Spectral Invariance of Besov-Bessel Subalgebras

TL;DR

This paper develops a systematic framework for constructing inverse-closed subalgebras of Banach algebras using smoothness space principles, with applications to matrix algebras exhibiting polynomial off-diagonal decay.

Contribution

It introduces a general method to generate inverse-closed subalgebras, including new classes of matrix algebras with polynomial off-diagonal decay, extending known results.

Findings

Inverse-closedness of Besov and Bessel potential algebras

Construction of polynomial off-diagonal decay matrix subalgebras

New classes of inverse-closed matrix subalgebras

Abstract

Using principles of the theory of smoothness spaces we give systematic constructions of scales of inverse-closed subalgebras of a given Banach algebra with the action of a d-parameter automorphism group. In particular we obtain the inverse-closedness of Besov algebras, Bessel potential algebras and approximation algebras of polynomial order in their defining algebra. By a proper choice of the group action these general results can be applied to algebras of infinite matrices and yield inverse-closed subalgebras of matrices with off-diagonal decay of polynomial order. Besides alternative proofs of known results we obtain new classes of inverse-closed subalgebras of matrices with off-diagonal decay .