Spectral properties of weakly asymptotically almost periodic semigroups
type: Item
instance of: scholarly article
title: Spectral properties of weakly asymptotically almost periodic semigroups
zbMATH Open document ID: 0882.22002
author: Edoardo Vesentini
published in: Advances in Mathematics
publication date: 1998-03-02T00:00:00Z
review text: Let \(E\) be a Banach space. By a definition of M. Fréchet, a continuous and bounded function \(f\) mapping \(\mathbb{R}_+\) to \(E\) is said to be asymptotically almost periodic if the set of all translates of \(f\) by elements of \(\mathbb{R}_+\) is relatively compact in \(C_b(\mathbb{R}_+,E)\). This paper investigates strongly continuous groups and semigroups \(T\) of operators on \(E\) for which all the functions \(\lambda\mapsto f(T(\lambda))\), for \(f\in E'\), are almost periodic on \(\mathbb{R}\) or asymptotically almost periodic, respectively. Several theorems show that in this situation there are strong conditions on the spectrum of the generator of the semigroup. There are stronger results than in the general case if \(E\) is weakly sequentially complete. Finally, much stronger results are obtained for the case, where \(E\) is a Hilbert space.
reviewed by: John W.Baker
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 43A60
zbMATH DE Number: 1044179
zbMATH Keywords: almost periodic function
zbMATH Keywords: spectrum
zbMATH Keywords: Banach space
zbMATH Keywords: strongly continuous groups
zbMATH Keywords: operators
zbMATH Keywords: Hilbert space
MaRDI profile type: Publication
full work available at URL: aima.1997.1613
OpenAlex ID: W2037589083
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DOI: 10.1006/AIMA.1997.1613
description: scientific article; zbMATH DE number 1044179