Pseudo Drazin inverses in associative rings and Banach algebras

Wang, Zhou; Chen, Jianlong

doi:10.1016/j.laa.2012.04.039

Cited by 53 publications

(48 citation statements)

References 17 publications

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“…Using this concept, Koliha and Patrício [6] In 2012, Wang and Chen [9] introduced an intermedium between Drazin inverse and generalized Drazin inverse. An element a ∈ R is said to be pseudo Drazin invertible if there exist b ∈ R and k ∈ N such that In 1965, Cline [1] showed that if ab is Drazin invertible then so is ba and in this case…”

Section: Introductionmentioning

confidence: 99%

“…It plays an important role in finding the Drazin inverse of a sum of two elements and that of a block matrix (see [8]). Generalizations of Cline's formula for generalized Drazin inverse and pseudo Drazin inverse were recently proved in [7] and [9], respectively. Their proof relied on the bridge "quasipolar" and "pseudopolar", respectively.…”

Section: Introductionmentioning

confidence: 99%

“…We also present Cline's formula for the pseudo Drazin inverse in this case. The proofs for the general case are more direct and slightly more technical than those in [7] and/or [9] for the special case b = c .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An extension of Cline''s formula for a generalized Drazin inverse

Lian¹,

Zeng²

2016

Turk J Math

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An extension of Cline''s formula for a generalized Drazin inverse

Lian¹,

Zeng²

2016

Turk J Math

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“…Clearly, comm 2 (a) ⊆ comm(a). This concept is closely related to quasipolar, perfectly clean, and pseudopolar elements (for details see [4,8,9,12]). We end this note by a more explicit result than [5,Theorem 10].…”

Section: Theorem 2 Let R Be a Ring Then The Following Are Equivalentmentioning

confidence: 99%

Uniquely strongly clean triangular matrices

Chen¹,

Gurgun²,

Kose³

2015

Turk J Math

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A ring R is uniquely (strongly) clean provided that for any a ∈ R there exists a unique idempotent e ∈ R ( e ∈ comm(a)) such that a − e ∈ U (R) . We prove, in this note, that a ring R is uniquely clean and uniquely bleached if and only if R is abelian, Tn(R) is uniquely strongly clean for all n ≥ 1 , i.e. every n × n triangular matrix over R is uniquely strongly clean, if and only if R is abelian, and Tn(R) is uniquely strongly clean for some n ≥ 1 . In the commutative case, more explicit results are obtained.

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“…If ind(a) = 1, then b is the group inverse of a and is denoted by a # . In 2012, Wang and Chen [16] introduced the notion of pseudo More results on (generalized) Drazin inverse can be found in [1][2][3][4][5][6]8,9,[11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Additive property of pseudo Drazin inverse of elements in Banach algebras

Zhu

Chen

2014

Filomat

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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 = 0),1 Any element b ∈ A satisfying the conditions above is called a p-Drazin inverse of a, denoted by a ‡ . The set of all p-Drazin invertible elements of A is denoted by A pD .By a Π = 1 − aa ‡ we mean the strongly spectral idempotent of a. It is well known that J(A ) ⊂ A qnil and a ∈ A qnil if and only if a n 1 n → 0 (n → ∞) in a Banach algebra A . Wei and Deng [17] presented some additive results on the Drazin inverse for square complex matrices that commute. Zhuang, Chen et al. [18] extended the results in [17] to

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Pseudo Drazin inverses in associative rings and Banach algebras

Cited by 53 publications

References 17 publications

An extension of Cline''s formula for a generalized Drazin inverse

An extension of Cline''s formula for a generalized Drazin inverse

Uniquely strongly clean triangular matrices

Additive property of pseudo Drazin inverse of elements in Banach algebras

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