In this paper we introduce the hybrid class of the so-called-hypoelliptic symbols, and consider the corresponding pseudo-differential operators. With any-elliptic pseudo-differential operator with positive order, we associate the minimal and maximal operators on. Further on, we prove that the minimal and maximal operators are equal and we compute their domains in terms of a family of suitable Sobolev spaces. In the last section, we show that an-elliptic pseudo-differential operator is Fredholm. Moreover, we discuss the essential spectra of-elliptic pseudo-differential operators with suitable orders.
