Spectral theory of a hybrid class of pseudo-differential operators

Abstract

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.

Keywords:

• SG-pseudo-differential operators
• M-hypoellipticity
• minimal and maximal operators
• Fredholm operators
• essential spectra

AMS Subject Classifications:

• Primary: 35S05
• 47G30
• Secondary: 47A53

Acknowledgments

We would like to thank the anonymous referees for the important comments on the results and the presentation of this manuscript.