Observations on some classes of operators on $C(K,X)$

Abstract

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 $T:C(K,X)\to Y$ is a strongly bounded operator with representing measure $m:\Sigma \to L(X,Y)$. We show that if $\hat{T}: B(K, X) \to Y$ 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 $T:C(K,X)\to Y$ is strongly bounded limited completely continuous (resp. limited $p$-convergent), then $m(A):X\to Y$ 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.