Dynamics of Weighted Backward Shifts on Certain Analytic Function Spaces

Abstract

We introduce the Banach spaces \(\ell ^p_{a,b}\) and \(c_{0,a,b}\), of analytic functions on the unit disc, having normalized Schauder bases consisting of polynomials of the form \(f_n(z)=(a_n+b_nz)z^n, ~~n\ge 0\), where \(\{f_n\}\) is assumed to be equivalent to the standard basis in \(\ell ^p\) and \(c_0\), respectively. We study the weighted backward shift operator \(B_w\) on these spaces, and obtain necessary and sufficient conditions for \(B_w\) to be bounded, and prove that, under some mild assumptions on \(\{a_n\}\) and \(\{b_n\}\), the operator \(B_w\) is similar to a compact perturbation of a weighted backward shift on the sequence spaces \(\ell ^p\) or \(c_0\). Further, we study the hypercyclicity, mixing, and chaos of \(B_w\), and establish the existence of hypercyclic subspaces for \(B_w\) by computing its essential spectrum. Similar results are obtained for a function of \(B_w\) on \(\ell ^p_{a,b}\) and \(c_{0,a,b}\).