Higher order isometric shift operator on the de Branges-Rovnyak space

TL;DR

This paper explores the structure of de Branges-Rovnyak spaces generated by nonextreme functions, showing they serve as models for certain expansive 2n-isometric operators and describing their invariant subspaces.

Contribution

It introduces a new model for expansive 2n-isometric operators using de Branges-Rovnyak spaces and characterizes the invariant subspaces of the associated shift operators.

Findings

de Branges-Rovnyak spaces model expansive 2n-isometric operators

Invariant subspaces of the shift operator on H(b) are characterized

H(b) spaces are invariant under the forward shift when b is nonextreme

Abstract

The de Branges-Rovnyak space $H(b)$ is generated by a bounded analytic function $b$ in the unit ball of $H^\infty$. When $b$ is a nonextreme point, the space $H(b)$ is invariant by the forward shift operator $M_z$. We show that the $H(b)$ spaces provide model spaces for expansive quasi-analytic $2n$-isometric operators $T$ with $T^*T - I$ being rank one. Then we describe the invariant subspaces of the $2n$-isometric forward shift operator $M_z$ on $H(b)$.