A generalization of Douglis–Nirenberg elliptic operators

Abstract

It is well known that the composition of two elliptic differential operators of orders ω and ω′ is again an elliptic operator of order ω+ω′. This fact is not true for the composition of two multi–order systems elliptic in the sense of Douglis and Nirenberg. We consider a new more general class of multi–order pseudodifferential operators. This class of multi–order essentially elliptic operators is closed with respect to the composition, coordinate and basis transformation. Moreover, every multi–order essentially elliptic operator has a parametrix belonging to the class and therefore is a Fredholm operator. Bibliography: 5 titles.