Uo-convergence and its applications to Ces\`aro means in Banach lattices

TL;DR

This paper investigates uo-convergence in Banach lattices, establishing its stability, applying it to Cesàro means, and developing new approaches to Banach-Saks properties with several improved results.

Contribution

It introduces the stability of uo-convergence under sublattices and applies this to extend results on Cesàro means and Banach-Saks properties in Banach lattices.

Findings

Established stability of uo-convergence under regular sublattices.

Extended and unified results on Cesàro means convergence.

Provided new proofs and generalizations of Banach-Saks related theorems.

Abstract

A net $(x_\alpha)$ in a vector lattice $X$ is said to uo-converge to $x$ if $|x_\alpha-x|\wedge u\xrightarrow{\rm o}0$ for every $u\ge 0$. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uo-convergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several results in [26,27]. In the second part, we use uo-convergence to study convergence of Ces\`aro means in Banach lattices. In particular, we establish an intrinsic version of Koml\'os' Theorem, which extends the main results of [35,16,31] in a uniform way. We also develop a new and unified approach to Banach-Saks properties and Banach-Saks operators based on uo-convergence. This approach yields, in particular, short direct proofs of several results in [21,24,25].