Absolutely convex sets of large Szlenk index

Let $X$ be a Banach space and $K$ an absolutely convex, weak${}^{\ast }$-compact subset of $X^{\ast }$. 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 $Y$ such that $Y$ has a basis and the Szlenk indices of $Y$ are comparable to the Szlenk indices of $X$. If $X$ is separable, then $X$ also admits subspace $Z$ such that the quotient $X\mathrm{/}Z$ has a basis and the Szlenk indices of $X\mathrm{/}Z$ are comparable to the Szlenk indices of $X$. 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 ${\mathrm{\ell }}_1$ to ${\mathrm{\ell }}_{\mathrm{\infty }}$ is a universal non-compact operator is then obtained as a corollary. Stronger results are obtained for operators having separable codomain.