A note on some classes of operators on C(K,X)

Abstract

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, Σ is the σ-algebra of Borel subsets of K , C (K, X) is the Banach space of all continuous X-valued functions (with the supremum norm), and T : C (K, X) → Y is a strongly bounded operator with representing measure m : Σ → L(X, Y). We show that if T is a strongly bounded operator and : B(K, X) → Y is its extension, then T* is pseudo weakly compact (resp. limited completely continuous, limited p-convergent, 1 ≤ p < ∞) if and only if has the same property.