Ergodicity and super weak compactness

TL;DR

This paper establishes the equivalence between super-weak compactness and super-ergodicity in Banach spaces, linking geometric properties with fixed point properties and the Banach-Saks property.

Contribution

It provides a new characterization of super weakly compact sets via ergodic properties and explores implications for fixed point properties and Banach space geometry.

Findings

Super weakly compact sets are characterized by super-ergodicity.

Super weakly compact sets have the fixed point property for continuous affine mappings.

M-fixed point property implies the Banach-Saks property.

Abstract

We prove that a closed convex subset of a Banach space is (super-)weakly compact if and only if it is (super)-ergodic. As a consequence we deduce that super weakly compact sets are characterized by the fixed point property for continuous affine mappings. We also prove that the M-(fixed point property for affine isometries) implies the Banach-Saks property.