Perturbations of globally hypoelliptic operators on closed manifolds

Inspired by results of A. Bergamasco on perturbations of vector fields defined on the two-dimensional torus, and of J. Delgado and M. Ruzhansky on properties of invariant operators with respect to an elliptic operator defined on a closed manifold, we give necessary and sufficient conditions to ensure that perturbations of a globally hypoelliptic operator defined on $\T\times M$, continue to be globally hypoelliptic, where $\T$ is the flat torus and $M$ is a closed smooth manifold. For this, we analyze the behavior, at infinity, of the sequences of eigenvalues generated by the family of matrices given by the restrictions of this on the eigenspaces of a fixed elliptic operator. As an application, we construct perturbations, invariant with respect to the Laplacian, of the vector field $D_t + \omega D_x$ on $\T^2$. In the case where these perturbations commute with the operator $D_x$, our examples recover and extend some results of Bergamasco. Additionally, we construct examples of low order perturbations that destroy the global hypoellipticity, in the presence of diophantine phenomena.