Luh hypercyclic vector for composition operator

TL;DR

This paper introduces Luh hypercyclic vectors for composition operators, demonstrating their dense existence and exploring the dynamics of cosine operator functions on various Banach spaces, with conditions for supercyclicity and concrete examples.

Contribution

It constructs Luh hypercyclic vectors for composition operators and analyzes the supercyclicity of cosine operator functions on Orlicz and Morrey spaces, providing new insights and examples.

Findings

Dense linear manifold of Luh hypercyclic vectors exists.

Sufficient conditions for supercyclicity of cosine operator functions are established.

Concrete examples of weighted translations satisfying supercyclicity conditions.

Abstract

In this paper, we deal with the construction of holomorphic functions on a simply connected domain satisfying that all its derivatives and antiderivatives under a composition operator have a dense orbit. Such functions will be called Luh hypercyclic vectors for the respective composition operator. We show that there is a dense linear manifold of Luh hypercyclic vectors. Moreover, we study the dynamics of cosine operator function generated by weighted composition operators on solid Banach function spaces, in particular on Orlicz and Morrey spaces, and we give sufficient conditions for supercyclicity of such cosine operator functions in terms of the corresponding weight function. Also, we give concrete examples of weighted translations satisfying these sufficient conditions.