An Atkinson-type theorem for B-Fredholm operators

Abstract

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)/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].

Index of B-Fredholm operators and generalization of a Weyl theorem

“…In the case of a bounded linear operator T acting on a Banach space X, it is well known that T is Drazin invertible if and only if it has a finite ascent and descent (Definition 2.1), which is also equivalent to the fact that [5]. We also show that if 0 is isolated in the spectrum of T , then T is a B-Fredholm operator of index 0 if and only if T is Drazin invertible.…”

“…This paper is a continuation of our previous works [2], [3], [4], [5]. We consider a Banach space X and L(X) the Banach algebra of bounded linear operators acting on X.…”

Generalized Weyl's theorem and hyponormal operators

“…It is apparent from [6] that the concept of Drazin invertibility plays an important role for the class of 5-Fredholm operators. If A is an algebra with a unit 1, following [16] we say that an element x of A is Drazin invertible if there is an element b of A and a nonnegative integer k such that Recall that the concept of Drazin invertibility was originally introduced by Drazin in [9] where elements satisfying relation (1) are called pseudo-invertible elements.…”

“…(See [16,Proposition 6] and [12, Corollary 2.2].) In [6,Theorem 3.4] it is shown that T is a B-Fredholm operator if and only if its projection in the algebra L(X)/F 0 (X) is Drazin 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 [4], 5-Weyl operators and B-Weyl spectrum were defined as follows: DEFINITION …”

WEYL'S TYPE THEOREMS FOR ALGEBRAICALLY (p, k)-QUASIHYPONORMAL OPERATORS

“…According to [27], we say that T satisfies a- Weyl In [12] Berkani define the class of B-Fredholm operators as follows. For each integer n, define T n to be the restriction of T to R(T n ) viewed as a map from R(T n ) into R(T n ) (in particular T 0 = T ).…”