de Branges-Rovnyak spaces and norm-constraint interpolation

TL;DR

This paper surveys the properties and recent applications of de Branges-Rovnyak spaces, highlighting their role in advanced function theory, operator theory, and interpolation problems, including multivariable and Kre
2b
4dn-space extensions.

Contribution

It introduces recent developments and applications of de Branges-Rovnyak spaces to complex interpolation, boundary behavior, and multivariable operator theory, expanding their theoretical framework.

Findings

Connections with $H^ty$-norm constrained interpolation

Parametrization of all solutions to interpolation problems

Extensions to multivariable and Kre
2b
4dn-space settings

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 ${\mathcal H}(S)$. A companion survey provides equivalent definitions and basic properties of these spaces as well as applications to function theory and operator theory. The present survey brings to the fore more recent applications to a variety of more elaborate function theory problems, including $H^\infty$-norm constrained interpolation, connections with the Potapov method of Fundamental Matrix Inequalities, parametrization for the set of all solutions of an interpolation problem, variants of the Abstract Interpolation Problem of Katsnelson, Kheifets, and Yuditskii, boundary behavior and boundary interpolation in de Branges-Rovnyak spaces themselves, and extensions to multivariable and Kre\u{\i}n-space settings.