The Feller Property for Absorption Semigroups

LetU=(U(t); t⩾0) be a substochastic strongly continuous semigroup onL1(X, m) whereXis locally compact andma Borel measure onX. We give conditions on absorption ratesVimplying that the (strong) Feller property carries over fromU* toU*V. These conditions are essentially in terms of the Kato class associated withU. Preparing these results we discuss the perturbation theory of strongly continuous semigroups and properties of one-parameter semigroups onL∞(m). In the symmetric case of Dirichlet forms we generalize the results to measure perturbations. For the case of the heat equation on Rdwe show that the results are close to optimal.