# Generalized Cline's Formula for G-Drazin Inverse

## TL;DR

This paper extends Cline's formula to the G-Drazin inverse in associative rings, providing new conditions under which the inverse exists and exploring spectral properties of operators on Banach spaces.

## Contribution

It generalizes Cline's formula for the G-Drazin inverse in rings, offering new theoretical insights and applications to operator spectral theory.

## Key findings

- Established conditions for the existence of G-Drazin inverse of bd
- Derived spectral properties of bounded linear operators
- Extended Cline's formula to pseudo Drazin and Drazin inverses

## Abstract

Let $R$ be an associative ring with an identity and suppose that $a,b,c,d \in R$ satisfy $bdb = bac,dbd = acd$. If $ac$ has generalized Drazin ( respectively, pseudo Drazin, Drazin) inverse, we prove that $bd$ has generalized Drazin (respectively, pseudo Drazin, Drazin) inverse. In particular, as applications, we obtain new common spectral properties of bounded linear operators over Banach spaces.