On the spectral identities and fundamental properties of one-sided Drazin inverses in Banach algebras

We establish several fundamental properties of one-sided (generalized) Drazin inverses in Banach algebras, including intertwining properties and reverse order laws. In particular, we introduce the concepts of one-sided strongly pi-regularity, which is shown to be equivalent to one-sided Drazin invertibility. By utilizing the Jacobson’s lemma for one-sided regularity, we prove the Jacobson’s lemma for one-sided (generalized) Drazin invertibility. These results allow us to derive the spectral identities for one-sided (generalized) Drazin invertible spectra in Banach algebras, as well as the spectral identities for Fredholm type operators acting on Banach spaces.