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

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, sigma is the sigma-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 :sigma -> L(X, Y).We show that if T is a strongly bounded operator and T circumflex expressionccent : B(K, X)-> Y is its extension, then T* is pseudo weakly compact (resp. limited completely continuous, limited p-convergent, 1 <= p < infinity) if and only if T circumflex expressionccent * has the same property.