  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 $f_n(z)=(a_n+b_nz)z^n, ~~n\geq0$, 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 $c_{0,a,b}$.
