${\mathcal F}$-Hypercyclic operators on Fr\' echet spaces

TL;DR

This paper explores a new class of hypercyclic operators called ${\mathcal F}$-hypercyclicity on Fr\'echet spaces, introducing lower $(m_{n})$-hypercyclicity and examining $q$-frequent hypercyclicity with numerous examples and open problems.

Contribution

It introduces the concept of lower $(m_{n})$-hypercyclicity and studies $q$-frequent hypercyclicity, expanding the understanding of hypercyclic operators on Fr\'echet spaces.

Findings

Introduction of lower $(m_{n})$-hypercyclicity concept.

Analysis of $q$-frequent hypercyclicity for $q\geq 1$.

Provision of many examples and open problems.

Abstract

In this paper, we investigate ${\mathcal F}$-hypercyclicity of linear, not necessarily continuous, operators on Fr\' echet spaces. The notion of lower $(m_{n})$-hypercyclicity seems to be new and not considered elsewhere even for linear continuous operators acting on Fr\' echet spaces. We pay special attention to the study of $q$-frequent hypercyclicity, where $q\geq 1$ is an arbitrary real number. We present several new concepts and results for lower and upper densities in a separate section, providing also a great number of illustrative examples and open problems.