论文标题
$ c_ {1，r} $类和量子环的特征和模型
Characterizations and models for the $C_{1,r}$ class and quantum annulus
论文作者
论文摘要
对于$ 0 <r <1 $，让$ a_r = \ {z \ in \ mathbb {c}：r <| z | <1 \} $是具有边界$ $ \ partial \ partial \ partial \ partline {a} _r = \ \ \ \ mathb { $ \ mathbb c $。具有$ \ ov {a} _r $作为频谱集的运算符称为$ a_r $ - \ textit {contraction}。另外，一个具有光谱的普通运算符位于边界$ \ partial \ overline {a} _r $中，称为\ textit {$ a_r $ -unitaly}。 \ textit {$ c_ {1，r} $ class}由Bello和Yakubovich引入以下方式：\ [c_ {1，r} = \ {t：t \ \ \ \ \ \ \ \ mbox {是可逆的，} \ leq 1 \}。 \ \] McCullough and Pascoe定义了\ textIt {量子Annulus} $ \ Mathbb Q \ Mathbb a_r $ by \ [\ Mathbb Q \ Mathbb Q \ Mathbb a_r = \ {t \ {t \，： \ leq 1 \}。 \]如果$ \ MATHCAL A_R $表示所有$ a_r $ - 合同的集合，则表示$ \ Mathcal a_r \ subsetneq c_ {1，r} \ subsetNeq \ subsetNeq \ subsetNeq \ mathbb q \ Mathbb a_r $。我们首先在$ c_ {1，r} $中找到一个操作员的模型，并以几种不同的方式在$ c_ {1，r} $中以$ c_ {1，r} $来表征运算符。我们证明$ c_ {1，r} $和$ \ mathbb q \ mathbb a_r $等效的类。然后，通过这种等价，我们在$ \ Mathbb Q \ Mathbb a_r $中获得了操作员的类似模型和特征。
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 $\ov{A}_r$ as a spectral set is called an $A_r$-\textit{contraction}. Also, a normal operator with its spectrum lying in the boundary $\partial \overline{A}_r$ is called an \textit{$A_r$-unitary}. The \textit{$C_{1,r}$ class} was introduced by Bello and Yakubovich in the following way: \[ C_{1, r}=\{T: T \ \mbox{is invertible and} \ \|T\|, \|rT^{-1}\| \leq 1\}. \] McCullough and Pascoe defined the \textit{quantum annulus} $\mathbb Q \mathbb A_r$ by \[ \mathbb Q\mathbb A_r = \{T \,:\, T \text{ is invertible and } \, \|rT\|, \|rT^{-1}\| \leq 1 \}. \] 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 $\mathbb Q \mathbb A_r$.
