfl
zk @&
the associated h e a r map is n
S : F'
4
E
t
S(Y*) =
C(Y*! / k ) z k 9
k=l
cE!l
or S = &@!k. The duality bracket between E @ F 4 T (F', E ) and its algebraic dual (E@F)' = L ( E , F') turns out to be the so-called trace duality between operators:
TRACE DUALITY: Let p E (E@F)*with associated map T E L ( E , F') and z E E @ F with associated S E T ( F ' , E ) . Then
.
(p,z ) = trE(S o T ) = t r p (T o S )
PROOF:By linearity, it is enough to check this for elementary tensors z = xo €9 yo but this is easy since (cp,
20
€9 Yo) = (Tzo,Yo) S O T = T * y o z o: E
E ToS=yo~Tzo:F*+F' +
.
0
Note that transposition z u* z' means going to the dual of the associated operator (see Ex 2.2.) - more or less: minor difficulties come from the domain and range spaces (see
Ex 2.3.). 2.7. The tensor product of operators is defined by the
PROPOSITION: Let E L(Ei, F,) for i = 1,2. Then there is a unique operator S E L(E18E2, F1@F2) with the p r o p e r t y S ( z l @ x t ) = 2'1+1@T222 for all ( z i , 2 2 ) E Ei xE2. The operator S i s denoted b y TI 8 T2, hence
PROOF: This follows from the diagram
and the universal property of tensor products. 0
PROPERTIES: (1) If
1'7
and
T2
are surjective, then TI@ T2 as Surjective.
2. The Algebraic Theory of Tensor Products
23
(2) If TI and T2 are injective, then TI @ T2 is injective. (3) In particular, if G c E, then G @ F is a subspace of E @ F . (4) The following formula holds: ker(T1 @ T2) = (ker TI) @ E2
+ El @ (ker T2) C El @ E2 .
PROOF: (1) is simple.(2) follows from the criterion 2.3.using a basis of the form zo@yp on El @ E2. To see (4) decompose
,
El = (ker 7'1) @ GI
E2
= (ker Tz) @ G2
and take z = Enz; @ z i E ker(T1 @ Tz) . Then z i = u6 Eng i @ g i = 0 ; therefore z
=
C(Ur,+ g:)
@ (ui
+ g f ) = C u:
+ g i and, by (2) and (3), 2
8 (ui + g f ) + C g k @un
n
n
E (kerT1) @ E2
n
+ El @ (kerT2) .
The other inclusion is obvious. 0 An interesting special case of TI @ T2 is given by zz E E' and T E L(F, G) :
x: @ T :E @F -+ 1K @ F = F
Exercises:
Ex 2.1. Show that the transposition map z
.r)
tt
defined by
is an isomorphism. Ex 2.2. Show that zEEt@F-2(E,F)3
$ 1
zt E
F @ E'
-
T
C(F', El) 3 T'
24
2. The Algebraic Theory of Tensor Producta
is commutative. Ex 2.3. Show that
5E 1
Z*
1 3
E " 8 E' = $(E', El) 3 T ' O K E I
commutes. This means in particular, that t f E l can be considered as being defined on the space S ( E , El'). Ex 2.4. (a) Take two separating dual systems (Ei, Fi). Then the mapping
is bilinear and defines a separating dual system
(b) In the case Fi = E: use E; 8 E,' C (El 8 E2)* to show that the duality bracket is just the restriction of the trace duality. This is why the pairing in (a) is also called trace duality. (c) For normed spaces E and F the dual system (E 8 F, El @ F') is separating. Ex 2.5. Let E and F be finite dimensional normed spaces, z E E 8 F = S(E', F) with associated operator S and p E ( E 8 F)* = 2 ( E ,F') with associated operator T.Then
Ex 2.6. Let E and F be normed. Show that E 8 F = {T E C(E', F ) I T weak-*-continuous,
rank(T)
< XI) .
& ( p ) be the space of p-integrable functions (on a measure space (a,p ) ) and & ( p ) the Banach space of equivalence classes of functions. Show that for every normed space E
Ex 2.7. Let
& ( p ) 8 E = {f : R -+ E I f Bochner integrable, dim span f(R) < m) Li(p) 8 E = & ( p ) 8 E / -
-
where stands for the "almost everywhere equal" equivalence relation. Ex 2.8. Show that for 1 5 p 5 00 and E normed
tp8 E = {(tn) E t P ( E )I dim span{z,} < XI}.
2. The Algebraic Theory of Tensor Products
Describe
co @ E
25
.
Ex 2.9. If E = El
@ E2, then
E €4 F = (El @ F ) @ (E2 €4 F ) ; more generally,
Is something like this true for infinite products? Ex 2.10. (a) If G is a Gvector space, define G to be the Gvector space with the same addition and the new scalar multiplication
Use this construction to show: If E l F,G are Gvector spaces and 0 : E x F + G biantilinear (i.e. biadditive and ~ ( A z ,y) = a(., Ay) = x@(zlY))~ then there is a unique antilinear T : El @ E2 -+ G such that
(b) Recall that the Riesz-identification RH : H H' of a complex Hilbert space with its dual is antilinear. Then there is a unique antilinear map
satisfying (RH€3 R K ) ( z @ y) = RHX@ RKY for all x E H and y E K. Ex 2.11. The formation of tensor products depends upon the underlying field. More precisely: Let Ec and Fc be complex vector spaces and ER, FR the same spaces considered as real ones. Show that there is a natural map ER @R FR -, E c Fc which is R-linear, and surjective but, in general, not injective. Ex 2.12. The mapping E -+ E @E defined by Q(z) := z@z is in some sense a typical quadratic form. (a) Show that Q is nearly injective in the following sense : if z+y # 0 and Q(z) = Q(y), then t = y. Hint: Use symmetric bilinear forms on E @ E , i.e. those for which (9,u @ v) = (p, v @ u), or think in terms of operators. (b) If p E %iI(E,E ) is symmetric, then the two canonical extensions to E" x E" (see 1.9.) coincide on the diagonal: ( A p ) A ( d l , d= ~ ) "(("(d',~'') for all x" E E". Hint: ("(P)~ = p" in this case. EX 2.13. Define a tensor product for n-linear mappings El x . .x En-+ G and denote it by El €4 + + - @ EShow n . that
-
for each 1 5 k
5 n-I
26
9. The Projective Norm
3. The Projective Norm The projective norm A on E 8 F has the following highly desirable universal property: For normed spaces E , F, G each bilinear mapping E x F G is continuous if and only if its linearization E 8=F + G is continuous.
3.1. Before going through the basic calculations, recall that if C c E is an absolutely convex subset of a vector space, its Minkowski gauge functional mc is defined by mc(z) := inf{A 2 0 I z E AC}
E [O,m]
It is easy to check that mc is a seminorm on the linear span of C which coincides with {z E E I mc(z) < 00) ; notation:
[q:= (span C, m c )
.
mc is a norm if C is linearly bounded (i.e. bounded on every 1-dimensional subspace). It follows that (*)
(1 - E)C C&C
c c B[C] c (1 + E ) C
for the open and closed unit balls and each E > 0 . If a Banach disc.
[qis a Banach space, C is called
If A1 C El and Az C EZ are subsets, then
This seems to be confusing since in the case where the Ai are subspaces this is not the tensor product of these two spaces - it is in general not linear! But in practice it will always be clear what is meant. 3.2. Take iI, E Bil(E,F; G) for normed spaces E, F, G and denote its linearization for the moment by OL E L(E 8 F, G). Then
27
3. The Projective Norm
DEFINITION: The projective norm A on the tensor product E 8 F of two normed spaces 0 0 E and F i s the Minkowski gauge functional of r ( B E 8 BF). The name “projective” will become clear in 3.8.. Notations: for z E E 8 F
E & F := ( E 8 F,*) and E&%Ffor the completion. Actually A is a norm and has an alternative and more convenient description (which the reader should always keep in mind):
PROPOSITION: A is a norm on E 8 F and can be calculated
c N
n(z;E , F ) = inf{
as follows:
N
lltnll llYnll
I N E JV , z = C zn 8 Yn}
=
n=l
n=l
Moreover,
% i [ ( E ,F ; G ) = 2 ( E
(1)
F, G )
holds isometrically - and A is the unique seminorm on E 8 F which has this property . In particular,
for G = K
a(%;E , F ) = m={I(S,
.>I
I s E 2 ( E ,F / ) ,IlSll I11
1
the duality bracket given by
(E
F)’ = %i[(E,F ) = 2 ( E ,F’)
,
which (by 2.6.) can be calculated b y trace duality. 0
0
(2) The open unit ball in E 8=F is r ( B E 8 BF).
(3) (4)
4t 8 Y;El F ) = 1141IlYll A
for all (2,Y) E E x F satisfies the metric mapping property: If Z E 2(Eil Fi)l then llT18 7” : El 8 r E2
(5)
A
-
FI 8=F2ll 5 111;11 llT’ll
is finitely generated, i.e. for all z E E 8 F
E , F ) = inf{*(z; M , N) I z E M 8 N ;M C E and N C F finite dimensional}.
A(%;
28
3. The Projective Norm 0
0
By definition r is a seminorm on E @ F = span r ( B E PROOF:
B F ) . The formula (**) for the norm in %iI(E, F; G) and property (*) of 3.l.for the unit balls of a Minkowski gauge functional imply that for each CP E BiI(E, F ; G ) @
in particular: %iK(E, F) = ( E @= F)' isometrically. To see that A is a norm note that by 2.4.(4) and the Hahn-Banach theorem the following holds: for every 0 # z E E @ F there exists an 2' @ g' E E' @ F' c %iK(E,F) with (2' @ g',z) = 1 (this was Ex 2.4.). Hence (E @ F, %iI(E,F))is a separating dual system, which implies that A is a norm. Clearly, every seminorm 11 on E @ F for which (E @ F, a)' = %iK(E, F) holds isometrically, has to be equal to A. So (1) has been proven. To see (2), observe first that, again by (*) of 3.1.,
Conversely, if z
= cE=, a k z k @ yk E&('I (1
for k = 1,...,n . Hence (1
+
b ~ )then , there is an E > 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
3. The Projective Norm
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-
9. The Projective Norm
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
3. The Projective Norm
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
...
3. The Projective Norm
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
3. The Projective Norm
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
9. The Projective Norm
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:
3. The Projective Norm
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
3. The Projective Norm
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 .
3. The Projective Norm
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
3. The Projective Norm
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
3. The Projective Norm
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
3. The Projective Norm
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
9. The Projective Norm
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
3. The Projective Norm
(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
3. The Projective Norm
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
4. The Injective Norm
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
4. The Injective Norm
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
4. The Injective Norm
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”
4 . The Injective Norm
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)
4 . The Injective Norm
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
4. The Injective Norm
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 ) .
5. The Approzimation Property
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
5. The Approximation Property
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).
5. The Approximation Property
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
5. The Approximation Property
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
5. The Approzimation Property
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
5. The Approzimation Property
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
5. The Approximation Property
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
6. Duality of the Projective and Injective Norm
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
6. Duality of the Projective and Injective Norm
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.
~
6. Duality of the Projective and Injective Norm
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:
6. Duality of the Projective and Injective Norm
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
6. Duality of the Projective and Injective Norm
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
6. Duality of the Projective and Injective Norm
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 $(F', E') , A E C(E', M I ) and suppose that M is finite dimensional. Then, given E > 0, there is an operator R E C ( M ,E ) such that IlRll _< (1 ~)llAlland R' o T = A o T . Hint: Apply the weak principle of local reflexivity to S := A' and N :=im(T). Ex 6.3. Check that the Johnson spaces C, have the metric approximation property if 1sp<0O. Ex 6.4. Lindenstrauss' compactness argument: Take Banach spaces E and F , an operator T E C(E,F ) and a net ( F a ) of subspaces of F , directed by inclusion, such that F,, := U, F, is dense in F . Let S, E Z(F,, E ) be operators such that TS,y = y for all y E F, and lim sup IlS,ll = A < 00 . (a) Define pa : E' x F, + K U ( 0 0 ) (onepoint compactification) by
+
Use the fact that (pa)is a net in the compact space (KU ( 0 0 ) ) ~ ' " ~ -(hence has a convergent subnet) to show that there is a (o E %i[(E',F,) such that IIp11 5 A and
for all s/ E F' and y E F,
.
7. The Natural Norm on the p-Integrable Functions
77
(b) Let U E C(E‘,F’) be the operator associated with 9. Show that UT‘ = i d p
.
(c) T’ is invertible on T’(F’) C E‘ and ll(T’)-lll 5 A ; the subspace T’(F’) is complemented in E’ with a projection of norm 5 AllTll . This result is quite relevant as the next exercise will show. The technique was refined to the so-called ultraproduct technique; see section 18, in particular Ex 18.9..
Ex 6.5. If F is a separable Banach space, then F’ is isometric to a 1-complemented subspace of each Johnson space C, . Hint: Choose an increasing sequence ( F k ) of finite dimensional subspaces and T n : Gn(k) -, Fk with IlTnll = 1 and llT;lll 5 l + l / n ; take in the foregoing exercise T = C Tn E 2(Cl ,F) and S, := T;l. This result is due to Johnson [130] . Ex
6.6. Use the principle of local reflexivity to show that a normed space E is an 2:,A-space if E” or the completion ,!? is f& . The full duality will be proved in section 23; for the converse of the statement about the completion see Ex 4.7..
Ex 6.7. Every abstract M-space is an 2$,1-space . EX 6.8. (a) Show that there are elements rn E P . , 8 Pm with
n(zn;e“,,e”,)1 f i
,
E(zn;e”,,P,)
5 1 and
Hint: Ex 4.3..
(b) If K and L are infinite compact sets, then there are llfnlloo = 1 and r(fn; C ( K ) ,C ( L ) )2 f i
fa
E C ( K ) 8 C ( L ) with
7. The Natural Norm on the p-Integrable Functions
The third type of natural norms for tensor products is the norm induced on L, 63 E from L , ( E ) . It behaves fundamentally different from E and n in one very important aspect: It does not satisfy the mapping property! Therefore, this section studies some additional conditions under which tensor product operators are continuous with respect to A, and gives a first look at the highly interesting consequences of this “failure”.
7.1. Let (52,p) be an arbitrary measure space, 1 5 p 5 00, and E a normed space. Then for p # 00 the space of (classes of a.e. equal) Bochner pintegrable functions R -+ E is denoted by L,(Pl@
and the space of (classes of locally a.e. equal) bounded p-measurable functions R
-+
,!?
78
by L,(p,
7. The Natural Norm on the p-Integmbte Functions
8),see Appendix B12.. Since the natural mapping LP(P)@E
f@z
4
L,@) f(-)z
is injective, the definitions
for 1 5 p < 00 and
A,(& L,, E ) := ess-sup
11f(.)ll~
give norms on Lp@ E; notation: L p ( p )@ A ~E and L,(P)&A$
for the completion.
PROPOSITION:
(1) (2)
(3)
A1 = a on Ll(p)@E . A , = € on L m ( p ) @ E . E < A , < ~on L , ( p ) @ E .
PROOF: (1) was one of the main examples for the projective norm (see 3.3.) and (2) was Ex 4.9.. For (3) observe first that for g @ z E L,@E the relation Ap(g@z)= [ ~ Y ~ ~ L ~ ~ implies A, 5 ?r by the triangle inequality. To see E 5 A, take h E BL,, (conjugate exponent), z' E BE' and fix f E L, @ E ; then
7.2. For 1 5 p < 00 the step functions
are dense in Lp(plk ) . This implies that L, @ E is dense in
L,(P)&,E
Lp(8)and therefore
=~ ~ ( p ) & A= , k~ p ( p ,
holds isometrically. For the product measure theorem shows that
p1 @ p2
on R1 x
$22
the Fubini-Tonelli
IZ~~E
7. The Natural Norm on the p-Integrable Functions
79
holds isometrically. Since the step functions in S(p1)@ S(p2) are dense in Lp(pl@p2), it follows that this injection has dense range and therefore isometrically 1
A
L ~ ( p l@ p2) = Lp(pl)~ApLp(p2)Lp(P1,Lp(P2))
(1 _< p c m). For p = 00 this is false (Ex 4.8. for discrete measures and Ex 7.2. for the general case). Recall from Ex 4.9. that, in general, Lm @ Eis not dense in Lm(k). Finally, it is convenient to define the “transpose”Ak of A, : 1
:=
E @ A ; Lp(p) x@f - the
L~(/l) @Ap
f@x
hu*
isometry being the transposition map. Clearly, Lp(p) @A; Lp(v)
1
= Lp(p)
@A,
Lp(v)
Lp(P @ v)
holds. is The norm in L,(Qi,p1) @A; LP(R2,p2)
and hence the continuous triangle inequality (see Appendix B5.) gives the
PROPOSITION: For measures p1 and p2 (dinerent from zero) and 1 5 q
[lid&
-
@ idLp : Lq(p1)@Aq Lp(p2)
5 p 5 00
Lq(p1)@A; Lp(p2)ll = 1 .
The case Am = E is trivial. This proposition has an interesting consequence for the continuity of operators S @ T : L, @ L, -, L, @ L,, see theorem 7.9..
7.3. Take operators S E 2 ( L p ( p ) ,Lp(v))and T E .C(E,F). Is it true that S 8 T Lp(p)@
AE ~
-
Lp(v) 8~~F
is continuous, in other words: does the mapping property which holds for s and E (see 3.2. and 4.1.) also hold for A, ? There are natural examples showing that this is not true for 1 c p < 00 (see 7.5. and 7.6.). But at least in some cases S @ T is continuous: Take S = idLp and T E 2 ( E ,F). Then
7. The Natuml Norm on the p-Integmble Functions
80
THEOREM: Let p and v be arbitrary measures, p , q E [l,m] and T E 2 ( E , F ) .
-
(1) If S E 2(Lp(p), Lq(v))is positive, then
= llsll ( 2 ) I f # is compact and S E 2 ( C ( K ) A((.)) , positive, then
11s@ T : L p b ) @Ap E
11s8 T : C ( K )8 c E
Lq(v) @Aq
FII
Lg(v) @Aq
Fll
-
= llsll
llTll
llTll
*
.
Since positive operators are involved, it is worthwhile t o note that the real and the complex case are treated jointly. Before beginning the proof recall the obvious (see also Ex 3.4.)
DENSITY LEMMA: Let 11 Ili be two norms on a vector space E with 11 112 5 cII 111for some c > 0 and T : E + F a linear map into a normed space F . IfT is 11 111-continuous and for some X 1 0
llT4lF I41412 holds for all z in a
11 Ill-dense
-
subset o f E , then
IIT : ( E , II 112)
Fll 5
-
PROOF of the proposition: Since S 63T = ( idLe(”)@ T )o (S@ i d E ) , it is enough t o take T = i d E . For G = L p ( p ) or C ( K ) ,clearly S@idE
G@=E-
Lq(v) @r E
Lq(v) @Aq E
is continuous, hence, by the density lemma, it suffices to check the continuity-inequality on a n-dense subset of G @= E . For (1) it is therefore enough to take step functions n m=l
with pairwise disjoint A , : Since ISx, I = Sx, , it follows that
7. The Natuml Norm on the p-Integrable Functions
81
- with obvious modifications if p = 00
(the case q = 00 follows also from 6 5 Ap ). For (2) take a partition ( p m ) of unity in K such that there are tm E K with p m ( t m ) = 1 and put n
f := C p m 8 Z m E C ( K )E~ m=l
(these functions form a dense subset of C ( K )@= E); then
and therefore
It is obvious that S@T is also continuous if the operator S E 2(Lp(p), L4(v)) is regular, i.e. S = S1 - S2 in the real case, or S = S1 - S2 i(S3 - S4) in the complex case, where the operators Sj are positive. The following result (due to Virot [275]) shows that this is best possible:
+
PROPOSITION: Let p , q E [l,001 with q < 00 and S E 2 ( L p ( p ) ,L,(v)) such that
i s continuous.
Then S i s regular.
The analoguous result holds for operators S E 2 ( C ( K ) ,L,(v)).The proof will show that both results also hold for q = 00 if v is localizable, i.e. L m ( v ) is order complete. Since A1 = A and A has the metric mapping property, the proposition shows that every Ll(v) is regular. See also Ex 7.19.. operator L1(p)
-
PROOF:Considering
cL=l
fk @ e k ,
it is obvious that continuity means that
The type of order completeness of L,(v) (see Appendix B5.) shows that S throws order bounded sets into order bounded sets. But this implies that S is regular (see e.g. Schaefer [244], p.229 and p.233). 0
7. The Natural Norm on the p-Integrable Functions
82
7.4. With regard to subspaces and quotients the norm Ap is well-behaved:
PROPOSITION: Let p be a measure and 1 5 p 5: 00. Then
respects metric injections and metric surjections.
PROOF:For p = 1 or 00 this was shown in 3.8., 3.10., 4.3. and 4.4. (recall that L , ( p ) = C ( K ) for some compact K). That L p ( p ) @A,, respects metric injections follows (for 1 < p < 00) directly from the definition of the norm. Now take Q : E --y F a metric surjection between normed spaces; by 7.3.
is continuous with norm 5 1 and clearly surjective. For every g E Lp @A, F and E it is necessary to find f E Lp @bP E with id @ Q(f) = g and
AP(f) I ( 1 +&)Ap(g) If g = Xk
X A @ ~
yk is a step function (the
>0
*
Ak pairwise disjoint), then there are . Hence f := XA,. @ zk covers
E E with Q(zk) = yk and 11zkl1 5 (1 + c)llykll
ci=l
g and
In other words, id@Q
sp(p) @Ap E
S p h ) @Ap F
(induced norms) is a metric surjection. The result now follows from the following lemma which is of independent interest (for the condition on the kernels see Ex 7.7.):
QUOTIENT LEMMA:Let E and F be n o m e d spaces, Q E C ( E ,F ) surjective, E, c E dense and Q0 := QIE. : E, + Q ( E , ) the (surjective) restriction. Then Q0 i s a meiric surjection i f and only if ker Qo = ker Q and Q is a metric surjection. PROOF: Assume first that Qo is a metric surjection and take z E kerQ. Then there are z, E E, with z, -+ z and, by assumption, z, E E , with Q0(zn) = &(zn) and
It follows that z,
- z,
E ker Qo and zn - z, + z,hence
k e r Q ~ m.
83
7. The Natural Norm on the p-Integrable Functions
Both implications of the claim now follow from the simple observation that
7.5. As was already mentioned, the operator s @ idE : L, @A, E 4 Lp @A, E is not always continuous. To see a concrete example, take S to be the Fourier transform 5 : L 4 R ) 4L2(R) defined by
[5(f)](t) := (2r)-’I2
Irn
exp(-ist) f ( s ) d s
-W
i f f has compact “support” - and then continuous extension; normed spaces E, consider the E-valued Fourier transform
5 is an isometry.
For
If H is a Hilbert space, a simple calculation with scalar products shows that 5~ is an isometry (see Ex 7.8.); considering H as some La, this also follows from 7.4.. But if E = 1 1 , the operator $11 is not even continuous: For the unit vectors en in el take
then
However, from
[z(x[a,b[)] ( t ) =
a
(exp(-ibt) - exp(-id))
it follows that
-
[exp(-i(m
(1” (C 6
1 --
1 7 -wt
“
rn=l
lexp(-i(rn
+ 1)t) - exp(-irnt)]
ernllildt)
+ 1)t) - exp(-imt)l)
112
2
dt)
112
=
=
84
7. The Natural Norm on the p-Zntegmble Functions
This shows that the operator 3cl = $ 8 idcl cannot be continuous. For the Fourier see Ex 7.18.. transform on &(Z) A celebrated result of Kwapieli [161] says that a Banach space E is isomorphic to a Hilbert space if and only if SE is continuous; this will be shown in section 30. 7.6. Another example is given by the discrete Ailbed transform A : tp(%) for 1 < p < 00 which is defined by
-+
lp(Z)
Due to a result of M. Riesz A is continuous (see for example Edwards-Gaudrey [62], theorem 6.7.4.), but A 8 idcl is not: Take zn := e k 8 e k E lp(zZ) 8 11; then A , ( ~ ~ ; 1 ~ ( Z ) , . t=1 )nilp and
n
n
-
Therefore
and since (l/n) log(n!/2)
+ 00
8 idtl cannot be continuous.
the operator
7.7. Though at first glance it is disappointing that Ap does not satisfy the mapping property, Kwapieli’s result shows that this behaviour gives new structural insights. It even can be said that this “misbehaviour” is an incredibly rich source for interesting properties of spaces and operators in modern Banach space theory: One of these sources is to study, for a fixed operator So E i ! ( L P ( p ) ,Lq(v)), the class of all those operators T E 2 ( E ,F) such that so
C3
-
: Lp(P) C ~ E A ~
Lq(v) @A, F
is continuous - and those spaces E for which idE is in this class.
To see an example of this kind, consider the Rademacher functions rn E &[O, 11defined in 1.5.(2). They form an orthonormal sequence, hence
85
7. The Natural Norm on the p-Intepble Functions
is a complemented subspace of L2[0,1]. If E is normed, define A2 on Rad B E to be the norm induced from Rad 8 E -+ L2[0,1] 8~~E . This means that
is the norm of f = C,"==l rncn in Rad @.a2E and not the norm coming from via the identification I :t 2 Rad (an)
-
e2
C3 E
Canrn
of Rad with t 2 . Actually it is this difference which counts: An operator T E C ( E ,F) is called of type p (for 1 5 p 5 2, hence Ip : tp4 t 2 L R a d is defined ) if 1 1 1 , C 3 T : ep @A, E
and of cotype q (for 2 5 q
llJ,
5 00,
-
L2[0,1] @A2 FII =: Tp(T) < I-'
hence J, : Rad-
8 T : Rad @ A 2 E
-
t,
Q
t, is defined) if
1, C ~ FII A ~=: C , ( T ) < 00
.
A normed space E is called of type p (resp. cotype q ) if idE is of type p (resp. of ) for the cotype). These notions are due cotype q ) and T,(E) := T p ( i d ~ (analogously to Maurey [285]and Hoffmann-Jfirgensen [106]. An appeal to the density lemma 7.3. shows that the inequalities for the mappings to be continuous need only be verified on r-dense subspaces of the original tensor product - for example, ep can be replaced by the dense subspace of finite sequences and E or Rad by any dense subspace. In particular, T E C ( E , F) is of type p if and only if there is some c 2 0 such that
for all
21,...,cn E
El and of cotype q if and only if
Since A, = A and Am = E , every operator/space has type 1 and cotype 00. In the exercises it will be shown that Hilbert spaces are of type 2 and cotype 2, for the converse see section 30 and also Ex 30.l.(b). Moreover, infinite dimensional C ( K ) have neither cotype < 00 nor type > 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
7. The Natural Norm on the p-Integrable Functions
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;
.
7. The Natural Norm on the p-Integrable Functions
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 < p 5 oo nor cotype q for 1 I q < 2 in the sense of the inequalities a t the end of 7.7.. Ex 7.16. Let E be a normed space. (a) For each finite dimensional subspace M c E and E > 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
<
~ l z n l l ~ ~ ) ”00 ~
n=l
(the notation will be used for finite sequences as well); in other words, (Zn)
E t p ( Q = tp&A,E
.
< m)
8. Absolutely and Weakly p-Summable Series and Averaging Techniques
91
p = 00 is just the case of bounded sequences: & ( E ) , and it is known (Ex4.8., Ex 7.2.)
that
L&E
= e&&,
E
c e&) #
.
The sequence ( z n ) is called weakly p-surnmable if
n=l
-
for all x' E E'; this means that the operator
T : E' 2'
lP
-((z',
zn))
is defined. The closed graph theorem shows that this operator T is bounded and hence wp(zn; E ) := wp(zn) := llTll < 00 :
defines a norm on the space q ( E ) of all weakly p-summable sequences in E . This formula shows that for each norming set D C BE'
Note that T' = Cen&znand T'(tPt)C B if 1 < p < 00 ; for p = 1 one has T'(c,) C k . The notation wp(zn)will also be used for finite sequences. The case p = 00 is, as before, the case of bounded sequences and w,(zn; E) = l m ( z n ; E ) .
PROPOSITION: If D C BE' is norming and 1 5 p _<
00,
then for all
21,
n
wp(zk; E ) =
SUP l l ( ( z ' , z k ) ) l l l ; Z'ED
=€
( C e k C?J X k ; $ , k=1
PROOF: Since
and
the relation
E
= wp on $ @ E follows from the definition of E . 0
E)
.
...,z n E E
92
8. Absolutely and Weakly p-Summable Series and Avemging Techniques
The following inequalities are simple but important: If T E C ( E ,F) and (zk)in E , then l p ( T z k ;F ) 5 IITllep(zs;E ) and wp(%; F) 5 llT1lwp(zk; E ) . 8.2. In order to describe the weak sequence spaces in terms of operators define the
closed subspace
$+'(E) := {(tn) E c ( E ) 1 W~((Z,);'N;E)
-
O for N
-* 00)
and give it the induced norm from c ( E ) . Clearly, Q o ( E )is the space c,(E) of zero sequences in E. It is easy to see that
E'&t, = f f ( E , ! , ) c C ( E , t , ) = t,(E') E'6eco = R ( E , c,) C C(E,c 0 ) = { ( x k ) E !,(El) ea&E = ff(&,E)c C ( t , , E ) = t , ( E )
I zk
+
0 weak*}
hold isometrically.
PROPOSITION: Let E be a Banach space. ( 1 ) The following relations hold isometrically: for l < p < 0 0
~ ( E ) = C ( ~ ~ ~ , E ) ~ R ( ~ ~ ~ , E ) = $ ~ O ( E ) = ~ ~ ~ ~ E
q ( E ) = C(c0,E ) 3 ff(c,, E ) = q ' " ( E ) = 4 & E . The identification is given by (xn)- Cz'I en&xn; the sum converges pointwise in C and with respect to the operator norm if it represents a compact operator. (2) Dually: c ( E ' ) = C(E,.tp)3 f f ( E , $ ) = c ? " ( E ' )= E'&eP hold isometrically for 1 5 p < 00; the identification is given b y convergence as in (1). ( 3 ) For p = 00 the following equalities hold isometrically: {(x,) E t , ( E ) I {z,} relatively compact} = e,&E c,(E) = co& E .
PROOF: That g ( E ) L, C(epl,E) (or C(co,E)) via S :=
llSll=
(zk)-+C;=l
xLsen,
= ff(t1, E )
en@,
and
wp(zn; E )
was already shown when defining wp(z,; E) in 8.1.. The norm relation also shows that S is compact if w P ( ( x n ) ~ ! , -+, ) 0. Conversely, take S 6 C(tpl ,E) (or C(c,, E)), define
x , := Sen and observe
8. Absolutely and Weakly p-Summable Series and Avemging Techniques
93
Therefore, C ( E )= C(t,/, E) holds isometrically; moreover,
t, atE
Q
C p o ( E ) ff(lp E/) ,= $&E Q
holds isometrically (tphas the approximation property, 5.3.) and (1) is proven.
For (2) take (z;) E C ( E ' ) . Then
this shows that
C ( E ' ) Q X ( E , t p ) and G+'(E')
L,
ff(E,tp)
hold isometrically. If T E C ( E , t p )and z; := T'ek (where the e', are the coordinate functionals), then, clearly, Tz = ((z;,~))and C ( E ' ) = X(E,tp)follows; the density of E' @tpin the space of compact operators gives, as in (l),the result. The first relation in (3) follows from
while the second is a consequence of the facts that
c,
at E
4
co(E)
holds isometrically (since both are isometrically in t , ( E ) ) and that c, @ E is dense in the Banach space c,(E): Just cut off the tail of the sequence! 0 One might be tempted to think that a description like that in (3) also holds for the space t p & E = c i o ( E ) if 1 5 p c m; but this is not the case. Take E = co and (2,) the following zero sequence in c,:
e1,2-1/pe212-'jPe2,3- '/peg, 3-'/pe3,3-'/~e3,
... ;
C(C,)
then (2,) E but ( z n ) 4 $*'(c,) . In Ex 8.4. a related question is treated: When is every (zn) E C (E) a zero sequence in E ? 8.3. A sequence (xn) in a normed space E is called unconditionally summable if for each bijective q : N -+ N the series C(zvcn,) converges; this is easily seen to be equivalent to the convergence of the net of finite sums
94
8. Absolutely and Weakly p-Summable Series and Averaging Techniques
where I runs through the directed set of finite subsets of N.
PROPOSITION: Let E be a Banach space. Then a sequence (xn) in E is unconditionally summable if and only if it is in G ' " ( E )= I ! ~ & ~. E PROOF: Observe first the simple fact (decomposition in positive, negative, real, imagAnl 5 1 for all finite inary parts) that if (An) is a sequence of scalars such that ICnEI I C N ,then C,"=lIAnl 4. Now assume (xn) to be unconditionally summable. Then there is an no E IV such that llxnEI Zn 5 E for all finite I C {no,no 1, ...} and therefore lCnEI(t', zn)l 5 E for all these I and x' E BE' : it follows that
+
11
m
Viceversa: If w l
((.n)r!,,)
5 E , then for all finite I c {no,no+ 1, ...}
Bessaga-Pekzyriski proved (see Diestel [54], p.45) that a Banach space does not contain co isomorphically if and only if each weakly summable sequence is unconditionally
summable. Hence, it was not by chance that the example at the end of 8.2. was constructed in the space co. See also Ex 8.4.. 8.4. The following powerful tool will be used frequently:
Take a probability measure p on a set 52 and an orthonormal sequence of functions fn E L ~ ( / such J ) that span{ fn} C L p ( p ) for all 1 5 p < 00 and the norms coming from the L p ( p ) are equivalent:
Clearly, when investigating sequence spaces or function spaces these inequalities may be very useful. Additionally, the orthogonality has another quite interesting aspect: If z = C:=, x, @ y, is an element in the tensor product of two normed spaces, then n
n
8. Abaolutely and Weakly p-Summable Series and Averaging Techniques
95
- and the element z is represented in a new way as a sum/integral of elementary tensors; note that the Bochner integral in this formula exists in the finite dimensional space span{zm}@ span{y,). It will be seen that in many cases this new representation, together with the norm equivalences turns out to be extremely fruitful! The method is called the averaging technique. As is already indicated by the definition of type and cotype, averages of the form
for elements 2 1 , ...,z, in a normed space E will pIay an important role. Note that for Hilbert spaces H biorthonormality gives
(see also Ex 7.8.(b)). For calculations it is very helpful if the functions f, are of a simple nature, e.g. if 52 is a Cartesian product, p a product measure and f, the coordinate functions: The probabilist would say in this case that the (f,) are “stochastically independent random variables” - and this is why the averaging technique is often labelled a probabilistic method. Anyhow, this observation allows one to use powerful theorems from probability theory. There will be two examples used for averaging: the Rademacher functions 8.5. and the Gauss functions 8.7.. The Rademacher functions attain only values +1 and are actually defined on the countable product of a discrete space (as will be seen in a moment); in the case of Gauss functions R is RN or fl with the Gauss measure - and hence invariant under orthogonal transformations. 8.5. Rademacher functions: For every n E N define on D, := { - 1 , l ) ” the probability measure p, by pn({w)) := 2-” for every w E D,;it follows that p, is the 61) on D1 = {-1, 1). On D := {-1, l)N, n-fold product measure of p1 = 2-l(6-1 the countable product of the p1 is a probability measure p , which has, by definition, the following property: If f : D -+ C depends only on the first n coordinates, i.e.
+
f :D
projection
fn D,-C,
then
{-I, 1) (this will cause no The coordinate projections Ek : D -+ {-I, 1) or E k : D, confusion) have norm 1 in Lz(p) and L z ( p , ) and are biorthonormal: Take 6 < !5 n; then -+
&k(w)EL(W)p(dW)
=
k, [L1 wf
=
L*
WkWfpn(d(W1,
1
Wkpl(dWk)
pl(dWL) = 0
*
---I
wn))
=
96
8. Absolutely and Weakly p-Summable Seriea and Averaging Techniques
are called discrete Rademacher functions.
The functions
REMARK: If r k are the Rademacher junctions on [0,1] with the Lebesgue measure (see 1.5.(2)), E a normed space and zl,...,z, E E , then f o r all 1 Ip < 00
This follows from the fact that all Rademacher functions r k are constant on the intervals [m2-”, (m 1)2-”[ . The same argument gives that (A the Lebesgue measure)
+
for all a1,...,an E R and this means, in the language of probability theory, that the measure p, on 0, C R” is the joint distribution of the “random variables’’ ( r l ,...,r,). The basic inequality is the
KHINTCHINE INEQUALITY: For 1 Ip < 00 there are constants up and bp 2 1 such that
for all n E LV and a1,...,a, E a’.
This is false for the L,-norm, see Ex 8.18. (which also treats the case 0 < p < 1). The best constants ap and bp were calculated by Haagerup [96] in 1982; they are the same if one considers only real a k (see Ex 26.9.):
The number p o
%
1,84742 is the solution of r ( ( p
+ 1)/2) = &/2
in ]1,2[.
PROOF: Since the following proof (taken from Stein [260], Appendix D) will not give the best constants anyway, it is enough to check the inequalities for real ( Y k , the left
8. Absolutely and Weakly p-Surnrnable Series and Averaging Techniques
inequality for p = 1 and the right one for natural numbers p and put
> 2.
Take
97
a: = 1
n k=l
Then, for each natural number p
> 2 and A > 0
Since
it follows that ~ An estimate for
Taking A := f i ,gives bp 5 is a consequence of Holder’s inequality:
5
~ f ~ ~ ~ (2p!)l/PA-’ , , ~ p n ~ exp(A2/2p).
a.
5
8.6. For the Rademacher functions on [0,1] the Khintchine inequality implies that on the subspace Rad:= span{rn}c &[O, 11 all norms coming from L,[O, 13 are equivalent (1 5 p < m) :
fact which was already used in section 1; for p = 00 see Ex 8.18.. For p > 1 Rad is even complemented in L,[O, 11, me Ex 8.17.. It might be interesting to know that the only continuous functions f in Rad are the multiples of the function 1 - 2t (see Chatterji [26]). What happens if the scalars a k are replaced by ck in a normed space -a
98
8. Absolutely and Weakly p-Summable Series and Averaging Techniques
E and the modulus by the norm? For p = 2 the definitions in 7.7. say that E is of cotype 2 if and only if the left inequality holds (for some constant) - and of type 2 if and only if the right one holds. Since there are spaces which have neither type 2 nor cotype 2 (for example C[O,11, see Ex 7.14.) there is no vector valued Khintchine inequality. However, the Kahane inequality says that there are constants K p such that for all normed spaces E and tl,...,zaE E
(for a proof see e.g. Lindenstrauss-Tzafriri [177], 11, p.74). By remark 8.5. the same holds for the discrete Rademacher functions. So it is important to have more examples of spaces with type and cotype. The Khintchine inequality will help. Moreover, in the following proof repeated use will be made of the “continuous triangle inequality” (see Appendix B5. and 7.2.)
PROPOSITION: Let ( $ 2 , ~be ) a measure space. (1) For 1 5 p 5 2 the space L p ( p ) i s of type p and cotype 2. ( 2 ) For 2 5 q < 00 the space L q ( p ) i s of cotype q and type 2. If L p ( p ) is infinite dimensional, then it is of no type larger than min{p, 2) and of no cotype smaller than max{p, 2}, see Ex 8.13..
PROOF: Discrete Rademacher functions will be used. A * will indicate the application of the “continuous triangle inequality”. If 1 5 p 5 2 and f1, ...,fa E L p ( p ) , then
which shows that Lp is of cotype 2; for type p
8. Absolutely and Weakly p-Summable Series and Averaging Techniques
99
For 2 5 q < 00 the same type of argument applies - but we prefer to write them down as a chain of mappings: Here Rad, denotes the closed span of the Rademacher functions with the norm induced from L,(D). The continuous triangle inequality * will be used in the form of 7.2.:
and it follows that C , ( i d ~ , = ) 1. That L, has type 2 follows from
and T z ( i d L q ) 5 b, from the Khintchine inequality. 0 Of course, it makes no difference whether one writes the proof as a chain of inequalities or as a chain of maps. This is only a question of mathematical taste - we sometimes prefer maps. The proof showed that for all n E N
C Z ( $ ) _< a, C , ( c )= 1
and TP($)= 1 and Tz(G) 5 b,
for 1 5 ~ 5 2 for 2 5 q
< 00.
Ex 7.12. implies that for every 2i,A-space E C z ( E ) 5 AQ, C,(E) 5 A
and T , ( E ) 5 A and T z ( E ) 5 Ab,
for 1 I p s 2 for 2 5 p < 0 0 .
For the remaining type and cotype constants of P;: see Ex 7.13., Ex 7.14. and Ex 8.13.. It is obvious, by the proof, that one could have used other averaging methods instead of the Rademacher one to obtain that L, (as above) is of some type/cotype with respect to this method; this may improve the final constants in calculations for L, which use type/cotype constants. 8.7. Ga uss functions: The Gauss measure y,, on K" is defined by
100
8. Absolutely and Weakly p-Summable Series and Avemging Techniques
where = (27r)-"12 in the real case and cn = (27r)-" in the complex case (i.e. the Gauss measures on and R2"coincide); 7n is a probability measure - clearly the n-fold product of y1 . On K N the Gauss measure 7 is the countable product of yl on K ; as before, 8.5., functions on K N depending only on the first n coordinates can be integrated with respect to yn The coordinate projections gk : K N -+ K or K" 4 K are called Gauss functions; for K = R they have norm 1 in La(7) or L2(yn) and they are biorthogonal: if 1 5 k < 1 5 n, then
.
gkgfd7 = L
N
kn
gkgfdyn =
wk
(L
wf7l(dwf))Tl(dwk) = 0
.
Therefore the ( g k ) form an orthonormal system in Lz(7). If K = C, the Qk are biorthogonal as well, but have norm 4 - however, for our purpose there is no need to normalize them. Note that for a = (01, ...,a n ) E (P
and, by the transformation formula for the Lebesgue measure,
1,.
f ( ~ w ) 7 n(dw) =
1,.
f(w)yn (dw)
for all isometries A : t?j -+ 5 : The Gauss measure is invariant under isometries. This is the great advantage of Gauss averaging - as can already be seen in the proof of the following result of Khintchine type.
PROPOSITION: For 1 5 p < 00 and 0 1 , . . . , a n E K
PROOF: For every isometry A : 5 -+
5 and a E K"
by the invariance of 7,, under isometries. Therefore it is enough to prove the result for a=el: , It is routine work to calculate the constants:
8. Absolutely and Weakly p-Summable Series and Avemging Techniques
101
moreover,
It turns out that the real and complex constants are different - in some sense contrary to the case of the Khintchine inequality for Rademacher functions (but recall also that the gk are different in the real and complex case). As in the case of Rademacher functions, there is no vector valued Khintchine result for Gauss functions, and there is a type of Kahane inequality saying that on the subspace [span {gk}]@E all norms coming from Lp(7, E) are equivalent (see Marcus-Pisier [183], p.44). It can be shown that for all tl,...,zn E E
but there is no converse estimate (see Ex 8.9.); it is a deep result of Maurey and Pisier [187] that exactly the Banach spaces E which do not contain the Pm uniformly permit the other estimate. 8.8. The averaging technique will now be used to calculate various norms of the element ek @ ek E K" @ IK" ; considered in L(K" ,K")this is the identity operator. The
Holder inequality gives
PROPOSITION:
PROOF:E 5
T,
hence r( ...) 2 1. For the converse inequality Rademacher averaging
will be used:
COROLLARY:
102
8. Absolutely and Weakly p-Summable Series and Avercrging Techniques
PROOF: The mapping property for T gives
k=l
k=l
Since
(C@ r q)'=
@c
GI holds isometrically,
These three estimates give the result. 0 8.9. The cotype estimates serve nicely when investigating the following question: In
which Banach spaces is every weakly r-summable sequence absolutely s-summable? In other words, when is 4V4cW) ? The question goes back to a result of Orlicz 1933 [195] stating that in Lp[O,l]for 1 5 p 5 2 each unconditionally convergent series (see 8.3.) is absolutely 2-summable. A closed graph argument shows that c ( E ) c t z ( E ) if and only if there is a constant c with !,(Zn;E)
I cwl(zn;E)
- and clearly it is enough to check this inequality for finite sequences. So the question is whether or not sup ]lid @ id : n
@c
E
-
Ell < 00 ?
If E is a Hilbert space, a nice direct argument shows (Ex 8.10.) that this supremum is 1. Orlicz used Rademacher functions in his proof for L p ; let us do the same: First observe that
8. Absolutely and Weaklg p-Summable Series and Averaging Technique8
Now if E has cotype 2 the Rademacher average (for rk or 12 (zf ; E), hence
&k)
103
dominates, by definition,
PROPOSITION(OrliC2): If E has cotype 2, then every unconditionally convergent series in E is absolutely 2-summable; more precisely, for all X I ,...,znE E
eZ(zk;E)5 C Z ( E ) w l ( z k ; E )
*
By 8.6. the L p ( p ) (for 1 5 p 5 2) are of cotype 2 - hence Orlicz’ original theorem was also obtained (the work for the result was done when proving that the L, have cotype 2!). Maurey and Pisier [187]showed that each Banach space with the “Orlicz property” (stated in the proposition) has cotype ( 2 E ) for all e > 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 < p < 00. Use this = if and only if p‘ > 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 .
> 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 < p < 00 the space Rad of Rademacher functions is complemented in Lp[O,I]. Hint: L, L* L2 -+ e 2 if p > 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
10. Integral Operators
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
10. Integral Operators
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
11. Absolutely p-Summing Operators
(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
11. Absolutely p-Summing Operators
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
11. Absolutely p-Summing Operators
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:
11. Absolutely p-Summing Operators
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
11. Absolutely p-Summing Operators
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,; <
E)P . 0
See also Ex 25.14.. Comparing this result with ll.l.(e), one might be tempted to think that $ 8~~F = vp(.$,,F) holds isometrically. This is not true for p # 1 - a fact which will follow from the results in Chapter 11: Ap fails to be a “tensor norm”. 11.9. The proposition can be used to prove the aforementioned
THEOREM: If E is a Banach space of dimension n, then Pz(idE) = f i PROOF: To show Then
.
5 fi,use the foregoing test and take S E Z ( e , E) with llSll 5 1. S:
Q
G/kers% E ,
where 11Q11 = 1 and llSoll= llSll I 1. Since c / k e r S is a Hilbert space lt with k it follows that P2(S) 5 IlQllP2(4:>IlSoll IdZ I J;; . Conversely, the Grothendieck-Pietsch factorization theorem gives a factorization idE
E - E
H
5 n,
11. Absolutely p-Summing Opemtors
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
11. Absolutely p-Summing Opemtors
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
11. Absolutely p-Summing Operators
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
11. Absolutely p-Summing Operators
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 .
11. Absolutely p-Summing Operators
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
11. Absolutely p-Summing Operators
(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)
vz,
for all finite Borel measures p on E; the infimum of these c is P P ( T ) .Hint: Choose a partition Dn of E with IIZ'zIIP - llTznllPl and 112 - zn[Ismall on Dn. (b) Extend (a) to u-finite Borel measures p. (c) If there is a u-finite Borel measure p on E with
for all 2' E El, then
11. Absolutely p-Summing Operetors
145
for each T E E(E, F). (d) Show that for ( x 1 , x 2 )E R2
Use this, (c) and the domination theorem to show that Pl(idR;) = r/2. This method was used by Gordon (see also Jameson [119]) for the calculation of various P , ( i d ~ ) . EX 11.20. Show that the class (qr,, ,P r , a ) of absolutely (r, s)-summing operators (see Ex 8.12.) is a Banach operator ideal. Moreover, T E E(E, F) is absolutely (r, s)summing if and only if i d @ T : e, @t E er @Ar F is continuous. In this case the norm of this mapping is Pr,a(T) . Ex 11.21. If r1 5 r2 and s1 5 s2 such that l/sl - l/rl 5 1/sz - l/rz, then Vr,,al C !J3r2,aa and Pr,,,, 5 P,,,,, . Hint: With l/r = l/rl - l/r2 and 1/s = 1/81 - l/sz it follows that
-
e+-1(akllT~kll) IPrl,sl(T)ea(ak)wa,(tk)
.
Ex 11.22. Use Rademacher functions and Ex 8.1. to show that for 2 5 p < 00 the identity map of any L p ( v ) is absolutely (p,l)-summing. Hint: CIfn(t)lP5 I CCn(W)fn(t)IPp(dW). For a more general result see 24.7.. Ex 11.23. (a) Take A E too and consider the diagonal operator Dx : co 4 co. Show that Dx is absolutely psumming if and only if A E ep. Hint: Ex 11.4. and unit vectors. (b) Take p , q, r, s E [l,00[ with l/p l/q = 1/r < l/s. Show with (a) that
+
Vqo'Pp
$ 'Pa
- 80 the index r in Pietsch's multiplication theorem Pqo qPc Vr is best possible. Hint: Dx o D,,= Dx,,. Ex 11.24. For the Gauss measure 7" on K" and 1 5 p < 00 define
and observe that Pp(idq) = c;,,/ci,, by theorem 11.10.. (a) Using polar coordinates as in 11.10., verify that
, coor(b) Use the Gauss-Khintchiee equality c:,,11z112 = ($ I(x,w ) l P y , ( d ~ ) ) ~ / Ppolar dinates and the calculation for (a) to show that
for every x E K", where qn-l is the normalized Lebesgue measure on the euclidean unit sphere S,-l of K " . The formula is taken from Pietsch [214], 22.1.2..
146
12. Definition and Ezamples
Chapter I1
Tensor Norms In this chapter the main topic of this book will be presented: The GrothendieckSchatten theory of tensor norms, with special emphasis on its interplay with the theory of operator ideals. Sections 12 - 15 and 20 give the basic definitions and properties of tensor norms, including Grothendieck’s fundamental theorem of the metric theory of tensor products -nowadays called Grothendieck’s inequality. Section 17 deals with the main theorem concerning the relationship between tensor norms and maximal Banach operator ideals - the representation theorem for maximal Banach operator ideals; the representation theorem for minimal Banach operator ideals will be given in section 22. Sections 18 and 19 study the (p,q)-factorable and (p, q)-dominated operators, the technique of ultraproducts, Maurey’s and Kwapieh’s factorization theorems. The central tool in all that follows will be the duality theory of tensor norms and later of operator ideals, in other words: a good understanding of trace duality is basic. Trace duality is “smooth” only for “accessible” tensor norms or under an additional hypothesis such as the metric approximation property. Due to the success of the notion of accessibility for tensor norms, accessible operator ideals will be defined and studied in section 21. It seems to us that the composition formulas for accessible operator ideals given in section 25 (which again are a sort of trace duality) fully justify the introduction of this notion. At the end of this chapter we hope that the reader is impressed with the power of also thinking in terms of tensor norms. In particular, she or he may have learned that it is natural to describe certain phenomena in terms of tensor norms, others in terms of operator ideals.
12. Definition and Examples In this section the definition and some basic examples of tensor norms are given. Two natural ways will be presented for extending a tensor norm from the class of finite dimensional spaces to all normed spaces - an inductive and a projective procedure. The inductive one (“the finite hull”) gives Grothendieck’s original definition of a tensor norm, the projective one (“the cofinite hull”) - essentially due to Lot2 - will turn out to be a useful tool for the understanding of various delicate phenomena related to the approximation property.
147
12. Definition and Ezamplea
12.1. A tensor norm a on the class NORM of all normed spaces assigns to each pair ( E lF) of normed spaces E and F a norm a( - ; E l F ) on the algebraic tensor product E @ F (shorthand: E Nu F and E& F for the completion) such that the following two conditions are satisfied: (1) Q is reasonable, i e . : E IQ 5 A . (2) a satisfies the meiric mapping property: If Ti E 2(Ei, Fi), then
llT1 @ T2 : El
@a
E2
-
Pl
@a F211
L 11TlIl llT2ll
‘
This definition goes back to Schatten [245] in 1943 (who called such an Q a uniform crossnorm) and Grothendieck [93] in 1954. It is clear that it can also be made for subclasses of all normed spaces: for the class FIN of all finite dimensional normed spaces, for the class BAN of all Banach spaces, for the class HILB of all Hilbert spaces or even for classes of pairs of normed spaces such as NORM x BAN or FIN x NORM. A tensor norm a being reasonable means
It can happen that all tensor norms are equivalent on some E @ F : the celebrated spaces P of Pisier, mentioned in 4.8., satisfy P P = P @I P isomorphically, and hence have this property. E and A are tensor norms since they have the metric mapping property (3.2. and 4.1.), but Ap is not a tensor norm:
PROPOSITION: For 1 < p and normed spaces F
< 00
there is no tensor norm Q such that for all spaces L p ( p )
L p b ) @A,
F =Lpb)
F
holds isomorphically. There exist, however, tensor norms “close” to Ap (see 15.10. and 15.11.) - and these tensor norms serve well for studying various phenomena related to Ap.
PROOF:The reason that no such tensor norm exists is that Ap does not satisfy the mapping property: For p = 2 it was seen in 7.5. that for the continuous Fourier transform 5 : L2(R) -,L 2 ( R ) the operator 5 @ idt, : L 2 ( R ) @ A ~ti -,L 2 ( R ) @A2 ti is not continuous, but 5 @ idt, : L 4 R ) @a t 1 -, L z ( R ) QU ti is continuous for all tensor norms a. For 1 < p < 00 the same argument applies using the continuous Hilbert transform tp(Z) tp(zZ)treated in 7.6.. 0 On the other hand, as was already shown in 7.1. and 7.3., Ap is at least a “one-sided” tensor norm: it is reasonable (i.e. E 5 Ap 5 T ) and llidLp @ T : Lp @ A ~E
for all T E C(E,F ) .
-
Lp @ A ~Fll=
IlTll
148
12. Definition and Examplea
12.2. The following test (formulated for NORM only) will be quite useful:
CRITERION: (Y is a tensor n o r m on NORM if and only if: ( 1 ) a( - ;E , F) is a seminorm on E €3 F f o r all pairs ( E ,F ) of nonned spaces. (2) a(1 c3 1; K ,K )= 1. ( 3 ) a satisfies the metric mapping property. Note the similarity with the criterion 9.4. for Banach operator ideals and also that the cases of real and complex scalars will be treated simultaneously.
PROOF: To show that the three properties suffice for a to be a tensor norm, one only has to check that a is reasonable, i.e. E Ia IA. For z' E E' and d E F'
-
x'@y':E@F-K@K=K, (x'@y',z)
hence l(z' €3 Y',
€3 d , 4; K ,K ) I11~'IlIldlla(z;E,F )
z)I =
by (2) and (3), and therefore e I a. To see that A dominates a,observe first that for x E E the operator T, : K -+ E defined by T,(A) := Az has norm 11211, hence
llG €3 q
:
€30
K
-
E
@a
Fll
I 1141llvll
9
and therefore (again by (2) and (3)),
ll4l llYll = 4 . €3 Y; E , F) I4
2
€3 Y;El F ) Illzll
which means that a ( z €3 y; E , F ) = 11x11
Ilvll.
IlYllQ(1 €3 1;K ,
w = 1141llvll
The triangle inequality gives a 5 A. 0
12.3. In general, tensor norms respect neither subspaces nor quotients isomorphically: this follows from the "bad" properties of E and A. The mapping property only implies continuity of the respective mappings. Clearly, if the subspace G c E is complemented and P : E + G a projection, then, again by the mapping property, 4 z ; E , F ) i 4 z ; G,F )IIIPlla(Z; E , F )
for all z E G €3 F. Recall the natural transposition map
12. Definition and Examples
149
If a is a tensor norm, then the transposed tensor norm at is defined by at(%; El F ) := “(2‘; F, E) . It is obvious that at is a tensor norm and art = a. The tensor norms symmetric, i.e. ct = E and rt = u.
E
and u are
12.4. Given a tensor norm a on the class FIN of all finite dimensional normed spaces, there are two natural ways of extending it to the claaa of all normed spaces. For this, denote for a normed space E
FIN(E) := {M C E I M E FIN} and COFIN(E) := {L c E I E/L E FIN}
,
the sets of finite dimensional subspaces and of finite codimensional, closed subspaces (with induced norms). Now take normed spaces E and F and define for z E E @I F + a ( z ; El F ) := inf{a(z; MIN) I M E FIN(E), N E FIN(F), z E M C3 N} t
a ( z ; El F) := sup{a(QE
@I Q r ( z ) ; E/K,
F/L) 1 K E COFIN(E), L E COFIN(F)}
(where QE : E * E / K is the canonical mapping); the arrows stem from the fact that the first procedure is an inductive, the second a projective one. By the metric mapping property it is enough to take cofinally many M, N resp. K, L in the definition. With the criterion it is very easy to verify that: 1
PROPOSITION: Let a be a tensor norm on FIN. Then the “Jinitehull” d o f a and the “cofinite hull” ‘Z of a are tensor norms on NORM wiih
Ifa as defined on NORM (and not only on FIN), then
holds on NORM. c
Clearly, ’Z 5 c /3 and d 5 c 7 whenever a 5 c p . Since E respects subspaces, E = T‘ and hence E = ?F; the description of the projective norm by an infimum yields that r = T‘ but it will be shown in 15.6. that u # f .A tensor norm a on NORM is called finitely generated if a = d and cofinitely generated if Q = %. The usual tensor norms are finitely generated - but the cofinite hull is natural as well and its consequent use (also for finitely generated tensor norms) helps to structure the theory, leads to a better understanding of various ideas and simplifies many proofs. In
150
12. Definition and Ezamples
his W u m 6 Grothendieck only considered finitely generated tensor norms - and this is quite understandable: It will be shown in 13.1. that
whenever E and F have the metric approximation property - and it was not until nearly 20 years after Grothendieck’s work when spaces without approximation properties were discovered (see section 5). It is obvious - but good to always bear in mind - that two finitely generated (or two cofinitely generated) tensor norms are equal if they are equal on finite dimensional spaces. This explains, for example, why these two types of tensor norms are often useful when investigating Banach spaces or operators in terms of finite dimensional subspaces.
12.5. Many of the interesting tensor norms can be obtained from the ones introduced by Laprest6 [166], who generalized those of Saphar [237], Chevet [27] and Cohen [31]. For p,q E [l,co]with l / p + l / q 2 1 take the unique r E [l,w] with l/r = l / p + l / q
-1
or equivalently,
1 = 1/r
+ l/p’ + I/q’
(where p’ is the conjugate of p) and define for normed spaces E and F and
(with the notation lp(*)and pup(-) from section 8). Obviously a1,1= a
t
E E@F
.
PROPOSITION: (1) ap,qis a finitely generated tensor norm on NORM. (2) apa,qa 5 Q P l , f l ifPl IPz and 41 I (3) a;,q = %,P *
4 2
*
PROOF:(1) By criterion 12.2.only the triangle inequality is not obvious: Take elements z1, z2 E E @I F and E > 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
(
2 a p , q ( z N )+ a p , q ( z ) ;
this implies P(z) 2 a,,,(%). 0
12.7. Special cases of ap,f-tensor norms are (1 5 p Im) gp := ap,1 dp := ( ~ 1 , ~ wp := ap,p' and therefore
g1=d1=n gp = d i wp
I gp
( g for "gauche") (d for "droite") (w for "weak")
, W1=dm ,w,=g,
,
wp =
1
WpI
I dp
153
1.2. Definition and Examples
For gp and dp the index r in the definition is p , for wp it is 00; therefore it is easy to see that n
gp(z; E , F ) = inf{lp(ti)wpt(yi)
I z=C
dp(z; El F ) = inf{ wp8(2i)lp(yi) I z =
x i QD yi}
i=l n
C zi QD
Vi}
i=l
Clearly, there is also a result for the completions which has the spirit of proposition 12.6.:
E l p ( E ) (in c,(E) i f p = m) and (yn) E T , ( F ) the series If, conversely, z E E g g ,F , then E l p ( E )(in c,(E) i f p = CO) and (yn) E c , * " ( F )with z = C,"=ltnQDPy,;
COROLLARY: (1) For C(Zn @
(tn)
yn) converges unconditionally in E g g PF .
there are ( t n ) moreover,
gp(z; E , F ) = inflp(xn)wp'(Yn)
,
where the infimum is taken over all such representations. (2) For 1 < p < 00 each series C(tn8 y n ) with (tn) E T * O ( E )and (yn) E T t ( F ) converges unconditionally in E g W p F . Conversely, each z E E g w p F has a representation z = Cr'l t n QD yn with (En) E $'po(E) and (yn) E T,"'(F); moreover, wp(z; E , F ) = infwp(xn)wp~(yn) ,
where the infimum is taken over all such representations. This follows immediately from the proposition - with the exception of the first part of (2): but this, as in the proof of 12.6., can be checked directly.
12.8. The following picture illustrates the situation:
1
2
P
Some of these tensor norms are equivalent (of course, two tensor norms called equivalent if there are c, d > 0 with CCY 5 p 5 da).
CY
and
p are
154
12. Definition and Examples
PROPOSITION: For p, q €11, co[with l/p
+ l / q 2 1,
where br is the constant from the Khintchine inequality. In particular, w2 and equiualeni if p, q €1 1,2].
Q
~ are , ~
PROOF: As the constants already indicate, Rademacher averaging will be used: For = Cy'l xi @ yi E E @ F take the new representation
z
Now the Khintchine inequality 8.5. gives
Consequently,
and therefore ap,q5 bq'bp'wz . 0 The tensor norms gp and dp cannot be estimated by w2; otherwise e.g. wm 5 gp 5 cw2 would follow which, by results to be presented in 23.2., would imply that Hilbert spaces are CLapaces, a contradiction. For the best constants satisfying ap,qI cw2 see Ex 24.7.. 12.9. It was shown in 8.2. that
Cso(E) =
E&tp m
(zn)
-Czn n=l
holds isometrically:
Ben
12. Definition and Examples
Since w p l ( e n ; t p )= 1 and e
155
5 w p , the representation 12.7. of E g W p t pimplies:
REMARK:The relations ~ ~ " (=EE&tp ) = E&,,tp c,(E) = E& C , = E g W mC ,
f o r 1 L p < 00
hold isometrically. In parlicular,
To obtain another useful description of wp consider the four-linear map
which can be linearized to a so-called tensor contraction
This mapping will turn out to be extremely useful; it mimics the composition of maps (see Ex 12.11.) and this has interesting consequences for theorems about the factorization of operators. A detailed treatment of tensor contractions will be presented in section 29; a first hint of its utility is given by the
COROLLARY: Let 1 5 p 5 00. Then the tensor contractions
are metric surjections. Moreover, the norms in the quotients can be calculated by only using elements in ( E m C t px) (tPlBCF ) or i t s completion, respectively. These statements also hold true if toois replaced by c,.
PROOF:Denote, for this proof, by sp c f?, the subspace of finite sequences with the induced norm. Then, by the definition of wp and the remark, the map
maps the Cartesian product of the open unit balls onto the open unit ball, hence
156
12. Definition and Ezamples
is a metric surjection; in particular, the completion of the bilinear mapping C is defined. Note that ( E QC sp)QT (sPt aCF ) is a dense subspace of (E&ep) a,, (eP#&F)if too is read as co. Since the second mapping in the statement is also surjective by the description 12.7.(2) of E & , ,F (and the remark), the quotient lemma 7.4. shows that both mappings are metric surjections. If 1, appears, note that both mappings have norm 5 1 by Ex 12,11.(c); the c,-case gives the result. 0 This result provides a description of the compact sets of E&,F, similar characterization holds for general ap,q, see Ex 35.10..
see Ex 12.12.. A
12.10. The last corollary, easy as it is, has quite interesting consequences. Take a tensor norm Q and consider the natural map C(K)&C(L)
-
C(K)&C(L)= C ( K x L )
for two compact sets K and L. For Q = ?r this map is injective since C ( K ) has the approximation property (see section 5 ) and it will be seen in 17.20. that this holds for all finitely generated a. What do the functions coming from C(K)&C(L) look like? For Q = ?r a function f E C(K x L) is in C ( K ) & C ( L ) if and only if there are fn E C ( K ) and gn E C ( L ) with
C,"=*Ilfnllmllgnllm < 00. For tr = wp the corollary above shows that c ( K , e p )x c(~,e,#) = (C(K)&~~,) x (ep,gcc(~))c ( K ) ~ L , c ( L ) c C ( K x L ) is onto - and this map is just (9, h) (g(.), h(.-))f,,l,, . To see this, calculate first for
-
and
u*
finite rank functions and then approximate. It follows that:
PROPOSITION: A function f E C ( K x L ) is in C(K)&,C(L) if and only if there are continuous functions g : K + 1, and h : L + e,, with for all ( s , t ) E K x L .
f ( s , t )= (!ds), h(t))f,,f,, Moreover, w P ( f ; C ( K ) , C ( L5 ) ) Ilgllmllhllm and for every can be chosen such that Ilgllmllhllm 5 wP(f)+ E .
E
>0
the functions g and h
Exercises:
Ex 12.1. (a) E @a K = E holds isometrically for all tensor norms a and normed spaces E.
12. Definition and Examples
157
(b) If Eo C E and Fo C F are dense subspaces, then E,, 8 F, is dense in E a0F for each tensor norm a. (c) If Ti E C(Ej,Fj) are isometries (onto), then
is an isometry for each tensor norm a. Is this true without surjectivity? Ex 12.2. If a is a tensor norm on FIN, then
Ex 12.3. Let a be a tensor norm on NORM; show that the “one-sided” finite and cofinite hull (also called right-finite and right-cofinite hull) a‘(%;
E, F) := inf {a(%; E, N) I N E FIN(F), t E E 8 N}
a‘(%;
El F ) := sup { a ( i d E 8 Q f ( z ) ;E , F / L ) I L E COFIN(F)}
are tensor norms such that ‘Z 5 a- 5 a 5 a- 5 d . Ex 12.4. Take S E f!(X,Y) with llSll = 1 and a and 9 , tensor norms (on NORM). Show that the class 2s of all T E 2(EIF) with
is a Banach operator ideal. Check exactly which of the tensor norm-properties are needed for the proof and note that a and /3 need only satisfy the metric mapping property on the right side of the tensor product. This means that X ma or Y @ p * could be replaced by L, @A, ; see also 9.6.(3), Ex 9.15., Ex 9.16. and 17.4.. Ex 12.5. Use Rademacher averaging to show that for all matrices (ajj) and all tensor norms a on FIN the relation n
n
holds. Hint: Iw((j):= ( ~ j ( w ) t j )for w E 0,defines an isometry. Ex 12.6. For each tensor norm a and p , q E [1,00]
i=l
where c := min{l/p
+ l / q , 1). Hint: 8.8..
158
12. Definition and Ezamples
Ex 12.7. For z E E @ F define the rank
of z as follows: rank(O):= 0 and for z
#0
(a) Show that rank(%)= rank(T,), where T, : E' -+ F is the associated linear operator. (b) For Banach spaces E, F and T E C ( E , F) the n-th approximation number a n ( T ) of T is defined by
I
an(T) := inf{llS - TI1 S E C ( E ,F),rank(S)
.
< n}
Show with the aid of these numbers that the set
I
Bn(E,F ) := { z E E @ F rank(%)5 n } is closed in E& F and in E @a F for each reasonable norm a on E @ F. If the map E&, F -+ E& F is injective, it is also closed in E& F. Hint: The u(E', E) - u(F, F')continuous operators in C(E', F) form a norm-closed subset, an is continuous and an(T) = 0 if and only if rank(T) < n. This result is due to Valdivia [274], see also [72]. Ex 12.8. Let l/p+ l/q 2 1. Then pp,4and 7p,qdefined by n
Pp,q(z; El F )
:= inf{ Il(ai,j) : GJ
+
I C
Cllwql(zj)wp'(Yi) z =
ai,jzj @ Yi}
i,j=l
Tp,q(z; E , P ) := inf{Il(ai,j) :
G,
-+
1
c n
~ l l e q ~ ( ~ j ) e p ' (z~= i)
ai,jzj 8 Yi}
i,j=l
are finitely generated tensor norms with PPA
Ia P , 9
and
PP,9 IIYPd
*
C,"".
@ Az : $l+nall _< Hint: Mimic the proof of 12.5. using the fact that (llA1ll' llA~l[r)l/r. These tensor norms will be studied in more detail in section 28. Ex 12.9. If (en) and (fn) are bases of Banach spaces E and F, respectively, then (el @ f 1 , e z ~ f ~ , e ~ @ f @fz,e3@ffi1e3@ffi, z,el ...) - the rectangular ordering - is a basis for E@,,F for each tensor norm a. Hint: Show that the expansion operators are uniformly bounded on E @a F and determine the coefficient functionals; use the fact that if the expansion operators of a biorthogonal system are uniformly bounded and converge on a dense subspace pointwise to the identity, then this biorthogonal system defines a basis. This result is due to Gelbaum-Gil de Lamadrid [80]. Ex 12.10. Let (%,A)be a normed operator ideal and define a ( . ; E F) , to be the induced norm from the injection E @ F L* Q(E', p). Then cr is a tensor norm on NORM; moreover, E @a F El' @a F holds isometrically. Which property does Q have if (Q, A) is injective? If (Q, A) is surjective?
+
-+
13. The Five Basic Lemmas
159
Ex 12.11. (a) Show that the tensor contraction
is the same as the composition map
(for p = 00 read co instead of lm). F ) + E 8 d p F is a metric surjection (1, can be (b) Show that (E tP)) BT ($ replaced by co). Hint: 12.7. and 12.9.. (c) For normed spaces E , F and G the tensor contraction
( E 8 G') 8 (G8 F )
-
E 8F
given by (z8 u') 8 (u 8 y) .r,(u', u)z 8 y is defined. Show that for an arbitrary tensor norm a ll(E 8 8 G') 8 r (G@a F ) E @a J'll I1 * +
[(En
-
Hint: En,m(uL, um)zn 8 Urn = u;@n) 8 id^] ( E m urn 8 ~ m ) Ex 12.12. Use corollary 12.9. and the description of compact subsets of the projective tensor product to show the following: For each compact subset K c E&,,,F there are a compact set D c L1 and zero sequences (zn)= ((zn(m))m)n in C * O ( E )and (yn) = ((yn(m))m)nin C ; " ( F ) such that for each z E K there is a (A,) E D with m
m
n=l
m=I
Note the case p = 1 : Each z, is an unconditionally convergent series and yn E c o ( F ) .
13. T h e Five Basic Lemmas This section contains five relatively simple lemmas which are basic for the understanding and use of tensor norms: the approximation, extension, embedding, density and local technique lemma. The power and importance of these devices will become clear while working with them.
19. The Five Basic Lemmas
160
13.1. Recall from section 5 that a Banach space E has the A-bounded approximation property if there is a net (T,) of finite rank operators E + E with llTqll5 X and T,z -+ z for all c E E .
APPROXIMATION LEMMA: Let Q and p be tensor norms (on NORM), E lF nomzed spaces, c 2 1 and on E@N ascp f o r cofinally many N E FJN(F). If F has ihe bounded approximation property with constant X 2 1, then a5Acp on E@F .
PROOF: It is easy to see that for each z E E €3 F
for the projective norm
A
and hence for all tensor norms. If 7 satisfies
and N is as in the hypothesis with T,,(F) C N , then, by the metric mapping property of tensor norms,
which implies the statement. 17 For the finite and cofinite hull of a tensor norm this lemma (and its transposed version) implies the following
PROPOSITION: Let a be a tensor norm (on FIN), E and F Banach spaces with the bounded approximation property with constants AE and X F , respectively. Then
I n particular, Z = d on E@ F i f both spaces have the metric approximation property. 13.2. Every 'p E (E@= F)' = 2 ( E ,F') has a canonical extension which satisfies the relation
'p"
E ( E @= F")'
19. The Five Basic Lemmas
161
where L, E C ( E ,F') is the linear operator associated with ifcpE(E@aF)'?
EXTENSION LEMMA:Let (on NORM). Then 'p
'p
E ( E Br F)' and let
E ( E @a F)'
i f and only i f
(Y
'p
(see 1.9.). What happens
be a finitely generated tensor norm
(p"
E ( E @ a F'I)'
.
PROOF:The metric mapping property implies IIE @a F Q E @a F"ll 5 1 and hence ll'pll... 5 ll'p"ll ... . Conversely, take z, E E @ F" and M E FIN(E) and N E FIN(F") such that zo E M @ N ; then the weak principle of local reflexivity (6.5.) gives for every E > 0 an R E Z ( N , F ) with IlRll _< 1 + such ~ that for ally" E N and z E M (Y", L , Z ) F ~ ~ ,= F '(RY", J?&Z)F,F! This means that (PA9 2 @ y") = ($7, ( i d E c3 R)(z@ Y'V
,
therefore (9" I 20) = (9, id^ 8 R(%,))
and hence I('p",zo)l
Ill'pll llRll4zo; E , N ) Ill'pll(1+
which implies the result since
(Y
&)"(to;
E ,N )
9
is finitely generated (see Ex 12.2.). CI
Sometimes the relation (*) is helpful. The reader may have noticed that the proof is nearly identical with the one given for cv = E in 6.7.. Strangely enough, we do not know whether the eztension lemma holds for cofinitely generated tensor norms. There is also a left-canonical extension "'p E (E" Br F)' of 'p E (E @ r F)' and it was shown in 1.9. that ("'p)" = " ( 9 " ) E (El'@,,F")' if and only if L, is weakly compact; clearly,
I I " ( ~ " ' I I ( E ~ ~ ~ ~ F ~ ~ ) =~ II("(PY"'I(Ew~F)J)J = IIPII(Es,F)~ by the lemma. Define two "natural" maps @j
: El' @ F"
C( E @ a F)"
162
13.
The Five Basic Lemmas
They obviously have norm 1 on E” @a FII . But we do not know which norms the Oj induce on E” @ F“. If the induced norm were a in reasonable situations, this would easily solve the problem of the bidual mappings which will be treated in Ex 17.16.. Clearly, (E €3 F”, (E @a F)’) is a separating duality system with the bracket (cp, 4 := (PA,4
*
The extension lemma and the bipolar theorem give the
COROLLARY: If a is a finitely generated tensor norm, then the unit ball BEB,F is u ( E €3 F N ,( E @a F)‘)-dense in the unit ball BE^,^ . 13.3. Tensor norms do not respect subspaces but the embedding into the bidual is usually respected:
EMBEDDING LEMMA:If a is a finitely or cofinztely generated tensor norm (on NORM), then idE69IcF : E B a F E m a F”
c ‘ .
is an isometry for all normed spaces E and F
.
See Ex 13.4. for a slight refinement of this result.
PROOF: The metric mapping property implies that for all z E E €3 F a(%;E , F”)
I a(%;E , F )
holds for all tensor norms a (the map idE €3 ICF will not always be written). (1) Let a be finitely generated. Then, by the extension lemma, a(%; E , F ) = SUP{I(V~,
%)II 9 E ( E ma F)‘, IIvII(EB,F)~ I 1) = ~ ~ P { J ( P I~P, E~ (I E ~ ~ F ) ~ , J l c p l lI ( ~1)~ , ~ y ISUP{ I($, %)II II, E ( E €3a F‘’)’, I I ~ I ( E ~ , F ~ ~ )IJ 1) = a(%;E , F”) ,
which is the reverse inequality. (2) If a is cofinitely generated and L E COFIN(F), then Loo (formed in F‘’) is in COFIN(F”) and the map KF/L :F/L
L(FIL)” = F”/Lo0
is isometric and surjective; moreover, Q f b o K F = K F / L o Q,“. Therefore, a(idE €3
QT(t.);E,F / L ) = a ( i d ~8 ( ~ F / L QT)(z); E , (F/L)“) = 0
id^ €3 K F ) ( J ) ; E , F”/Loo) 5 < Z ( i d E €3 K F ( Z ) ; E,F”) = Z(Z; E , F”) . = a ( ( i d E €3 Q$o)
o
IS. The Five Basic Lemmas
163
Taking the supremum over all L gives the missing inequality. 0
13.4. In addition, since F and its completion
have the same biduals:
COROLLARY: I f a is finitely or cofinitely generated, then
is an isometric and dense embedding f o r all normed spaces E and F . Density follows from Ex 12.1.. In particular, finitely or cofinitely generated tensor norms respect dense subspaces. This implies that the density lemma 7.3.can be applied to give the more specialized
DENSITYLEMMA: Let a be a finitely or cofinitely generated tensor norm, E and F normed spaces and E, and F, dense subspaces of E and F, respectively. I f G is a locally conuez space and T E Z(E @r F , G ) i s such that
then T E Z ( E 8, F , G)
.
Recall that T : L -+ G is continuous if each S o T : L + D is continuous, where D is normed and S E 2(G,D) . A particularly interesting special case is:
COROLLARY: Let a and ,d be tensor norms such that cr is finitely or cofinitely generated. Z E E(Ei, Fi) and Gi C Ei are dense subspaces such that
If
then TI @ T2 E Z ( E 1 @a E2, FI @g F2) . Since
Ti @ T2 : El Br E2
-
Fi Br F2
2 FI @ p F2 =: G
is continuous, the proof is obvious.
13.5. The local techniques for Zz-spaces (see 3.13. for the definition) are condensed in the following lemma; for E and ?r the argument has already been used several times:
Z P - L o c ~TECHNIQUE ~ LEMMA:Let a and ,d be tensor norms, c 2 0 , and E a normed space such that a
on
E@$
164
19. The Five Basic Lemmas
for all n E N. Then
a < ~ c 7 j + on E @ F for each i$x-space F
.
For a generalization of this from T = idE to arbitrary T see Ex 13.7.; see also 23.1. and Ex 23.13..
PROOF: For N E FIN(F) take a factorization
Then, for every z E E @ N,
this implies the statement. 0 Recalling from Ex 12.3. that a’ a-
is the right-finite hull of a,one notes that actually
5 Acp-
on E @ F
has been shown. Since wp = E on E @ $ (see 12.9.), it follows that E S W ~ ~ A on E E@F for each Cp,X-space F and arbitrary normed space E . Consequently, E6& F = E6&, F holds isomorphically whenever F is an 2;-space - and, e.g., Ex 12.12. gives a nice description of the compact subsets of E& F, which has an interesting application to compact operators (see Ex 13.6.). In section 23 it will be shown that E wp on @ F actually characterizes the Z;-spaces.
-
Exercises:
Ex 13.1. Let a be a tensor norm and E and F normed spaces with the metric approximation property. Then E
@a
F
c,
E @a F” holds isometrically. Hint: Use
z.
165
19. The Five Baaic Lemmas
Ex 13.2. Let El F, G be normed spacesl S E 2(E’, F’) and CY a finitely generated tensor norm. Then there is a linear S : (G @a E)‘ -, (G 80 F)‘ of norm llSll (if G # { 0)) such that
id,i@S
I
commutes; S is unique. Hint: Extension lemma. Ex 13.3. If a is finitely or cofinitely generated and E”&$ + C(E’, F ) is injective, then E&F -+ C(E’,F ) is injective. Is the converse true? Hint: Section 5. Ex 13.4. Check the proofs of 13.2.and 13.3. in order to show that: (a) If a = a* (i.e. a is finitely generated from the right, see Ex 12.3.), then the extension lemma holds for a. (b) If a = ‘ a or a = a+,then the embedding lemma holds for a. Ex 13.5. Use the special version Ex 13.1. of the embedding lemma and the tensor norm from Ex 12.10.to demonstrate the following statement: If @ , A ) is a normed operator ideal, E a reflexive and F an arbitrary Banach space such that E’ and F have the metric approximation property, then
A(T : E
-+
F)=A
( K ~o
T : E -,P I )
for all T E $(E, F ) . This is a sort of “regularity property” of finite rank operators with respect to all normed operator ideals. Ex 13.6. (a) Let E and F be Banach spaces, F an 2;-space and K C $ ( E , F ) a compact set of approximable operators. Then there is a compact D C PI , sequences (&) in $(E,Pp) and (Sn)in g(Pp,F ) with ll&ll + 0 and IISnll 0 such that for each T E K there is a (An) E D with
-
m
T=
AnSno R,,
,
-
for p = m replace t, by c, . Hint: e wp and a modification of Ex 12.12.. It will be shown later on (21.6.)that all 2g-spaces F have the approximation property, hence $(E, F) = R(E, F ) holds. (b) Show that every approximable (=compact) operator with values in an C;-space factors compactly through Pp (or c, if p = m). Hint: 12.9.. See Ex 23.9.for more information. (c) Use (b) to show the following result of Terzioglu [270]: Every compact operator t,(I’). factors compactly through a subspace of c,. Hint: E -,F
-
166
14. Grothendieck’s Inequality
Ex 13.7. Cp-local technique for operators: Let a and /3 be tensor norms and a be finitely generated. If T E C ( E ,F) satisfies Ilidt; @ T :
c No E
for all n E N ,then
-
5 @ p J’ll
llidc @T :G @ aE - G a p
5c
FIJ5 Xc
for each C;,A-space G.
14. Grothendieck’s Inequality Grothendieck’s “thbrkme fondamental de la thbrie mdtrique des produits tensoriels” ([93], 54’ thbrkme 1) states that the identity operator on a Hilbert space is in the operator ideal (3Jur)ini(see section 20); the operators in this ideal were called “pr6intdgral” by Grothendieck. A remarkable achievement of Lindenstrauss and Pelczydski in their celebrated 1968 paper [173] was to have noticed that this theorem can be reformulated in the form of an inequality concerning n x n matrices and Hilbert spaces - as well as in the form 2 ( t l , t , )= p~(t!l,tz). This is why the fundamental theorem is nowadays called Grothendieck’s inequality - in matrix form, in tensor form, or in operator form in order to highlight its various glamorous facets. Our path of entry to the many equivalent formulations of the inequality is establishing the fact that 7r and w2 are equivalent tensor norms on loo @loo; we follow Krivine’s proof [155], which was smoothed out by Pisier in his book [225] and polished into its final form by Pelczydski. Simple as it is now, the proof is in some sense dual to Grothendieck’s original one, but has still the same ingredients.
14.1. The idea of the proof is to take a function
f E C ( K )@w2 C ( K ) (where K = (1, ..., n}), factor it as in 12.10. through some Hilbert space
e,i.e.
n
with 11zk11 = llgllll = 1, and use an integral representation of (z,y) for elements in the euclidean unit sphere to obtain a new representation
167
14. Grothendieck's Inequality
with the functions g and h being uniformly bounded on Rn (with values in C ( K ) ) . However, the formula for (z,y) reads = sin [JR....dyn] , so it is better to represent sinf instead of f as in (*). In order to be able to define sinf (with control of the C ( K ) is a Banach algebra: norm) one needs the fact that C ( K )BW2
LEMMA: If K and L are compact, l / p + l / q 2 1 , then C ( K )8%
C(L)
i s a normed algebra (under the pointwise multiplication coming f r o m C ( K x L ) ) , i.e., :ffi,fa E C ( K )8 C ( L ) ,then aP,q(fl
I O b , q ( f 1 ; C ( W ,W m p , q ( f 2 ;
f 2 ; C(K)l C(JW
PROOF: Represent
fi
C ( K ) ,C ( L ) )
N
= En=l ALgi 8 hk. Then
c N
fl
f2
=
A:A;
(g:g;)
8 @Ah;) E C ( K )8 C ( L )
.
n,m=l
and likewise for 2 ~ ~ 1 ( ( h A h & )the ~ , ~result ) , is obvious. 0 In particular, the space PW&,a which is nothing more than (z1 * -72, ek
Pw is a Banach algebra under pointwise multiplication
8 e f ) = (z11ek 8 e f ) ( 1 2 , ek 8 e f )
in tensor notation. If C(amtm) is a power series with infinite radius of convergence, then E,"=,a,zm is defined in the Banach algebra & &i for each z E & 8 & and 00 W
for all k,A!= 1, ...,n by continuity of the point evaluation ek 8 el E (PwBW,PW)'. 14.2. The second ingredient in the proof will be factorization:
LEMMA: For z E Pw 8 PW with w2(z;P,,PW) < 1 there is an rn E EV and elements ...)2,, y1, ...,f/n E g of n o m 1 svcA that
21,
n
168
14. Grothendieck's Inequality
PROOF: Each representation z = allows one to define functions
xzl
Ui
8~ with W Z ( U ~ ; %< ) 1 and Z O ~ ( V ~ ; % ) < 1
The construction in this proof is the same as in 12.9. and 12.10..
14.3. Denote the sign function R + {-1, 0, +1} by sign. The main tool for the proof of Grothendieck's inequality is the following formula for the scalar product (=duality bracket) of vectors in the euclidean unit sphere of R".
where 7n is the Gaussian measure on
R"
.
PROOF: There is a 0 E [0, 7r] such that ( 2 ,y) = cost9 = sin (5 - e) . Let T : R" -, R" be a linear isometry (with respect to the euclidean norm) such that
Tx = e l and Ty = (cos 8, sin 8,0, ...0) . Since the Gaussian measure is invariant under isometries (8.7.),it follows that 7r
5
sign(2, w)sign(y, w ) m ( d w ) = J R n
7r
=Z J R m sign(e1,w)sign((cm 0, sin 8,0, ...0), w)7n(dw)= 1 -I . J J sign s .sign(s cos e + t sine) exp(--(s2 + t2))dsdt = 2 2 x R R 2
=a 1" Jdm
r2 sign(r cos 'p) sign(r cos 'p cos e + r sin psin e) exp(--)rdrd'p
=$l"
2
-
sign(cos 'p) sign(cos(9 - O))&
.
=
14. Grothendieck'a Inequality
169
The integration of this simple step function gives 1
=4 [ ( 2~ 28) - 281= 2 -8 , f
hence the result. 0 14.4. Now everything is prepared for proving
GROTHENDIECK'S INEQUALITY
(tensor form): There is a constant c such that
n(.;qL,CJI c
. w z ( * ; ~ , ~ )
for all n. If KG is the best such constant, 2hen (real theory) (complet iheory).
PROOF:The complex case will follow from the real one, which will be treated first: Take a := ln(1 + < 1 which satisfies sinh(a) = $(exp(a) - exp(-a)) = 1, and z E Pm @ P , with w ~ ( ~ ; P , ~ e " , < ) a . It has to be shown that n(z;Pa,Pm) 5 n/2 . The element
a)
22k+1
m
sin z := C ( - l ) k (2k + l)! E C @ W , e " , k=O
is defined in the Banach algebra from .&) and satisfies
BW2 P , (by 14.1., recall: pointwise multiplication m
= sinh [wz(.z;P , ,e",)]< sinh(a) = 1 . Lemma 14.2. provides an rn f I? and 2 1 , ...,zn, yl, ...,y, E (zk,
Since, by formula 14.3.,
yi)
= (sin 2, ek 8 e l ) = sin [ ( z , e k 8 e l ) ]
of norm 1 such that
.
14. Grothendieck's Inequality
170
it follows that
This implies that
which implies the inequality in the real case. To treat the complex case the notation @x,aK and w r will be used. It has to be shown that ac<2K$w? on GacG,
Kz
where denotes Kn with the l,-norm. The R-linear identity map I : 0 ,4Ez has norm 1 and its inverse has norm &. The space ( G ) R is nothing but iZ& considered as a real space; it was shown (Ex 2.11. and Ex 3.15.) that the natural map
has norm one (it is even a metric surjection). For the w2-norm observe first that for PmEx 8.1. implies that
21,...,zmE
NOW take z0 E G@'P,with w f ( z o ; G , G ) < 1. Then there are elements 21,...,zrn and yl, ...,ym E G with zo = C?=:=, zhmcyk and wf(zk) 5 1 as well as wf(yk) 1. Therefore
<
rn
21
:=
EzL aR yk E (G)R
@R
(G)R
k=l
satisfies O(z1) = zo and w f ( z 1 ;
( E ) R(G)R) , < 1. It follows that
171
14. Grothendieck'8 Inequality
14.5. By 1-complementation it is clear that r 5 Kcw2 on Pm 8 hence the &-local technique lemma 13.5. gives that:
COROLLARY 1: If E
is a CL,x-space and
for all n, rn E M ,
F a E&,,-space, then on
E 8 F .
In particular,
COROLLARY 2: For compact spaces K and L
PROOF: Since, by the preceding corollary, a 5 K G W on ~ C ( K )8 C ( L ) ,it follows that the natural map C ( K ) & , , C ( L )= C ( K ) & w , C ( L )4C ( K ) & C ( L ) = C ( K x L ) is injective. By 12.10, the assumption on the function f means that f E C ( K ) & , , , C ( L ) and wZ(f;C ( K ) ,C ( L ) )5 Ilglloollhlloo,which implies the statement. 0 Recall that f E C(K)&,,C(L)means that f has an absolutely convergent representation
which easily gives a representation f = (9,h) as in the corollary.
COROLLARY 3: If H i s
a Hilbed space and X I ,...,xn,y1, ...,yn
E H , then
172
14. Grothendieck'a Inequality
or For H = t z or LZ it does not matter whether one writes the scalar product (-1 the duality bracket ( ., ..), since, clearly, there are elements #j with 11#j:jll= llyjll and . a )
(zilyj)
= (zi,#j)
*
An immediate consequence is the Lindenstrauss-Pekzyriski formulation:
GROTHENDIECK'S INEQUALITY(matrix form): Let (ai,j) E C(Kn,K")be an n x n matrix. Then for each H i l b e r i space H
PROOF:Define cp E (& Br C)' by
then
Hence for zi, yj E BH
by the preceding corollary. This is the claim.
14.6. Note that each of the last four statements was deduced directly from the former one with the same constant; hence the best constants in the last four statements are non-increasing. Actually they are the same:
14. Grothendieck'a Inequality
173
PROPOSITION: The best constants in - Grothendieck's inequalitiy, tensor form - Grothendieck's inequalitiy, function form (corollary 2) - corollary 3 - Grothendieck's inequalitiy, matriz form all coincide. In particular, it will be shown that all of the last four statements are equivalent formulations of the same fact.
which is a metric surjection: For wz(z;Pm,Pm) ...,znand y1, ...y, E B12 with
<
1 there are (by 14.2. or 12.9.)
21,
=C
( z i , Yjtj)ei 8 e j
9
i ,j
hence ~ ( zPm, ; &) 5 KG , which implies the tensor form with the constant from the matrix form. 0
14.7. Considerable effort has been made to find the exact values of the Grothendieck constants K$ and K$ but they are still unknown. A lower estimate (due to Grothendieck) can be obtained as follows. Denote for the moment by KLG the best constant in the little Grothendieck theorem (11.11.); then: PROPOSITION:K ~ I GKG
PROOF: For E > 0 there is an ra E N and T E Z(P,,&) with JlTll= 1 and P Q ) > 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 <
A
+ 4)= 1.78221...
2 1 41 A
- ln(l+fi)
= 3.56442...
Pisier [220] showed that K,$ _< exp(1 - 7) = 1.5262... (where y is Euler's constant) and Haagerup [98] improved this to KE 5 1.40491 using a representation of the scalar product
for z,y E e 2 (complex) with 1 1 ~ 1 = 1 ~ llyl12 = 1 (sign denotes the complex sign function) which he deduced from the representation of the real scalar product. The difficulty is that the function
is much more complicated to handle than the inverse of the sine function, but apart from these tough analytical calculations, the proof runs along the same lines as the proof of the real case since W 1can be expressed as a power series (in X and I). In particular, these estimates show that KZ < K Z ; Krivine [154] showed directly by an estimate of the norm of the complexification of a real operator Pm -, that K g 5 f i K $ (see Ex 14.4. for a slightly worse estimate). Concerning the lower bounds it seems that Krivine showed that Kg 2 1.676 and Davie that K g 2 1.3381, but these results are unpublished (see Pisier [225], p.68, Jameson [119], p.114 and Konig [I481 ).
14.8. We resist the temptation of giving the operator form of Grothendieck's inequality now (see 17.14.) and prefer instead to continue with the theory of tensor norms in order to discover the precise connection between operator ideals and tensor norms. Nevertheless, we prove a result, which will be seen to be nothing but c(tl,tz)= Vl(t1,ez).
175
14. Grothendieck’s Inequality
PROPOSITION: a I K c d , on e;t 84 for all n E N.
PROOF:For ti E e with
ti
n = Cj,l ai,jej
Hence, for every z = CEl Zi 8 Vi E
84
i=l
j=1
i=l
by Grothendieck’s inequality (matrix form). Taking the infimum over all representations of z gives a(z;G,G)5 K c d , ( z ; G , G ) . 0 It will turn out that the best constant in this relation is also the Grothendieck constant K c (see 17.14.and 20.17.). For other versions of Grothendieck’s inequality see 17.14., 19.5., 20.17.,22.4., 23.10.and Ex 23.18.. Grothendieck’s original formulation will be presented in 20.17..The so-called “finite dimensional” Grothendieck constants will be treated briefly in 20.17.~ well. Other proofs of Grothendieck’s inequality will be given in Ex 14.1.,31.6. and Ex 32.7..
Exercises:
Ex 14.1. Take elements t 1 , ...,tn,yl, ...,yn E R” of euclidean norm 1 and denote by 0(ti,yj) E [0, a] the angle between ti and yj. (a) Prove that for all k E IV
176
14. Grothendieck's Inequality
Hint: 14.1. and 14.3.. (b) Check that
(c) Prove that A(
2
A
(zi,yj)ei @ e j ; c , c )
5 sinh(-)2
.
iJ=l
(d) Show that these arguments prove Grothendieck's inequality in tensor form with 5 sinh(~/2).Hint: 14.6.. This is a slight modification of Grothendieck's original proof of the "th6orhme fondamental" . Ex 14.2. Let K and L be compact sets, (o E BiI(C(K),C ( L ) )and g k E C ( K ) ,hk E C ( L )be given. Then
Kg
What happens if (o comes from a signed measure on K x L? Ex 14.3. Let Sn-lbe the unit sphere of q and sn(z,y) := (sly) for (a) Show that sn E C(Sn-l) 8 C(Sn-l) and
2 ,y
f
5 1 , A(&; c(sn-I),c(sn-1))5 K G . yl, ...,yn E Sn-l (b) Prove that for all 21, ...,zn, W ( s n ; C(sn-l),
c(&-1))
n
A(
C (zilYj)ei 8 e j ; c ' G )
5 ?r(sn;~(~n-l),~(~n-l))
-
i,j=l
Hint: A ( C i , j g ( z i ) h ( y j ) e i 8 ej;&,&) 5 Ilgllmllhllw for all g , h E C(Sn-1). (c) Conclude that li%+m r(sn;C(Sn-l), C(Sn-l))= KG . Hint: Corollary 3. For a more precise statement of this result see Ex 20.16. (and 20.15.). Ex 14.4. Recall from Ex 11.19.(d) that for every A E CT 1A1=
1'" 0
.
Ih(eitA)ldt
(a) If A : Pw-+ Ry is an R-linear operator and Ac its complexification then
llA": Cw
-+
A q I I 5 5llA : Rk
-+
-+
q,
Ry(I .
See also 26.3. and Ex 28.14. for the complexification of operators.
(b) Show that K g (c) For
A :=
(i )1:
5
5 KE . Hint:
Matrix form.
: RL + R: the following holds: llA"ll
the best constant in hence obtains KF 5
is 2
f i .Krivine
= ./211All. This shows that
[154] calculated that it is actually
f i and
25. Dual Tensor Norms
177
15. Dual Tensor Norms
The purpose of this section is to study the norm which is induced on E’@ F’ by its embedding into (E @p F)‘. This investigation is quite fundamental for applying tensor product methods to operator ideals, as will be seen shortly. It also gives new insight into the importance of the cofinite hull of a tensor norm. Moreover, some notions (“accessibility”) concerning the relationship between and are defined and tensor norms close to Ap are investigated.
15.1. For two separating dual pairings ( E i ,Fi) of normed spaces the bilinear mapping
(El 8 E 2 ) x (Fl 8 F2) N
M
n=l
m=l
N
K
M
n=l m = l
gives a separating dual pairing (El 8 E z ,FI @ Fz) according to Ex 2.4.. This simple and natural pairing is also called trace duality since it is the restriction of the trace duality studied in 2.6.: Clearly, the duality bracket induces a mapping 0 CP : Fi @ Fz
-
(El @ E2)’ = L(E1, E ; )
which is injective (since the pairing is separating) and the elements of the range of 0 are certain finite rank operators El + E i . For u E El @ Ez and v E F1 @ FZ let Tu E $(,Ti, E2) be the operator associated with u and L, E $(Ez, E i ) the operator associated with w := C P ( V ) ~ E (E2 8 E l ) * . Then (21%
4 ~ ~ c x m=,(@(v), ~0~ ~ (~ E ~ @ E ~ ) * ,= E ~tm2 B E(T~’
-
0
L,) = t r E ; ( L ,
0
T,) .
Note again that transposing z E E @ F 4 2 ( E ‘ , F) means passing to the dual of the associated operator T, - more or less: Tl = ICE o T,t . 15.2. The interesting cases for the (restricted) trace duality are the injections
E @F E l @F E’@F’
(E‘ @t F’)’ ( E QDC F’”’ (E&F)
(E‘ F’)’ ( E @o F’)’ c (E@pF)’
(E’ & F‘)’ ( E ~F’)‘ x (E@rF)’
(p any tensor norm). Which norm is induced on E @F by (E’@pF’)’? TOanswer this question dual tensor norms will be introduced. Recall that M
@N
= (MI@* N’)’
178
15. Dual Tensor Norms
holds algebraically for all M, N E FIN.
PROPOSITION: Let a be
a iensor norm on
FIN. Then a' defined by
I
a ' ( z ; M ,N) := sup{((z~,~)Ia ( u ; M',N') 5 1) (for all z E M 8 N ) is a iensor norm on FIN.
PROOF: To apply the criterion 12.2. (for FIN), observe first that a'(-;
r) is a norm,
(2) is obvious and the mapping property (3) follows from
((Tl @ Tz)z, u) = (2,(T:63 Ti)u)' In other words,
M
@at
N := (M'
*
0
ma N')'
~-
A
holds isometrically by definition. The finite hull a' of a' will be called the d a d tensor norm a' (on NORM) of the tensor norm a (on FIN or NORM). 15.3. The following properties are obvious: (1) If a 5 cp, then 5 ca'.
( 2 ) a = a" on FIN and = a". (3) a =a ' on NORM if and only if a i s finiiely generaied. By dualization, the relation
and hence
E
5 a' 5 R implies for a = E that
d = E and E' = R .
This is part of the duality relation between the projective and injective norm which was treated in section 6, where, in fact, much more was achieved. Note that for two finitely generated tensor norms a and p the relation /3 = a' is just a statement about finite dimensional spaces. It is obvious that (a')' = (a')' and this norm is denoted by a*;it is called the adjoint or contragradient tensor norm of a.At this point it may not be apparent why one would want to consider a*.The reason will become clear in 17.9., where the relationship to adjoint operator ideals will be explained. 15.4. There is an interesting connection between Saphar's tensor norms gp and its duals:
REMARK: g; = db 5 gpj for 1 5 p 5
00
(where p' i s the conjugate exponent of p ) .
179
15. Dual Tenaor Norms
PROOF:From dl = g1 =
?r
it follows that d: =
1 < p < 00 it has to be shown that
llM @gp, N
(M’
E
5
g m and db,
5
?r
= 91. For
a d p N’)’ll 5 1
for all M , N E FIN since it is enough to check d; spaces. In other words, it has to be shown that
I gp’ on finite dimensional Banach
I(u, 41 I dp(u; M’, “)gp’(Z;
M ,N )
for allu E M ’ @ N ‘ and z E M @ N . Take = Enz;@d and z = Em zm@vy, (finite representations with all entries different from zero); then Holder’s inequality gives
n
m
I epj (zm )wp’(zk )wp (3/m )ep ( A1
*
Passing to the infimum over all representations of z and u yields, by the representation 12.7. of dp and gP’, the desired inequality. 0 At the end of this section this remark will be used for determining the tensor norms close to Ap. The inequality g;‘ 5 g p will be improved in a certain sense in 20.14..
15.5. It is clearly of fundamental importance to know in which sense the isometric embedding for finite dimensional spaces N‘
L( M
N)’
extends to infinite dimensional spaces. The answer is given by the
DUALITYTHEOREM: Let a be a tensor norm (on FIN). Then for all normed spaces E and F the following natural mappings are isometn’es:
15. Dual Tenaor Norms
180
PROOF: To prove (3), observe first that FIN(E') = { K o l K E COFIN(E)} and that, for ( K , L ) E COFIN(E)x COFIN(F)
c E' @ F'. Now,by the valid duality relation
whenever z E E @ F and u E KO @ Lo for finite dimensional spaces
and this is exactly (3). The extension lemma 13.2. and the commutative diagram
imply (2) - and (1) follows in the same way. 0 The proof showed that the duality theorem is nothing but a reformulation of the definition of the cofinite hull. The duality theorem indicates that the cofinite hull of a tensor norm a will be an extremely helpful device. Since & 5 a,the theorem implies in the theorem (8: replaced by BU)are conin particular that all mappings @a + tinuous and of norm 1 (if E and F are different from (0)). The approximation lemma 13.1. gives = a on E @ F if both spaces have the metric approximation property, 1 (El F')' in this case. But what is the situation in general? hence E BU F
-
a
c
z
15.6. To get information in this direction recall first from section 13 that 5a5 holds for all tensor norms a on NORM. A tensor norm a is called right-accessible if
z
+
a(.; M , F ) = Z(.; M , F )
for all ( M , F) E F I N x NORM, left-accessible if a ' is right-accessible, and accessible if it is right- and left-accessible. a is called totally accessible if = i.e. if a is finitely and cofinitely generated. The injective norm E is totally accessible (this was already
z z,
181
15. Dual Tensor Norm8
mentioned in 12.4.), the projective norm A is accessible by theorem 6.4. (duality of and E ) and the duality theorem - but it is not totally accessible: The mapping
E’63rF’-W g(E,F’)
4
?r
J ( E , F’) = ( E B CF)’
is in general not injective (see section 5 about the approximation property and also 6.2.), hence A # by the duality theorem. It will be shown in section 21 that all ap,q are even totally accessible. are accessible and that all Is every finitely generated tensor norm accessible? This problem seems to be quite hard since - by the approximation lemma - the non-accessibility of a tensor norm can only appear on spaces without the metric approximation property. In section 31 Pisier’s space P (which does not have the approximation property, see 5.2.(1)) will be used to construct a tensor norm which is neither left- nor right-accessible. It is obvious from the definitions that the presence of accessibility facilitates the investigation of questions involving trace duality. It will turn out that in some situations an accessibility assumption is even necessary, see 25.4., proposition 1. In section 21 the accessibility properties of tensor norms will be “translated” to conditions on operator ideals. Concerning the distinction between right- and left-accessible it is worthwhile to note that in specific situations it is sometimes enough (and easier) to only check that a tensor norm is accessible from one side - and that there are even tensor norms which are only accessible from one side (see 31.6., corollary).
PROPOSITION: Let a be a tensor norm on NORM. (1) a is right-accessible if and only if a’ is right-accessible. (2) If a is accessible, then the transposed tensor norm a t , the dual tensor norm a‘ and the adjoint tensor norm a’ are accessible.
If a is totally accessible, a‘ is accessible (by (2)) but in general not totally accessible: take a = E as an example.
PROOF:Clearly, only (1) needs
a proof. Take a right-accessible; then the duality
theorem gives
M I aallF’
A M I a;
F’
A M I a:
F’
A (M
F)’
for finite dimensional MI hence
+
+
-
which means that a’ = a’ = a‘ on M @J F (again by the duality theorem). This shows that a’ is right-accessible. Conversely, if a‘ is right-accessible, then a’’= Q is right-accessible and this is the same as a being right-accessible. 0
182
15. Dual Tensor Norms
15.7. Recall that it is an immediate consequence of the approximation lemma 13.1. that a = fl on E 8 F whenever a = p on M 8 F for all M E FIN(E) and E has the metric approximation property. This observation and the duality theorem show that for a tensor n o r m a the relations
E @a F
1
A E 8-a F and E&F
(El
F‘)‘
hold in each of the following three cases: (1) E and F have the metric approximation property. ( 2 ) a i s right-accessible and E has fhe metric approximation property. (2’) a is left-accessible and F has the metric approximation property. ( 3 ) a i s totally accessible. So, “two ingredients” are necessary in order for a “good” relationship to hold between
a and a’.For the bounded approximation property the relations hold isomorphically. It will be seen in the next section that for a = T the metric approximation property is in some sense even necessary for the “good” duality. Note also that each of the conditions (1) - (3) (with “metric” replaced by “bounded”) implies that the natural map E&F E&F is injective, but this observation will be improved in 17.20.. The definition of the dual tensor norm (as the finite hull of what one might have expected) and the duality theorem (explaining the role of the cofinite hull) appear innocent enough. However, in the forties Schatten defined the dual norm by using the embedding E 8 F C (G ma F‘)‘ and ran into severe difficulties; it was Grothendieck in his Rksumk who used the definition of the dual tensor norm as a finite hull - and it turned out that this is the appropriate notion for the investigations he had in mind, of a for example, for the study of operator ideals. By the way: the cofinite hull (see [93], p.11). tensor norm a is, by the duality theorem, Grothendieck’s norm 11 Nowadays, when it seems natural to study Banach spaces and operators in terms of ++finite dimensional spaces, the definitions of a,a and a‘ fit easily into the accepted philosophy - a successful philosophy whose beginning was marked by Grothendieck’s definitions (and results) in his %sum6 1954.
-
15.8. Two tensor norms being equivalent on some E 8 F is always a fruitful situation. Up to now, Khintchine’s and Grothendieck’s inequalities have produced the most powerful equivalences. In 13.5. the relation E 5 wp 5 XE on E 8 G was proven to hold whenever G is an 2;,Aapace. Dually: PROPOSITION: Let E and G be normed spaces, G an 2;,A-space. Then w;
5 f 5 Xw;
PROOF: For NE FIN it follows from E = wp#on N‘ 8
e;:
N = (q,8 t N‘)’=
($1
-
onG8E.
c,that
Bw;, N’)’ - (GI QW,
Nt)I = q Bw;N
.
15. Dual Tensor Norms
183
The &-local technique lemma 13.5. implies the result since wh is, by definition, finitely generated. 0 The trace on E‘ @ E is always continuous with respect to the projective norm z. Even more holds in the above situation: The proposition implies that the traces
and
(for a Hilbert space H) are continuous. It will be seen in section 23 that the statement of the proposition is actually characteristic for C;-spaces.
15.9. A nice analytic application is the following: In 8.8. the values of
were calculated explicitly by using some averaging and the knowledge of the &-norm of - e k 8 e k (which is the operator norm of id : $, -+ To obtain a result of this type for general tensor norms a define for the moment
q).
and the operators
S(Z1,- - - I Mw(21,
2,) -.-I
:= (
2,)
~ 2 1 ~ - .3. I 1t n l ~ 1 )
:= ( E I ( W ) Z I , - - - , E n ( W ) Z n )
for w E D, = {-l,+l}n, where the ~k are the Rademacher functions from 8.5.. It is easy to see that
for all z E K” 8 Kn (duality in (Kn8 K”,Kn@ Kn)).If z = t o:= ELzla k k e k 8 ek is the associated “diagonal” element, then
a i j e j 8 e j and
184
15. Dual Tensor Norms
(where pn is the "Rademacher" measure on Dn)and n
C sk8 sk(t0) = t r ( t ) Jn) = ( z , Jn))Jn) .
(3)
k=l
This is obvious for z = ei @ ej and the general case follows from linearity. Now everything is prepared for demonstrating the following result whose idea is due to Garling and Gordon [79] (see also Jameson [119], p.72):
PROPOSITION: Let E and F be Kn eqvipped with n o m s such lhat llMwII = llSll = 1 (in E(E, E ) and 2 ( F , F ) ) for all w E Dn.Then
k=l
k=l
for all tensor norms a.
The conditions on E and F are satisfied for the tp-norms and more generally for Orlicanorms on K". Note further that the result implies a relationship between certain norms of the identity operators E' -+ F and E --+ F' (see section 17). A result which is a bit more general will be treated in Ex 15.4..
PROOF: Since n = (u("), d"))5 a(u(");E , F)a'(u(");E', F'), one inequality is obviF')' = E a a F holds isometrically; ous. To see the other one, observe first that (E' therefore there is a t E E 8 F with a ( z ;E, F) = 1 and (t,d"))= a'(u(");El, F'). Then, by (2) and (3) a(u(");E , F)a'(u(");E', F') = ( 2 , d")) . a(u(");E , F ) = n
8 sk(Z,); E , F ) I n +,;
=
E,F ) I
k=l
a(Mw8 M W ( z ) ;E , F)pn(dw) L n
by the mapping property of a and pn(Dn) = 1 . 0 15.10. The duality relation for the "one-sided" tensor norm Ap is exactly what one would expect:
PROPOSITION: For 1 5 p 5 00, any normed space E and any measure p
15. Dual Tensor Norms
185
PROOF:Since A1 = A and Am = E (see 7.1.), the claim for p = l,00 follows from the duality theorem because e and A are accessible and L1 and L, have the metric approximation property. For 1 < p < 00 the assertion is an immediate consequence of the isometric embedding
(see Appendix B12.). 0
COROLLARY 1: For every measure p , every normed space E and every 1 5 p 5 00 the following inequalities hold: on L p ( p ) @ E . dp 5 A, 5 s;‘ PROOF: Since d, = w1 = E on L, @ E and A = w< = g& on L1 @ E (see 13.5. and 15.8.), the result holds for p = 1 and p = 03. Let 1 < p < 00. Since dp and Ap are smaller than the projective norm, the relation dp 5 Ap only needs to be verified for step functions n
the (Am) being pairwise disjoint and of positive measure (by the density lemma 7.4.). 0bviously, Ap(f) = l p ( p ( A m ) l / P t m ; E ) and
dp(f; L p , E ) 5 ~p’(p(Am)-l’PXA,; Lp)ep(p(Am)l’PZm; E ) -
For g E B L ~ ,
and it follows that w ~ ’ ( ~ ( A ~ ) - ~ / P x A 5 ,1.) This shows that dp 5 Ap. To see that the other inequality holds use duality: By the proposition and the duality theorem there are isometries
t86
15. Dual Tensor Norms
and the vertical mapping has norm 5 1 by what was already shown. Therefore, c
Ap 5
4
1
5 dil = giI
and the result is proven. 0 Using g i l
gp from 15.4., the Cheuet-Persson-Saphar inequalities $8
I dp I Ap I $ + I ‘9p
on
L, 8 E
are obtained (due to [27], [208], [242]), where E is an arbitrary normed space.
COROLLARY 2: d;, = dp = Ap = A; = g;l = gp on L p ( f 2 ~ , p 8 l ) Lp(R2,p2) for arbitra y measures pi.
PROOF:Since dil 5 dp 5 Ap 5 g;l 5 gp on L p ( p l ) 8 E and gp 5 A; 5 dil on F 8 Lp(p2)by the Chevet-Persson-Saphar inequalities, the result follows. 0 Later on (25.10.) the spaces E such that Ap = dp (and so on) on Lp 8 E will be characterized.
15.11. To finish this section it will be shown that dp and norms closest to Ap, a result of Gordon-Saphar [87].
are actually the tensor
PROPOSITION: Let a be a tensor norm, p a measure such that L p ( p ) i s infinite dimensional and c 2 1. (1) If a 5 cA, on L p ( p )8 E for all normed spaces E , then a I cdp. (2) If Q i s finitely generated and Ap 5 ca on Lp(p)8 E for all normed spaces E , then g;1
I CQ.
Actually, the condition that a is finitely generated in (2) is superfluous, see Ex 15.6.. See also 25.10. and 29.11. to understand how close dp and Ap are.
PROOF: Since lpis “nicely” 1-complemented in L p ( p ) (see Ex 7.1.), it is enough to consider tPinstead of L p ( p ) . (1) For the tensor contractions Ci (see 12.9.) the following diagram commutes
187
15. Dual Tensor Norma
Ex 12.11. showed that (71is a metric surjection and llC2ll I 1; since the left vertical map has, by assumption, norm I c, it follows that the right one also has norm I c. This means that a 5 cd, on G @ E. Note that E could be fixed.
(2) If Ap 5 c a on for A,)
+
and therefore a' c
ep @ E, then
I cApt on 4,)
a'
I c$,
on
M 8 E for all M
giI
Ica. 0
(by the duality theorem for a and the duality 15.10.
@
El. From the first part of the proof it follows that
on M' @ E' for all M E FIN. Dualizing again yields
E FIN. Since gi, and a are finitely generated, this implies that
For a fixed normed apace E the proof shows that a 5 cdp on all G @ E if (Y 5 CA, on L p ( p ) @ E for some infinite dimensional L,(p); using the fact that g i l is totally accessible (this will be proven in section 21), statement (2) reads: gi, G 8 E if Ap I c a on L p ( p )@ E (where a is finitely generated).
I c a on
all
15.12. A neat application of the fact that there are tensor norms close to A, is the following alternative proof of the extension of Beckner's result given in 7.9. (we omit the case of subspaces El @A; E2) :
PROPOSITION: Let 1 5 q I p 5 some measures p and v. Then
00
and T E 2 ( E , L p ( p ) )and S E 2 ( L , ( v ) , F ) for
PROOF: For z E E @ L, the Chevet-Persson-Saphar inequalities and 15.4. imply that
This result does not hold for p < q : This is the content of Maurey's factorization theorem which will be treated in 18.8.. In 26.3. the reader will find an explicit counterexample and other positive results in this direction. See also section 32.
15. Dual Tensor Norms
188
Exercises: Ex 15.1. Let a be a tensor norm on NORM and k E HV. Then a('), defined by a(')(z;E, F )
:= sup{l(u, z)I
I
II
E E' 8 F', rank(u) 5 k,a ( u ;E', F') 5 1) ,
is a tensor norm on NORM. If a is finitely generated, then c
lim a ( ' ) ( ~ E l; F) = a'(z;E , F )
k-wx
for all normed E l F and z E E 8 F. Hint: Duality theorem. Ex 15.2. Recall that there is a natural continuous mapping
J : E&&F
-+
E&F
C(E',F ) .
(a) Take z E EgaF.Then (p6&idp(z) = 0 E K 8 F = F for all cp E E' if and only if the associated operator J ( z ) : El -* F is zero. (b) Formulate conditions on the spaces El F and the tensor norm a in terms of accessibility and the bounded approximation property such that the following holds for each z E E g a F : If ( ~ & i d ~ ( z=) 0 for all cp E E', then J = 0. Hint: 15.7.. For another weaker condition see Ex 17.11.. Ex 15.3. (a) Show for E = that for every E > 0 there are R E C(E,$) and S E C ( G , E ) w i t h S o R = i d E and
C
(c n
Ils((llRll 5 w2
ek
8 e k ; E', E) (1 4- €1.
k=l
Hint: 12.9.. (b) Show that the trace on f!, @ l p t is not continuous with respect to w i if p # 2. Hint: z ) this would imply wZ(u("); $,, 5 c for some c and hence, by With (tr,z) = (d"), (a), d ( c ,6 )5 2c; but d($, 4) 00 (a well-known fact, see also Ex 26.4.). This is a proof by local technique. Later on (23.2.) a more natural result will be given: if the trace on E is wi-continuous, then E is isomorphic to a Hilbert space. Ex 15.4. Using the ideas of 15.9.show that: If E and F are lK" equipped with some norms, then
C)
-+
where
k=l
is the symmetry constant of the canonical basis
(ek)
of E .
15. Dual Tensor Norms
189
Ex 15.5. Show that L z ( p ; H) = L z ( p ) 6 d a H = L z ( ~ ) holds ~ ~ ~isometrically H for all Hilbert spaces. Hint: 7.2.. Ex 15.6. Use the fact that g; is totally accessible (to be proven in section 21) to show that for each tensor norm a the inequality Ap 5 c a on all .tp 8 E implies that gp+/5 ca.
E x 15.7. Is the natural duality ( E g 0 F ,E’ @ F’) separating? Ex 15.8. Show by using the duality theorem that
Ex 15.9. If E is an Ci,x-space, then dp 5 Asp+, on E 8 F for all normed spaces F. Hint: 13.5.. Ex 15.10. Let a be a tensor norm. (a) If A c E&F is bounded, then A is u(EI&F, E’ 8 F‘)-bounded. (b) The converse of (a) is true for a = E , but not for a = ?r. Hint: Uniform boundedness principle. (c) If a = e or E’ @ F’ is norm-dense in (EGQF)’,then every U ( E & I ~E‘F@ , F‘)convergent norm-bounded sequence in E g 0 F is weakly convergent. Hint: 4.6.. Note that if for two locally convex topologies on a vector space the convergent sequences coincide, then the Cauchy sequences also coincide (proof?). A subset A of a normed space is called weakly conditionally compact if every sequence in A has a weakly Cauchy subsequence. Assume now for (d), (e) and (f) that E and F are Banach spaces, F separable and a is such that every norm-bounded, a(E&F, E‘ 8 F’)-convergent sequence in E&F is weakly convergent. This implies, in particular, that the duality C(E‘, F ) is injective). system (E&F, E’B F’) is separating (and 80 E&F (d) A bounded subset A c E&F is weakly conditionally compact if it satisfies (denote J1 : EGaF C(E’, F ) and JZ : F & I ~ ~ C(F’, E E ) the canonical maps) (1) ([J~z](z’)~z E A} c F is weakly relatively compact for each 2’ E E‘. (2) ([J&](y‘)lz E A} c E is weakly conditionally compact for each y’ E F‘. Hint: Take a countable D c F‘ which is u(F’,F)-dense, hence dense for the Mackey topology p(F’, F ) . For a given sequence in A choose a subsequence (2”) for which ([JzzA](y‘))converges for all y’ E D. For 2: 8 yb E E‘ 8 F‘ use (1) and the p(F’, F)density t o show that ( z ” ,2: @ 6,)= ([J1zn](z~), y’,) = ([Jz.z!,](y’,),2:) converges. (e) If, additionally, E and F are reflexive, EGa F is reflexive if and only if it is weakly sequentially complete. (f) Rosenthal’s theorem (see Diestel [54]) states that .t, is not contained isomorphically in E if and only if every bounded subset of E is weakly conditionally compact. Use this to show that, if F # (0) is in addition reflexive, then .tl is not contained isomorphically in E g 0 F if and only if it is not in E .
-
-
-
16. The Bounded Approzimation Property
190
(g) Using only the fact that E &
-
respects subspaces, show that (d) and (e) hold for a = E if F is not necessarily separable. These results are due to Lewis [167] and Rivera Ortun [232].There are many interesting situations in which E'gaF' = (E&F)' holds isometrically, hence the condition on E&F is satisfied, see section 33.
16. The Bounded Approximation Property This section resumes the study of the bounded approximation property introduced in section 5. A characterization in terms of T = 'ir will be given. With the aid of the Radon-Nikodim property (see Appendix D) the full duality of E and T will be investigated - as well as the duality of the spaces of compact, nuclear, and all continuous operators.
16.1. According to definition 5.1. a normed space E has the &bounded approximation oS.property (for X 1 1) whenever idE E A B ~ ( E , E ), where T,, stands for the topology of uniform convergence on the absolutely convex, compact subsets of E; in other words, there is a net (Tq)of finite rank operators of norm 5 A such that Tqz + z uniformly on absolutely convex compact sets. There are two other locally convex topologies on C ( E ,F) which will be needed: r, := the topology of pointwise convergence rw := u ( C ( E ,F ) , E
8 F'), the topology of pointwise weak convergence.
LEMMA:Let E and F be normed spaces. (1) On bounded subsets of C ( E ,F ) the topologies rcoand r8 coincide. (2) ( C ( E ,F ) , T ~ ) = ' E 8 F' with the obvious duality bracket n
n
k=l
k=l
(3) For bounded, conuet subsets A
c 2 ( E ,F ) the
closures with respect t o rco,rr and
rw coincide: -rce
A
1 . =A =xY~C(E,F).
26. The Bounded Approzimation Property
191
PROOF: (1) is simple and was already used in 5.1.(2). Tosee (2) take (o E ( C ( E ,F ) , r,)‘; then there are 21, ...,znE E with
and hence, by the Hahn-Banach theorem, (o factors through
rv
Since (F“)’ = (F‘)”, the result follows (this was Ex 5.1.). But now = -7. A (for convex A) is an immediate consequence of the following well-known fact: if (G, H) is a separating dual system, then the closures of a convex set A c G coincide with respect to all locally convex topologies 7 on G satisfying (G, 7)’ = H. This, together with (1) gives (3). 0
REMARK: A normed space E has the A-bounded approcimation property if and only if idE
E JBZ(E,E)‘,
16.2. Since A is accessible, it follows that A = f holds on E 8 F whenever one of the spaces is finite dimensional. The approximation lemma 13.1. gives A 5 A f on E @ F if E or F has the &bounded approximation property. The key for proving a sort of a converse to this observation is the following lemma - which will even be stated for arbitrary tensor norms. It characterizes a 5 A= in terms of approximating operators in ( E & F‘)‘ c C ( E ,F”) by finite rank operators E -+ F :
LEMMA:Let a be a t e n s o r n o r m , E and F n o n n e d spaces. T h e n a 5 A ‘ Z o n E 8 F’ if and only if
c
B ( E ~ , F J ) ~ABEIB,,F~
c C ( E ,F”) ,
where u := a ( C ( E ,F”), E 8 F’).
PROOF: This sounds more complicated than it is. Clearly, both E‘8 F and ( E & F’)’ are regarded as subspaces of C ( E ,F”). For each z E E 8 F’ the duality theorem 15.5. gives c a (z;E , F‘) = S U P I(T, .)I T E BE’O,,F}
{
1
and it is straightforward that “(z;
{
E , F’) = S U P I(T %)I
IT
E qEOoF’)’}
*
Now the result is obvious if one applies the following simpIe fact to the separating dual system ( C ( E ,F“), E 8 F’) : If (G,H) is a separating dual system, A and B are
192
16. The Bounded Approximation Property
subsets of G , then A C XBoo if and only if Bo c AAO. Just take A := B ( E B , ~ )and , B := BE#B,,Fand apply the bipolar theorem. 0 The proof shows that the result also holds with the topology u replaced by 5 := U ( ( E @a F I Y ,E ~ x ) . In section 21 this lemma will be used to investigate the so-called bounded a-approximation property. For the moment only Q = ?r is interesting. From T' = E it follows that the relevant spaces in the lemma are
(operator norm in both cases), and hence
?r
5 A% on E 8 F' if and only if
c AB%(E,F)*
B~(E,FII)
(with u := u ( C ( E ,F"), E @ F')). In particular, i d E E B C ( E , E ) c A B 3 ( E , E ) * n & ( E ,E )
A
5 A% on E 8 E' implies
= A B 3 ( E , E ) " Y = AB3(E,E)
2.2.
(by 16.1.), hence E has the A-bounded approximation property. Collecting all this, the following result has been proved:
THEOREM: A normed space E has the A-bounded approximation property i f and only if ?rsAf
onE@F
for all normed spaces F (or only F = El). In particular, E has the metric approtimation property af and only af c A = T onE8F for all normed spaces F (or only F = El).
16.3. From the structural point of view this result is very important. Some immediate consequences are:
COROLLARY 1: If for u normed space E the dual E' has the bounded approximation property, then E has the bounded approximation property as well, with the same constant. PROOF:By the easy part of the theorem (which was just the approximation lemma) T 5 A% on E' @ E . Since A = A' the other part of the theorem gives the result. In the following statement the constants for the bounded approximation property are omitted - since they will be clear anyhow.
16. The Bounded Approximation Property
193
COROLLARY 2: Let E be a normed space. (1) E has the metric (resp. bounded) approximation property if and only if
holds isometrically (resp. isomorphically) for all normed spaces F (or only F = E l ) ; moreover, this is equivalent to saying that
holds isometrically (resp. isomorphically) for all normed spaces F (or only F = E). (2) E' has the metric (resp. bounded) approximation property if and only i f
holds isometrically (resp. isomorphically) for all F (or only F = E"); this is also equivalent to the statement:
holds isometrically (resp. isomorphically) for all F (or only F = El).
PROOF:This is an immediate consequence of the theorem and the duality theorem 15.5.. 0 Clearly, these injections extend to the completions isometrically, resp. isomorphically. For the embedding of the nuclear into the integral operators
this implies: If El o r F has the A-bounded approximation property, then
for all T E %(ElF ) . Under what conditions is % ( E ,F ) = J ( E , F ) ? To answer this question the Radon-Nikodfm property is the appropriate tool. 16.4. A Banach space E possesses the Radon-Nikodim property if for every finite measure p every operator T : Ll(R, p ) + E is representable, i.e. there exists a bounded p-measurable function g : R + E with
T f = J f g dP
for all f E L l ( R , p )
16. The Bounded Appron'mation Property
194
The results needed concerning representable operators and the Radon-Nikodfm property can be found in Appendix C and D, as well as the notion of Pietsch integral operators. The Radon-Nikodfm property has powerful consequences for the questions treated in this section. For tensor norms a variant of the Radon-Nikodfm property will be investigated in Chapter 111, section 33.
COROLLARY 3: Let E be a Banach space with the Radon-Nikody'm property which is A-complemented in its bidual (i.e. there is a projection P : E" --y E with llPll 5 A). If E has the approzimation property, then it has the bounded approzimation property with constant A. PROOF: It follows from Appendix D7.that %(E,E) coincides isometrically with the space 'J?3(E,E) of Pietsch integral operators. The factorization properties of integral and Pietsch integral operators and the existence of a projection E" -+ E imply that 3 ( E , E) = V 3 ( E ,E) and PI(T) 5 AI(T). Since E has the approximation property, it follows that 1 E'&E V 3 ( E ,E ) = 3 ( E , E ) ( E El)' and hence, by the duality theorem, T(Z;
E', E ) = PI(T,) 5 AI(T,) = A?i(z; E', E ) ;
therefore, the theorem implies corollary 3. 0 Reflexive spaces and separable dual spaces have the Radon-Nikodfm property and are clearly 1-complemented in their biduals, hence:
COROLLARY 4: If E is reflexive or a dual space which is separable, then E has the metric approximation property if it has the approzimation property. 16.5. The Radon-Nikodfm property is intimately connected with the duality of the tensor norms E and T .
PROPOSITION: Let F be a Banach space. Then F' has the Radon-Nikody'm property i f and only i f the canonical mapping J E : E'& F'
-
( E @= F)'
is surjective f o r all Banach spaces E (or only E = C ( K ) with K compact, or only L,(p) for arbitrary finite measures). In this case, J E is even a metric surjection for all Bonach spaces E .
PROOF: Surjectivity of JE means that %(E, F') = 3 ( E , F') by the very definitions of nuciear and integral operators, so the result is nothing but a reformulation of corollary 1 in Appendix D7..0
195
16. The Bounded Approximation Property
The result cited from the appendix even implies that in order for F' to have the RadonNikodgm property it is enough to take only E = C[O,11or only E = L, [0,11(Lebesgue measure) in the proposition. 16.6. Combining the results of 16.4. and 16.5. easily gives the following result on the duality of the projective and injective norm:
THEOREM: Let E and F be Banach spaces. (1) If E' or F' has the approximation property and: E' or F' has the Radon-Nikodim property, then
E'& F' = ( E @< F)'
isometrically.
( 2 ) Conversely: If E'&G' = ( E G)' (as vector spaces) for all Banach spaces G , then E' has the Radon-Nikoddm property and metric approximation property.
PROOF: Since E' or F' has the approximation property, the canonical map J : E'GXF'
-
(E
F)'
-
is injective (see 5.6.), hence proposition 16.5. gives (1). For the converse, proposition 16.5. again shows that E' has the Radon-Nikodim property. The injectivity of the mapping E'&G' (E G)' for all G implies that E' has the approximation property (again by theorem 5.6.) and hence the metric approximation property by corollary 3 from 16.4.. 0 16.7. These results have nice applications for spaces of operators. From section 5 on the approximation property it is known that
n ( E ,F )
E'&F
and f f ( E F , ) A E'&F
if E' or F have the approximation property. Proposition 16.5. gives
% ( E ,F') = (E&F)' if E' or F' has the Radon-Nikodgm property and, clearly,
C ( E ,F') = (EGXF)'. The duality is always the natural trace duality.
PROPOSITION: Let E and F be Banach spaces such that E" or F' has the approximation property and: El' or F' has the Radon-Nikody'm property. Then ( f f ( EF))' , = %(El,F') ( f f ( EF))" , = (%(El,F'))' = C(E", F")
isometrically isometrically .
196
16. The Bounded Approzirnation Property
The duality brackets are given by m
n=l m
if T = C,"==l 2," @I b, E %(E', F') = E"&F'. The natural embedding of f f ( E F , ) into its bidual ( f f ( EF))" , = C(E",F") ff(E,F ) is passing to the bidual: S-S"
.
A
PROOF: The statements ff' A % and %' 2 follow from what was said before. The duality brackets originate from the natural embedding
and the identification
(%(El,F'))' = (E"@I~F')' = C(E", F")
.
It also follows (check for elementary tensors) that for S E ff and T E %
which is nothing but the statement that ff
LI
ff" = 2.0
I f E" has the approximation property, then the duality brackets are
An immediate consequence is
COROLLARY: Let E and F be Banach spaces (f { 0 } ) , one of which has the approzimation property. Then E&F is reflexive if and only if E and F are reflexive and every T E C(E,F') as compact. PROOF: If E and F are reflexive and C(E,F') = R(E,F'), then
( f f ( EF'))'' , = C(E,F') = R(E,F')
16. The Bounded Approximation Property
197
by the proposition, therefore C(E,F’) = (E6=F)’is reflexive, hence also E&F. Conversely, if E&=F is reflexive, then E and F are reflexive (as complemented subspaces of E&F), moreover C ( E ,F’) = (E&F)’ and its subspace f f ( E ,F’) are also reflexive. The proposition gives
R(E, F’)= ( f f ( EF’))‘’ , = C(E,F‘) and the result is proven. 0 For an example, recall that Pitt’s theorem (see e.g. Kothe [152], 11, p.208) says that
if 1 5 g < p
< 00. It follows that for 1 < q < p < 00 the space
is reflexive. Recall that 1 2 6 r t 2 is not reflexive (Ex 3.9.). For some more details in this direction, see Ex 16.4. - 16.8.; in section 33 the reflexivity of other operator ideals will be treated. 16.8. Finally an answer to the following question concerning dual nuclear mappings is obtained: If T E E ( E , F ) is such that T’ is nuclear, T is nuclear as well? In 5.9. it was shown that the answer is positive if E‘ has the approximation property. But in general this is false; this can be seen as follows: Take a Banach space E with the approximation property such that E‘ has the Radon-Nikodim property (e.g.: E’ is separable). Clearly
commutes. The Radon-Nikodim property (16.5. or Appendix D7.) implies that 1 2 is an isometry. Since the subspace of dual operators is closed in L!(E’,E’), the space G is closed in %(El,E’). So if11 is surjective, the closed graph theorem yields that 11 is a topological isomorphism and hence J is as well: Corollary 2 in 16.3. now implies that E has the bounded approzimation property. However, the Figiel-Johnson example 5.2.(5) is a Banach space E with the approximation property and such that E’ has the Radon-Nikodim property (since E‘ is separable), but without the bounded approximation property. It follows that:
PROPOSITION: There is a non-nuclear operator the dual of which is nuclear.
198
16. The Bounded Approzimation Property
Note that the counterexample is an operator T E C ( E ,E ) , where E has the approximation property and E’ is separable. For another counterexample, see 22.9..
16.9. There is a simple characterization of the bounded approximation property which has interesting consequences for the factorization of finite rank operators:
PROPOSITION: A normed space E hos the A-bounded approximation property if and only if for every M E FIN(E) and E > 0 there is an R E S ( E ,E ) with IlRll 5 A(1+ e) ond Rz = 2 for all 2 E M . PROOF: Clearly, the condition is sufficient (see also Ex 5.7.). Conversely, take a basis ( e l , . . . , e n ) of M with coefficient functionals eh E E’ and choose T E S ( E , E ) with IlTll 5 A and IITek - ekll 5 6 for all k = 1, ...,n. Then
R := T +
n
eiB(ek - Tek) k=1
COROLLARY: Let E and F be normed spaces, T E $(El F ) and E > 0 . (1) If F has the A-bounded approximation property, then there is an R E S ( F , F ) with llRll
+
((R’oI E E ) z , T ‘ ~ / = ) (Sc,T’y’) for all z E E and y’ E F‘. This implies T o S = T . 0
Exercises: Ex 16.1. A Banach space F has the A-bounded approximation property if and only if for every Banach space E each operator T E & ( E , F ) can be approximated with respect to the topology of pointwise weak convergence by finite rank operators of norm 5 A.
16. The Bounded Approximation Property
199
Ex 16.2. (a) If Go and G are normed spaces and I E C(Go,G)is of norm 5 1 such that ~ ( B G ,is) norming for G’,then I’ : G’-+ G‘, is an isometry, hence f : &, 4 (? is a metric surjection. C ( E @c F)’ is norming for (E QCF)”. (b) Use (a) to decide whether or not BE’ @ B F ~ Ex 16.3. (a) If K is compact and F a Banach space the dual of which has the Radon-Nikodjm property, then ( C ( K ,F))’ = M(K)@,*F’, where M ( K ) is the space of signed regular Bore1 measures on K.In particular, for every (o E ( C ( K ,F))’there are measures pn E B q q , elements y’, E B p and (An) E .f, such that
for all f E C ( K ,F). Hint: Recall that M ( K ) has the metric approximation property by Ex 5.6.. (b) Is statement (a) true for all Banach spaces F? More explicitly, show that
C([0,11)‘63,*C([O,11)’
5 C([O,112)’
Hint: C([O,11)‘ does not have the Radon-Nikodjrm property and C[O,11 is a “testspace” for this property, see Appendix D7.,proposition (2). The following results (Ex 16.4. - Ex 16.8.) are due to Holub [114]; in section 33 the reflexivity of other operator ideals will be treated. Ex 16.4. Let E and F be Banach spaces one of which has the approximation property. Then E& F is reflexive if and only if El& F’ is reflexive. Ex 16.5. Show that for p , q E [l,oo]: (a) ep6,*.fq is reflexive if and only if 1 < q‘ < p < 00 . (b) e p & ’ , is reflexive if and only if 1 < q < p’ < 00 . Ex 16.6. If E is an infinite dimensional Banach space with the approximation property, then neither E& E‘ nor E& E‘ is reflexive. Ex 16.7. Lp[O,l]&Lq[O, 11 is not reflexive if p, q E [l,001 and a = ?r or E . Hint: The Rademacher functions are complemented in Lp[O, 11 if 1 < p < 00, see Ex 8.17.. Ex 16.8. Use the fact (to be proved in 26.1.) that every operator from a n C:-space into an Zi-space factors through a Hilbert space (if 1 5 p 5 2 5 q 5 00) and Pitt’s theorem to show that W , I LP(P)) is reflexive if 1 < p 5 2
< q < 00,
and Z(Lq(P),.fP>
200
17. The Representation Theorem for Maximal Operator Ideals
is reflexive if 1 < p < 2 5 q < 00 . What does this imply for the reflexivity of Lq&d, and La&& ? E x 16.9. If T E 2 ( E ,F ) has a nuclear dual, then it is integral. E x 16.10. It may happen that N(T’) < N(T) even for finite rank operators. Hint: Look at the diagram in 16.8.. E x 16.11. An operator ideal a is called regular if T E % ( E ,F ) whenever the operator K F o T is in a ( E ,F”). Using 16.8., show that the ideal ?TI of nuclear operators is not regular. For positive results in this direction see 22.8.. E x 16.12. If .C(tm, F) = !l31(tm, F), then F is finite dimensional. Hint: The assump1 tion implies that t16bCF 3(&, F ) c* (t, mr F’)‘ ; 11.3., corollary 3.
-
17. The Representation Theorem for Maximal
Operator Ideals
This section deals with the one-to-one correspondence between finitely generated tensor norms and maximal Banach operator ideals - a simple, but extremely powerful correspondence, which will be central to the investigations which follow. It links together thinking in terms of operators with “tensorial” thinking - and as a result both theories, the theory of tensor products and the theory of Banach operator ideals, are incredibly enriched (and very often simplified). There will be two types of characterizations for operators T which are in a maximal operator ideal: firstly, in terms of the associated bilinear form and, secondly, in terms of the continuity behaviour of T @ id : @a + @ p for appropriate tensor norms.
17.1. If
(a,A) is a normed operator ideal, then M@* N
: A rn(M‘,N)
defines a tensor norm a on FIN; in other words, if z E M @ N and T, E 2 ( M ’ , N) is the associated operator, then a(%;M ,N) := A(T, : M’+ N )
.
That a is a tensor norm is an immediate consequence of the criterion 12.2.: the m e t r i c mapping property of a i s the s a m e as the ideal property for % (for finite dimensional spaces) since for z E M 8 N
R o T, o S‘ = Ts@qt)
($1
17. The Repmeentation Theorem for Mazimal Operator Ideals
201
holds for all S E E(M, M I ) and R E E(N, N1). Note that the criteria 12.2. and 9.4. for tensor norms and operator ideals are quite similar.
17.2. Conversely, let a be a tensor norm on FIN. Define, for finite dimensional Banach spaces M and N, a norm A on C ( M ,N ) by A(T : M
+
N ) := ~ ( z TM', ; N),
where ZT E M' @ N = C ( M ,N) is the associated element in the tensor product. It follows from formula (*) that in this way a normed operator ideal (%,A)for finite dimensional spaces is defined. How to extend it to all Banach spaces? (Recall that operator ideals were only defined for Banach spaces, not for general normed spaces.) Of course there are several different ways to do this - for our purposes the smallest and the largest extensions will be the most interesting ones. The smallest will be treated in section 22; for the largest take the following
DEFINITION: Let (a,A ) be a (quasi-) normed operator ideal. (1) For T E E(E,F ) define Amaz(T):= sup {A(@ o T o I g ) %"""
E FIN(E) I LM COFIN(F) E
:= {T E E I A"""(T) < a}
and call (a,A)""" := (%mazIA""") the maximal hull of (%, A). (2) (%, A) is called maximal if (a,A ) = (%maz, Amas).
It is clear that (%"'az, Amas)is always a maximal (quasi-) normed operator ideal. It is also immediate from criterion 9.4. that maximal (quasi-) normed ideals are necessarily (quasi-) Banach ideals. The following observation is simple, but useful. REMARK: Let (n,A) be a maximal quasinormed operator ideal and (23, B) a quasinonned operator ideal which coincides with (%, A) for finite dimensional spaces. Then 23 c 2l and A _< B. In particular, two maximal ideals coincide if they coincide on finite dimensional spaces. Returning to the tensor norm a on FIN, for T E C ( E ,F) define
A(T : E
-+
1
F) := sup a ( ~ ~ ; ~M', ~ F ~ / ,L ;) ;
M E FIN(E) E COFIN(F)
and a := { T E E I A(T) < a}. It is clear that ( % , A )is a normed operator ideal, that % ( M ,N) = M' & N holds isometrically for MIN E FIN, and hence that (a,A) is even a maximal normed operator ideal.
202
17. The Repreaentation Theorem for Mazimal Operator Ideals
17.3. Both finitely generated tensor norms and maximal Banach operator ideals are, by their definition, uniquely determined by their behaviour on FIN.
DEFINITION: A finitely generated tensornom a (on NORM) and a maximal Banach operator ideal (a,A ) are said t o be associated, notation: (%,A) a
if for all M ,N E FIN % ( M ,N ) = M‘ @a N holds isometrically: A(T, : M + N ) = a(%;M’, N). The constructions in 17.1. and 17.2. show that in this way a one-to-one correspondence between macimal Banach operator ideals (24, A ) and finitely generated tensor norms a is established. This correspondence is basic to the understanding of all that follows. 17.4. Before examining this correspondence more precisely, the maximality of various operator ideals will be checked. Obviously, (C,II 11) is maximal and (ff, 11 11) is not. A quick look at the definition shows that the absolutely psumming operators ppare maximal - but there is a nice, rather general result which gives this and a great variety of other maximal Banach operator ideals. For this, take a normed space G and a tensor norm a. Then (1) E 5 a 5 ?r on G @E f o r all normed E ; (2) a(i& @ T ( z ) ;G, F ) 5 llTlla(z;G, E ) for all T E C(E,F). If G is a subspace of some L p ( p ) and a = Ap is the norm induced on G 8 E by L p ( p ) @A, El then (1) and (2) also hold by the results in 7.3.. This is why we adopt a more general notion and call a a right-tensor norm (with respect t o a fixed normed space G ) if it is defined on all G 8 E with E E NORM, and satisfies (1) and (2). We repeat: we do not want to make a fuss over this definition - it is just more practical to treat a and Ap jointly. A right-tensor norm is finitely generated (from the right) if
a(z;G, E) = inf{a(z; G, N) I N E FIN(E), z E G @ N } for all E E NORM and cofinitely generated (from the right) if
a(z;G, E) = sup(a(idc @ Qf(z); G, E / L ) I L E COFIN(E)}
,
Clearly, finitely/cofinitely generated tensor norms are finitely/cofinitely generated from the right (see Ex 12.3.) and it was shown in Ex 7.16. that Ap is finitely and cofinitely generated from the right with respect to every subspace G c Lp((cl).
PROPOSITION: Let G, H be normed spaces, S E C ( G , H ) with llSll = 1, a a finitely generated right-tensor norm with respect t o G and fl a cofinitely generated right-tensor norm with respect t o H . Then the ideal (.Cz*p,Lzlp)defined by
17. The Representation Theorem for Maximal Opemtor Ideals
203
is a m a t i m a l Banach operator ideal.
The superscripts a and p will be omitted if they are clear from the context.
PROOF: Clearly, LS is a norm on each &(E, F); the ideal property and the relation L s ( i d K ) = llSll = 1 are obvious as well. Maximality follows from
I
Ls(T) = sup {p(S @ T ( z ) ;H , F ) a(%;G , E ) < l} =
-
Note that H @r is cofinitely generated if H has the metric approximation property (since ?r is accessible) and hence the proposition implies that the following ideals are maximal: - the ideal Qp of absolutely p-summing operators (see ll.l.(d)) - the ideal Q(,,) of absolutely (r, s)-summing operators (see Ex 11.20.) - the ideal 3 of integral operators (see Ex 17.1.) - the ideal 2, of type p operators (see the definition 7.7.) - the ideal Cq of cotype g operators (see the definition 7.7.) - the ideal of K-convex operators (see Ex 9.16.) where 5 : L 2 ( R ) + &(R)is the Fourier transform (see 7.5.) - the ideal
- the ideal C2r1Ap,where 3 : t!,(ZZ) 00
-+
t,(ZZ))is the Hilbert transform and 1 < p <
(see 7.6.).
Actually, it is much easier (and more natural) to deduce that the integral operators form a maximal ideal by using the following representation theorem. 17.5. If a maximal operator ideal (%, A) is associated with a finitely generated tensor norm a,then % ( M ,N ) = M' @a N = ( M @a# N')'
holds isometrically for all M , N E FIN. The extension of this to infinite dimensional spaces is the important
REPRESENTATION THEOREM FOR MAXIMALOPERATOR IDEALS:Let (%,A) be a
-
maximal normed ideal and a a finitely generated tensor n o r m which are associated with each other: (a,A) a. Then for all Banach spaces E and F the relations
% ( E ,F') = ( E
F)'
204
17. The Representation Theorem for Mazimal Operator Ideals
and
Q(E,F ) = ( E @at F')' n S ( E , F )
hold isometrically.
By definition 10.1. of the integral operators, 3(E, F) = ( E @c F')' n S ( E ,F) holds isometrically. In particular, 3 ( M , N) A ( M&C N')' A M' @= N and hence A 3""";
-
the representation theorem gives
F*"(E, F ) A ( E @c FI)' n C ( E ,F )
-
J(E,F )
-
and therefore 3 = 3""" and A 3 (see also Ex 17.2.). This example explains why operators in 2i are sometimes called a-integral operators whenever Q a. This theorem is due to Lotz [180]. His approach to tensor norms was different from ours and was very influential in the development of the theory of operator ideals: Roughly speaking, he took the representation theorem as a definition for tensor norms and consequently emphasized the one-to-one correspondence between maximal normed operator ideals and tensor norms; see also Ex 17.2..
PROOF: The second formula will be proven first: it needs to be shown for T E C ( E ,F) that T E %(E,F) if and only if the associated bilinear form P l c F 0is~ a'-continuous: P s F o ~E
( E Bat F')'
(with equal norms). But this is easy: T E Q(E,F) and A(T) 5 c if and only if
for all (M, L) E FIN(E)xCOFIN(F); by B ( M , F / L ) = (M Bat Lo)' this is equivalent to ((Pfi~oT .)I , = I(pQ:oT.Izt z)l 5 ca'(z; M , L o ) for all z E hf @ Lo Therefore, the result follows since a' is finitely generated. To see that the first formula holds just look at the diagram
and use the extension lemma. 0
17.6. An obvious combination of the representation theorem and the duality theorem 15.5. gives the
17. The Representation Theorem for Maximal Operator Ideals
EMBEDDING THEOREM: If (a,A)
-
a are associated, then the relations
E' 8F' Q
a(E , F')
E @I-Q F
%(El,F )
E'
% ( E ,F )
@+Q
205
F
hold isometrically. In particular, the extension
E'&F
-
%(ElF )
of E' @a F + %(E,F) is well-defined and has norm 1 (if E and F are not trivial). In 16.3. this map was studied for the case a = T and was useful in understanding the relationship between nuclear and integral operators. More generally, in section 22 the embedding theorem will be crucial for the investigation of minimal operator ideals.
-
17.7. An essential goal of the theory is to compare different tensor norms/maximal operator ideals. The very definition of (%,A) a using finite dimensional spaces shows the
-
-
REMARK: Let (%,A) a and (%,B) /3 6e associated, only i f A 5 c B . In this case, 23 C %.
c
2 1. Then a 5 c p i f and
Very interesting phenomena can be deduced from estimates on special Banach spaces. The representation and embedding theorems
E' 8F' Q
CZL(E,F')
( E @IQ, F)'
imply the following:
-
-
COROLLARY 1: Let (%,A) a and (%,B) p be associated, Banach spaces. Consider the following conditions: (a) p' 5 ca' on E 8 F. (b) B ( E ,F') C % ( E ,F') and A 5 c B . (c) z Then
c
2 1 and E and F
5 c F on E' 8 F'.
(1) ~a)c+(b)-+)* (2) If E' and F' have the metric approximation property, or: a and /3 are accessible and E' or F' has the metric approximation property or: a and p' are totally accessi6le, then (a), ( b ) and (c) are equivalent.
206
17. The Representation Theorem for Mazimal Operator Ideals
PROOF: (1) follows from the observation above. So it remains to show that (c) implies (a) under the conditions of (2). But this follows from the diagram
c
since 'Z = a on E'@ F' and p' = p' on E @F under the assumptions of (2) (use 15.7., 15.6. and 16.3.). 17 This result, although simple, is often quite useful and will be fundamental in the understanding of the interplay between maximal ideals and tensor norms. Since it transfers statements about operator ideals to inequalities involving tensor norms and vice-versa, it will be referred to as the transfer argument. Note that (2) gives conditions under which: a 5 cp on E' @ F' if and only if p' ca' on E @ F.
17.8. The representation theorem has simple, but important consequences for the behaviour of maximal operator ideals. Looking at
Q(E,F )
(E
F')'
A %(ElF") ,
gives the
COROLLARY 2: Maximal operator ideals (a,A) are regular, which means: T E C ( E ,F ) is in ?2I if (and only if) K F o T E % ( E ,F"); moreover, A(T : E
--+
F ) = A( ICFo T : E
-+
F")
.
Recall from 9.9. that Qdual := {T E C I T' E a} and AdUar(T) := A(T').
COROLLARY 3: If (Q, A) is a maximal normed operator ideal associated with a finitely generated tensor norm a, then (Qduar,Adual) is maximal and i s associated with the transposed tensor norm at :
Adual)w a* .
PROOF:Denote by (23,B)the maximal normed operator ideal associated with a'. Then, by the representation theorem, F')' n 2 ( E ,F ) = %(I?, F ) = ( E = { T E 2 ( E ,F ) E (F'
= {T
1 E C ( E ,F ) I T' E %(F',
E)'} = E ' )}
17. The Representation Theorem for Maximal Operator Ideals
207
with B(T)= A(T'). In other words, (23, B) = (adua', AdUa'); this was the claim. 0 Since (!21dua')dua'
-
ati = a
-
2l, one obtains
COROLLARY 4: Let (24, A ) be a maximal normed operator ideal. Then (24dua')dua' = 2f holds isometrically. In other words, T E 24 if and only if T" E a; in this case, A(T) = A(T") . Another way to formulate this is: T is &-integral if and only if T' is &integral.
17.9. Let
-
(a,A ) a be associated.
Then for M , N E FIN
N)' = ( N Bat M')' = N'
( % ( M ,N))' = (M'
M
holds isometrically. Hence, if (a*,A*)is the maximal normed operator ideal associated with (ai)' = a*,one obtains
(2l(M, N))' = 24*(N,M )
- the duality bracket given by (T,S)a. ,a = t r N ( S o T ) = trM(T o S) whenever T E 2l*(N,M) and S 6 O ( M ,N). This is why (a*,A*) is called the adjoint maximal normed operator ideal of (a,A).In section 22 it will be seen how the duality relation is extended to infinite dimensional spaces. When is an operator T E C(E, F) in a*?Observe first that, by duality,
sup
{
M E FIN(E) L E COFIN(F) A(S : F / L M ) 5 1
-
( t r M ( S o Q: o T o
E-F
M
T
FIL
+---------
.
208
A*(T : E
17. The Representation Theorem for Maximal Operator Ideals
-+
M E FIN(E) L E COFIN(F) A(S : F/L + M )5 1
{
F) := sup ltrM(S o Q[ o T o I:)I
17.10. The maximal normed operator ideal associated with the tensor norm ap,*of Laprest6 (see 12.5.) is the ideal ('cp,q, LP,*)of ( p , q)-factorable operators; recall that l/p l / q 2 1 and 1/r := l/p l / q - 1. If for finite dimensional M , N
+
+
T E zp,*(M,N ) = M' @Qp,p N , then, by definition of the norm, Lp,*(T)= a , , , ( z ~ ;M',N), there is for every e representation
>0 a
n
T
M-N I
with llRll llDxll llSll that
t
5 LP,*(T)(l + E ) . Reading this argument backwards, it follows LPJT) = infllRll
llW list1
17. The Representation Theorem for Mazimai Operator Ideals
209
where the infimum is taken over all such factorizations. An example: Since llid : TI + $11 = rillr, it follows that
id : cl -+ 5 )= nilr . These factorizations help to explain the name of the ideal; it will be proved in section 18 that an operator T is in Cp,q(E, F) if and only if it factors
. T
1
note that q’ 2 p : if q’ > p, then the measure p is understood to be finite and I is the canonical inclusion, whereas in the case q’ = p the measure may be arbitrary and I is just the identity map. There are several interesting special cases: the ideal (Cp,Lp)associated with wp = ( Y ~ , is ~ I the ideal of all p-factorable operators. By what was just said these are the operators T such that n~ o T factors through some L p ( p ) . Since A = q 1 , the (1,1)-factorable operators are exactly the integral operators and the factorization theorem mentioned above is already known from 10.5.; more generally, the ideal (Jp,Ip) := (Cp,l, LP,1)associated with gp = aP,lis called the ideal of p-integral operators. Note that 3, = .I& since , wm = gm . It is important to reformulate the factorization of T E C,,,(M, N‘) = (M @a;,QN)’ in terms of bilinear forms: The diagonal operator Dx E f!(ql,$) = (qlBX q l ) I corresponds to the “diagonal” bilinear form n
6x :=
C
Xkek
QD ek
E
$8)‘
.
k=l
Therefore, for every
’p
E (M@* N ) ‘ , Il’pIl(M@.;,*N)’
= inf IPll IlSll 1164
9
where the infimum is taken over all factorizations
where R E C ( M , q , ) , S E f!(N,$,) and X E K ” . Clearly, it is enough to take only X E lK“ with non-negative components.
17.11. Take p, q E [l,m] such that l / p + l / q 5 1. Then l/p’ + l/q’ 2 1 and ( Y , I , ~ I is defined. The ideal (9p,q, DP,,) associated with (Y:~,~, = ( Y ~ ~ , is, ~ the maximal normed
17. The Representation Theorem for Mazimal Opemtor Ideals
210
operator ideal of so-called (p, q)-dominated operators. To describe the operators in 9p,q, define r E [1,00] by 1/r := l / p + l/q ,
+
+
so that l/p l / q l/r' = 1. The representation theorem for maximal operator ideals if and only if the associated bilinear form is Q ~ I , ~ ~ shows that T E &(E,F ) is in continuous: PICPOT E ( E 8 a p t , , ,F')' - with equal norms. This means that DP,q(T)5 c if and only if for every element z = Cr=lA k x k 8 4 E E 8 F' ~ ( P I C F O T I= Z )( ~e A k ( d , T z k ) l5 k=l
C
e,l(Ak)wp(Xk;E)wq(3/I;F') ,
which is equivalent to c
er((!h,T2k))
for all c
21,
...,x,,
E E and
wp(xk;E)wp(d;F')
&, ...,b, E F'; clearly, DP,q(T)is the
minimum of all such
2 0.
-
Special cases: (SP,Dp) wit = ail, is the maximal operator ideal of p-dominated operators; the operators T E DP(E,F? = !I)p,p~(E, F ) are characterized by
4((dVTXk)) I
c
Wp(xk;E)wp'(d;F')
For q = 00 one obtains r = p and the operators in !I)p,m(E,F) are those satisfying
44d,TXk))5 c wp(zk;E)%o(3/I;F') or equivalently (choose appropriate
4EBp)
ep(llTxkll)
I
c
wp(w
E)
for all 21, ...,x,, E E. These are exactly the absolutely p-summing operators from section 11: ('Pplpp)
= (Bp,m,Dp,m)
N
(~;t,i
= $*'
*
In section 19 the Grothendieck-Pietsch domination theorem for ppwill be generalized to JDPlqr thus explaining the name (p, q)-dominated. 17.12. Before going into some examples showing the interplay between maximal operator ideals and tensor norms, it might be helpful to have a list of the operator ideals connected with Q ~ , It~ is . clear from the definition of the adjoint operator ideal and a**= Q (if (Y is finitely generated) that
=
!Dptrqt
and
hold isometrically for p , q E [l,001 with l/p
= &,
+ l / q 5 1 . Hence
- -
17. The Representation Theorem for Maximal Operator Ideals
211
all operators integral operators (p, q)-factorable operators (p‘, q’)-dominated operators p-factorable operators p’-dominated operators p-integral operators absolutely p’-summing operators , where qJm := l:, which is a reasonable definition from several points of view.
17.13. Transfering results from tensor norms to operator ideals and viceversa, is a sort of “game” whose value stems from the fact that it is quite often easier (or much more natural) to prove a result in one form than in the other. We are going to illustrate this now, although not always with the intention of getting the best result; this will be done later. The Khintchine equality gives that aP,(Ibq’bp~wzwhenever p , q E]l,oo[ (see 12.8.). From the very definition of rzL being associated with a it follows that Q(M,N)
M’rXlaN,
hence l:2(E, F
) c Cp,q(E, F ) and Lp,q(T) I bqJ,lLZ(T) for P, Ell,I..
for all Banach spaces E and F. Since w2 that
I ap,q,whenever p, q E]1,2], it follows even
.C2(E,F ) = l : p , q ( E , F ) and LZ(T) 5 Lp,q(T) IbqJJp’L2(T) for p , 4J €11921 * The relevance of these results stems from the fact that the 2-factorable operators are exactly those which factor through a Hilbert space (to be shown in the next section), thus emphasizing the very special role of Hilbert spaces in the theory. What does g,; 5 gp (see 15.4.) imply? Since v p
-
$1
5 Sp
-
this clearly means (again by the definition of 2t sional spaces) that
Jp
-
9
a and extension to infinite dimen-
Jp(E,F) c Vp(E,F) and Pp(T) I Ip(T) (this inclusion will also follow easily from the factorization behaviour of p-integral and absolutely psumming operators). These results were obtained by using the simple transfer argument : a 5 cp if and only if A 5 cB.
212
17. The Representation Theorem for Maximal Operator Ideals
For special pairs of spaces the method of arguing with the representation and embedding theorem is the transfer argument from corollary 1 in 17.7.. An example: From the Chevet-Persson-Saphar inequalities (see 15.10.) it is known that dp 5 gi, on L, @ * which implies (Lp @g;, E’)’ll 5 1 ll(Lp @ d , El)’
-
-
-
Since d i = g; Y,I and (g;,)’ = g;l 3Pa‘(by the description of the dual operator ideal in 17.8.), it follows from the representation theorem for maximal operator ideals that Yp1(Lp, E ) c 3$’”’(L,, E) and I,I(T’) = q a ’ ( T )5 PPd(T). For a “transposed” result see Ex 17.7., for p = 2 see Ex 17.8.. The spaces F which satisfy this property of F = Lp will be characterized in 25.9..
17.14. As has already been mentioned, Grothendieck’s fundamental theorem of the metric theory of tensor products (section 14) has powerful consequences for operators; at this point we could examine the general versions in terms of C:-spaces, but we will postpone this task to section 23 and restrict ourselves for the moment to considering the sequence spaces e, for p = 1,2, and 00. Since T 5 K G W on ~ loo €31co and w~ 5: dz (see 12.5.), it follows that
by the representation theorem, and therefore:
THEOREM: Every operator T E C(e,,tl) is 2-dominated and (hence) absolutely 2summing; moreover, pZ(T) 5 DZ(T) 5 KGllTll * In 20.18. and Ex 20.14. it will be shown that this result is also equivalent to Grothendieck’s inequality: For 2 = ID2 with the same constant (this is obvious), for C = ’&, with another one; see also Ex 25.5.. It is customary nowadays to call another operator form of Grothendieck’s inequality
THEOREM: Every T E C(41,ez) is absolutely l-summing and GROTHENDIECK’S
PROOF: It was shown in 14.8. that T 5 KGd, on 8 and hence, by l-complementation, on @I q for all m,n. The density lemma (or the C!-local technique lemma) implies that this holds on 1 1 @&, hence C(e1 ,t z ) = (el €3IX f2)’ = (!I ad,,,ez)’ = P1(.f1,&) since PI g& = d k . 0
-
17. The Representation Theorem for Maximal Operator Ideal8
213
20.17.concerning the fact that the best constant in Grothendieck’s theorem is the Grothendieck constant KG. The little Grothendieck theorem (11.11.) says that
See
whenever E = 4, or
el.
Since
it follows:
PROPOSITION: If E = tl or .too, then
17.15. Apart from the representation theorem there are other extremely useful characterizations of a-integral operators T € C ( E ,F) in terms of tensor product mappings idc @ T : G@pE
-
G a yF
with appropriate tensor norms p and 7 - as has already been seen in the characterization 10.7.of integral operators and 11.1.of absolutely p-summing operators. Actually the following theorem will be proved along the lines of the proof of 10.7.; the key to that proof waa to use two simple formulas (check for elementary tensors) which connect TEC(E,F)andT@idG: (1) For ‘pE ( F & G)’ and z E E @ G
THEOREM: Let (%,A)N a be associated and T E C ( E , F ) . Then the foflowing two statements are equivaleni: (a) T E %(E,F ) . (b) For all Banach spaces G (or only G = F’ or G = L if L‘ = F isometrically)
is continuous. In this case,
214
17. The Repmaentotion Theorem for Maximal Opemtor Ideals
-
We prefer to formulate the theorem for idG 8 T instead of T 8 i d c , even though the proof will be for T 8 id0 : @.(In 8=.
L, o T E a ( E ,G’) = ( E
G)’
and formula (1) implies that
which shows that
r ( T @ i d G ( 2 ) ; F, G)5 A(T)a‘(z;E , G ) .
Conversely, assume (b) to be satisfied for G = F’. Since 2l is regular (1?.8.), it is enough to show that PspoT E %(El F“) = ( E @an F’)‘ . For z E E 8 F’ formula (2) implies that
-
J(PKpO z)I~=, J(trF,T8 i d ~ f ( z ) 5 ) ) r ( T 8 i d d z ) ;F, F‘) 5 5 llT @ i d p :Bun QD,*lla’(z; E , F’) which gives T E %(ElF ) with the norm estimate. For F = L’ the proof is nearly the same for checking that pT E (E L)‘ . 0 It is important to note that (b) is actually a statement about the composition of operators
S ( G , E )= G‘ 8 E S
id@T
G ’ 8 F = S(G,F ) ToS
(see also 17.19. and Ex 17.6.).
17.16. In order to obtain characterizations involving E - which will turn out to be a sort of a dualization of the last theorem - the following lemma will be needed; the reader will find it to be a simple consequence of the metric mapping property of tensor norms and the fact that for every M E FIN and E > 0 there is a 1-complemented subspace N of the Johnson space C, (see 6.3.) with d ( M , N) 5 1 E .
+
LEMMA: Let p and 7 be tensor norms, P finitely generated, c 2 0 and T E 2 ( E , F). (1) If for a normed space G
17. The Repwsentotion Theorem for Mazimal Operator Ideals
215
for cofinally many M E FIN(G), then
( 2 ) If for some Johnson space Cp Ilidc, @ T : Cp @p
E -* Cp 8-yF l l 5
-
then IlidG @ T : G @ p
E
c
9
G @7 F i l s c
for all normed spaces G . Everything is now prepared for the
COROLLARY: Let (24,A )
-
Q be associated and T E C ( E ,F ) . I f a i s a right-accessible and finitely generated tensor norm or if F has the metric approximation property, then the following statements are equivalent:
(a) T E %(E,F ) . (b) For all Banach spaces G (or only G a Johnson space Cp for some 1
5 p 5 cm)
is continuous. In this case,
-
at and in this case A(T) = Adua'(T') PROOF:T E 24 if and only if T' E holds. So the theorem and the lemma imply that A(T) 5 c if and only if
for all M E FIN. Since the map (see 6.4. for the duality of E and
is the restriction to M'
T)
F' of the dual of
it is clear from the lemma that (b) implies (a), and the converse is true if the relation 1 F holds for all M E FIN; but this is true if a is right-accessible or M @a F = M approximation property. 0
216
17. The Representation Theorem for Maximal Operator Ideals
Note that the assumption about the accessibility of Q or the assumption on F was only used for the proof that (a) implies (b).
-
17.17. An example: For the ideal (cfp1Ip) gp of p-integral operators the following three statements are equivalent: (a) T E E(E, F ) is p-integral. (b) For all Banach spaces G (or only G = F')
is continuous. (c) For all Banach spaces G
is continuous.
For (a) or (b) implies (c) the fact that gp is accessible is used - a fact which will be proved in section 21. However, this will not be needed in the following nice application of this result.
PROPOSITION: Every positive operator T E 2 ( E ,L p ( p ) ) , where E is a C ( K ) or some L,(v), is p-integral and I,(T) = IlTll.
It has already been seen in 11.2. that these operators are absolutely p-summing and P,(T) = IlTll; see also Ex 18.3..
PROOF: It is known from theorem 7.3. that for all Banach spaces G IlidG €4 T : G @c E
-*
G @A;
Lpll
= llidGII llTll
I
Since gp 5 A: by the Chevet-Persson-Saphar inequality 15.10., it follows from the corollary (since L, has the metric approximation property) that T is p-integral and IP(T) = IlTIl * 0
17.18. Sometimes it is possible to find better test spaces than G = C, in corollary 17.16.. Using the fact that gp, is accessible (this will be proved in section 21), it is clear that for T E gp(ElF) and G = t, the map
is continuous by the corollary. Since A,, 5 g i , on t, @ F, it follows from proposition 11.l.that this actually characterizes psumming operators:
17. The Representation Theorem for Maximal Opemtor Ideals
217
REMARK:T E C ( E ,F ) is absolutely p-summing if and only if
Another description of Q p ( E , F) is given in Ex 17.9. with the help of Kwapieli’s characterization 11.8.. 17.19. The theorem (and its corollary) also has structural consequences.
PROPOSITION 1: Let (a,A ) be a maximal normed operator ideal such that the associated tensor norm a is accessible. Then Q* o
C3
and
I(T o S) 5 A * ( T ) A ( S );
the composition T o S of operators S E Q and T E a* is integral.
In 25.4. this result will be generalized to the so-called “cyclic composition theorem”; it will also be shown in 25.4. that Q* 0 % C 3 implies that a is necessarily right-accessible. In section 21 the accessibility of a will be explained in terms of the operator ideal a. If a is not accessible (recall that there is an example known), the proof will also show that % * ( F , G ) o Q ( E , F )C 3 ( E , G ) , with the norm inequality whenever F has the metric approximation property. In 19.4. the reader will find examples in which a* o Q consists of nuclear operators only. For another “converse” of the proposition, see Ex 17.6., Ex 17.15. and Ex 25.10..
-
-
PROOF:Since a a and a* a*,the theorem and its corollary show that for S E n ( E ,F) and T E Q’(F, G ) the composition
-
has norm 5 A*(T)A(S); since ?r 3 and integral and I(T o S) 5 A*(T)A(S).0
A*
= E , the theorem implies that T o S is
If dim E = n, then I ( i d E ) = N ( i d E ) = n (see 10.4. and 3.7.). It follows that for each isomorphism T E C ( E , F ) A(T : E
+ F)A*(T-’ : F -+
E) 2 n
for each normed operator ideal ( Q , A )(maximal or not: E and F are finite dimensional). For special cases there is equality: Recall the investigation 15.9. of certain
218
17. The Representation Theorem for Maximal OpeMtOr Ideals
norms on K" which are invariant under the isomorphism M, (multiplication of the components by f l ) and the cyclic shift S. In terms of operators, 15.9. reads as follows:
PROPOSITION 2: Let E and F be K" equipped with norms such that llM, an C(E', E') and C(F, F ) for all w E 0,.Then A ( i d ~ :-E for all normed operator ideals
-+
F ) A * ( i d K m: F
+E)
11 = llSll = 1
=n
(a,A).
xi=,
PROOF: The element u(") := ek 8 ek corresponds to the identity operator. If a is the tensor norm associated with am"*, then A(id : E
+F ) =
A*(id : F
+E)
a ( u ( " ) ;E', F )
= a*(u("); F', E ) = o'(u("); El F')
- and the result follows from a(u(");E', F)a'(u(");E l F') = n which was shown in 15.9.. 0 This implies, for example, that the results in 8.8.about the nuclear norm of the identity id : $ -+ can be directly calculated from the operator norm of id : -+ $ since C E and E* = A N 3, which is n on finite dimensional spaces. The exact calculation of P l ( i d t ; ) in 11.10. gives the exact values of the norm of idt; in 'Qt = 3, = ,C,. that is, the exact values of the oo-factorable norm:
-
It follows from corollary 11.10. that
These results are quite interesting since it turns out that L,(idE) constant of E whenever E is finite dimensional, see 23.6..
is the projection
17.20. The final topic of this section is the following question: Under what conditions is the natural map
injective? If a is totally accessible, then the duality theorem 15.5. for tensor norms implies that
17. The Representation Theorem for Mazimal Operotor Ideals
E h Q F= E g - F I*
1
(E’c&JF’)’
219
C(E’, F“)
hence I is injective.
PROPOSITION: If a is a finitely generated tensor norm and E and F are Banach spaces, one of which has the approximation property, then the natural map
is injective.
PROOF: Assume that F has the approximation property, z E E g a F and I(z) = 0. It has to be shown that (cp,z) = 0 for all
By theorem 17.15. (and obviously also by the correspondence between maximal operator ideals and tensor norms)
is continuous. The lower map in the diagram
is injective by the approximation property of F (see 5.6.), hence
Formula (2) in 17.15. implies that
which was to be shown. 0 The diagram
220
17. The Representation Theorem for Mozimal Operator Ideals
shows that:
-
COROLLARY: If (%,A)w a, then the natural map
E'haF
%(E,F )
is injective if (Y is totally accessible or i f E'
or
F has the approximation property.
See Ex 25.4. and Ex 17.11. - Ex 17.14. for more results connected with the injectivity Ofg3a+g)r.
17.21. The components Q ( E ,F) of a maximal operator ideal are not in general closed in C(E,F) (with the operator norm) (why?) - however the unit balls are. Even more is true:
PROPOSITION: Let (%,A) be a maximal normed operator ideal. Then the closed unit ball of g ( E ,F ) is closed in the space C ( E ,F ) with respect t o the weak operator topology u ( C ( E , F ) ,E 8 F'). In other words: If (T,)is a net in % ( E ,F ) which converges pointwise weakly to T E C ( E , F ) and sup, A(T,) < 00, then T E % and A(T) 5 liminfA(T,) .
PROOF:Since the unit ball of % ( M ,N) is closed for u(...,M @ N') (finite dimensional spaces), the second assertion follows from the definition of Amax . 0
Exercises: Ex 17.1. (a) Let {(%A, A ~ ) l ) rE A) be a family of maximal normed operator ideals. Then the class of all T E 2 with
A(T) := SUP Ax(T) < 00 A€A
is a maximal normed operator ideal. (b) Deduce from (a) and 10.7. that the ideal (3, I) of integral operators is maximal. Ex 17.2. A normed operator ideal (a,A) is maximal if and only if there is a finitely generated tensor norm p such that
B ( E ,F )
( E @p F')'
n C ( E ,F )
17. The Representation Theorem for Maximal Operator Ideals
221
for all Banach spaces E and F. In this case,
( E8 3 8 F)'
%(ElF')
.
Ex 17.3. Deduce directly from the representation theorem that T E Q if and only if T" E Q whenever Q is maximal. Hint: Embedding lemma and "9. Ex 17.4. Let (a,A) a and E, F be Banach spaces. Assume that a is totally accessible or: a accessible and E or F' has the metric approximation property or: E and F' have the metric approximation property. Then an operator T E C ( E ,F) is in the adjoint operator ideal Q* if and only if
-
Is
AA(T) := sup{lirE(S 0 T ) I
E S(F, E ) ,A(S) 5 1) < 00 .
In this case this number is A * ( T ) .Hint: Embedding theorem. Gordon-Lewis-Retherford [85] studied the ideal QA of all T with AA(T) < 00 under the name conjugate ideal of Q. Ex 17.5. Let (%,A)be a normed operator ideal. Show that T E C ( E ,F) is in Q* if and only if there is a c 3 0 such that for all M, N E FIN
for all R E C ( N ,M),S E C ( M , E) and U E C ( F ,N). Ex 17.6. Use the embedding theorem and
/IT' @ idE : F'
E
-
E'
@ r Ell
= A(T)
to show that: If (Q, A) a are associated, a is accessible, E and F are Banach spaces and E has the metric approximation property, then T E Q ( E ,F) if and only if there is a c 2 0 such that I(S o T ) 5 cA*(S) N
for all S E 5 ( F , E). In this case, A(") is the minimum of these constants c. See also Ex 17.15.. Ex 17.7. Show that !$Fa'(E,L p ) c ' J , ( E ,L p ) . Hint: Chevet-Persson-Saphar inequality. See 25.9. for more information about this relation. Ex 17.8. Use the result in 11.12. to show that: (a) There are no constants c1 and c2 with dz 5 clgz and 9 2 5 czdz. (b) A Banach space E is a Hilbert space if and only if $2 5 cgz on E @ & for some c > 1. Ex 17.9. Use Kwapieri's characterization 11.8. of absolutely p-summing operators to , if and only if show that T E V p ( E F)
idL,, 8 T I : ePl @dpl F'
-
epI@r
E'
222
17. The Representation Theorem for Mozimal Opemtor Ideals
is continuous.
-
Ex 17.10. If (a,A) a and a is accessible, then the following three statements are equivalent for each Banach space E and c 2 0 : E E space(%) and A ( i d ~5) c , (b) T 5 ca+ on G @ E for all Banach spaces G (or only G = El). (c) a 5 CE on G 8 E for all Banach spaces G (or only some Johnson space Cp). Ex 17.11. Let E or F have the approximation property and z E E&F for some finitely generated tensor norm, If (o&idi;.(r) = 0 for all (o 6 E l , then z = 0. Hint: (a)
17.20. and Ex 15.2..
Ex 17.12. Let E C(Ei,Fi) be two injective operators between Banach spaces and a a finitely generated tensor norm. If El or E2 has the approximation property, then T1&T2 E C(E1&,&, Fl&F2) is injective. Hint: 17.20. and this property for a = E . Ex 17.13. The canonical map L P ( p ) 8 * , E -, LP(p)&E is injective. Hint: Use 15.10. - or prove it directly for p < 00 using the fact that every f E L p ( E ) has separable range.
Ex 17.14. If a is a finitely generated tensor norm and a 5 Ap on L p ( p )@ El then LP(p)6~,E, LP(p)&E is injective. Also: if a 2 Ap on Lp(p) @ E , the mapping LP(p)&E + LP(p)&A,E is injective. Ex 17.15. Use the representation theorem and corollary 17.16. to prove the converse of proposition 1 in 17.19.: Let (a,A) be a maximal normed operator ideal the associated tensor norm of which is accessible. Then T E C ( E ,F ) is in 24 if and only if for all Banach spaces G and operators S E B'(F, G ) the composition S o T is integral. See also Ex 25.10.. Ex 17.16. Let S E C ( X ,Y)and T E C ( E ,F ) be continuous operators between Banach spaces and a,P two tensor norms such that
is continuous. Question: Is S @ T" : X E" -,Y @p F" also continuous? Unfortunately, we do not have a complete answer, even in the case where p = A. Assume a to be finitely generated. Show that the answer is positive under each of the following conditions: (a) T is weakly compact. Hint: [(S@ T)'(oJ'' = (o o (SQD T*)for all (o E (Y F)' if F : E" 4 F is the astriction of TI'. (b) p 5 cp' on Y 8 F" for some c 2 0 (the right-cofinite hull, see Ex 12.3.) Hint:
Proposition 17.4.. (c) O , is totally accessible. (d)
p is accessible and: Y or F" has the bounded approximation property.
(e) Y and F" have the bounded approximation property.
223
18. (p,q)-Factorable Operators
Ex 17.17. Density lemma for maximal normed operator ideals: Let (a,A) be a normed operator ideal, E and F Banach spaces, E, c E and Go C F' norm-dense subspaces, C C FIN(E,) and D c FIN(G,) cofinal subsets. Then AmaX (T
: E -+
F ) = SU~{A(Q;o T o
12)I M E C ,LO
E D) .
Hint: Representation theorem, density lemma 13.4. and cx' finitely generated.
18. (p,q)-Factorable Operators
The purpose of this section is to show that every ak,q-continuous functional on
E 8 F factors through the integrating functional on Lg1(p)8 Lp1(p) for some measure p. In terms of operators this means that every (p, q)-factorable operator T E E(E, F)
has the property that ICF o T factors through the embedding Lgt(p)4 Lp(p) . The most important special cases are the p-factorable and p-integral operators. In these cases, the proofs are actually simpler than in the general case. A central tool for the demonstrations will be the ultraproduct technique, the basic properties of which will be presented as well.
18.1. The maximal normed operator ideal (.Cp,q,Lp,g)of (p,9)-factorable operators is associated with the tensor norm aPIg. Using the definition of Q ~ directly, , ~ it was shown in 17.10. that for finite dimensional Banach spaces M and N the L,,,-norm of T E E ( M , N) is given by the factorizations
M-N
T
where Dx is a diagonal operator the entries of which can clearly be assumed to be non-negative: LP,,(T) = infllSll IlDXll IlRll * It is a simple matter of concentration (and this will be done in 18.10.) to show that Dx factors through an embedding Lq1(p)+ Lp(p)for a finite measure with control of the
224
18. (p,q)-Factorable Opemtors
norms. Since the operator ideal .CP,, is maximal, every T E Alp,,is determined by its finite dimensional components - and a tricky “fitting together” technique (ultraproducts), representation theorems for abstract Lp-spaces and a characterization of positive operators L , I ( ~ ) Lp(v)will give the desired characterization: T E .Cp,q(E,F) if and only if IEF o T factors through some embedding L,i(p) c* L p ( p ) (see 18.11.); recall from 17.8. that T E .Cp,q if and only if KF o T E .CP,(.
-
18.2. Not T : E -+ F but KF o T : E + F” can be factored. Therefore it is more F)’ rather than operators. reasonable to treat functionals in (E Using the fact that 6 = wp, on N‘@$,,it was shown in 15.8. that ?r = w; on L p ( p ) @F for every measure p (not necessarily finite), and hence the integrating functional P I : Lp(cL)
8 LP’W
fBg
--
K Jfgdp
(which is nothing but the trace) is wL-continuous. In other words, for every measure p the identity map on L p ( p ) is in the ideal .Cp of pfactorable operators associated with wp. For the ideal .Cp,q ap,qof ( p , q)-factorable operators the following result holds (the integrating functional is defined for l / p + l / q > 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
18. (p,q)-Factorable Operators
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 <
00
the following three
statements are equivalent: ( a ) T is p-integral.
(b) There are a Borel-Radon measure p on a compact K, operators R E E ( E , C ( K ) ) and S E C(L,(p), F”) such that
commutes; I the canonical map C ( K )+ Lp(p).
(c) There are a finite measure p and operators R E E(E, L,(p)) and S E C(LP(p), F”) such that K F O T S =o I o R . I n this case, Ip(T)= minllSll IlI(( llRll, where the minimum is taken over all factorizations (IS in (b) o r as in (c). I
Thus, the natural map C(K)-L,(K, p ) is not only the typical absolutely p-summing but also the typical pintegral operator - only the form of the factorization is different: for absolutely p-summing operators into !,(BF,),for pintegral operators into F” (for p = 1 this is theorem 10.5.). For p = 00 the ideals , 3 and C , coincide. The factorization cannot be into F (instead of F“) since there are (l-)integral operators which are not Pietsch integral (see Appendix D9.).However, for p = 2, it follows as in 18.6. (corollary 2) that the operator S in the factorization can be chosen to have values in F, hence, T E &(ElF) if and only if there is a factorization
232
18. (p,q)-Foctomble Operators
Moreover, Iz(T) = minllRll llIll IlSll. But this is exactly the factorization of the absolutely 2-summing operators (see 11.3., corollary 2), hence
Since g ;
-
! & = 32
-
g2,
it follows that:
COROLLARY 4: 92 = g ; . In Ex 18.3. it will be shown that V p ( E ,F ) = J P ( E ,F) holds isometrically if E = C ( K ) or L,. If F is 1-complemented in F”, then obviously T E J p ( E ,F ) if and only if T R I factors E-C-L,&F and &(T) = minllRll llIll IlSll, the minimum taken over these factorizations. In particular, this holds if F is a dual space or if F has the metric extension property; example: F = 4, (r).It follows from the Grothendieck-Pietsch factorization theorem that
Vp(E,&Q(r)) J p ( E , 4 m ( W holds isometrically. Since (VP, Pp)is an injective operator ideal, this implies:
< 00 the injeciive hull of the Vp of absolutely p-summing operators:
COROLLARY 5: For 1 5 p is the ideal
(Vp,Pp)= (J?,
ideal Jp of p-integral operators
Iy).
For the associated tensor norm one obtains 1
1
N Bg;,e”, = V p ( N ’ , e ” , )= JP(N’,e”,) N
Bgp
.
Since g;, and g p are finitely generated, it follows that:
COROLLARY 6: g;, = g p on E 8 P , f o r all normed spaces E. We shall come back to this result in 20.14.. 18.8. For p-factorable operators the factorization theorem followed directly from theorem 18.5., for pintegral operators the Grothendieck-Pietsch factorization theorem was used. This suggests that the general case of (p,q)-factorable operators is even more complicated - and it is. Starting from the fact that for l/p l/q > 1 every
+
233
18. (p,q)-Factorable Operators
( 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
18. (p,q)-Factorable Operators
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
19. (p,q)-Dominated Operators
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
19. (p,q)-Dominated Operators
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
19. (p,q)-Dominated Operators
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
20. Projective and Injective Tensor Norms
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 )
20. Projective and Injective Tensor Norms
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
20. Projective and Injective Tensor Norms
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
20. Projective and Injective Tensor Norms
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
20. Projective and Injective Tensor Norms
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
80. Projective and Injective Tensor Norms
(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/.
20. Projective and Injective Tensor Norms
258
<
(d) Finally, take any right-projective tensor norm ,8 2 a;it follows that a/ p on NORM x FIN by the definition of a/.Since p = p’ on NORM by lemma 20.2.(1), it follows that a/ = (a/)+5 8 ,‘ = ,8. This ends the proof. 0 Consider the natural metric surjection
Q : ei(Bil(r))A el(r) ; then there is a lifting T : 41(I’) + 41(Bcl(r)) of idt,(r) (i.e. Q llTll 5 1 E . Therefore, for any normed space E, the diagram
+
o
T = idtl(r)) with
+
shows that a/ 5 (1 €)aon E 8 tl(r).Since a 5 a/ always holds, it follows:
COROLLARY 1: If E is a nonned space, then a =a/
on E@ll(I’)
for every set I’.
For obvious reasons \a := ( ( a t ) / ) t
is called the lefl-projective associate of a .
COROLLARY 2: Let a be a tensor norm. Then
is called the projective associate of a; it is the unique smallest projective tensor norm
2 a, is finitely generated and
1
f l ( B E ) @a L l ( B F )
E
@\a/
F
is a meiric surjection whenever E and F are Banach spaces.
The proof follows easily from the “transitivity of metric surjections” and the theorem. 20.7. Fortunately, the injective case is simpler.
120. Projective and Injective Tensor Norms
259
THEOREM: Let a be a tensor norm on NORM. Then there is a unique right-injective tensor norm a\ 5 a on NORM such that /3 a\ for all right-injective tensor norms /3 5 a. For all n o m e d spaces E and F
<
is a metric injection. a\ is finitely generated from the right.
The tensor norm a\ is called the right-injective associate of a .
PROOF:Define a\ on E @ F to be the norm induced by E@FLE@utrn(BF'). Then & 5 a\ _< n on NORM since E is right-injective. To see the metric mapping property, take S E C(E1, E2) and T E C(F1,F2). By the metric extension property of t,(I') there is an extension T of T such that
commutes. This easily implies the conclusion. a\ is right-injective: If I : F c, G is a metric injection, then the metric extension property again gives a mapping i such that
The definition of a\ immediately implies that /3 5 a\ whenever p Clearly, /3 = /3' for every right-injective tensor norm. 0
As in the projective case, / a := ( ( d ) \ ) t
5 a is right-injective.
260
20. Projective and Injective Tensor Norms
is called the left-injective associate of a and
is the injective associate which is the unique largest injective tensor norm smaller than a. It follows that E F &(BE#)& ~W(BF') .
r'.
Clearly, every injective tensor norm is finitely generated. The notation, with all the various slashes, may seem confusing at first, but it is really quite natural and not that difficult to memorize: Start with the right-projective associate a/ (coming from quotients FIG), - then take its counterpart a\ for the rightinjective associate - and finally the rest by symmetry.
COROLLARY: If the Banach space F has the A-extension property, then a\sa
onE@F
for all normed spaces E.
PROOF:F is A-complemented in . t w ( B p ) .0 20.8. The following is a special case.
PROPOSITION: For every tensor norm a , normed space E and n E N
Now the 2p-local technique lemma 13.5. can be applied.
COROLLARY: Let a be a tensor norm and E a normed space. (1) If F is an 2:,x-space, then asa/sAa'
on E @ F .
( 2 ) If F is an 2&,x-space, then a\
on E B F .
Note that a- 5 p a on E @ F if F has the p-bounded approximation property by the approximation lemma (C;-spaces have the bounded approximation property, see 21.6.) and a = ' a if a is finitely generated.
20. Projective and Injective Tensor Norms
261
PROOF:The local technique lemma showed, in case (l), that the proposition implies that a/ 5 (a/)* _< Xa’ on E @ F (see the remark after the proof in 13.5.). In case (2) it follows that a- 5 X(a\)+ = Xa\ . 0 20.9. The following gives a simple test for recognizing whether or not a tensor norm
p is the projective or injective associate of a given tensor norm a : PROPOSITION: Let a and p be tensor norms. (1) If
P
is right-projeciive, then the following staiemenis are equivalent:
(4 P = a/
*
(b) For all E E NORM and n E N
E @p (2)
= E @a
q
isometrically.
If P is right-injeciive, ihen the following siatements are equivalent: (4 P = a\ *
(b) For all E E NORM and n E N
E @p P , = E @& P,
isometrically.
(3) If a and p are finiiely generated, then in both cases it is enough t o tesi only for finite dimensional E. (4) If P is projective, then ,B = \a/ if and only if f o r all n E N
q @p q
q
=
isometrically.
( 5 ) If p is injective, then p = /a\ i f and only if for all n E N P, @ p P , =P ,
Pm
isometrically.
PROOF: That (a) implies (b) (in (1) and (2)) follows from the last proposition. Assume (l)(b). Then by the local technique lemma, ‘p = a+ on E @ l l ( r )and hence p = a on E @ l , ( r )by the approximation lemma: the fact that P is right-projective along with property (*) in theorem 20.6. gives (a). The remaining implication in (2), as well as (4) and (5), follow in the same fashion. Statement (3) is obvious. 0 It is worth noting that these characterizations (with the same proofs) also hold if constants are involved - and this will be used. For example: I f a and p are tensor norms, P injective, then c p L /a\ 5 d p if and only if for all n E N cp
Q
5 dp
on P,
@P, .
20.10. The following formulas contain many of the fundamental properties of projective/injective associates and finitelcofinite hulls; they create a type of “calculus” which will be helpful when dealing with accessibility in the next section.
20. Projective and Injective Tensor Norms
262
PROPOSITION: Let a be a tensor norm on NORM. -c
(1) (Z)\= a\
and (S)/=
3.
c
(2) (=)\ = a\ , but, in general, (Z)/# (3) (a/)’ = (a’)\ and (a\)‘ = (a’)/ . (4) ( a / ) *= /(a*)and (a\)* = \(a*) .
a’/ .
PROOF: (1) By 20.8. it follows that
+
Since p := a\ is right-injective by proposition 20.3.(1), the test 20.9. gives
The same argument applies to the right-projective associate. (3) and (4) also follow from the preceding test: a’and (a/)‘ are finitely generated and clearly, N @ u ~ -c (N’ @a = (N’ mu/q)‘= N @(a/)#e”,
e)’
hence, (a’)\= (a/)’, which implies all the formulas in (3) and (4). c
(2) Note first that a\ is right-injective by proposition 20.3.(1). Since (by (3) and 20.6., corollary 1)
E’&# J ! ~ ( B = F ~E’@(a\y ) ti(BF’) and (by the duality theorem 15.5.)
F d+E @z~,(BF’) @(3\ [email protected] F a\
E @ - !,(BF#)
(E‘ @at 1
J!~(BF~))’
( E ’ @ ( a \ ) f t i ( B ~ f ,) ) ’
a\
c
one obtains (=)\ = a\. The related formula for the right-projective associate fails to hold: In fact, f = T on E @ .tl(I’) and hence, c-
(%)/ = ?f/ = s # f = ( T / ) since u is right-projective. The first statement of the proposition implies the
COROLLARY: If a is a finitely generated tensor norm on NORM, then all of its ciates a / ,a\, \a, /a,/a\ and \a/ are finitely generated as well.
asso-
80. Projective and Injective Tensor Norms
263
20.11. If a is a finitely generated tensor norm and (%,A) its associated maximal operator ideal, what do the ideals associated with a/,a\,/a,\a,... look like? Finding an answer to this question will be very fruitful for the study of both tensor norms and ideals. Start with the right-injective associate a\ of a,denote by (93,B)the ideal associated with a\ and fix T E Z ( E , F). Since (&,(Bp))'is an L l ( p ) (by Kakutani's representation theorem, Appendix A7.) and hence an X;,l-space, it follows that
If /&- and P I ~ Tare the functionals associated with T and I o T (where I : F r-, t O o ( B p ) ) , then
commutes. Since id @ I' is a metric surjection, pT is continuous if and only if PI,,T is continuous (with the same norms); the representation theorem for maximal operator ideals yields that T E 93 if and only if I o T E %; moreover, B ( T ) = A ( I o T). Hence, the definition 9.7. of the injective hull % j n j of an operator ideal gives the first part of the
THEOREM: Let a
-
( % , A ) be associated. ( 1 ) The injective hull (gi"j,A'"') of (%, A) is maximal and associated with a\. In particular, a is right-injective if and only if (a,A) is an injective operator ideal. ( 2 ) The surjective hull (a8"', AbUr)of (%, A) is macimal and associated with /a.In particular, a is left-injective if and only if (a,A) is a surjective operator ideal. 1
PROOF: To see (2) take the canonical quotient map Q :~ ~ ( B E ) - and E T E Z ( E , F); look at the diagram PToO
to obtain the result, as before. I 3
To quickly draw one consequence from these relationships:
K
264
20.
Projective and Injective Tensor Norms
COROLLARY: If (9,A) is a maximal normed operator ideal, then
PROOF: This is just (at)\ = (/a)'.0
20.12. The projective associates give rise to factorization theorems for the corresponding operator ideals. Denote by (a/,A/) the ideal associated with a/ and take T E C ( E ,F ) . Then
If a-I. is continuous, then the Hahn-Banach theorem gives an extension p with the same norm; it follows that K F o T = I' o L , and, by the representation theorem for maximal operator ideals, A(&) = A / ( T ) .So part of the following result has already been shown.
-
-
-
(a,A) and denote by (a/,A/) a/ and (\a,\A) \a the maximal operator ideals associated with the projective associates; take T E f!(E,F ) :
PROPOSITION: Let a
(1) T E !Z/(E,F ) i f and only if there exist a measure p , operators R E % and S E 2 such that
E
T
F
F"
I n this case, A/(T) = minA(R)IIS)), where the minimum is taken over all such factorizations and i s attained for a metric surjection S : L1(p) -* F" for a B o d - R a d o n measure p. (2) T E \Q(E, F ) if and only if there exist a compact set K , operators R E C and S E !2l such that
20. Projective and Injective Tensor Norms
E
T
F
i=-+
265
F"
In this case, \A(T) = minIIRIIA(S), where the minimvm is taken over all such factorizations and is attained for a metric injection R.
The space C ( K ) in (2) can be replaced by L , ( p ) .
PROOF:For T E Q l / ( E F), , II~'IIA(LIp)= A / V )
,
and it remains only to use Kakutani's representation theorem Appendix A7., which gives that &,(Bp)' = L1 ( p ) for some Borel-Radon measure. 1
For the converse, note first that L1(p) = L1(po)for some Borel-Radon measure po (by Kakutani's representation theorem), therefore, Ll(p)' L,(po). Now 20.8. and 20.10. imply that E @(?I)/ Li = E L i and hence A(R) = A/(R) by the representation theorem for maximal operator ideals. To see (2) apply the same type of argumentation using the diagram
A
teT
E
F'
In Ex 20.7. the associated operator ideal for the projective associate \a/ is described. Ex 20.6. discusses extension and lifting properties related to 2l/ and \%.
-
-
20.13. The characterization of T E QlZL'"J a\ and T E /a in terms of idG @ T given in 17.15. and 17.16. can be modified to be in terms of a (instead of a\ and /a). We omit stating and proving this now: In 29.10. it will be a very easy consequence of the calculus of traced tensor norms. 20.14. The simple relationship between the injective associates of a and surjective and injective operator ideals provides nice information about the associates of ap,p:
PROPOSITION: Let 1 5 p 5 00. Then (1) gp\ = g;, = 4, . (2) \gi = gpt and d i / = dpt . (3) \(gp\) = gp and W P ) / = dP * (4) T\ = g& = w k = wi = d&, and
E/
= d , = w1 .
20. Projective and Injective Tensor Norma
266
-
Jy
-
PROOF: Corollary 5 in 18.7. states that = ppholds isometrically whenever 1 5 p < 00; since gp Jp and g;# '&, theorem 20.11. (31'"j a\) implies that gp\ = g;# . For p = 00 it is known from 12.9. that
-
gm=woo=E
onE@P,
and therefore 20.9. implies that goo\ = E . Since g2 = a2,1 is right-injective (20.4.), the last statement follows from the first (note that 92 = g2f was already shown in 18.7.). All the other statements come from (1) as well, using transposition and dualization (see 20.10.). 0 20.15. The behaviour of the aP,* is particularly interesting when one of the involved indices is 2: in such a case, the discussion is about tensor norms/operator ideals con-
nected with Hilbert spaces. The following list exhibits some of the most important examples:
The symbol
-
indicates that the norms are equivalent.
PROOF:Only the statements for 9 2 and w2 will be proven - the others follow by transposition, dualization, and from g2 = g; . Since g2 is left-projective and right-injective, it follows that g2 = \g2 = g2\. Moreover, w2 is injective, hence w2 = /w2\ = w2\ = 1.12. The remaining statements are a consequence of the little Grothendieck theorem: In Ex 20.10. the reader is asked to show (with the factorizations coming from the characterization of \%, %/ and \%/) that
S 92 5 KLG\% \w2/ 5 w; 5 K:G(\w2/) 9 2 / 5 w; 5 KLGg2/ \w2
Moreover, this implies that
= (\W2)t 5 d2 5 KLGW2/ = 192 2 w2 KLGW2 = KLGw;* 2 ( S Z / ) * = / & )
w2/
20. Projective and Injective Tensor Norms
Since \g2/ = g2/ and
/g2\
267
= 192, all the relations have been demonstrated. 0
n Tp,q(Z;
F ) := inf{ (I(ak,L) : qj
-t
$lllqt(zL)lpt(Yk)
Z
=
a k , L z L @ gk}
k ,L=
1
(for l/p+ l/q 3 1); as in 20.4., it is easily seen that -yp,c is projective (Ex 20.2.) and it turns out that b;,q/
= Y9,P
- in particular, wz = y2,2. Everything has been prepared for giving a proof of this result but we prefer to postpone it to section 28. 20.17. Certainly the most exciting relationship between tensor norms is GROTHENDIECK'S INEQUALITY (tensor form):
PROOF: Since w2 is injective, it follows from the test 20.9. that the first relation is nothing but Grothendieck's inequality 14.4.. The inequality with the projective associate \e/ follows by duality. 0
-
-
-
Since r 3 (the integral operators), w2 2 2 (the operators factoring through a Hilbert space, see 18.6.),ir\ J'"J 91 (the absolutely summing operators, see 11.3. or 18.7.) and /r\ = /(r\) = (IT)\, this can be reformulated as
A
GROTHENDIECK 'S INEQUALITY (operator form): 9;ur
= ( j s u + ) i n j = gZ
L2(T) 5 P;"'(T) = (IaU')inj(T) 5 K G L ~ ( T. ) Moreover, Grothendieck's constant KG is the best constant i n this inequality. Or - as Grothendieck formulated it orginally in the Usumk:
COROLLARY: If H i s a Hilberl space, then P f ' ' ( i d ~ ) 5 K G . Since for T E Z ( t , , l , ) the relation
268
20. Projective and Injective Tensor Norma
holds, it follows what was called "Grothendieck's theorem" in 17.14.:
<
In particular, the best constant c in this inequality is PIu'(idf,) I KG; for an operator T E 2 2 factorization shows that Pi"'(T) 5 cL2(T) and hence KG 5 c. It follows that c = Pd,u'(idf,)= KG.Moreover, the maximality of the operator ideal ?J3:"' (and an obvious complementation argument) implies that
KG = Pfu'(idt,)= sup Piu'(idp) n
and Pfu'(id~) = KG for all infinite dimensional Hilbert spaces H (see also Ex 20.12.).
PROPOSITION: The number KG(n) := Pf"'(idp) is the smallest constant c 2 1 such that Grothendieck's inequality an matrix form is satisfied for m x m matn'z (m E N arbitrary) such that
5: Let
(ai,j)yj=i be a
Then
PROOF: For the moment, call the smallest constant c in the matrix inequality MG(n). As in the proof of 14.8., it follows that H 5 MG(n)dm on el 8 6 ,which means that PI(T:l l + 5)5 MG(n)IITII. In particular, KG(n) = P y ( i d q ) _< MG(n) Conversely, define for n E M and z E M 8 N = L(M', N) (finite dimensional Banach spaces)
(this is not a tensor norm!). Since
20. Projective and Injective Tensor Norms
269
holds isometrically, it follows that r 5 Ko(n)wp) on em,@E for all m E N.Moreover, if 2; and yj are in the unit ball of 4, then m
m
have norm
5 1 and factor z, := C G = l ( x i l pj)ei @I e j
E
@
If (ai,,) is an m x m matrix satisfying the assumption, then
has norm _< 1 in
em,, which means that
‘p
defined by
(em, @Ir em,)’, and hence
In the real case Krivine [154] proved:
KE(2) = K z ( 3 ) < 1.517
Recall that r / 2 5 K$ from 14.7.. Using Krivine’s method and its complex modification by Haagerup (see 14.7.), Konig [148] estimated some “finite dimensional” Grothendieck constants in the complex case
1.1526 1.2108 1.2413 1.2600 1.2984 1.3181 1.3300 1.3381
5 K,$(2) 5 Kg(3) 5 K (4) 5 K %(5) 5 KG(lO) 8 5 Ks(20) 5 Ks(50) 5 Ks
5 5 5
1.2157 1.2744 1.3048 5 1.3236 5 1.3628 5 1.3834 5 1.3962 < 1.4049
and conjectured slightly better upper bounds. For other descriptions of K G ( ~see ) Ex 20.16., Ex 28.6., Ex 28.13. and Ex 28.14..
2 70
20. Projective and Injective Tensor Norms
20.18. There are more Banach spaces E which satisfy Grothendieck’s theorem
(they are sometimes called G.T. spaces). In fact, in 23.10. it will be shown that all C;-spacea do - this is rather simple. For a long time it was not known whether or not there are any other spaces satisfying Grothendieck’s theorem. In 1976,Kisliakov [142] and Pisier [219]independently constructed spaces which do: for example, quotients L1(p)/R where R is a reflexive subspace. Bourgain [16]showed that L I / H ’ (where H’ is the Hardy space) satisfies Grothendieck’s theorem. We refer the reader to Pisier’s book [225],ch.6, for more details on this topic and give only the following characterization, which appeared first in [225]in a slightly different form (see Ex 20.14.), but is in some sense also contained in Lindenstrauss-Pelczyriski’s famous 1968 analysis of Grothendieck’s inequality; see also Ex 20.13.and Ex 20.14..
PROPOSITION: For every Banach space E the following two statements are equivalent: (a) q E ’ , l 2 ) = Pl(E’,t 2 ) (b) C ( E ,41) = 9 2 ( E ,ti) . The constants in (a) and (b) are the same. That (b) holds for E = co can be shown as in 17.14.using also section 23 on Xg-spaces.
?r
5 K ~ w 2on
c, @ c,; see
PROOF: Observe first (by the representation theorem 17.5.for maximal ideals and the &-local technique lemma 13.5.) that (a) means that there is a best constant c1 < 00 such that P1(T) 5 c1llTll for all T E C(E’,G), and (b) that there is a best c2 < 00 such that D2(T) 5 c211Tll for all T E C ( E , e ) . Assume (a) and take T E C ( E , e ) . Then, by 17.9. and since 9: = 2 2 ,
-
Hence the factorization of L2 from 18.1.,T = T o QferT, the fact that w i 2; and (a) give
-
ppa’
gg
=
since Jtr(Ro S)l 5 A*(R)A(S) for operators between finite dimensional spaces. Conversely, take T € C(E’,G) and, in order to apply Kwapieri’s characterization 11.8., S E 2(l&,,El). Then 2* = 3 implies that
20. Projective and Injective Tensor Norms
271
In Ex 20.14. it will be seen that P2 5 D2 5 K L G P on ~ 92(E,t,) = q2(E,tl). Therefore, one can replace 9 2 by ' 3 2 in this proposition - at the price of an additional constant in one implication.
COROLLARY: Let E be a Banach space and c 2 1. Then the following are equivalent: (a) ?r 5 c d , on E ' @ & . (b) 7 r 5 c w 2 o n E @ c c , . 20.19. There is a quite useful result due to Saphar [242] which is in the same spirit as
the previous one:
PROPOSITION: Let E be a Banach space, c >_ 0 and 1 5 p three statements are equivalent: (a) For every M E FIN and T E C ( E ,M )
5 00.
Then the following
P I ( T ) 5 CIp#(T). (b) For every M E FIN and T E C ( E ,M )
PROOF: Observe first, that (by 15.7. and the embedding theorem 17.6.)
since g;/ is right-injective and hence right-accessible by 20.3.. This implies that (c) is equivalent to 5 CE on ti' @ E and therefore, by 20.9. (recall the remark at the end of 20.9. concerning constants) and 20.14.
\$ I c(\c) = cg,
on M @ E.
272
20. Projective and Injective Tensor Norma
By the representation theorem for maximal ideals, (a) is equivalent to $1
on M 8 E.
Icgo3
Since goo is left-projective, it follows again from 20.9. that \g;, 5 cgw on M @ E. Hence, the equivalence of (a) and (c) is shown. Since gp = \gi, by 20.14., the same arguments give the equivalence with (b). 0 In 25.6. a much more general statement of this type will be given. For p = 2 the proposition shows that P l ( T ) Ic P z ( T ) for every T E E(E, F) whenever E satisfies (c) and F is an arbitrary Banach space. The little Grothendieck theorem 11.11. shows the following:
COROLLARY: E v e y absolutely 2-summing operator T : 12 t r a y Banach space) is absolutely l-summing and
+F
(where F is an arbi-
P l ( T ) 5 KLGPZ(T) * In 11.9. it was shown that Pz(idl;) = fi.It follows that n-'12P1(idL;)
5 KLG
for all n E N.But lim n-llzPl(idm;) =
n-w
6 2
lim n-'12P1(idq) = -
fi
n-+w
by 11.10., hence the best constants in the little Grothendieck theorem are actually those stated in 11.11.. 20.20. One may ask whether or not there exists a tensor norm Q which is at the same time injective and projective! Existence would imply, by 20.17.,
g;
5 wz'
-
\6/
5 5 /.\ Q
w2
I
hence 2 2 C f ) ? ~which is impossible since id12 is not power compact. More generally, and at a much deeper level:
PROPOSITION: There is no tensor norm which is right-injective and right-projective. PROOF:This would imply that
20. Projective and Injective Tensor Norms
273
and hence that c 21.But this is not true by a result of Gordon and Lewis [84] which solved an old problem from Grothendieck's R&umC ([93],p.72, question 2). 0
Exercises: Ex 20.1. Given a normed space E and a tensor norm a such that for all G E NORM and F C G the space E aU F is an isomorphic subspace of E @u G, then there is a constant A 3 1 such that a ( z ;E , G)5 a(%;E , F ) 5 A a ( z ; El G)
for all F c G and z E E @ F. Ex 20.2. Use the method of proof from 20.4. to show that the tensor norms 5,,,from are right-injective. Ex 12.8. are projective and the tensor norms Ex 20.3. Show that a = a\ = /a = /a\ on C ( K )@ C ( L ) for all compact K and L and all tensor norms a. Ex 20.4. (a) Show that the ideal J of integral operators is neither injective nor surjective. (b) Show that = (Jpdua')surholds isometrically. Ex 20.5. (a) (a/)\3 a if a is right-injective. In particular, ((T\)/)\ = T\. (b) Show that (\a)\ 5 \(a\) for every tensor norm a. Hint: Use the description in terms of L ~ ( B Eand ) l,(Bp). Ex 20.6. Assume (%,A) a. (a) a is left-projective if and only if for all ElG E BAN, each subspace F C E and each T E %(F,G) there exists an ezlension f' E %(ElG")of n c o T with A(T) = A@). (b) a is right-projective if and only if for all Banach spaces El metric surjections Q : F -+ G and all T E B(E, G) there exists a lifting T E g ( E , F") of KG o T (i.e. Q" o T = KG o T) such that A(?') = A(T). Hint: It is enough to show that a is right-projective on FIN. Ex 20.7. For a given tensor norm a denote by (\%/,\A/) the maximal normed operator ideal associated with \a/.Show that T E C ( E , F) is in \%/ if and only if there are a compact K ,a strictly localizable measure /A on some set 52, operators R,S E C and U E % such that
!J3ra'
-
T
E - F .
FN
20. Projective and Injective Tensor Norms
274
Moreover, \A/(T) = min I1RIIA(V)llSll,the minimum being taken over all such factorizations.
Ex 20.8. Let E be a Banach space and 1 5 p 5
The following three statements
00.
are equivalent: (a) idE E inJ.
xrr
(b) E is isomorphic to a subspace of a quotient of some L p ( p ) . (c) E is isomorphic to a quotient of a subspace of some L p ( p ) .Hint: cZLSur
= %'"j 'w.
Ly
Ex 20.9. The class of all Banach spaces E such that (idE) = 1 is stable under formation of isometric subspaces and quotients. Hint: (/wp\)\ = /wp\ ... Ex 20.10. Use the little Grothendieck theorem to show that \.C2 = 32,\Cz/ = ,pa' o ?2(= 9 2 ) and 321 = 9 2 with equivalent norms. More precisely,
E x 20.11. Show that (\w2)\ = \w2 and /
( w 4 = w2/. Hint: Consider the associate ideals, 20.11. and 20.12.. E x 20.12. Show that Iltrt2 :12 @ ( /&q ~Kll = K G .
-
Ex 20.13. (a) If a is left-accessible, E and N are normed spaces, N finite dimensional, then
(E
N)" = E"
@a
N
holds isometrically. (b) Use (a) and the fact that g& is accessible (to be proven in the next section) to verify that the bidual E" of a Banach space satisfies Grothendieck's theorem if and only if E does.
E x 20.14. (a) Show that P2 5 D2 5 KLGP2 on I)z(E,tl) = % ( E , e l ) for every Banach space E. (b) Formulate (a) in terms of tensor norms on E @ c,,. Ex 20.15. (a) Show that fi 5 P l ( i d p ) 5 K ~ f iHint: . 11.9.. (b) Deduce from this that llid@ id : 6 @c e;l
E x 20.16. Let Sn-lbe the unit sphere of S,-l x
If a
N
-
6 @ r qll 5 K
G ~ . and sn the scalar product restricted to
(%,A), then
Hint: Use the isometries z-(z 1): and z-(2 C(Sn-l)). case K G ( ~=) ~ ( 8 , ;C(Sn-i),
I .) of t?j into C(Sn-l). Note the special
21. Accessible Tensor Norms and Operator Ideals
275
21. Accessible Tensor Norms and Operator Ideals
The natural mapping
is a metric injection if E and F have the metric approximation property (see section 15). In this section other conditions are investigated under which I is an isomorphic injection - or at least injective. Some criteria will be presented in order to show that a given tensor norm is totally accessible. With the help of certain “accessibility” notions for operator ideals it will turn out that all a& are totally accessible. In the second part of this section some more general approximation properties will be analyzed.
21.1. The duality theorem 15.5. for a tensor norm a,which is nothing else but the embedding theorem 17.6. for the associated maximal operator ideal a, states that
Since Z 5 a 5 d always holds for the finite and cofinite hull a,the problem under investigation is to find conditions which ensure that Q 5 ?F on E C3 F for a given finitely generated tensor norm and Banach spaces E and F. Recall from 15.6. that a tensor norm Q is called right-accessible if M, F) = %(-;MIF) for all M E FIN and F E NORM (or only E BAN by 13.4.), left-accessible if its transpose a‘ is rightaccessible, accessible if it is both right- and left-accessible and totally accessible if c a =+ a. With the notions of injectivity and projectivity of tensor norms it is possible to verify accessibility in many cases. The following result is basic to what will be done; it is due to Saphar [241].
z(.;
PROPOSITION: Let a be a tensor norm. (1) The right-projective associate a/ and the right-injective associate a\ are right-
accessible. ( 2 ) If a is left-accessible, then a\ is tofally accessible. ( 3 ) The tensor norms (\a)\,/(a/), and the injective associate /a\ are totally accessible. In particular, injective tensor norms are totally accessible.
PROOF:The first statement was already proven in 20.3.(4). For the proof of (2) take normed spaces E and F. Since & , ( B p ) has the metric approximation property and a
276
21. Accessible Tensor Norms and Operator Ideals
is left-accessible, the approximation lemmashows that 'Z = d on E @.te,(Bp). Now the formulas from 20.10. imply that for z E E 8 F 4
a\(%;E , F ) = + a \(z; E , F ) = Z ( ZE,&,(Bp)) ; = c
= a (%; E , & ( B p ) ) =
t
\(Z;
t
E , F ) = a\(%;E , F ) .
(3) is a simple consequence of (1) and (2). 0
There is no statement like (2) for the projective associate a/:Just take the projective norm ir = T / as an example. But even if Q is totally accessible, a/ need not be; this follows from the following two facts which will be shown in a moment: w i is totally accessible (see 21.5.) but w i / is not (see 21.6.).
COROLLARY: The tensor n o m s g;,
w2
and 92 are totally accessible.
PROOF: Since gpJis left-projective and g; = gpJ\ (see 20.14.), it follows that g;
= gpJ\ = (\99')\
is totally accessible. Moreover, g i = gp and wp is injective. 0
21.2. It turns out that it is sometimes easier to check the accessibility of a given finitely generated tensor norm through its associated maximal operator ideal. A quasi-Banach operator ideal ( 0 ,A) is called right-accessible, if for all (M, F) E FIN x BAN, operators T E C ( M , F) and E > 0 there are N E FIN(F) and S E C ( M ,N ) such that
M-F
T
\ J'5
and A(S) 5 (1
+ E ) A ( T ).
N
It is called left-accessible, if for all (E, N ) E BAN x FIN, operators T E C ( E ,N ) and E > 0 there are L E COFIN(E) and S E C ( E / L , N ) such that T
E-N 0;
1 /$
and A(S)
5 (1 + &)A(T).
A left- and right-accessible ideal is called accessible. Moreover, ( 0 ,A) is totally accessible, if for every finite rank operator T E S ( E ,F) between Banach spaces and E > 0 there are ( L ,N ) E COFIN(E)xFIN(F) and S E 2 ( E / L ,N ) such that
21. Accessible Tensor Norms and Operator Ideals
E-F
277
T
-
and A(S)
5 (1 + E ) A ( T .)
JI:
EIL
N
S
Obviously, every injective quasi-Banach ideal is right-accessible and every surjective ideal is left-accessible. The canonical factorization T
E-F
I -
im T
ElkerT
shows that a surjective and injective quasi-Banach ideal is even totally accessible. For example, the injective and surjective ideal 2, of type p operators (see Ex 9.15.) is totally accessible and the injective ideal C9 of cotype q operators is right-accessible. We do not know whether or not C9 is left-accessible (being injective it would even be totally accessible - by 21.1.(2) and the following proposition, or by a direct proof). In Ex 21.1. it will be checked that the factorizations in the definitions of accessibility also hold when only the metric approximation property is assumed. Reisner [230] called accessible operator ideals “semi-tensorial”.
21.3. For a maximal Banach operator ideal associated with a (finitely generated) tensor norm a the notions of accessibility coincide; this will follow from the embedding theorem E‘Q-a F ~(E,F).
c ‘ .
PROPOSITION: A finitely generated tensor norm is right-accessible (resp. left-accessible, accessible, totally accessible) if and only if its associated maximal Banach ideal is.
PROOF:It will be shown that a is totally accessible if and only if (%,A)has this property; all other proofs are similar. Assume that a is totally accessible and let T E 5 ( E , F ) . Then
which implies that there are (M, N) E FIN(E’) x FIN(F) and u E M 8 N with a(u;M , N ) 5 (1
+ &)A(T)
and
+
1%’€3 I s ( u ) = ZT
.
Hence, T,, E C ( E / M o ,N) satisfies A(T,,) 5: (1 E)A(T) and 1; o T,,o Q&,,= T . Conversely, let (a,A) be totally accessible. By the embedding lemma 13.3. it suffices to check that a(*; E‘, F) 5 Z(*; E‘, F )
21. Accessible Tensor Norms and Operator Ideals
278
for all Banach spaces E and F. For z E E' 8 F and E > 0 there are L E COFIN(E) and N E FIN(F) as well a8 an operator S E C ( E / L ,N) such that
-
A(S) 5 (1 + E)A(T,) and 1; o S o Qf = T, . For 2s E Lo @a
1 N = % ( E / L ,N) it follows that
N = (E/L)'
L (1 + E)%(z; E', F )
cr(zs; Lo, N ) = A(S)
and If: 8 IE(zs) = z
,
which completes the proof. 0
-
Since 21dUa' a* and a* a* (see 17.8. and 17.9.) for. the dual and adjoint operator ideals, the proposition 15.6. gives the N
COROLLARY: Let ( Q , A ) be a maximal normed operator ideal. (1) (adu0', Ad"'') is right-accessible (resp. left-accessible, totally accessible) if and only if (a,A) is left-accessible (resp. right-accessible, totally accessible). ( 2 ) (%*,A*)is right-accessible (resp. lejl-accessible) if and only if (%,A) is leftaccessible (resp. right-accessible).
21.4. The following result will be quite useful:
Let (a,A) and (%, B) be quasi-Banach ideals, (a,A) injective and PROPOSITION: left-accessible, (%, B) totally accessible. Then (% o a,B o A) is totally accessible.
PROOF: Take T E $(El F) and E > 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
PROPOSITION 2: For 1 < p < 00 a Banach space E has the bounded gp-approzimation if (and only i f ) gp 5 A& holds on F @ E for all reflexive Banach spaces F. property PROOF: Take an arbitrary Banach space F and z E F @ffv E. One has to show that for every cp E (FQffVE)' = v p r ( E ,F') of Ppr-norm one the inequality I(cp0, .)I
IG
( z ;F, E )
holds. The Grothendieck-Pietsch factorization theorem 11.3. shows that the operator T E V p , ( EF') , associated with cp factors through operators R E Vp1(E,G) and S E L!(G, F'), where G is a subspace of some Lpt, hence reflexive, llSll = 1 and PP1(T) = PPr(R).It follows that there is a yo E (G' @gv E)' of norm one with (9, z ) = ( P o ,
[(s' K F ) @ i d E ]
(2))
*
21. Accessible Tensor Norma and Operator Ideala
283
Therefore, I(Pp,Z)l
s g p ( [ ( s ’ O K F ) ~ i d E ] ( % ) ; G ’ , EA) & s (’*’;G’,E)s
5 xlls’ 0 KFll
llidE11&(Z; F, E ) 5 A&(%; F, E )
by the metric mapping property. The proof works neither for p = 1 nor for p = 00; it is not clear to us whether the statement holds for p = 1, which is the usual bounded approximation property (probably not, by the foliowing.corollary).
21.10. For 1 5 p
< 00 an operator T E C ( E ,F) is called Pietsch p-integral if it factors T
E-F
Ts I
C(K)
L, (PI
, on K. With for some compact K and Borel-Radon measure u
the class ‘$333, of all Pietsch pintegral operators becomes a Banach operator ideal. Clearly, ‘$333, = q33 and ‘$3& = 332 = ‘$32 hold isometrically (see 18.7.). Moreover,
F’)
A 3,(E, F‘)
by the factorization theorem for p-integral operators. As for p = 1 (see Appendix D9.),the fact that there are (for p # 2 ) Banach spaces without the gp-approximation property was used by Reinov [228] in order to show that there are p-integral operators which are not Pietsch p-integral. It will be shown in 33.6. that
F’c!&, E = ‘WP( F, E ) holds isometrically if F‘ has the Radon-Nikodym property and F‘638,E injective. This will be used to prove the
F’& E is
COROLLARY: Let 1 < p < 00 and E a Banach space which i s A-complemented in its bidual. If E has ihe gp-approximation property, then it has the bounded gp-approxirnation property with constant A. Note that for p = 1 only the weaker statement 16.4. is known.
284
21. Accessible Tensor Norms and Opemtor Ideats
PROOF: By proposition 2 it is enough to show that gp 5 A& on F' 8 E for reflexive -+ E the associated operator and let P : E" E be a projection. Since, as a reflexive space, F has the Radon-Nikodym property, it follows that
spaces F. Take z E F' 8 E , denote by T, : F
by the regularity of Jp and the embedding theorem. 0
21.11. It is good to be able to verify the a-approxim'ation property with the help of the coincidence of operator ideals: PROPOSITION: Let E be a Banach space and a and /3 finitely generated, accessible tensor norms with associated operator ideals 24 and % such that %(ElF ) = %(ElF ) for all reflexive Banach spaces F. (1) Then for all even duals E(2k)of E and all Banach spaces F 24(E(2k),F ) = % ( E ( 2 k )F, )
.
(2) If p' is totally accessible,, then the space E and all its even dual spaces E ( 2 k )have the a' -approximation property.
( 3 ) If a is totally accessible, then all the odd dvals of E have the p'-appro;cimation property.
PROOF: Since (EGa,F)' = % ( E ,F') = % ( E ,F') = ( E g p F ) ' , it follows that a' and p' are equivalent norms on E 8F for all reflexive spaces F. Taking the Johnson space C2 for F, lemma 17.16.shows that there are c1 2 0 and cp 2 0 satisfying c1a' 5 p' c2a' on E @ M for all M E FIN. Since ( E @-, N)" = E" @y N if 7 is accessible (see Ex 20.13.), the fact that the tensor norms are finitely generated implies that E ( ~ ~ ) &F, = , , E ( ~ ~ )F & ~ , for all Banach spaces F ; in particular, (2) holds. The regularity of 24 and B ! implies c
(1). To see (3), observe first that the embedding theorem implies that % and p are equivalent norms on all E' @ F' for reflexive F. Since a and p are accessible, it follows c that a = and ,O = p on E' 8 M for all M E FIN - and the same arguments as before show that E ( 2 k t 1 ) & a F= E(2kt+')&3pF for all Banach spaces F; this implies (3).
21. Accessible Tensor Norma and Opemtor Ideals
285
Taking Q = g; and 0 = g p i , the fact that g;l is totally accessible, corollary 21.10. and proposition 21.7.(5) give the THEOREM:Let 1 < p < 00 and E a Banach space such that
for all reflexive Banach spaces F . Then: (1) For all Banach spaces F and all even duals of E
(2) E and all its even duals have the metric g,-approximation property. (3) The odd duals of E have the d,#-approximation property.
(1) is due to Kisliakov [143] and (2) to Bourgain-Reinov [17]. H” satisfies the assumption of the theorem for all 1 < p < 00 : This was shown by Kisliakov [143] and Gordon-Reisner [86] - both using the fact that H m is complemented in the bidual of the disk algebra A and a result of Mityagin-Pelczyriski (see [202], p.16) stating that %$(A’ *) = JP(A’ -1. It follows in particular that H” has the metric gp-approximation property (BourgainReinov); the reader is reminded that it is not known whether or not H a has the approximation property.
21.12. Here is a summary of the results which were obtained in this section about the accessibility of Laprestb’s tensor norms Q ~ : , ~ (1) All Q ~ and , ~all a;,q as well as all their projective and injective associates are accessible (21.5. and 21.1.). (2) Totally accessible are: w2, d 2 , g 2 , qq and ap,2(by 21.5. and 21.1. since ~ 2 is , right-inj ect ive) . (3) Not totally accessible are: d p and g p for 1 5 p < 00 and p # 2 (by Reinov’s counterexample, 21.7.)’ all w i / for p # 2 (see 21.6.). We do not know whether or not goo and wp (for 1 _< p 5 00 and p # 2) are totally accessible.
Exercises:
Ex 21.1. Let (a,A) be a quasi-Banach operator ideal. Use the factorization of finite rank operators proved in 16.9. to show the following results:
~
21. Accessible Tensor Norms and Operator Ideals
286
(a) If M E FIN and F has the bounded approximation property with constant A 2 1, then for every T E 2 l ( M , F ) and E > 0 there is a subspace N E FIN(F) such that A(T: M + N ) 5 X(l+E)A(T: M + F ) . (b) If N E FIN and E' has the bounded approximation property with constant X 2 1, then there is an L E COFIN(E) and To E Q ( E / L , N ) such that T = To o Qf and A(To) IX(1 E)A(T). (c) If 2l is left-accessible and F has the bounded approximation property with constant A 2 1 (or: rz( is right-accessible and E' has the bounded approximation property with constant A 2 l), then for every T E $ ( E , F ) and 6 > 0 there are subspaces L E COFIN(E) and N E FIN(F) and To E C ( E / L ,N) such that T = If,, o T o o Q f and A(T') 5 A(1+ e)A(T). (d) If Q is left-accessible, then Q'"j is totally accessible. (e) If 2l is right-accessible, then 2l'"' is totally accessible.
+
If (%,A) is maximal normed, note that all these statements follow from 21.3., the embedding theorem, the approximation lemma and the respective properties for tensor norms. Ex 21.2. (a) If ('2, A) and (B,B)are both right-accessible (resp: both left-accessible), then ('13 o IZL, B o A) is right-accessible (resp: left-accessible). be two quasi-Banach operator ideals with 2l C '13 and (b) Let (%,A) and ('13,B) B 5 A such that (%,A) = ('13,B) on FIN. Then (%,A)is right-accessible (resp. left-accessible, totally accessible) whenever ('13, B) is. Ex 21.3. Principle of local reflexivity for operator ideals: Let (%,A) be a rightaccessible quasi-Banach operator ideal, F a Banach space, M € FIN and N € FIN(F'). Then for every T E C ( M , F") and E > 0 there is an operator R E C ( M ,F) such that
(1) I (1 + W T ) , (2) ( 3 / , R z )= ( T z , d )for all ( 2 , ~ ' )E M x N, (3) Rz = Tz whenever Tz E E, (4) If T is injective, R can be chosen to be injective and satisfy [[R-lIlI (l+~)llT-'[[. Hint: Use 6.6.. If (a,A) is maximal normed, the weak principle (i.e. only (1) and (2)) follows from Q ( M , E)" = %(M, E") (see Ex 20.13.), by Helly's lemma. Ex 21.4. (a) Given E > 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].
22. Minimal Operator Ideals
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)
22. Minimal Opemtor Ideals
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
22. Minimal Operator Ideals
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
22. Minimal Opemtor Ideals
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
22. Minimal Opemtor Ideals
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
22. Minimal Opemtor Ideals
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.
22. Minimal Opemtor Ideals
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
22. Minimal Operator Ideals
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
22. Minimal Operator Ideals
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 ...}.
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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.
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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) .
<
Corollary 1 implies, in particular, that every operator CS, -+ Cp"(for 1 p 5 2 ) factors through a Hilbert space. More generally, it will be shown in 26.1. that every operator C: -+ C; (with 1 _< p 5 2 5 q I 00) has this property. A nice consequence of this result is a characterization, due to Grothendieck, of Hilbert spaces.
3: Let 1 5 p 5 2 5 q 5 00. A Banach space E is isomorphic t o a Hilberi COROLLARY space if and only if it is isomorphic to a quotient of an 2:-space and isomorphic t o a subspace of an 2;-space. PROOF: If T : 2 : -W E 4 C;, then T factors through a Hilbert space; this easily implies that E is isomorphic to a Hilbert-space. Conversely, since pz' is an isometric subspace of some LP(pn) (for example with Gauss functions), the ultraproduct technique gives that every Hilbert space H is isometric to a subspace of an ultraproduct of L,, which is an L, by 18.4. (this was Ex 18.7.(c)). Dualizing the embedding H 4 L 4 t ( p ) gives L h ) H*0 +
312
23. Cg-Spaces
Exercises: Ex 23.1. C;-spaces have the metric approximation property if 1 < p < 00. Hint: Reflexivity. Ex 23.2. There is no constant c such that ww 5 cw2. Hint: End of 23.2.. Ex 23.3. A Banach space is an ZG-space if and only if its bidual is isomorphic to a complement