Applications of the quantification of super weak compactness

TL;DR

This paper introduces a new measure of super weak noncompactness for operators in Banach spaces, providing a characterization of subspaces of Hilbert generated spaces and exploring their structure.

Contribution

It defines the measure $ ext{Gamma}$ for super weak noncompactness and uses it to characterize certain Banach space substructures, extending previous research.

Findings

Characterization of Banach spaces as subspaces of Hilbert generated spaces.

Introduction of the measure $ ext{Gamma}$ for super weak noncompactness.

Connections established between super weak compactness and Banach-Saks properties.

Abstract

We introduce a measure of super weak noncompactness $\Gamma$ defined for bounded linear operators and subsets in Banach spaces that allows to state and prove a characterization of the Banach spaces which are subspaces of a Hilbert generated space. The use of super weak compactness and $\Gamma$ casts light on the structure of these Banach spaces and complements the work of Argyros, Fabian, Farmaki, Godefroy, H\'ajek, Montesinos,\linebreak Troyanski and Zizler on this subject. A particular kind of relatively super weakly compact sets, namely uniformly weakly null sets, plays an important role and exhibits connections with Banach-Saks type properties.