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 × 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 ) .