Operators on spaces of bounded vector-valued continuous functions with strict topologies
Summary: Let \(X\) be a completely regular Hausdorff space, and let \((E,\|\cdot\|_E)\) and \((F,\|\cdot\|_F)\) be Banach spaces. Let \(C_b(X,E)\) be the space of all \(E\)-valued bounded, continuous functions defined on \(X\), equipped with the strict topologies \(\beta_z\), where \(z= \sigma,p,\tau,t\). General integral representation theorems of \((\beta_z,\|\cdot\|_F)\)-continuous linear operators \(T: C_b(X,E)\to F\) with respect to the corresponding operator-valued measures are established. Strongly bounded and \((\beta_z,\|\cdot\|_F)\)-contiguous operators \(T: C_b(X,E)\to F\) are studied. We extend to ``the completely regular setting'' some classical results concerning operators on the spaces \(C(X,E)\) and \(C_0(X,E)\), where \(X\) is a compact or a locally compact space.
scientific article; zbMATH DE number 6474205
scientific article