Abstract
In this article, we investigate additive properties on the Drazin inverse of elements in rings. Under the commutative condition of ab = ba, we show that a + b is Drazin invertible if and only if 1 + a D b is Drazin invertible. Not only the explicit representations of the Drazin inverse (a + b) D in terms of a, a D , b and b D , but also (1 + a D b) D is given. Further, the same property is inherited by the generalized Drazin invertibility in a Banach algebra and is extended to bounded linear operators.
Acknowledgements
We would like to thank Prof. R.B. Bapat and the referee for their useful comments which greatly improved the presentation of this article. J.L. Chen is supported by the National Natural Science Foundation of China under grant 10971024, the Specialized Research Fund for the Doctoral Program of Higher Education under grant 200802860024, and the Natural Science Foundation of Jiangsu Province under grant BK2010393. Y. Wei is supported by the National Natural Science Foundation of China under grant 10871051, Doctoral Program of the Ministry of Education under grant 20090071110003 and Shanghai Education Committee (Dawn Project).