The generalized and pseudo n-strong Drazin inverses in rings

Abstract

We introduce new classes of generalized Drazin inverses in a ring, which are called the generalized n-strong Drazin inverse and pseudo n-strong Drazin inverse for arbitrary $n\in N$. Some basic properties of these inverses are studied. We prove extensions of Cline's formula for these inverses and Jacobson's lemma for generalized and pseudo strong Drazin inverse. In a Banach algebra, we define and characterize the weighted generalized strong Drazin inverse and weighted pseudo strong Drazin inverse.

COMMUNICATED BY:

• Ravindra B. Bapat

Keywords:

• Generalized strong Drazin inverse
• generalized Drazin inverse
• Cline's formula
• Jacobson's lemma
• rings

2010 Mathematics subject classifications:

• 16B99
• 15A09
• 16U99

Acknowledgments

The author is grateful to referees for constructive comments and careful reading of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The author is supported by the Ministry of Education, Science and Technological Development (Ministarstvo Prosvete, Nauke i Tehnološkog Razvoja), Republic of Serbia [grant no. 174007].