On spaces of weak$^{*}$ to weak continuous compact operators

摘要

This paper is concerned with the space $mathcal K_{w^{*}}(X^{*}, Y)$ of weak$^{*}$ to weak continuous compact operators from the dual space $X^{*}$ of a Banach space $X$ to a Banach space $Y$. We show that if $X^{*}$ or $Y^{*}$ has the Radon-Nikod'ym property, $mathcal C$ is a convex subset of $mathcal K_{w^{*}}(X^{*}, Y)$ with $0 in mathcal C$ and $T$ is a bounded linear operator from $X^{*}$ into $Y$, then $Tin overline{mathcal C}^{au_{c}}$ if and only if $Tin overline{{Sin mathcal C : |S|leq |T| }}^{au_{c}}$, where $au_{c}$ is the topology of uniform convergence on each compact subset of $X$, moreover, if $T in mathcal K_{w^{*}}(X^{*}, Y)$, here $mathcal C$ need not to contain $0$, then $Tin overline{mathcal C}^{au_{c}}$ if and only if $Tin overline{mathcal C}$ in the topology of the operator norm. Some properties of $mathcal K_{w^{*}}(X^{*}, Y)$ are presented.