  We characterize the longterm behavior of a semiflow on a compact space $K$ by
asymptotic properties of the corresponding Koopman semigroup. In particular, we
compare different concepts of attractors, such as asymptotically stable
attractors, Milnor attractors and centers of attraction. Furthermore, we give a
characterization for the minimal attractor for each mentioned property. The
main aspect is that we only need techniques and results for linear operator
semigroups, since the Koopman semigroup permits a global linearization for a
possibly non-linear semiflow.
