Factorization, Riemann-Hilbert problems and the corona problem

TL;DR

This paper explores the solvability of Riemann-Hilbert boundary value problems with Wiener-Hopf type factorizations, connecting them to the corona problem and providing constructive criteria for specific matrix classes.

Contribution

It extends the understanding of Riemann-Hilbert problems with general diagonal middle factors and links them to the corona problem, offering new factorization criteria.

Findings

Established solvability conditions for Riemann-Hilbert problems with bounded outer factors.

Connected Riemann-Hilbert problems to the corona problem in the almost periodic setting.

Derived constructive factorization criteria for certain triangular matrices.

Abstract

The solvability of the Riemann-Hilbert boundary value problem on the real line is described in the case when its matrix coefficient admits a Wiener-Hopf type factorization with bounded outer factors but rather general diagonal elements of its middle factor. This covers, in particular, the almost periodic setting. Connections with the corona problem are discussed. Based on those, constructive factorization criteria are derived for several types of triangular 2-by-2 matrices.