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

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 $Q{A}_r$ 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.

arXiv commentJournal of Operator Theory, To appear