Wiener's Lemma for Infinite Matrices II

TL;DR

This paper extends Wiener's lemma to a class of infinite matrices associated with Beurling algebras, proving they form an inverse-closed subalgebra of bounded operators on weighted sequence spaces.

Contribution

It introduces a new class of infinite matrices related to Beurling algebras and proves their inverse-closedness on weighted sequence spaces, generalizing Wiener's lemma.

Findings

The class of matrices is inverse-closed in ${ m B}( extstyleigl( extstyleigoplus_{n eq 0} extstyleigl r^{d}igr)_{w,q})$.

The result applies to all weights in the Muckenhoupt $A_q$ class.

The work extends classical Wiener's lemma to a broader setting.

Abstract

In this paper, we introduce a class of infinite matrices related to the Beurling algebra of periodic functions, and we show that it is an inverse-closed subalgebra of ${\mathcal B}(\ell^q_w)$, the algebra of all bounded linear operators on the weight sequence space $\ell^q_w$, for any $1\le q<\infty$ and any discrete Muckenhoupt $A_q$-weight $w$.