scholarly journals Jacobson’s lemma for the generalized Drazin inverse

2012 ◽  
Vol 436 (3) ◽  
pp. 742-746 ◽  
Author(s):  
Guifen Zhuang ◽  
Jianlong Chen ◽  
Jian Cui
Keyword(s):  
Drazin Inverse ◽  
Generalized Drazin Inverse ◽  
Jacobson’S Lemma

Jacobson’s Lemma via Gröbner-Shirshov Bases

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.

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 .

Weighted generalized Drazin inverse in rings

2016 ◽  
Vol 23 (4) ◽  
pp. 587-594 ◽  
Author(s):  
Dijana Mosić ◽  
Dragan S. Djordjević
Keyword(s):  
Drazin Inverse ◽  
Generalized Drazin Inverse

AbstractIn this paper, we introduce and investigate the weighted generalized Drazin inverse in rings. We also introduce and investigate the weighted EP elements

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.

On the Generalized Drazin Inverse of the Sum of Two Operators

2013 ◽  
Vol 846-847 ◽  
pp. 1286-1290
Author(s):  
Shi Qiang Wang ◽  
Li Guo ◽  
Lei Zhang
Keyword(s):  
Banach Space ◽  
Explicit Representation ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Generalized Drazin Inverse

In this paper, we investigate additive properties for the generalized Drazin inverse of bounded linear operators on Banach space . We give explicit representation of the generalized Drazin inverse in terms of under some conditions.

scholarly journals The Expression of the Generalized Drazin Inverse ofA−CB

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Xiaoji Liu ◽  
Dengping Tu ◽  
Yaoming Yu
Keyword(s):  
Banach Spaces ◽  
Drazin Inverse ◽  
Generalized Drazin Inverse

We investigate the generalized Drazin inverse ofA−CBover Banach spaces stemmed from the Drazin inverse of a modified matrix and present its expressions under some conditions.

scholarly journals New extensions of Cline’s formula for generalized inverses

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 1973-1980 ◽  
Author(s):  
Qingping Zeng ◽  
Zhenying Wu ◽  
Yongxian Wen
Keyword(s):  
Banach Space ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Space Operator ◽  
Cline’S Formula ◽  
Generalized Drazin Inverse

In this paper, Cline?s formula for the well-known generalized inverses such as Drazin inverse, pseudo Drazin inverse and generalized Drazin inverse is extended to the case when ( acd = dbd dba = aca. Also, applications are given to some interesting Banach space operator properties like algebraic, meromorphic, polaroidness and B-Fredholmness.

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.

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