Vector measures and Mackey topologies
Let \((\Omega,\Sigma)\) be a measurable space, and consider the Banach lattice \(B(\Sigma)\) of all bounded and \(\Sigma\)-measurable real functions endowed with the Mackey topology \(\tau=\tau(B(\Sigma),ca(\Sigma))\). In this paper, \((\tau, \xi)\)-continuous linear operators from \(B(\Sigma)\) to a quasi complete locally convex space \((E,\xi)\) are studied. In Section 2, the interesting class of the \(\sigma\)-smooth operators, which has been already studied by the author in other papers, is introduced in order to find a useful characterization of the continuity of linear maps quoted above (Proposition 2.1 and also Proposition 3.1). After that, in Theorem 2.3, a characterization of relatively compact sets in the space of \((\tau,\xi)\)-continuous linear maps is given. In Section 3, integral representation of operators by means of vector measures provides a convenient tool for analysing \((\| \cdot\|,\tau)\)-continuous operators. This allows to prove a Vitali-Hahn-Saks theorem for operators from \(B(\Sigma)\) to \(E\) (Theorem 3.2) and a Nikodym type theorem (Theorem 3.3). As an application it is shown that the space \((B(\Sigma),\tau)\) has the strict Dunford-Pettis property. Finally, in Section 4, a Yosida-Hewitt type decomposition theorem for weakly compact operators on \(B(\Sigma)\) is shown (Theorem 4.3).
scientific article; zbMATH DE number 6009291
scientific article