Some results on the Drazin invertibility and idempotents

Cvetković-Ilić, Dragana S.; Deng, Chunyuan

doi:10.1016/j.jmaa.2009.05.062

Cited by 32 publications

(21 citation statements)

References 12 publications

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“…In our proof we succeed in avoiding the use of Cline's formula (ba) D = b((ab) D ) 2 a originally employed in [4], which would require a proof for the generalized Drazin inverse.…”

Section: Lemma 23 Let a ∈ A Let F Be A Function Holomorphic In An mentioning

confidence: 99%

“…We observe that [4] states incorrectly that a ∈ A is Drazin invertible if and only if 0 is not an accumulation point of σ (a). Some of the proofs use this characterization, and those results are then in fact proved for the generalized Drazin inverse rather than for the Drazin inverse.…”

Section: Introductionmentioning

confidence: 98%

“…In this paper we highlight the fact that for Drazin and generalized Drazin invertibility we have the equivalence of the three conditions of p − q ∈ A D , p + q ∈ A D and 1 − pq ∈ A D , with a view to correcting the results of [4].…”

Section: Introductionmentioning

confidence: 98%

“…Consequently, this topic received attention in a number of papers, extending the ordinary invertibility to the Drazin invertibility, for instance [2,1,[4][5][6][7]13], and the difference and sum to a linear combination of idempotents [2,1,4,13]. These references by no means exhaust the literature on the topic.…”

Section: Introductionmentioning

confidence: 99%

“…Some of the proofs use this characterization, and those results are then in fact proved for the generalized Drazin inverse rather than for the Drazin inverse. This means that some results in [4] are proved only for the generalized Drazin inverse, and the rest only for the Drazin inverse. We intend to rectify this by establishing the results in both cases.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Generalized Drazin invertibility of combinations of idempotents

Koliha

Cvetković-Ilić

Deng

2012

Linear Algebra and its Applications

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The paper serves as a correction to J. Math. Anal. Appl. 359 (2009) 731-738 dealing with the Drazin invertibility of combinations of idempotents p, q in a Banach algebra. As the arguments employed to do this work equally well for both the usual and generalized Drazin inverse, the latter is included in the discussion which covers the equivalence of the Drazin invertibility of 1 − pq, p − q, p + q and in general of αp + βq with αβ = 0, as well as the equivalence of the Drazin invertibility of the commutator pq − qp and anticommutator pq + qp of p, q.

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“…In our proof we succeed in avoiding the use of Cline's formula (ba) D = b((ab) D ) 2 a originally employed in [4], which would require a proof for the generalized Drazin inverse.…”

Section: Lemma 23 Let a ∈ A Let F Be A Function Holomorphic In An mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%