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Compact Weighted Composition Operators and Fixed Points in Convex Domains
Fixed Point Theory and Applications volume 2007, Article number: 028750 (2007)
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.
References
Caughran JG, Schwartz HJ: Spectra of compact composition operators. Proceedings of the American Mathematical Society 1975,51(1):127–130. 10.1090/S0002-9939-1975-0377579-7
Cowen CC, MacCluer BD: Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, Fla, USA; 1995:xii+388.
MacCluer BD: Spectra of compact composition operators on . Analysis 1984,4(1–2):87–103.
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-z
Gunatillake G: Spectrum of a compact weighted composition operator. Proceedings of the American Mathematical Society 2007,135(2):461–467. 10.1090/S0002-9939-06-08497-8
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.2004.11.014
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)00215-5
Hammond C: On the norm of a composition operator, Ph.D. thesis. University of Virginia, Charlottesville, Va, USA; 2003.
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.
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.
Conway JB: A Course in Functional Analysis, Graduate Texts in Mathematics. Volume 96. 2nd edition. Springer, New York, NY, USA; 1990:xvi+399.
Krantz SG: Geometric Analysis and Function Spaces, CBMS Regional Conference Series in Mathematics. Volume 81. American Mathematical Society, Washington, DC, USA; 1993:xii+202.
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.
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.
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Clahane, D.D. Compact Weighted Composition Operators and Fixed Points in Convex Domains. Fixed Point Theory Appl 2007, 028750 (2007). https://doi.org/10.1155/2007/28750
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DOI: https://doi.org/10.1155/2007/28750