Finite Sections of Band-Dominated Operators: lp-Theory

We propose an algebraic approach to the stability problem for the finite sections of general band-dominated operators acting on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^{p} = l^{p}({\mathbb{Z}})$$\end{document} for 1 < p < ∞. This approach allows us to get new results which previously were known mainly for p = 2. One of the main results shows that a band-dominated operator is Fredholm if and only if the approximation numbers of its finite sections have a special behavior.