Operators on the space of vector-valued totally measurable functions
Several classes of linear operators from the space \(B(\Sigma, X)\) of \(X\)-valued totally measurable functions into a Banach space \(Y\) are characterized in this article. A bounded linear operator \(T:B(\Sigma, X) \to Y\) is called \(\sigma\)-smooth if \((T(f_n))\) converges to \(0\) in \(Y\) whenever \((f_n)\) is a sequence in \(B(\Sigma, X)\) with \((\| f_n(\cdot )\|_X)\) order convergent to \(0\) in \(B(\Sigma)\). The author proves that a bounded linear operator \(T:B(\Sigma, X) \to Y\) is \(\sigma\)-smooth exactly when its representing measure \(m\) is variationally semi-regular (that is, if \((\| m\| (A_n))\) converges to \(0\) whenever the sequence \((A_n)\) of \(\Sigma\)-measurable sets decreases to \(\varnothing\) where \(\| m\|\) is the semivariation of \(m\)). As a consequence, the space \(L_{\sigma s}(B(\Sigma,X), Y)\) of \(\sigma\)-smooth operators from \(B(\Sigma,X)\) to \(Y\) endowed with the strong operator topology is sequentially complete. With \(B(\Sigma,X)^{\widetilde{ }}_C\) denoting \(L_{\sigma s}(B(\Sigma,X), \mathbb{R})\) and \(\tau(B(\Sigma,X), B(\Sigma,X)^{\widetilde{ }}_C)\) denoting the Mackey topology on \(B(\Sigma,X)\), the author also proves that, if \(Y\) contains no isomorphic copy of \(c_0\), then \(T:B(\Sigma,X) \to Y\) is \(\sigma\)-smooth if and only if \(T\) is \((\tau(B(\Sigma,X), B(\Sigma,X)^{\widetilde{ }}_C), \|\cdot\|_Y)\)-continuous. More generally, the \((\tau(B(\Sigma,X), B(\Sigma,X)^{\widetilde{ }}_C), \|\cdot\|_Y)\)-continuous operators \(T:B(\Sigma,X) \to Y\) are characterized in terms of countable additivity of the measure \(m\) representing \(T\).
scientific article; zbMATH DE number 5362161
scientific article