A generalization of Douglis-Nirenberg elliptic operators

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.