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We introduce the Banach spaces $\ell^p_{a,b}$ and $c_{0,a,b}$, of analyticfunctions on the unit disc, having normalized Schauder bases consisting ofpolynomials of the form $f_n(z)=(a_n+b_nz)z^n, ~~n\geq0$, where $\{f_n\}$ isassumed to be equivalent to the standard basis in $\ell^p$ and $c_0$,respectively. We study the weighted backward shift operator $B_w$ on thesespaces, and obtain necessary and sufficient conditions for $B_w$ to be bounded,and prove that, under some mild assumptions on $\{a_n\}$ and $\{b_n\}$, theoperator $B_w$ is similar to a compact perturbation of a weighted backwardshift on the sequence spaces $\ell^p$ or $c_0$. Further, we study thehypercyclicity, mixing, and chaos of $B_w$, and establish the existence ofhypercyclic subspaces for $B_w$ by computing its essential spectrum. Similarresults are obtained for a function of $B_w$ on $\ell^p_{a,b}$ and $c_{0,a,b}$.

Dynamics of weighted backward shifts on certain analytic function spaces