Operator positivity and analytic models of commuting tuples of operators
Volume 232 / 2016
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 exist Hilbert spaces $\mathcal E$ and $\mathcal E_*$ and a partially isometric multiplier $\theta \in \mathcal M(H^2(\mathcal E), A^2_m(\mathcal E_*))$ such that \[ \mathcal H \cong \mathcal Q_{\theta} = A^2_m(\mathcal E_*) \ominus \theta H^2(\mathcal E) \quad \mbox{and} \quad T \cong P_{\mathcal Q_{\theta}} M_z|_{\mathcal Q_{\theta}}, \] where $A^2_m(\mathcal E_*)$ is the $\mathcal E_*$-valued weighted Bergman space and $H^2(\mathcal E)$ is the $\mathcal E$-valued Hardy space over the unit disc $\mathbb{D}$. We then proceed to study 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 the 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 Engliš, over the unit polydisc $\mathbb{D}^n$.
