On the spectral identities and fundamental properties of one-sided Drazin inverses in Banach algebras
Kai Yan
TL;DR
This work develops a comprehensive framework for one-sided (generalized) Drazin inverses in Banach algebras, introducing one-sided strongly $\pi$-regularity and proving a one-sided Jacobson’s lemma that underpins spectral identities for one-sided Drazin spectra. It establishes intertwining-based equivalences and explicit inverse constructions under natural intertwinings, and derives spectral identities for products and Fredholm-type operators, including B-Fredholm spectra via the quotient map to $B(X)/F(X)$. The results extend classical Drazin theory to the one-sided setting, yielding practical tools for analyzing spectral properties of operators in Banach spaces and their Banach-algebra framework.
Abstract
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.