On the equality of De Branges-Rovnyak and Dirichlet spaces

TL;DR

The paper addresses when a de Branges–Rovnyak space $H(b)$ equals a harmonically weighted Dirichlet space $\mathcal{D}_\mu$ for finitely supported measures $\mu$, extending prior results by removing the rationality constraint on the Pythagorean mate. The authors derive complete characterizations of the embeddings $H(b)\hookrightarrow \mathcal{D}_\mu$ and $\mathcal{D}_\mu\hookrightarrow H(b)$ and establish exact criteria for the full equality $H(b)=\mathcal{D}_\mu$ in terms of a factorization $a=(\prod_j(z-\zeta_j))\,g$ and the boundary spectrum $\sigma(b)$, using potential theory, Corona, and Ball–Kriete theory. They further develop a polynomial-type refinement, connect $H(b)$ with $H(p_b)$ via Fejér–Riesz, and provide concrete examples, including the boundary-continuous case and the explicit instance $b(z)=e^{z^N-1}$ with $\mu$ supported on the $N$-th roots of unity, for which $H(b)=\mathcal{D}_\mu$. These results yield explicit structural descriptions of $H(b)$ in terms of $a$, $b$, and $\mu$ and have implications for Clark measures and boundary behavior. Overall, the work advances the understanding of when operator-model spaces align with Dirichlet-type spaces, enabling explicit descriptions of $H(b)$ in broader settings.

Abstract

This work is devoted to the comparison of de Branges--Rovnyak $H(b)$ spaces harmonically weighted Dirichlet spaces $\mathcal{D}_μ$. We completely characterize which $H(b)$ spaces are also harmonically weighted Dirichlet spaces $\mathcal{D}_μ$, when $μ$ is a finite sum of atoms. This is a generalization of a previous result by Costara--Ransford \cite{costara2013}: we make no assumptions on the Pythagorean pair $(b,a)$, and we produce new examples.

Theorems & Definitions (23)

• Theorem 2.1: Costara--Ransford
• Remark
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 3.1
• Theorem 3.2
• Theorem 3.3
• proof