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http://dx.doi.org/10.4134/JKMS.2011.48.5.969

HYPERCYCLICITY OF WEIGHTED COMPOSITION OPERATORS ON THE UNIT BALL OF ℂN  

Chen, Ren-Yu (Department of Mathematics Tianjin University)
Zhou, Ze-Hua (Department of Mathematics Tianjin University)
Publication Information
Journal of the Korean Mathematical Society / v.48, no.5, 2011 , pp. 969-984 More about this Journal
Abstract
This paper discusses the hypercyclicity of weighted composition operators acting on the space of holomorphic functions on the open unit ball $B_N$ of $\mathbb{C}^N$. Several analytic properties of linear fractional self-maps of $B_N$ are given. According to these properties, a few necessary conditions for a weighted composition operator to be hypercyclic in the space of holomorphic functions are proved. Besides, the hypercyclicity of adjoint of weighted composition operators are studied in this paper.
Keywords
hypercyclic operator; weighted composition operator; linear fractional map; generalized Cayley transform; Heisenberg transform; Denjoy-Wollf point;
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1 B. Yousefi and H. Rezaei, Some necessary and sufficient conditions for Hypercyclicity Criterion, Proc. Indian Acad. Sci. Math. Sci. 115 (2005), no. 2, 209-216.   DOI
2 K. Zhu, Spaces of Holomorphic functions in the Unit Ball, Graduate Texts in Mathematics, 226. Springer-Verlag, New York, 2005.
3 N. S. Feldman, The dynamics of cohyponormal operators, Trends in Banach spaces and operator theory (Proc. Conf., Memphis, TN, 2001), 71-85.
4 R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), no. 2, 281-288.   DOI   ScienceOn
5 G. Godefroy and J. H. Shapiro, Operators with dense invariant cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229-269.   DOI
6 C. Kitai, Invariant closed sets for linear operators, Thesis, Univ. of Toronto, 1982.
7 K. G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345-381.   DOI
8 K. G. Grosse-Erdmann, Recent developments in hypercyclicity, Rev. R. Acad. Cien. Serie A. Mat. 97 (2003), no. 2, 273-286.
9 L. Jiang and C. Ouyang, Cyclic behavior of linear fractional composition operators in the unit ball of ${\mathbb{C}^N$, J. Math. Anal. Appl. 341 (2008), no. 1, 601-612.   DOI   ScienceOn
10 B. D. MacCluer, Iterates of holomorphic self-maps of the unit ball in ${\mathbb{C}^N$, Michigan Math. J. 30 (1983), no. 1, 97-106.   DOI
11 S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22.
12 W. Rudin, Function Theory in the Unit Ball of ${\mathbb{C}^n$, Grundlehren Math. Wiss., vol. 241, Springer-Verlag, New York, 1980.
13 H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), no. 3, 993-1004.   DOI   ScienceOn
14 B. Yousefi and H. Rezaei, Hypercyclic property of weighted composition operators, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3263-3271.   DOI   ScienceOn
15 F. Bayart, A class of linear fractional maps of the ball and their composition operators, Adv. Math. 209 (2007), no. 2, 649-665.   DOI   ScienceOn
16 F. Bracci, M. D. Contreras, and S. Diaz-Madrigal, Classification of semigroups of linear fractional maps in the unit ball, Adv. Math. 208 (2007), no. 1, 318-350.   DOI   ScienceOn
17 C. Bisi and F. Bracci, Linear fractional maps of the unit ball: A geometric study, Adv. Math. 167 (2002), no. 2, 265-287.   DOI   ScienceOn
18 P. S. Bourdon and J. H. Shapiro, Cyclic composition operators on $H^2$, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988), 43-53, Proc. Sympos. Pure Math., 51, Part 2, Amer. Math. Soc., Providence, RI, 1990.
19 P. S. Bourdon and J. H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125 (1997), no. 596, x+105 pp.
20 K. C. Chan and J. H. Shapiro, The cyclic behavior of translation operators on Hilbert spaces of entire functions, Indiana Univ. Math. J. 40 (1991), no. 4, 1421-1449.   DOI
21 X. Chen, G. Cao, and K. Guo, Inner functions and cyclic composition operators on $H^2(B_n)$, J. Math. Anal. Appl. 250 (2000), no. 2, 660-669.   DOI   ScienceOn
22 J. B. Conway, Function of One Complex Variable, Springer-Verlag, 1973.
23 C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Roton, 1995.
24 C. C. Cowen and B. D. MacCluer, Linear fractional maps of the ball and their composition operators, Acta Sci. Math. (Szeged) 66 (2000), no. 1-2, 351-376.