de Branges–Rovnyak Spaces: Basics and Theory

Joseph A. Ball () and Vladimir Bolotnikov ()
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Joseph A. Ball: Virginia Tech, Department of Mathematics
Vladimir Bolotnikov: The College of William and Mary, Department of Mathematics

Chapter 31 in Operator Theory, 2026, pp 799-847 from Springer

Abstract: Abstract 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.

Date: 2026
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DOI: 10.1007/978-3-032-16356-1_6

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