  We study analytic models of operators of class $C_{\cdot 0}$ with natural
positivity assumptions. In particular, we prove that for an
$m$-hypercontraction $T \in C_{\cdot 0}$ on a Hilbert space $\mathcal{H}$,
there exists a Hilbert space $\mathcal{E}$ and a partially isometric multiplier
$\theta \in \mathcal{M}(H^2(\mathcal{E}), A^2_m(\mathcal{H}))$ such that
\[\mathcal{H} \cong \mathcal{Q}_{\theta} = A^2_m(\mathcal{H}) \ominus \theta
H^2(\mathcal{E}), \quad \quad \mbox{and} \quad \quad T \cong
P_{\mathcal{Q}_{\theta}} M_z|_{\mathcal{Q}_{\theta}},\]where $A^2_m$ is the
weighted Bergman space and $H^2$ is the Hardy space over the unit disc
$\mathbb{D}$. We then proceed to study and develop analytic models for doubly
commuting $n$-tuples of operators and investigate their applications to joint
shift co-invariant subspaces of reproducing kernel Hilbert spaces over
polydisc. In particular, we completely analyze doubly commuting quotient
modules of a large class of reproducing kernel Hilbert modules, in the sense of
Arazy and Englis, over the unit polydisc $\mathbb{D}^n$.
