Let \(X\) be an infinite-dimensional complex Banach space and \(L(X)\) be the algebra of all bounded linear operators acting on \(X\). For \(T\in L(X)\), denote by \(N(T)\) and \( R(T)\) the null space of \(T\) and the range of \(T\), respectively. \(T\) is said to be quasi-Fredholm if there exists a positive integer \(d\) such that \( R(T^{d+1})\) is closed and \(N(T)\cap R(T^n)=N(T)\cap R(T^{n+1})\) for all \(n\geq d\). \(T\) is said to be semi-regular if \(R(T)\) is closed and \(N(T)\subset R(T^n)\) for every \(n\). An operator \(A\) is called lower semi Saphar if \(A\) is semi-regular and \(R(A)\) is complemented. \(T\) is said to be generalized lower semi B-Fredholm if \(T\) is quasi-Fredholm and \(N(T^d)+R(T)\) is complemented.
In the paper under review, it is proved that an operator \(T\in L(X)\) is generalized lower semi B-Fredholm if and only if there exist two \(T\)-invariant subspaces \(M\) and \(N\) such that \(M\) is closed, \(X = M \oplus N\), \(T|_M\) is lower semi Saphar and \(T|_N\) is nilpotent. Symmetrically, a characterization of generalized upper semi B-Fredholm operators is obtained.