Exponential factorizations of holomorphic maps

Abstract

We show that any element of the special linear group prefixSL2false(prefixRfalse) is a product of two exponentials if the ring prefixR is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential factor is not enough since the exponential map corresponding to prefixSL2false(double-struckCfalse) is not surjective. Our result extends to the linear group prefixGL2false(prefixRfalse), where it is sharp as well.

On the Factorization of Matrices Over Commutative Banach Algebras

“…Further, it was shown in [KS,Thm.2] that e 2 (A ) = 2 if A is the disk algebra, and it was recently proved in [L,Thm. 1] (answering a question posed in [KS]) that the same is valid for algebras of holomorphic functions on closed bordered Riemann surfaces.…”

On holomorphic matrices on bordered Riemann surfaces

“…Let be the algebra of all complex matrices, and the group of its invertible elements. Then, in the same way as in [11, Corollary 1], the following corollary can be deduced from Theorem 1.1. Corollary Let be continuous on , holomorphic in , and null‐homotopic.…”

On holomorphic matrices on bordered Riemann surfaces

“…The problem is that X need not be simply connected. Our proof of Theorem 1.1 is nevertheless some adaption of that proof in [8].…”

“…Let D be the closed unit disk in C. For X = D, Theorem 1.1 was recently proved by Kutzschebauch and Studer [8,Theorem 2]. In [8] also the question is asked if Theorem 1.1 is true in general, and it is noted that there is some problem to adapt in a straightforward way the proof of [8] to the general case. The problem is that X need not be simply connected.…”