Li–Yorke chaos translation set for linear operators

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Abstract

In order to study Li–Yorke chaos by the scalar perturbation for a given bounded linear operator T on a Banach space X, we introduce the Li–Yorke chaos translation set of T, which is defined by \(S_{LY}(T)=\{\lambda \in {\mathbb {C}};\lambda +T \text { is Li--Yorke chaotic}\}\). In this paper, some operator classes are considered, such as normal operators, compact operators, shift operators, and so on. In particular, we show that the Li–Yorke chaos translation set of the Kalisch operator on the Hilbert space \(\mathcal {L}^2[0,2\pi ]\) is a simple point set \(\{0\}\).

Keywords

Li–Yorke chaos Translation set Shift Kalisch operator 

Mathematics Subject Classification

Primary 47A16 Secondary 37D45 

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Acknowledgements

The authors would like to thank the referee for his/her careful reading of the paper and helpful comments and suggestions.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsJilin UniversityChangchunPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsXidian UniversityXi’anPeople’s Republic of China
  3. 3.School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China

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