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关于$*$-$n$-仿正规算子的一个注记
 A Note on $*$-$n$-paranormal Operators

左飞1, 申俊丽2
 ZUO  Fei1, SHEN  JunLi2

1. 河南师范大学数学与信息科学学院,新乡, 河南,453007;2. 新乡学院数学系,新乡, 河南,453000
1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan, 453007, P. R. China; 2. Department of Mathematics, Xinxiang University, Xinxiang, Henan, 453000, P. R. China

出版日期: 2013-04-25
DOI: 10.11845/sxjz.2013.42.02.0153

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摘要 设$n$为正整数,称$T$为$*$-$n$-仿正规算子,若$||T^{1+n}x||^{\frac{1}{1+n}}\geq||T^{*}x||$对$H$中的每个单位向量$x$都成立;称$T$为$*$-$\hat{n}$-仿正规算子,若$||T^{1+i}x||^{\frac{1}{1+i}}\geq||T^{*}x||$对$H$中的每个单位向量$x$及$i\geq n$都成立.若对任意$\lambda\in \mathbb{C}$,$T-\lambda$都是$*$-$\hat{n}$-仿正规算子,则称$T$为完全$*$-$\hat{n}$-仿正规算子.若$T$是$*$-$n$-仿正规算子,它的近似点谱和联合近似点谱是相等的.另外证明了若$T$或者$T^{*}$是完全$*$-$\hat{n}$-仿正规算子,则Weyl定理对$f(T)$成立,其中$f\in H(\sigma(T))$, 还证明了若$T^{*}$是完全$*$-$\hat{n}$-仿正规算子,则$\alpha$-Weyl定理对$f(T)$成立.
关键词 $*$-$n$-仿正规算子 Weyl定理 $\alpha$-Weyl定理 $\alpha$-Browder定理    
Abstract:Let $n$ be a positive integer. An operator $T$ belongs to class $*$-$n$-paranormal if $||T^{1+n}x||^{\frac{1}{1+n}}\geq||T^{*}x||$ for unit vector $x$. An operator $T\in B(H)$ is said to be $*$-$\hat{n}$-paranormal if$||T^{1+i}x||^{\frac{1}{1+i}}\geq||T^{*}x||$ for unit vector $x$ and $i\geq n$. An operator $T\in B(H)$ is said to be totally $*$-$\hat{n}$-paranormal, if $T-\lambda$ is $*$-$\hat{n}$-paranormal for every $\lambda\in \mathbb{C}$. It is showed that if $T$ belongs to class $*$-$n$-paranormal operators, then its approximate point spectrum and joint approximate point spectrum are identical. We also prove that if either $T$ or $T^{*}$ is totally $*$-$\hat{n}$-paranormal, then Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma(T))$, and also $\alpha$-Weyl's theorem holds for $f(T)$ if $T^{*}$ is totally$*$-$\hat{n}$-paranormal.
Key words Weyl's theorem    $\alpha$-Weyl's theorem    $\alpha$-Browder's theorem
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