Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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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)
  • Received : 2010.02.26
  • Published : 2011.09.01
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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.

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References

  1. F. Bayart, A class of linear fractional maps of the ball and their composition operators, Adv. Math. 209 (2007), no. 2, 649-665. https://doi.org/10.1016/j.aim.2006.05.012
  2. C. Bisi and F. Bracci, Linear fractional maps of the unit ball: A geometric study, Adv. Math. 167 (2002), no. 2, 265-287. https://doi.org/10.1006/aima.2001.2042
  3. 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.
  4. P. S. Bourdon and J. H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125 (1997), no. 596, x+105 pp.
  5. 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. https://doi.org/10.1016/j.aim.2006.02.010
  6. 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. https://doi.org/10.1512/iumj.1991.40.40064
  7. 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. https://doi.org/10.1006/jmaa.2000.7091
  8. J. B. Conway, Function of One Complex Variable, Springer-Verlag, 1973.
  9. C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Roton, 1995.
  10. 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.
  11. N. S. Feldman, The dynamics of cohyponormal operators, Trends in Banach spaces and operator theory (Proc. Conf., Memphis, TN, 2001), 71-85.
  12. 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. https://doi.org/10.1090/S0002-9939-1987-0884467-4
  13. G. Godefroy and J. H. Shapiro, Operators with dense invariant cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229-269. https://doi.org/10.1016/0022-1236(91)90078-J
  14. K. G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345-381. https://doi.org/10.1090/S0273-0979-99-00788-0
  15. K. G. Grosse-Erdmann, Recent developments in hypercyclicity, Rev. R. Acad. Cien. Serie A. Mat. 97 (2003), no. 2, 273-286.
  16. 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. https://doi.org/10.1016/j.jmaa.2007.10.031
  17. C. Kitai, Invariant closed sets for linear operators, Thesis, Univ. of Toronto, 1982.
  18. 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. https://doi.org/10.1307/mmj/1029002792
  19. S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22.
  20. W. Rudin, Function Theory in the Unit Ball of ${\mathbb{C}^n$, Grundlehren Math. Wiss., vol. 241, Springer-Verlag, New York, 1980.
  21. H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), no. 3, 993-1004. https://doi.org/10.2307/2154883
  22. B. Yousefi and H. Rezaei, Hypercyclic property of weighted composition operators, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3263-3271. https://doi.org/10.1090/S0002-9939-07-08833-8
  23. 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. https://doi.org/10.1007/BF02829627
  24. K. Zhu, Spaces of Holomorphic functions in the Unit Ball, Graduate Texts in Mathematics, 226. Springer-Verlag, New York, 2005.

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