On the essential spectrum of elliptic differential operators

Abstract

Let $\mathcal{A}$ be a $C^{\ast }$-algebra of bounded uniformly continuous functions on $X={\mathbb{R}}^d$ such that $\mathcal{A}$ is stable under translations and contains the continuous functions that have a limit at infinity. Denote ${\mathcal{A}}^{†}$ the boundary of $X$ in the character space of $\mathcal{A}$. Then the crossed product $\mathcal{A}=\mathcal{A}⋊X$ of $\mathcal{A}$ by the natural action of $X$ on $\mathcal{A}$ is a well defined $C^{\ast }$-algebra and to each operator $A\in \mathcal{A}$ one may naturally associate a family of bounded operators $A_{\varkappa }$ on $L^2(X)$ indexed by the characters $\varkappa \in {\mathcal{A}}^{†}$. We show that the essential spectrum of $A$ is the union of the spectra of the operators $A_{\varkappa }$. The applications cover very general classes of singular elliptic operators.

25 pages. In version 6 some typing errors in version 5 are corrected and a four lines comment page 20 is added