The fredholm property and essential spectra of pseudodifferential operators on non-compact manifolds and limit operators

The paper is devoted to the Fredholm theory and the essential spectrum of pseudodifferential operators (psdo’s) onC^∞ non-compact manifolds Q with a conical structure at infinity. We consider psdo’s which in the local coordinates have symbols in the Hörmander class S_1,0^m (Rⁿ). For the study of the Fredholm property and the essential spectra of psdo’s, we apply the local theory and the limit operators method. The main result of the paper is: a pseudodifferential operator A acting from H^s(Q, E) into H^s−m(Q, E) where H^s (Q, E) is a Sobolev space of sections of a Hermitian vector bundle p : E → Q is a Fredholm operator if and only if: (i) A is elliptic at every point x ∈ Q; (ii) all limit operators of A are invertible. We apply these results to a description of the essential spectra of a realization of uniformly elliptic psdo of positive order as unbounded operators in L²(Q, E).