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On the Drazin inverse for upper triangular operator matrices. (English) Zbl 1312.47003
Summary: In this paper we investigate the stability of Drazin spectrum $$\sigma_D(.)$$ for upper triangular operator matrices $$M_C= \left[\begin{matrix} A& C\\ 0 & B\end{matrix}\right]$$ using tools from local spectral theory. We show that $$\sigma_D(M_C) \cup [\mathcal{S}(A^\ast)\cap\mathcal{S}(B)]=\sigma_D(A)\cup\sigma_D(B)$$ where $$\mathcal{S}(.)$$ is the set where an operator fails to have the SVEP. As application we explore how the generalized Weyl’s theorem survives for $$M_C$$.

##### MSC:
 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A11 Local spectral properties of linear operators
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