Long time semiclassical Egorov theorem for $\hbar$-pseudodifferential systems

In the Heisenberg picture, we study the semiclassical time evolution of a bounded quantum observable $Q^w(x,\hbar D_x)$ associated to a $(m\times m)$ matrix-valued symbol $Q$ generated by a semiclassical matrix-valued Hamiltonian $H\sim H_0+\hbar H_1$. Under a non-crossing assumption on the eigenvalues of the principal symbol $H_0$ that ensures the existence of almost invariant subspaces of $L^{2}(\mathbb R^n)\otimes \mathbb C^m$, and for a class of observables that are semiclassically block-diagonal with respect to the projections onto these almost invariants subspaces, we establish a long time matrix-valued version for the semiclassical Egorov theorem valid in a large time interval of Ehrenfest type $T(\hbar)\simeq log(\hbar^{-1})$.