This is an announcement for the paper “The Szlenk index of $L_p(X)$ and $A_p$” by Ryan M. Causeyhttps://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1.

Abstract: Given a Banach space $X$, a $w^*$-compact subset of $X^*$, and $1<p<\infty$, 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 $A: X\rightarrow Y$, we provide an estimate of the Szlenk index of the "pointwise $A$" operator $A_p: L_p(X)\rightarrow L_p(Y)$ in terms of the Szlenk index of $A$.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1701.06226