| 高福根,李晓春.以仿正规算子为参数的初等算子的性质[J].数学年刊A辑,2015,36(1):111~118 |
| 以仿正规算子为参数的初等算子的性质 |
| Properties of an Elementary Operator with Paranormal Operators as Parameters |
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| DOI： |
| 中文关键词: 仿正规算子, 初等算子, 广义Weyl定理 |
| 英文关键词:Paranormal operators, Elementary operator, Generalized Weyl's theorem |
| 基金项目:国家自然科学基金(No.11271112, No.11301155), 河南省高校科研创新团队(No.14IRTS THN023),
河南省教育厅科学技术研究重点项目(No.13B110077), 河南师范大学国家级项目培育基金,
河南师范大学青年基金和河南师范大学博士科研启动费支持课题(No.qd12102) |
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| 中文摘要: |
| 若对$\forall x\in \mathcal{H}$, $\|Tx\|^{2} \leq \|T^{2}x\|\|x\|$, 则称$T$是仿正规算子. $d_{AB}$表示$\delta_{AB}$或$\triangle_{AB}$,
其中$\delta_{AB}$和$\triangle_{AB}$分别表示Banach空间$B(\mathcal{H})$上
的广义导算子和初等算子, 其定义为$\delta_{AB}X=AX-XB$, $\triangle_{AB}X=AXB-X$, $\forall X \in B(\mathcal{H})$.
若$A$和$B^{\ast}$是仿正规算子, 则可证$d_{AB}$是polaroid算子, $\forall f\in
H(\sigma
(d_{AB}))$, $f(d_{AB})$满足广义Weyl定理,
$f(d_{AB}^{\ast})$满足广义$a$-Weyl定理, 其中$H(\sigma
(d_{AB}))$表示在$\sigma
(d_{AB})$的某邻域上解析的函数全体. |
| 英文摘要: |
| An operator $T$ is called paranormal if $\| Tx\|^{2}\leq
\| T^{2}x\|
\| x \|$ for all $x\in \mathcal{H}$. Let
$d_{AB}$ denote either $\delta_{AB}$ or $\triangle_{AB}$, where $\delta_{AB}$ and
$\triangle_{AB}$ denote the generalized derivation and the elementary operator on a Banach
space $B(\mathcal{H})$ defined by $\delta_{AB}X=AX-XB$ and $\triangle_{AB}X=AXB-X,\ \forall X \in B(\mathcal{H})$, respectively. If $A$ and $B^{\ast}$ are paranormal
operators, it is shown that $d_{AB}$ is polaroid and the generalized
Weyl's theorem holds for $f(d_{AB})$, the generalized $a$-Weyl's theorem holds for
$f(d_{AB}^{\ast})$ for every $f\in
H(\sigma
(d_{AB}))$, where $H(\sigma
(d_{AB}))$ denotes the set of all analytic functions in a neighborhood
of $\sigma
(d_{AB})$. |
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