De Branges–Rovnyak Spaces: Basics and Theory

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.KeywordsHilbert SpaceHardy SpaceToeplitz OperatorReproduce Kernel Hilbert SpaceContraction OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.