Measure-preserving semiflows and one-parameter Koopman semigroups

Abstract

For a finite measure space X, we characterize strongly continuous Markov lattice semigroups on L p (X) by showing that their generator A acts as a derivation on the dense subspace D(A) ∩ L ∞ (X). We then use this to characterize Koopman semigroups on L p (X) if X is a standard probability space. In addition, we show that every measurable and measure-preserving flow on a standard probability space is isomorphic to a continuous flow on a compact Borel probability space.In this article we address mainly the following two issues. First, we characterize strongly continuous Markov lattice semigroups (T (t)) t≥0 on L p (X) by properties of their generators for a finite measure space X = (X, Σ, µ). We will show that a strongly continuous semigroup on L p (X) is a Markov lattice semigroup if and only if its generator A acts as a derivation on D(A) ∩ L ∞ (X), 1 ∈ D(A) and the semigroup is locally bounded on L ∞ (X). Similar results have been established by R. Nagel and R. Derndinger in [2, Satz 2.5] for semigroups on C(K), see also [8,, and recently by T. ter Elst and M. Lemańczyk in [4] for unitary groups on L 2 (X). Second, we show that such semigroups are always similar to a semigroup of Koopman operators. More precisely, we construct a compact space K and a Borel measure ν such that L 1 (X, Σ, µ) is isometrically Banach lattice isomorphic to L 1 (K, ν) and, via this isomorphism, the semigroup (T (t)) t≥0 is similar to a semigroup of Koopman operators on L 1 (K, ν) induced by a continuous semiflow (ϕ t ) t≥0 on K. Furthermore, in case that the space L 1 (X, Σ, µ) is separable, we show that K can be chosen to be metrizable. Similar results have been already obtained for strongly continuous representations of locally compact groups on L p (X) as bi-Markov embeddings, see [7, Theorem 5.14]. The article is organized as follows. In the second part of the introduction, we specify our notation and recall some basic facts we use throughout the article. In Section 1, we prove our main result, Theorem 1.1, that characterizes strongly Markov semigroups of lattice homomorphisms by the condition that their generator acts as a derivation, followed by a version for semigroups that are not necessarily Markov. In Section 2 we then use these results to characterize Koopman semigroups and in particular obtain [4, Theorem 1.1] in which ter Elst and Lemańczyk proved a corresponding result for unitary operator groups as Corollary 2.6. In Section 3, we turn to the construction of topological models. Finally, in Section 4, we consider ergodic, measure-preserving flows and give a new proof for the fact that they contain at most countably many non-ergodic mappings, provided that their induced group of Koopman operators is strongly continuous on L 2 (X). This has previously been proven in [9, Theorem 1] for R k -actions.

Relatively uniformly continuous semigroups on vector lattices

“…In Section 6 this extension theorem is used to identify relatively uniformly continuous Koopman semigroups on C(R) through their semiflows. Such semigroups have been studied by [Koo31], [KN42], [Amb42], [BMM12], [EL15], [MG16], [BKR17], [EGK18] and by others. In Section 4 we study standard constructions of new relatively uniformly continuous semigroups from a given one.…”