Uniformly factoring weakly compact operators

Abstract

Let $X$ and $Y$ be separable Banach spaces. Suppose $Y$ either has a shrinking basis or $Y$ is isomorphic to $C(2^{\mathbb{N}})$ and $A$ is a subset of weakly compact operators from $X$ to $Y$ which is analytic in the strong operator topology. We prove that there is a reflexive space with a basis $Z$ such that every $T\in A$ factors through $Z$. Likewise, we prove that if $A\subset L(X,C(2^{\mathbb{N}}))$ is a set of operators whose adjoints have separable range and is analytic in the strong operator topology then there is a Banach space $Z$ with separable dual such that every $T\in A$ factors through $Z$. Finally we prove a uniformly version of this result in which we allow the domain and range spaces to vary.

19 pages, comments welcome