On holomorphic matrices on bordered Riemann surfaces

Abstract

Let D be the unit disk. Kutzschebauch and Studer [8] recently proved that, for each continuous map A : D → SL(2, C), which is holomorphic in D, there exist continuous maps E, F : D → sl(2, C), which are holomorphic in D, such that A = e E e F . Also they asked if this extends to arbitrary compact bordered Riemann surfaces. We prove that this is possible.MSC 2020: 47A56, 15A54, 15A16, 30H50. Keywords: holomorphic matrices, bordered Riemann surfaces, exponentials. 1 In the sense of [1, II.3A], which includes that X is connected. For example, X can be the closure of a bounded smooth domain X in the complex plane C.2 For X = D, this is well-known [7]. I do not know if this is already known for non-simply connected X.

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

“…As noted in [6], to find such B with values in sl(2, C) is impossible already by the fact that not every matrix in SL(2, C) has a logarithm in sl(2, C). 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 SL(n, C) in place of SL(2, C), for arbitrary n 2 (see [2, Example 1.4 (1)]).…”