超常算子和$\ast $-超常算子交换元组的Weyl定理

N. Bala, G. Ramesh
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引用次数: 0

摘要

在本文中,我们证明了具有拟三角形性质的$\ast$ -超常算子$T_1$和$T_2$的交换对$T=(T_1,T_2)$满足Weyl定理1,即$$\sigma_T(T)\setminus\sigma_{T_W}(T)=\pi_{00}(T)$$;超常算子的交换对满足Weyl定理2,即$$\sigma_T(T)\setminus\omega(T)=\pi_{00}(T),$$,其中$\sigma_T(T),\, \sigma_{T_W}(T),\,\omega(T)$和$\pi_{00}(T)$是Taylor谱,Taylor Weyl谱,联合Weyl谱和由孤立特征值组成的集$T$具有有限多重性。此外,我们证明了Weyl定理ii对于$f(T)$成立,其中$T$是一个超常算子的交换对,$f$是一个在$\sigma_T(T)$的邻域中的解析函数。
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Weyl’s theorem for commuting tuples ofparanormal and $\ast $-paranormal operators
In this article, we show that a commuting pair $T=(T_1,T_2)$ of $\ast$-paranormal operators $T_1$ and $T_2$ with quasitriangular property satisfy the Weyl's theorem-I, that is $$\sigma_T(T)\setminus\sigma_{T_W}(T)=\pi_{00}(T)$$ and a commuting pair of paranormal operators satisfy Weyl's theorem-II, that is $$\sigma_T(T)\setminus\omega(T)=\pi_{00}(T),$$ where $\sigma_T(T),\, \sigma_{T_W}(T),\,\omega(T)$ and $\pi_{00}(T)$ are the Taylor spectrum, the Taylor Weyl spectrum, the joint Weyl spectrum and the set consisting of isolated eigenvalues of $T$ with finite multiplicity, respectively. Moreover, we prove that Weyl's theorem-II holds for $f(T)$, where $T$ is a commuting pair of paranormal operators and $f$ is an analytic function in a neighbourhood of $\sigma_T(T)$.
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