Semigroups of Linear Operators and Applications

This is a heuristic treatment of semigroups of linear operators and linear Cauchy problems with a great many applications to classical approximation theory, probability theory, and mathematical physics as well as to the Feynman path integral and the mean ergodic theorem. The first part deals with the Hille-Yosida generation theorem along with adjoint semigroups, analytic semigroups, perturbation theory and applications. The second part of the volume covers linear Cauchy problems — both homogeneous and inhomogeneous — nonlinear equations, parabolic problems, a spectral theorem (every self-adjoint operator is unitarily equivalent to a generalized multiplication operator of the form \(\mathcal{M}_{id}\)), second order Cauchy problems, symmetric hyperbolic systems along with an example from acoustics, discussion higher order Cauchy problems (without proofs), regular and singular perturbation problems, discussion on the elliptic boundary value problems (without proofs), time dependent abstract Cauchy problems with sufficient conditions for well-posedness and abstract scattering theory.

Throughout the book, the presentation is clear, concise and motivational. Historical notes and remarks for each of the two parts are hugely extensive and provide the relevance of each section and its sources for the enthusiastic reader. An over 50 pages of references is phenomenal, although the author also cites in the reference their own book on A (More-or-Less) Complete Bibliography of Semigroups of Operators Through 1984, Tulane University, New Orleans, Louisiana, 1985. There are exercises at the end of each section. Many exercises are nontrivial, and so they even are research results that deserved to be included.

The last part of the volume presents a set of wonderful notes with references of the five lectures delivered by the author at the Workshop on Operator Semigroups and Evolution Equations in Blaubeuren, Germany on October 30–November 3, 1989. The titles of the five lectures in the order of their presentation are On Hardy-Landau-Littlewood Inequalities, The Feynman-Kac Formula with an Application to the Heat Equation with a Singular Potential, Scattering Theory and Equipartition of Energy, The Navier-Stokes Equations, and Singular Perturbations.

In summary, the book is suitable for a thorough understanding of the linear semigroup theory and its applications to several areas in sciences, mathematics and engineering.

Dhruba Adhikari is Associate Professor of Mathematics at Kennesaw State University, Marietta, Georgia.