On Exponential Factorizations of Matrices over Banach Algebras

TL;DR

This paper proves that invertible matrices over certain Banach algebras, specifically those of holomorphic functions on Riemann surfaces, can be factored into two exponentials, extending previous results for 2x2 matrices.

Contribution

It extends exponential factorization results to matrices over Banach algebras of holomorphic functions on Riemann surfaces.

Findings

Invertible matrices over these algebras can be expressed as a product of two exponentials.

The result generalizes earlier findings for 2x2 matrices.

Provides a new factorization approach in the context of 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 $2\times 2$ matrices.