Journal of Function Spaces

Journal of Function Spaces / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 767568 | 8 pages | https://doi.org/10.1155/2015/767568

Formulae for the Generalized Drazin Inverse of a Block Matrix in Banach Algebras

Academic Editor: Seppo Hassi
Received15 Jun 2015
Accepted01 Oct 2015
Published21 Oct 2015

Abstract

We present some new representations for the generalized Drazin inverse of a block matrix in a Banach algebra under conditions weaker than those used in resent papers on the subject.

1. Introduction

The Drazin inverse has applications in a number of areas such as control theory, Markov chains, singular differential and difference equations, and iterative methods in numerical linear algebra. Representations for the Drazin inverse of block matrices under certain conditions were given in the literature [15]. Generalized inverses of block matrices have important applications in automatics, probability, statistics, mathematical programming, numerical analysis, game theory, econometrics, control, and so on [6, 7]. In 1979, Campbell and Meyer proposed the problem of finding a formula for the Drazin inverse of a matrix in terms of its various blocks, where the blocks on the diagonal are required to be square matrices [7]. At the present time, there is no known complete solution to this problem.

In this paper, we present the representation for the Drazin inverse of under the conditions that and and that and (cf. Theorem 4). And we also give several representations for the Drazin inverse of under some weaker conditions.

Let be a complex Banach algebra with the unit. The sets of all Drazin invertible and quasinilpotent elements of will be denoted by and , respectively.

The generalized Drazin inverse of (introduced by Koliha in [8]) is the element which satisfiesIf there exist the generalized Drazin inverse, then the generalized Drazin inverse of is unique and is denoted by .

Let and let be an idempotent (). We denote . Then we can write

Every idempotent induces a representation of an arbitrary element given by the following matrix:

Let and be the spectral idempotent of corresponding to . It is well known that can be represented in the following matrix form [9, Ch. 2]:where , is invertible in the algebra , is its inverse in , and is quasinilpotent in the algebra . Thus, the generalized Drazin inverse of can be expressed as Obviously, if , then is generalized Drazin invertible and .

2. Preliminary Results

First we start the following result which is proved in [10] for matrices, extended in [2] for a bounded linear operator and in [11] for arbitrary elements in a Banach algebra.

Lemma 1 (see [11, Theorem  2.3]). Let be a Banach algebra, , and an idempotent. Assume that and are represented as (i)If and , then and are generalized Drazin invertible, andwhere(ii)If and , then , and and are given by (7) and (8).

Lemma 2. Let , .(1)[11, Corollary  3.4] If , then and (2)[12, Theorem  2.2] If , then and

Lemma 3. Let be a Banach algebra and let have the representation (4), where . If and for some , then (in particular, if , then ). If and for some , then (in particular, if , then ).

Proof. From we get , , and . There exists such that . Since , we obtain . The proof for is similar.

3. Main Results

In [13, Theorem  3.2] authors gave an explicit representation for under conditions and . Here we replace the last condition by the condition ; we will get a much simpler expression for .

Theorem 4. Let be a Banach algebra and let . If there exist , , such that and , then and

Proof. Let . We can represent as in (4), where is invertible in the subalgebra and is quasinilpotent. Hence,Let us writeFrom and we get . Thus, and can be represented asBy Lemma 1 and , we get andAn elementary computation from (17) and leads to Since , we get . An induction argument proves for any . Let us denote and observe that since , then is a subsequence of . Since is quasinilpotent, we get Hence . By Lemma 2, we have that , because , , and . Also, Lemma 2 allows us to obtainAgain, from Lemma 1 and (16), we get that andwhere Observe that since , then . Hence, the expression of reduces to From we get . Hence So, we get Observe that (22) and yieldWe will express using (21) in terms only of and . We have from (17) Hence from (21) we obtainAlso we have (we have written with an asterisk any entry whose exactly expression is not necessary)From (27), (29), and (30), it follows (11). The inequality (12) trivially follows from (11).

If is a Banach algebra, then we can define another multiplication in by . It is trivial that is a Banach algebra. If we apply Theorem 4 to this new algebra, we can immediately establish the following result.

Theorem 5. Let be a Banach algebra and let . If there exist , , such that and , then and

Observe that we can obtain paired results as Theorems 4 and 5. We will not write explicitly these (trivially obtained) results.

Theorem 6. Let be a Banach algebra and let be such that . If there exist , , such that and , then and

Proof. Let us consider the matrix representations of , , and given in (4), (13), and (14) relative to the idempotent . As in the proof of Theorem 4, from , we get .
From we have Therefore, and . From Lemma 3 we get . Hence Lemma 2 implies that because , , and . Also, we obtainIn this situation, we obtainSince , we have . Thus (35) reduces toFrom we have . Therefore, so we get We haveAlso,From (36), (38), (41), and (42), it follows (32).
The proof is finished.

As we have commented before, we can obtain a paired result by considering the Banach algebra with the product . The key hypothesis of this new result will be and .

Theorem 7. Let be a Banach algebra and let be such that . If there exist , , such that and , then and

Theorem 8. Let be a Banach algebra and let be such that . If there exist , , such that and , then and

Proof. Let us consider the matrix representations of , , and given in (4), (13), and (14) relative to the idempotent . From , we haveThe second equality of (45) implies for any . By Lemma 3 we get . Hence Therefore, if denotes and denotes , then . Recall that is quasinilpotent, and thus . We have just proved or, equivalently, . Similarly we obtain .
Thus, and can be represented asSince , by the proof of Theorem 4, we get . From Lemma 2, we have , because , , and . Also, Lemma 2 allows us to obtain We have if and only if . But . If , then The theorem is just proved.

Theorem 9. Let be a Banach algebra and let . If there exist , , such that and , then and

Proof. Let us consider the matrix representations of , , and given in (4), (13), and (14) relative to the idempotent . Since , we get and ; as in the proof of Theorem 4, we have . From Lemma 3 we get .
From we have Therefore, , , and . From Lemma 3 we get and . Hence Lemma 2 implies that because , , and ; as in the proof of Theorem 6, we obtainIn this situation, we obtainObserve that since , then . Hence, the expression of reduces to From and , we have So we get Hence Thus (55) reduces toWe haveFrom (56), (62), and (63), it follows (50). The inequality (51) trivially follows from (50).
The proof is finished.

As we have commented before, we can obtain a paired result by considering the Banach algebra with the product . The key hypothesis of this new result will be and .

Theorem 10. Let be a Banach algebra and let . If there exist , , such that and , then and

In the rest of the paper, we will use some weaker conditions than in [11, Theorem  4.1]. For example, if we assume that and instead of and , we will get a much simpler expression for .

Theorem 11. Let be a Banach algebra and let . If there exist , , such that and , then and

Proof. Let us consider the matrix representations of , , and given in (4), (13), and (14) relative to the idempotent . We will use the condition : since we get and ; as in the proof of Theorem 4, we have . From Lemma 3 we get .
From we have Therefore, . Hence where The proof is finished.

Theorem 12. Let be a Banach algebra and let . If there exist , , such that and , then and

Proof. Let us consider the matrix representations of , , and given in (4), (13), and (14) relative to the idempotent . As in the proof of Theorem 11, from , we get and .
Since , we get . As in the proof of Theorem 11, we have where From , we get The proof is finished.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11361009), the Guangxi Provincial Natural Science Foundation of China (no. 2013GXNSFAA019008), and Science Research Project 2013 of the China-ASEAN Study Center (Guangxi Science Experiment Center) of Guangxi University for Nationalities. The authors would like to thank the referees and the editors for their suitable comments.

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Copyright © 2015 Xiaoji Liu and Xiaolan Qin. 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.


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