Factorization in a torus and Riemann–Hilbert problems

A new concept of meromorphic Σ-factorization, for Hölder continuous functions defined on a contour Γ that is the pullback of R ˙ (or the unit circle) in a Riemann surface Σ of genus 1, is introduced and studied, and its relations with holomorphic Σ-factorization are discussed. It is applied to study and solve some scalar Riemann–Hilbert problems in Σ and vectorial Riemann–Hilbert problems in C , including Wiener–Hopf matrix factorization, as well as to study some properties of a class of Toeplitz operators with 2 × 2 matrix symbols.