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.