Quantification of Banach-Saks properties of higher orders

TL;DR

This paper explores quantitative measures of Banach-Saks properties of higher orders, establishing new relations, introducing a measure of weak compactness, and illustrating limitations with examples.

Contribution

It provides a quantitative framework for Banach-Saks properties of higher orders, including characterizations, relations, and a new measure of weak compactness.

Findings

Quantitative characterization of weak ξ-Banach-Saks sets using ℓ₁^{ξ+1}-spreading models

Relations between ξ-Banach-Saks sets, weak ξ-Banach-Saks sets, and compactness properties

Introduction of a new measure of weak compactness

Abstract

We investigate possible quantifications of Banach-Saks sets and weak Banach-Saks sets of higher orders and their relations to other quantities. We prove a quantitative version of the characterization of weak $\xi$-Banach-Saks sets using $\ell_1^{\xi+1}$-spreading models and a quantitative version of the relation of $\xi$-Banach-Saks sets, weak $\xi$-Banach-Saks sets, norm compactness and weak compactness. We further introduce a new measure of weak compactness. Finally, we provide some examples showing the limitations of these quantifications.