Research Article | Open Access

Xiaoji Liu, Dengping Tu, Yaoming Yu, "The Expression of the Generalized Drazin Inverse of ", *Abstract and Applied Analysis*, vol. 2012, Article ID 412872, 10 pages, 2012. https://doi.org/10.1155/2012/412872

# The Expression of the Generalized Drazin Inverse of

**Academic Editor:**OndΕej DoΕ‘lΓ½

#### 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,Theoremββ4.2]). 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, [1β5, 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, Theoremββ2.3]). * Let be the generalized Drazin invertible. If , then is generalized Drazin invertible and
*

Lemma 1.2 (see [7, Theoremββ5.1]). *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, Theoremββ2.4]). *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, Theoremββ4.3]). *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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#### Copyright

Copyright Β© 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.