One sided invertibility of matrices over commutative rings, corona problems, and Toeplitz operators with matrix symbols

TL;DR

This paper investigates conditions under which Toeplitz operators with matrix symbols share invertibility and Fredholm properties with scalar symbol operators, using criteria from commutative ring theory and corona problems.

Contribution

It introduces new criteria for one-sided invertibility of matrices over commutative rings and applies these to analyze Toeplitz operators with matrix symbols.

Findings

Shared Fredholmness between Toeplitz operators with matrix and scalar symbols

Criteria for one-sided invertibility of matrices over commutative rings

Connections between invertibility properties and corona problems

Abstract

Conditions are established under which Fredholmness, Coburn's property and one- or two-sided invertibility are shared by a Toeplitz operator with matrix symbol $G$ and the Toeplitz operator with scalar symbol $\det G$. These results are based on one-sided invertibility criteria for rectangular matrices over appropriate commutative rings and related scalar corona type problems.