An Expansion Theorem for State Space of Unitary Linear System Whose Transfer Function is a Riemann Mapping Function

  • Subhajit Ghosechowdhury
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 3)

Abstract

A power series W(z) with complex coefficients which represents a function bounded by one in the unit disk is the transfer function of a canonical unitary linear system whose state space D(W)is a Hilbert space. If the power series has constant coefficient zero and coefficient of z positive, and if it represents an injective mapping of the unit disk, it appears as a factor mapping in a Löwner family of injective analytic mappings of the disk. The Löwner differential equation supplies a family of Herglotz functions. Each Herglotz function is associated with a Herglotz space of functions analytic in the unit disk. There exists an associated extended Herglotz space. An application of the Löwner differential equation is an expansion theorem for the starting state space in terms of the extended Herglotz spaces of the Löwner family. A generalization of orthogonality called complementation is used in the proof.

Keywords

Hilbert Space Power Series Unit Disk Space Versus Orthogonal Complement 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Aronszajn, Theory of Reproducing Kernels, Trans. Amer. Math. Soc. 68 (1950), 337 404.MathSciNetGoogle Scholar
  2. 2.
    L. de Branges, Coefficient estimates, J. Math. Anal. Appl. 82 (1981), 420–450.Google Scholar
  3. 3.
    L. de Branges, Grunsky spaces of analytic functions,Bull. Sci. Math. 105 (1981), 401416.Google Scholar
  4. 4.
    L. de Branges, Löwner expansions, J. Math. Anal. Appl. 100 (1984), 323–337.Google Scholar
  5. 5.
    L. de Branges, A proof of the Bieberbach conjecture,Acta Math. 154 (1985), 137–152.Google Scholar
  6. 6.
    L. de Branges, Powers of Riemann mapping functions, Mathematical Surveys, vol. 21, Amer. Math. Soc., Providence, 1986, 51–67Google Scholar
  7. 7.
    L. de Branges, Unitary linear systems whose transfer functions are Riemann mapping functions, Operator Theory: Advances and Applications, vol. 19, Birkhauser Verlag, Basel, 1986, 105–124Google Scholar
  8. 8.
    L. de Branges, Underlying concepts in the proof of the Bieberbach conjecture, Proceedings of the International Congress of Mathematicians, Amer. Math. Soc., Providence, 1987, 25–42Google Scholar
  9. 9.
    L. de Branges, Complementation in Krein spaces,Trans. Amer. Math. Soc. 305 (1988), 277–291.Google Scholar
  10. 10.
    L. de Branges, Square Summable Power Series,Bieberbach Conjecture Edition, Springer-Verlag, Heidelberg (to appear).Google Scholar
  11. 11.
    L. de Branges and J. Rovnyak, Canonical models in quantum scattering theory, Perturbation Theory and its Applications in Quantum Mechanics, Wiley, New York, 1966, 295–392Google Scholar
  12. 12.
    L. de Branges and J. Rovnyak, Square Summable Power Series, Holt, Rinehart and Winston, New York, 1966.Google Scholar
  13. 13.
    S. Ghosechowdhury, Löwner expansions, Dissertation, Purdue University, 1997.Google Scholar
  14. 14.
    S. Saitoh, One approach to Some General Integral Transforms and its applications, Integral Transforms and Special Functions 3, No. 1 (1995), 49–84.Google Scholar
  15. 15.
    S. Saitoh, Theory of reproducing kernels and its applications, Pitman Research Notes in Math. Series 189, Longman Scientific and Technical, Essex, England, 1988.Google Scholar
  16. 16.
    S. Saitoh, Integral transforms, reproducing kernels and their applications, Pitman Research Notes in Math. Series 369, Longman Scientific and Technical, Essex, England, 1997.Google Scholar
  17. 17.
    L. Shulman, Perturbations of unitary transformations,Amer. J. Math. 91 (1969), 267288.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Subhajit Ghosechowdhury
    • 1
  1. 1.Department of MathematicsPurdue UniversityUSA

Personalised recommendations