Abstract and Applied Analysis

Volume 2012 (2012), Article ID 629178, 22 pages

doi:10.1155/2012/629178

Research Article

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

Xiaoji Liu,¹ Yonghui Qin,² and Hui Wei³

¹Faculty of Science, Guangxi University for Nationalities and Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis, Nanning 530006, China

²College of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning 530006, China

³Department 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

where

and and are taken as Theorem 3.1.

Proof. Similar to the proof of Theorem 3.1 the details are omitted. A simple proof is given as follows.

By (2.45), (2.46), and (2.47), we obtain

In view of (3.17), we easily have

It shows from (3.18) that

where

Therefore, we complete the proof.

Similar to Theorems 3.1 and 3.3. The proof of the following theorem follows from Theorem 2.15.

Theorem 3.4. 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 2.15, then

where

and and are taken as Theorem 3.1.

Acknowledgments

X. Liu is supported by the NSFC grant (11061005, 61165015), the Ministry of Education Science and Technology Key Project under Grant 210164 and Grants (HCIC201103) of Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis Open Fund and Key issues for Department of Education of Guangxi (201202ZD031). H. Wei is supported by 973 Program (Project no. 2010CB327900).

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