Additive property of Drazin invertibility of elements in a ring |
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| Authors: | Guifen Zhuang Dragana S Cvetkovi?-Ili? Yimin Wei |
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| Institution: | 1. Department of Mathematics , Southeast University , Nanjing 210096 , P.R. China;2. School of Science , Jiangnan University , Wuxi 214122 , P.R. China;3. Department of Mathematics, Faculty of Sciences and Mathematics , University of Ni? , P.O. Box 224, Vi?egradska 33, 18000 Ni? , Serbia;4. School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Sciences , Fudan University , Shanghai 200433 , P.R. China |
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| 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. |
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| Keywords: | Drazin inverse generalized Drazin inverse ring Banach algebra |
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