Factorization of Asplund operators

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 ${\mathfrak{G}}_{\xi }$.