Topics in the Theory of Gibbs Semigroups

Book excerpt: One-parameter semigroup theory started to be an important branch of mathematics in the thirties when it was realized that the theory has direct applications to partial differential equations, random processes, infinite dimensional control theory, mathematical physics, etc. It is now generally accepted as an integral part of contemporary functional analysis. Compact strongly continuous semigroups have been an important research subject since a long time, as in almost all applications of partial differential equations with bounded domains the semigroups turn out to be compact. From this point of view, the present volume of the Leuven Notes in Mathematical and Theoretical Physics emphasizes a special subclass of these semigroups. In fact, the focus here is mainly on semigroups acting on a Hilbert space H with values in the trace class ideal C1(H) of bounded operators on H. Historically, this class of semigroups is closely related to quantum statistical mechanics.