Markov Operators, Positive Semigroups and Approximation Processes

This research monograph gives a detailed account of a theory which is mainly concerned with certain classes of degenerate differential operators, Markov semigroups and approximation processes. These mathematical objects are generated by arbitrary Markov operators acting on spaces of continuous functions defined on compact convex sets; the study of the interrelations between them constitutes one of the distinguishing features of the book.

Among other things, this theory provides useful tools for studying large classes of initial-boundary value evolution problems, the main aim being to obtain a constructive approximation to the associated positive C₀-semigroups by means of iterates of suitable positive approximating operators. As a consequence, a qualitative analysis of the solutions to the evolution problems can be efficiently developed.

The book is mainly addressed to research mathematicians interested in modern approximation theory by positive linear operators and/or in the theory of positive C₀-semigroups of operators and evolution equations. It could also serve as a textbook for a graduate level course.

F. Altomare, M. C. Montano, U. of Bari, It.; V. Leonessa, U.d.S. della Basilicata, Potenza, It.; I. Rasa, T.U. of Cluj-Napoca, Rom.

"The authors successfully succeed in demonstrating that the relationship between approximation theory and the theory of C₀-semigroups of linear operators and evolution equations represents a fascinating and powerful mathematical theory that succeeds in solving problems related to potential theory, partial differential equations, and mathematical physics. [...]The material, forming a perfect integral whole, offers information that puts the reader at the forefront of current research and determines fruitful directions for future advanced studies. The architecture of the monograph also includes substantial motivation and examples for key results. Clarity of the text and rigorous proofs represent other features of this book. We believe that the reader will also appreciate the final section of each chapter named notes and comments. [...] In closing, it is worth mentioning that the monograph may clearly serve as a perfect basis both for researchers and graduate students in pure and applied mathematics who are interested in a modern approach to the theory of positive C₀-semigroups of operators and their connections with approximation processes." Mathematical Reviews

"This book is a welcome contribution to the topic of approximation theory for solutions of partial differential equations with degenerate coefficients. It is written as – and meant to be – a standard reference for researchers working in this field, and it will certainly become one." Zentralblatt für Mathematik