A spectral criterion of asymptotic almost periodicity for uniformly continuous representations of Abelian semigroups
type: Item
instance of: scholarly article
title: A spectral criterion of asymptotic almost periodicity for uniformly continuous representations of Abelian semigroups
zbMATH Open document ID: 0699.22006
DOI: 10.1007/BF02209170
author: Vu Kuok Fong
author: Yu. I. Lubich
published in: Journal of Soviet Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let S be a topological abelian semigroup with unit and T a representation of S as linear operations in a Banach space B. T is supposed continuous with respect to the strong topology of linear operations in B and is bounded (i.e. \(\sup_{s}\| T(s)\| \leq \infty)\). We make S a directed space by assuming \(s\geq t\) iff \(s=tu\) for some \(u\in S\). This induces a direction in \(\{\) T(s)\(\}\). We call T asymptotically almost periodic (a.a.p.) iff every directed subset of \(\{\) T(s)\(\}\) contains a strongly convergent directed sequence. This is a weakened form of almost periodicity of T - a condition which means that for every \(x\in B\) the orbit \(O(x)=\{T(s)x\}\) is relatively compact. According to a result of \textit{M. Yu. Lyubich} and \textit{Yu. I. Lyubich} [Teor. Funkts., Funkts. Anal. Prilozh. 45, 69-84 (1986; Zbl 0629.47033)] the a.a.p. property of T is equivalent to the possibility of representing B as the direct sum \(B_ o+B_ 1\) of the ``interior spectrum'' \(B_ 0=\{x:\) \(\lim_{s} T(s)x=0\}\) and the ``boundary spectrum'' \(B_ 1\) which is a topological direct sum of all \(V_{\chi}=\{x:\) \(x\in B\), \(T(s)x=\chi (s)x\) (s\(\in S)\}\) where \(\chi\) denotes a unitary character of S such that \(V_{\chi}\neq \{0\}\). (The set of all such characters is called the unitary asymptotic spectrum or unitary a-spectrum of T). The author infers therefrom that, if T is a.a.p., and \(\chi\) a unitary character of S then the spaces \(V_{\chi}\) and \(W_{\chi}=\{f:\) \(f\in B^*\), \(T^*(s)f=\chi (s)f\) (s\(\in S)\}\) are total sets of linear functionals for each other (in other words: T satisfies the spectral duality condition). The main result of the paper consists in some kind of an inverse theorem: suppose that 1) T is uniformly continuous (i.e. continuous with respect to the operator norm), 2) its unitary a-spectrum is countable and 3) T satisfies spectral duality. Then t is a.a.p.. Instead of assuming spectral duality for T one can assume in this theorem that T…
reviewed by: Stanisław Hartman
Mathematics Subject Classification ID: 22A25
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 20M30
Mathematics Subject Classification ID: 20M20
zbMATH DE Number: 4146282
zbMATH Keywords: topological abelian semigroup
zbMATH Keywords: linear operations
zbMATH Keywords: Banach space
zbMATH Keywords: strong topology
zbMATH Keywords: asymptotically almost periodic
zbMATH Keywords: strongly convergent directed sequence
zbMATH Keywords: almost periodicity
zbMATH Keywords: orbit
zbMATH Keywords: unitary character
zbMATH Keywords: unitary asymptotic spectrum
zbMATH Keywords: total sets of linear functionals
zbMATH Keywords: spectral duality
zbMATH Keywords: representation
MaRDI profile type: Publication
cites work: Q3802105
cites work: Q3766358
cites work: Applications of almost periodic compactifications
cites work: Ergodentheorie und fastperiodische Funktionen auf Halbgruppen
cites work: Q3671556
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full work available at URL: https://doi.org/10.1007/bf02209170
OpenAlex ID: W2332546733
description: scientific article; zbMATH DE number 4146282