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

Chandler-Wilde, S. N. ORCID: https://orcid.org/0000-0003-0578-1283 and Lindner, M. (2008) Sufficiency of Favard's condition for a class of band-dominated operators on the axis. Journal of Functional Analysis, 254 (4). pp. 1146-1159. ISSN 0022-1236 doi: 10.1016/j.jfa.2007.09.004

Abstract/Summary

The purpose of this paper is to show that, for a large class of band-dominated operators on $\ell^\infty(Z,U)$, 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 $\ell^p(Z,U)$, regardless of $p\in[1,\infty]$, as the union of point spectra of its limit operators considered as acting on $\ell^p(Z,U)$.