Monotone convergence of operator semigroups and the dynamics of infinite particle systems
type: Item
instance of: scholarly article
title: Monotone convergence of operator semigroups and the dynamics of infinite particle systems
zbMATH Open document ID: 0574.46049
DOI: 10.1016/0021-9045(85)90101-7
published in: Journal of Approximation Theory
publication date: 1985-01-01T00:00:00Z
review text: Convergence of strongly continuous contraction semigroups on a Banach space X is considered. The starting point is a family \(\{A_{\gamma}\}_{\gamma \in \Gamma}\) of infinitesimal (semigroup) generators, indexed by a directed set \(\Gamma\). If there is a dense subspace of vectors x such that \(A_{\gamma}(x)\) is defined for \(\gamma\) sufficiently large, and \(\lim_{\gamma}A_{\gamma}(x)\) exists, then additional conditions are considered which ensure the existence of an infinitesimal generator which may be regarded as the limit of the net \(\{A_{\gamma}\}\). In case X is known to have a complete order structure, a monotone convergence theorem of a general nature is proved and it is shown how it applies to a particular existence problem for the dynamical semigroup in lattice gasses of classical statistical mechanics. A second type of results is also proved. These results are based on resolvent convergence and are applied to the corresponding existence problem in quantum statistical mechanics. Here the \(C^*\)-algebraic formalism is introduced, and the dynamics is given by a strongly continuous one-parameter group of *-automorphisms. Finally, a distinguished, and canonical, extension operator is obtained, and its infinitesimal generator properties are analyzed.
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 82B05
Mathematics Subject Classification ID: 82B10
Mathematics Subject Classification ID: 46N99
zbMATH DE Number: 3917021
zbMATH Keywords: Convergence of strongly continuous contraction semigroups on a Banach space
zbMATH Keywords: existence of an infinitesimal generator
zbMATH Keywords: complete order structure
zbMATH Keywords: monotone convergence theorem
zbMATH Keywords: existence problem for the dynamical semigroup in lattice gasses of classical statistical mechanics
zbMATH Keywords: resolvent convergence
zbMATH Keywords: existence problem in quantum statistical mechanics
zbMATH Keywords: \(C^*\)- algebraic formalism
zbMATH Keywords: strongly continuous one-parameter group of *- automorphisms
zbMATH Keywords: extension operator
author: Palle Jorgensen
MaRDI profile type: Publication
full work available at URL: 0021-9045(85)90101-7
OpenAlex ID: W2024373956
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description: scientific article; zbMATH DE number 3917021