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

It is shown that two classes of generalized elliptic pseudo-differential operators, GEL(X) and REL(X), selected by the author from the class of classical linear pseudo-differential operators coincide. It is also shown that for any operators A, B ∈GEL(X) their composition AB and their global parametrices P_A, P_B belong to GEL(X). An operator A belongs to GEL(X) independently of the choice of a basis in E and of the weighted order of A. Some properties of the classes EFL(U) and REL(U) arising in microalocal analysis of generalized elliptic operators are studied. Bibliography: 12 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 215–269. Translated by N. A. Karazeeva.