Examples of de Branges-Rovnyak spaces generated by nonextreme functions

TL;DR

This paper explores specific de Branges-Rovnyak spaces generated by nonextreme functions, providing explicit descriptions and properties of these spaces when associated with functions defined by fractional powers.

Contribution

It characterizes de Branges-Rovnyak spaces generated by nonextreme functions $b_eta$ with explicit formulas, expanding understanding of these spaces beyond the extreme case.

Findings

Explicit descriptions of $ ext{H}(b_eta)$ spaces for $eta > 0$

Identification of nonextreme functions $b_eta$ in the unit ball of $H^$

Analysis of the structure and properties of these spaces

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 $\partial \mathbb D$.