Open Access

Compact Weighted Composition Operators and Fixed Points in Convex Domains

Fixed Point Theory and Applications20072007:028750

DOI: 10.1155/2007/28750

Received: 18 April 2007

Accepted: 24 June 2007

Published: 17 October 2007


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.


Authors’ Affiliations

Department of Mathematics, University of California


  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


© Dana D. Clahane. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.