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

For $0<r<1$, let ${\mathbb{A}}_r=\{z\in \mathbb{C}:r<\mathrm{\mid }z\mathrm{\mid }<1\}$ be the annulus with boundary $\mathrm{\partial }{\bar{\mathbb{A}}}_r=\mathbb{T}\cup r\mathbb{T}$, where $\mathbb{T}$ is the unit circle in the complex plane $\mathbb{C}$. We study the class of operators $$C_{1,r}=\{T:T\text{ is invertible and }\mathrm{\parallel }T\mathrm{\parallel },\mathrm{\parallel }r{T}^{-1}\mathrm{\parallel }\le 1\},$$ introduced by Bello and Yakubovich. Any operator $T$ for which the closed annulus ${\bar{\mathbb{A}}}_r$ is a spectral set is in $C_{1,r}$. The class $C_{1,r}$ is closely related to the $\mathit{\text{quantum}}\mathit{\text{annulus}}$ which is given by $$Q{A}_r=\{T:T\text{ is invertible and }\mathrm{\parallel }rT\mathrm{\parallel },\mathrm{\parallel }r{T}^{-1}\mathrm{\parallel }\le 1\}\mathrm{.}$$ McCullough and Pascoe proved that an operator in $Q{A}_r$ admits a dilation to an operator $S$ satisfying $(r^{-2}+r^2)I-S^{\ast }S-S^{-1}{S}^{-\ast }=0$. 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.