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 $ξ\in (0, ω_1)\setminus\{ω^η: η<ω_1\text{\ a limit ordinal}\}$, there exists a Banach space $\mathfrak{G}_ξ$ having a shrinking basis and Szlenk index $ω^ξ$ such that for any separable Banach space $X$ and any operator $A:X\to Y$ having Szlenk index less than $ω^ξ$, $A$ factors through a subspace and through a quotient of $\mathfrak{G}_ξ$, and if $X$ has a shrinking FDD, $A$ factors through $\mathfrak{G}_ξ$.