Lindner, Marko : The finite section method and stable subsequences

Author(s):

Lindner, Marko

Title:

The finite section method and stable subsequences

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 15, 2008

Mathematics Subject Classification:

47N40			[ Applications in numerical analysis ]
47L40			[ Limit algebras, subalgebras of $C^*$-algebras ]
65J10			[ Equations with linear operators ]

Abstract:

The purpose of this note 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 $\ell^p(\Z^N,X)$. We hereby generalize previous results into several directions: We generalize the subsequence theorem from dimension $N=1$ (see [Rabinovich/Roch/Silbermann 2006]) to arbitrary dimensions $N\ge 1$. 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 $N=2$. Finally, we allow the truncations to be taken by homothetic copies of very general starlike geometries $\Omega\in\R^N$ rather than convex polytopes.

Keywords:

finite sections, stability, limit operator, band-dominated operator

Language:

English

Publication time:

8 / 2008