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

Abstract

In this paper we consider nonautonomous elliptic operators $\mathcal{A}$ with nontrivial potential term defined in $I\times {\mathbb{R}}^d$, where $I$ is a right-halfline (possibly $I=\mathbb{R}$). We prove that we can associate an evolution operator $(G(t,s))$ with $\mathcal{A}$ in the space of all bounded and continuous functions on ${\mathbb{R}}^d$. 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 $C_0({\mathbb{R}}^d)$.