Compactness and invariance properties of evolution operators associated with Kolmogorov operators with unbounded coefficients

In this paper we consider nonautonomous elliptic operators A with nontrivial potential term defined in I×Rd, where I is a right-halfline (possibly I=R). We prove that we can associate an evolution operator (G(t,s)) with A in the space of all bounded and continuous functions on Rd. We also study the compactness properties of the operator G(t,s). Finally, we provide sufficient conditions guaranteeing that each operator G(t,s) preserves the usual Lp-spaces and C0(Rd).