Fredholm theory for band-dominated and related operators: A survey

This paper presents the Fredholm theory on lp-spaces for band-dominated operators and important subclasses, such as operators in the Wiener algebra. It particularly closes several gaps in the previously known results for the case p=∞ and addresses the open questions raised by Chandler-Wilde and Lindner in [S.N. Chandler-Wilde, M. Lindner, Limit operators, collective compactness and the spectral theory of infinite matrices, Mem. Amer. Math. Soc. 210 (989) (2011)]. The main tools are provided by the limit operator method and an algebraic framework for the description and adaption of Fredholmness and convergence. A comprehensive overview of this approach is given.