Developing Reverse Order Law for the Moore–Penrose Inverse with the Product of Three Linear Operators

Qi, Yang; Xiaoji, Liu; Yaoming, Yu

doi:https://doi.org/10.1155/2021/6585951

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Research Article | Open Access

Volume 2021 | Article ID 6585951 | https://doi.org/10.1155/2021/6585951

Developing Reverse Order Law for the Moore–Penrose Inverse with the Product of Three Linear Operators

Yang Qi,¹Liu Xiaoji,²and Yu Yaoming³

Academic Editor: Valerii Obukhovskii

Received16 Jun 2021

Accepted04 Oct 2021

Published31 Oct 2021

Abstract

In this paper, we study the reverse order law for the Moore–Penrose inverse of the product of three bounded linear operators in Hilbert spaces. We first present some equivalent conditions for the existence of the reverse order law . Moreover, several equivalent statements of and are also deducted by the theory of operators.

1. Introduction

Let denote the set of all bounded linear operators from Hilbert space to Hilbert space . Especially, it is briefly written as when . For , the symbols , , and denote null space, range, and Moore–Penrose inverse of , respectively.

The Moore–Penrose inverse of is the operator that satisfies the Penrose equations, i.e.,

If the Moore–Penrose inverse of exists, then it is unique. exists if and only if is closed in (see [1, 18] for details).

The reverse order law for the generalized inverse of a product of matrices or bounded linear operators yields a class of interesting fundamental problems in the theory of the generalized inverse of matrices or operators. It has been widely studied since the middle 1960s. Greville [7] first discussed the reverse order law for the Moore–Penrose inverse of the product of two matrices and gave a necessary and sufficient condition for the law. Hartwig et al. [8] studied the reverse order law for the Moore–Penrose inverse of the product of three matrices and derived the equivalent conditions for the existence of the reverse order law. Ji and Wei [10] studied the weighted Moore–Penrose inverse and extended the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Recently, Panigrahy and Mishra presented some properties for the weighted Moore–Penrose inverse for an arbitrary order tensor via the Einstein product [12]; they also proposed the expression for the Moore–Penrose inverse of the product of two tensors via the Einstein product [13, 14]. Until now, some reverse order laws for the known generalized inverses have been widely studied and extended into a complex number field, a Hilbert space, or an associative ring (see [2, 3, 5, 6, 9, 11, 15–17, 19–23] and references therein).

Recently, based on the results in [8], Dinčić and Djordjević [4] extended some finite dimensional results to infinite dimensional settings and obtained new results related to the mixed-type reverse order law for the Moore–Penrose inverse of various products of three operators in Hilbert space. This motivates us to have a further study concerning the reverse order law of three operators in Hilbert space and to extend some results in [4, 8].

In the remainder of this section, we will give some denotations. In Section 2, we will present several lemmas. In Section 3, we will investigate some equivalent conditions for the existence of . Moreover, several equivalent statements of and are also deducted by the theory of operators.

Now, we describe several denotations. If , then the symbol stands for the commutator of and , i.e., . Also, for , the symbols and are obtained. Throughout this paper, we always assume that , where , denote arbitrary Hilbert spaces. And then, will be well-defined in what follows.

2. Lemmas

First of all, we list two lemmas, which come from [4, 6], and the first is a rewritten version of Lemmas 1.1 and 1.2 in [6].

Lemma 2.1 (see [6]). Let have a closed range. Let and be closed and mutually orthogonal subspaces of such that (here, represents direct sum). Let and be closed and mutually orthogonal subspaces of such that . Then, the operator has the following matrix representations with respect to the orthogonal sums of subspaces and :(a)Representations with respect to the orthogonal sums of subspaces are where maps into itself and (meaning is invertible). Also,(b)Representations with respect to the orthogonal sums of subspaces are where maps into itself and (meaning is invertible). Also,

Here, , denote different operators in any of these two cases.

In particular,where is invertible. Also,

Lemma 2.2 (see [4]). Let be Hilbert spaces. Let have a closed range, and let be Hermitian and invertible. Then, if and only if .

Next, the following lemma can guarantee the existence of the Moore–Penrose inverses of various operators appearing in what follows.

Lemma 2.3. Let , and have closed ranges. Then , , , , , , , , , and exist.

Proof. Since , . Furthermore, we have , , and exist since and have closed ranges. Then,Hence, exists.
Similarly, we can obtain , , , , , , and exist.
Finally, the following two lemmas are used for proving some results in the next section. Here, the symbol , which denotes the commutator of and .

Lemma 2.4. Let be invertible and have a closed range and , , and exist. Then,(i) if and only if if and only if if and only if (ii) if and only if if and only if if and only if (iii) if and only if and if and only if and

Proof. Obviously, due to the invertibility of . Then, has a closed range since is closed. Hence, exists. So does . And then, exists. Similarly, exists. Hence,(i)Clearly , , and .Note that holds if and only if , namely, , which is equivalent to . Hence, the proposition is true by the definition of the Moore–Penrose inverse.
The proofs of (ii) and (iii) are similar.

Lemma 2.5. Let be Hermitian and invertible and have a closed range. Then, for arbitrary rational numbers and , we have the following properties:(i) if and only if (ii) if and only if (iii) implies (iv) implies

Proof. For arbitrary rational numbers , and exist by Lemma 2.4, since and are obviously invertible.(i)By Lemma 2.2, if and only if if and only if , i.e., if and only if .(iii)By Lemma 2.2, leads to . Thus, and therefore .The proofs of (ii) and (iv) are similar.

Remark 1. If matrix is Hermitian and invertible, let be eigenvalues for ; then, the matrix is similar to a diagonal matrix. And, it can be written as follows:where is a unitary matrix. Therefore, we havewhere denotes arbitrary rational numbers.

3. Reverse Order Laws

In this section, we will prove the result concerning the reverse order law for the Moore–Penrose inverse of various products of bounded linear operators in Hilbert spaces.

Firstly, we shall show some equivalent conditions for the existence of the reverse order law , which extends Theorem 1 in [8].

Theorem 3.1. Let , and have closed ranges. Then, the following statements are equivalent:(i)(ii), , and are Hermitian(iii), , and are Hermitian(iv), , and are Hermitian(v), , and are Hermitian

Proof. By Lemma 2.3, the Moore–Penrose inverses , , , and exist.
Set in Lemma 2.1(b). By Lemma 2.1(b),where maps into itself and ,Set in Lemma 2.1(b). Then,Since has a closed range, by Lemma 2.1, we can obtain thatThus, is invertible. Furthermore, we haveand then,Similarly, we can obtainIn addition, since is Hermitian, we haveAnd, is Hermitian, soMultiplying by and , respectively, on the left and right sides of equation (22) yields (i)(ii): similar to the proof of Theorem 1 in [8]. (ii)(iii): by (12), we haveThen, implies , and therefore, by (17) and (24),By (21) and (25),and then,As a result, by Lemma 2.4(i), we haveHence, by (18), (25), and (28), we can obtain (iii) (v): since implies By (18), (24), and (31), we have and . Consequently, by Lemma 2.4(ii), we have Combined with (19), (31), and (33), we can further obtain (v) (iv): since using (19) and , we can obtain Combining (21) and (36), we have and then, . Hence, by Lemma 2.4(i), we have and therefore, by (16), (38), and (36), we have (iv) (ii): since by (16), implies Combining (23) and (41), we have and then, by Lemma 2.4(ii), we can obtain that Accordingly, by (17), (41), and (43), we haveIn [4], Dinčić and Djordjević showed some equivalent statements of and . Here, we will present other equivalent statements.

Theorem 3.2. Let , and have closed ranges. Then, the following statements are equivalent:(i)(ii)(iii)(iv)(v)(vi)(vii) and

Proof. By Lemma 2.3, we can obtain that the Moore–Penrose inverse , , and exist. : multiplying and on the left and the right side of , respectively, we then have where . : multiplying and on the left and the right side of , respectively, we thus have namely, Thus, by (12), obviously holds. : multiplying and on the left and the right side of , respectively, then we have By (12) and (16), and then easily holds. : by (16), we have Therefore, is equivalent to By Lemma 2.4(iii), we can obtain that (50) holds if and only if the following equations are satisfied: Namely, On the other hand, by Lemma 2.2, (v) is equivalent to . : the equation can be deduced and then, the equivalence between and is similarly proved.In addition, we can deduce a key equation, similar to (16), and therefore obtain the equivalence of , , and .
By Theorems 2.1 and 2.2 in [4], we can prove the following theorem, similar to the proof of Theorem 3.2.

Theorem 3.3. Let , and have closed ranges. Then, the following statements are equivalent:(i)(ii)(iii)(iv)(v)(vi)(vii) and Next, we will further deduce some statements, each of which is equivalent to and .

Theorem 3.4. Let , and have closed ranges. Then, the following statements are equivalent:(i)(ii)(iii)(iv)(v)(vi)(vii)(viii)

Proof. By Lemma 2.3, we can obtain that , , , , , , , , and exist. By (12), we haveSince has a closed range, by Lemma 2.1, we havewhich is invertible, andand then,Similarly, we can get that (i)(viii): using (56) and (61), we have Now, (i) holds; i.e., holds if and only if By Lemma 2.4, (67) is equivalent to Since is invertible, by Lemma 2.4, we have In addition, is equivalent to whenever . From this, we can get that (68) is equivalent to namely, By Lemma 2.2, (71) holds if and only if Namely, (i) is equivalent to . (viii)(vii): statement (viii) implies that which is equivalent to Thus, and then, the left equation in (75) implies By Lemma 2.5, we have Hence, by Lemma 2.4, we have . From this, the right equation in (75) becomes and then Therefore, we have . Consequently, (76) becomes Thus, combining Lemma 2.4, (78), and (79), we have (vii)(vi): by (60) and (65), (vii) holds if and only if Then, (81) is equivalent to By Lemmas 2.4 and 2.5, the first equation in (82) is equivalent to Thus, and therefore, . The second equation in (82) can be further rewritten as follows: Consequently, (82) becomes and then, , namely, (vi) holds. (vi)(v): by (59) and (64), we can obtain that (vi) is equivalent to or, equivalently, By Lemma 2.4 and 2.5, the right equation in (88) is equivalent to By Lemma 2.5, the left equation in (88) is equivalent to As a result, (88) is equivalent to if and only if , namely, (v) holds. (v)(iv): by (56) and (63), (v) holds if and only if Using Lemma 2.4, we have By Lemma 2.5, the right equation in (93) is equivalent to and implies . Thus, by Lemma 2.4, we have . Accordingly, (93) becomes Similarly, combining Lemmas 2.5 and 2.4, from the left equation in (94), it follows that . Consequently, (94) becomes if and only if , namely, (iv) holds. (iv)(iii): by (56) and (62), (iv) holds if and only if Notice that the left equation in (96) holds if and only if with Lemma 2.5. Hence, and then, by Lemma 2.4, we have . Thus, (96) becomes Hence, and therefore, (iii) holds. (iii)(ii): by (58) and (61), (iii) holds if and only if Similarly, and if and only if and . Therefore, if and only if (ii) holds. (ii)(i): by (57) and (61), (ii) is equivalent to Therefore, by Lemma 2.4, we have if and only if , i.e., (i) holds.By Theorems 3.1–3.3, we can obtain the following theorem.

Theorem 3.5. Let , and have closed ranges. The following statements are equivalent:(i)(ii)(iii)(iv) and (v) and (vi) and (vii) and (viii) and (ix)(x) and (xi) and (xii) and (xiii) and

Proof. By Lemma 2.3, those Moore–Penrose inverses exist. (i)(ii): if (i) holds, then and then The other Penrose equations are similarly verified. Hence, (ii) holds. Conversely, the proof is same. (i)(iii): this is similar to the proof above. (i)(iv): by Theorem 3.1, we have and (iv)(v): by Theorems 3.2 and 3.3, we have and (v)(vi) and (vi)(vii): This is similar to the proof of “(iv) (v).” (vii)(i): The proof is obvious. (i)(viii): if (i) holds, then we can obtain that is Hermitian by (13). Thus, it is easily verified that holds by the definition of the Moore–Penrose inverse. Therefore, Conversely, the following is apparent: (i)(ix), (i)(x), and (i)(xi): these are analogous to the proof above. (i)(xii): since (i), (viii), and (ix) are equivalent, (i) implies (xii).Conversely, and then,Thus, the first and third Penrose equations are satisfied. Similarly, the other Penrose equations are satisfied: (i)(xiii): the proof is the same as the above.

Data Availability

Data are available upon request to the corresponding author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 72061007), the Guangxi Special Fund for Talents (Grant no. 2020AC19203), the Guangxi Key Laboratory of Cryptography and Information Security at Guilin University of Electronic Technology (Grant no. GCIS201929), and Guangxi Young Scholar Research Ability Enhancement Program (Grant no. 2020KY05017).

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