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The Königs Problem and Extreme Fixed Points

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Recent Advances in Operator Theory in Hilbert and Krein Spaces

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 198))

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Abstract

We consider a linear-fractional mapping \( \mathcal{F}_A \) of the unit operator ball, which is generated by a triangular operator. Under the assumption that \( \mathcal{F}_A \) has an extreme fixed point C and under some natural restrictions on one of the diagonal elements of the operator block-matrix A, we prove the KE-property of \( \mathcal{F}_A \). In this case, the structure of the other diagonal element is studied completely.

We consider specific cases in which for C one can take any arbitrary point of the unit sphere.

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References

  1. V. Khatskevich, S. Reich, and D. Shoikhet, Schröder’s Functional Equation and the Königs Embedding Property. Nonlin. Anal. 47 (2001), 3977–3988.

    Article  MATH  MathSciNet  Google Scholar 

  2. V. Khatskevich, S. Reich, and D. Shoikhet, Abel—Schröder Equations for Linear Fractional Mappings and the Königs Embedding Problem. Acta Sci. Math. (Szeged) 69 (2003), 67–98.

    MATH  MathSciNet  Google Scholar 

  3. V. Khatskevich and V. Senderov, The Königs Problem for Linear-Fractional Mappings. Dokl. RAN 403 (2005), no. 5, 607–609 (in Russian).

    MathSciNet  Google Scholar 

  4. M. Elin and V. Khatskevich, The Königs Embedding Problem for Operator Affine Mappings. Contemporary Math. 382 (2005), 113–120.

    MathSciNet  Google Scholar 

  5. M. Elin and V. Khatskevich, Triangular Plus-Operators in Banach Spaces: Applications to the Königs Embedding Problem. J. Nonlinear and Convex Analysis 6 (2005), no. 1, 173–185.

    MATH  MathSciNet  Google Scholar 

  6. N. Danford and J. Schwartz, Linear Operators. Pt. 1: General Theory. New York, London: Intersci. Publ., 1958.

    Google Scholar 

  7. T.Ya. Azizov and I.S. Iokhvidov, Linear Operators in Spaces with Indefinite Metric. Moscow: Nauka, 1986.

    MATH  Google Scholar 

  8. V. Khatskevich and V. Senderov, The Abel-Schröder Equations for Linear Fractional Maps of Operator Balls. Dokl. Ross. Akad. Nauk 379 (4), 455–458 (2001).

    MathSciNet  Google Scholar 

  9. V. Khatskevich and V. Senderov, Abel—Schröder Type Equations for Maps of Operator Balls. Funct. Different. Equats. 10 (1/2), 239–258 (2003).

    MATH  MathSciNet  Google Scholar 

  10. V. Khatskevich and V. Senderov, Basic Properties of Linear Fractional Mappings of Operator Balls: Scröder’s Equation. Fields Inst. Communs. 25, 331–344 (2000).

    MathSciNet  Google Scholar 

  11. Paul R. Halmos, A Hilbert Space Problem Book. Toronto: London, 1967.

    Google Scholar 

  12. G. Königs, Recherches sur les int′egrales de certaines équations fonctionnelles. Annales Sci. École Normale Sup. (Sér 3), 1, supplément, 3–41 (1884).

    Google Scholar 

  13. P. Lévy, Fonctions à croissance régulière et itération d’ordre fractionnaire. Ann. Mat. Pura Appl. 5, 269–298 (1928).

    Article  MathSciNet  Google Scholar 

  14. J. Hadamard, Two Works on Iteration and Related Questions. Bull. Amer. Math. Soc. 50, 67–75 (1944).

    Article  MATH  MathSciNet  Google Scholar 

  15. T.E. Harris, Some Mathematical Models for Branching Processes. in 2nd Berkeley Symposium (1951), pp. 305–328.

    Google Scholar 

  16. I.N. Baker, Fractional Iteration near a Fixed Point of Multiplier 1. J. Australian Math. Soc. 4, 143–148 (1964).

    Article  MATH  Google Scholar 

  17. S. Karlin and J. McGregor, Embedding Iterates of Analytic Functions with Two Fixed Points into Continuous Groups. Trans. Amer. Math. Soc. 132, 137–145 (1968).

    Article  MATH  MathSciNet  Google Scholar 

  18. C.C. Cowen, Iteration and the Solution of Functional Equations for Functions Analytic in the Unit Disk. Trans Amer. Math. Soc. 265, 69–95 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  19. A.G. Siskakis, Weighted composition semigroups on Hardy spaces. Linear Algebra Appl. 84, 359–371 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  20. C.C. Cowen and B.D. MacCluer, Linear Fractional Maps of the Ball and Their Composition Operators. Acta Sci. Math. (Szeged) 66, 351–376 (2000).

    MATH  MathSciNet  Google Scholar 

  21. D. Alpay and V. Khatskevich, Linear Fractional Transformations: Basic Properties, Applications to Spaces of Analytic Functions and Schröder’s Equation. Internat. J. Appl. Math. 2, 459–476 (2000).

    MATH  MathSciNet  Google Scholar 

  22. Maria J. Martin, Composition Operators with Linear Fractional Symbols and Their Adjoints. in First Advanced Course in Operator Theory and Complex Analysis. University of Seville, June 2004.

    Google Scholar 

  23. T.Ya. Azizov, A.I. Barsukov, and A. Dijksma, The Cauchy Problem Associated with an (ω,W)-Dissipative Operator. Methods of Funct. Anal. and Topology 10 (3), 1–6 (2004).

    MATH  MathSciNet  Google Scholar 

  24. T.Ya. Azizov, A.I. Barsukov, and A. Dijksma, Decompositions of a Krein Space in Regular Subspaces Invariant under a Uniformly Bounded C 0-Semigroup of Bicontractions. J. of Funct. Anal. 211 (2), 324–354 (2004).

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. ORT Braude Academic College, College Campus, P.O. Box 78, Karmiel, 21982, Israel

    V. A. Khatskevich

  2. Pyatnitskoe highway 23-2-156, Moscow, 125430, Russia

    V. A. Senderov

Authors
  1. V. A. Khatskevich
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  2. V. A. Senderov
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Editor information

Editors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Jussi Behrndt  & Karl-Heinz Förster  & 

  2. Institut für Mathematik, Technische Universität Ilmenau, Postfach 100565, 98684, Ilmenau, Germany

    Carsten Trunk

Additional information

Dedicated to Peter Jonas

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© 2009 Birkhäuser Verlag Basel/Switzerland

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Khatskevich, V.A., Senderov, V.A. (2009). The Königs Problem and Extreme Fixed Points. In: Behrndt, J., Förster, KH., Trunk, C. (eds) Recent Advances in Operator Theory in Hilbert and Krein Spaces. Operator Theory: Advances and Applications, vol 198. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0180-1_12

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