Compactness of Bochner representable operators on Orlicz spaces
Let \(L^\Phi\) be an Orlicz space over the \(\sigma\)-finite measure space \((\Omega,\Sigma,\mu)\) and \(X\) a Banach space. An operator \(T\) from \(L^\Phi\) to \(X\) is said to be Bochner representable if there is a function \(g\) belonging to the Orlicz--Bochner space \(L^{\Phi^*}(X)\) (where \(\Phi^*\) is the complementary function of \(\Phi\)) such that \(T(u)=\int_\Omega u(\omega) g(\omega) \, d \mu\), \(u \in L^\Phi\). In this paper, some results on compactness for this kind of operators are given. Starting from the original results of \textit{J.\,J.\thinspace Uhl jun.}\ [Stud.\ Math.\ 40, 17--22 (1971; Zbl 0228.47020)] for continuous operators with respect to the norm topologies, the results of the present paper extend them to the following version. Consider the mixed topology \(\gamma_{L^\Phi}\) on \(L^\Phi\) (the finest locally convex topology on \(L^\Phi\) which agrees with the topology of convergence in measure on \(\| \cdot\|_\Phi\)-bounded subsets). It is proved that every Bochner representable operator \(T:L^\Phi \to X\) is \((\gamma_{L^\Phi},\| \cdot \|_X)\)-compact (Theorem 2.3). As an application, an improvement of a particular case of the results of J.\,J.\thinspace Uhl quoted above is obtained; namely, every Bochner representable operator \(T:L^\infty \to X\) is \((\tau(L^\infty,L^1),\|\cdot\|)\)-compact (Theorem 3.3).
scientific article; zbMATH DE number 5534422
scientific article