The order-type Banach-Saks properties

TL;DR

This paper investigates order Banach-Saks properties in Banach function spaces, providing conditions and characterizations, especially in rearrangement invariant and Orlicz spaces, with implications for applications like financial mathematics.

Contribution

It introduces new sufficient conditions and characterizations for order Banach-Saks properties, including in Orlicz spaces, and establishes their hereditary equivalence.

Findings

Sufficient conditions for (weak) order Banach-Saks property

Characterization of the property in Orlicz spaces

Equivalence of the property with its hereditary version

Abstract

The study of the Banach-Saks property in Banach spaces has a long and illustrious history. Of late, motivated by applications in financial mathematics, interest has arisen in the Banach-Saks type properties with respect to order convergence. This paper presents a study of order Banach-Saks properties in Banach function spaces, and in particular in rearrangement invariant spaces. Among the results obtained, we provide some sufficient conditions for the (weak) order Banach-Saks property. We also characterize the (weak) order Banach-Saks property in Orlicz spaces. It is also shown that the (weak) order Banach-Saks property is equivalent to its hereditary version.