An isomorphic property in spaces of compact operators and some classes of operators on $C(K,X)$ (Q2416452)

Let $K$ be a compact Hausdorff space and let $X, Y$ be a real Banach spaces. Let $B(K,X)$ denote the set of $X$-valued bounded, Borel measurable functions on $K$ with separable range, equipped with the supremum norm. Consider the canonical inclusion $C(K,X) \subset B(K,X) \subset C(K,X)^{\ast\ast}$. A bounded linear operator $T: C(K,X) \rightarrow Y$ is said to be strongly bounded if the representing vector measure $m$ is ${\mathcal{L}}(X,Y)$-valued (as opposed to being ${\mathcal{L}}(X,Y^{\ast\ast})$-valued). In this case, $T$ has an extension $T':B(K,X) \rightarrow Y$. The author studies several sequential continuity properties of $T$ that lift to $T'$ or from $T^\ast$ to $T'^\ast$. We recall that $T$ is said to be $p$-convergent if it maps weakly $p$-summable sequences to norm null sequences. In this article, the author shows that, for $1\leq p <\infty$, $T^\ast$ is $p$-convergent if and only if $T'^\ast$ is $p$-convergent.