On linear chaos in the space of convergent sequences

TL;DR

The paper investigates the nature of linear chaos in the space of convergent sequences, showing that it cannot be achieved by simple extensions of known operators, and introduces new chaotic operators with detailed spectral analysis.

Contribution

It introduces a new sufficient condition for linear chaos and constructs novel chaotic operators in the space of convergent sequences.

Findings

Linear chaos cannot be obtained by extending weighted backward shifts from $c_0$ to $c$.

New chaotic operators in $c( N)$ are constructed via conjugation with operators in $c_0( Z_+)$.

Spectral structures of the constructed chaotic operators are explicitly characterized.

Abstract

We show that linear chaos in the space $c(\mathbb{N})$ of convergent sequences cannot be arrived at by merely extending the weighted backward shifts in the space $c_0(\mathbb{N})$ of vanishing sequences. Applying a newly found sufficient condition for linear chaos, we furnish concise proofs of the chaoticity of the foregoing operators along with their powers and also itemize their spectral structure. We further construct bounded and unbounded linear chaotic operators in $c(\mathbb{N})$ as conjugate to the chaotic backward shifts in $c_0(\mathbb{Z}_+)$ via a homeomorphic isomorphism between the two spaces.