De Branges–Rovnyak spaces and Dirichlet spaces

Sarason has shown that the local Dirichlet spaces Dλ may be considered as manifestations of de Branges–Rovnyak spaces H(b), and has used this identification to give a new proof that the spaces Dλ are star-shaped. We investigate which other Dirichlet spaces D(μ) arise as de Branges–Rovnyak spaces, and which other de Branges–Rovnyak spaces H(b) are star-shaped. We also prove a transfer principle which represents H(b)-spaces inside Dλ.