Iterates of systems of operators in spaces of w-ultradifferentiable functions

Given two systems P = (Pj(D))(j=1)(N) and Q = (Q(j)(D))(j=1)(M) of linear partial differential operators with constant coefficients, we consider the spaces epsilon(P)(w) and epsilon(Q)(sigma) of weighted-ultradifferentiable functions with respect to the iterates of the systems P and Q respectively. We find necessary and sufficient conditions, on the systems and on the weights w(t) and sigma(t), for the inclusion epsilon(P)(w) subset of epsilon(Q)(sigma). As a consequence we obtain a generalization of the classical Theorem of the Iterates.