Perturbations of globally hypoelliptic operators on closed manifolds

Analyzing the behavior at infinity of the sequence of eigenvalues given by matrix symbol of a invariant operator with respect to a fixed elliptic operator, we obtain necessary and sufficient conditions to ensure that perturbations of globally hypoelliptic operators continue to have this property. As an application, we recover classical results about perturbations of constant vector fields on the torus and extend them for more general classes of perturbations. Additionally, we construct examples of low order perturbations that destroy the global hypoellipticity, in the presence of diophantine phenomena.