Drazin Spectrum and Weyl's Theorem for Operator Matrices
Xiaohong Cao, Guo Mao-zhen · Journal of Mathematical Research and Exposition · 2006
A∈B(H)is called Drazin invertible if A has finite ascent and descent.Letσ_D(A)={λ∈C∶A-λI is not Drazin invertible}be the Drazin spectrum.This paper shows that if M_C=(?)is a 2×2 upper triangular operator matrix acting on the Hilbert space H⊕K,then the passage fromσ_D(A)∪σ_D(B)toσ_D(M_C)is accomplished by removing certain open subsets ofσ_D(A)∩σ_D(B)from the former,that is,there is equalityσ_D(A)∪σ_D(B)=σ_D(M_C)∪G, where G is the union of certain holes inσ_D(M_C)which happen to be subsets ofσ_D(A)∩σ_D (B). Weyl's theorem and Browder's theorem are liable to fail for 2×2 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 2×2 upper triangular operator matrices on the Hilbert space.