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

$Ain B(H)$ is called Drazin invertible if $A$ has finite ascent and descent. Let $sigma_D(A)={lambdain{Bbb C}: A-lambda I$ is not Drazin invertible $}$ be the Drazin spectrum. This paper shows that if $M_C=left( begin{array} {cccc}A&C0&Bend{array} right)$ is a $2times 2$ upper triangular operator matrix acting on the Hilbert space $Hoplus K$, then the passage from $sigma_D(A)cupsigma_D(B)$ to $sigma_D(M_C)$ is accomplished by removing certain open subsets of $sigma_D(A)capsigma_D(B)$ from the former, that is, there is equality $sigma_D(A)cupsigma_D(B)=sigma_D(M_C)cup {mathcal{G}},$ where $mathcal{G}$ is the union of certain holes in $sigma_D(M_C)$ which happen to be subsets of $sigma_D(A)capsigma_D(B)$. Weyl's theorem and Browder's theorem are liable to fail for $2times 2$ 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 $2times 2$ upper triangular operator matrices on the Hilbert space.