Representing compact sets of compact operators and of compact range vector measures
type: Item
instance of: scholarly article
title: Representing compact sets of compact operators and of compact range vector measures
zbMATH Open document ID: 0645.46007
DOI: 10.1007/BF01210716
author: William H. Graves
author: Wolfgang M. Ruess
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: Let X\({\tilde \otimes}_{\epsilon}Y\) be the completed injective tensor product of Banach spaces X, Y. If X is an \({\mathcal L}_ 1\)- or \({\mathcal L}_{\infty}\)-space, then X has the approximation property, and so X\({\tilde \otimes}_{\epsilon}Y\) can be identified with \(K_{W\quad *}(X\) *,Y), the space of compact weak *-weakly continuous linear operators from X * into Y with the usual operator norm. The authors prove the following theorem: Assume that X is an \({\mathcal L}_ 1\)-space (respectively, an \({\mathcal L}_{\infty}\)-space). Then a subset H of X\({\tilde \otimes}_{\epsilon}Y\) is relatively compact if and only if there exist a sequence \((x_ i)\in c_ 0(X)\) (respectively, \((x_ i)\in \ell\) 1(X)) and a relatively compact subset \(\{\) y h:h\(\in H\}\) of \(\ell\) 1(Y) (respectively, of \(c_ 0(Y))\) such that \(\lim_{n\to \infty}\sup_{h\in H}\| h-\sum^{n}_{1}x_ i\otimes y\) \(h_ i\| =0.\) The proof is based on some results of \textit{P. Saphar} on tensor norms [Stud. Math. 38, 71-100 (1970; Zbl 0213.142)] and an earlier result of the second named author on compactness in \(K_{w\quad *}(X\) *,Y) for general Banach spaces X, Y [J. Math. Anal. Appl. 84, 400-417 (1981; Zbl 0477.47027)]. Some consequences are derived concerning series representations and factorizations of compact sets of compact operators, whose domain is an \({\mathcal L}_ 1\)- or \({\mathcal L}_{\infty}\)-space, and of compact sets of compact range vector measures.
Mathematics Subject Classification ID: 46A32
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 46A50
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 47L05
zbMATH DE Number: 4052291
zbMATH Keywords: completed injective tensor product of Banach spaces
zbMATH Keywords: approximation property
zbMATH Keywords: series representations
zbMATH Keywords: factorizations of compact sets of compact operators
zbMATH Keywords: compact sets of compact range vector measures
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01210716
OpenAlex ID: W2085741599
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description: scientific article; zbMATH DE number 4052291