Algebraic Properties of Generalized Inverses

Cvetković-Ilić, Dragana S.; Wei, Yimin

doi:10.1007/978-981-10-6349-7

Cited by 64 publications

(25 citation statements)

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“…Proof. We know (7) ⇔ (8) by Theorem 2.2 and Corollary 2.1, so we only need to prove the equivalence of (1)- (7).…”

Section: Corollary 23 If a Bmentioning

confidence: 99%

“…In general, the previous equality doesn't hold when the ordinary inverse is replaced by generalized inverse. Since the reverse order law for the generalized inverse is a useful computational tool in applications and it is significant from the theoretical point of view, many papers characterized the reverse order laws, such as [2][3][4][5][6][7][8][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

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Reverse order laws for Hirano inverses in rings

Chen

Zou

2019

Filomat

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In this paper, various kinds of reverse order laws for the Hirano inverse are characterized in a ring with the unit 1. Under some of conditions aba = a 2 b, ab 2 = bab, a h ab 2 = ba h ab, a 2 bb h = abb h a instead of the condition ab = ba, respectively, we present some equivalent conditions of the reverse order laws for Hirano inverses. Definition 1.1. Let a ∈ R. If there exists b ∈ R such that (2) bab = b, (5) ab = ba, (7) a 2 − ab ∈ N(R), then b is called the Hirano inverse of a, and a is Hirano invertible. Remark 1. For a ∈ R, the set of elements satisfying the (i)th equation in Definition 1.1 is denoted by a{i}, where i ∈ {2, 5, 7}. And the set of Hirano invertible elements in R is denoted by R h. Definition 1.2. Let a ∈ R. If there exists b ∈ R such that (2) bab = b, (5) ab = ba, (7) a − a 2 b ∈ N(R), then b is called the Drazin inverse of a, and a is Drazin invertible.

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“…Proof. We know (7) ⇔ (8) by Theorem 2.2 and Corollary 2.1, so we only need to prove the equivalence of (1)- (7).…”

Section: Corollary 23 If a Bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Reverse order laws for Hirano inverses in rings

Chen

Zou

2019

Filomat

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show abstract

“…There are several papers on the core inverse and core-EP inverse [1][2][3][4][5][7][8][9]. Recently, there are recent monographs [10][11][12] on the generalized inverse.…”

Section: Introductionmentioning

confidence: 99%

Perturbation theory for core and core-EP inverses of tensor via Einstein product

Wang

2019

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“…(1) axa = a, (2) xax = x, (3) (ax) * = ax, (4) (xa) * = xa, if it exists, is called the Moore-Penrose inverse of a and is denoted by a † . From the definition of the Moore-Penrose inverse, we conclude that both a † a and aa † are projections, where by a projection we mean an element e ∈ A which is a hermitian idempotent, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Notice that one generalization on Hartwig's result is given in [4] for the case of closed-range bounded linear operators on infinite dimensional Hilbert spaces based on operator matrices. For huge number of different reverse order laws see [2]. Also, some interesting results on the reverse order law can be founded in the following papers [11][12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%