Topics in Operator Semigroups [Hardcover]

This monograph is concerned with the interplay between the theory of operator semigroups and spectral theory. The basics on operator semigroups are concisely covered in this self-contained text. Part I deals with the Hille--Yosida and Lumer--Phillips characterizations of semigroup generators, the Trotter--Kato approximation theorem, Katos unified treatment of the exponential formula and the Trotter product formula, the Hille--Phillips perturbation theorem, and Stones representation of unitary semigroups. Part II explores generalizations of spectral theorys connection to operator semigroups.

Concerned with the interplay between the theory of operator semigroups and spectral theory, this self-contained text first discusses the basics of operator semigroups. It then explores generalizations of spectral theorys connection to 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 authors 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 theclassicHillePhillipsmonograph[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 IIII], [Kat1], [RS], [Y], and many others).General Theory.- Basic Theory.- The Semi-Simplicity Space for Groups.- Analyticity.- lñ