The Cowen-Douglas class and de Branges-Rovnyak spaces

TL;DR

The paper addresses how de Branges-Rovnyak spaces $\\mathcal{H}(b)$ interact with the Cowen-Douglas operator class by studying the backward shift $X_b$ on these spaces. It establishes a sharp dichotomy: $X_b$ lies in the rank-one Cowen-Douglas class $B_1(\\mathbb{D})$ precisely when $b$ is non-extreme, and never belongs to any $B_n(\\Omega)$ when $b$ is extreme. The authors develop a curvature-based framework, giving explicit formulas for the curvature $\\mathcal{K}_{X_b}$ in terms of the pythagorean pair $a,b$ through $\\Phi=b/a$, and derive unitary equivalence criteria for $X_b$ (and for the shift $S_b$) in terms of these invariants. They further connect these invariants to Carathéodory angular derivatives and provide a concrete example illustrating unitary equivalence vs. equivalence of the corresponding spaces, highlighting the nuanced relationship between algebraic, geometric, and function-theoretic aspects of the problem.

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.

Theorems & Definitions (14)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof