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

TL;DR

This paper characterizes Dirichlet-type and $n$-isometric shifts in finite rank de~Branges--Rovnyak spaces, linking rationality of associated functions to atomic measures and providing methods to compute reproducing kernels.

Contribution

It establishes conditions under which the shift operators have finite rank defect operators and characterizes the structure of the corresponding function spaces.

Findings

Rationality of $B$ corresponds to finitely atomic measures.

Finite rank defect operators occur precisely for finitely atomic measures.

Provides a method to compute reproducing kernels for these 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 $\vec{\mu} = (\frac{|dz|}{2\pi}, \mu_1, \ldots, \mu_{n-1})$ 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 $\vec{\mu}$.