Lower order perturbation and global analytic vectors for a class of globally analytic hypoelliptic operators

In this work we return to the class of globally analytic hypoelliptic Hörmander's operators defined on the N-dimensional torus introduced by Cordaro and Himonas and prove that if P is any operator in this class, then a perturbation of P by an analytic pseudodifferential operator with degree smaller than the subelliptic index of P remains globally analytic hypoelliptic. We also study the Gevrey regularity of the Gevrey vectors for such a class and at the end we also show that Cordaro and Himonas's result can be extended to a similar class of operators now defined in a product of compact Lie group by a compact manifold.