Local structure of generalized elliptic pseudodifferential operators and method of Gauss

In this paper we study the class of operators dominant parts of which admits elliptic factorization in some conique domain $U$ from $T'(x)$, i.e., they can be represented as compozition of diagonal operators and elliptic at $U$ operators of order zero. We denote this class by $ETF^\circ(U)$. It arrises in microlocalization of notion “generalized ellipticity.” We are interested in the problem of simplest factorization of dominant part of the operator BAC where $\mathscr A\in EFL^\circ(U)$ and operators $B$ and $C$ are choosen from class $EL^\circ(U_q)$ (elliptic operators in some neighborhood $U_q$ of the point $q\in U$). For the operators $\mathscr A$ from subclass $BEL^\circ(U)$ the dominant part $BAC$ can be reduced to one diagonal operator. It turns out that for operators from the whole class $EFL^\circ(U)$ there is no such a representation but the representation in which the dominant part $BAC$ is composition of finite number of diagonal operators, permutation matrices and lower triangular matrices with identity in general diagonal exists olwais. We prove this theorem by analog of Gauss method which we introduce in algebra of pseudodifferential operators.