On the equality of De Branges-Rovnyak and Dirichlet spaces

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.