Operator ideals and three-space properties of asymptotic ideal seminorms

Causey, Ryan; Draga, Szymon; Kochanek, Tomasz

doi:10.1090/tran/7759

Operator ideals and three-space properties of asymptotic ideal seminorms
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by Ryan M. Causey, Szymon Draga and Tomasz Kochanek PDF

Trans. Amer. Math. Soc. 371 (2019), 8173-8215 Request permission

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 $\max \{p,q\}$.

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