Abstract and Applied AnalysisVolume 2012 (2012), Article ID 412872, 10 pagesdoi:10.1155/2012/412872

Research Article

The Expression of the Generalized Drazin Inverse of

Xiaoji Liu,1,2 Dengping Tu,¹ and Yaoming Yu³

¹School of Science, Guangxi University for Nationalities, Nanning 530006, China²Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis, Nanning 530006, China³School of Mathematical Sciences, Monash University, Caulfield East, VIC 3800, Australia

Received 5 August 2011; Accepted 5 December 2011

Academic Editor: Ondřej Došlý

Copyright © 2012 Xiaoji Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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

1. Introduction

Let and be Banach spaces. We denote the set of all bounded linear operators from to by . In particular, we write instead of .

For any , and represent its range and null space, respectively. If , the symbols and stand for its spectrum and the set of all accumulation points of , respectively.

Recall the concept of the generalized Drazin inverse introduced by Koliha [1] that the element is called the generalized Drazin inverse of provided it satisfies If it exists then it is unique. The Drazin index of is the least positive integer if , and otherwise .

From the definition of the generalized Drazin inverse, it is easy to see that if is a quasinilpotent operator, then exists and . It is well known that the generalized Drazin inverse of exists if and only if (see [1,Theorem  4.2]). If is generalized Drazin invertible, then the spectral idempotent of corresponding to 0 is given by .

The generalized Drazin inverse is widely investigated because of its applications in singular differential difference equations, Markor chains, (semi-) iterative method numerical analysis (see, for example, [1–5, 7], and references therein).

In this paper, we aim to discuss the generalized Drazin inverse of over Banach spaces. This question stems from the Drazin inverse of a modified matrix (see, e.g., [6]). In [3], Deng studied the generalized Drazin inverse of . Here we research the problem under more general conditions than those in [3]. Our results extend the relative results in [3, 4].

In this section, we will list some lemmas. In next section, we will present the expressions of the generalized Drazin inverse of . In final section, we illustrate a simple example.

Lemma 1.1 (see [4, Theorem  2.3]). Let be the generalized Drazin invertible. If , then is generalized Drazin invertible and

Lemma 1.2 (see [7, Theorem  5.1]). If and are generalized Drazin invertible and , then is also generalized Drazin invertible and where

2. Main Results

We start with our main result.

Theorem 2.1. Let be the generalized Drazin invertible, , and . Suppose that there exists a such that and . If and are generalized Drazin invertible, then is generalized Drazin invertible and where and the symbols , if .

Proof. Let and . Then since and . So, by Lemma 1.1, Next, we will give the representations of , , and . In order to obtain the expression of , rewrite as Since , and then for since . So exists and . By (2.3), and then . So, by Lemma 1.1, and then Since and for , From , it is easy to verify that Hence, Therefore, we reach (2.1).

When , we have the following corollary.

Corollary 2.2. Let be generalized Drazin invertible. , and . Suppose that there exists a such that and . If and are generalized Drazin invertible and and , then is generalized Drazin invertible and where and the symbols , if .

If an operator is quasinilpotent, and . So, the following corollary follows from Theorem 2.1.

Corollary 2.3. Let be generalized Drazin invertible, , and . Suppose that there exists a such that and . If is generalized Drazin invertible and is a quasinilpotent operator, then is generalized Drazin invertible and where .

Theorem 2.4. Let be generalized Drazin invertible, , and . Suppose that there exists an idempotent such that and . If is generalized Drazin invertible, then is generalized Drazin invertible and

Proof. Since , we have and can write in the following matrix form: The condition , therefore, yields the matrix form of as follows: From and the hypothesis that exists, and are generalized Drazin invertible since if and only if and . And, by Lemma 1.2, where is some operator. Since exists and To use Theorem 2.1 to complete the proof, let . So and are generalized Drazin invertible. And from the conditions and , we can obtain and . Thus, by Theorem 2.1, we have where .Since and and then and . So . Note that and then and . Thus it follows from (2.21) that Since , and . Note that and . Substituting and in (2.22) yields (2.15).

Adding the condition in Theorem 2.4 yields a result below.

Corollary 2.5. Let be generalized Drazin invertible, , and . Suppose that there exists an idempotent such that , , and . If is generalized Drazin invertible, then is generalized Drazin invertible and

Adding the condition in Theorem 2.4 yields . So similar to the proof of in Theorem 2.4, we can gain .

Corollary 2.6. Let be generalized Drazin invertible, , and . Suppose that there exists an idempotent such that , , and ; then is generalized Drazin invertible and

Analogously, we can deduce Theorem 2.7 and Corollary 2.9 below.

Theorem 2.7. Let be generalized Drazin invertible, , and . Suppose that there exists an idempotent such that and . If is generalized Drazin invertible, then is generalized Drazin invertible and

Remark 2.8 (see [4, Theorem  2.4]). It is a special case of Theorem 2.7.

Corollary 2.9. Let be generalized Drazin invertible, , and . Suppose that there exists an idempotent such that , , and ; then is generalized Drazin invertible and

Similar to Theorem 2.1 and Corollary 2.2, we can show the following two results.

Theorem 2.10. Let be generalized Drazin invertible, , and . Suppose that there exists a such that and . If and are generalized Drazin invertible, then is generalized Drazin invertible and where and the symbols , if .

Corollary 2.11. Let be generalized Drazin invertible. , and . Suppose that there exists a such that and . If and are generalized Drazin invertible and and , then is generalized Drazin invertible and where and the symbols , if .

When and in Theorem 2.10, we can obtain the following result since .

Corollary 2.12 (see [3, Theorem  4.3]). Let be the generalized Drazin invertible, , and . Suppose that there exists an idempotent commuting with such that . If is generalized Drazin invertible, then is the generalized Drazin invertible and where .

3. Example

Before ending this paper, we give an example as follows.

Example 3.1. Let Then We will compute the Drazin inverse of . To do this, we choose the matrix Apparently, is not idempotent and . But and Obviously, . Computing we have . So, by Corollary 2.2,

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (11061005), the Ministry of Education Science and Technology Key Project under Grant no. 210164, and Grants HCIC201103 of Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis Open Fund.

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