Topics in Operator Semigroups

The theory of operator semigroups was essentially discovered in the early 1930s. Since then, the theory has developed into a rich and exciting area of functional analysis and has been applied to various mathematical topics such as Markov processes, the abstract Cauchy problem, evolution equations, and mathematical physics. This self-contained monograph focuses primarily on the theoretical connection between the theory of operator semigroups and spectral theory. Divided into three parts with a total of twelve distinct chapters, this book gives an in-depth account of the subject with numerous examples, detailed proofs, and a brief look at a few applications. Topics include:* The Hille'Yosida and Lumer'Phillips characterizations of semigroup generators* The Trotter'Kato approximation theorem* Kato's unified treatment of the exponential formula and the Trotter product formula* The Hille'Phillips perturbation theorem, and Stone's representation of unitary semigroups* Generalizations of spectral theory's connection to operator semigroups* A natural generalization of Stone's spectral integral representation to a Banach space setting With a collection of miscellaneous exercises at the end of the book and an introductory chapter examining the basic theory involved, this monograph is suitable for second-year graduate students interested in operator semigroups.