Towards a Koopman theory for dynamical systems on completely regular spaces

The Koopman linearization of measure-preserving systems or topological dynamical systems on compact spaces has proven to be extremely useful. In this article, we look at dynamics given by continuous semiflows on completely regular spaces, which arise naturally from solutions of PDEs. We introduce Koopman semigroups for these semiflows on spaces of bounded continuous functions. As a first step we study their continuity properties as well as their infinitesimal generators. We then characterize them algebraically (via derivations) and lattice theoretically (via Kato's equality). Finally, we demonstrate-using the example of attractors-how this Koopman approach can be used to examine properties of dynamical systems. This article is part of the theme issue 'Semigroup applications everywhere'.