Positivity of perturbed Ornstein-Uhlenbeck semigroups on \(C_{b}(H)\) (Q1780022)

The authors develop the abstract theory of bi-continuous semigroups. The bi-continuous semigroups are semigroups of linear operators in a Banach space, where the usual conditions of strong continuity with respect to the basic topology are replaced with strong continuity with respect to a certain locally convex topology which is weaker than the basic one. The results are applied to show the positivity for a class of perturbed Ornstein-Uhlenbeck semigroups on the Banach lattice \(C_b({\mathcal H})\) of bounded continuous functions on a separable Hilbert space \({\mathcal H}\). The perturbations \(B\) are of the form \[ (Bf)(x)=\langle F(x),(-A)^\gamma Df\rangle, \] where \(F:{\mathcal H}\to {\mathcal H}\) is a bounded continuous function, \(A\) is the generator of a \(C_0\)-semigroup on \({\mathcal H}\), \(\gamma\in [0,1/2)\), and \(D\) is the Fréchet derivative.