A theory of strongly continuous semigroups in terms of Lie generators
type: Item
instance of: scholarly article
title: A theory of strongly continuous semigroups in terms of Lie generators
zbMATH Open document ID: 0848.22005
author: Yanyan Li
published in: Journal of Functional Analysis
publication date: 1996-10-21T00:00:00Z
review text: For a separable complete metric space \(X\) denote by \({\mathcal F} (X)\) the set of all continuous transformations of \(X\) into itself and by \({\mathcal C} (X)\) the linear space of all bounded continuous real-valued functions from \(X\). A function \(T: [0, \infty)\to {\mathcal F} (X)\) is called jointly continuous semigroup in \(X\) if \(T(0)\) is the identity, \(T(t) T(s)= T(t+ s)\) for all \(t,s \geq 0\), \(T(-)\) is continuous from \([0, \infty)\) to \(X\) for every \(x\in X\), and the mapping \((t, x)\mapsto T(t) x\) from \([0, \infty) \times X\) to \(X\) is jointly continuous. The Lie generator of \(T\) is the linear operator \(A\) in \({\mathcal C} (X)\) consisting of all pairs \((f, g)\) of functions such that \(g(x)= \lim_{t\leftarrow 0} {1\over t} [f(T (t)x)- f(x)]\) for any \(x\in X\). A necessary and sufficient condition for a linear operator in \({\mathcal C} (X)\) to be a Lie generator of a jointly continuous semigroup is given. Examples and applications are discussed.
Mathematics Subject Classification ID: 22A15
zbMATH DE Number: 871739
zbMATH Keywords: strongly continuous semigroup
zbMATH Keywords: Lie generator
zbMATH Keywords: metric space
zbMATH Keywords: continuous transformations
zbMATH Keywords: linear space
zbMATH Keywords: linear operator
zbMATH Keywords: jointly continuous semigroup
reviewed by: Václav Koubek
MaRDI profile type: Publication
OpenAlex ID: W2001309777
DOI: 10.1006/JFAN.1996.0023
description: scientific article; zbMATH DE number 871739