De Branges-Rovnyak spaces and local Dirichlet spaces of higher order

TL;DR

This paper characterizes local Dirichlet spaces of higher order using derivatives, explores their relation to de Branges-Rovnyak spaces generated by specific functions, and examines properties of associated shift operators.

Contribution

It generalizes known results on local Dirichlet spaces, provides explicit formulas for generating functions, and analyzes the properties of shift operators in these spaces.

Findings

Characterization of local Dirichlet spaces of order m via m-th derivatives.

Explicit formulas for functions b when oincides with local Dirichlet space.

Property of wandering vectors of the restricted shift operator.

Abstract

We discuss de Branges-Rovnyak spaces $\mathcal H(b)$ generated by nonextreme and rational functions $b$ and local Dirichlet spaces of order $m$ introduced in [6]. In [6] the authors characterized nonextreme $b$ for which the operator $Y=S|_{\mathcal H(b)}$, the restriction of the shift operator $S$ on $H^2$ to $\mathcal H(b)$, is a strict $2m$-isometry and proved that such spaces $\mathcal H (b)$ are equal to local Dirichlet spaces of order $m$. Here we give a characterization of local Dirichlet spaces of order $m$ in terms of the $m$-th derivatives that is a generalization of a known result on local Dirichlet spaces. We also find explicit formulas for $b$ in the case when $\mathcal H(b)$ coincides with local Dirichlet space of order $m$ with equality of norms. Finally, we prove a property of wandering vectors of $Y$ analogous to the property of wandering vectors of the restriction of $S$ to harmonically weighted Dirichlet spaces obtained by D. Sarason in [11].