Diffusion semigroups in spaces of continuous functions with mixed topology

We study transition semigroups and Kolmogorov equations corresponding to st ochastic semilinear equations on a Hilbert space II. It is shown that the t ransition semigroup is strongly continuous and locally equicontinuous in th e space of polynomially increasing continuous functions on H when endowed w ith the so-called mixed topology. As a result we characterize cores of cert ain second older differential operators in such spaces and show that they h ave unique extensions to generators of strongly continuous semigroups. (C) zool academic Press.