Topics in Operator Semigroups - Kantorovitz, Shmuel; SpringerLink (Online service) (Boston : Birkhäuser Boston, 2010)
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.
General Theory -- Basic Theory -- The Semi-Simplicity Space for Groups -- Analyticity -- The Semigroup as a Function of its Generator -- Large Parameter -- Boundary Values -- Pre-Semigroups -- Integral Representations -- The Semi-Simplicity Space -- The Laplace-Stieltjes Space -- Families of Unbounded Symmetric Operators -- A Taste of Applications -- Analytic Families of Evolution Systems -- Similarity.
Keywords: MATHEMATICS; ALGEBRA; GROUP THEORY; OPERATOR THEORY; MATHEMATICS; OPERATOR THEORY; GROUP THEORY AND GENERALIZATIONS; ALGEBRA