Uniformly Factoring Weakly Compact operators and Parametrized Dualization

TL;DR

This paper investigates conditions under which collections of weakly compact operators between separable Banach spaces can be factored through a single reflexive Banach space with a Schauder basis, and explores related descriptive set theoretical properties.

Contribution

It establishes the existence of a universal reflexive Banach space with a Schauder basis for factoring certain classes of weakly compact operators and analyzes the Borel structure of the dualization map.

Findings

Existence of a reflexive space Z with a Schauder basis for factoring weakly compact operators.

Borel measurability of the dualization map for Borel subsets of weakly compact operators.

Factorization results for operators into L_1 spaces.

Abstract

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 $\mathcal B\to \mathcal L$.