Skip to content

Advertisement

  • Research Article
  • Open Access

Compact Weighted Composition Operators and Fixed Points in Convex Domains

Fixed Point Theory and Applications20072007:028750

https://doi.org/10.1155/2007/28750

  • Received: 18 April 2007
  • Accepted: 24 June 2007
  • Published:

Abstract

Let be a bounded, convex domain in , and suppose that is holomorphic. Assume that is analytic, bounded away from zero toward the boundary of , and not identically zero on the fixed point set of . Suppose also that the weighted composition operator given by is compact on a holomorphic, functional Hilbert space (containing the polynomial functions densely) on with reproducing kernel satisfying as . We extend the results of J. Caughran/H. Schwartz for unweighted composition operators on the Hardy space of the unit disk and B. MacCluer on the ball by showing that has a unique fixed point in . We apply this result by making a reasonable conjecture about the spectrum of based on previous results.

Keywords

  • Differential Geometry
  • Composition Operator
  • Computational Biology
  • Convex Domain
  • Weight Composition

[1234567891011121314]

Authors’ Affiliations

(1)
Department of Mathematics, University of California, Riverside, CA 92521, USA

References

  1. Caughran JG, Schwartz HJ: Spectra of compact composition operators. Proceedings of the American Mathematical Society 1975,51(1):127–130. 10.1090/S0002-9939-1975-0377579-7MATHMathSciNetView ArticleGoogle Scholar
  2. Cowen CC, MacCluer BD: Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, Fla, USA; 1995:xii+388.Google Scholar
  3. MacCluer BD: Spectra of compact composition operators on . Analysis 1984,4(1–2):87–103.MATHMathSciNetView ArticleGoogle Scholar
  4. Clahane DD: Spectra of compact composition operators over bounded symmetric domains. Integral Equations and Operator Theory 2005,51(1):41–56. 10.1007/s00020-003-1250-zMATHMathSciNetView ArticleGoogle Scholar
  5. Gunatillake G: Spectrum of a compact weighted composition operator. Proceedings of the American Mathematical Society 2007,135(2):461–467. 10.1090/S0002-9939-06-08497-8MATHMathSciNetView ArticleGoogle Scholar
  6. Clifford JH, Dabkowski MG: Singular values and Schmidt pairs of composition operators on the Hardy space. Journal of Mathematical Analysis and Applications 2005,305(1):183–196. 10.1016/j.jmaa.2004.11.014MATHMathSciNetView ArticleGoogle Scholar
  7. Shapiro JH, Smith W: Hardy spaces that support no compact composition operators. Journal of Functional Analysis 2003,205(1):62–89. 10.1016/S0022-1236(03)00215-5MATHMathSciNetView ArticleGoogle Scholar
  8. Hammond C: On the norm of a composition operator, Ph.D. thesis. University of Virginia, Charlottesville, Va, USA; 2003.Google Scholar
  9. Abate M: Iteration Theory of Holomorphic Maps on Taut Manifolds, Research and Lecture Notes in Mathematics. Complex Analysis and Geometry. Mediterranean Press, Rende, Italy; 1989:xvii+417.Google Scholar
  10. Vigué J-P: Points fixes d'applications holomorphes dans un domaine born convexe de [Fixed points of holomorphic mappings in a bounded convex domain in ]. Transactions of the American Mathematical Society 1985,289(1):345–353.MATHMathSciNetView ArticleGoogle Scholar
  11. Conway JB: A Course in Functional Analysis, Graduate Texts in Mathematics. Volume 96. 2nd edition. Springer, New York, NY, USA; 1990:xvi+399.Google Scholar
  12. Krantz SG: Geometric Analysis and Function Spaces, CBMS Regional Conference Series in Mathematics. Volume 81. American Mathematical Society, Washington, DC, USA; 1993:xii+202.Google Scholar
  13. Vigué J-P: Sur les points fixes d'applications holomorphes [On the fixed points of holomorphic mappings]. Comptes Rendus de l'Académie des Sciences. Série I. Mathématique 1986,303(18):927–930.MATHGoogle Scholar
  14. Huang XJ: A non-degeneracy property of extremal mappings and iterates of holomorphic self-mappings. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1994,21(3):399–419.MATHMathSciNetGoogle Scholar

Copyright

Advertisement