Non-denseness of factorable matrix functions

Abstract

It is proved that for certain algebras of continuous functions on compact abelian groups, the set of factorable matrix functions with entries in the algebra is not dense in the group of invertible matrix functions with entries in the algebra, assuming that the dual abelian group contains a subgroup isomorphic to Z 3 . These algebras include the algebra of all continuous functions and the Wiener algebra. More precisely, it is shown that infinitely many connected components of the group of invertible matrix functions do not contain any factorable matrix functions, again under the same assumption. Moreover, these components actually are disjoint with the subgroup generated by the triangularizable matrix functions.

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

“…Quite a few partial results were obtained in this direction, showing that the problem is indeed intriguing and complicated. An interested reader may consult [3] for a coherent description of the state of affairs as of about ten years ago, and [5,7,8] for some more current results.…”

Factorizations, Riemann-Hilbert problems and the corona theorem

“…Added in revision. After this paper was submitted for publication, its results were used in [9] to further develop AP factorization criteria and in [4]…”