  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.
