The BD property in spaces of compact operators

Abstract

For Banach spaces $X$ and $Y$, let $K_{w^{\ast }}(X^{\ast },Y)$ denote the space of all $w^{\ast }-w$ continuous compact operators from $X^{\ast }$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $\mathit{BD}$ property if every limited subset of $X$ is relatively weakly compact. We prove that if $X$ has the Gelfand–Phillips property and $Y$ has the $\mathit{BD}$ property, then $K_{w^{\ast }}(X^{\ast },Y)$ has the $\mathit{BD}$ property.