One sided invertibility of matrices over commutative rings, corona problems, and Toeplitz operators with matrix symbols (Q401168)

The Coburn property for a class of linear bounded operators acting on some Banach space means that every Fredholm operator from this class has trivial kernel or trivial cokernel, i.e., is one-side invertible. The authors give conditions under which the Fredholmness, the Coburn property and one- or two-sided invertibility of a Toeplitz operator with matrix symbol \(G\) and a Toeplitz operator with scalar symbol \(\det G\) arise simultaneously. The mentioned results are based on a criterion of one-sided invertibility for rectangular matrices with entries in a commutative ring, as well as on results dealing with corona type problems.