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Powers of Hypercyclic Functions for Some Classical Hypercyclic Operators

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Abstract.

We show that no power of any entire function is hypercyclic for Birkhoff’s translation operator on \(\mathcal{H}(\mathbb{C})\). On the other hand, we see that the set of functions whose powers are all hypercyclic for MacLane’s differentiation operator is a Gδ-dense subset of \(\mathcal{H}(\mathbb{C})\).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Kent, OH, 44242, USA

    R. M. Aron

  2. Departament de Matemàtica Aplicada and IMPA-UPV, F. Informàtica, Universitat Politècnica de València, E-46022, València, Spain

    J. A. Conejero

  3. Departament de Matemàtica Aplicada and IMPA-UPV, E.T.S. Arquitectura, Universitat Politècnica de València, E-46022, València, Spain

    A. Peris

  4. Facultad de Ciencias Matemáticas, Departamento de Análisis Matemático, Universidad Complutense de Madrid, Plaza de las Ciencias 3, E-28040, Madrid, Spain

    J. B. Seoane–Sepúlveda

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  1. R. M. Aron
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  2. J. A. Conejero
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  3. A. Peris
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  4. J. B. Seoane–Sepúlveda
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Corresponding author

Correspondence to A. Peris.

Additional information

The second author was supported in part by GVA, grant CTESPP/2005.

The second and third authors were supported in part by MEC and FEDER, Project MTM2004-02262 and Research Net MTM2006-26627-E.

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Aron, R.M., Conejero, J.A., Peris, A. et al. Powers of Hypercyclic Functions for Some Classical Hypercyclic Operators. Integr. equ. oper. theory 58, 591–596 (2007). https://doi.org/10.1007/s00020-007-1490-4

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  • DOI: https://doi.org/10.1007/s00020-007-1490-4

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