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

Full-text PDF

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.

**[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*, 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*, 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 -continuous*, Thesis, Univ. Federal do Rio de Janeiro, 1993

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
46B07,
46B28,
46G20,
47A68

Retrieve articles in all journals with MSC (1991): 46B07, 46B28, 46G20, 47A68

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