Some properties of strong Feller semigroups on \(L^ 2\)-spaces (Q1107803)

Let S be a topological state space which is homeomorphic to a separable metric space, \(\mu\) be a probability measure on the Borel algebra of S with dense support in S and let \(T_ t\), \(t\geq 0\), be a \(C_ 0\)- semigroup on \(L^ 2(S,\mu)\) associated with a Markov transition function \(P_ t(x,\cdot)\). Then \(\mu\) is invariant w.r.t. \(P_ t(x,\cdot)\) if and only if \(\| T_ t\|_{op}\equiv 1\). Furthermore, if \(T_ t\) has the strong Feller property, then \(P_ t(x,dy)\) is absolutely continuous w.r.t. \(\mu\) (dy).