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].