scholarly journals Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 105 ◽  
Author(s):  
Yonghui Qin ◽  
Xiaoji Liu ◽  
Julio Benítez
Keyword(s):  
Banach Algebra ◽  
Drazin Inverse ◽  
Symmetric Representation ◽  
Generalized Drazin Inverse

Based on the conditions a b 2 = 0 and b π ( a b ) ∈ A d , we derive that ( a b ) n , ( b a ) n , and a b + b a are all generalized Drazin invertible in a Banach algebra A , where n ∈ N and a and b are elements of A . By using these results, some results on the symmetry representations for the generalized Drazin inverse of a b + b a are given. We also consider that additive properties for the generalized Drazin inverse of the sum a + b .

Expressions for the G-Drazin Inverse of Additive Perturbed Elements

2011 ◽  
Vol 88-89 ◽  
pp. 509-514
Author(s):  
Li Guo ◽  
Yu Jing Liu
Keyword(s):  
Banach Algebra ◽  
Explicit Representation ◽  
Drazin Inverse ◽  
Generalized Drazin Inverse

To study the properties of the generalized Drazin inverse in a Banach algebra, an explicit representation of the generalized Drazin inverse under the some conditions. Thus some results are generalized.

scholarly journals Expressions for the g-Drazin inverse in a Banach algebra

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3845-3854
Author(s):  
Huanyin Chen ◽  
Marjan Sheibani
Keyword(s):  
Banach Algebra ◽  
Complex Number ◽  
Block Matrix ◽  
Explicit Representation ◽  
Drazin Inverse ◽  
Operator Matrices ◽  
Nonzero Complex Number ◽  
Generalized Drazin Inverse

We explore the generalized Drazin inverse in a Banach algebra. Let A be a Banach algebra, and let a,b ? Ad. If ab = ?a?bab? for a nonzero complex number ?, then a + b ? Ad. The explicit representation of (a + b)d is presented. As applications of our results, we present new representations for the generalized Drazin inverse of a block matrix in a Banach algebra. The main results of Liu and Qin [Representations for the generalized Drazin inverse of the sum in a Banach algebra and its application for some operator matrices, Sci. World J., 2015, 156934.8] are extended.

scholarly journals Generalized cline’s formula for g-Drazin inverse in a ring

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2573-2583
Author(s):  
Huanyin Chen ◽  
Marjan Abdolyousefi
Keyword(s):  
Banach Algebra ◽  
Spectral Property ◽  
Banach Spaces ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear ◽  
Group Inverses ◽  
Cline’S Formula ◽  
Generalized Drazin Inverse ◽  
Weaker Conditions

In this paper, we give a generalized Cline?s formula for the generalized Drazin inverse. Let R be a ring, and let a, b, c, d ? R satisfying (ac)2 = (db)(ac), (db)2 = (ac)(db), b(ac)a = b(db)a, c(ac)d = c(db)d. Then ac ? Rd if and only if bd ? Rd. In this case, (bd)d = b((ac)d)2d: We also present generalized Cline?s formulas for Drazin and group inverses. Some weaker conditions in a Banach algebra are also investigated. These extend the main results of Cline?s formula on g-Drazin inverse of Liao, Chen and Cui (Bull. Malays. Math. Soc., 37(2014), 37-42), Lian and Zeng (Turk. J. Math., 40(2016), 161-165) and Miller and Zguitti (Rend. Circ. Mat. Palermo, II. Ser., 67(2018), 105-114). As an application, new common spectral property of bounded linear operators over Banach spaces is obtained.

scholarly journals Explicit formulae for the generalized drazin inverse of block matrices over a Banach algebra

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5907-5917
Author(s):  
Daochang Zhang ◽  
Dijana Mosic
Keyword(s):  
Banach Algebra ◽  
Block Matrix ◽  
Drazin Inverse ◽  
Block Matrices ◽  
Explicit Formulae ◽  
Generalized Drazin Inverse

In this paper, we give expressions for the generalized Drazin inverse of a (2,2,0) block matrix over a Banach algebra under certain circumstances, utilizing which we derive the generalized Drazin inverse of a 2x2 block matrix in a Banach algebra under weaker restrictions. Our results generalize and unify several results in the literature.

scholarly journals Additive results for the generalized Drazin inverse in a Banach algebra

2006 ◽  
Vol 418 (1) ◽  
pp. 53-61 ◽  
Author(s):  
Dragana S. Cvetković-Ilić ◽  
Dragan S. Djordjević ◽  
Yimin Wei
Keyword(s):  
Banach Algebra ◽  
Drazin Inverse ◽  
Generalized Drazin Inverse

scholarly journals On the Generalized Drazin inverse of the Sum in a Banach Algebra

2018 ◽  
Vol 42 (10) ◽  
pp. 1335-1344 ◽  
Author(s):  
Dijana Mosić ◽  
Daochang Zhang
Keyword(s):  
Banach Algebra ◽  
Drazin Inverse ◽  
Generalized Drazin Inverse
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