| 曹小红,郭懋正,孟彬.Drazin谱和算子矩阵的Weyl定理[J].数学研究及应用,2006,26(3):413~422 |
| Drazin谱和算子矩阵的Weyl定理 |
| Drazin Spectrum and Weyl's Theorem for Operator Matrices |
| 投稿时间：2004-11-08 |
| DOI：10.3770/j.issn:1000-341X.2006.03.001 |
| 中文关键词: Weyl定理 a-Weyl定理 Browder定理 a-Browder定理 Drazin谱. |
| 英文关键词:Weyl's theorem a-Weyl's theorem Browder's theorem a-Browder's theorem Drazin spectrum. |
| 基金项目:the National Natural Science Foundation of China (10571099) |
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| 中文摘要: |
| $A\in B(H)$称为是一个Drazin可逆的算子, 若$A$有有限的升标和降标. 用$\sigma_D(A)=\{\lambda\in{\Bbb C}:\ A-\lambda I$不是Drazin可逆的 $\}$表示Drazin谱集. 本文证明了对于Hilbert空间上的一个$2\times 2$上三角算子矩阵$M_C=\left(\begin{array} {cccc}A&C\\0&B\\\end{array} \right)$, 从$\sigma_D(A)\cup\sigma_D(B)$到$\sigma_D(M_C)$的道路需要从前面子集中移动$\sigma_D(A)\cap\sigma_D(B)$中一定的开子集, 即有等式:$$\sigma_D(A)\cup\sigma_D(B)=\sigma_D(M_C)\cup {\mathcal{G}},$$ 其中$\mathcal{G}$为$\sigma_D(M_C)$中一定空洞的并, 并且为$\sigma_D(A)\cap\sigma_D(B)$的子集. $2\times 2$算子矩阵不一定满足Weyl定理, 利用Drazin谱,我们研究了$2\times 2$上三角算子矩阵的Weyl定理, Browder定理, a-Weyl定理和a-Browder定理. |
| 英文摘要: |
| $A\in B(H)$ is called Drazin invertible if $A$ has finite ascent and descent. Let $\sigma_D(A)=\{\lambda\in{\Bbb C}:\ A-\lambda I$ is not Drazin invertible $\}$ be the Drazin spectrum. This paper shows that if $M_C=\left( \begin{array} {cccc}A&C\\0&B\\\end{array} \right)$ is a $2\times 2$ upper triangular operator matrix acting on the Hilbert space $H\oplus K$, then the passage from $\sigma_D(A)\cup\sigma_D(B)$ to $\sigma_D(M_C)$ is accomplished by removing certain open subsets of $\sigma_D(A)\cap\sigma_D(B)$ from the former, that is, there is equality $$\sigma_D(A)\cup\sigma_D(B)=\sigma_D(M_C)\cup {\mathcal{G}},$$ where $\mathcal{G}$ is the union of certain holes in $\sigma_D(M_C)$ which happen to be subsets of $\sigma_D(A)\cap\sigma_D(B)$. Weyl's theorem and Browder's theorem are liable to fail for $2\times 2$ operator matrices. By using Drazin spectrum, it also explores how Weyl's theorem, Browder's theorem, a-Weyl's theorem and a-Browder's theorem survive for $2\times 2$ upper triangular operator matrices on the Hilbert space. |
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