J-class sequences of linear operators

TL;DR

This paper introduces the extended limit set for sequences of linear operators on Banach spaces and explores its connection to hypercyclicity and topological transitivity.

Contribution

It defines the extended limit set for operator sequences and establishes its relationship with hypercyclicity and topological transitivity in linear dynamics.

Findings

Extended limit set characterizes hypercyclicity of operator sequences.

Strong relation between extended limit set and topological transitivity.

Hypercyclicity is equivalent to the existence of a cyclic vector with full extended limit set.

Abstract

In this paper we first introduce the extended limit set $J_{\{T^n\}}(x)$ for a sequence of bounded linear operators $\{T_n\}_{n=1}^{\infty}$ on a separable Banach space $X$ . Then we study the dynamics of sequence of linear operators by using the extended limit set. It is shown that the extended limit set is strongly related to the topologically transitive of a sequence of linear operators. Finally we show that a sequence of operators $\{T_n\}_{n=1}^{\infty}\subseteq \mathcal{B}(X)$ is hypercyclic if and only if there exists a cyclic vector $x\in X$ such that $J_{\{T^n\}}(x)=X$.