de Branges–Rovnyak Spaces: Basics and Theory

For S a contractive analytic operator-valued function on the unit disk 𝔻 $$\mathbb{D}$$ , de Branges and Rovnyak associate a Hilbert space of analytic functions ℋ ( S ) $$\mathcal{H}(S)$$ and related extension space 𝒟 ( S ) $$\mathcal{D}(S)$$ consisting of pairs of analytic functions on the unit disk 𝔻 $$\mathbb{D}$$ . 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 ℋ ( S ) $$\mathcal{H}(S)$$ , de Branges–Rovnyak spaces 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 𝒟 ( S ) $$\mathcal{D}(S)$$ 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.