On properties of the generalized elliptic pseudo-differential operators on closed manifolds (Q1129796)

This paper deals with some properties of generalized elliptic pseudodifferential operators \((\psi\).d.o.) on closed manifolds. The author considers the set of classical \(\psi\).d.o. of weighted order \((s,t)\), namely \(L^{(s,t)}(X,E,E)= L^{(s,t)}(X)\) acting between the sections of the vector bundle \(E\) over the closed manifold \(X\). He defines the subset \(\text{GEL}^{(s,t)}(X)\) of generalized elliptic operators on \(X\) and the subset \(\text{REL}^{(s,t)}(X)\) of \(\psi\).d.o. microlocally reducible to (micro)local elliptic operators of order \((s,t)\). It is proved that these two subsets coincide. It is also shown that if \(A,B\in\text{GEL}(X)\) then their product \(AB\) belongs to \(\text{GEL}(X)\). Moreover, each operator \(A\in \text{GEL}(X)\) possesses a global parametrix belonging to the same class, and \(\text{GEL}(X)\) is invariant under the choice of the fasis in \(E\).