Bulletin of the
Korean Mathematical Society BKMS

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 $T\in \overline{\mathcal C}^{\tau_{c}}$ if and only if $T\in \overline{\{S\in \mathcal C : \|S\|\leq \|T\| \}}^{\tau_{c}}$, where $\tau_{c}$ 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 $T\in \overline{\mathcal C}^{\tau_{c}}$ if and only if $T\in \overline{\mathcal C}$ in the topology of the operator norm. Some properties of $\mathcal K_{w^{*}}(X^{*}, Y)$ are presented.

Keywords: weak$^{*}$ to weak continuous compact operator, dual of operator space, the topology of compact convergence, approximation properties

MSC numbers: Primary 47A05; Secondary 46B28