This article devoted to study periodic M-hypoelliptic pseudo-differential operators on torus More precisely, we consider toroidal M-hypoelliptic symbols and we construct the minimal and maximal extension on for periodic M-hypoelliptic pseudo-differential operators. We obtain some results about the minimal and maximal operators of periodic M-hypoelliptic pseudo-differential operators. For periodic M-hypoelliptic pseudo-differential operators on with constant coefficients, the spectrum and essential spectrum are computed. Finally, we will apply the periodic theory developed in this article to the Cauchy problem and heat operator on torus.
