This is an announcement for the paper “Concerning $q$-summable Szlenk index” by Ryan M. Causeyhttps://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1.

Abstract: For each ordinal $\xi$ and each $1\leq q<\infty$, we define the notion of $\xi$-$q$-summable Szlenk index. When $\xi=0$ and $q=1$, this recovers the usual notion of summable Szlenk index. We define for an arbitrary weak$^*$-compact set a transfinite, asymptotic analogue $a_{\xi, p}$ of the martingale type norm of an operator. We prove that this quantity is determined by norming sets and determines $\xi$-Szlenk power type and $\xi$-$q$-summability of Szlenk index. This fact allows us to prove that the behavior of operators under the $a_{\xi, p}$ seminorms passes in the strongest way to injective tensor products of Banach spaces. Furthermore, we combine this fact with a result of Schlumprecht to prove that a separable Banach space with good behavior with respect to the $a_{\xi, p}$ seminorm can be embedded into a Banach space with a shrinking basis and the same behavior under $a_{\xi, p}$, and in particular it can be embedded into a Banach space with a shrinking basis and the same $\xi$-Szlenk power type. Finally, we completely elucidate the behavior of the $a_{\xi, p}$ seminorms under $\ell_r$ direct sums. This allows us to give an alternative proof of a result of Brooker regarding Szlenk indices of $\ell_p$ and $c_0$ direct sums of operators.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1801.00033