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

Suppose $X$ and $Y$ are Banach spaces, $K$ is a compact Hausdorff space, $\mathrm{\Sigma }$ 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 $m:\mathrm{\Sigma }\to L(X,Y)$. We show that if $\widehat{T}:B(K,X)\to Y$ is its extension, then $T$ is weak Dunford-Pettis (resp. weak${}^{\ast }$ Dunford-Pettis, weak $p$-convergent, weak${}^{\ast }$ $p$-convergent) if and only if $\widehat{T}$ 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 $A\in \mathrm{\Sigma }$. We also prove that the above implications become equivalences when $K$ is a dispersed compact Hausdorff space.