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arXiv · 1307.7232

Additive property of pseudo Drazin inverse of elements in a Banach algebra

Huihui Zhu·Jianlong Chen
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Abstract

We study properties of pseudo Drazin inverse in a Banach algebra with unity 1. If $ab=ba$ and $a,b$ are pseudo Drazin invertible, we prove that $a+b$ is pseudo Drazin invertible if and only if $1+a^‡b$ is pseudo Drazin invertible. Moreover, the formula of $(a+b)^‡$ is presented . When the commutative condition is weaken to $ab=λba ~(λ\neq 0)$, we also show that $a-b$ is pseudo Drazin invertible if and only if $aa^‡(a-b)bb^‡$ is pseudo Drazin invertible.

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Huihui Zhu, Jianlong Chen. 2013-07-27. Additive property of pseudo Drazin inverse of elements in a Banach algebra. https://arxiv.org/abs/1307.7232

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