On doubly commuting operators in $C_{1, r}$ class and quantum annulus

TL;DR

The paper addresses the dilation and decomposition of finitely many doubly commuting operator tuples belonging to the $C_{1,r}$ class and the quantum annulus $QA_r$. It proves dilation characterizations: membership in $C_{1,r}$ (or $QA_r$) is equivalent to the existence of a dilation to a tuple in $\mathcal{C}_{1,r}$ (or $\mathcal{Q}A_r$) with the defining relations $(1+r^2)I - J_m^*J_m - r^2 J_m^{-1}J_m^{-*}=0$ (or $(r^{-2}+r^2)I - J_m^*J_m - J_m^{-1}J_m^{-*}=0$). It then develops a canonical decomposition for $C_{1,r}$-contractions and extends it to doubly commuting tuples in $QA_r$, yielding a $2^d$-fold decomposition into subspaces on which each component is either in the corresponding class or c.n.u. The results extend single-operator dilation theory to the doubly commuting setting and connect dilation theory with spectral-set analysis on annular domains.

Abstract

For $ 0 < r < 1 $, let $ \mathbb{A}_r = \{ z \in \mathbb{C} : r < |z| < 1 \} $ be the annulus with boundary $ \partial \overline{\mathbb{A}}_r = \mathbb{T} \cup r\mathbb{T} $, where $ \mathbb{T} $ is the unit circle in the complex plane $\mathbb C$. We study the class of operators \[ C_{1,r} = \{ T : T \text{ is invertible and } \|T\|, \|rT^{-1}\| \leq 1 \}, \] introduced by Bello and Yakubovich. Any operator $T$ for which the closed annulus $\overline{\mathbb{A}}_r$ is a spectral set is in $C_{1,r}$. The class $C_{1, r}$ is closely related to the \textit{quantum annulus} which is given by \[ QA_r = \{ T : T \text{ is invertible and } \|rT\|, \|rT^{-1}\| \leq 1 \}. \] McCullough and Pascoe proved that an operator in $ QA_r $ admits a dilation to an operator $ S $ satisfying $(r^{-2} + r^2)I - S^*S - S^{-1}S^{-*} = 0$. An analogous dilation result holds for operators in $ C_{1,r}$ class. We extend these dilation results to doubly commuting tuples of operators in quantum annulus as well as in $C_{1,r}$ class. We also provide characterizations and decomposition results for such tuples.

Theorems & Definitions (23)

• Lemma 1.1: N-S1, Lemma 1.5
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1: Pas-McCull, Proposition 3.2
• Theorem 2.2: Pas-McCull, Theorem 1.1
• Proposition 2.3
• Theorem 2.4
• Theorem 2.5
• proof
• Definition 3.1