Spectral theory of a hybrid class of pseudo-differential operators

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.