An Atkinson-type theorem for B-Fredholm operators

M. Berkani; M. Sarih

Studia Mathematica (2001)

  • Volume: 148, Issue: 3, page 251-257
  • ISSN: 0039-3223

Abstract

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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].

How to cite

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M. Berkani, and M. Sarih. "An Atkinson-type theorem for B-Fredholm operators." Studia Mathematica 148.3 (2001): 251-257. <http://eudml.org/doc/284831>.

@article{M2001,
abstract = {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].},
author = {M. Berkani, M. Sarih},
journal = {Studia Mathematica},
keywords = {Fredholm operator; Atkinson-type theorem; Drazin invertible elements},
language = {eng},
number = {3},
pages = {251-257},
title = {An Atkinson-type theorem for B-Fredholm operators},
url = {http://eudml.org/doc/284831},
volume = {148},
year = {2001},
}

TY - JOUR
AU - M. Berkani
AU - M. Sarih
TI - An Atkinson-type theorem for B-Fredholm operators
JO - Studia Mathematica
PY - 2001
VL - 148
IS - 3
SP - 251
EP - 257
AB - 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].
LA - eng
KW - Fredholm operator; Atkinson-type theorem; Drazin invertible elements
UR - http://eudml.org/doc/284831
ER -

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