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Supercyclicity of weighted composition operators on spaces of continuous functions

  • M. J. Beltrán-Meneu
  • E. JordáEmail author
  • M. Murillo-Arcila
Article
  • 33 Downloads

Abstract

Our study is focused on the dynamics of weighted composition operators defined on a locally convex space \(E\hookrightarrow (C(X),\tau _p)\) with X being a topological Hausdorff space containing at least two different points and such that the evaluations \(\{\delta _x:\ x\in X\}\) are linearly independent in \(E'\). We prove, when X is compact and E is a Banach space containing a nowhere vanishing function, that a weighted composition operator \(C_{w,\varphi }\) is never weakly supercyclic on E. We also prove that if the symbol \(\varphi \) lies in the unit ball of \(A(\mathbb {D})\), then every weighted composition operator can never be \(\tau _p\)-supercyclic neither on \(C(\mathbb {D})\) nor on the disc algebra \(A(\mathbb {D})\). Finally, we obtain Ansari–Bourdon type results and conditions on the spectrum for arbitrary weakly supercyclic operators, and we provide necessary conditions for a composition operator to be weakly supercyclic on the space of holomorphic functions defined in non necessarily simply connected planar domains. As a consequence, we show that no composition operator can be weakly supercyclic neither on the space of holomorphic functions on the punctured disc nor in the punctured plane.

Keywords

Weighted composition operator Weak supercyclicity Disc algebra Space of holomorphic functions 

Mathematics Subject Classification

47A16 47B33 46E15 

Notes

Acknowledgements

The authors are very thankful to the referee for his/her careful reading of the manuscript and his/her valuable comments and observations. The first and the second author were supported by MEC, MTM2016-76647-P. The third author was supported by MEC, MTM2016-75963-P and GVA/2018/110.

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Copyright information

© Universitat de Barcelona 2019

Authors and Affiliations

  1. 1.Universitat Jaume ICastellón de la PlanaSpain
  2. 2.Universitat Politècnica de ValènciaValenciaSpain

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