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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.

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