A class of bilateral weighted shift operators, and linear dynamics

This article aims to initiate a study of bilateral weighted backward shift operators defined on the spaces ${\mathrm{\ell }}_{a,b}^p({\mathrm{\Omega }}_{r,R})$ and $c_{0,a,b}({\mathrm{\Omega }}_{r,R})$ which are Banach spaces of analytic functions on a suitable annulus in the complex plane, having a normalized Schauder basis of the form, $$f_n(z):=(a_n+b_{n}z)z^n,n\in \mathbb{Z}\mathrm{.}$$ 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 ${\mathrm{\ell }}^p(\mathbb{Z})$. 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 ${\mathrm{\ell }}^p(\mathbb{Z})$. Most of our results are obtained using the matrix form of $B_w$.