Generalized Jacobson's lemma for Drazin inverses and its applications
Kai Yan, Qingping Zeng, Yu-Can Zhu · Linear and Multilinear Algebra · 2018
Let R be any associative ring with unit 1 and suppose that a, b, c, d∈R satisfy acd=dbd,dba=aca; we prove that 1−ac is Drazin (respectively, generalized Drazin, pseudo-Drazin) invertible if and only if 1−bd is Drazin (respectively, generalized Drazin, pseudo-Drazin) invertible. In other words, we give an affirmative answer to the conjecture of D. Mosić. Moreover, some applications to Banach algebra elements and Banach space operators are also given.