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Uniform factorization for compact
sets of operators


Authors: R. Aron, M. Lindström, W. M. Ruess and R. Ryan
Journal: Proc. Amer. Math. Soc. 127 (1999), 1119-1125
MSC (1991): Primary 46B07; Secondary 46B28, 46G20, 47A68
DOI: https://doi.org/10.1090/S0002-9939-99-04619-5
MathSciNet review: 1473654
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove a factorization result for relatively compact subsets of compact operators using the Bartle and Graves Selection Theorem, a characterization of relatively compact subsets of tensor products due to Grothendieck, and results of Figiel and Johnson on factorization of compact operators. A further proof, essentially based on the Banach-Dieudonné Theorem, is included. Our methods enable us to give an easier proof of a result of W.H. Graves and W.M. Ruess.


References [Enhancements On Off] (What's this?)

  • [AP] R. M. Aron, J. B. Prolla, Polynomial approximation of differentable functions on Banach spaces, J. Reine Angew. Math. 313 (1980), 195-216 MR 81c:41078
  • [BG] R. Bartle, L. Graves, Mappings between function spaces, Trans. Amer. Math. Soc. 72 (1952), 400-413 MR 13:951i
  • [D] J. Dazord, Factoring operators through $c_0$, Math. Ann. 220 (1976), 105-122 MR 54:8332
  • [Di] J. Diestel, Sequences and Series in Banach Spaces, Springer, New York, 1984 MR 85i:46020
  • [Fe] C. Fernandez, A counterexample to the Bartle-Graves selection theorem for multilinear maps, Proc. Amer. Math. Soc. 126 (1998), 2687-2690. CMP 98:13
  • [F] T. Figiel, Factorization of compact operators and applications to the approximation property, Studia Math. 45 (1973), 191-210 MR 49:1070
  • [G] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Amer. Math. Soc. Memoirs 16, Providence, RI, 1955 MR 17:763c
  • [GR] W. Graves, W. Ruess, Representing compact sets of compact operators and of compact range vector measures, Arch. Math. 49 (1987), 316-325 MR 89a:46145
  • [J] W.B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345 MR 44:7318
  • [LT] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces, Lecture Notes in Math., vol. 338, Springer, Berlin, 1973 MR 54:3344
  • [R1] D. Randtke, A structure theorem for Schwartz spaces, Math. Ann. 201 (1973), 171-176 MR 48:4691
  • [R2] D. Randtke, A compact operator characterization of $l_1$, Math. Ann. 208 (1974), 1-8 MR 49:3507
  • [Ru1] W.M. Ruess, Compactness and collective compactness in spaces of compact operators, J. Math. Analysis Appl. 84 (1981), 400-417 MR 83h:47032
  • [Ru2] W.M. Ruess, [Weakly] Compact operators and DF spaces, Pacific J. Math. 98 (1982), 419-441 MR 83h:47034
  • [T] E. Toma, Aplicacoes holomorfas e polinomios $\tau$-continuous, Thesis, Univ. Federal do Rio de Janeiro, 1993

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Additional Information

R. Aron
Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44240
Email: aron@mcs.kent.edu

M. Lindström
Affiliation: Department of Mathematics, Åbo Akademi University, FIN-20500 Åbo, Finland
Email: mlindstr@ra.abo.fi

W. M. Ruess
Affiliation: Fachbereich Mathematik, Universität Essen, D-45117 Essen, Germany
Email: mate00@sp2.power.uni-essen.de

R. Ryan
Affiliation: Department of Mathematics, University College Galway, Galway, Ireland
Email: Ray.Ryan@UCG.IE

DOI: https://doi.org/10.1090/S0002-9939-99-04619-5
Keywords: Banach spaces, compact factorization, tensor products, Michael's selection theorem, Banach-Dieudonn\'e theorem
Received by editor(s): July 25, 1997
Additional Notes: This note was written while the second and the fourth authors were visiting Kent State University to which thanks are acknowledged. The research of Mikael Lindström was supported by a grant from the Foundation of Åbo Akademi University Research Institute.
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1999 American Mathematical Society

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