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

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 $\overline {A}_r$ as a spectral set is called an $A_r$ - contraction . Also, a normal operator with its spectrum lying in the boundary $\partial \overline {A}_r$ is called an $A_r$ -unitary . The $C_{1,r}$ class was introduced by Bello and Yakubovich in the following way: $$\begin{align*}C_{1, r}=\{T: T \ \text{is invertible and} \ \|T\|, \|rT^{-1}\| \leq 1\}. \end{align*}$$ McCullough and Pascoe defined the quantum annulus $\mathbb Q \mathbb A_r$ by $$\begin{align*}\mathbb Q\mathbb A_r = \{T \,:\, T \text{ is invertible and } \, \|rT\|, \|rT^{-1}\| \leq 1 \}. \end{align*}$$ 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 $\mathbb Q \mathbb A_r$ .