Additive Property of Drazin Invertibility of Elements

TL;DR

This paper explores the additive properties of the Drazin inverse in rings and algebras, providing conditions under which sums and differences of elements are Drazin invertible and explicit formulas for their inverses.

Contribution

It establishes new additive properties of the Drazin inverse under weak commutativity and derives explicit formulas for the inverse of sums of elements.

Findings

$a-b$ is Drazin invertible iff $aa^{D}(a-b)bb^{D}$ is Drazin invertible under weak commutativity.

Explicit representation of $(a+b)^{D}$ in terms of $a, b, a^{D}, b^{D}$ under specific conditions.

Conditions $a^{3}b = ba$ and $b^{3}a = ab$ enable explicit formulas for the Drazin inverse of sums.

Abstract

In this article, we investigate additive properties of the Drazin inverse of elements in rings and algebras over an arbitrary field. Under the weakly commutative condition of $ab = \lambda ba$, we show that $a-b$ is Drazin invertible if and only if $aa^{D}(a-b)bb^{D}$ is Drazin invertible. Next, we give explicit representations of $(a+b)^{D}$, as a function of $a, b, a^{D}$ and $b^{D}$, under the conditions $a^{3}b = ba$ and $b^{3}a = ab$.