Additive property of Drazin invertibility of elements in a ring

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...

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Published in:Linear & multilinear algebra 2012-08, Vol.60 (8), p.903-910
Main Authors: Zhuang, Guifen, Chen, Jianlong, Cvetković-Ilić, Dragana S., Wei, Yimin
Format: Article
Language:English
Subjects:
Additives
Algebra
Banach algebra
Drazin inverse
generalized Drazin inverse
Inverse
Linear operators
Representations
ring
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description 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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subjects Additives
Algebra
Banach algebra
Drazin inverse
generalized Drazin inverse
Inverse
Linear operators
Representations
ring
title Additive property of Drazin invertibility of elements in a ring
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