Monotone convergence of operator semigroups and the dynamics of infinite particle systems (Q1063836)

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.