Uniformly factoring weakly compact operators

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.

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