Drazin谱和算子矩阵的Weyl定理

A∈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σ_D(A)={λ∈C：A-λI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2×2上三角算子矩阵M_C=■,从σ_D(A)∪σ_D(G)到σ_D(M_C)的道路需要从前面子集中移动σ_D(A)∩σ_D(B)中一定的开子集,即有等式：σ_D(A)∪σ_D(B)=σ_D(M_C)∪G,其中G为σ_D(M_C)中一定空洞的并,并且为σ_D(A)∪σ_D(B)的子集．2×2算子矩阵不一定满足Weyl定理,利用Drazin谱,我们研究了2×2上三角算子矩阵的Weyl定理,Browder定理,a-Weyl定理和a-Browder定理．

Drazin Spectrum and Weyl's Theorem for Operator Matrices

$A\in B(H)$ 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.