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STRONG HYPERCYCLICITY OF BANACH SPACE OPERATORS

  • Received : 2019.12.18
  • Accepted : 2020.07.17
  • Published : 2021.01.01

Abstract

A bounded linear operator T on a separable infinite dimensional Banach space X is called strongly hypercyclic if $$X{\backslash}\{0\}{\subseteq}{\bigcup_{n=0}^{\infty}}T^n(U)$$ for all nonempty open sets U ⊆ X. We show that if T is strongly hypercyclic, then so are Tn and cT for every n ≥ 2 and each unimodular complex number c. These results are similar to the well known Ansari and León-Müller theorems for hypercyclic operators. We give some results concerning multiplication operators and weighted composition operators. We also present a result about the invariant subset problem.

Keywords

Acknowledgement

We are very grateful to the referees for their careful reading, suggestions and corrections which improved the quality of our paper.

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