Operator Positivity and Analytic Models of Commuting Tuples of Operators

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 $\theta \in \mathcal{M}(H^2(\mathcal{E}),A_m^2(\mathcal{H}))$ such that $$\mathcal{H}\cong {\mathcal{Q}}_{\theta }=A_m^2(\mathcal{H})\ominus \theta H^2(\mathcal{E}),andT\cong P_{{\mathcal{Q}}_{\theta }}{M}_z{\mathrm{\mid }}_{{\mathcal{Q}}_{\theta }},$$where $A_m^2$ 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 ${\mathbb{D}}^n$.