Let X be a Banach space and let T be a bounded linear operator acting on X. Atkinson's well known theorem says that T is a Fredholm operator if and only if its projection in the algebra L(X)/F₀(X) is invertible, where F₀(X) is the ideal of finite rank operators in the algebra L(X) of bounded linear operators acting on X. In the main result of this paper we establish an Atkinson-type theorem for B-Fredholm operators. More precisely we prove that T is a B-Fredholm operator if and only if its projection in the algebra L(X)/F₀(X) is Drazin invertible. We also show that the set of Drazin invertible elements in an algebra A with a unit is a regularity in the sense defined by Kordula and Müller [8].
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm148-3-4,
author = {M. Berkani and M. Sarih},
title = {An Atkinson-type theorem for B-Fredholm operators},
journal = {Studia Mathematica},
volume = {147},
year = {2001},
pages = {251-257},
zbl = {1005.47012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm148-3-4}
}
M. Berkani; M. Sarih. An Atkinson-type theorem for B-Fredholm operators. Studia Mathematica, Tome 147 (2001) pp. 251-257. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm148-3-4/