Abstract and Applied Analysis

Volume 2013 (2013), Article ID 789896, 6 pages

http://dx.doi.org/10.1155/2013/789896

Research Article

The Group Inverse of the Combinations of Two Idempotent Operators

Shunqin Wang¹ and Chunyuan Deng²

¹School of Mathematics and Statistics, Nanyang Normal University, Nanyang 473061, China

²School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received 1 August 2013; Accepted 7 October 2013

Academic Editor: Jaan Janno

Copyright © 2013 Shunqin Wang and Chunyuan Deng. 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 present some inverses and group inverses results for linear combinations of two idempotents and their products.

1. Introduction

Let be a complex Hilbert space. Denote by the Banach algebra of all bounded linear operators on . and represent the range and the null space of , respectively. The identity onto a Hilbert space is denoted by or if there does not exist confusion. For , the group inverse [1] of is the unique element such that

exists if and only if has finite ascent and descent such that [2]. When ind, the group inverse reduces to the standard inverse; that is, . In particular, if and if for a scalar . One of the most important applications of group inverses is to derive some closed-form formulas for general solutions to operator equations. An operator is said to be idempotent if . If is group invertible, then is closed and the spectral idempotent is given by . The operator matrix form of with respect to the space decomposition is given by , where is invertible on [1].

Idempotents are a type of simplest operators. Various expressions or equalities consisting of idempotents occur in operator theory and its applications. Some previous work on linear combinations of idempotents in statistics can be found in [3]. There have been several papers devoted to the invertibility of a linear combination of two idempotent operators in a Hilbert space or in a -algebra. In [4], Buckholtz studied the idempotency of the difference of two operators in a Hilbert space. Du and Li in [5] had established the spectral characterization of generalized projections. In [6], the invertibility of the difference of two orthogonal projectors in a -algebra was studied. Li in [7] had investigated how to express Moore-Penrose inverses of products and differences. In [8], J. K. Baksalary and O. M. Baksalary discussed the invertibility of a linear combination of idempotent matrices. This paper was improved by Koliha and Rakočević [9] by showing that the rank of a linear combination of two idempotents is constant. Du et al. [10] extended the conclusion on an infinite-dimensional Hilbert space.

The purpose of this note is to characterize the invertibility and the group invertibility of the linear combinations of idempotents and their products , , , , , and . These linear combinations were studied by some authors in recent years [2, 4, 6–9, 11–18]. Some formulas for the group inverse of a sum of two bounded operators under some conditions were given (see [7, 11, 14, 19, 20]). Here, we will find group invertibility for a linear combination of two idempotents under the condition . A previous study of the group invertibility of two idempotents was made under the conditions or or or . It is clear that these conditions are special cases of our results.

2. Main Results and Proofs

We start by discussing some lemmas. Let and be two idempotents. Now, we consider the invertibility of . This problem is the subject of Buckholtz’s papers [4] and Koliha and Rakočević’s paper [12].

Lemma 1 (see [12]). Let and be two idempotents.

(i) is invertible if and only if and are invertible.(ii)If is group invertible, then and are group invertible.

Proof. Since the properties in the lemma are similarly invariant, without loss of generality, we can assume that is an orthogonal projection. In this case, and have the operator matrix representations as follows

with respect to the space decomposition , respectively. Since ,

So

It is clear that item (i) holds, and if is group invertible, then is group invertible, which implies that and are group invertible.

As for -idempotents, we have the following decomposition.

Lemma 2 (see [21, Theorem 2.3]). Let . Then if and only if

(i);(ii)there exists a resolution set of the identity and an invertible operator such that

where denotes the orthogonal direct sum, is an orthogonal projection with , and if , , .

Lemma 3. Let be such that is invertible. Then is invertible if and only if the Schur complement is invertible.

Lemma 4 (see [1, Theorem ]). Let be such that exists. Then exists if and only if exists and . In this case,

where .

Lemma 5 (see [19, Theorem 3.1]). Let be such that is invertible and the Schur complement is group invertible. Then is group invertible if and only if is invertible.

As we know, an operator is said to be involutory if , to be anti-idempotent if , and to be tripotent if . Obviously, involutory, idempotent, and anti-idempotent are special cases of tripotent. For linear combinations of two commutative tripotents and their products, we have the following result.

Theorem 6. Let satisfy , , and . For any scalar , and , let

(i)If , then is invertible. In this case, .(ii) is always group invertible and .

The and , in items and are defined as

Proof. Since , by Lemma 2, and there exists an invertible operator such that . We consider a partition conforming with . Since , can be written as , where . In a similar way, , can be written as . Let . Now, we get

Let and , be defined as in (8). By (9), is a diagonal block matrix such that the th diagonal element is the identity and the remaining diagonal elements are , . Moreover, , , and . We get . Hence, if , is invertible and . If , is group invertible and .

The matrix case of Theorem 6 was first investigated by Tian [17]. The commutative relations ensure that idempotents (or tripotents) have simple block matrix forms. It is natural to ask whether this kind of combinational properties still hold when a pair of idempotents is noncommutative. Next, let , , and . It is easy to verify that this condition includes some specific cases:

(i), (ii) (see [14]),(iii), (iv), (v), (vi) (see [17]),(vii)  (see [14]).

Applying Lemmas 3 and 4, we get the following main result.

Theorem 7. Let and be two idempotents such that . For any scalar and with , and , let

(i) is invertible if and only if is invertible.(ii) is always group invertible.

Proof. Let and have the forms as in (2). Then

If , then . Moreover, by (3), and . These imply that , , and . So and can be rewritten as block matrix forms as

with respect to the space decomposition , respectively. From , by (12), we deduce that

By (2) and (12) we get as an operator on which can be represented as operator matrix form:

Denote

Since , then and are invertible. The Schur complement has the form

Hence, by Lemma 3, is invertible if and only if (see (12)) is invertible if and only if (see (2)) is invertible, which is equivalent to that

is invertible.

If the idempotent operator is not invertible, by Lemma 4, is group invertible:

where the omitted element can be got by Lemma 4. Note that and are invertible. By Lemma 5, is group invertible.

If , we get the following main result.

Theorem 8. Let and be two idempotents such that . For any scalar , and with , and , let be defined as in (10). Then

(i) is invertible if and only if is invertible;(ii)if any one of and is invertible, then is always group invertible.

Proof. We use the notations from the proof of Theorem 7. By (14), if , then the upper left submatrix fails to be invertible. Perturb it a little. as an operator on can be written as

Denote

The Schur complement of in (19) has the structure

Hence, by Lemma 3, is invertible if and only if idempotents and (see (12)) are invertible if and only if idempotents and are invertible, which is equivalent to the fact that (or ; see (4)) is invertible by Lemma 1.

If is invertible, then is invertible by (12). Since is idempotent, then . By Lemma 4, is group invertible:

If is invertible, then is invertible by (12). Since is idempotent, then . By Lemma 4, is group invertible:

So, if any one of and is invertible, then . By Lemma 5, is invertible. Hence is group invertible.

3. Concluding Remarks

In Theorems 7 and 8, the inverse and the group inverse formulae can be obtained by using the results in Lemma 3 and [19, Theorem 3.1], respectively. This is a trivial and redundant work. If is group invertible and , by definition (1), it is also trivial to check

Hence, is always group invertible. In particular, if is invertible (see [18, Theorem 3.1]), then

From Theorems 7 and 8 we know that in (10) is always group invertible if . It seems very difficult to find the minimum requirements that guarantee that in (10) is group invertible, which could be the topic of some future research. Hence, we suggest the following question: what are the minimum requirements that guarantee that in (10) is group invertible?

Conflict of Interests

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

Acknowledgments

Shunqin Wang is supported by the Basic and Advanced Research Program of Henan Science Committee (no. 102300410145) and the Research Award for Teachers in Nanyang Normal University (nynu200749). Chun Yuan Deng is supported by the National Natural Science Foundation of China under Grant 11171222 and the Doctoral Program of the Ministry of Education under Grant 20094407120001.

References

S. L. Campbell and C. D. Meyer Jr., Generalized Inverses of Linear Transformations, Dover Publications, New York, NY, USA, 1991.

A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York, NY, USA, 2nd edition, 1980.

F. A. Graybill, Introduction to Matrices with Applications in Statistics, Wadsworth Publishing Company Inc., Belmont, Calif, USA, 1969.

D. Buckholtz, “Inverting the difference of Hilbert space projections,” The American Mathematical Monthly, vol. 104, no. 1, pp. 60–61, 1997.

H. K. Du and Y. Li, “The spectral characterization of generalized projections,” Linear Algebra and Its Applications, vol. 400, pp. 313–318, 2005.

J. J. Koliha and V. Rakočević, “On the norm of idempotents in -algebras,” The Rocky Mountain Journal of Mathematics, vol. 34, no. 2, pp. 685–697, 2004.

Y. Li, “The Moore-Penrose inverses of products and differences of projections in a -algebra,” Linear Algebra and Its Applications, vol. 428, no. 4, pp. 1169–1177, 2008.

J. K. Baksalary and O. M. Baksalary, “Nonsingularity of linear combinations of idempotent matrices,” Linear Algebra and Its Applications, vol. 388, pp. 25–29, 2004.

J. J. Koliha and V. Rakočević, “The nullity and rank of linear combinations of idempotent matrices,” Linear Algebra and Its Applications, vol. 418, no. 1, pp. 11–14, 2006.

H. Du, X. Yao, and C. Deng, “Invertibility of linear combinations of two idempotents,” Proceedings of the American Mathematical Society, vol. 134, no. 5, pp. 1451–1457, 2006.

J. Benítez and N. Thome, “Idempotency of linear combinations of an idempotent matrix and a -potent matrix that commute,” Linear Algebra and Its Applications, vol. 403, pp. 414–418, 2005.

J. J. Koliha and V. Rakočević, “Invertibility of the difference of idempotents,” Linear and Multilinear Algebra, vol. 51, no. 1, pp. 97–110, 2003.

J. J. Koliha and V. Rakočević, “Invertibility of the sum of idempotents,” Linear and Multilinear Algebra, vol. 50, no. 4, pp. 285–292, 2002.

X. Liu, L. Wu, and Y. Yu, “The group inverse of the combinations of two idempotent matrices,” Linear and Multilinear Algebra, vol. 59, no. 1, pp. 101–115, 2011.

Y. Tian and Y. Takane, “Some properties of projectors associated with the WLSE under a general linear model,” Journal of Multivariate Analysis, vol. 99, no. 6, pp. 1070–1082, 2008.

Y. Tian and Y. Takane, “On -orthogonal projectors associated with a semi-norm,” Annals of the Institute of Statistical Mathematics, vol. 61, no. 2, pp. 517–530, 2009.

Y. Tian, “A disjoint idempotent decomposition for linear combinations produced from two commutative tripotent matrices and its applications,” Linear and Multilinear Algebra, vol. 59, no. 11, pp. 1237–1246, 2011.

K. Zuo, “Nonsingularity of the difference and the sum of two idempotent matrices,” Linear Algebra and Its Applications, vol. 433, no. 2, pp. 476–482, 2010.

C. Bu, M. Li, K. Zhang, and L. Zheng, “Group inverse for the block matrices with an invertible subblock,” Applied Mathematics and Computation, vol. 215, no. 1, pp. 132–139, 2009.

D. S. Cvetković-Ilić, D. S. Djordjević, and Y. Wei, “Additive results for the generalized Drazin inverse in a Banach algebra,” Linear Algebra and Its Applications, vol. 418, no. 1, pp. 53–61, 2006.

C. Y. Deng, Q. H. Li, and H. K. Du, “Generalized -idempotents and hyper-generalized -idempotents,” Northeastern Mathematical Journal, vol. 22, no. 4, pp. 387–394, 2006.