Iterates of systems of operators in spaces of $omega$-ultradifferentiable functions

Given two systems $P=(P_j(D))_{j=1}^N$ and $Q=(Q_j(D))_{j=1}^M$ of linear partial differential operators with constant coefficients, we consider the spaces ${mathcal E}_omega^P$ and ${mathcal E}_omega^Q$ of $omega$-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 $omega(t)$ and $sigma(t)$, for the inclusion ${mathcal E}_omega^Psubseteq{mathcal E}_sigma^Q$. As a consequence we have a generalization of the classical Theorem of the Iterates.