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

摘要

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