Perturbation Bound of the Group Inverse and the Generalized Schur Complement in Banach Algebra
Xiaoji Liu,1Yonghui Qin,2and Hui Wei3
Academic Editor: Patricia J. Y. Wong
Received05 Apr 2012
Revised26 Jun 2012
Accepted11 Jul 2012
Published30 Aug 2012
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], BenΓtez 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], BenΓtez 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 Γ 2 block complex square matrix in (1.4) with a singular generalized Schur complement was considered in [4β6], where
For the expression of a 2 Γ 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.
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, Theoremββ2.3]). 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, Corollaryββ3.4]). 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, Theoremββ4.2]). 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, Theoremββ4.2]). 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 2.8ββ 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, Theoremββ3.1]). 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 [29β31]. 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
andββ and are Schur complement of in and Schur complement ofββ 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 , andββ 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 , andββ 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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