Uniform ergodic theorems for semigroup representations

TL;DR

This paper explores the relationships between spectral properties, ergodic behavior, and quasi-compactness of semigroup representations on Banach spaces, extending classical theorems to broader contexts.

Contribution

It generalizes the Niiro-Sawashima theorem to semigroup representations, linking spectral, ergodic, and quasi-compactness properties in Banach lattices.

Findings

Positive, bounded semigroup representations with finite-dimensional fixed space are quasi-compact if uniformly mean ergodic.

Established connections between spectral properties and ergodic behavior for semigroup representations.

Extended classical theorems to more general semigroup contexts.

Abstract

We consider a bounded representation $T$ of a commutative semigroup $S$ on a Banach space and analyse the relation between three concepts: (i) properties of the unitary spectrum of $T$, which is defined in terms of semigroup characters on $S$; (ii) uniform mean ergodic properties of $T$; and (iii) quasi-compactness of $T$. We use our results to generalize the celebrated Niiro-Sawashima theorem to semigroup representations and, as a consequence, obtain the following: if a positive and bounded semigroup representation on a Banach lattice is uniformly mean ergodic and has finite-dimensional fixed space, then it is quasi-compact.