On properties of the generalized elliptic pseudo-differential operators on closed manifolds

The coincidence of two classes of the generalized elliptic pseudo-differential operators — the class $GEL(X)$ and the class $REL(X)$ — selected by author from the class of all the linear classical pseudo-differential operators $L(X;E,E)$ (Theorem 5.2) is proved. It is also proved that the composition $AB$ of any operators $A$ and $B$ from $GEL(X)$ and global parametrix $P_A$ and $P_B$ belongs to $GEL(X)$ (Theorems 3.4 and 3.6.1). The belonging of $A$ to $GEL(X)$ does depend neither the choosing of basis in $E$ nor the weight-orders of $A$ (Theorems 3.2.2, 3.7.1, and 3.7.2).
In § 2 and § 4, some properties of the classes $EFL(U)$ and $REL(U)$, which arise from microlocal analysis of the generalized elliptic operators, are studied.