On the essential spectrum of elliptic differential operators

Let A be a C⁎-algebra of bounded uniformly continuous functions on a finite dimensional real vector space X such that A is stable under translations and contains the continuous functions that have a limit at infinity. Denote A† the boundary of X in the character space of A. Then to each operator A in the crossed product A⋊X one may naturally associate a family of bounded operators Aϰ on L2(X) indexed by the characters ϰ∈A†. We show that the essential spectrum of A is the union of the spectra of the operators Aϰ. The applications cover very general classes of singular elliptic operators.