On hypercyclicity and linear chaos in a nonclassical sequence space and beyond

TL;DR

This paper investigates hypercyclicity and chaos of weighted backward shifts in a nonclassical sequence space related to l_1, constructing new operators with these properties and analyzing their spectral structure.

Contribution

It introduces new hypercyclic and chaotic operators in a nonclassical sequence space and l_1, expanding understanding of linear dynamics in these spaces.

Findings

Constructed new bounded and unbounded hypercyclic operators

Identified operators that are hypercyclic but not chaotic

Analyzed spectral properties of these operators

Abstract

We analyze the hypercyclicity, chaoticity, and spectral structure of (bounded and unbounded) weighted backward shifts in a nonclassical sequence space, which the space $l_1$ of summable sequences is both isometrically isomorphic to and continuously and densely embedded into. Based on the weighted backward shifts, we further construct new bounded and unbounded linear hypercyclic and chaotic operators both in the nonclassical sequence space and the classical space $l_1$, including those that are hypercyclic but not chaotic.