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 op-erator T : C_b(X;E) → F is weak Dunford-Pettis. It is shown that every absolutely summing operator T : Cb(X;E) → F is dominated.

Mathematics Subject Classication (2010): 46G10, 46E40, 46A70, 47B10.

Key words: Spaces of vector-valued continuous functions, strict topologies, operator mea-
sures, strongly bounded operators, dominated operators, absolutely summing operators.