Extension of generalized strong Drazin inverse
Dijana Mosić, Hong in Zou · Operators and Matrices · 2021
As an extension of the generalized strong Drazin inverse, we present a new generalized inverse for Banach algebra elements based on a g-Drazin invertible element rather than on a quasinilpotent element in the definition of the generalized strong Drazin inverse. Because of that, our new inverse will be called an extended gs-Drazin inverse. Some characterizations of this inverse are given using idempotents and tripotents. We also study extensions of Cline's formula to the case of extended gs-Drazin inverse. Applying these results, we introduce and investigate an extended s-Drazin inverse.