Operator ideals and three-space properties of asymptotic ideal seminorms

TL;DR

This paper develops asymptotic versions of classical Banach space properties like type and cotype, extending key local theory results and analyzing associated operator ideals, with applications to Szlenk power types of twisted sums.

Contribution

It introduces asymptotic analogues of type and cotype, extending classical results and analyzing operator ideals in the asymptotic setting, with applications to Szlenk power types.

Findings

Extended classical local theory results to asymptotic setting

Established properties of operator ideals related to asymptotic subtype/subcotype

Proved that twisted sums of Banach spaces have Szlenk power type equal to the maximum of the summands' types

Abstract

We introduce asymptotic analogues of the Rademacher and martingale type and cotype of Banach spaces and operators acting on them. Some classical local theory results related, for example, to the `automatic-type' phenomenon, the type-cotype duality, or the Maurey-Pisier theorem, are extended to the asymptotic setting. We also investigate operator ideals corresponding to the asymptotic subtype/subcotype. As an application of this theory, we provide a sharp version of a result of Brooker and Lancien by showing that any twisted sum of Banach spaces with Szlenk power types $p$ and $q$ has Szlenk power type $\max\{p,q\}$.