Dominated and absolutely summing operators on the space \(\,C_{rc}(X,E)\) of vector-valued continuous functions

Abstract

Let X be a completely regular Hausdorff space and E and F be Banach spaces. Let \(C_{rc}(X,E)\) denote the Banach space of all continuous functions \(f:X\rightarrow E\) such that f(X) is a relatively compact set in E, and \(\beta _\sigma \) be the strict topology on \(C_{rc}(X,E)\). We characterize dominated and absolutely summing operators \(T:C_{rc}(X,E)\rightarrow F\) in terms of their representing operator-valued Baire measures. It is shown that every absolutely summing \((\beta _\sigma ,\Vert \cdot \Vert _F)\)-continuous operator \(T:C_{rc}(X,E)\rightarrow F\) is dominated. Moreover, we obtain that every dominated operator \(T:C_{rc}(X,E)\rightarrow F\) is absolutely summing if and only if every bounded linear operator \(U:E\rightarrow F\) is absolutely summing.