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\(\Gamma \)-Supercyclicity for Strongly Continuous Semigroups

Abstract

We characterize the subsets \(\Gamma \) of the complex plane \(\mathbb {C}\) for which the notion of \(\Gamma \)-supercyclicity for strongly continuous semigroups coincides with that of hypercyclicity. In addition, we characterize the sets \(\Gamma \subset {\mathbb {C}}\) such that for every strongly continuous semigroup of operators \({\mathcal {T}}=(T_t)_{t\geqslant 0}\) on a Banach space X, a vector \(x\in X\) is hypercyclic for \({\mathcal {T}}\) if and only if \({\mathrm {Orb}}(\Gamma x,\mathcal {T})\) is somewhere dense in X. We derive a characterization of \(C_0\)-semigroup versions of 1-dimensional hypercyclic subsets and Bourdon–Feldman subsets. We finally prove a multi-\(\Gamma \)-supercyclicity result for \(C_0\)-semigroups.

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Acknowledgements

I gratefully acknowledge many helpful suggestions, comments and remarks of Stéphane Charpentier and Evgeny Abakumov during the preparation of the paper. I also would like to thank Romuald Ernst for reading this paper and for his support. I am grateful to the referee for helpful comments and suggestions.

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Correspondence to Arafat Abbar.

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Communicated by Isabelle Chalendar.

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Abbar, A. \(\Gamma \)-Supercyclicity for Strongly Continuous Semigroups. Complex Anal. Oper. Theory 13, 3923–3942 (2019). https://doi.org/10.1007/s11785-019-00941-y

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Keywords

  • \(C_0\)-semigroup
  • Hypercyclicity
  • Supercyclicity

Mathematics Subject Classification

  • 47A16