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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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