New additive results for the generalized Drazin inverse in a Banach algebra

Liu, Xiaoji; Qin, Xiaolan; Benítez, Julio

doi:10.2298/fil1608289l

Cited by 15 publications

(7 citation statements)

References 7 publications

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“…By the following examples, we observe that the conditions aba π = 0 (which appears in [5]) and aba π = a π b π bab π a π are independent. Precisely, in the first example, the condition aba π = 0 holds, but the condition aba π = a π b π bab π a π is not satisfied.…”

Section: Additive Resultsmentioning

confidence: 90%

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The generalized Drazin inverse of the sum in a Banach algebra

Mosić¹,

Zou²,

Chen³

2017

Ann. Funct. Anal.

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Section: Additive Resultsmentioning

confidence: 90%

“…Using the assumptions a π b = b and aba π = 0, the representation for (a + b) d was presented in [1]. In [5], the formula for (a + b) d was given under conditions which involve aba π = 0.…”

Section: Introductionmentioning

confidence: 99%

The generalized Drazin inverse of the sum in a Banach algebra

Mosić¹,

Zou²,

Chen³

2017

Ann. Funct. Anal.

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“…See [14]. The motivation for this article is [15]. In the paper, the authors considered some conditions on a, b∈Λ(Let Λ be a complex Banach algebra with the unit 1)that allowed them to express (a + b) d in terms of a, ad, b, bd.…”

Section: Cb π =mentioning

confidence: 99%

A Note on the Formulas for the Drazin Inverse of the Sum of Two Matrices and Its Applications

Qin¹,

Luo²

2019

PAMJ

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The Drazin inverse has applications in a number of areas such as control theory, Markov chains, singular differential and difference equations, and iterative methods in numerical linear algebra. The study on representations for the Drazin inverse of block matrices stems essentially from finding the general expressions for the solutions to singular systems of differential equations, and then stimulated by a problem formulated by Campbell. In 1983, Campbell (Campbell et al. (1976)) established an explicit representation for the Drazin inverse of a 2 × 2 block matrix M in terms of the blocks of the partition, where the blocks A and D are assumed to be square matrices. Special cases of the problems have been studied. In 2009, Chunyuan Deng and Yimin Wei found an explicit representation for the Drazin inverse of an anti-triangular matrix M, where A and BC are generalized Drazin invertible, if A π AB=0 and BC (I-A π)=0. Afterwards, several authors have investigated this problem under some limited conditions on the blocks of M. In particular, a representation of the Drazin inverse of M, denoted by M d. In this paper, we consider the Drazin inverse of a sum of two matrices and we derive additive formulas under the conditions of ABA π =0 and BA π =0 respectively. Precisely, for a block matrix M, we give a new representation of M d under some conditions that AB=0 and DCA π =0. Moreover, some particular cases of this result related to the Drazin inverse of block matrices are also considered.

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“…The popularity of the Drazin inverse is confirmed by many recently published papers [1,3,4,23,24,30,31,[37][38][39], and by its applications in singular differential and difference equations, Markov chains, eigenvalues, and semi-iterative methods [8,9,26,27,29].…”

Section: Introductionmentioning

confidence: 99%

The Drazin inverse of anti-triangular block matrices

Zhang

Jin

Mosić

2021

J. Appl. Math. Comput.

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The aim of this paper is to establish formulae for the Drazin inverse of anti-triangular block matrices under new assumptions in literature. Precisely, we consider the Drazin inverse of three kinds of anti-triangular block matrices. Applying these results, we present new expressions for the Drazin inverse of an arbitrary block matrix. In this way, we generalize a list of earlier results and illustrate it with three examples.

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New additive results for the generalized Drazin inverse in a Banach algebra

Abstract: In this paper, we investigate additive properties of the generalized Drazin inverse in a Banach algebra A. We find explicit expressions for the generalized Drazin inverse of the sum a + b, under new conditions on a, b ∈ A.

Cited by 15 publications

References 7 publications

The generalized Drazin inverse of the sum in a Banach algebra

The generalized Drazin inverse of the sum in a Banach algebra

A Note on the Formulas for the Drazin Inverse of the Sum of Two Matrices and Its Applications

The Drazin inverse of anti-triangular block matrices

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