Abstract and Applied Analysis

Volume 2012 (2012), Article ID 736929, 9 pages

http://dx.doi.org/10.1155/2012/736929

## Proper Splitting for the Generalized Inverse and Its Application on Banach Spaces

School of Science, Guangxi University for Nationalities and Guangxi Key Laborarory of Hybrid Computational and IC Design Analysis, Nanning 530006, China

Received 7 December 2011; Accepted 16 March 2012

Academic Editor: Jean Michel Combes

Copyright © 2012 Xiaoji Liu and Shaowu Huang. 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

A possible type of the operator splitting is studied. Using this operator splitting, we introduce some properties and representations of generalized inverses as well as iterative method for computing various solutions of the restricted linear operator system , where and *T* is an arbitrary but fixed subspace of **X**.

#### 1. Introduction

The subject of splitting was investigated by numerous authors. And there are several papers concerning the iterative methods of the general formas follows: where is a splitting of , and is an arbitrary closed subspaces of (see [1–4]). Particular results concerning the computation of the Drazin inverse and the Moore-Penrose inverse can be investigated in [5, 6].

In this paper, we will consider the linear operator system , where , , and . The concept of a operator splitting can be used in characterizations of the generalized inverse and in iterative method as follows: for solving linear operator system . In particular, let , the authors in [7] gave the iterative method to solve linear system .

Now, we introduce some notations and terminologies.

Let be Banach spaces and be the set of all bounded operators from to . We use and , respectively, to denote the range, the null space, and the spectral radius for an operator . Suppose that there exists an operator , such that Then is usually denoted by . Recall that the splitting is called a proper splitting of if and .

Let if, for some nonnegative integer , there exists such that then is called the Drazin inverse of , and the smallest is called the index of and will be denoted by . If an operator has a Drazin inverse, then it is unique and is denoted by (see [8] for details). Particularly, when , is called the group inverse of and is denoted by .

The paper is organized as follows. In the remainder of this section, we will introduce some lemmas which are useful in the proofs. In Section 2, We express some properties and representations of generalized inverses based on the operator splitting. Moreover, by using these representations, we introduce iterative method for computing various solutions of linear operator system. In Section 3, we give a numerical example to demonstrate one of the results in Section 2.

Basic auxiliary results are summarized in the following lemmas.

Lemma 1.1 (see [9, page 43]). *Let and be closed subspaces of and such that and . Then has the following matrix form:
**
where is invertible. Moreover,
*

Lemma 1.2 (see [9, lemma 3.5.2]). *Let . and are closed subspaces of such that and is a projection from onto parallel to . Then, one has *(i)* if and only if ,*(ii)* if and only if .*

Lemma 1.3 (see [6]). *Let be factorized in the following form:
**
where is an nonsingular matrix, and and are permutation matrices. Then
**
(i.e., is a proper splitting) if and only if
**
where is a nonsingular matrix of order . *

Lemma 1.4 (see [10]). *Let with and be partitioned as
**
where and are the same as Lemma 1.3. Then the group inverse exists if and only if
*

#### 2. Main Results

In this section, the representation of the generalized inverse of a linear operator is studied. Moreover, we consider the fundamental problem of solving a general linear operator equation of the type , where .

Now we are ready to present the representation theorem.

Theorem 2.1 (representation theorem). *Let be given and and , respectively, be closed subspaces of and such that there exists the generalized inverse . And let with , , be a proper splitting of , that is, , . Then, one has
**
where is a subspace of .*

*Proof. *By Lemma 1.2, it is easy to verify that
Now, we will show that is invertible. Notice that
Since is a projection from onto parallel to and is an invertible operator from to , we get that is invertible. Thus, the proof is completed.

Now, we are in position to state the main result of this section.

Theorem 2.2. *Under the hypotheses of Theorem 2.1, we have that
**
converges to for every if and only if . Then we get that has the error estimation
**
Moreover, one has
**
where is a subspace of .*

*Proof. *“” Suppose that . According to Representation theorem, we obtain that
By the hypothesis and (2.7), it is to see that converges to for every .

“” A simple computation shows that
it follows that as , which implies that . Now the results follow immediately.

From Theorem 2.2, we can immediately obtain the following theorem.

Theorem 2.3. *Under the hypotheses of Theorem 2.1, we have that
**
converges to (the unique solution of linear operator system .) for every if and only if . Then we get that has the error bounded
**
Moreover, one has
**
where is a subspace of .*

*Proof. *The proof is similar to that of Theorem 2.2.

Similar to [7, Theorems 3.1 and 3.3]. In the case when , the previous theorems reduce to the following corollary.

Corollary 2.4. *Let , and be the same as Theorem 2.1 and let , and . Let be a proper splitting of , that is,
**
Then, one has*(i)*,*(ii)*, and*(iii)*the iteration converges to for each if and only if . *

Now, we will consider the proper splitting of .

Theorem 2.5. *Let be given and and , respectively, be closed subspaces of and , such that there exists the generalized inverse . And let with , , be a proper splitting of , that is, , . Then, one has
**
where is a subspace of .*

Theorem 2.6. *Under the hypotheses of Theorem 2.5, we have that
**
converges to for every if and only if . Then we get that has the error estimation as follows:
**
Moreover, consider that
**
where is a subspace of .*

It is well known that . Let ; thus, the following result follows from Corollary 2.4.

Corollary 2.7. *Let , , be closed subspaces of and such that exists with Ind. Suppose that and be a proper splitting of , that is,
**
Then the iteration
**
or
**
converges to or for every if and only if .*

*Remark 2.8. *In [11], the author present splitting for computing the generalized inverse , particular result concerning the computation Drazin inverse. Let , and be a splitting of . Then the iteration
converges to for every if and only if . Moreover, the iteration (2.20) can be reduced to

The splitting (2.17) is more practical than (2.21). Since in (2.17) can easily be calculated by Lemmas 1.3 and 1.4, while in (2.21) is more difficult to be getten.

#### 3. Illustration Example

The following matrix is from [11]. Let where ind, rank.

We will use Corollary 2.7 to compute the unique solution of the restricted linear equation and . Take which satisfies the conditions of Corollary 2.7. Therefore, by Lemmas 1.3 and 1.4,

Denote that and . We have Table 1 for the norm , and from which we conclude that is an approximation of the exact solution of .

#### 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, 61165015), the Ministry of Education Science and Technology Key Project under Grant (210164), Grants HCIC201103 of Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis Open Fund, Key issues for Department of Education of Guangxi (201202ZD031).

#### References

- B. Načevska, “Iterative methods for computing generalized inverses and splittings of operators,”
*Applied Mathematics and Computation*, vol. 208, no. 1, pp. 186–188, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. S. Djordjević and Y. Wei, “Outer generalized inverses in rings,”
*Communications in Algebra*, vol. 33, no. 9, pp. 3051–3060, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Wei and H. Wu, “$\{T,S\}$ splitting methods for computing the generalized inverse ${A}_{T,S}^{(2)}$ and rectangular systems,”
*International Journal of Computer Mathematics*, vol. 77, no. 3, pp. 401–424, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Chen, W. Wang, and Y. Song, “Splitting based on the outer inverse of matrices,”
*Applied Mathematics and Computation*, vol. 132, no. 2-3, pp. 353–368, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Chao and G. Chen, “Index splitting for the Drazin inverse of linear operator in Banach space,”
*Applied Mathematics and Computation*, vol. 135, no. 2-3, pp. 201–209, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Berman and M. Neumann, “Proper splittings of rectangular matrices,”
*SIAM Journal on Applied Mathematics*, vol. 31, no. 2, pp. 307–312, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Wang and Y. Wei, “Proper splittings for restricted linear equations and the generalized inverse ${A}_{T,S}^{(2)}$,”
*Numerical Mathematics*, vol. 7, no. 1, pp. 1–13, 1998. View at Google Scholar · View at Zentralblatt MATH - D. H. Cai, “The Drazin generalized inverses of linear operators,”
*Journal of Mathematics. Shuxue Zazhi*, vol. 5, no. 1, pp. 81–88, 1985 (Chinese). View at Google Scholar - D. S. Djordjević and V. Rakočević,
*Lectures on Generalized Inverses*, Faculty of Sciences and Mathematics, University of Niš, Niš, Serbia, 2008. - A. Ben-Israel and T. N. E. Greville,
*Generalized Inverses: Theory and Applications*, John Wiley & Sons, New York, NY, USA, 1974. - D. S. Djordjević, “Iterative methods for computing generalized inverses,”
*Applied Mathematics and Computation*, vol. 189, no. 1, pp. 101–104, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH