The g-Drazin inverse of the sum in Banach algebras

ABSTRACT

Let $\mathcal{A}$ be a complex Banach algebra. An element $a\in \mathcal{A}$ has g-Drazin inverse if there exists $b\in \mathcal{A}$ such that $b=bab,ab=ba,a-a^{2}b\in A^{qnil}.$ New additive results for the g-Drazin inverse in a Banach algebra are presented. These extend the main results of Deng and Wei (J. Math. Analysis Appl., 379(2010), 313–321) and Wang et al. (Filomat, 30(2016), 1185–1193).

Keywords:

• g-Drazin inverse
• multiplicative property
• additive property
• perturbation
• Banach algebra

AMS Subject Classifications:

• 15A09
• 47L10
• 32A65

Acknowledgments

The authors would like to thank the referee for his/her helpful suggestions for the improvement of this paper. The very detailed comments improve many proofs of the paper, e.g. Theorem 3.5.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was supported by the Natural Science Foundation of Zhejiang Province, China (No. LY17A010018).