To complete the study of Fredholm type operators of 10 and 11, we define in this paper the classes of left and right semi-B-Fredholm operators (Definition 3.1). Then we prove that an operator T ? L(X), X being a Banach space, is a left (resp. right) semi-B-Fredholm operator if and only if T is the direct sum of a left (resp. right) semi-Fredholm operator and a nilpotent one. This result extend the earlier characterization of B-Fredholm operators as the direct sum of a Fredholm operator and a nilpotent one obtained in 4, Theorem 2.7 and extend the Kato decomposition 14, Theorem 4 for these new classes of operators.
Hamdan et al. (Wed,) studied this question.