
Journal of Convex Analysis 15 (2008), No. 1, 165178 Copyright Heldermann Verlag 2008 Linear Operators on VectorValued Function Spaces with Mackey Topologies Marian Nowak Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Szafrana 4A, 65516 Zielona Góra, Poland M.Nowak@wmie.uz.zgora.pl [Abstractpdf] \newcommand{\si}{\sigma} \newcommand{\Si}{\Sigma} \newcommand{\om}{\omega} \newcommand{\Om}{\Omega} \newcommand{\ps}{\rightarrow} \newcommand{\wf}{\widetilde{f}} \newcommand{\wg}{\widetilde{g}} \newcommand{\cl}{{\cal L}} 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 WOTsequentially complete, and every SOTcompact subset of $\cl_\tau(E(X),Y)$ is $\,(\tau(E(X),E'(X^*)),\\cdot\_Y)$equicontinuous. Moreover, a VitaliHahnSaks type theorem for $\cl_\tau(E(X),Y)$ is obtained. Keywords: Vectorvalued function spaces, Mackey topologies, strongly Mackey topologies, weak compactness, RadonNikodym property, Asplund spaces, sequential completeness, convex compactness property, weak operator topology, strong operator topology, linear operator MSC: 46E40, 46E30, 46A20, 46A70 [ Fulltextpdf (169 KB)] for subscribers only. 