Representing compact sets of compact operators and of compact range vector measures
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.
completed injective tensor product of Banach spaces
factorizations of compact sets of compact operators
compact sets of compact range vector measures
scientific article; zbMATH DE number 4052291
scientific article