One sided invertibility of matrices over commutative rings, corona   problems, and Toeplitz operators with matrix symbols
  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. 
Maria Cristina C\^amara, Leiba Rodman, Ilya M. Spitkovsky
math.FA