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.
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