Operator Positivity and Analytic Models of Commuting Tuples of Operators

TL;DR

This paper develops analytic models for certain classes of operators with positivity properties, including hypercontractions and doubly commuting tuples, with applications to subspace analysis in multivariable operator theory.

Contribution

It introduces new analytic models for hypercontractions and doubly commuting operator tuples, extending the understanding of their structure and applications in multivariable operator theory.

Findings

Existence of a Hilbert space and a partially isometric multiplier representing hypercontractions.

Development of analytic models for doubly commuting operator tuples.

Complete analysis of quotient modules in reproducing kernel Hilbert spaces over polydiscs.

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 $\mathbb{D}^n$.