Dynamics of Weighted Backward Shifts on Certain Analytic Function Spaces

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