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

For fixed $0<r<1$, let $A_r=\{z \in \mathbb{C} : r<|z|<1\}$ 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 $\mathbb Q \mathbb A_r$.
Comment: 14 pages, Thoroughly revised, Title changed, New results added