Absolutely convex sets of large Szlenk index

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 ξ the class of operators whose Szlenk index is not an ordinal less than or equal to ξ admits a universal element if and only if ξ < ω 1 ; W.B. Johnson's theorem that the formal identity map from ℓ 1 to ℓ ∞ is a universal non-compact operator is then obtained as a corollary. Stronger results are obtained for operators having separable codomain.