Uniformly factoring weakly compact operators and parametrised dualisation

This article deals with the problem of when, given a collection 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 factors through Z (or through a subspace of Z). In particular, we show that there exists a reflexive space Z with a Schauder basis so that for each separable Banach space X, each weakly compact operator from X to factors through Z. We also prove the following descriptive set theoretical result: Let be the standard Borel space of bounded operators between separable Banach spaces. We show that if is a Borel subset of weakly compact operators between Banach spaces with separable duals, then for , the assignment can be realised by a Borel map .