Gibbs Semigroups

This book focuses on the theory of the Gibbs semigroups, which originated in the 1970s and was motivated by the study of strongly continuous operator semigroups with values in the trace-class ideal. The book offers an up-to-date, exhaustive overview of the advances achieved in this theory after half a century of development. It begins with a tutorial introduction to the necessary background material, before presenting the Gibbs semigroups and then providing detailed and systematic information on the Trotter-Kato product formulae in the trace-norm topology. In addition to reviewing the state-of-art concerning the Trotter-Kato product formulae, the book extends the scope of exposition from the trace-class ideal to other ideals. Here, special attention is paid to results on semigroups in symmetrically normed ideals and in the Dixmier ideal. By examining the progress made in Gibbs semigroup theory and in extensions of the Trotter-Kato product formulae to symmetrically normed and Dixmier ideals, the book shares timely and valuable insights for readers interested in pursuing these subjects further. As such, it will appeal to researchers, undergraduate and graduate students in mathematics and mathematical physics. Front Matter ....Pages i-xv Semigroups and their generators (Valentin A. Zagrebnov)....Pages 1-40 Classes of compact operators (Valentin A. Zagrebnov)....Pages 41-79 Trace inequalities (Valentin A. Zagrebnov)....Pages 81-98 Gibbs semigroups (Valentin A. Zagrebnov)....Pages 99-142 Product formulae for Gibbs semigroups (Valentin A. Zagrebnov)....Pages 143-208 Product formulae in symmetrically-normed ideals (Valentin A. Zagrebnov)....Pages 209-243 Product formulae in the Dixmier ideal (Valentin A. Zagrebnov)....Pages 245-256 Back Matter ....Pages 257-319