Since the characterization of generators of C0 semigroups was established in the 1940s, semigroups of linear operators and its neighboring areas have developed into an abstract theory that has become a necessary discipline in functional analysis and differential equations. This book presents that theory and its basic applications, and the last two chapters give a connected account of the applications to partial differential equations.
Advanced graduate-level treatment of semigroup theory explores semigroups of linear operators and linear Cauchy problems. The text features challenging exercises and emphasizes motivation, heuristics, and further applications. 1985 edition.
This book explores the theory of strongly continuous one-parameter semigroups of linear operators. A special feature of the text is an unusually wide range of applications such as to ordinary and partial differential operators, to delay and Volterra equations, and to control theory. Also, the book places an emphasis on philosophical motivation and the historical background.
The asymptotic behaviour, in particular "stability" in some sense, is studied systematically for discrete and for continuous linear dynamical systems on Banach spaces. Of particular concern is convergence to an equilibrium with respect to various topologies. Parallels and differences between the discrete and the continuous situation are emphasised.
The theory of semigroups of operators is one of the most important themes in modern analysis. Not only does it have great intellectual beauty, but also wide-ranging applications. In this book the author first presents the essential elements of the theory, introducing the notions of semigroup, generator and resolvent, and establishes the key theorems of Hille–Yosida and Lumer–Phillips that give conditions for a linear operator to generate a semigroup. He then presents a mixture of applications and further developments of the theory. This includes a description of how semigroups are used to solve parabolic partial differential equations, applications to Levy and Feller–Markov processes, Koopmanism in relation to dynamical systems, quantum dynamical semigroups, and applications to generalisations of the Riemann–Liouville fractional integral. Along the way the reader encounters several important ideas in modern analysis including Sobolev spaces, pseudo-differential operators and the Nash inequality.
In recent years important progress has been made in the study of semi-groups of operators from the viewpoint of approximation theory. These advances have primarily been achieved by introducing the theory of intermediate spaces. The applications of the theory not only permit integration of a series of diverse questions from many domains of mathematical analysis but also lead to significant new results on classical approximation theory, on the initial and boundary behavior of solutions of partial differential equations, and on the theory of singular integrals. The aim of this book is to present a systematic treatment of semi groups of bounded linear operators on Banach spaces and their connec tions with approximation theoretical questions in a more classical setting as well as within the setting of the theory of intermediate spaces. However, no attempt is made to present an exhaustive account of the theory of semi-groups of operators per se, which is the central theme of the monumental treatise by HILLE and PHILLIPS (1957). Neither has it been attempted to give an account of the theory of approximation as such. A number of excellent books on various aspects of the latter theory has appeared in recent years, so for example CHENEY (1966), DAVIS (1963), LORENTZ (1966), MEINARDUS (1964), RICE (1964), SARD (1963). By contrast, the present book is primarily concerned with those aspects of semi-group theory that are connected in some way or other with approximation.
The book offers a direct and up-to-date introduction to the theory of one-parameter semigroups of linear operators on Banach spaces. The book is intended for students and researchers who want to become acquainted with the concept of semigroups.
Advanced graduate-level treatment of semigroup theory explores semigroups of linear operators and linear Cauchy problems. The text features challenging exercises and emphasizes motivation, heuristics, and further applications. 1985 edition.
The book contains a unitary and systematic presentation of both classical and very recent parts of a fundamental branch of functional analysis: linear semigroup theory with main emphasis on examples and applications. There are several specialized, but quite interesting, topics which didn't find their place into a monograph till now, mainly because they are very new. So, the book, although containing the main parts of the classical theory of Co-semigroups, as the Hille-Yosida theory, includes also several very new results, as for instance those referring to various classes of semigroups such as equicontinuous, compact, differentiable, or analytic, as well as to some nonstandard types of partial differential equations, i.e. elliptic and parabolic systems with dynamic boundary conditions, and linear or semilinear differential equations with distributed (time, spatial) measures. Moreover, some finite-dimensional-like methods for certain semilinear pseudo-parabolic, or hyperbolic equations are also disscussed. Among the most interesting applications covered are not only the standard ones concerning the Laplace equation subject to either Dirichlet, or Neumann boundary conditions, or the Wave, or Klein-Gordon equations, but also those referring to the Maxwell equations, the equations of Linear Thermoelasticity, the equations of Linear Viscoelasticity, to list only a few. Moreover, each chapter contains a set of various problems, all of them completely solved and explained in a special section at the end of the book.The book is primarily addressed to graduate students and researchers in the field, but it would be of interest for both physicists and engineers. It should be emphasised that it is almost self-contained, requiring only a basic course in Functional Analysis and Partial Differential Equations.
