Non-spectral Asymptotic Analysis of One-Parameter Operator Semigroups

verfasst von: Eduard Yu. Emel’yanov

Verlag: Birkhäuser Basel

Buchreihe : Operator Theory: Advances and Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

In this book, non-spectral methods are presented and discussed that have been developed over the last two decades for the investigation of asymptotic behavior of operator semigroups. This concerns in particular Markov semigroups in L1-spaces, motivated by applications to probability theory and dynamical systems. Recently many results on the asymptotic behaviour of Markov semigroups were extended to positive semigroups in Banach lattices with order-continuous norm, and to positive semigroups in non-commutative L1-spaces. Related results, historical notes, exercises, and open problems accompany each chapter.

Inhaltsverzeichnis

Frontmatter

Chapter 1. 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.

Chapter 2. Positive semigroups in ordered Banach spaces

Abstract

In this chapter, we deal with one-parameter positive semigroups in ordered Banach spaces. Firstly, we discuss the notion of ideally ordered Banach spaces and uniformly order convex Banach spaces. Both classes include L^p-spaces (1 ≤ p < ∞) as well as preduals of von Neumann algebras. We prove several theorems about positive semigroups in such Banach spaces. Then we consider positive semigroups in Banach lattices and investigate several types of asymptotic regularity of these semigroups. In the last section of this chapter, we deal with relations between the geometry of Banach lattices and mean ergodicity of bounded positive semigroups in them.

Chapter 3. Positive semigroups in L1-spaces

Abstract

In this chapter, we investigate asymptotic properties of one-parameter positive semigroups in L¹(Ω, Σ, μ), where (Ω, Σ, μ) is a measure space with a σ-finite measure μ. In the last section, we shall also consider the theory of Markov semigroups in so-called non-commutative L¹-spaces. For one-parameter positive semigroups in L¹-spaces, there is a rich theory, which includes many results on the existence of invariant densities, criteria for asymptotic stability, decomposition theorems, etc. (cf. [71]).

The choice of results presented in this chapter is motivated mainly by the author’s research interests, and it does not reflect the present state of the very broad asymptotic theory of positive semigroups in L¹-spaces. We send the reader for many other important aspects of this theory and for their applications to books of Foguel [43], Krengel [67], Lasota and Mackey [71], and Schaefer [110].

Backmatter

Titel: Non-spectral Asymptotic Analysis of One-Parameter Operator Semigroups
verfasst von: Eduard Yu. Emel’yanov
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-7643-8114-1
Print ISBN: 978-3-7643-8095-3
DOI: https://doi.org/10.1007/978-3-7643-8114-1