The Fredholm Index of Locally Compact Band-dominated Operators on $$L^p \,(\user2{\mathbb{R}})$$

Abstract

We establish a necessary and sufficient criterion for the Fredholmness of a general locally compact band-dominated operator A on \(L^p \,(\user2{\mathbb{R}})\) and solve the long-standing problem of computing its Fredholm index in terms of the limit operators of A. The results are applied to operators of convolution type with almost periodic symbol.