Operators on the space of bounded strongly measurable functions
Let \(\Sigma\) be a \(\sigma\)-algebra of subsets of a set \(\Omega\), \(X,Y\) be Banach spaces, and \(\mathcal L^\infty(\Sigma,X)\) be the space of all strongly measurable bounded functions \(f: \Omega \to X\) equipped with the supremum norm. A bounded linear operator \(T: \mathcal L^\infty(\Sigma,X) \to Y\) is called \(\sigma\)-smooth if it maps uniformly bounded sequences pointwise convergent to zero to norm convergent ones. The paper consists of four sections. The first one contains introduction and terminology. In the second one, a characterization of \(\sigma\)-smooth operators \(T: \mathcal L^\infty(\Sigma,X) \to Y\) in terms of their representing measure is given and the following Banach-Steinhaus type theorem for \(\sigma\)-smooth operators is proved: if a sequence of \(\sigma\)-smooth operators \(T_n: \mathcal L^\infty(\Sigma,X) \to Y\) converges pointwise on \(L^\infty(\Sigma,X)\), then this sequence is uniformly \(\sigma\)-smooth, and the limiting operator is also \(\sigma\)-smooth. The third section deals with the space of all \(\sigma\)-smooth functionals on \(\mathcal L^\infty(\Sigma,X)\), and in the fourth section two sufficient conditions for an operator \(T: \mathcal L^\infty(\Sigma,X) \to Y\) to be \(\sigma\)-smooth are established.
scientific article; zbMATH DE number 6005697