Topological Wiener-Wintner theorems for amenable operator semigroups
type: Item
instance of: scholarly article
zbMATH Open document ID: 1320.37004
author: Marco Schreiber
published in: Ergodic Theory and Dynamical Systems
publication date: 2014-11-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1210.0108
Mathematics Subject Classification ID: 37A25
Mathematics Subject Classification ID: 37C30
zbMATH DE Number: 6366535
MaRDI profile type: Publication
OpenAlex ID: W2963344922
arXiv ID: 1210.0108
cites work: Topological Wiener–Wintner ergodic theorems via non-abelian Lie group extensions
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cites work: On diffraction by aperiodic structures
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cites work: Topological Wiener–Wintner ergodic theorems and a randomL2ergodic theorem
cites work: On abstract mean ergodic theorems
cites work: On Uniform Convergence in the Wiener-Wintner Theorem
DOI: 10.1017/ETDS.2013.14
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title: Topological Wiener-Wintner theorems for amenable operator semigroups
review text: The paper unifies and extends some known Wiener-Wintner type theorems [\textit{E. A. Robinson jun.}, J. Lond. Math. Soc., II. Ser. 49, No. 3, 493--501 (1994; Zbl 0827.28010); \textit{P. Walters}, Ergodic Theory Dyn. Syst. 16, No. 1, 179--206 (1996; Zbl 0861.28013); \textit{S. I. Santos} and \textit{C. Walkden}, ibid. 27, No. 5, 1633--1650 (2007; Zbl 1124.37002); \textit{D. Lenz}, Contemp. Math. 485, 91--112 (2009; Zbl 1200.78008); \textit{D. Lenz}, Commun. Math. Phys. 287, No. 1, 225--258 (2009; Zbl 1178.37011)] to the case of amenable semigroups of Markov operators on the space \(C(K)\) where \(K\) is a compact topological space.NEWLINENEWLINEOne of the major results of the paper (Theorem 2.7) is a characterization of mean ergodicity for an amenable representation \(\{S_g : g\in G\}\) of a semitopological semigroup \(G\) as Markov operators, i.e., as positive operators satisfying the condition \(S_g\mathbf{1}=\mathbf{1}\), for all \(g\in G\). In proving this result the author builds upon his previous work [Semigroup Forum 86, No. 2, 321--336 (2013; Zbl 1273.43009)] where the mean ergodicity of a more general bounded (not necessarily Markov) amenable representations has been characterized. The latter result already has a very interesting corollary which the author points out in the current paper (Corollary 2.3.): a bounded amenable semigroup \(\mathcal{S} = \{S_g : g\in G\}\) of operators is uniquely ergodic if and only if it is mean ergodic and Fix \(\mathcal{S} = \mathbb{C} \cdot \mathbf{1}\).NEWLINENEWLINEIf \(\chi : G\to \mathbb{T}\) is a character of \(G\), one can consider a semigroup \(\chi \mathcal{S}\) formed out of \(\mathcal{S}\). The mean ergodicity question of this semigroup has been studied in the literature (e.g., in [loc. cit.]). In the second part of the paper, the author studies amenable representations \(\mathcal{S}\) of semitopological semigroups as Koopman operators (i.e., operators of type \(T:f\to f\circ \phi \) on \(C(K)\) where \(K\) is comp…
reviewed by: Azer Akhmedov
description: scientific article; zbMATH DE number 6366535