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 x R-d. 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 R-d. 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 L-p-spaces and C-0(R-d). (C) 2010 Elsevier Inc. All rights reserved.