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On semi B-Fredholm operators. (English) Zbl 0995.47008
Summary: An operator $$T$$ on a Banach space is called ‘semi B-Fredholm’ if for some $$n\in\mathbb{N}$$ the range $$R(T^n)$$ is closed and the induced operator $$T_n$$ on $$R(T^n)$$ semi-Fredholm. Semi B-Fredhom operators are stable under finite rank perturbation, and subject to the spectral mapping theorem; on Hilbert spaces they decompose as sums of nilpotent and semi-Fredholm operators. In addition some recent generalizations of the punctured neighborhood theorem turn out to be consequences of Grabiner’s theory of ‘topological uniform descent’.

##### MSC:
 47A53 (Semi-) Fredholm operators; index theories 47A55 Perturbation theory of linear operators
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