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.

Valentin A. Zagrebnov is an Emeritus Professor of Aix-Marseilles University. He is a member of the Institut de Mathématiques de Marseille - UMR 7373 and its research group on Analysis, Geometry and Topology. 

Semigroups and their generators.- Classes of compact operators.- Trace inequalities.- Gibbs semigroups.- Product formulae for Gibbs semigroups.- Product formulae in symmetrically-normed ideals.- Product formulae in Dixmier ideal.- Appendix A. Spectra of closed operators.- Appendix B. Inequalities in ideals.- Appendix B. More inequalities.-  Appendix C. Kato functions.- Appendix D. Lie-Trotter-Kato product formulae: comments on the bibliography.- Bibliography.- Index