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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