Trotter-Kato Product Formulæ

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.

- Part I Preliminaries.- Semigroups and their generators.- Linear Evolution Equations.- Quasi-sectorial contractions and operator-norm convergence.- Part II: Trotter-Kato product formulæ for self-adjoint semigroups.- Product approximations of self-adjoint semigroups.- Trotter-Kato product formulæ: strong operator topology.- Trotter-Kato product formulæ: operator-norm topology.- Trotter-Kato product formulae: operator-norm topology and error bounds.- Part III: Trotter-Kato product formulæ for non-self-adjoint semigroups.- Operator-norm approximation theory `a la Cherno .- Product formulæ for non-self-adjoint semigroups.- Operator-norm Trotter product formula on Banach spaces.- Part IV: Time-dependent product formulæ. - Time-dependent product formulæ: Banach space.- Time-dependent product formulæ: Hilbert space.- Part V: Unitary and Zeno product formulæ.- Unitary product formulæ. - Zeno product formulæ.

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. His research interests include Functional Analysis and Semigroup Theory. In particular they are concentrated around Gibbs semigroups, Trotter-Kato product formulæ and semigroup approximations developing the Chernoff method; non-autonomous Cauchy problem and the product formula construction of solution operators.

Hagen Neidhardt  worked from 1975 to 1991 as a research assistant at the Karl-Weierstraß-Institut für Mathematik of the Academy of Sciences of the GDR in Berlin, in the group "Operator Theory and Mathematical Physics" led by Professor Hellmut Baumgärtel. In 1992-1993 Neidhardt worked as a research assistant in the Mathematics Department at the Technical University of Berlin and then in the Mathematics Department at the University of Potsdam (1994-1999). From 2000 and until his professional retirement in 2016, he returned as scientific staff member to the Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS) in Berlin. Neidhardt was a world-renowned expert in the areas of functional analysis, operator theory and mathematical physics, where he made a number of highly original contributions. Several mathematical concepts were named after Hagen Neidhardt. They are the Koplienko-Neidhardt trace formula and also the Howland-Evans-Neidhardt formula and approach to evolutionary equations in Hilbert space.

Takashi Ichinose  is an Emeritus Professor of Kanazawa Universoty, Kanazawa, Japan. His research interests include all sorts of problems in functional analysis, operator theory, partial differential equations and functional integration appearing in mathematical physics.