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Monotone convergence of operator semigroups and the dynamics of infinite particle systems
Journal article   Open access   Peer reviewed

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

Palle E.T Jørgensen
Journal of Approximation Theory, Vol.43(3), pp.205-230
1985
DOI: 10.1016/0021-9045(85)90101-7
url
https://doi.org/10.1016/0021-9045(85)90101-7View
Published (Version of record) Open Access

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.

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