On Dirichlet-type and $n$-isometric shifts in finite rank de~Branges--Rovnyak spaces

Abstract

This paper studies the function spaces $\mathcal{D}(\mu )$ by Richter and Aleman, and ${\mathcal{D}}_{\vec{\mu }}$ by the second author. It is known that the forward shift $M_z$ is bounded and expansive on $\mathcal{D}(\mu )$, and therefore $\mathcal{D}(\mu )$ coincides with a de~Branges--Rovnyak space $\mathcal{H}[B]$. We show that such a $B$ is rational if and only if $\mu$ is finitely atomic, and this happens exactly when the corresponding defect operator has finite rank. We also outline a method for calculating the reproducing kernel of $\mathcal{D}(\mu )$ for finitely atomic $\mu$. Similarly, we characterize the allowable tuples $\vec{\mu }=(\frac{|dz|}{2\pi },{\mu }_1,\dots ,{\mu }_{n-1})$ such that $M_z$ on ${\mathcal{D}}_{\vec{\mu }}$ is expansive with finite rank defect operator. This investigation provides many interesting examples of normalized allowable tuples $\vec{\mu }$.

Minor revision of mathematical content. Major revision of presentation and disposition