Sufficiency of Favard’s condition for a class of band-dominated operators on the axis

The purpose of this paper is to show that, for a large class of band-dominated operators on ∞(Z,U), with U being a complex Banach space, the injectivity of all limit operators of A already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at infinity of A, which, on the other hand, is often equivalent to the Fredholmness of A. As a consequence, for operators A in the Wiener algebra, we can characterize the essential spectrum of A on p(Z,U), regardless of p ∈ [1,∞], as the union of point spectra of its limit operators considered as acting on ∞(Z,U). © 2007 Elsevier Inc. All rights reserved.