Linear dynamics of the adjoint of a unilateral weighted shift operator

TL;DR

This paper investigates the dynamics of the adjoint of weighted shift operators on specific function spaces, providing conditions for continuity, spectral properties, and chaotic behavior, with new insights into their hypercyclicity and periodic vectors.

Contribution

It offers new criteria for the continuity and spectral analysis of adjoint weighted shift operators and explores their chaotic dynamics on specialized function spaces.

Findings

Operators similar to compact perturbations of weighted shifts

Conditions for hypercyclicity, mixing, and chaos of the adjoint

Existence of non-trivial periodic vectors without hypercyclicity

Abstract

This paper is a sequel to our work in \cite{Das-Mundayadan}. Here, we primarily study the dynamics of the adjoint of a weighted forward shift operator $F_w$ on the analytic function space $\ell^p_{a,b}$ having a normalized Schauder basis of the form $\{(a_n+b_nz)z^n:~n \geq 0\}$. We obtain sufficient conditions for $F_w$ to be continuous, and show, under certain conditions, that the operator $F_w$ is similar to a compact perturbation of a weighted forward shift on $\ell^p(\mathbb{N}_0)$. This also allows us to obtain the essential spectrum of $F_w$. Further, we study when adjoint $F_w^*$ is hypercyclic, mixing, and chaotic, and provide a class of chaotic operators that are compact perturbations of weighted shifts on $\ell^p(\mathbb{N}_0)$. Finally, it is proved that the adjoint of a shift on the dual of $\ell^p_{a,b}$ can have non-trivial periodic vectors, without being even hypercyclic. Also, the zero-one law of orbital limit points fails for $F_w^*$.