A∈B(H) is called Drazin invertible if A has finite ascent and descent. Let σD (A)={λ∈ C : A -λI is not Drazin invertible } be the Drazin .spectrum. This paper shows that if Mc =(A C 0 B)is a 2 × 2 upp...
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A∈B(H) is called Drazin invertible if A has finite ascent and descent. Let σD (A)={λ∈ C : A -λI is not Drazin invertible } be the Drazin .spectrum. This paper shows that if Mc =(A C 0 B)is a 2 × 2 upper triangular operator matrix acting on the Hilbert space H + K, then the passage from OσD(A) U σD(B) to σD(Mc) is accomplished by removing certain open subsets of σD(A)∩σD(B) from the former, that is, there is equality
σD(A)∪σD(B)=σD(MC)∪G
where G is the union of certain holes in σD (Me) which happen to be subsets of σD (A)∩σD (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.
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