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 Tˆ : 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 Tˆ∗ has the same property.