Valentin A. Zagrebnov - Böcker

Dedicated to Tosio Kato’s 100th birthday, this book contains research and survey papers on a broad spectrum of methods, theories, and problems in mathematics and mathematical physics.

Dedicated to Tosio Kato’s 100th birthday, this book contains research and survey papers on a broad spectrum of methods, theories, and problems in mathematics and mathematical physics.

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.

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 captures a fascinating snapshot of the current state of results about the operator-norm convergent Trotter-Kato Product Formulæ on Hilbert and Banach spaces. It also includes results on the operator-norm convergent product formulæ for solution operators of the non-autonomous Cauchy problems as well as similar results on the unitary and Zeno product formulæ.After the Sophus Lie product formula for matrices was established in 1875, it was generalised to Hilbert and Banach spaces for convergence in the strong operator topology by H. Trotter (1959) and then in an extended form by T. Kato (1978).  In 1993 Dzh. L. Rogava discovered that convergence of the Trotter product formula takes place in the operator-norm topology. The latter is the main subject of this book, which is dedicated essentially to the operator-norm convergent Trotter-Kato Product Formulæ on Hilbert and Banach spaces, but also to related results on the time-dependent, unitary and Zeno product formulæ.The book yields a detailed up-to-date introduction into the subject that will appeal to any reader with a basic knowledge of functional analysis and operator theory. It also provides references to the rich literature and historical remarks.