Matricial topological ranks for two algebras of bounded holomorphic functions

Let N and D be two matrices over the algebra H ∞ of bounded analytic functions in the disk, or its real counterpart . Suppose that N and D have the same number n of columns. In a generalization of the notion of topological stable rank 2, it is shown that N and D can be approximated (in the operator norm) by two matrices Ñ and , so that the Aryabhatta–Bezout equation admits a solution. This has particular interesting consequences in systems theory. Moreover, in case that N is a square matrix, X can be chosen to be invertible in the case of the algebra H ∞, but not always in the case of .