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

For fixed $0<r<1$, let $A_r=\{z\in \mathbb{C}:r<\mathrm{\mid }z\mathrm{\mid }<1\}$ be the annulus with boundary $\mathrm{\partial }{\bar{A}}_r=\mathbb{T}\cup r\mathbb{T}$, where $\mathbb{T}$ is the unit circle in the complex plane $\mathbb{C}$. An operator having $\text{ov}{A}_r$ as a spectral set is called an $A_r$-$\mathit{\text{contraction}}$. Also, a normal operator with its spectrum lying in the boundary $\mathrm{\partial }{\bar{A}}_r$ is called an $A_r\mathit{\text{-unitary}}$. The $C_{1,r}\mathit{\text{class}}$ was introduced by Bello and Yakubovich in the following way: $$C_{1,r}=\{T:Tisinvertibleand\mathrm{\parallel }T\mathrm{\parallel },\mathrm{\parallel }r{T}^{-1}\mathrm{\parallel }\le 1\}\mathrm{.}$$ McCullough and Pascoe defined the $\mathit{\text{quantum}}\mathit{\text{annulus}}$ $\mathbb{Q}{\mathbb{A}}_r$ by $$\mathbb{Q}{\mathbb{A}}_r=\{T:T\text{ is invertible and }\mathrm{\parallel }rT\mathrm{\parallel },\mathrm{\parallel }r{T}^{-1}\mathrm{\parallel }\le 1\}\mathrm{.}$$ If ${\mathcal{A}}_r$ denotes the set of all $A_r$-contractions, then ${\mathcal{A}}_r⊊C_{1,r}⊊\mathbb{Q}{\mathbb{A}}_r$. 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 $\mathbb{Q}{\mathbb{A}}_r$.