Elementary theory of one-parameter semigroups

Abstract

In the first chapter of the book, we give an introduction to the theory of one-parameter operator semigroups. We begin with the splitting theorem of Jacobs- Deleeuw-Glicksberg and the Eberlein mean ergodic theorem for a one-parameter operator semigroup. Then we present the elementary theory of C₀-semigroups and discuss some relations between spectral properties of the generator of a C₀- semigroup and its asymptotic behavior. We follow the standard textbooks [13], [48], [57], [67], [74], [80], [130], and send the reader for other deep and delicious topics of this theory to those books and to [17], [67], [87], [41], [89]. In the last section, we discuss the asymptotically finite-dimensional semigroups. We use frequently well-known results from operator theory and functional analysis, and send the reader to standard textbooks [2], [74], [105], and [130] for them.