Weak precompactness and property $(V^*)$ in spaces of compact operators

Ioana Ghenciu¹

¹ Department of Mathematics University of Wisconsin-River Falls River Falls, WI 54022, U.S.A.

Colloquium Mathematicum, Tome 138 (2015) no. 2, pp. 255-269.

Résumé

We give sufficient conditions for subsets of compact operators to be weakly precompact. Let $L_{w^*}(E^*,F)$ (resp. $K_{w^*}(E^*,F)$) denote the set of all $w^*$-$w$ continuous (resp. $w^*$-$w$ continuous compact) operators from $E^*$ to $F$. We prove that if $H$ is a subset of $K_{w^*}(E^*,F)$ such that $H(x^*)$ is relatively weakly compact for each $x^* \in E^*$ and $H^*(y^*)$ is weakly precompact for each $y^* \in F^*$, then $H$ is weakly precompact. We also prove the following results: If $E$ has property $(wV^*)$ and $F$ has property $(V^*)$, then $K_{w^*}(E^*,F)$ has property $(wV^*)$. Suppose that $L_{w^*}(E^*,F)=K_{w^*}(E^*,F)$. Then $K_{w^*}(E^*,F)$ has property $(V^*)$ if and only if $E$ and $F$ have property $(V^*)$.

Détail

DOI : 10.4064/cm138-2-10

Keywords: sufficient conditions subsets compact operators weakly precompact * * resp * * denote set * w continuous resp * w continuous compact operators * prove subset * * * relatively weakly compact each * * * * weakly precompact each * * weakly precompact prove following results has property * has property * * * has property * suppose * * * * * * has property * only have property *