On semi B-Fredholm operators. (English) Zbl 0995.47008

Berkani, M.; Sarih, M.

Glasg. Math. J. 43, No. 3, 457-465 (2001).

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

Keywords:

semi B-Fredholm operators; topological uniform descent; finite rank perturbation; spectral mapping theorem; sums of nilpotent and semi-Fredholm operators