On the equality of De Branges-Rovnyak and Dirichlet spaces

TL;DR

This paper characterizes when de Branges-Rovnyak spaces coincide with harmonically weighted Dirichlet spaces, generalizing previous results and providing new examples without restrictions on the Pythagorean pair.

Contribution

It completely characterizes the equality of de Branges-Rovnyak and Dirichlet spaces for finite atomic measures, extending prior work and removing assumptions on the Pythagorean pair.

Findings

Identified conditions for $H(b)$ to be a Dirichlet space with atomic measures

Generalized previous results without restrictions on $(b,a)$

Produced new examples of space equivalences

Abstract

This work is devoted to the comparison of de Branges--Rovnyak $H(b)$ spaces harmonically weighted Dirichlet spaces $\mathcal{D}_\mu$. We completely characterize which $H(b)$ spaces are also harmonically weighted Dirichlet spaces $\mathcal{D}_\mu$, when $\mu$ 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.