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

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.

**[AP]**R. M. Aron and J. B. Prolla,*Polynomial approximation of differentiable functions on Banach spaces*, J. Reine Angew. Math.**313**(1980), 195–216. MR**552473**, 10.1515/crll.1980.313.195**[BG]**Robert G. Bartle and Lawrence M. Graves,*Mappings between function spaces*, Trans. Amer. Math. Soc.**72**(1952), 400–413. MR**0047910**, 10.1090/S0002-9947-1952-0047910-X**[D]**Jean Dazord,*Factoring operators through 𝑐₀*, Math. Ann.**220**(1976), no. 2, 105–122. MR**0420318****[Di]**Joseph Diestel,*Sequences and series in Banach spaces*, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR**737004****[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 problem*, Studia Math.**45**(1973), 191–210. (errata insert). MR**0336294****[G]**Alexandre Grothendieck,*Produits tensoriels topologiques et espaces nucléaires*, Mem. Amer. Math. Soc.**No. 16**(1955), 140 (French). MR**0075539****[GR]**William H. Graves and Wolfgang M. Ruess,*Representing compact sets of compact operators and of compact range vector measures*, Arch. Math. (Basel)**49**(1987), no. 4, 316–325. MR**913163**, 10.1007/BF01210716**[J]**William B. Johnson,*Factoring compact operators*, Israel J. Math.**9**(1971), 337–345. MR**0290133****[LT]**Joram Lindenstrauss and Lior Tzafriri,*Classical Banach spaces*, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR**0415253****[R1]**Daniel J. Randtke,*A structure theorem for Schwartz spaces*, Math. Ann.**201**(1973), 171–176. MR**0326347****[R2]**Daniel J. Randtke,*A compact operator characterization of 𝑙₁*, Math. Ann.**208**(1974), 1–8. MR**0338743****[Ru1]**Wolfgang Ruess,*Compactness and collective compactness in spaces of compact operators*, J. Math. Anal. Appl.**84**(1981), no. 2, 400–417. MR**639673**, 10.1016/0022-247X(81)90177-3**[Ru2]**Wolfgang Ruess,*[Weakly] compact operators and DF spaces*, Pacific J. Math.**98**(1982), no. 2, 419–441. MR**650020****[T]**E. Toma,*Aplicacoes holomorfas e polinomios -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:
http://dx.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