Propagation of Semiclassical Wave Front Sets for Non-Self-Adjoint Matrix-Valued Pseudo-Differential Operators

We study the semiclassical wave front set of vector-valued microlocal solutions to the pseudo-differential system with a non-Hermitian matrix-valued symbol. The main result establishes the propagation of the semiclassical wave front sets under a generalized microhyperbolicity condition for the real part of the symbol and the non-negativity for the imaginary part. The proof is based on Hörmander’s positive commutator method.