de Branges spaces on compact Riemann surfaces and a Beurling-Lax type theorem

TL;DR

This paper explores de Branges-Rovnyak spaces on compact Riemann surfaces using operator vessels, leading to a Beurling-Lax type theorem for Hardy spaces on finite bordered surfaces.

Contribution

It introduces a novel framework connecting operator vessels with de Branges-Rovnyak spaces on Riemann surfaces, extending classical theorems to this geometric setting.

Findings

Established a Beurling-Lax type theorem for Hardy spaces on finite bordered Riemann surfaces.

Connected operator vessel theory with sections of line bundles on Riemann surfaces.

Extended classical results to a geometric and operator-theoretic context.

Abstract

Using the notion of commutative operator vessels, this work investigates de Branges-Rovnyak spaces whose elements are sections of a line bundle of multiplicative half-order differentials on a compact real Riemann surface. As a special case, we obtain a Beurling-Lax type theorem in the setting of the corresponding Hardy space on a finite bordered Riemann surface.