The Szlenk index of 𝐿_{𝑝}(𝑋) and 𝐴_{𝑝}

Abstract

Given a Banach space X, a w *- compact subset of X*, and 1 < p < infinity, we provide an optimal relationship between the Szlenk index of K and the Szlenk index of an associated subset of L-p(X)*. As an application, given a Banach space X, we prove an optimal estimate of the Szlenk index of L-p(X) in terms of the Szlenk index of X. This extends a result of H ' ajek and Schlumprecht to uncountable ordinals. More generally, given an operator A : X -> Y, we provide an estimate of the Szlenk index of the "pointwise A" operator Ap : Lp(X) -> Lp(Y) in terms of the Szlenk index of A.