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HYPERCYCLICITY FOR TRANSLATIONS THROUGH RUNGE'S THEOREM

  • Hallack Andre Arbex (Departamento de Matematica Universidade Federal de Juiz de Fora)
  • Published : 2007.02.28

Abstract

In this paper, we first adapt Runge's Theorem to work on certain domains in any complex Banach space. Then, using this result, we extend Birkhoff's Theorem on the hypercyclicity of translations on $H(\mathbb{C})$ and Costakis' and Sambarino's result on the existence of common hypercyclic functions for uncountable families of translations on $H(\mathbb{C})$ to subs paces of $H_b(E)$ (in some cases all of $H_{b}$(E)), E being in a large class of Banach spaces.

Keywords

hypercyclic operators;hypercyclicity

References

  1. R. Aron, Weakly uniformly continuous and weakly sequentially continuous entire func-tions, Advances in holomorphy (Proc. Sem. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), North-Holland Math. Studies, 34. North-Holland, Amsterdam (1979), 47-66
  2. R. Aron and D. Markose, On universal functions, J. Korean Math. Soc. 41 (2004), no. 1, 65-76 https://doi.org/10.4134/JKMS.2004.41.1.065
  3. G. Birkhoff, Demonstration d'un theoreme elementaire sur les fonctions entieres, Cr. R. Acad. Sci. Paris 189 (1929), 473-475
  4. G. Costakis and M. Sambarino, Genericity of wild holomorphic functions and common hypercyclic vectors, Adv. Math. 182 (2004), no. 2, 278-306 https://doi.org/10.1016/S0001-8708(03)00079-3
  5. 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
  6. J. Mujica, Complex Analysis in Banach Spaces, North-Holland Mathematics Studies 120, North-Holland Publishing Co., 1986
  7. J. Conway, A course in functional analysis, Graduate Texts in Mathematics, 96. Springer-Verlag, New York, 1990