Abstract. An operator T on a Banach space is called 'semi B-Fredholm' if for some n 2 N the range RðT n Þ of T n is closed and the induced operator T n on RðT n Þ semi-Fredholm. Semi B-Fredholm 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'.1991 Mathematics Subject Classification. 47A53, 47A55.1. Introduction. The first author [1] has studied B-Fredholm operators on Banach spaces, defined as operators for which some power T n has closed range RðT n Þ, on which the restriction T n is Fredholm, in the sense of having null space NðT Þ of finite dimension ðT Þ and range RðT Þ of finite codimension ðT Þ; the difference indðT Þ ¼ ðT Þ À ðT Þ is known as the index. In this note we extend our study to ''semi B-Fredholm '' operators, for which T n is either upper or lower semiFredholm, in the sense that either NðT Þ is finite dimensional and RðT Þ closed, or RðT Þ is closed of finite codimension. We shall see that the semi B-Fredholm operators SBFðX Þ on a Banach space X in general properly contain the semi-Fredholm operators SFðX Þ and we prove that each semi B-Fredholm operator is a quasiFredholm operator in the sense defined by Mbekhta and Muller in [10], a definition which coincides with the definition given in the case of operators acting on a Hilbert space by Labrousse in [9] . Conversely a quasi-Fredholm operator for which there exists an integer n such that NðT Þ \ RðT n Þ is of finite dimension or NðT n Þ þ RðT Þ is of finite codimension is a semi B-Fredholm operator.In Theorem 2.6 and in the case of operators acting on a Hilbert space H we prove that T 2 LðHÞ is a semi B-Fredholm operator if and only if T ¼ Q È F, where Q is a nilpotent operator and F is a semi-Fredholm operator. But we do not know if this characterization is still valid in the case of operators acting on a Banach space. In Proposition 2.7, we prove that if T 2 LðX Þ is a semi B-Fredholm operator and if F 2 LðX Þ is a finite dimensional operator then T þ F is also a semi B-Fredholm operator.

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].

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We study the new properties (b) and (gb), which we had introduced in [13], for an operator having the SVEP on the complement of distinguished parts of its spectrum. Classes of operators are considered as illustrating examples. Mathematics Subject Classification (2010). 47A53, 47A10, 47A11. Keywords. property (b), property (gb), B-Fredholm operator, generalized Browder's theorem, generalized a-Browder's theorem, single valued extension property.

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