A semigroup analogue of the Fonf--Lin--Wojtaszczyk ergodic characterization of reflexive Banach spaces with a basis

TL;DR

This paper characterizes finite-dimensional and reflexive Banach spaces with a basis through properties of mean ergodic semigroups, extending ergodic theory analogies to a semigroup setting.

Contribution

It provides new semigroup-based characterizations of finite-dimensionality and reflexivity in Banach spaces with a basis, generalizing previous results.

Findings

Finite-dimensionality characterized by uniform mean ergodicity of all bounded, uniformly continuous, mean ergodic semigroups.

Reflexivity characterized by all bounded strongly continuous or uniformly continuous semigroups being mean ergodic.

Extends ergodic characterization results to a semigroup framework for Banach spaces.

Abstract

In analogy to a recent result by V. Fonf, M. Lin, and P. Wojtaszczyk, we prove the following characterizations of a Banach space $X$ with a basis. (i) $X$ is finite-dimensional if and only if every bounded, uniformly continuous, mean ergodic semigroup on $X$ is uniformly mean ergodic. (ii) $X$ is reflexive if and only if every bounded strongly continuous semigroup is mean ergodic if and only if every bounded uniformly continuous semigroup on $X$ is mean ergodic.