On the Factorization of Matrices Over Commutative Banach Algebras

Abstract

We study exponential factorization of invertible matrices over unital complex Banach algebras. In particular, we prove that every invertible matrix with entries in the algebra of holomorphic functions on a closed bordered Riemann surface can be written as a product of two exponents of matrices over this algebra. Our result extends similar results proved earlier in [KS] and [L] for 2 × 2 matrices.

Factorization by elementary matrices, null-homotopy and products of exponentials for invertible matrices over rings

“…Corollary 6 applies to the disk algebra and also to the rings O(C) and O(D) of holomorphic functions. Indeed, the identity bsr(O(Ω)) = 1 for an open Riemann surface follows from the strengthening of the classical Wedderburn lemma (see [19,Chapter 6,Section 3]; see also [10] or [2]). However, for R = O(C) and R = O(D), the number 4 is not optimal; see Section 4.4 below.…”

On holomorphic matrices on bordered Riemann surfaces

“…NOTE : After this paper was written and the preprint was posted in the arXiv [12], I got to know the preprint [2, Theorem 1.3] with a substantial generalization of Theorem 1.1. This generalization, in particular, contains Theorem 1.1 with in place of , for arbitrary (see [2, Example 1.4 (1)]).…”