This is an announcement for the paper “Absolutely convex sets of large Szlenk index” by Philip A.H. Brookerhttps://arxiv.org/find/math/1/au:+Brooker_P/0/1/0/all/0/1.

Abstract: Let $X$ be a Banach space and $K$ an absolutely convex, weak$^*$-compact subset of $X$. We study consequences of $K$ having a large or undefined Szlenk index, and subsequently derive a number of related results concerning basic sequences and universal operators. We show that if $X$ has a countable Szlenk index then $X$ admits a subspace with a basis and with Szlenk indices comparable to the Szlenk indices of $X$. If X is separable, then $X$ also admits a quotient with these same properties. We also show that for a given ordinal $\xi$ the class of operators whose Szlenk index is not an ordinal less than or equal to $xi$ admits a universal element if and only if $xi<\omega_1$; W.B. Johnson's theorem that the formal identity map from $\ell_1$ to $\ell_{\infty}$ is a universal non-compact operator is then obtained as a corollary. Stronger results are obtained for operators having separable codomain.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.08127