A Family of Almost Invertible Infinite Matrices

TL;DR

This paper introduces a new algebraic framework for Fredholm matrices, a class of infinite matrices, along with a measure of their invertibility proximity, and studies its algebraic properties.

Contribution

It develops an algebraic analogue of Fredholm operators for row and column finite matrices and defines a measure of their distance from invertibility.

Findings

The measure respects matrix multiplication.

It is invariant under certain perturbations.

It remains unchanged under conjugation by invertible matrices.

Abstract

An algebraic analogue of the family of Fredholm operators is introduced for the family of row and column finite matrices, dubbed "Fredholm matrices." In addition, a measure is introduced which indicates how far a Fredholm matrix is from an invertible matrix. It is further shown that this measure respects multiplication, is invariant under perturbation by a matrix from $M_\infty(K)$, and is invariant under conjugation by an invertible row and column finite matrix.