Jacobson's lemma for the generalized \(n\)-strong Drazin inverses in rings and in operator algebras

In this paper, we extend Jacobson's lemma for Drazin inverses to the generalized \(n\)-strong in a ring, and prove that \(1-ac\) is invertible if only \(1-ba\) invertible, provided \(a(ba)^{2}=abaca=acaba=(ac)^{2}a\). addition, left right Fredholm operators, furthermore, consistent invertibility spectral property index are investigated.