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. This book is based on lecture notes from a second-year graduate course, and is a greatly expanded version of our previous monograph [K8]. We expose some aspects of the theory of semigroups of linear operators, mostly (but not only) from the point of view of its meeting with that part of spectral theory which is concerned with the integral representation of families of operators. This approach and selection of topics di?erentiate this book from others in the general area, and re?ect the author's own research directions. There is no attempt therefore to cover thoroughly the theory of semigroups of operators. This theory and its applications are extensively exposed in many books, from theclassicHille–Phillipsmonograph[HP]tothemostrecenttextbookofEngel and Nagel [EN2] (see [A], [BB], [Cl], [D3], [EN1], [EN2], [Fat], [G], [HP], [P], [Vr], and others), as well as in chapters in more general texts on Functional Analysis and the theory of linear operators (cf. [D5], [DS I–III], [Kat1], [RS], [Y], and many others). Front Matter....Pages 1-11 Front Matter....Pages 1-1 Basic Theory....Pages 3-48 The Semi-Simplicity Space for Groups....Pages 49-61 Analyticity....Pages 63-69 The Semigroup as a Function of its Generator....Pages 71-86 Large Parameter....Pages 87-112 Boundary Values....Pages 113-130 Pre-Semigroups....Pages 131-138 Front Matter....Pages 140-140 The Semi-Simplicity Space....Pages 141-160 The Laplace–Stieltjes Space....Pages 161-175 Families of Unbounded Symmetric Operators....Pages 177-190 Front Matter....Pages 194-195 Analytic Families of Evolution Systems....Pages 195-201 Similarity....Pages 203-218 Back Matter....Pages 1-47
