Spectral theory of a hybrid class of pseudo-differential operators

In this paper we introduce the hybrid class of the so-called SG-M-hypoelliptic symbols, and consider the corresponding pseudo-differential operators. With any SG-M-elliptic pseudo-differential operator with positive order, we associate the minimal and maximal operators on L-p(R-n), 1 < p < infinity. 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 SG-M-elliptic pseudo-differential operator is Fredholm. Moreover, we discuss the essential spectra of SG-M-elliptic pseudo-differential operators with suitable orders.