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 D_μ. We completely characterize which $H(b)$ spaces are also harmonically weighted Dirichlet spaces D_μ, when μ is a finite sum of atoms. This is a generalization of a previous result by Costara--Ransford {costara2013}: we make no assumptions on the Pythagorean pair $(b,a)$, and we produce new examples.