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Applications of Banach ideals of operators


Author: J. R. Retherford
Journal: Bull. Amer. Math. Soc. 81 (1975), 978-1012
MSC (1970): Primary 46-02, 47-02, 46C05, 46B10, 46B15, 46E05, 46E15, 46E30, 47B10
DOI: https://doi.org/10.1090/S0002-9904-1975-13881-X
MathSciNet review: 0412834
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  • 1. 1. J. Lindenstrauss and L. Tzafriri, The classical Banach spaces, Lecture Notes in Math., Springer-Verlag, Berlin and New York, 1973. MR 415253
  • 1. 2. A. Pietsch, Theorie der Operatorenideale, Friedrich-Schiller-Universität, Jena 1972. MR 361822
  • 1. 3. J. R. Retherford, Operator characterizations of ${\cal L}\sbp$-spaces, Israel J. Math. 13 (1972), 337-347. MR 328620
  • 1. 4. J. R. Retherford, D. R. Lewis and Y. Gordon, Banach ideals of operators with applications to the finite dimensional structure of Banach spaces, Israel J. Math. 13 (1972), 348-360. MR 343054
  • 1. 5. J. R. Retherford, D. R. Lewis and Y. Gordon, Banach ideals of operators with applications, J. Functional Analysis 14 (1973), 85-129. MR 380488
  • 1. 6. C. Stegall and D. R. Lewis, Banach spaces whose duals are isomorphic to $l\sb{1}(\Gamma )$J. Functional Analysis 12 (1973), 177-187. MR 342987
  • 2. 1. J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $L\sbp$-spaces and their applications, Studia Math. 29 (1968), 275-326. MR 37 # 6743. MR 231188
  • 2. 2. P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 13 (1973), 309-317. MR 402468
  • 2. 3. A. M. Davis, The approximation problem for Banach spaces, Bull. London Math. Soc. 5 (1973), 261-266. MR 338735
  • 2. 4. J. S. Morrell and J. R. Retherford, p-trivial Banach spaces, Studia Math. 43 (1972), 1-25. MR 47 #2321. MR 313767
  • 3. 1. D. Hubert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. IV, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. 1905 157-227.
  • 3. Erhard Schmidt, Zur Theorie der linearen und nicht linearen Integralgleichungen Zweite Abhandlung, Math. Ann. 64 (1907), no. 2, 161–174 (German). MR 1511432, https://doi.org/10.1007/BF01449890
  • 3. F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), no. 1, 116–229. MR 1503275, https://doi.org/10.2307/1968693
  • 3. 4. J. von Neumann and R. Schatten, The cross-space of linear transformations. II, III, Ann. of Math. (2) 47 (1946), 608-630; (2) 49 (1948), 557-582. MR 8, 31; 10, 256. MR 16533
  • 3. 5. J. W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math. (2) 42 (1941), 839-873. MR 3, 208. MR 5790
  • 3. 6. R. Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und inhr Grenzgebiete, N.F., Heft 27, Springer-Verlag, Berlin, 1960. MR 22 #9878. MR 119112
  • 3. 7. 1. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators in Hilbert space, "Nauka", Moscow, 1965; English transl., Transl. Math. Monographs, vol. 18, Amer. Math. Soc. Providence, R. L, 1969. MR 36 #3137; 39 #7447. MR 246142
  • 3. 8. W. Oosterbrink, Normed ideals, Dissertation, Univ. of Groningen, 1970.
  • 3. Friedrich Riesz, Über lineare Funktionalgleichungen, Acta Math. 41 (1916), no. 1, 71–98 (German). MR 1555146, https://doi.org/10.1007/BF02422940
  • 3. 10. S. Kakutani, Iteration of linear operations in complex Banach spaces, Proc. Imp. Acad. Tokyo 14 (1938), 295-300.
  • 3. 11. A. F. Ruston, On the Fredholm theory of integral equations for operators belonging to the trace class of a general Banach space, Proc. London Math. Soc. (2) 53 (1951), 109-124. MR 13, 138. MR 42612
  • 3. 12. A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955). MR 17, 763. MR 75539
  • 3. 13. T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261-322. MR 21 #6541. MR 107819
  • 3. 14. R. Schatten, A theory of cross-spaces, Ann. of Math. Studies, no. 26, Princeton Univ. Press, Princeton, N.J., 1950. MR 12, 186. MR 36935
  • 3. 15. A. Pietsch, Absolut summierende Abbildungen in lokalkonvexen Räumen, Math. Nachr. 27 (1963), 70-103. MR 28 # 1473. MR 158247
  • 3. 16. A. Pietsch, Quasinukleäre Abbildungen in normiertenRäumen, Math. Ann. 165 (1966), 76-90. MR 33 #6412. MR 198253
  • 3. 17. A. Pietsch, Absolut p-summierende Abbildungen in normierten Räumen, Studia Math. 28 (1966/67), 333-353. MR 35 #7162. MR 216328
  • 3. 18. A. Pietsch, Hilbert-Schmidt-Abbildungen in Banach-Räumen, Math. Nachr. 37 (1968), 237-245. MR 38 #2617. MR 234300
  • 3. 19. A. Pietsch, Ideal von Operatoren in Banachräumen, Mitt. Math. Ges. DDR (1968), 1-13.
  • 3. 20. A. Pietsch, Ideale von S, Studia Math. 38 (1970), 59-69. MR 412833
  • 3. 21. A. Pietsch, l, Acta Sci. Math. (Szeged) 31 (1970), 117-123. MR 42 #889. MR 265980
  • 3. 22. A. Pietsch, Adjungierte normierte Operatorenideale, Math. Nachr. 48 (1971), 189-211. MR 44 #7307. MR 290122
  • 4. 1. B. S. Brudovskiĭ, Associated nuclear topology, mappings of type s, and strongly nuclear spaces, Dokl. Akad. Nauk SSSR 178 (1968), 271-273=Soviet Math. Dokl. 9 (1968), 61-63. MR 37 #1952. MR 226362
  • 4. 2. E. Dubinsky and M. S. Ramanujan, Inclusion theorems for absolutely λ-summing maps, Math. Ann. 192 (1971), 177-190. MR 45 #2371. MR 293294
  • 4. 3. E. Dubinsky and M. S. Ramanujan, λ-nuclear spaces, Mem. Amer. Math. Soc. No. 128 (1972). MR 420215
  • 4. 4. A. Pietsch, Nukleare lokalkonvexe Räume, Akademie-Verlag, Berlin, 1965. MR 31 #6114. MR 181888
  • 4. 5. L. Schwartz, Séminaire, Applications radonifiantes, Paris, 1969/70.
  • 4. 6. L. Schwartz, Méasures cylindriques et applications radionifiantes dans les espaces du suites, Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), Univ. of Tokyo Press, Tokyo, 1970, pp. 41-59. MR 44 #6932. MR 289744
  • 4. 7. L. Schwartz, Applications p-radonifiantes et théorème de dualité, Studia Math. 38 (1970), 203-213. MR 46 #321. MR 301163
  • 4. 8. A. Badrikian, Séminaire sur les fonctions aléatoires linéaires et les mésures cylindriques, Lecture Notes in Math., vol. 139, Springer-Verlag, Berlin and New York, 1970. MR 43 #4994. MR 279271
  • 5. 1. Same as [3.22].
  • 5. 2. Same as [1.5].
  • 6. 1. C. Stegall and J. R. Retherford, Fully nuclear and completely nuclear operators with applications to ${\cal L}\sb{1}-$ and ${\cal L}\sb{\infty }$-spaces, Trans. Amer. Math. Soc. 163 (1972), 457-493. MR 415277
  • 6. 2. Same as [1.6].
  • 6. 3. I. M. Gel'fand, Abstrakte Funktionen und lineäre Operatoren, Mat. Sb. 4 (1938), 235-286.
  • 6. 4. A. Pełczyński, On Banach spaces containingL1(μ), Studia Math. 30 (1968), 231-246. MR 38 #521. MR 232195
  • 6. 5. J. Lindenstrauss and H. P. Rosenthal, The ${\cal L}\sbp$ spaces, Israel J. Math. 7 (1969), 325-349. MR 42 #5012. MR 270119
  • 6. 6. J. Cohen, Absolutely p-summing, p-nuclear operators and their conjugates, Dissertation, University of Maryland, College Park, Md., 1969.
  • 6. 7. J. R. Holub, A characterization of subspaces ofLp(μ), Studia Math. 42 (1972), 265-270. MR 46 #2402. MR 303264
  • 6. 8. W. B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345. MR 44 #7318. MR 290133
  • 6. 9. S. Kwapien, On operators factorizable through L, Bull. Soc. Math. France Mém. 31732 (1972), 215-225. MR 397464
  • 6. 10. D. R. Lewis, Integral operators on ${\scr L}\sbp$-spaces, Pacific J. Math. 46 (1973), 451-456. MR 328651
  • 6. 11. A. Persson, On some properties of p-nuclear and p-integral operators, Studia Math. 33 (1969), 213-222. MR 40 #769. MR 247504
  • 6. 12. Same as [1.5].
  • 6. 13. D. R. Lewis and Y. Gordon, Banach ideals on Hilbert spaces, Studia Math, (to appear). MR 388149
  • 7. 1. A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1-79 (1956), MR 20 #1194. MR 94682
  • 7. 2. Same [2.1].
  • 7. 3. A. Grothendieck, Sur certaines classes des suites dans les espaces de Banach et le théorème de Dvoretzky-Rogers, Bol. Soc. Mat. São Paulo 8 (1953), 81-110 (1956). MR 20 #1195. MR 94683
  • 7. 4. B. Maurey, Une nouvelle demonstration d'un théorème de Grothendieck, Séminaire Maurey-Schwartz 1972/73. Exposé 22.
  • 7. 5. K. H. Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962. MR 24 #A2844. MR 133008
  • 7. 6. Same as [2.4].
  • 8. 1. S. Kwapien, On a theorem of L. Schwartz and its applications to absolutely summing operators, Studia Math. 38 (1970), 193-201. MR 43 #3822. MR 278090
  • 8. 2. P. Saphar, Applications p décomposantes et p-absolument sommantes, Israel J. Math. 11 (1972), 164-179. MR 45 #5779. MR 296720
  • 8. 3. P. Saphar, Une charactérisation des sous-espaces de L$^p$ et ses applications, Séminaire Maurey-Schwartz 1972/73. Exposé 14. MR 355569
  • 8. 4. Same as [6.6].
  • 8. 5. D. J. H. Garling, Lattice bounding, radonifying and absolutely summing mappings, Proc. Cambridge Philos. Soc. 77 (1974), 327-333. MR 361882
  • 8. 6. N. J. Neilson, On Banach ideals determined by Banach lattices and their applications, Dissertationes Math. 109 (1973). MR 487385
  • 9. 1. E. Dubinsky, A. Peteyński and H. Rosenthal, On Banach spaces X for which $\Pi_2(L_\infty,X)=B(L_\infty,X)$, Studia Math. 54 (1972), 617-648. MR 365097
  • 9. 2. W. Orlicz, Über unbedingte Konvergenz in Funktionenräumen. I, Studia Math. 4 (1933), 33-37; II, Studia Math. 4 (1933), 41-47.
  • 10. 1. H. P. Rosenthal, On subspaces of L, Ann. of Math. (2) 97 (1973), 344-373. MR 47 #784. MR 312222
  • 10. 2. J. Bretagnolle and D. Dacunha-Castelle, Application de l'étude de certaines formes linéaires aléatories au plongement d'espaces de Banach dans les espaces L, Ann. Sci. École Norm. Sup. (4) 2 (1969), 437-480. MR 42 #839. MR 265930
  • 10. 3. B. Maurey, Un lemma de H. P. Rosenthal, Séminaire Maurey-Schwartz 1972/73, Exposé 21. MR 399809
  • 10. 4. Same as [1.1].
  • 10. 5. Same as [2.1].
  • 11. 1. Y. Gordon and D. R. Lewis, Absolutely summing operators and local unconditional structures, Acta Math. 133 (1974), 27-48. MR 410341
  • 11. 2. A. Pełczyński, Some new isomorphic properties of the Banach spaces of holomorphic functions A and H (to appear).
  • 11. 3. D. R. Lewis, An isomorphic characterization of the Schmidt class(to appear). MR 374968
  • 11. 4. W. B. Johnson, On finite dimensional subspaces of Banach spaces with local unconditional structure, Studia Math. 51, (1974), 223-238. MR 358306
  • 11. 5. T. Figiel, W. B. Johnson and L. Tzafriri, On Banach lattices and spaces having local unconditional structure with applications to Lorenz function spaces(to appear). MR 367624
  • 11. 6. R. C. James, Some self-dual properties of normed linear spaces, Ann. of Math. Studies, no. 69, Princeton Univ. Press, Princeton, N.J., 1972, pp. 159-175. MR 454600
  • 11. 7. R. C. James, Super-reflexive Banach spaces, Canad. J. Math. 24 (1972), 896-904. MR 47 #9248. MR 320713
  • 11. 8. P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13 (1972), 281-288. MR 336297
  • 11. 9. L. Tzafriri, On Banach spaces with unconditional bases, Israel J. Math. 17 (1974), 84-93. MR 348456
  • 11. 10. M. M. Day, Normed linear spaces, 3rd ed., Springer-Verlag, Berlin and New York, 1972.
  • 11. 11 W. Johnson and L. Tzafriri, On the local structure of subspaces of Banach lattices (to appear). MR 420210
  • 12. 1. Y. Gordon, On p-absolutely summing constants of Banach spaces, Israel J. Math. 7 (1969), 151-163. MR 40 #3269. MR 250028
  • 12. 2. Y. Gordon, Asymmetry and projection constants of Banach spaces, Israel J. Math. 14 (1973), 50-62. MR 47 #7388. MR 318842
  • 12. 3. Y. Gordon and D. J. H. Garling, Relations between some constants associated with finite-dimensional Banach spaces, Israel J. Math. 9 (1971), 346-361. MR 412775
  • 12. 4. Same as [11.1].
  • 12. 5. Y. Gordon and D. R. Lewis, Absolutely summing, L, Bull. Amer. Math. Soc. 79 (1973), 1270-1273. MR 336416
  • 12. 6. A. Pietsch. Absolutely p-summing operators in L. I, II, Séminaire Goulaouic-Schwartz 1970/71.
  • 12. 7. V. I. Gurariĭ, M. I. Kadec and V. I. Macaev, On Banach-Mazur distance between certain Minkowski spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 719-722. MR 32 #8113. MR 190701
  • 12. 8. Same as [1.5].
  • 12. 9. S. Kwapien and A. Pełczyński, The main triangle projection in matrix spaces and its applications, Studia Math. 34 (1970), 43-68. MR 270118
  • 12. 10. M. Ĭ. Kadeć and M. G. Snobar, Certain junctionals on the Minkowski compactum, Mat. Zametki 10 (1971), 453-457=Math. Notes 10 (1971), 694-696. MR 45 #861. MR 291770
  • 12. 11. Same as [1.5].
  • 12. 12. Same as [11.3].
  • 12. 13. Same as [6.13].
  • 12. 14. A. Pełczyński, A characterization of Hilbert-Schmidt operators, Studia Math. 28 (1966/67), 355-360. MR 35 #7163.
  • 12. 15. D. J. H. Garling, Absolutely p-summing operators in Hubert space, Studia Math. 38 (1970), 319-331. MR 43 #6769. MR 281050
  • 13. 1. J.-P. Kahane, Some random series of functions, Heath, Lexington, Mass., 1968. MR 40 #8095. MR 254888
  • 13. 2. B. Maurey, Probabilités cylindriques, type et ordre. Applications radonifiantes, Séminaire Maurey-Schwartz 1972/73, Exposé l. MR 397821
  • 13. 3. B. Maurey, Probabilités cylindriques stables sur les espaces L, Séminaire Maurey-Schwartz 1972/73, Exposé 5.
  • 13. 4. B. Maurey, Espaces de cotype $p$, $0 < p łe 2$, Séminaire Maurey-Schwartz 1972/73, Exposé 7.
  • 13. 5. B. Maurey, Type et cotype dans les espaces munis de structures locales unconditionelles, Séminaire Maurey-Schwartz 1973/74, Exposé 24-25. MR 399796
  • 13. 6. G. Pisier, Sur les espaces qui ne contiennent pas de l, Séminaire Maurey-Schwartz, Annexe 1972/73.
  • 13. 7. G. Pisier, "Type" des espaces normes, Séminaire Maurey-Schwartz 1973/74, Exposé 3. MR 342989
  • 13. 8. G. Pisier, Sur les espaces qui ne contiennent pas de $l_n^\infty$ uniformément, Séminaire Maurey-Schwartz 1973/74, Exposé 7. MR 415276
  • 13. 9. G. Pisier, Une propriété du type p-stable, Séminaire Maurey-Schwartz 1973/74, Exposé 8. MR 399811
  • 13. 10. A. Pełczyński, On unconditional bases and Rademacher averages, Lecture, Kent State University, 1973.
  • 13. 11. J. L. Krivine, Théorème de factorisation dans les espaces réticulés, Séminaire Maurey-Schwartz 1973/74, Exposé 22-23. MR 440334
  • 13. 12. N. Tomczak-Jaegerman, The modulus of smoothness and convexity and the Rademacher averages of trace classes $S\sbp(1łeq p<\infty )$, Studia Math. 50 (1974), 163-182.
  • 13. 13. G. Nordlander, On sign-independent and almost sign-independent convergence in normed linear spaces, Ark. Mat. 4 (1962), 287-296. MR 25 #4330. MR 140916
  • 13. 14. B. Maurey, These, École Polytechnique, Paris, 1973.
  • 13. 15. R. C. James, Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), 542-550. MR 30 #4139. MR 173932
  • 14. 1. J. R. Holub, Schauder bases and norm ideals of compact operators, Proc. Amer. Math. Soc. 38 (1973), 343-348. MR 47 #5560. MR 317012
  • 14. 2. J. R. Holub, Schauder bases and norm ideals of compact operators. II, Indiana Univ. Math. J. 24 (1974), 555-564. MR 350441
  • 14. 3. J. R. Holub, A characterization of the norm ideals of compact operators on Hilbert space, J. Math. Anal. Appl. 50 (1975), 596-606. MR 372638
  • 14. 4. B. S. Mitjagin, Normed ideals of intermediate type, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 819-832; English transl., Amer. Math. Soc. Transl. (2) 63 (1967), 180-194. MR 30 #4142. MR 173935
  • 15. 1. C. A. McCarthy, cp, Israel J. Math. 5 (1967), 249-271. MR 37 #735. MR 225140
  • 15. 2. Same as [13.12].
  • 15. 3. J. R. Holub, On subspaces of separable norm ideals, Bull. Amer. Math. Soc. 79 (1973), 446-448. MR 47 #2432. MR 313880
  • 15. 4. J. R. Holub, On the metric geometry of ideals of operators on Hilbert spaces, Math. Ann. 201 (1973), 157-163. MR 48 #4757. MR 326413
  • 15. 5. J. Arazy and J. Lindenstrauss, Some linear topological properties of the spaces C(to appear).
  • 15. 6. P. Saphar and M. Feder, Spaces of compact operators and their dual spaces, Israel J. Math, (to appear). MR 377591
  • 15. 7. H. P. Rosenthal, A characterization of Banach spaces containing l, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411-2413. MR 358307
  • 15. 8. T Figiel and W. B. Johnson, The approximation property does not imply the bounded approximation property, Proc. Amer. Math. Soc. 41 (1973), 197-200. MR 341032
  • 15. 9. W. B. Johnson, A complementary universal conjugate Banach space and its relation to the approximation problem, Israel J. Math. 13 (1972), 301-310 (1973). MR 48 #4700. MR 326356
  • 15. 10. A. Pełczyński and R. I. Ovsepian, The existence in every separable Banach space of a fundamental, total and bounded biorthogonal sequence and related constructions of uniformly bounded orthonormal systems in L, Séminaire Maurey-Schwartz 1973/74, Exposé 20.
  • 15. 11. Same as [1.2].
  • 15. 12. A. Pietsch, p-majorisierbare vektorwertige Masse, Wiss. Z. Fredrich-Schiller-Univ. Jena/Thüringen 18 (1969), Heft 2, 243-247. MR 41 #8627. MR 264028

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DOI: https://doi.org/10.1090/S0002-9904-1975-13881-X

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