Characterizations and models for the $C_{1,r}$ class and quantum annulus

TL;DR

This work develops a comprehensive dilation-based framework for the $C_{1,r}$ class and its quantum counterpart. By proving a model-cum-characterization of $C_{1,r}$ through a pair consisting of an $A_r$-unitary and a self-adjoint unitary, it reduces analysis to structured normal operators on a larger space and connects $C_{1,r}$ with the quantum annulus via a simple scaling equivalence. The authors then transfer the model to the quantum annulus, obtaining parallel characterizations in terms of normal operators with spectrum on the annulus boundary and spectral-set conditions. Collectively, the results unify these annulus-related operator classes under a common dilation paradigm and provide explicit operator-pair models for both $C_{1,r}$ and $\\mathbb{Q} \\mathbb{A}_r$.

Abstract

For fixed $0<r<1$, let $A_r=\{z \in \mathbb{C} : r<|z|<1\}$ be the annulus with boundary $\partial \overline{A}_r=\mathbb{T} \cup r\mathbb{T}$, where $\mathbb T$ is the unit circle in the complex plane $\mathbb C$. An operator having $\ov{A}_r$ as a spectral set is called an $A_r$-\textit{contraction}. Also, a normal operator with its spectrum lying in the boundary $\partial \overline{A}_r$ is called an \textit{$A_r$-unitary}. The \textit{$C_{1,r}$ class} was introduced by Bello and Yakubovich in the following way: \[ C_{1, r}=\{T: T \ \mbox{is invertible and} \ \|T\|, \|rT^{-1}\| \leq 1\}. \] McCullough and Pascoe defined the \textit{quantum annulus} $\mathbb Q \mathbb A_r$ by \[ \mathbb Q\mathbb A_r = \{T \,:\, T \text{ is invertible and } \, \|rT\|, \|rT^{-1}\| \leq 1 \}. \] If $\mathcal A_r$ denotes the set of all $A_r$-contractions, then $\mathcal A_r \subsetneq C_{1,r} \subsetneq \mathbb Q \mathbb A_r$. We first find a model for an operator in $C_{1,r}$ and also characterize the operators in $C_{1,r}$ in several different ways. We prove that the classes $C_{1,r}$ and $\mathbb Q\mathbb A_r$ are equivalent. Then, via this equivalence, we obtain analogous model and characterizations for an operator in $\mathbb Q \mathbb A_r$.

Theorems & Definitions (34)

• Theorem 1.1: Dmitry, Theorem 1.1
• Theorem 1.2: Pas-McCull, Theorem 1.1
• Theorem 1.3
• Theorem 1.4
• Lemma 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 2.1: Ando, Ando
• Lemma 2.2