Left inverses of matrices with polynomial decay

TL;DR

This paper investigates the properties of Schur operators with polynomial decay on metric spaces, establishing conditions under which left-invertibility in one l^p space implies invertibility in others and existence of a left-inverse within the algebra.

Contribution

It extends previous work by proving that left-invertibility in one l^p space guarantees a left-inverse in the weighted Schur algebra for operators with polynomial decay.

Findings

Left-invertibility in one l^p space implies bounded below in all q spaces.

Such operators admit a left-inverse within the weighted Schur algebra.

The results apply to metric spaces with the doubling property.

Abstract

The algebra of Schur operators on l^2 is known not to be inverse-closed. When l^2=l^2(X) where X is a metric space, we can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra for a strictly polynomial weight is inverse-closed. Here, we prove a result dealing with left-invertibility. Namely, if such an operator is bounded below in l^p for some p, then it is bounded below for all q, and it admits a left-inverse in the weighted Schur algebra.