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

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.