Affine semigroups on Banach spaces
type: Item
instance of: scholarly article
title: Affine semigroups on Banach spaces
zbMATH Open document ID: 0675.47028
author: Shinnosuke Oharu
author: Andrew Vogt
author: Jerome A. Goldstein
published in: Hiroshima Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: On a Banach space \(X\) the one-parameter family \(\{S(t),\,t\geq 0\}\) of affine transformations with the properties \((S_ 1)\) \(S(0)=I\), \(S(t+S)=S(t)\cdot S(s)\), \(s,t\geq 0\); and \((S_ 2)\) \(\forall x\in X\), \(t\mapsto S(t)x\) is a continuous function from \([0,\infty)\) into \(X\) is considered. Such a family is called an affine semigroup. An affine semigroup \(S(t)\) decomposes into a linear part \(T(t)\) and a translation part \(z(t)\) by formulas \[ z(t)=S(t)0,\quad T(t)x=S(t)x-S(t)0. \] Here \(T(t)\) is a \(C_ 0\)-semigroup of linear operators and the function \(z(t): [0,\infty)\to X\) satisfies the equality \[ z(t+s)=T(t)z(s)+z(t),\quad t,s\geq 0. \] On a Banach space \(\hat X=X\times R\) the augmented semigroup \[ \hat T(t)[x,\tau]=[T(x)x+\tau z(t),\tau],\quad [x,\tau]\in X \] associated with \(S(t)\) is constructed. The relations between the infinitesimal generators \(A,U,\hat U\) of the semigroups \(S(t)\), \(T(t)\), \(\hat T(t)\), respectively, and the linear and translation parts of the semigroup \(S(t)\) are established. The necessary and sufficient conditions for the function \(z(\cdot):[0,\infty)\to X\) to be the translation part of an affine semigroup and for the operator \(A: X\to X\) to be the infinitesimal generator of such a semigroup are studied. Analogous results for the affine cosine functions are obtained. The criteria of compactness and analyticity for affine semigroups which are analogous to the linear case are formulated. The affine group is also considered.
reviewed by: Y. Eidelman
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47D99
zbMATH DE Number: 4105664
zbMATH Keywords: affine semigroup
zbMATH Keywords: \(C_ 0\)-semigroup of linear operators
zbMATH Keywords: augmented semigroup
zbMATH Keywords: infinitesimal generators
zbMATH Keywords: affine cosine functions
zbMATH Keywords: compactness
zbMATH Keywords: analyticity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206129734
OpenAlex ID: W1524020625
Wikidata QID: Q128870019
DOI: 10.32917/HMJ/1206129734
description: scientific article; zbMATH DE number 4105664