  We prove that a family of linear bounded evolution operators $({\bf
G}(t,s))_{t\ge s\in I}$ can be associated, in the space of vector-valued
bounded and continuous functions, to a class of systems of elliptic operators
$\bm{\mathcal A}$ with unbounded coefficients defined in $I\times \Rd$ (where
$I$ is a right-halfline or $I=\R$) all having the same principal part. We
establish some continuity and representation properties of $({\bf G}(t,s))_{t
\ge s\in I}$ and a sufficient condition for the evolution operator to be
compact in $C_b(\Rd;\R^m)$. We prove also a uniform weighted gradient estimate
and some of its more relevant consequence.
