1 results
Characterizations and models for the
$C_{1,r}$ class and quantum annulus
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 07 February 2025, pp. 1-16
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- Article
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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 
$\overline {A}_r$ as a spectral set is called an 
$A_r$-contraction. Also, a normal operator with its spectrum lying in the boundary 
$\partial \overline {A}_r$ is called an 
$A_r$-unitary. The 
$C_{1,r}$ class was introduced by Bello and Yakubovich in the following way: 
$$\begin{align*}C_{1, r}=\{T: T \ \text{is invertible and} \ \|T\|, \|rT^{-1}\| \leq 1\}. \end{align*}$$McCullough and Pascoe defined the quantum annulus
$\mathbb Q \mathbb A_r$ by 
$$\begin{align*}\mathbb Q\mathbb A_r = \{T \,:\, T \text{ is invertible and } \, \|rT\|, \|rT^{-1}\| \leq 1 \}. \end{align*}$$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 
$\mathbb Q \mathbb A_r$.
