Linear dynamics of the adjoint of a unilateral weighted shift operator

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 ${\mathrm{\ell }}_{a,b}^p$ having a normalized Schauder basis of the form $\{(a_n+b_{n}z)z^n:n\ge 0\}$. 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 ${\mathrm{\ell }}^p({\mathbb{N}}_0)$. This also allows us to obtain the essential spectrum of $F_w$. Further, we study when the adjoint $F_w^{\ast }$ is hypercyclic, mixing, and chaotic, and provide a class of chaotic operators that are compact perturbations of weighted shifts on ${\mathrm{\ell }}^p({\mathbb{N}}_0)$. Finally, it is proved that the adjoint of a shift on the dual of ${\mathrm{\ell }}_{a,b}^p$ can have non-trivial periodic vectors, without being even hypercyclic. Also, the zero-one law of orbital limit points fails for $F_w^{\ast }$, which means that, under certain conditions, the adjoint $F_w^{\ast }$ is non-hypercyclic, but it has an orbit possessing non-zero norm limit points.