Almost periodic regularized groups, semigroups, and cosine functions
type: Item
instance of: scholarly article
title: Almost periodic regularized groups, semigroups, and cosine functions
zbMATH Open document ID: 0888.47020
author: Quan Zheng
author: L. Liu
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-05-21T00:00:00Z
review text: Let \(T(t)\), \(t\in {\mathbf J}\), be a strongly continuous family of linear bounded operators over a Banach space \(X\) such that \[ T(0) = C, \] and \[ T(t + s)C = T(t)T(s), \quad\forall t,s\in {\mathbf J}, \] where \(C\) -- called regulator operator -- is a linear bounded operator in \(X\). Then \(T(t)\) is called a C-regularized semigroup in case \({\mathbf{J}} = {\mathbf{R}}_{+}\), and C-regularized group in the case \({\mathbf{J}} = {\mathbf{R}}\). This paper studies the almost periodicty of regularized groups, semigroups, cosine functions, and the corresponding sine functions. The authors obtain a characterization of the generator of an almost periodic regularized group and give conditions under which weak almost periodicity (resp. almost periodicity) of a regularized group implies its almost periodicty (resp. uniform almost periodicity). Almost periodic regularized semigroups as well as regularized cosine functions and sine functions are then characterized. Finally, periodicity of regularized groups, cosine functions and sine functions are characterized. The case in which the regulator operator \(C\) is some power of the generator is also studied as well the applications to distribution semigroup.
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47D09
zbMATH DE Number: 902429
zbMATH Keywords: strongly continuous family of linear bounded operators
zbMATH Keywords: regular operator
zbMATH Keywords: \(C\)-regularized semigroup
zbMATH Keywords: \(C\)-regularized group
zbMATH Keywords: almost periodicity
zbMATH Keywords: cosine functions
zbMATH Keywords: sine functions
zbMATH Keywords: distribution semigroups
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0009
OpenAlex ID: W1985733304
DOI: 10.1006/JMAA.1996.0009
description: scientific article; zbMATH DE number 902429