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arXiv · 2504.04068

The Cowen-Douglas class and de Branges-Rovnyak spaces

Emmanuel Fricain·Jaydeb Sarkar
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Abstract

We establish a connection between the de Branges-Rovnyak spaces and the Cowen-Douglas class of operators which is associated with complex geometric structures. We prove that the backward shift operator on a de Branges-Rovnyak space never belongs to the Cowen-Douglas class when the symbol is an extreme point of the closed unit ball of $H^\infty$ (the algebra of bounded analytic functions on the open unit disk). On the contrary, in the non extreme case, it always belongs to the Cowen-Douglas class of rank one. Additionally, we compute the curvature in this case and derive certain exotic results on unitary equivalence and angular derivatives.

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Emmanuel Fricain, Jaydeb Sarkar. 2025-04-05. The Cowen-Douglas class and de Branges-Rovnyak spaces. https://arxiv.org/abs/2504.04068

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