de Branges-Rovnyak spaces: basics and theory

TL;DR

This survey explores the foundational aspects and various formulations of de Branges-Rovnyak spaces, emphasizing their role in modeling contraction operators and extending to broader classes with connections to function theory.

Contribution

It provides a comprehensive overview of de Branges-Rovnyak spaces, including three equivalent definitions and their application to modeling non-isometric contraction operators.

Findings

Three equivalent formulations of ${ m extbf{H}}(S)$ are described.

The role of ${ m extbf{H}}(S)$ in modeling contraction operators is analyzed.

Extensions to nonunitary contraction operators are discussed.

Abstract

For $S$ a contractive analytic operator-valued function on the unit disk ${\mathbb D}$, de Branges and Rovnyak associate a Hilbert space of analytic functions ${\mathcal H}(S)$ and related extension space ${\mathcal D(S)}$ consisting of pairs of analytic functions on the unit disk ${\mathbb D}$. This survey describes three equivalent formulations (the original geometric de Branges-Rovnyak definition, the Toeplitz operator characterization, and the characterization as a reproducing kernel Hilbert space) of the de Branges-Rovnyak space ${\mathcal H}(S)$, as well as its role as the underlying Hilbert space for the modeling of completely non-isometric Hilbert-space contraction operators. Also examined is the extension of these ideas to handle the modeling of the more general class of completely nonunitary contraction operators, where the more general two-component de Branges-Rovnyak model space ${\mathcal D}(S)$ and associated overlapping spaces play key roles. Connections with other function theory problems and applications are also discussed. More recent applications to a variety of subsequent applications are given in a companion survey article.