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 $T:C(K,X)\to Y$ is a strongly bounded operator with representing measure $m:\Sigma \to L(X,Y)$.

We  show that if $T$ is a strongly bounded operator and $\hat{T}: B(K, X) \to Y$ is its extension, then $T$ is limited if and only if its extension $\hat{T}$ is limited, and that $T^*$ is completely continuous (resp. unconditionally converging) if and only if  $\hat{T}^*$ is completely continuous (resp. unconditionally converging).

We prove that if  $K$ is a dispersed compact Hausdorff space and $T$ is a strongly bounded operator, then $T$ is limited (resp. weakly precompact, has a completely continuous adjoint, has an unconditionally converging adjoint) whenever $m(A):X\to Y$ is  limited (resp. weakly precompact, has a completely continuous adjoint, has an unconditionally converging adjoint) for each $A \in \Sigma$.