Hypercyclicity of weighted translations on locally compact Hausdorff spaces

Abstract

Let G be a locally compact second countable Hausdorff space with a positive regular Borel measure and ω is a weight on G. In this article, we provide necessary and sufficient conditions for the hypercyclic weighted translations acting on the weighted space $L^p(G,\omega )(1\le p<\mathrm{\infty })$  in two different cases. Also, some examples are given to illustrate that the results in the first case generalize the characterizations on hypercyclicity for unilateral weighted shifts studied by Salas [16], and the results in the second case generalize Chen and Chu's work in [8]. Furthermore, we give characterizations of hypercyclicity for adjoint operators of these weighted translations.

Keywords:

• Hypercyclicity
• weighted translation
• locally compact Hausdorff space

2010 Mathematics Subject Classifications:

• 47A16
• 47B38
• 46E15

Acknowledgments

The authors deeply thank the referees for a careful reading and numerous helpful suggestions to improve this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported in part by the National Natural Science Foundation of China [grant numbers 11801402, 11771323].