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 ℓ p ( Z N , 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 ∈ R N rather than convex polytopes.