Factorization of Asplund operators

TL;DR

This paper characterizes when operators on Banach spaces with shrinking FDDs can be factored through spaces with matching Szlenk indices, introducing specific Banach spaces for each ordinal to facilitate such factorizations.

Contribution

It provides necessary and sufficient conditions for factorization through spaces with matching Szlenk indices and constructs new Banach spaces for each relevant ordinal to support these factorizations.

Findings

Operators with shrinking FDDs can be factored through spaces with the same Szlenk index.

Existence of Banach spaces with prescribed Szlenk indices for each ordinal.

Factorization results apply to all operators with Szlenk index below certain ordinals.

Abstract

We give necessary and sufficient conditions for an operator $A:X\to Y$ on a Banach space having a shrinking FDD to factor through a Banach space $Z$ such that the Szlenk index of $Z$ is equal to the Szlenk index of $A$. We also prove that for every ordinal $\xi\in (0, \omega_1)\setminus\{\omega^\eta: \eta<\omega_1\text{\ a limit ordinal}\}$, there exists a Banach space $\mathfrak{G}_\xi$ having a shrinking basis and Szlenk index $\omega^\xi$ such that for any separable Banach space $X$ and any operator $A:X\to Y$ having Szlenk index less than $\omega^\xi$, $A$ factors through a subspace and through a quotient of $\mathfrak{G}_\xi$, and if $X$ has a shrinking FDD, $A$ factors through $\mathfrak{G}_\xi$.