Examples of de Branges-Rovnyak spaces generated by nonextreme functions

Abstract

We describe de Branges-Rovnyak spaces $\mathcal{H}(b_{\alpha })$, $\alpha >0$, where the function $b_{\alpha }$ is not extreme in the unit ball of $H^{\infty }$ on the unit disk $\mathbb{D}$, defined by the equality $b_{\alpha }(z)/a_{\alpha }(z)=(1-z{)}^{-\alpha }$, $z\in \mathbb{D}$, where $a_{\alpha }$ is the outer function such that $a_{\alpha }(0)>0$ and $|a_{\alpha }{|}^2+|b_{\alpha }{|}^2=1$ a.e. on $\partial \mathbb{D}$.