Set Theory & Analysis

Three-space properties of asymptotic ideal seminorms

Tomasz Kochanek

Institute of Mathematics

Polish Academy of Sciences

Polish Academy of Sciences

Tuesday, 19. September 2017 - 10:00 to 12:00

We introduce two "flavors" of seminorms of Banach spaces (or, more generally, operators acting on them), each having both type and cotype version, and each of these having both weak and weak* version, resulting in eight families of seminorms. The intuition behind these new quantities is to define asymptotic versions of the classical Rademacher and Pisier (martingale) types and cotypes. We will describe our seminorms in terms of asymptotic structures, discuss some ideal and duality properties. As an application of this theory, we will prove that having Szlenk power type at most p is a three-space property. This strengthens and, in fact, gives a sharp version of an earlier result by Brooker and Lancien (J. Math. Anal. Appl. 2013) who proved that any twisted sum of Banach spaces with Szlenk power types p and q has power type at most pq. The talk is based on a joint work with R.M. Causey and S. Draga.

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