{"title":"Uniform ergodic theorems for semigroup representations","authors":"Jochen Glück, Patrick Hermle, Henrik Kreidler","doi":"arxiv-2408.08961","DOIUrl":null,"url":null,"abstract":"We consider a bounded representation $T$ of a commutative semigroup $S$ on a\nBanach space and analyse the relation between three concepts: (i) properties of\nthe unitary spectrum of $T$, which is defined in terms of semigroup characters\non $S$; (ii) uniform mean ergodic properties of $T$; and (iii)\nquasi-compactness of $T$. We use our results to generalize the celebrated Niiro-Sawashima theorem to\nsemigroup representations and, as a consequence, obtain the following: if a\npositive and bounded semigroup representation on a Banach lattice is uniformly\nmean ergodic and has finite-dimensional fixed space, then it is quasi-compact.","PeriodicalId":501373,"journal":{"name":"arXiv - MATH - Spectral Theory","volume":"20 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Spectral Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.08961","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.