Abstract and Applied Analysis

Volume 2012, Article ID 629178, 22 pages

http://dx.doi.org/10.1155/2012/629178

## Perturbation Bound of the Group Inverse and the Generalized Schur Complement in Banach Algebra

^{1}Faculty of Science, Guangxi University for Nationalities and Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis, Nanning 530006, China^{2}College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006, China^{3}Department of Computer Science, Fudan University, Shanghai 200433, China

Received 5 April 2012; Revised 26 June 2012; Accepted 11 July 2012

Academic Editor: Patricia J. Y. Wong

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 relative perturbation bound of the group inverse and also consider the perturbation bound of the generalized Schur complement in a Banach algebra.

#### 1. Introduction

Let denote a Banach algebra with unit 1. The symbols , , , , , , and stand for the sets of all invertible, Drazin invertible, generalized Drazin invertible, group invertible, nilpotent, quasinilpotent, and idempotent elements of a Banach algebra , respectively.

Some definitions will be given in the following.

Letting , there is an unique element such that Then is called the Drazin inverse of , denoted by . The smallest nonnegative integer which satisfies (1.1) is called the index of , denoted by . If , then .

Let , if the conditions (1.1) are replaced by Then is called the group inverse of , denoted by . If the conditions (1.1) are replaced by Then is called the generalized Drazin inverse of , denoted by .

Some notations of the Schur complement are given in the following.

For a block complex matrix is defined as where , , , and . If is nonsingular, then the classical Schur complement of in is given as follows (see [1]):

In [2], Benítez and Thome considered the expression and is called the generalized Schur form of the matrix given in (1.4) being for some fixed generalized inverses , , where is called generalized Schur complement of in . In [2, Theorem 2], Benítez and Thome investigated the expression of the group inverse of in (1.4) by the generalized Schur complement, where (1.5) is replaced by Similar results also were given by Sheng and Chen in [3, Theorem 3.2]. The Drazin inverse of a 2 × 2 block complex square matrix in (1.4) with a singular generalized Schur complement was considered in [4–6], where For the expression of a 2 × 2 block operator matrix was investigated by Deng and Wei in [7].

Some notations for the block matrix form of a given element are introduced in [8]. Let and (see [8, Chapter VII]) which denotes the set of all idempotent elements in . Then we write and use the notations For a representation of arbitrary element is given as the following matrix form:

In this paper, we will consider some results on the relative perturbation bounds of group inverse and also give the perturbation bounds of the generalized Schur complement of an element under some certain conditions in a Banach algebra.

#### 2. Perturbation Bound of in Banach Algebra

In recent years, perturbation theory for the Drazin inverse of a given matrix and its applications have been considered in [9–20]. In [12], Yimin and Guorong gave a perturbation result for the Drazin inverse under condition () (see [12] for details). In [8, Ch 5], Djordjević and Rakočević extended the perturbation bound of Yimin and Guorong [12] to Banach algebra. In [13], Wei had discussed the upper perturbation bound of with and had answered the question of Campbell and Meyer [21] when . In [14], Wei and Wu presented the perturbation upper bounds of under the weaker condition and completely answered the question of Campbell and Meyer in [21]. In [16], Wei derived a relative perturbation upper bound of by Jordan canonical of . In [5], Li gave sharper upper bounds for under weaker conditions: and . In [17], Wei et al. derived constructive perturbation bound of the Drazin inverse of a square matrix by using a technique proposed by Stewart and based on perturbation theory for invariant subspaces. In [18], Xu et al. gave some upper bounds for only under the condition that is a stable perturbation of . In [22], González and Koliha investigated the perturbation of the Drazin inverse of a closed linear operator and derived explicit bounds for the perturbations under certain restrictions on the perturbing operators. In [23], González and Vélez-Cerrada analyzed the perturbation of the Drazin inverse and also gave explicit upper bounds of and and obtained a result on the continuity of the group inverse for operators on Banach space.

In this section, we will investigate the relative perturbation bound of the group inverse in Banach algebra.

At first, we will give some concepts and lemmas as follows.

For , let and (see [24]): where is invertible and is quasinilpotent.

For any , we write , , and for the spectrum, the resolvent set, and the spectral radius of , respectively. For and let . If is an isolated point of , then the spectral idempotent corresponding to the set is defined by where is a small circle surrounding and separating from .

Some lemmas will be useful for the following proof in this paper.

Lemma 2.1 (see [24, Theorem 2.3]). *Let , and let . Assume that
*(i)*If and , then and
where . *(ii)*If and , then and is given by (2.4). *

Lemma 2.2 (see [24, Corollary 3.4]). *If are generalized Drazin invertible, is quasinilpotent, and , then is generalized Drazin invertible and
*

The following lemma is a generalization of [25, Theorem 1].

Lemma 2.3. *Let such that . Then if and only if . In this case
*

Now we will state a lemma for the representation of the group inverse of an element with block form in Banach algebra (see [26, Theorem ] and [23, Theorem 2.2.] which were established for a finite dimensional case and partitioned operators matrix, resp.).

Lemma 2.4. *Let , and it has the block matrix form as , where is an idempotent element, is invertible in , and . Let . Then is group invertible if and only if is an invertible element in . In this case
*

Let be a perturbation element of . According to (2.1), we obtain where .

Theorem 2.5. *Let and be a perturbation element of , and which are defined as (2.1) and (2.8), respectively. If , then is invertible in subalgebra . Furthermore let and . Then is group invertible if and only if is invertible and is invertible if and only if is invertible. In this case,
**
where
*

*Proof. *Let . Then , , and have the matrix form as (2.1) and (2.8), respectively, where is invertible in and is quasinilpotent in .

It follows from the hypothesis that . Thus, it implies that is invertible. It is easy to see that . Let ; that is, we have . Therefore, we have
From the previous equations, we get that is invertible if and only if is invertible. Since and by Lemma 2.4, we obtain that is group invertible if and only if is invertible.

In the following, we consider the upper bound of .

Applying Lemma 2.4, we obtain
where .

Note that
where .

It shows from (i.e., ) that

From (2.13), (2.14), (2.15), and by , we obtain that

It follows from (2.12) that

Therefore, according to (2.14), (2.15), and (2.16), we obtain
Since and by (2.18), it is easy to see that the conclusion holds.

Thus, we complete the proof.

Let be both bounded linear operators with on Banach space, where denotes Banach space. If is satisfied, (it implies that and ), then we have the remark.

*Remark 2.6 (see [23, Theorem 4.2]). *Let be Drazin invertible and group invertible, respectively. If , then

Let and ; if we put , then is invertible in when . From the Proposition 2.2 (5) of [20], we have when for . Therefore, for , we arrive at [20, Theorem 4.2]. In fact, the following remark implies that Theorem 2.5 improves the upper bound of of [20, Theorem 4.2].

*Remark 2.7 (see [20, Theorem 4.2]). *Let and let with . Assume that . Then and
where and .

Theorem 2.8. *Let be generalized Drazin invertible and satisfy the conditions
**
Then exists if and only if is group invertible. In this case,
*

*Proof. *Since exists, is defined as (2.1). Let have the block matrix form as

Applying the condition , we have and
It follows from (2.24) that is invertible, , and
Combining (2.1) and (2.25), we obtain

The condition implies in the subalgebra . Therefore, we conclude that is invertible and . According to (2.26) and by Lemma 2.1, one observes that exists if and only if also. Thus, exists if and only if is group invertible.

If is group invertible and by Lemma 2.1, we obtain
where .

Since and is quasinilpotent, by Lemma 2.2, we obtain

From , one easily has
It follows from (2.27) and (2.29) that

Combining (2.27), (2.28), and (2.29), we obtain

From (2.31), we derive
Moreover, by (2.32) we get

Finally, from (2.33) we easily finish the proof.

Corollary 2.9. *Let and let . If satisfy the conditions
**
then exists if and only if is group invertible. In this case,
*

The conditions of Theorem 2.8 are weaker than the conditions () (see [12, Theorem 3.2] for finite dimensional cases and [8, Theorem and Corollary ] for Banach algebra). According to , we obtain that (2.26) holds. However, in view of (), we have Thus, by the conditions (), we know that and have the same Drazin invertible property (see [12, Theorem 3.1]). Thus, if is group invertible, then is group invertible. It is easy to see that , are weaker than the conditions (). From [8, Theorem and Corollary ], we easily state the following remark.

*Remark 2.10. *Let and let . If satisfy the condition ()
then is group invertible and

Theorem 2.11. *Let be generalized Drazin invertible and satisfy the conditions
**
Then exists if and only if is group invertible. In this case,
*

*Proof. *The notations are taken as Theorem 2.8, and the rest of proof of theorem is similar to Theorem 2.8. Now, we only consider the perturbation of . From (2.28) and the first condition of (2.39), we have and
Thus, from (2.41) we completed the proof.

Theorem 2.12. *Let be generalized Drazin invertible and satisfy the conditions
**
Then exists if and only if is group invertible. In this case,
*

*Proof. *Letting , and it is similar to Theorem 2.8, we obtain that , , and have the matrix forms as (2.1). Here is taken as (2.23) in the proof of Theorem 2.8. The condition implies that
Thus, according to (2.44), we obtain , and . Because is invertible in subalgebra , we have . Thus, , have the matrix forms as follows:

It follows from the condition that . Thus, it shows from that is invertible in subalgebra . Therefore, easily we observe that is Drazin invertible if and only if is Drazin invertible. That is, exists if and only if is group invertible.

In the following, we will consider the perturbation of .

Let be group invertible. The condition implies that holds. Since is quasinilpotent in subalgebra and by Lemma 2.3, we get

By virtue of , we get that

It follows from (2.46) and (2.47) that

Next, according to (2.48), we obtain

Finally, using (2.49) the proof is finished.

Corollary 2.13. *Let and let . If satisfy the conditions
**
Then exists if and only if is group invertible. In this case,
*

Let with , and let If (see [10, Theorem 2.1] ), then where is invertible and is quasinilpotent (it follows that ). It follows from (2.53) that implies that , (i.e., ). If is group invertible, then is group invertible and where .

By virtue of and (see [10]), we give the following remark.

*Remark 2.14 (see [10, Theorem 3.1]). *Let with . Then
If , then

Theorem 2.15. *Let be generalized Drazin invertible and satisfy the conditions
**
Then exists if and only if is group invertible. In this case,
*

*Proof. *Similarly to Theorem 2.12, we have that the formulas (2.45) hold. The details will be omitted. In the following we only give the simple proof.

By the condition
it shows that and . Thus, the first result shows that is invertible. In view of Lemma 2.1, one concludes that is Drazin invertible if and only if is Drazin invertible. That is, exists if and only if is group invertible.

In the following, we consider the perturbation of .

After application of the hypothesis , we find that . It follows from Theorem 2.12 and Lemma 2.3 that

It follows from the condition and
It implies that and

Therefore, combining (2.49) with (2.62), we have
Thus, by (2.63), we complete the proof.

#### 3. Perturbation Bound of the Generalized Schur Complement

The perturbation bounds of the Schur complement are investigated in [29–31]. In [29] Stewart gave perturbation bounds for the Schur complement of a positive definite matrix in a positive semidefinite matrix. In [30] Wei and Wang generalized the results in [29] and enrich the perturbation theory for the Schur complement. In [31] the authors derived some new norm upper bounds for Schur complements of a positive semidefinite operator matrix. In this section, we consider the perturbation bounds of the generalized Schur complement in Banach algebra.

Some notations of the generalized Schur complement over Banach algebra will be stated in the following.

Let , and let it be written in the form as follows: It has the following matrix form: where is idempotent element in and is taken as (1.10).

The formulas (1.5) and (1.6) are written in Banach algebra, respectively: Similarly, the generalized Schur complement in (1.7) and (1.8) is defined in the following over Banach algebra, respectively: where denotes the generalized Schur complement of in .

Theorem 3.1. *Let be given as (3.2) let and
**
be perturbed version of , and the following conditions are satisfied:
**
where . If , and satisfy the conditions of Theorem 2.8, then
**
where
**
and and are Schur complement of in and Schur complement of in , respectively.*

*Proof. *Since , and satisfy Theorem 2.8, according to (2.31), we obtain
Therefore, it is easy to see that

From (3.11) and by the conditions (3.6), we obtain
where
Thus, we finish the proof.

Similar to Theorem 3.1. It follows from the proof of Theorem 2.11 that the results are given as follow.

Theorem 3.2. *Let and be taken as Theorem 3.1, and let the relations in (3.6) be satisfied, where . If , and satisfy the conditions of Theorem 2.11, then
**
where and are taken as Theorem 3.1.*

Theorem 3.3. *Let and be taken as Theorem 3.1, and let the relations in (3.6) be satisfied, where . If , and satisfy the conditions of Theorem 2.12, then
*