An isomorphic property in spaces of compact operators and some classes of operators on C(K,X)

Abstract

Let $${K_{w^{*}}(X^{*},Y)}$$ denote the set of all w*−w continuous compact operators from X* to Y. We investigate whether the space $${K_{w^{*}}(X^{*},Y)}$$ has property RDP * ( $${1\le p < \infty}$$ ) when X and $${Y}$$ have the same property. 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 p-convergent if and only if $${\hat{T}^{*}}$$ is p-convergent, for $${1\le p < \infty}$$ .