Dominated operators, absolutely summing operators and the strict topology

Abstract

Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let C_b(X, E) be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study dominated and absolutely summing operators T : C_b(X, E) → F . We derive that if X is a locally compact Hausdorff space and E has the Dunford-Pettis property, then every dominated operator T : C_b(X, E) → F is weak Dunford-Pettis. It is shown that every absolutely summing operator T : C_b(X, E) → F is dominated.