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.
D. Alpay, A. Pinhas, and V. Vinnikov. de Branges spaces on compact Riemann surfaces and a Beurling-Lax type theorem. Advances in Math., 373:107315, 2020.
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