Tensor norms and operator ideals
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| 100 | 1 | |a Defant, Andreas |d 1953- |e Verfasser |0 (DE-588)1018502793 |4 aut | |
| 245 | 1 | 0 | |a Tensor norms and operator ideals |c Andreas Defant ; Klaus Floret |
| 264 | 1 | |a Amsterdam [u.a.] |b North-Holland |c 1993 | |
| 300 | |a XI, 566 S. | ||
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| 490 | 1 | |a North-Holland mathematics studies |v 176 | |
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Datensatz im Suchindex
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| _version_ | 1823051885849870336 |
| adam_text | Contents V
Contents
Introduction 1
Chapter I: Basic Concepts 7
1. Bilinear Mappings 7
1.2. Continuous bilinear mappings, 1.5. non validity of Hahn Banach the¬
orem, 1.7. non validity of open mapping theorem, 1.8. canonical extension
to the bidual.
2. The Algebraic Theory of Tensor Products 15
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.
3. The Projective Norm 26
3.1. Minkowski gauge functional, 3.2. basic properties, 3.3. Bochner inte
grable functions, theorem of Dunford Pettis, 3.4. compact sets, 3.6. nuclear
operators, 3.7. trace, 3.9. it does not respect subspaces, 3.10. extension
property, 3.11. Grothendieck s characterization of L , 3.12. lifting prob¬
lems, 3.13. £| spaces, Ex 3.24. Radon Nikodym theorem for operator val¬
ued measures.
4. The Injective Norm 46
4.1. Basic properties, 4.2. examples, 4.3. t 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.
5. The Approximation Property 58
5.2. Survey about counterexamples, 5.3. compact operators, 5.4. character¬
ization 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.
compactly approximable operators.
6. Duality of the Projective and Injective Norm 70
6.3. Dense sequences of finite dimensional Banach spaces, Johnson spaces
Cp, 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.
vi Contents
7. The Natural Norm on the p Integrable Functions 77
7.1. Bochner p integrable functions, Ap, 7.2. continuous triangle inequality,
7.3. positive operators, density lemma, 7.4. Ap 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 Lp.
8. Absolutely and Weakly p Summable Series and Averaging Tech 90
niques
8.1. Absolutely p summable and weakly p summable sequences, 8.2. rep¬
resentations of operators on or into £p, 8.3. unconditional summability, 8.4.
general scheme of averaging, 8.5. Rademacher functions, Khintchine inequal¬
ity, 8.6. type and cotype of Lp, 8.7. Gauss functions, 8.9. Orlicz property,
Ex 8.9. Rademacher versus Gauss averaging, Ex 8.12. absolutely (r, s)
summing operators.
9. Operator Ideals 108
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. .fiT convex operators.
10. Integral Operators 118
10.3. Characterization with the trace, 10.4. examples, 10.5. factorization,
10.7. characterization with T ® idc 10.8. and 10.9. summability of the
diagonal of infinite matrices, Ex 10.4. extension and lifting properties of
integral operators.
11. Absolutely p—Summing Operators 127
11.1. Basic characterizations, 11.2. positive operators, 11.3. Grothendieck
Pietsch domination theorem and factorization, 11.4. Dvoretzky Rogers the¬
orem, 11.5. composition, 11.6. Hilbert Schmidt operators, 11.7. Pietsch
lemma, 11.8. Kwapieri s test, 11.9. absolutely 2 summing norm of idg,
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 be¬
tween finite dimensional Banach spaces and the Kadec Snobar result about
projections, Ex 11.18. factorization of Hilbert Schmidt operators.
Contents vii
Chapter II: Tensor Norms 146
12. Definition and Examples 146
12.1. Reasonable norms and the metric mapping property, 12.2. criterion,
12.4. finite and cofinite hull, 12.5. Lapreste s tensor norms aPi9, 12.6. com¬
pletion with respect to ctp,q, 12.8. the diagram of Lapreste s tensor norms,
12.9. tensor product representation of weakly p summable sequences, Ex
12.7. tensors of finite rank.
13. The Five Basic Lemmas 159
13.1. Approximation lemma, 13.2. extension lemma, 13.3. embedding
lemma, 13.4. density lemma, 13.5. £p local technique lemma.
14. Grothendieck s Inequality 166
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 Kg, Ex
14.1. the original proof more or less.
15. Dual Tensor Norms 177
15.1. Trace duality, 15.2. dual norms, 15.5. duality theorem, 15.6. acces¬
sibility 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 Ap, 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.
16. The Bounded Approximation Property 190
16.1. Topologies on S,(E, F), 16.2. characterization with the cofinite hull *¥
of the projective norm, 16.4. results involving the Radon Nikodym property,
16.5. and 16.6. duality of e and tt, 16.7. duality of the operator ideals £,91,
and £, 16.8. non nuclear operators with nuclear dual, Ex 16.4. Ex 16.8.
reflexivity of 2 {E, F) for special spaces.
17. The Representation Theorem for Maximal Operator Ideals . . . 200
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. Grothen¬
dieck s theorem, 17.15. and 17.16. characterization with tensor product
operators T®ida, 17.19. ideal norms of identity operators in symmetric fi¬
nite dimensional spaces, 17.20. injective embedding of E®aF into the space
of operators, 17.21. unit ball of 2t(.E, F), Ex 17.16. continuity of S®T , Ex
17.17. density lemma for maximal normed operator ideals.
viii Contents
18. (p, g) Factorable Operators 223
18.2. The norm of the integrating functional, 18.4. ultraproducts, 18.5. fac¬
torization through positive functionals, 18.6. p factorable operators, 18.7.
p integral operators, 18.9. Maurey s factorization theorem, 18.11. the fac¬
torization theorem, Ex 18.4. Ex 18.11. properties of ultraproducts.
19. (p, q)~Dominated Operators 241
19.2. The basic estimates, 19.3. Kwapien s factorization theorem, 19.4.
composition of factorable and dominated operators, 19.6. Grothendieck in¬
equality for C* algebras.
20. Projective and Injective Tensor Norms 250
20.3. Duality relations, 20.4. examples, 20.6. the projective associate, 20.7.
the injective associate, 20.9. finite dimensional characterization, 20.10. gen¬
eral rules for associates, 20.11. 20.13. relations with operator ideals, 20.14.
results about gp, 20.15. table for wi,dii9i and their adjoints, 20.17. Gro¬
thendieck s inequality in its original formulation (operator version), finite
dimensional Grothendieck constants, 20.18. Banach spaces satisfying Gro
thendieck s theorem, 20.19. a result of Saphar and the best constants in the
little Grothendieck theorem.
21. Accessible Tensor Norms and Operator Ideals 275
21.2. Accessible operator ideals, 21.4. total accessibility of certain com¬
position ideals, 21.5. accessibility of Lapreste s tensor norms, 21.6. a re¬
sult about the bounded approximation property, 21.7. the a approximation
properties, Reinov s results in the case a = gp, 21.11. some results of Kisl
jakov and Bourgain Reinov, H°°, Ex 21.3. principle of local reflexivity for
operator ideals.
22. Minimal Operator Ideals 287
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
Vimin{E,F), 22.7. weak * continuous operators in Wnin(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.
23. £» Spaces 300
23.1. Local techniques, 23.2. various characterizations, 23.3. relations
with the £p spaces of Lindenstrauss Pelczynski, 23.4. dual characterization,
23.5. £ , and £f spaces, 23.6. the projection constant, 23.7. quasinuclear
operators, 23.8. characterization of space ideals with integral operators, 23.9.
coincidence of absolutely 1 summing and nuclear operators, Hardy spaces,
23.10. Grothendieck s theorem for £jj spaces, a characterization of Hilbert
spaces, Ex 23.8. the extension norm of an operator.
Contents ix
24. Stable Measures 314
24.1. The linear dimension of £p and Lq, 24.2. positive definite functions and
Bochner s theorem, 24.3. moments of stable measures, 24.4. Levy s theorem
about the embedding of £p into Lq (Levy embeddings), 24.5. embeddings
into Lq, 24.6. 24.7. results due to Saphar, Kwapien and Maurey about
absolutely ^ summing operators with values in £p 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 £,PiJ
and S,p, in particular, of the associated space ideals — results of Kwapien,
25.10. isomorphic characterization of subspaces, quotients, etc. of Lp, 25.11.
the minimal kernel of the injective resp. surjective hull of an operator ideal.
26. More About Lp and Hilbert Spaces 344
26.1. Inequalities about aPi9 coming from the Khintchine and Grothen
dieck inequalities, 26.2. factorization through Hilbert spaces of the identity
mapping (% —? £J?, 26.3. continuity of operators between spaces of Bochner
p integrable functions, complexification of operators, 26.5. Kwapieri s re¬
sult about the factorization of operators Lr —? L, through Lp, 26.6. tensor
norms and ideals on Hilbert spaces, 26.7. the Hilbert Schmidt tensor norm
r, 26.8. Schatten s result about self adjoint, symmetric extensions of r to
Banach spaces, 26.10. limit orders of tensor norms, Puhl s result, 26.11. un¬
conditional bases in £i®a£i, Ex 26.6. unconditionally summable sequences
in Lp(n ® v).
27. Grothendieck s Fourteen Natural Norms 361
27.2. Grothendieck s diagram, 27.3. the original notations.
Chapter III: Special Topics 365
28. More Tensor Norms 365
28.1. Three new classes of tensor norms, 28.3. description of the projective
associate of a* ?! 28.4. characterization of operators in £J, | *ur, 28.5. a
characterization of operators factoring through a Hilbert space, 28.6. 28.8.
description of the composition ideals £p o £q 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, 2], 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
21 S 23, 29.9. — 29.11. tensor products of special tensor norms, Ex 29.8.
ultrastable ideals.
30. The Vector Valued Fourier Transform 394
30.1. Fourier operators, 30.3. their characterization, 30.5. Rademacher and
Gauss type and cotype, Kwapien s type/cotype theorem, characterization of
Hilbert spaces, 30.6. the main theorem, 30.8. type and cotype with respect
to orthonormal bases.
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 oper¬
ators 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 characteri¬
zation, 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 99?^, , 32.7. Maurey s
splitting theorem, 32.10. and 32.11. relation with absolutely (r, s) summing
operators, 32.12. a finite dimensional result.
33. The Radon Nikodym Property for Tensor Norms and Reflexivity 430
33.1. Duality of e and it revisited, 33.3. Lewis theorem, 33.4. permanence
properties, 33.5. Lapreste s tensor norms, 33.6. coincidence of p nuclear and
p integral operators, 33.7. p ^strong operators, 33.8. 33.11. reflexivity of
tensor products and components of operator ideals.
34. Tensorstable Operator Ideals 445
34.1. /? tensorstable and metrically ^ tensorstable operator ideals, 34.2.
strongly /? tensorstable operator ideals, 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 Khint
chine inequality, stability of tensor products of Hilbert spaces, 34.9. e and
x stability of (p, g) factorable operators and their relatives, 34.10. distribu¬
tion of eigenvalues, Pietsch s tensor product trick, 34.11. a result of Kwapieri
unifying Orlicz s, Littlewood s and Grothendieck s inequalities, 34.12. im¬
proving weak inequalities with tensor products, mixing operators l —? £p.
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. probleme des topologies, Taskinen s counterexample,
35.10. injective tensor product of (£)ir) spaces.
Appendices:
A. Some Structural Properties of Banach Spaces 489
Al. Subspaces and quotients, A2. dual systems, A3. lemma of Ky Fan, A4.
bases, A5. Banach algebras, A6. lattices, A7. abstract Lp spaces and their
representation.
B. Integration Theory 493
Extension procedure and the basic theorems: Bl. Daniell 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. du¬
ality, Radon Nikodym theorem, Segal s localization theorem, Lebesgue de¬
composition, B7. strictly localizable measures; Borel—Radon measures
and Riesz representation theorem: B8. r continuity and Bourbaki s
extension procedure, Kolzow s theorem, B9. representation of Borel Radon
measures, BIO. L is complemented in its bidual; Bochner integration:
Bll. measurability of Banach space valued functions, B12. p(n,E), B14.
Pettis integrability, B15. variation lemma.
C. Representable Operators 508
Grothendieck s characterization: Cl. Riesz densities, C3. nuclear op¬
erators, C4. factorization through £%; The Dunford—Pettis theorem: C6.
a general result about representability, C7. strong version of the Dunford
Pettis theorem.
D. The Radon Nikodym Property 517
Basic properties and examples: Dl. reduction to the Lebesgue measure,
D3. examples, D4. dualof Lp(n, E); Pietsch integral operators: D5. and
D6. relation with other operator ideals; The Radon Nikodym property
and operator ideals: D7. and D8. characterizations in terms of (Pietsch)
integral=nuclear, D9. integral operators which are not Pietsch integral.
Bibliography 527
List of Symbols 545
Index 555
|
| any_adam_object | 1 |
| author | Defant, Andreas 1953- Floret, Klaus 1941-2002 |
| author_GND | (DE-588)1018502793 (DE-588)172077206 |
| author_facet | Defant, Andreas 1953- Floret, Klaus 1941-2002 |
| author_role | aut aut |
| author_sort | Defant, Andreas 1953- |
| author_variant | a d ad k f kf |
| building | Verbundindex |
| bvnumber | BV007464565 |
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| id | DE-604.BV007464565 |
| illustrated | Not Illustrated |
| indexdate | 2025-02-03T16:44:36Z |
| institution | BVB |
| isbn | 0444890912 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-004845399 |
| oclc_num | 246656956 |
| open_access_boolean | |
| owner | DE-12 DE-739 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-824 DE-11 DE-188 DE-20 |
| owner_facet | DE-12 DE-739 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-824 DE-11 DE-188 DE-20 |
| physical | XI, 566 S. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | North-Holland |
| record_format | marc |
| series | North-Holland mathematics studies |
| series2 | North-Holland mathematics studies |
| spellingShingle | Defant, Andreas 1953- Floret, Klaus 1941-2002 Tensor norms and operator ideals North-Holland mathematics studies Tensor (DE-588)4184723-4 gnd Norm Mathematik (DE-588)4172021-0 gnd Operatorenideal (DE-588)4284995-0 gnd |
| subject_GND | (DE-588)4184723-4 (DE-588)4172021-0 (DE-588)4284995-0 |
| title | Tensor norms and operator ideals |
| title_auth | Tensor norms and operator ideals |
| title_exact_search | Tensor norms and operator ideals |
| title_full | Tensor norms and operator ideals Andreas Defant ; Klaus Floret |
| title_fullStr | Tensor norms and operator ideals Andreas Defant ; Klaus Floret |
| title_full_unstemmed | Tensor norms and operator ideals Andreas Defant ; Klaus Floret |
| title_short | Tensor norms and operator ideals |
| title_sort | tensor norms and operator ideals |
| topic | Tensor (DE-588)4184723-4 gnd Norm Mathematik (DE-588)4172021-0 gnd Operatorenideal (DE-588)4284995-0 gnd |
| topic_facet | Tensor Norm Mathematik Operatorenideal |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004845399&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003247 |
| work_keys_str_mv | AT defantandreas tensornormsandoperatorideals AT floretklaus tensornormsandoperatorideals |