Weak precompactness and property $(V^*)$ in spaces of compact operators
Volume 138 / 2015
Colloquium Mathematicum 138 (2015), 255-269
MSC: Primary 46B20, 46B28; Secondary 28B05.
DOI: 10.4064/cm138-2-10
Abstract
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 $(V^*)$.
