Note on the Invariance Properties of Operator Products Involving Generalized Inverses

Abstract

We investigate further the invariance properties of the bounded linear operator product and its range with respect to the choice of the generalized inverses and of bounded linear operators. Also, we discuss the range inclusion invariance properties of the operator product involving generalized inverses.

1. Introduction

Throughout this paper, by “an operator” we mean “a bounded linear operator over Hilbert spaces.” Let the symbol denote the set of all bounded linear operators from Hilbert space to Hilbert space . In particular, . For , the symbols , , and , respectively, denote its adjoint, range, and nullspace.

Recall that an operator is called the Moore-Penrose inverse of if satisfies the following operator equations: If such an operator exists, then it is unique and is denoted by (see, e.g., [1–4] for details). It is well known that the Moore-Penrose inverse exists if and only if is closed. Let . If satisfies the equation , then is an -inverse of and is written as . The set of all -inverses of is denoted by . Obviously it is well defined that , and write when .

In 1971, Rao and Mitra [2] first discussed the invariance property of a matrix product with respect to any choice of and presented the necessary and sufficient conditions for such a matrix product to be invariant without respect to the choice of . From then on, the invariance property has attracted more and more researchers to investigate it and showed its importance in theoretic research of many aspects, such as range invariance (see [5]), rank invariance (see [6]), invariance of the eigenvalues, singular values, and norms of matrix products (see [7]). The representational results, for example, have necessary and sufficient conditions for the invariance properties of (see [5]) and of rank (see [6]).

Recently in [8], the authors discussed the invariance of expressions of the matrix product . In [9], using the method of extremal ranks, the authors study the range inclusion invariance of the triple matrix product involving generalized inverses. And in [10], exploiting the matrix form of a bounded linear operator, the authors researched the invariance properties of the bounded linear operator product with respect to the choice of the generalized inverse of a bounded linear operator.

In this paper, we investigate further the invariance properties of the bounded linear operator product and its range with respect to the choice of the generalized inverses and of bounded linear operators. Also, we discuss the range inclusion invariance properties of the operator product involving generalized inverses. The paper is organized as follows. In Section 2, we introduce some lemmas. In Section 3, we present the equivalent conditions of the operator product being invariant without respect to the choice of and , which involve some inclusive relations among ranges of operators mentioned, the reverse order law for the -inverses of and , and being invariant without respect to the choice of and . And we also establish the relationship between invariance properties of and its range under some condition. In Section 4, we deduce the range inclusion invariance properties of the operator product involving -inverses, -inverses, and the some inclusive relations among ranges of operators mentioned.

2. Lemmas

In the section, we will introduce several lemmas as follows. The following lemma, with respect to generalized inverses of an operator, is similar to [1, Corollaries 2.1, 2.3, and 2.4], [2, Page 28], and [9, ] for a matrix.

Lemma 1. Let .(i)If , then (ii)Consider (iii)If , then .(iv)If , then .

Lemma 2 (see [11, Lemmas 1.1 and 1.2]). Let have a closed range. Let and be closed and mutually orthogonal subspaces of , such that . Let and be closed and mutually orthogonal subspaces of , such that . Then the operator A has the following matrix representations with respect to the orthogonal sums of subspaces and .(i)Consider  where is invertible on . Also, (ii)Consider  where is invertible on . Also, In particular, where is invertible. Moreover,

The next lemma, with respect to generalized inverses of an operator, is similar to [8, Lemma 3] for matrices.

Lemma 3. Let and have closed ranges. Then for every if and only if or .

Proof. If either or is zero, then for every .
Assume . Then, by Lemma 2, and can be represented, respectively, as the following matrix forms: where and are invertible in and , respectively. Then take because and define Thus if , then and therefore because of the invertibility of and . This leads to a contradiction. Hence and then the result holds.

Lemma 4. Let have a closed range, . Then

Proof. “” Since is closed, exists and then . Thus, there exists an such that . Hence .
“” Since , .

3. Invariance Properties of Operator Product

In the section, we first present the main result of invariance properties.

Theorem 5. Let nonzero operators , , , and have closed ranges. Then the first three statements below are equivalent. Moreover, suppose that has closed range. Then the following statements are equivalent:(i) does not depend on the choice of and ;(ii), , ;(iii) does not depend on the choice of and for every and ;(iv) does not depend on the choice of and .

Proof. (i)(ii): Since does not depend on the choice of and , we have
Take and , and put them into (13). Then ; namely, By the arbitrariness of and , By Lemma 3, So if , then and . Thus , which contradicts . Similarly, implies , which also leads to a contradiction. Hence
Next take and , and put them into (13). Then ; namely, By the arbitrariness of and and Lemma 3, By the above equation and (17), Namely, , , .
(ii)(iii): The inclusion implies and then by the definition of the Moore-Penrose inverse. So the two equations lead to and therefore and . Consequently, for every and , That is, .
Note that and imply and for certain operators , respectively. Then, by Lemma 1, That is, does not depend on the choice of .
(iii)(i): It is obvious.
Now we consider the situation under having closed range. It is evident that (i) (iv). We will show that (iv)(ii).
Obviously, holds for every and . Then has closed range and therefore
Putting and into (23) yields where Putting and into (23) yields where By the arbitrariness of and , (24) and (26) lead to . From these, as the argument in (i)(ii) above, we can reach (ii).

Note that if and only if . Thus, we have the following corollary.

Corollary 6. Let nonzero operators , , , and have closed ranges. Then the following statements are equivalent:(i) holds for every and ;(ii), , , ;(iii) holds for every and for every and .

Corollary 7 (see [10, Theorem 2.1]). Let nonzero operators , and have closed ranges. Then the following statements are equivalent:(i)the operator product does not depend on the choice of ;(ii), .

When in Theorem 5, we have the next result.

Corollary 8. Let nonzero operators , and have closed ranges. Then the following statements are equivalent:(i) does not depend on the choice of and ;(ii);(iii) for every and .

Next we will research the situation with respect to -inverses.

Theorem 9. Let nonzero operators , , and have closed ranges. Then the first three statements below are equivalent. Moreover, suppose that has closed range. Then the following statements are equivalent:(i) does not depend on the choice of and ;(ii), , ;(iii) does not depend on the choice of and for every and ;(iv) does not depend on the choice of and .

Proof. (ii)(i): and imply for some and for some , respectively. By Lemma 1, and where . So, by Theorem 5, and then (i) is true.
(i)(ii): Since does not depend on the choice of and , we have
First of all, we will show by contradiction. For this, assume . Take . Then putting and into (29) leads to Namely, By the arbitrariness of , , and , the condition that and , and Lemma 3, By (32), and then since . Thus, by (33), and then , which contradicts . Hence .
Now take and , and put them into (29). Then By the arbitrariness of and and Lemma 3, So if , then and . Thus , which contradicts . Similarly, implies , which also leads to a contradiction. Hence
Next take and , and put them into (29). Then By the arbitrariness of and and Lemma 3, So if , then and . Thus , which contradicts . Similarly, implies . But, by (36), since , which also leads to a contradiction. Hence Consequently, by (36) and (39), Namely, and .
Take and , and put them into (29). Then, by (36) and (39), Since and , then ; by the arbitrariness of and and Lemma 3, we have .
(ii)(iii): By Theorem 5, Statement (iii) is obvious.
(iii)(ii): When replacing and in Statement (i) by and , respectively, we immediately get and in view of the equivalence between (i) and (iii).
Now we will show . Since , for any and , by Lemma 1, Using Corollary 8, we have .
The remainder is to discuss the situation under having closed range. It is clear that (i)(iv). Following the process of the proof of (iv)(ii) in Theorem 5, we can also turn out (iv)(ii) by the argument in the proof of (i)(ii).

Remark 10. Obviously Theorems 5 and 9 are equivalent. If either and is zero, then it must be false that does not depend on the choice of and except for or . But it is always true that does not depend on the choice of and because is the only -inverse of .

Finally we will investigate the situations with respect to -inverses and -inverses.

Theorem 11. Let nonzero operators , , , and have closed ranges. Then the first two statements below are equivalent. Suppose that has closed range. Then the following statements are equivalent:(i) does not depend on the choice of and ;(ii) and , or and ;(iii) does not depend on the choice of and .

Proof. (i)(ii): Clearly, . Putting and into this equation leads to for any and . Then Thus and , or and . Hence (ii) is true.
(ii)(i): By Lemma 4, Statement (ii) implies and , or . So . Since and , . Now, by Lemma 1, That is, Statement (i) is true.
Now consider the situation under having closed range. Apparently, (i) implies (iii). We will show (iii)(ii).
Obviously, holds for every and . Then has closed range and therefore Similar to the argument above, putting and , the above equation leads to and then (ii) can be reached.

Corollary 12. Let nonzero operators , , , and have closed ranges. Then holds for every and if and only if , , and , or and .

Corollary 13 (see [10, Theorem 3.1]). Let nonzero operators , , and have closed ranges. Then the following statements are equivalent:(i)the operator product does not depend on the choice of ;(ii).

Note that if and only if . Thus we have the next theorem by Theorem 11 and then the following corollaries.

Theorem 14. Let nonzero operators , , , and have closed ranges. Then the first two statements below are equivalent. Suppose that has closed range. Then the following statements are equivalent:(i) does not depend on the choice of and ;(ii) and , or and ;(iii) does not depend on the choice of and .

Corollary 15. Let nonzero operators , , and have closed ranges. Then holds for every and if and only if , , and , or and .

Corollary 16 (see [10, Theorem 3.4]). Let nonzero operators , and have closed ranges. Then the following statements are equivalent:(i)the operator product does not depend on the choice of ;(ii).

4. Range Inclusion Invariance Properties of Operator Product

In the section, we will discuss the range inclusion invariance properties of the operator product involving generalized inverses. The following theorem is concerning the equivalent conditions of the topic.

Theorem 17. Let nonzero operators , , , and have closed ranges. Then the following statements are equivalent:(i) holds for every ;(ii) holds for every ;(iii); or , and .

Proof. (i)(ii): It is obvious.
(ii)(iii): By Lemma 2, we have the following matrix form: where and are invertible in , and , respectively. So, by Lemma 1, where , .
The inclusion is equivalent to ; that is, Thus, from the two equations above and the arbitrariness of and ,
If , then, by (52) and (54), Namely, . Thus .
If , then and . Therefore the former implies and the latter, by (51) and (53), implies Namely, , and .
It is clear that . As a result, we reach (iii).
(iii)(i): If , then . Since , . Since and , for some and for some . Then and therefore .

Remark 18. When , , , and are matrices, where symbol denotes any matrix with rows such that coincides with the orthogonal complement of for (see [9] for details). So by the argument in [9, page 97], Theorem 17(iii) is equipollent to [9, Theorem 2(ii)(c)] for matrices when .

In the following theorem, we state the range inclusion invariance involving -inverse and -inverse.

Theorem 19. Let nonzero operators , , , and have closed ranges. Then the following statements are equivalent:(i) holds for every ;(ii), .

Proof. (i)(ii): By Lemmas 1 and 2, we get We know that is equivalent to ; namely, Thus, from the two equations and the arbitrariness of ,
If , then, by (62), If , then and therefore, by (61), we also get (63). Hence, from (63), ; namely, .
It is clear that . Consequently, (ii) is reached.
(ii)(i): Since and , for some and for some . Then and therefore .

Concerning -inverses, we have the following result.

Theorem 20. Let nonzero operators , , and have closed ranges. Then the following statements are equivalent:(i) holds for every ;(ii), or and .

Proof. (i)(ii): By Lemmas 1 and 2, The condition is equivalent to ; namely, Thus, from the two equations and the arbitrariness of ,
If , then, by the four equations above, Hence ; namely, .
If , then and .
It is clear that . Consequently, (ii) is reached.
(ii)(i): Since and , for some and for some . Then and therefore .
If , then clearly .

Conflict of Interests

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

Acknowledgments

Xiaoji Liu and Miao Zhang were supported by the Guangxi Natural Science Foundation (2013GXNSFAA019008), the Key Project of Education Department of Guangxi (201202ZD031), Project supported by the National Science Foundation of China (11361009), and Science Research Project 2013 of the China-ASEAN Study Center (Guangxi Science Experiment Center) of Guangxi University for Nationalities.