Completely continuous operators and the strict topology

Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X,E) be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study weakly completely continuous and completely continuous operators T:Cb(X,E)→F. We establish the relationship between these classes of operators and the corresponding Borel operator measures given by the Riesz representation theorem. Some applications concerning the coincidence among these classes of operators are derived. It is shown that if X is a k-space and E is a Schur space, then the space (Cb(X,E),β) has the strict Dunford-Pettis property. Moreover, it is proved that if X is a paracompact k-space and E contains no isomorphic copy of ℓ1, then the space (Cb(X,E),β) has the Dieudonné property.