Factorization in a torus and Riemann-Hilbert problems

TL;DR

This paper introduces a new meromorphic factorization concept on Riemann surfaces of genus 1, linking it to classical problems like Riemann-Hilbert and Wiener-Hopf factorizations, with applications to Toeplitz operators.

Contribution

It develops the theory of meromorphic $ ext{Sigma}$-factorization on genus 1 Riemann surfaces and applies it to solve scalar and vector Riemann-Hilbert problems, extending classical factorization methods.

Findings

Introduced meromorphic $ ext{Sigma}$-factorization for functions on genus 1 Riemann surfaces.

Connected $ ext{Sigma}$-factorization with holomorphic factorization and classical Riemann-Hilbert problems.

Applied the theory to Wiener-Hopf matrix factorization and Toeplitz operators with 2x2 symbols.

Abstract

A new concept of meromorphic $\Sigma$-factorization, for H\"{o}lder continuous functions defined on a contour $\Gamma$ that is the pullback of $\dot{\mathbb{R}}$ (or the unit circle) in a Riemann surface $\Sigma$ of genus 1, is introduced and studied, and its relations with holomorphic $\Sigma$-factorization are discussed. It is applied to study and solve some scalar Riemann-Hilbert problems in $\Sigma$ and vectorial Riemann-Hilbert problems in $\mathbb{C}$, including Wiener-Hopf matrix factorization, as well as to study some properties of a class of Toeplitz operators with $2 \times 2$ matrix symbols.