Skip to main content
SpringerLink
Log in
Book cover

Operator Theory pp 1–36Cite as

de Branges–Rovnyak Spaces and Norm-Constrained Interpolation

  • 319 Accesses

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 ∞-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 Kreĭn-space settings.

This is a preview of subscription content, log in via an institution.

References

  1. Agler, J., McCarthy, J.E.: Pick Interpolation and Hilbert Function Spaces. In: Graduate Studies in Mathematics, vol. 44. Amer. Math. Soc., Providence (2002)

    Google Scholar 

  2. Ahern, P.R., Clark, D.N.: Radial limits and invariant subspaces. Amer. J. Math. 92, 332–342 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aronszajn, N.: Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337–404 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  4. Arov, D.Z., Grossman, L.Z.: Scattering matrices in the theory of unitary extensions of isometric operators. Soviet Math. Dokl. 270, 17–20 (1983)

    MathSciNet  Google Scholar 

  5. Arov, D.Z., Grossman, L.Z.: Scattering matrices in the theory of unitary extensions of isometric operators. Math. Nachr. 157, 105–123 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ball, J.A., Bolotnikov, V.: Interpolation problems for Schur multipliers on the Drury–Arveson space: from Nevanlinna–Pick to abstract interpolation problem. Integr. Equ. Oper. Theory 62(3), 301–349 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ball, J.A., Bolotnikov, V.: Canonical transfer-function realization for Schur–Agler-class functions on domains with matrix polynomial defining functions in \(\mathbb{C}^{n}\). In: Ball, J.A., Curto, R., Grudsky, S., Helton, W., Quiroga-Barranco, R., Vasilevski, N. (eds.) Recent Progress in Operator Theory and Its Applications. Oper. Theory Adv. Appl., vol. 220, pp. 23–55. Birkhäuser, Basel (2012)

    Google Scholar 

  8. Ball, J.A., Bolotnikov, V.: Canonical transfer-function realization for Schur-Agler-class functions of the polydisk. In: Dym, H., Kaashoek, M.A., Lancaster, P., Langer, H., Lerer, L. (eds.) A Panorama of Modern Operator Theory and Related Topics. The Israel Gohberg Memorial Volume. Oper. Theory Adv. Appl.,l vol. 218, pp. 75–122. Birkhäuser, Basel (2012)

    Chapter  Google Scholar 

  9. Ball, J.A., Bolotnikov, V.: Canonical transfer-function realization for Schur multipliers on the Drury–Arveson space and models for commuting row contractions. Indiana Univ. Math. J. 61, 665–716 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ball, J.A., Bolotnikov, V.: de BrangesRovnyak spaces: basics and theory. arXiv:1405.2980

    Google Scholar 

  11. Ball, J.A., Bolotnikov, V., ter Horst, S.: Interpolation in de Branges–Rovnyak spaces. Proc. Amer. Math. Soc. 139(2), 609–618 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ball, J.A., Bolotnikov, V., ter Horst, S.: Abstract interpolation in vector-valued de Branges–Rovnyak spaces. Integr. Equ. Oper. Theory 70(2), 227–263 (2011)

    Article  MATH  Google Scholar 

  13. Ball, J.A., Vinnikov, V.: Formal reproducing kernel Hilbert spaces: the commutative and noncommutative settings. In: Alpay, D., Vinnikov, V. (eds.) Operator Theory, System Theory and Scattering Theory: Multidimensional Generalizations. Oper. Theory Adv. Appl., vol. 134, pp. 77–134. Birkhäuser, Basel (2003)

    Google Scholar 

  14. Beatrous, F., Burbea, J.: Positive-definiteness and its applications to interpolation problems for holomorphic functions. Trans. Amer. Math. Soc. 284(1), 247–270 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  15. Bolotnikov, V.: Interpolation for multipliers on reproducing kernel Hilbert spaces. Proc. Amer. Math. Soc. 131(5), 1373–1383 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Bolotnikov, V., Dym, H.: On degenerate interpolation, entropy and extremal problems for matrix Schur functions. Integr. Equ. Oper. Theory 32(4), 367–435 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bolotnikov, V., Dym, H.: On boundary interpolation for matrix valued Schur functions. Mem. Amer. Math. Soc. 181(856) (2006)

    Google Scholar 

  18. Bolotnikov, V., Kheifets, A.: A higher order analogue of the Carathéodory–Julia theorem. J. Funct. Anal. 237(1), 350–371 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Bolotnikov, V., Kheifets, A.: The higher order Carathéodory–Julia theorem and related boundary interpolation problems. In: Recent Advances in Matrix and Operator Theory. Oper. Theory Adv. Appl. vol. 179, pp. 63–102. Birkhäuser, Basel (2008)

    Google Scholar 

  20. Bolotnikov, V., Kheifets, A.: Carathéodory–Julia type theorems for operator valued Schur functions. J. Anal. Math. 106, 237–270 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Bolotnikov, V., Kheifets, A.: Carathéodory–Julia type conditions and symmetries of boundary asymptotics for analytic functions on the unit disk. Math. Nachr. 282(11), 1513–1536 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. de Branges, L.: Perturbation theory. J. Math. Anal. Appl. 57(2), 393–415 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  23. de Branges, L., Rovnyak, J.: Canonical models in quantum scattering theory. In: Wilcox, C. (ed.) Perturbation Theory and Its Applications in Quantum Mechanics, pp. 295–392 Holt, Rinehart and Winston, New York (1966)

    Google Scholar 

  24. de Branges, L., Rovnyak, J.: Square Summable Power Series. Holt, Rinehart and Winston, New York (1966)

    MATH  Google Scholar 

  25. de Branges, L., Shulman, L.: Perturbations of unitary transformations. J. Math. Anal. Appl. 23, 294–326 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  26. Derkach, V.A.: On the indefinite abstract interpolation problem. Methods Funct. Anal. Topol. 7(4), 87–100 (2001)

    MATH  MathSciNet  Google Scholar 

  27. Derkach, V.A.: On the indefinite Schur–Nevanlinna–Pick interpolation problem. Ukrain. Mat. Zh. 55(10), 1299–1313 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Derkach, V., Dym, H.: Bitangential interpolation in generalized Schur classes. Complex Anal. Oper. Theory 4(4), 701–765 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  29. Dewilde, P.: Dym, H.: Lossless inverse scattering, digital filters, and estimation theory. IEEE Trans. Inform. Theory 30(4), 644–662 (1984)

    Google Scholar 

  30. Douglas, R.G.: On majorization, factorization, and range inclusion of operators on Hilbert space. Proc. Amer. Math. Soc. 17, 413–415 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  31. Dubovoj, V.K.: Indefinite metric in Schur’s interpolation problem for analytic functions. I. Teor. Funktsi Funktsional. Anal. i Prilozhen. 37, 14–26 (1982)

    MATH  Google Scholar 

  32. Dubovoj, V.K.: Indefinite metric in Schur’s interpolation problem for analytic functions. IV. Teor. Funktsi Funktsional. Anal. i Prilozhen. 42, 46–57 (1984)

    MATH  Google Scholar 

  33. Dym, H.: J -contractive matrix functions, reproducing kernel Hilbert spaces and interpolation. In: CBMS Regional Conference Series in Mathematics, vol. 71, Providence, RI (1989)

    Google Scholar 

  34. Fricain, E., Mashreghi, J.: Boundary behavior of functions in the de Branges–Rovnyak spaces. Complex Anal. Oper. Theory 2(1), 87–97 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  35. Helson, H.: Lectures on Invariant Subspaces. Academic Press, New York (1964)

    MATH  Google Scholar 

  36. Katsnelson, V.E.: Methods of J -Theory in Continuous Interpolation Problems of Analysis. Part I. Hokkaido University, Sapporo (1985)

    Google Scholar 

  37. Katsnelson, V.E.: On transformations of Potapov’s fundamental matrix inequality. In: Dym, H., et al. (eds.) Topics in Interpolation Theory. Oper. Theory Adv. Appl., vol. 95, pp. 253–281. Birkhäuser, Basel (1997)

    Chapter  Google Scholar 

  38. Katsnelson, V., Kheifets, A., Yuditskii, P.: An abstract interpolation problem and extension theory of isometric operators. In: Dym, H, et al. (eds.) Operators in Function Spaces and Problems in Function Theory (Russian), pp. 83–96. “Naukova Dumka”, Kiev, 1987; English transl. in: Topics in Interpolation Theory Oper. Theory Adv. Appl., vol. 95, pp. 283–298. Birkhäuser, Basel (1997)

    Google Scholar 

  39. Kheifets, A.: The abstract interpolation problem and applications. In: Axler, S., McCarthy, J.E., Sarason, D. (eds.) Holomorphic Spaces, pp. 351–379. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  40. Kheifets, A., Yuditski, P.: An analysis and extension of V. P. Potapov’s approach to interpolation problems with applications to the generalized bi-tangential Schur–Nevanlinna–Pick problem and J-inner–outer factorization. In: Matrix and Operator-Valued Functions. Oper. Theory Adv. Appl., vol. 72, pp. 133–161. Birkhäuser, Basel (1994)

    Google Scholar 

  41. Kovalishina, I.V.: J -expansive matrix-valued functions in the Carathódory problem. Akad. Nauk Armjan. SSR Dokl. 59, 129–135 (1974)

    MathSciNet  Google Scholar 

  42. Kovalishina, I.V.: J -expansive matrix-valued functions, and the classical problem of moments. Akad. Nauk Armjan. SSR Dokl. 60(1), 3–10 (1975)

    MATH  MathSciNet  Google Scholar 

  43. Kovalishina, I.V.: The Carathódory–Julia theorem for matrix-functions, Teor. Funktsi Funktsional. Anal. i Prilozhen. 43, 70–82 (1985)

    MATH  Google Scholar 

  44. Kovalishina, I.V., Potapov, V.P.: An indefinite metric in the Nevanlinna–Pick problem. Akad. Nauk Armjan. SSR Dokl. 59, 17–22 (1974)

    MATH  MathSciNet  Google Scholar 

  45. Kovalishina, I.V., Potapov, V.P.: Integral representation of Hermitian Positive Functions. Hokkaido University, Sapporo (1982)

    Google Scholar 

  46. Nikolskii, N.K., Vasyunin, V.I.: A unified approach to function models, and the transcription problem. In: Dym, H., et al. (eds.) The Gohberg Anniversary Collection, OT41, vol. 2, pp. 405–434. Birkhäuser, Basel (1989)

    Chapter  Google Scholar 

  47. Rosenblum, M., Rovnyak, J.: Hardy Classes and Operator Theory. Oxford Mathematical Monographs, Oxford University Press, Oxford (1985)

    MATH  Google Scholar 

  48. Sarason, D.: Sub-Hardy Hilbert Spaces in the Unit Disk. Wiley, New York (1994)

    MATH  Google Scholar 

  49. Shapiro, J.H.: Composition Operators and Classical Function Theory. Springer, New York (1993)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Joseph A. Ball

  2. Department of Mathematics, The College of William and Mary, Williamsburg, VA, USA

    Vladimir Bolotnikov

Authors
  1. Joseph A. Ball
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladimir Bolotnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joseph A. Ball .

Editor information

Editors and Affiliations

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

    Daniel Alpay

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Basel

About this entry

Cite this entry

Ball, J.A., Bolotnikov, V. (2014). de Branges–Rovnyak Spaces and Norm-Constrained Interpolation. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0692-3_5-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-0692-3_5-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Basel

  • Online ISBN: 978-3-0348-0692-3

  • eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Chapter history

  1. Latest

    Applications of de Branges Spaces of Vector Valued Functions
    Published:
    01 August 2015

    DOI: https://doi.org/10.1007/978-3-0348-0692-3_1-2

  2. Applications of de Branges Spaces of Vector Valued Functions
    Published:
    04 November 2014

    DOI: https://doi.org/10.1007/978-3-0348-0692-3_1-1

  3. Original

    de Branges–Rovnyak Spaces and Norm-Constrained Interpolation
    Published:
    04 November 2014

    DOI: https://doi.org/10.1007/978-3-0348-0692-3_5-1

Search

Navigation