Uniform ergodic theorems for semigroup representations

Jochen Glück, Patrick Hermle, Henrik Kreidler
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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.
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半群表示的均匀遍历定理
我们考虑了巴纳赫空间上交换半群$S$的有界表示$T$,并分析了三个概念之间的关系:(i) $T$的单元谱性质,它是根据半群特征$S$定义的;(ii) $T$的均匀平均遍历性质;以及 (iii) $T$的准紧凑性。我们利用我们的结果将著名的二郎-川岛定理推广到半群表征上,并由此得到:如果巴拿赫网格上的有界正半群表征是均值遍历的,并且具有有限维的固定空间,那么它就是准紧凑的。
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