Operators on the Banach space of $p$-continuous vector-valued functions

Abstract

Let $X$, $Y$, and $Z$ be Banach spaces, and let $\alpha$ be a tensor norm. Let a bounded linear operator $S\in\mathcal{L}(Z,\mathcal{L}(X,Y))$ be given. We obtain (necessary and/or sufficient) conditions for the existence of an operator $U\in\mathcal{L}(Z\hat{\otimes}_{\alpha}X,Y)$ such that $(Sz)x = U(z\otimes x)$, for all $z\in Z$ and $x\in X$, i.e., $S= U^{#}$, the associated operator to $U$. Let $\Omega$ be a compact Hausdorff space and denote by $\mathcal{C}(\Omega)$ the space of continuous functions from $\Omega$ into $\mathbb{K}$. We apply these results to $S\in\mathcal{L}(\mathcal{C}(\Omega),\mathcal{L}(X, Y))$ for characterizing the existence of an operator $U\in\mathcal{L}(\mathcal{C}_{p}(\Omega,X),Y)$ such that $U^{#}=S$, where $\mathcal{C}_{p}(\Omega,X)$ is the space of $p$-continuous $X$-valued functions, $1\leq p \leq \infty$.