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A note on Drazin invertibility for upper triangular block operators. (English) Zbl 1304.47005
A bounded linear operator \(A\) acting on a Banach space \(X\) is said to be an upper triangular block operator of order \(n\), denoted \(A\in\mathcal{U}\mathcal{T}_n(X)\), if there exists a decomposition of \(X=X_1\oplus\cdots\oplus X_n\) and an \(n\times n\) matrix operator \((A_{i,j})_{1\leq i,j\leq n}\) such that \(A=(A_{i,j})_{1\leq i,j\leq n}\), \(A_{i,j} =0\) for \(i>j\). For these operators, the author obtains several conditions on the entries \(A_{i,j}\), \(j>i\) or \(i=j\), for the equality \(\sigma_D(A)=\bigcup_{i=1}^n\sigma_D(A_{i,i})\), where \(\sigma_D(.)\) is the Drazin spectrum. Also, some applications concerning the Fredholm theory and meromorphic operators are given.

MSC:
47A10 Spectrum, resolvent
47A11 Local spectral properties of linear operators
47A53 (Semi-) Fredholm operators; index theories
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