Absolutely convex sets of large Szlenk index - Brooker , Philipp A. H. (North-Western European Journal of Mathematics, 2019)
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 \(Y\) 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 \(X/Z\) has a basis and the Szlenk indices of \(X/Z\) are comparable to the Szlenk indices of \(X\). 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 \(\ell_1\) to \(\ell_∞\) is a universal non-compact operator is then obtained as a corollary. Stronger results are obtained for operators having separable codomain.