The finite section method and stable subsequences

The purpose of this paper is to prove a sufficient and necessary criterion on the stability of a subsequence of the finite section method for a so-called band-dominated operator on ℓp(ZN,X). We hereby generalize previous results into several directions: We generalize the subsequence theorem from dimension N=1 (see Rabinovich, Roch and Silbermann (2008) [18]) to arbitrary dimensions N⩾1; and even for the case of the full sequence, our result is new in dimensions N>2 and it corrects a mistake in the literature for N=2. Moreover, we allow the truncations to be taken by homothetic copies of very general starlike geometries Ω∈RN rather than convex polytopes.