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de Branges–Rovnyak Spaces: Basics and Theory

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Operator 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.

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Notes

  1. 1.

    This last point was missed in [22]; the normalization condition (32) was not mentioned explicitly.

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  1. Department of Mathematics, Virginia Tech, 24061-0123, Blacksburg, VA,USA

    Joseph A. Ball

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  1. Ben-Gurion University of the Negev, Department of Mathematics, Beer-Sheva, Israel

    Daniel Alpay

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Ball, J.A., Bolotnikov, V. (2014). de Branges–Rovnyak Spaces: Basics and Theory. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0692-3_6-1

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