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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
MathSciNet review: 0412834
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  • 1. Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
  • 1. Albrecht Pietsch, Theorie der Operatorenideale (Zusammenfassung), Friedrich-Schiller-Universität, Jena, 1972 (German). Wissenschaftliche Beiträge der Friedrich-Schiller-Universität Jena. MR 0361822
  • 1. J. R. Retherford, Operator characterizations of \cal𝐿_{𝑝}-spaces, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 337–347 (1973). MR 0328620
  • 1. Y. Gordon, D. R. Lewis, and J. R. Retherford, Banach ideals of operators with applications to the finite dimensional structure of Banach spaces, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 348–360 (1973). MR 0343054
  • 1. Y. Gordon, D. R. Lewis, and J. R. Retherford, Banach ideals of operators with applications, J. Functional Analysis 14 (1973), 85–129. MR 0380488
  • 1. D. R. Lewis and C. Stegall, Banach spaces whose duals are isomorphic to 𝑙₁(Γ), J. Functional Analysis 12 (1973), 177–187. MR 0342987
  • 2. J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in 𝐿_{𝑝}-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 0231188
  • 2. Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309–317. MR 0402468
  • 2. A. M. Davie, The approximation problem for Banach spaces, Bull. London Math. Soc. 5 (1973), 261–266. MR 0338735
  • 2. J. S. Morrell and J. R. Retherford, 𝑝-trivial Banach spaces, Studia Math. 43 (1972), 1–25. MR 0313767
  • 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, 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, 10.2307/1968693
  • 3. Robert Schatten and John von Neumann, The cross-space of linear transformations. II, Ann. of Math. (2) 47 (1946), 608–630. MR 0016533
  • 3. 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 0005790
  • 3. Robert Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete. N. F., Heft 27, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0119112
  • 3. I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR 0246142
  • 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, 10.1007/BF02422940
  • 3. 10. S. Kakutani, Iteration of linear operations in complex Banach spaces, Proc. Imp. Acad. Tokyo 14 (1938), 295-300.
  • 3. 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 0042612
  • 3. Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955), 140 (French). MR 0075539
  • 3. Tosio Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322. MR 0107819
  • 3. Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, no. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
  • 3. Albrecht Pietsch, Absolut summierende Abbildungen in lokalkonvexen Räumen, Math. Nachr. 27 (1963), 77–103 (German). MR 0158247
  • 3. Albrecht Pietsch, Quasinukleare Abbildungen in normierten Räumen, Math. Ann. 165 (1966), 76–90 (German). MR 0198253
  • 3. A. Pietsch, Absolut 𝑝-summierende Abbildungen in normierten Räumen, Studia Math. 28 (1966/1967), 333–353 (German). MR 0216328
  • 3. Albrecht Pietsch, Hilbert-Schmidt-Abbildungen in Banach-Räumen, Math. Nachr. 37 (1968), 237–245 (German). MR 0234300
  • 3. 19. A. Pietsch, Ideal von Operatoren in Banachräumen, Mitt. Math. Ges. DDR (1968), 1-13.
  • 3. Albrecht Pietsch, Ideale von 𝑆_{𝑝}-Operatoren in Banachräumen, Proceedings of the International Colloquium on Nuclear Spaces and Ideals in Operator Algebras (Warsaw, 18–25 June, 1969), 1970, pp. 59–69 (German). MR 0412833
  • 3. Albrecht Pietsch, 𝑙_{𝑝}-faktorisierbare Operatoren in Banachräumen, Acta Sci. Math. (Szeged) 31 (1970), 117–123 (German). MR 0265980
  • 3. Albrecht Pietsch, Adjungierte normierte Operatorenideale, Math. Nachr. 48 (1971), 189–211 (German). MR 0290122
  • 4. B. S. Brudovskiĭ, Associated nuclear topology, type 𝑠 mappings and strongly nuclear spaces, Dokl. Akad. Nauk SSSR 178 (1968), 271–273 (Russian). MR 0226362
  • 4. Ed Dubinsky and M. S. Ramanujan, Inclusion theorems for absolutely 𝜆-summing maps, Math. Ann. 192 (1971), 177–190. MR 0293294
  • 4. Ed Dubinsky and M. S. Ramanujan, On 𝜆-nuclearity, American Mathematical Society, Providence, R. I., 1972. Memoirs of the American Mathematical Society, No. 128. MR 0420215
  • 4. Albrecht Pietsch, Nukleare lokalkonvexe Räume, Schriftenreihe der Institute Für Mathematik bei der Deutschen Akademie der Wissenschaften zu Berlin. Reihe A, Reine Mathematik, Heft 1, Akademie-Verlag, Berlin, 1965 (German). MR 0181888
  • 4. 5. L. Schwartz, Séminaire, Applications radonifiantes, Paris, 1969/70.
  • 4. Laurent Schwartz, Mesures cylindriques et applications radonifiantes dans les espaces de suites, Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), Univ. of Tokyo Press, Tokyo, 1970, pp. 41–59 (French). MR 0289744
  • 4. Laurent Schwartz, Applications 𝑝-radonifiantes et théorème de dualité, Proceedings of the International Colloquium on Nuclear Spaces and Ideals in Operator Algebras (Warsaw, 1969), 1970, pp. 203–213. (errata insert) (French). MR 0301163
  • 4. Albert Badrikian, Séminaire sur les fonctions aléatoires linéaires et les mesures cylindriques, Lecture Notes in Mathematics, Vol. 139, Springer-Verlag, Berlin-New York, 1970 (French). MR 0279271
  • 5. 1. Same as [3.22].
  • 5. 2. Same as [1.5].
  • 6. C. P. Stegall and J. R. Retherford, Fully nuclear and completely nuclear operators with applications to \cal𝐿₁- and \cal𝐿_{∞}-spaces, Trans. Amer. Math. Soc. 163 (1972), 457–492. MR 0415277, 10.1090/S0002-9947-1972-0415277-3
  • 6. 2. Same as [1.6].
  • 6. 3. I. M. Gel'fand, Abstrakte Funktionen und lineäre Operatoren, Mat. Sb. 4 (1938), 235-286.
  • 6. A. Pełczyński, On Banach spaces containing 𝐿₁(𝜇), Studia Math. 30 (1968), 231–246. MR 0232195
  • 6. J. Lindenstrauss and H. P. Rosenthal, The \cal𝐿_{𝑝} spaces, Israel J. Math. 7 (1969), 325–349. MR 0270119
  • 6. 6. J. Cohen, Absolutely p-summing, p-nuclear operators and their conjugates, Dissertation, University of Maryland, College Park, Md., 1969.
  • 6. J. R. Holub, A characterization of subspaces of 𝐿^{𝑝}(𝜇), Studia Math. 42 (1972), 265–270. MR 0303264
  • 6. William B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337–345. MR 0290133
  • 6. Stanislaw Kwapień, On operators factorizable through 𝐿_{𝑝} space, Actes du Colloque d’Analyse Fonctionnelle de Bordeaux (Univ. de Bordeaux, 1971) Soc. Math. France, Paris, 1972, pp. 215–225. Bull. Soc. Math. France, Mém. No. 31–32. MR 0397464
  • 6. D. R. Lewis, Integral operators on ℒ_{𝓅}-spaces, Pacific J. Math. 46 (1973), 451–456. MR 0328651
  • 6. Arne Persson, On some properties of 𝑝-nuclear and 𝑝-integral operators, Studia Math. 33 (1969), 213–222. MR 0247504
  • 6. 12. Same as [1.5].
  • 6. Y. Gordon and D. R. Lewis, Banach ideals on Hilbert spaces, Studia Math. 54 (1975), no. 2, 161–172. MR 0388149
  • 7. A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1–79 (French). MR 0094682
  • 7. 2. Same [2.1].
  • 7. A. Grothendieck, Sur certaines classes de suites dans les espaces de Banach et le théorème de Dvoretzky-Rogers, Bol. Soc. Mat. São Paulo 8 (1953), 81–110 (1956) (French). MR 0094683
  • 7. 4. B. Maurey, Une nouvelle demonstration d'un théorème de Grothendieck, Séminaire Maurey-Schwartz 1972/73. Exposé 22.
  • 7. Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR 0133008
  • 7. 6. Same as [2.4].
  • 8. S. Kwapień, On a theorem of L. Schwartz and its applications to absolutely summing operators, Studia Math. 38 (1970), 193–201. (errata insert). MR 0278090
  • 8. Pierre Saphar, Applications 𝑝 décomposantes et 𝑝 absolument sommantes, Israel J. Math. 11 (1972), 164–179 (French). MR 0296720
  • 8. Pierre Saphar, Une caractérisation des sous-espaces de 𝐿^{𝑝} et ses applications, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), A35–A38 (French). MR 0355569
  • 8. 4. Same as [6.6].
  • 8. D. J. H. Garling, Lattice bounding, radonifying and summing mappings, Math. Proc. Cambridge Philos. Soc. 77 (1975), 327–333. MR 0361882
  • 8. N. J. Nielsen, On Banach ideals determined by Banach lattices and their applications, Dissertationes Math. (Rozprawy Mat.) 109 (1973), 62. MR 0487385
  • 9. Ed Dubinsky, A. Pełczyński, and H. P. Rosenthal, On Banach spaces 𝑋 for which Π₂(\cal𝐿_{∞},𝑋)=𝐵(\cal𝐿_{∞},𝑋), Studia Math. 44 (1972), 617–648. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI. MR 0365097
  • 9. 2. W. Orlicz, Über unbedingte Konvergenz in Funktionenräumen. I, Studia Math. 4 (1933), 33-37; II, Studia Math. 4 (1933), 41-47.
  • 10. Haskell P. Rosenthal, On subspaces of 𝐿^{𝑝}, Ann. of Math. (2) 97 (1973), 344–373. MR 0312222
  • 10. Jean Bretagnolle and Didier Dacunha-Castelle, Application de l’étude de certaines formes linéaires aléatoires au plongement d’espaces de Banach dans des espaces 𝐿^{𝑝}, Ann. Sci. École Norm. Sup. (4) 2 (1969), 437–480 (French). MR 0265930
  • 10. B. Maurey, Un lemme de H. P. Rosenthal, Séminaire Maurey-Schwartz Année 1972–1973: Espaces 𝐿^{𝑝} et applications radonifiantes, Exp. No. 21, Centre de Math., École Polytech., Paris, 1973, pp. 11. MR 0399809
  • 10. 4. Same as [1.1].
  • 10. 5. Same as [2.1].
  • 11. Y. Gordon and D. R. Lewis, Absolutely summing operators and local unconditional structures, Acta Math. 133 (1974), 27–48. MR 0410341
  • 11. 2. A. Pełczyński, Some new isomorphic properties of the Banach spaces of holomorphic functions A and H (to appear).
  • 11. D. R. Lewis, An isomorphic characterization of the Schmidt class, Compositio Math. 30 (1975), no. 3, 293–297. MR 0374968
  • 11. William B. Johnson, On finite dimensional subspaces of Banach spaces with local unconditional structure, Studia Math. 51 (1974), 225–240. MR 0358306
  • 11. T. Figiel, W. B. Johnson, and L. Tzafriri, On Banach lattices and spaces having local unconditional structure, with applications to Lorentz function spaces, J. Approximation Theory 13 (1975), 395–412. Collection of articles dedicated to G. G. Lorentz on the occasion of his sixty-fifth birthday, IV. MR 0367624
  • 11. Robert C. James, Some self-dual properties of normed linear spaces, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967) Princeton Univ. Press, Princeton, N.J., 1972, pp. 159–175. Ann. of Math. Studies, No. 69. MR 0454600
  • 11. Robert C. James, Super-reflexive Banach spaces, Canad. J. Math. 24 (1972), 896–904. MR 0320713
  • 11. Per Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 281–288 (1973). MR 0336297
  • 11. L. Tzafriri, On Banach spaces with unconditional bases, Israel J. Math. 17 (1974), 84–93. MR 0348456
  • 11. 10. M. M. Day, Normed linear spaces, 3rd ed., Springer-Verlag, Berlin and New York, 1972.
  • 11. W. B. Johnson and L. Tzafriri, On the local structure of subspaces of Banach lattices, Israel J. Math. 20 (1975), no. 3-4, 292–299. MR 0420210
  • 12. Yehoram Gordon, On 𝑝-absolutely summing constants of Banach spaces, Israel J. Math. 7 (1969), 151–163. MR 0250028
  • 12. Y. Gordon, Asymmetry and projection constants of Banach spaces, Israel J. Math. 14 (1973), 50–62. MR 0318842
  • 12. D. J. H. Garling and Y. Gordon, Relations between some constants associated with finite dimensional Banach spaces, Israel J. Math. 9 (1971), 346–361. MR 0412775
  • 12. 4. Same as [11.1].
  • 12. Y. Gordon and D. R. Lewis, Absolutely summing, 𝐿₁ factorizable operators and their applications, Bull. Amer. Math. Soc. 79 (1973), 1270–1273 (1974). MR 0336416, 10.1090/S0002-9904-1973-13410-X
  • 12. 6. A. Pietsch. Absolutely p-summing operators in L. I, II, Séminaire Goulaouic-Schwartz 1970/71.
  • 12. V. I. Gurariĭ, M. Ĭ. 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 (English, with Russian summary). MR 0190701
  • 12. 8. Same as [1.5].
  • 12. S. Kwapień and A. Pełczyński, The main triangle projection in matrix spaces and its applications., Studia Math. 34 (1970), 43–68. MR 0270118
  • 12. M. Ĭ. Kadec′ and M. G. Snobar, Certain functionals on the Minkowski compactum, Mat. Zametki 10 (1971), 453–457 (Russian). MR 0291770
  • 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. D. J. H. Garling, Absolutely 𝑝-summing operators in Hilbert space, Studia Math. 38 (1970), 319–331. (errata insert). MR 0281050
  • 13. Jean-Pierre Kahane, Some random series of functions, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968. MR 0254888
  • 13. B. Maurey, Probabilités cylindriques, type et ordre. Applications radonifiantes, Séminaire Maurey-Schwartz Année 1972–1973: Espaces 𝐿^{𝑝} et applications radonifiantes, Exp. No. 1, Centre de Math., École Polytech., Paris, 1973, pp. 12 (French). MR 0397821
  • 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. B. Maurey, Type et cotype dans les espaces munis de structures locales inconditionnelles, Séminaire Maurey-Schwartz 1973–1974: Espaces 𝐿^{𝑝}, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 24 et 25, Centre de Math., École Polytech., Paris, 1974, pp. 25 (French). MR 0399796
  • 13. 6. G. Pisier, Sur les espaces qui ne contiennent pas de l, Séminaire Maurey-Schwartz, Annexe 1972/73.
  • 13. Gilles Pisier, Type des espaces normés, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A1673–A1676 (French). MR 0342989
  • 13. G. Pisier, Sur les espaces qui ne contiennent pas de 𝑙^{∞}_{𝑛} uniformément, Séminaire Maurey-Schwartz Année 1972–1973: Espaces 𝐿^{𝑝} et applications radonifiantes, Annexe, Centre de Math., École Polytech., Paris, 1973, pp. 9 (French). MR 0415276
  • 13. G. Pisier, Une propriété du type 𝑝-stable, Séminaire Maurey-Schwartz 1973–1974: Espaces 𝐿^{𝑝}, applications radonifiantes et gèmétrie des espaces de Banach, Exp. No. 8, Centre de Math., École Polytech., Paris, 1974, pp. pp10 pp. (errata, p. E.1) (French). MR 0399811
  • 13. 10. A. Pełczyński, On unconditional bases and Rademacher averages, Lecture, Kent State University, 1973.
  • 13. J. L. Krivine, Théorèmes de factorisation dans les espaces réticulés, Séminaire Maurey-Schwartz 1973–1974: Espaces 𝐿^{𝑝}, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. 22 et 23, Centre de Math., École Polytech., Paris, 1974, pp. 22 (French). MR 0440334
  • 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. Göte Nordlander, On sign-independent and almost sign-independent convergence in normed linear spaces, Ark. Mat. 4 (1962), 287–296 (1962). MR 0140916
  • 13. 14. B. Maurey, These, École Polytechnique, Paris, 1973.
  • 13. Robert C. James, Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), 542–550. MR 0173932
  • 14. J. R. Holub, Schauder bases and norm ideals of compact operators, Proc. Amer. Math. Soc. 38 (1973), 343–348. MR 0317012, 10.1090/S0002-9939-1973-0317012-2
  • 14. J. R. Holub, Schauder bases and norm ideals of compact operators. II, Indiana Univ. Math. J. 24 (1974/75), 555–564. MR 0350441
  • 14. J. R. Holub, A chacterization of the norm ideals of compact operators on Hilbert space, J. Math. Anal. Appl. 50 (1975), 596–606. MR 0372638
  • 14. B. S. Mitjagin, Normed ideals of intermediate type, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 819–832 (Russian). MR 0173935
  • 15. Charles A. McCarthy, 𝑐_{𝑝}, Israel J. Math. 5 (1967), 249–271. MR 0225140
  • 15. 2. Same as [13.12].
  • 15. J. R. Holub, On subspaces of separable norm ideals, Bull. Amer. Math. Soc. 79 (1973), 446–448. MR 0313880, 10.1090/S0002-9904-1973-13208-2
  • 15. J. R. Holub, On the metric geometry of ideals of operators on Hilbert space, Math. Ann. 201 (1973), 157–163. MR 0326413
  • 15. 5. J. Arazy and J. Lindenstrauss, Some linear topological properties of the spaces C(to appear).
  • 15. Moshe Feder and Pierre Saphar, Spaces of compact operators and their dual spaces, Israel J. Math. 21 (1975), no. 1, 38–49. MR 0377591
  • 15. Haskell P. Rosenthal, A characterization of Banach spaces containing 𝑙¹, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413. MR 0358307
  • 15. 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 0341032, 10.1090/S0002-9939-1973-0341032-5
  • 15. William B. Johnson, A complementary universal conjugate Banach space and its relation to the approximation problem, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 301–310 (1973). MR 0326356
  • 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. Albrecht Pietsch, 𝑝-majorisierbare vektorwertige Maße, Wiss. Z. Friedrich-Schiller-Univ. Jena/Thüringen 18 (1969), no. Heft 2, 243–247 (German). MR 0264028

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