Perturbations of Positive Semigroups with Applications: (Springer Monographs in Mathematics)

This book is devoted predominantly to one particular way of looking at the evolution of the system in which we describe time changes as transitions from one state to another. This leads in a natural way to the semigroup theory that has been developed in the last fifty years. There are many applications coming from kinetic theory, fragmentation theory, mathematical biology and many other fields, where the assumptions of standard perturbation theory are not satisfied. In this book the authors concentrate on perturbation results that take advantage of the fact that in many applications the operators involved are positive (as only nonnegative data and solutions make sense) and dissipative. The main part of the book is devoted to linking the properties of the generator with the properties of the semigroup, that are important in applications. In the final part of the book the authors discuss applications of the developed theory to a variety of problems ranging from the classical birth-and-death type problems of population dynamics, through fragmentation models in both conservative and mass loss regimes, to kinetic models. The authors also discuss a few miscellaneous applications that do not fit into their theory exactly but nevertheless their treatment draws substantially from it.

Table of Contents:
Basic Facts from Functional Analysis and Banach Lattices.- An Overview of Semigroup Theory.- Some Classical Perturbation Results.- Positive Perturbations of Positive Semigroups.- Substochastic Semigroups and Generator Characterization.- Applications to Birth-and-death Problems.- Applications to Pure Fragmentation Problems.- Fragmentation with Growth and Decay.- Applications to Kinetic Theory.- Singularly Perturbed Inelastic Collision Models.

Review :
From the reviews: "Several books of different sorts have been devoted in recent years to semigroup theory applied to many types of problems. The book under review tries to answer some of the most important questions which arise in the above problems ... . The references are very detailed ... . the very accurate proofs of the theorems and the choices of the examples and applications make it quite readable. In my opinion, it is a very interesting and useful tool for both pure and applied mathematicians." (Silvia Totaro, Mathematical Reviews, Issue 2006 i) "This book is a self-contained mathematical monograph devoted to the theory of semigroups of linear operators ... . The book includes a systematic treatment of classical and recent results in this area as well as a comprehensive survey of the vast literature ... . Each chapter contains a preliminary section with clear references to the bibliography and with technical remarks. ... Throughout the book, the style is clear and elegant. ... this book can be highly recommended to anybody who works in this area." (Adina Luminita Sasu, Zentralblatt MATH, Vol. 1097 (23), 2006) "More precisely, as correctly described in the title, [BA] concentrates on the perturbation of positive semigroups and its applications to birth and death processes, fragmentation processes ... and to kinetic theory. ... Obviously [BA] should be in the hand of every researcher who is interested in any of these applications. It also seems to be the only book available that deals with relatively bounded positive perturbations of positive semigroups and with general questions of their stochasticity and honesty." (H. R. Thieme, SIAM Review, Vol. 49 (2), 2007)