  We study the backward shift operator $B$ acting on the Hilbert space of
analytic functions on the unit disc, having an orthonormal basis consisting of
polynomials of the form $f_n(z)=(a_n+b_nz)z^n, ~~n\geq0$. We obtain necessary
and sufficient conditions for We study the backward shift operator $B$ acting
on the Hilbert space $\mathcal{H}_{a,b}$ of analytic functions on the unit
disc, having an orthonormal basis consisting of polynomials of the form
$f_n(z)=(a_n+b_nz)z^n, ~~n\geq0$. We obtain necessary and sufficient conditions
for $B$ to be bounded, and prove that, under some mild assumptions on $\{a_n\}$
and $\{b_n\}$, the operator $B$ is unitarily equivalent to a compact
perturbation of a weighted backward shift on $\ell^2$. Further, we characterize
the hypercyclicity, mixing, and chaos of $\lambda B$ for a non-zero scalar
$\lambda$, and establish the existence of hypercyclic subspaces for $\lambda B$
by computing its essential spectrum. We also provide vector valued versions of
our results for $B$ when it acts on a reproducing kernel space corresponding to
matrix valued kernels.

Functional Analysis