On doubly commuting operators in $C_{1, r}$ class and quantum annulus (2503.23754v1)

Abstract: 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.