The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle

Dariusz Partyka

Banach Center Publications (1995)

  • Volume: 31, Issue: 1, page 303-310
  • ISSN: 0137-6934

Abstract

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This paper provides sufficient conditions on a quasisymmetric automorphism γ of the unit circle which guarantee the existence of the smallest positive eigenvalue of γ. They are expressed by means of a regular quasiconformal Teichmüller self-mapping φ of the unit disc Δ. In particular, the norm of the generalized harmonic conjugation operator A γ : ℍ → ℍ is determined by the maximal dilatation of φ. A characterization of all eigenvalues of a quasisymmetric automorphism γ in terms of the smallest positive eigenvalue of some other quasisymmetric automorphism σ is given.

How to cite

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Partyka, Dariusz. "The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle." Banach Center Publications 31.1 (1995): 303-310. <http://eudml.org/doc/262865>.

@article{Partyka1995,
abstract = {This paper provides sufficient conditions on a quasisymmetric automorphism γ of the unit circle which guarantee the existence of the smallest positive eigenvalue of γ. They are expressed by means of a regular quasiconformal Teichmüller self-mapping φ of the unit disc Δ. In particular, the norm of the generalized harmonic conjugation operator $A_γ:ℍ → ℍ$ is determined by the maximal dilatation of φ. A characterization of all eigenvalues of a quasisymmetric automorphism γ in terms of the smallest positive eigenvalue of some other quasisymmetric automorphism σ is given.},
author = {Partyka, Dariusz},
journal = {Banach Center Publications},
keywords = {quasisymmetric automorphisms; harmonic conjugation operator; quasiconformal mappings; eigenvalues and spectral values of a linear operator; Teichmüller mappings; quasisymmetric automorphism; Teichmüller mapping},
language = {eng},
number = {1},
pages = {303-310},
title = {The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle},
url = {http://eudml.org/doc/262865},
volume = {31},
year = {1995},
}

TY - JOUR
AU - Partyka, Dariusz
TI - The smallest positive eigenvalue of a quasisymmetric automorphism of the unit circle
JO - Banach Center Publications
PY - 1995
VL - 31
IS - 1
SP - 303
EP - 310
AB - This paper provides sufficient conditions on a quasisymmetric automorphism γ of the unit circle which guarantee the existence of the smallest positive eigenvalue of γ. They are expressed by means of a regular quasiconformal Teichmüller self-mapping φ of the unit disc Δ. In particular, the norm of the generalized harmonic conjugation operator $A_γ:ℍ → ℍ$ is determined by the maximal dilatation of φ. A characterization of all eigenvalues of a quasisymmetric automorphism γ in terms of the smallest positive eigenvalue of some other quasisymmetric automorphism σ is given.
LA - eng
KW - quasisymmetric automorphisms; harmonic conjugation operator; quasiconformal mappings; eigenvalues and spectral values of a linear operator; Teichmüller mappings; quasisymmetric automorphism; Teichmüller mapping
UR - http://eudml.org/doc/262865
ER -

References

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  1. [A] L. V. Ahlfors, Lectures on Quasiconformal Mappings, D. Van Nostrand, Princeton, 1966. 
  2. [BS] S. Bergman and M. Schiffer, Kernel functions and conformal mapping, Compositio Math. 8 (1951), 205-249. Zbl0043.08403
  3. [B1] B. Bojarski, Homeomorphic solution of Beltrami systems, Dokl. Akad. Nauk SSSR 102 (1955), 661-664 (in Russian). 
  4. [B2] B. Bojarski, Generalized solutions of a system of differential equations of the first order and elliptic type with discontinuous coefficients, Mat. Sb. N.S. 43 (1957), 451-503 (in Russian). 
  5. [G] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. Zbl0469.30024
  6. [K] J. G. Krzyż, Quasicircles and harmonic measure, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), 19-24. Zbl0563.30016
  7. [KP] J. G. Krzyż and D. Partyka, Generalized Neumann-Poincaré operator, chord-arc curves and Fredholm eigenvalues, Complex Variables, Theory Appl. 21 (1993), 253-263. Zbl0793.30031
  8. [Kü] R. Kühnau, Wann sind die Grunskyschen Koeffizientenbedingungen hinreichend für Q-quasikonforme Fortsetzbarkeit?, Comment. Math. Helv. 61 (1986), 290-307. Zbl0605.30023
  9. [L] O. Lehto, Univalent Functions and Teichmüller Spaces, Graduate Texts in Math. 109 Springer, New York, 1987. Zbl0606.30001
  10. [LV] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, 2nd ed., Grundlehren Math. Wiss. 126, Springer, New York, 1973. Zbl0267.30016
  11. [Ł] K] J. Ławrynowicz and J. G. Krzyż, Quasiconformal Mappings in the Plane: Parametrical Methods, Lecture Notes in Math. 978, Springer, Berlin, 1983. 
  12. [P1] D. Partyka, Spectral values of a quasicircle, Complex Variables Theory Appl., to appear. Zbl0708.30022
  13. [P2] D. Partyka, Generalized harmonic conjugation operator, Proceedings of the Fourth Finnish-Polish Summer School in Complex Analysis at Jyväskylä, Ber. Univ. Jyväskylä Math. Inst. 55 (1993), 143-155. 
  14. [P3] D. Partyka, Spectral values and eigenvalues of a quasicircle, preprint. Zbl0826.30014
  15. [S] M. Schiffer, The Fredholm eigenvalues of plane domains, Pacific J. Math. 7 (1957), 1187-1225. Zbl0138.30003
  16. [St1] K. Strebel, Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises I, Comment. Math. Helv. 36 (1962), 306-323. 
  17. [St2] K. Strebel, Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises II, ibid. 39 (1964), 77-89. 

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