Weakly almost periodic semigroups of operators
type: Item
instance of: scholarly article
title: Weakly almost periodic semigroups of operators
zbMATH Open document ID: 0655.47033
published in: Pacific Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: We address the question as to when a motion or almost-orbit u of a strongly continuous semigroup \((S(t))_{t\geq 0}\) of operators in a Banach space X will be weakly almost periodic in the sense of Eberlein. In particular, we show (a) that this is the case in practice exactly when u uniquely decomposes as the sum \(u=S(\cdot)y+\phi\) of an almost periodic motion S(\(\cdot)y:{\mathbb{R}}\) \(+\to X\) of \((S(t))_{t\geq 0}\) and a function \(\phi\) :\({\mathbb{R}}\) \(+\to X\) that vanishes at infinity in a certain weak sense, and (b) that an almost-orbit u of a uniformly bounded \(C_ 0\)-semigroup of linear operators will be weakly almost periodic provided only that u has weakly relatively compact range. Our results on existence and representation are then applied to a qualitative study of asymptotic behavior of solutions to the abstract Cauchy problem in which the focus is on almost periodicity properties and ergodic theorems.
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 4069644
zbMATH Keywords: motion
zbMATH Keywords: almost-orbit
zbMATH Keywords: strongly continuous semigroup
zbMATH Keywords: weakly almost periodic in the sense of Eberlein
zbMATH Keywords: uniformly bounded \(C_ 0\)-semigroup of linear operators
zbMATH Keywords: results on existence and representation
zbMATH Keywords: qualitative study of asymptotic behavior of solutions to the abstract Cauchy problem
author: Wolfgang M. Ruess
author: William H. Summers
MaRDI profile type: Publication
full work available at URL: pjm.1990.143.175
OpenAlex ID: W1984429854
DOI: 10.2140/PJM.1990.143.175
description: scientific article; zbMATH DE number 4069644