Generalized Jacobson’s lemma for generalized Drazin inverses

Huanyin Chen; Marjan Abdolyousefi

doi:10.2298/fil2107267c

Generalized Jacobson’s lemma for generalized Drazin inverses

Filomat ◽

10.2298/fil2107267c ◽

2021 ◽

Vol 35 (7) ◽

pp. 2267-2275

Author(s):

Huanyin Chen ◽

Marjan Abdolyousefi

Keyword(s):

Drazin Inverse ◽

Multilinear Algebra ◽

Jacobson’S Lemma

We present new generalized Jacobson?s lemma for generalized Drazin inverses. This extends the main results on g-Drazin inverse of Yan, Zeng and Zhu (Linear & Multilinear Algebra, 68(2020), 81-93).

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Jacobson’s lemma for the generalized Drazin inverse

Linear Algebra and its Applications ◽

10.1016/j.laa.2011.07.044 ◽

2012 ◽

Vol 436 (3) ◽

pp. 742-746 ◽

Cited By ~ 16

Author(s):

Guifen Zhuang ◽

Jianlong Chen ◽

Jian Cui

Keyword(s):

Drazin Inverse ◽

Generalized Drazin Inverse ◽

Jacobson’S Lemma

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On gs-Drazin inverses in a ring

Journal of Algebra and Its Applications ◽

10.1142/s0219498820500292 ◽

2019 ◽

Vol 19 (02) ◽

pp. 2050029 ◽

Cited By ~ 1

Author(s):

Huanyin Chen ◽

Mete Burak Calci

Keyword(s):

Drazin Inverse ◽

Cline’S Formula ◽

Jacobson’S Lemma

An element [Formula: see text] in a ring [Formula: see text] has a gs-Drazin inverse if there exists [Formula: see text] such that [Formula: see text]. In this paper, we extend Cline’s formula and Jacobson’s Lemma for gs-Drazin inverses. Various additive properties of gs-Drazin inverses are thereby obtained.

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Jacobson’s Lemma via Gröbner-Shirshov Bases

Algebra Colloquium ◽

10.1142/s1005386717000189 ◽

2017 ◽

Vol 24 (02) ◽

pp. 309-314

Author(s):

Xiangui Zhao

Keyword(s):

Generalized Inverses ◽

Drazin Inverse ◽

Generalized Drazin Inverse ◽

Jacobson’S Lemma

Let R be a ring with identity 1. Jacobson’s lemma states that for any [Formula: see text], if 1− ab is invertible then so is 1 − ba. Jacobson’s lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse, generalized Drazin inverse, and inner inverse. In this note we give a constructive way via Gröbner-Shirshov basis theory to obtain the inverse of 1 − ab in terms of (1 − ba)−1, assuming the latter exists.

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Cline’s formula for g-Drazin inverses in a ring

Filomat ◽

10.2298/fil1908249c ◽

2019 ◽

Vol 33 (8) ◽

pp. 2249-2255

Author(s):

Huanyin Chen ◽

Marjan Abdolyousefi

Keyword(s):

Operator Theory ◽

Spectral Properties ◽

Associative Ring ◽

Linear Operators ◽

Drazin Inverse ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Cline’S Formula

It is well known that for an associative ring R, if ab has g-Drazin inverse then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula is so-called Cline?s formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory. In this paper, we generalize Cline?s formula to the wider case. In particular, as applications, we obtain new common spectral properties of bounded linear operators.

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