Uniformly factoring weakly compact operators

TL;DR

This paper proves that under certain conditions, sets of weakly compact operators between Banach spaces can be uniformly factored through a reflexive space with a basis, extending to variable domain and range spaces.

Contribution

It establishes a uniform factorization result for weakly compact operators with analytic properties, generalizing previous results to variable spaces and broader classes.

Findings

Existence of a reflexive space through which all operators in the set factor

Extension of factorization to operators with separable adjoint range

Uniform factorization when domain and range spaces vary

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.