Kato Decomposition for Left and Right semi-B-Fredholm Operators

Abstract

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.