slideshow 3

Set Theory & Analysis

Three-space properties of asymptotic ideal seminorms

Tomasz Kochanek
Institute of Mathematics
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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