Factorization of Asplund operators

Abstract

We give necessary and sufficient conditions for an operator A:X→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∈(0,ω1)∖{ωη:η<ω1a limit ordinal}, there exists a Banach space Gξ having a shrinking basis and Szlenk index ωξ such that for any separable Banach space X and any operator A:X→Y having Szlenk index less than ωξ, A factors through a subspace and through a quotient of Gξ, and if X has a shrinking FDD, A factors through Gξ.