Uniformly Factoring Weakly Compact operators and Parametrized Dualization

This paper deals with the problem of when, given a collection $\mathcal C$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space $Z$ with a Schauder basis so that every element in $\mathcal C$ factors through $Z$ (or through a subspace of $Z$). A sample result is the existence of a reflexive space $Z$ with a Schauder basis so that for each separable Banach space $X$, each weakly compact operator from $X$ to $L_1$ factors through $Z$. We also prove the following descriptive set theoretical result: Let $\mathcal L$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if $\mathcal B$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then the assignment $A\in \mathcal B\to A^*$ can be realized by a Borel map $\mathcal B\to \mathcal L$.