We prove that a family of linear bounded evolution operators (G(t, s)) t≥s∈I can be associated, in the space of vector-valued bounded and continuous functions, to a class of systems of elliptic operators A with unbounded coefficients defined in I × R d (where I is a right-halfline or I = R) all having the same principal part. We establish some continuity and representation properties of (G(t, s)) t≥s∈I and a sufficient condition for the evolution operator to be compact in C b (R d ; R m ). We prove also a uniform weighted gradient estimate and some of its more relevant consequence.2000 Mathematics Subject Classification. 35K45; 35K58, 47B07, 60H10, 91A15.Remark 2.4. (i) Hypothesis 2.2(i) can be replaced with the weaker requirement that K η,ε is bounded from below in J × R d , uniformly with respect to η ∈ ∂B 1 , for any bounded interval J ⊂ I. Indeed, in this case, for any J as above, let c J > 0 be such that K η,ε ≥ −c J in J × R d for any η ∈ ∂B 1 . The change of unknowns v(t, x) := e −cJ (t−s)/4 u(t, x) transforms the elliptic operator A into the operator A − c J /4, which satisfies Hypothesis 2.2(i) and, clearly, the uniqueness of v is equivalent to the uniqueness of u. (ii) In the scalar case when the elliptic operator in (1.1) is A = Tr(QD 2 ) + b, ∇ + c and c is bounded from above (otherwise, Proposition 2.5 fails in general), taking ε = 1 and κ = c, one easily realizes that Hypothesis 2.2(i) is trivially satisfied. Moreover, Hypothesis (2.2)(ii) reduces to require the existence of a Lyapunov function for the operator A+ c, for any bounded interval J ⊂ I. This condition seems to be much more general than that typically assumed
