In this paper, we discuss the common properties for the products ac and ba in various categories under the condition a(ba) 2 = abaca = acaba = (ac) 2 a. We prove that generalized Jacobson's lemma and Cline's formula are suitable for generalized n-strong Drazin invertible in rings, and generalized Jacobson's lemma is suitable for left and right Fredholm operator on Banach spaces.

Let R be an associative ring with unit 1, and a, b, c ∈ R satisfy a(ba) 2 = abaca = acaba = (ac) 2 a, this paper proves that 1 − ac has generalized Drazin inverse (Drazin inverse, pseudo Drazin inverse, respectively) if and only if 1 − ba has generalized Drazin inverse (Drazin inverse, pseudo Drazin inverse, respectively). In particular, we obtain new common spectral properties for ac and ba in Banach algebras. As applications, new extension of Jacobson's lemma for B-Fredholm elements and generalized Fredholm elements in rings is established.

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

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