Linear Operators on Vector-Valued Function Spaces with Mackey Topologies

Let $\,E\,$ be an ideal of $\,L^0\,$ over a $\,\sigma$-finite measure space $\,(\Om,\Si,\mu)\,$ and let $E'$ be the K\"othe dual of $\,E$. Let $\,(X,\|\cdot\|_X)\,$ be a real Banach space, and $\,X^*\,$ the Banach dual of $\,X$. Let $\,E(X)\,$ be a subspace of the space $\,L^0(X)\,$ of $\mu$-equivalence classes of all strongly $\Si$-measurable function $\,f:\Om\ps X$, and consisting of all those $\,f\in L^0(X)\,$ for which the scalar function $\wf$, defined by $\,\wf(\om)=\|f(\om)\|_X\,$ for $\,\om\in\Om$, belongs to $E$. Assume that a Banach space $\,X\,$ is an Asplund space. It is shown that a subset $C$ of $\,E'(X^*)\,$ is relatively $\,\si(E'(X^*),E(X))$-compact iff the set $\,\{\wg:g\in E'(X^*)\}\,$ in $E'$ is relatively $\,\si(E',E)$-compact. We consider the topology $\,\overline{\tau(E,E')}\,$ on $E(X)$ associated with the Mackey topology $\,\tau(E,E')\,$ on $E$. It is shown that $\,\overline{\tau(E,E')}\,$ is strongly Mackey topology; hence $\,\overline{\tau(E,E')}\,$ coincides with the Mackey topology $\,\tau(E(X),E'(X^*))$. Moreover, $\,E'(X^*)\,$ is $\,\si(E'(X^*), E(X))$-sequentially complete whenever $E'$ is perfect. We examine the space $\cl_\tau(E(X),Y)$ of all $\,(\tau(E(X),E'(X^*)),\|\cdot\|_Y)$-continuous linear operators from $\,E(X)\,$ to a Banach space $\,(Y,\|\cdot\|_Y)$, equipped with the weak operator topology (briefly WOT) and the strong operator topology (briefly SOT). It is shown that if $E$ is perfect, then $\cl_\tau(E(X),Y)$ is WOT-sequentially complete, and every SOT-compact subset of $\cl_\tau(E(X),Y)$ is $\,(\tau(E(X),E'(X^*)),\|\cdot\|_Y)$-equicontinuous. Moreover, a Vitali-Hahn-Saks type theorem for $\cl_\tau(E(X),Y)$ is obtained.