Weighted Shift operators on spaces of analytic functions over an annulus (2412.05509v2)

Abstract: This article aims to initiate a study of the forward weighted shift operator, denoted by $F_w$, defined on the Banach spaces $\ell^p_{a,b}(\mathtt{A})$ and $c_{0,a,b}(\mathtt{A})$. Here, $\ell^p_{a,b}(\mathtt{A})$ and $c_{0,a.b}(\mathtt{A})$, respectively, are the Banach spaces of analytic functions on a suitable annulus $\mathtt{A}$ in the complex plane, having a normalized Schauder basis of the form ${(a_n+b_nz)z^n, ~~n\in \mathbb{Z}}$, which is equivalent to the standard basis in $\ell^p(\mathbb{Z})$ and $c_0(\mathbb{Z})$. We first obtain necessary and sufficient conditions for $F_w$ to be bounded, and show that, under certain conditions on ${a_n}$ and ${b_n}$, the operator $F_w$ is similar to a compact perturbation of a bilateral weighted shift on $\ell^p(\mathbb{Z})$. This also allows us to obtain the essential spectrum of $F_w$. Further, we study when $F_w$ and its adjoint $F_w^*$ are hypercyclic, supercyclic, and chaotic, and provide a class of chaotic operators that are compact perturbations of weighted shifts on $\ell^p(\mathbb{Z})$. Finally, it is proved that the adjoint of a shift on the dual of $\ell^p_{a,b}(\mathtt{A})$ can have non-trivial periodic vectors, without being even hypercyclic. Also, the zero-one law of orbital limit points fails for $F_w^*$.