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Remarks on multiples of distributionally chaotic operators

Volume 243 / 2018

Zongbin Yin, Yu Huang Studia Mathematica 243 (2018), 25-52 MSC: Primary 47A16; Secondary 47B37, 54H20. DOI: 10.4064/sm170222-31-7 Published online: 2 March 2018

Abstract

This paper investigates distributional chaos and the existence of common distributionally irregular vectors for multiples of linear operators on Banach spaces. We focus on the topological property of the set $DC_T : = \{\lambda \gt 0: \lambda T$ is distributionally chaotic$\}$ for a given operator $T$. For any open set $U \subset (0, \infty )$ which is bounded away from zero, we prove that there is a bounded operator $T$ on $l^{p} (1\leq p \lt \infty )$ such that $U= DC_T$. As a consequence, there exists an operator $T_1$ such that $T_1$ and $3 T_1$ are distributionally chaotic but $2 T_1$ is not. We also construct an invertible operator $T$ such that $DC_T$ is a singleton. Furthermore, sufficient conditions for the existence of common distributionally irregular vectors for the family of operators $\{\lambda T : \lambda \in DC_T\}$ are provided.

Authors

  • Zongbin YinSchool of Mathematics
    Sun Yat-sen University
    510275 Guangzhou, P. R. China
    e-mail
  • Yu HuangSchool of Mathematics
    Sun Yat-sen University
    510275 Guangzhou, P. R. China
    e-mail

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