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Interpolation Theory, Systems Theory and Related Topics pp 67–97Cite as

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Notes on Interpolation in the Generalized Schur Class. I. Applications of Realization Theory

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Part of the Operator Theory: Advances and Applications book series (OT,volume 134)

Abstract

Realization theory is used to study Nevanlinna-Pick and Carathéo­dory-Fejér interpolation problems for generalized Schur classes. In the first part of the paper, conditions are given for the existence of a solution of a factorization problem that includes Nevanlinna-Pick interpolation and factor­ization problems of Leech type for operator-valued functions. In the second part, an analysis is made of the numbers of positive and negative eigenvalues of classical matrices which arise in coefficient problems. The complete solution of an indefinite Carathéodory-Fejér problem is obtained.

Keywords

  • Hilbert Space
  • Negative Eigenvalue
  • Interpolation Problem
  • Selfadjoint Operator
  • Partial Isometry

These 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.

To Harry Dym: teacher, colleague and friend, in appreciation and with best wishes.

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References

  1. V.M. Adamjan, D.Z. Arov, and M.G. Krein, Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem, Mat. Sb. (N.S.) 86(128) (1971), 34–75.

    MathSciNet  Google Scholar 

  2. D. Alpay, Some remarks on reproducing kernel Krein spaces, Rocky Mountain J. Math. 21 (1991), no. 4, 1189–1205.

    MathSciNet  MATH  Google Scholar 

  3. D. Alpay, A. Dijksma, J. Rovnyak, and H.S.V. de Snoo, Schur functions, operator colligations,and reproducing kernel Pontryagin spaces, Oper. Theory Adv. Appl., vol. 96, Birkhäuser, Basel, 1997.

    Google Scholar 

  4. AReproducing kernel Pontryagin spaces, Holomorphic Spaces (S. Axler, J.E. McCarthy, and D. Sarason, eds.), MSRI Publications, vol. 33, Cambridge University Press, Cambridge, 1998, pp.425–444.

    Google Scholar 

  5. B Realization and factorization in reproducing kernel Pontryagin spaces, Operator theory, system theory and related topics. The Moshe Livsic anniversary volume (D. Alpay and V. Vinnikov, eds.), Oper. Theory Adv. Appl., vol. 123, Birkhäuser, Basel, 2001, pp. 43–65.

    Google Scholar 

  6. J.A. Ball, I. Gohberg, and L. Rodman, Interpolation of rational matrix functions, Oper. Theory Adv. Appl., vol. 45, Birkhäuser, Basel, 1990.

    Google Scholar 

  7. J.A. Ball and J.W. Helton, A Beurling-Lax theorem for the Lie group U which contains most classical interpolation theory(m, J. Operator Theory 9 (1983), no. 1, 107­142.

    MathSciNet  MATH  Google Scholar 

  8. J.A. Ball and T.T. Trent, Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables, J. Funct. Anal. 157 (1998), no. 1, 1–61.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. T. Constantinescu and A. Gheondea, On the indefinite trigonometric moment prob­lem of I. S. Iohvidov and M. G. Krein, Math. Nachr. 171 (1995), 79–94.

    Google Scholar 

  10. T. Constantinescu and A. Gheondea, Representations of Hermitian kernels by means of Krein spaces, Publ. Res. Inst. Math. Sci. 33 (1997), no. 6, 917–951.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. T. Constantinescu and A. Gheondea, On the Carathéodory type problem of M. G. Krein and H. Langer, C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), no. 3, 243–247.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. M.A. Dritschel and J. Rovnyak, Operators on indefinite inner product spaces, Lec­tures on operator theory and its applications (Waterloo, ON, 1994), Fields Institute Monographs, vol. 3, Amer. Math. Soc., Providence, RI, 1996, pp. 141–232, Supple­mentary material and errata are available athttp://wsrv.clas.virginia.edu/j1r5m/papers/papers.html

    Google Scholar 

  13. M.A. Dritschel and J. Rovnyak, Operators on indefinite inner product spaces, Lectures on operator theory and its applications (Waterloo, ON, 1994), Fields Institute Monographs, vol. 3, Amer. Math. Soc., Providence, RI, 1996, pp. 141-232, Supplementary material and errata are available at I.S. Iokhvidov, Hankel and Toeplitz matrices and forms,algebraic theory, Birkhäuser, Boston, 1982.

    Google Scholar 

  14. I.S. Iokhvidov and M.G. Krein, Spectral theory of operators in spaces with indefinite metric. II, Trudy Moskov. Mat. Obsc. 8 (1959), 413–496, English transl.: Amer. Math. Soc. Trans’. (2) 34 (1963), 283–373.

    Google Scholar 

  15. I.S. Iokhvidov, M.G. Krein, and H. Langer, Introduction to the spectral theory of op­erators in spaces with an indefinite metric, Mathematical Research, vol. 9, Akademie-Verlag, Berlin, 1982.

    Google Scholar 

  16. A. Kheifets, The abstract interpolation problems and applications, Holomorphic Spaces (S. Axler, J. E. McCarthy, and D. Samson, eds.), MSRI Publications, vol. 33, Cambridge University Press, Cambridge, 1998, pp. 351–379.

    Google Scholar 

  17. M.G. Krein and H. Langer, Über die verallgemeinerten Resolventen und die charak­teristische Funktion eines isometrischen Operators im Raume H, Hilbert space op­erators and operator algebras (Proc. Internat. Conf., Tihany, 1970), North-Holland, Amsterdam, 1972, pp. 353–399. Colloq. Math. Soc. János Bolyai, 5.

    MathSciNet  Google Scholar 

  18. M.G. Krein and H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Ilk zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187–236.

    MathSciNet  MATH  Google Scholar 

  19. A.A. Nuclei/man, A generalization of classical interpolation problems, Dokl. Akad. Nauk SSSR 256 (1981), no. 4, 790–793.

    MathSciNet  Google Scholar 

  20. J. Rovnyak, Methods of Krein space operator theory,this volume.

    Google Scholar 

  21. T. Takagi, On an algebraic problem related to an analytic theorem of Carathéodory and Fejér, Japan J. Math. 1 (1924), 83–93, ibid. 2 (1925), 13–17.

    MATH  Google Scholar 

  22. H. Woracek, An operator-theoretic approach to degenerated Nevanlinna-Pick inter­polation,Math. Nachr. 176 (1995), 335–350.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. H. Woracek, Nevanlinna-Pick interpolation: the degenerated case, Linear Algebra Appl. 252 (1997), 141–158.

    CrossRef  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, P. O. Box 653, 84105, Beer-Sheva, Israel

    D. Alpay

  2. Programs in Mathematical Sciences, University of Texas at Dallas, Box 830688, Richardson, 75083-0688, TX, USA

    T. Constantinescu

  3. Department of Mathematics, University of Groningen, P. O. Box 800, 9700 AV, Groningen, The Netherlands

    A. Dijksma

  4. Department of Mathematics, University of Virginia, P. O. Box 400137, VA, 22904-4137, Charlottesville, USA

    J. Rovnyak

Authors
  1. D. Alpay
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  2. T. Constantinescu
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  3. A. Dijksma
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  4. J. Rovnyak
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva, 84105, Israel

    Daniel Alpay & Victor Vinnikov & 

  2. School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978, Israel

    Israel Gohberg

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Alpay, D., Constantinescu, T., Dijksma, A., Rovnyak, J. (2002). Notes on Interpolation in the Generalized Schur Class. I. Applications of Realization Theory. In: Alpay, D., Vinnikov, V., Gohberg, I. (eds) Interpolation Theory, Systems Theory and Related Topics. Operator Theory: Advances and Applications, vol 134. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8215-6_5

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  • DOI: https://doi.org/10.1007/978-3-0348-8215-6_5

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