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Two-sided Tangential Interpolation for Hilbert-Schmidt Operator Functions on Polydisks

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Recent Advances in Operator Theory

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 124))

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Abstract

Bitangential interpolation problems are formulated for the class of Hilbert-Schmidt operator-valued functions, which are analytic on a polydisk and have square summable power series. A procedure is described for reduction of the problems in d-disk to the analogous problems in (d - 1)-disk. Using this procedure, for the case of the bidisk, the minimal norm solutions are explicitly described in terms of the interpolation data, and formulas for the general solution are obtained.

Dedicated with respect and affection to Professor Israel Gohberg on the occasion of his 70-th birthday

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References

  1. J. Agler, On the representation of certain holomorphic functions defined on a polydisk, Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel 48 (1990)47–66.

    Google Scholar 

  2. D. Alpay and V. Bolotnikov Two sided interpolation for matrix functions with entries in the Hardy space Linear Algebra and its Applications 223/224 (1995), 31–56.

    Article  MathSciNet  Google Scholar 

  3. D. Alpay and V. Bolotnikov, On tangential interpolation in reproducing kernel Hilbert space modules and applications, Operator Theory: Advances and Applications 95 (1997), 37–68.

    MathSciNet  Google Scholar 

  4. D. Alpay and V. Bolotnikov, On the tangential interpolation problem for matrix-valued H 2-functions of two variables, Proceedings of the Amer. Math. Soc 127 (1999), 1789–1799.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Alpay, V. Bolotnikov and Ph. Loubaton, On two-sided residue interpolation for matrix-valued h 2-functions with symmetries, Journal of Mathematical Analysis and Applications 200 (1996), 76–105.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Alpay, V. Bolotnikov and L. Rodman, Tangential interpolation with symmetries and two-points interpolation for matrix valued H 2-functions, Integral Equations and Operator Theory 32 (1998), 1–28.

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Alpay, V. Bolotnikov and L. Rodman, One-sided tangential interpolation for operator-valued Hardy functions on polydisks, Integral Equations and Operator Theory (to appear).

    Google Scholar 

  8. J.A. Ball, I. Gohberg and L. Rodman, Interpolation of Rational Matrix Functions, OT 45 Birkhäuser Verlag, Basel 1990.

    MATH  Google Scholar 

  9. J.A. Ball, T. Trent, Unitary colligations, reproducing kernel Hilbert spaces and Nevanlinna-Pick interpolation in several variables, J. of Functional Analysis 157 (1998), 1–61.

    Article  MathSciNet  MATH  Google Scholar 

  10. J.A. Ball and V. Vinnikov, Zero-pole interpolation for meromorphic matrix functions on an algebraic curve and transfer functions of 2D systems, Acta AppL Math 45 (1996), 239–316.

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Dym, J Contractive Matrix Functions, Reproducing Kernel Spaces and Interpolation, CBMS Lecture Notes. Amer. Math. Soc, Rhodes Island 1989.

    Google Scholar 

  12. H. Dym, Book review: The commutant lifting approach to interpolation problems, by Ciprian Foiaş and Arthur E. Frazho, Bull. of the Amer. Math. Soc 31 (1994), 125–140.

    Article  MathSciNet  Google Scholar 

  13. C. Foiaş and A.E. Frazho The Commutant Lifting Approach to Interpolation Problems, OT 44 Birkhäuser Verlag, Basel 1990.

    MATH  Google Scholar 

  14. C. Foiaş, A.E. Frazho, I. Gohberg and M.A. Kaashoek, Metric Constrained Interpolation, Commutant Lifting, and Systems, OT 100, Birkhäuser Verlag, Basel 1998.

    MATH  Google Scholar 

  15. P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd edition, Academic Press, Orlando 1985.

    MATH  Google Scholar 

  16. R.A. Penrose, A generalized inverse for matrices, Proceedings of Cambridge Phil. Soc 51 (1955), 406–413.

    Article  MathSciNet  MATH  Google Scholar 

  17. W. RudinFunction Theory in Polydiscs W.A. Benjamin, Inc., New York—Amsterdam, 1969.

    MATH  Google Scholar 

  18. I. Schur, Über die Potenzreihen, die im Innern des Einheitkreises Beschrankt rind, Journal für die reine and angewandte Mathematik 147 (1917), English translation in: I. Schur methods in operator theory and signal processing. Operator theory: Advances and Applications Birkhauser Verlag, Basel OT 18 (1986), 31–59; 61–88.

    Google Scholar 

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Author information

Authors and Affiliations

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

    Daniel Alpay

  2. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Vladimir Bolotnikov & Leiba Rodman

Authors
  1. Daniel Alpay
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  2. Vladimir Bolotnikov
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  3. Leiba Rodman
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Editor information

Editors and Affiliations

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

    A. Dijksma

  2. Division of Mathematics and Computer Science Faculty of Sciences, Vrije Universiteit, De Boelelaan 1081a, 1081 HV, Amsterdam, The Netherlands

    M. A. Kaashoek  & A. C. M. Ran  & 

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© 2001 Springer Basel AG

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Alpay, D., Bolotnikov, V., Rodman, L. (2001). Two-sided Tangential Interpolation for Hilbert-Schmidt Operator Functions on Polydisks. In: Dijksma, A., Kaashoek, M.A., Ran, A.C.M. (eds) Recent Advances in Operator Theory. Operator Theory: Advances and Applications, vol 124. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8323-8_2

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  • DOI: https://doi.org/10.1007/978-3-0348-8323-8_2

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9516-3

  • Online ISBN: 978-3-0348-8323-8

  • eBook Packages: Springer Book Archive

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