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 H∞ R . Suppose that N and D have the same number n of columns. In a generalisation 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 e N and e D, so that the Aryabhatta-Bezout equation X e N + Y e D = In 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 H∞ R . 12.1.2009 Let R be a commutative unital ring with unit element e and let Rm×n be the set of matrices over R with m rows and n columns. The identity matrix of size n× n will be denoted by In. If M = (ai,j) ∈ X m×n is a matrix over a normed ring(1) (X, ‖ · ‖) then ‖M‖op = √√√√ m ∑