| Drazin Spectrum and Weyl's Theorem for Operator Matrices |
| Received:November 08, 2004 |
| Key Words:
Weyl's theorem a-Weyl's theorem Browder's theorem a-Browder's theorem Drazin spectrum.
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| Fund Project:the National Natural Science Foundation of China (10571099) |
| Author Name | Affiliation | | CAO Xiao-hong | LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
College of Math. \& Info. Sci., Shaanxi Normal University, Xi'an 710062, China | | GUO Mao-zheng | LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China | | MENG Bin | LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China |
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| Abstract: |
| $A\in B(H)$ 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. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2006.03.001 |
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