Operator Positivity and Analytic Models of Commuting Tuples of Operators

Abstract

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 $\mathbb{D}^n$.