Monotone convergence of operator semigroups and the dynamics of infinite particle systems

Abstract

Convergence of strongly continuous contraction semigroups on a Banach space X is considered. The starting point is a family { A γ } γϵΓ of infinitesimal (semigroup) generators, indexed by a directed set Γ. If there is a dense subspace of vectors x such that A γ ( x) is defined for γ sufficiently large, and lim γ A γ ( 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 γ }. 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. At both levels (the classical and quantum), the solution to the time-Cauchy problem is obtained as a natural operator extension of the given partially defined, unbounded, infinitesimal operator. The extensions reflect particular boundary conditions for the problems under consideration. Finally, a distinguished, and canonical, extension operator is obtained, and its infinitesimal generator properties are analyzed.