An Atkinson-type theorem for B-Fredholm operators

M. Berkani¹ ; M. Sarih²

¹ Département de Mathématiques Faculté des Sciences Université Mohammed I Oujda, Maroc
² Département de Mathématiques Faculté des Sciences Université Ibn Tofail Kénitra, Maroc

Studia Mathematica, Tome 148 (2001) no. 3, pp. 251-257

Résumé

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 $L(X)/ F_{0}(X)$ is invertible, where $ F_{0}(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 $L(X)/ F_{0}(X)$ 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].

Détail

DOI : 10.4064/sm148-3-4

Keywords: banach space bounded linear operator acting atkinsons known theorem says fredholm operator only its projection algebra invertible where ideal finite rank operators algebra bounded linear operators acting main result paper establish atkinson type theorem b fredholm operators precisely prove b fredholm operator only its projection algebra drazin invertible set drazin invertible elements algebra unit regularity sense defined kordula ller