Linear dynamics of the adjoint of a unilateral weighted shift operator

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 the 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 $F_w^*$, which means that, under certain conditions, the adjoint $F_w^*$ is non-hypercyclic, but it has an orbit possessing non-zero norm limit points.