A class of bilateral weighted shift operators, and linear dynamics

Abstract

This article aims to initiate a study of bilateral weighted backward shift operators defined on the spaces $\ell^p_{a,b}(\Omega_{r,R})$ and $c_{0,a,b}(\Omega_{r,R})$ which are Banach spaces of analytic functions on a suitable annulus in the complex plane, having a normalized Schauder basis of the form, $$ f_n(z):= (a_n+b_{n}z)z^{n},\hskip 0.5cm n\in \mathbb{Z}. $$ We obtain necessary and sufficient conditions for a weighted shift $B_w$ to be bounded, and find conditions so that $B_w$ is similar to a compact perturbation of a weighted shift on $\ell^p(\mathbb{Z})$. In addition, we study when $B_w$ is hypercyclic, supercyclic, and chaotic. It shown that the zero-one law of orbital limit points does not hold for $B_w$, which is in contrast to the case of weighted shifts on $\ell^p(\mathbb{Z})$. Most of our results are obtained using the matrix form of $B_w$.