Observations 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 \(T \colon C(K,X)\to Y\) is a strongly bounded operator with representing measure \(m \colon \Sigma \to L(X,Y)\). We show that if \(\hat{T} \colon B(K, X) \to Y\) is its extension, then T is weak Dunford--Pettis (resp.weak^* Dunford--Pettis, weak p-convergent, weak^* p-convergent) if and only if \(\hat{T}\) has the same property.

We prove that if \(T \colon C(K,X)\to Y\) is strongly bounded limited completely continuous (resp. limited p-convergent), then \(m(A) \colon X\to Y\) is limited completely continuous (resp. limited p-convergent) for each \(A\in \Sigma\). We also prove that the above implications become equivalences when K is a dispersed compact Hausdorff space.