Some properties and applications of equicompact sets of operators
Volume 181 / 2007
Abstract
Let $X$ and $Y$ be Banach spaces. A subset ${\rm M}$ of ${\cal K}(X,Y)$ (the vector space of all compact operators from $X$ into $Y$ endowed with the operator norm) is said to be equicompact if every bounded sequence $(x_n)$ in $X$ has a subsequence $(x_{k(n)})_n$ such that $(Tx_{k(n)})_n$ is uniformly convergent for $T\in{\rm M}$. We study the relationship between this concept and the notion of uniformly completely continuous set and give some applications. Among other results, we obtain a generalization of the classical Ascoli theorem and a compactness criterion in ${\cal M}_{\rm c}({\cal F},X)$, the Banach space of all (finitely additive) vector measures (with compact range) from a field ${\cal F}$ of sets into $X$ endowed with the semivariation norm.
