On some classes of operators on \(C(K,X)\) (Q2787110)

Let \(X\) and \(Y\) be Banach spaces, \(K\) a compact Hausdorff space, \(\Sigma\) the \(\sigma\)-algebra of Borel sets of \(K\), \(C(K, X)\) the Banach space of all continuous \(X\)-valued functions equipped with the supremum norm, \(T: C(K, X) \to Y\) a strongly bounded operator with the representing measure \(m: \Sigma \to L(X, Y)\), and \(\hat{T}: B(K, X) \to Y\) its extension.NEWLINENEWLINEIn the paper under review, it is shown that \(T\) is limited if and only if \(\hat{T}\) is limited, and \(T^{*}\) is completely continuous (resp., unconditionally converging) if and only if \(\hat{T^{*}}\) is completely continuous (resp., unconditionally converging). Moreover, if \(K\) is dispersed, 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 \(A \in \Sigma\).