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Volume 2021 |Article ID 5528089 | https://doi.org/10.1155/2021/5528089

Xiaopeng Li, Junjie Huang, Alatancang Chen, "Hypo-EP Matrices of Adjointable Operators on Hilbert -Modules", Complexity, vol. 2021, Article ID 5528089, 8 pages, 2021. https://doi.org/10.1155/2021/5528089

# Hypo-EP Matrices of Adjointable Operators on Hilbert -Modules

Accepted27 Apr 2021
Published25 May 2021

#### Abstract

This paper introduces and studies hypo-EP matrices of adjointable operators on Hilbert -modules, based on the generalized Schur complement. The necessary and sufficient conditions for some modular operator matrices to be hypo-EP are given, and some special circumstances are also analyzed. Furthermore, an application of the EP operator in operator equations is given.

#### 1. Introduction and Preliminaries

The EP matrix, as an extension of the normal matrix, was proposed by Schwerdtfeger; a square matrix over the complex field is said to be an EP matrix if and share the same range [1, 2]. The notion of EP matrices was extended by Campbell and Meyer to operators with closed range on a Hilbert space in . Let be a complex Hilbert space and the collection of all bounded linear operators on . Let . Recall that is called an EP operator if its range, , is closed, and . It is well known that is closed if and only if the Moore–Penrose inverse of exists and that is an EP operator if and only if . Sharifi  provided a generalization of the result for EP operators on Hilbert -modules. This has been studied by many other authors, see, e.g.,  and references therein. More generally, is said to be a hypo-EP operator if . In fact, is a hypo-EP operator if and only if is closed and . It is also shown that is a hypo-EP operator if and only if . The hypo-EP operator is our focus of attention in this paper, and it has been studied in [10, 11]. The EP operator can be applied to the solution of operator equations, see Section 3 of this article. The properties of hypo-EP and EP operators can find applications also in reverse order law  and core partial order  and will be useful in some other applied fields [14, 15]. In this note, we investigate the hypo-EP operators on Hilbert -modules, and then we formulate some results of hypo-EP matrices of adjointable operators on Hilbert -modules. As an application, the solvability conditions, and the general expression for the EP solution to the operator equations are given.

Since the finite-dimensional spaces, Hilbert spaces, and -algebras can all be regarded as Hilbert -modules, one can study hypo-EP modular operators in a unified way in the framework of Hilbert -modules. Let us briefly recall some basic knowledge about Hilbert -modules and adjointable operators. Throughout this paper, is a -algebra. A Hilbert -module is a right -module equipped with an -valued inner product such that is complete with respect to the induced norm . Suppose that and are Hilbert -modules, and let be the set of all maps for which there is a map such that for and . It is well known that an arbitrary element of must be a bounded linear operator, which is also -linear in the sense of for any and . We call the set of adjointable operators from to . We use to denote the -algebra . Let be the set of Hermitian elements of . For , the range and the null space of are denoted by and , respectively. An operator is said to be if there is an operator satisfying ; is called an (or -inverse) of . It is easy to prove that is regular if and only if is closed. The -inverse of is not unique in general.

In this paper, we use the generalized inverse to the generalized Schur complement as defined in . Suppose is a modular operator matrix of the formwhere . Then, the generalized Schur complement of in iswhere is an inner inverse of . Similarly, the generalized Schur complement of in iswhere is an inner inverse of . The formulas (2) and (3) have previously appeared in papers dealing with generalized inverses of partitioned matrices (cf. ).

Definition 1. (see ). Let . The Moore–Penrose inverse of (if exists) is an element in which satisfies(a)(b)(c)(d)These equations imply that will be uniquely determined if it exists, and and are both orthogonal projections. Moreover, , , , and . Clearly, the Moore–Penrose inverse of exists if and only if is closed; is Moore–Penrose invertible if and only if is Moore–Penrose invertible, and in this case, . Obviously, the Moore–Penrose inverse of is one of inner inverses of .
Similar to , Lemma 2.2.4, and , Lemma 2.2, we have the following conclusions on Hilbert -modules.

Lemma 1. Let and . If has an inner inverse then(i) if and only if (ii) if and only if

Lemma 2. Let be a modular operator matrix of form (1) with and . If has an inner inverse then is regular if and only if is regular, where . In this case, the inner inverse of is given by

From Lemma 2, we can obtain the following corollary.

Corollary 1. Let be a modular operator matrix of form (1) with , , , and . Ifand are closed, then the Moore–Penrose inverse of can be expressed as

Remark 1. The preceding result given in , Theorem 1, was proved for finite matrices.

Lemma 3 (see ). Let , where , and . If and are closed, then if and only if and .

Proof. The proof is similar to that in , Corollary 12, for Hilbert space operators.

Definition 2. (see ). Let be a Hilbert -module. An operator is called EP if .

Definition 3. Let be a Hilbert -module. An operator is called hypo-EP if .
Obviously, the range of an EP or a hypo-EP operator on Hilbert -modules is not necessarily closed, and we further have the following properties.

Proposition 1 (see ). Let be a Hilbert -module and with closed range. Then, the following conditions are equivalent:(i) is an EP operator(ii)(iii) is Moore–Penrose invertible and

Proposition 2. Let be a Hilbert -module and with closed range. Then, the following conditions are equivalent:(i) is a hypo-EP operator(ii)(iii) is Moore–Penrose invertible and

Remark 2. The class of all hypo-EP operators contains the class of all EP operators on Hilbert -modules. Meanwhile, the EP operator with closed range is an extension of the invertible operator and the normal operator with closed range. In the case of finite dimensional situation, EP and hypo-EP are the same.

#### 2. Main Results and Proofs

First, using generalized Schur complements, we study the hypo-EP property of matrices of adjointable operators on Hilbert -modules.

Theorem 1. Let be a modular operator matrix of the form (1) with , , , and . Suppose that and are closed. Then, the following conditions are equivalent:(i) is a hypo-EP operator matrix with closed range(ii) and are hypo-EP operators

Proof. Let be a hypo-EP operator matrix with closed range. Since and are closed, let us consider the operator matriceswhere . Obviously, and are invertible. By using Lemma 1 and by assumptions and , it is clear that can be factorized as . Hence, . Since is a hypo-EP operator matrix with closed range, . By using Lemma 1 again, it is immediate thatholds for every inner inverse of . In particular, forwe have from relation (7) thatThen, implies . Hence, is a hypo-EP operator. Since , substituting intoyields . This implies . Thus, is a hypo-EP operator.
Conversely, according to the assumptions , , , and , the Moore–Penrose inverse of exists, and is given byby Corollary 1. Using and , by Lemma 1, is described asSimilarly, by using , , and Lemma 1, it is given thatThen,Since and are hypo-EP operators with closed range,Thus, . Therefore, is a hypo-EP operator matrix with closed range.
The following conclusion is a natural extension of , Theorem 3.1, on Hilbert -modules.

Corollary 2. Let and . If is closed, then is a hypo-EP operator matrix with closed range if and only if is a hypo-EP operator.

If using the generalized Schur complement of in , similar to Theorem 1, one can get the following results.

Theorem 2. Let be a modular operator matrix of form (1) with , , , and . Suppose that and are closed. Then, the following conditions are equivalent:(i) is a hypo-EP operator matrix with closed range(ii) and are hypo-EP operators

Corollary 3. Let and . If is closed, then is a hypo-EP operator matrix with closed range if and only if is a hypo-EP operator.

Next, using the properties of generalized inverses, we study upper triangular hypo-EP matrices of adjointable operators on Hilbert -modules.

Theorem 3. Let with and , where , and . If and are closed, then is a hypo-EP operator matrix with closed range if and only if and are hypo-EP operators.

Proof. Let be a hypo-EP operator matrix with closed range. We writeObviously, is invertible. By Lemma 1 and assumption , it is clear that can be decomposed as . Hence, . Since is a hypo-EP operator matrix with closed range, . By Lemma 1, it is immediate that , where is given byThis givesHence, implies . Thus, is a hypo-EP operator. From , it follows that . Therefore, is a hypo-EP operator.
Conversely, suppose and are hypo-EP operators. Since and are closed and and , by Lemma 3, the Moore–Penrose inverse of exists andSince , by Lemma 1, is described asSimilarly, by Lemma 1, leads toThen,Since and are hypo-EP operators with closed range, and . Thus, . Therefore is a hypo-EP operator matrix with closed range.

Corollary 4. Let with and , where , , and . If and are closed, then is an EP operator matrix with closed range if and only if and are EP operators.

Proof. Let be an EP operator matrix with closed range. In view of Theorem 3, to prove the necessity, it is enough to show and . Since is an EP operator matrix with closed range, by the proof of Theorem 3, . Applying Lemma 1, we have , i.e.,Hence, and imply and , respectively.
Conversely, let and be EP operators. Since and are closed, , and , we get . Thus,Therefore, is an EP operator matrix with closed range.

Corollary 5. Let and . If and are closed, then is a hypo-EP operator matrix with closed range if and only if and are hypo-EP operators.

Corollary 6. Let and . If is closed, then is a hypo-EP operator matrix with closed range if and only if is a hypo-EP operator.

Remark 3. The Hilbert space version of the preceding four conclusions is given by , and the conditions of closed range can be naturally omitted there. Moreover, the alternative proofs of the conclusions in Hilbert space setting can be found in section 3 of . In addition, these results originated from the research of the EP property of block matrices, according to Hartwig .
Finally, the following are devoted to investigating the hypo-EP property of antitriangular block matrices of adjointable operators on Hilbert -modules.

Lemma 4. Let . If and are closed, then if and only if and .

Proof. Sufficiency: since and are closed, and are Moore–Penrose invertible. From and , it follows that and . We write . A direct calculation shows thatBy Definition 1, as desired.
Necessity: sinceare self-adjoint, we have and . From , we get . Therefore, and .

Lemma 5. Let with and . If and are hypo-EP operators with closed ranges, then is a hypo-EP operator matrix with closed range.

Proof. Since and are closed, , and , by Lemma 4, the Moore–Penrose inverse of is given byUsing and , by Lemma 1, we haveBy Definition 1, we have and . Since and are hypo-EP operators with closed ranges, and . Then,Therefore, is a hypo-EP operator with closed range.

Corollary 7. Let with and . If and are EP operators with closed ranges, then is an EP operator matrix with closed range.

Proof. According to the assumption, as with Lemma 5, we have equation (28). Since and are EP operators with closed ranges, and . Then,Thus, is an EP operator matrix with closed range.

Theorem 4. Let with and . If and are closed, then is a hypo-EP operator matrix with closed range if and only if and are hypo-EP operators.

Proof. The sufficiency is clear by Lemma 5. Now, we suppose that is a hypo-EP operator matrix with closed range. We writeIn the similar way as in the proof of Theorem 3,
we have , and hence, , since is a hypo-EP with closed range. This means by Lemma 1, i.e.,Hence, , which together with , implies . Thus, is a hypo-EP operator. Similarly, it follows from and that , and therefore, is a hypo-EP operator.

Corollary 8. Let and with . If and are closed, then is a hypo-EP operator matrix with closed range if and only if and are hypo-EP operators.

Corollary 9. Let with and . If and are closed, then is an EP operator matrix with closed range if and only if and are EP operators.

Proof. By Corollary 7, we only need to show the necessity, which can be easily verified according to the proofs of Corollary 4 and Theorem 4.

Corollary 10. Let and with . If and are closed, then is an EP operator matrix with closed range if and only if and are EP operators.

Remark 4. In Hilbert space case, the conditions of closed range in Theorem 4 and Corollary 9 can be naturally omitted in Theorem 3.8 and Theorem 3.9 of , and the alternative proofs of Theorem 4 and Corollary 9 can be, respectively, found in Theorem 3.8 and Theorem 3.9 of .

#### 3. The Application of EP Operators

In this section, let , and be Hilbert spaces. We establish the solvability conditions and the general expression for the EP solution to the operator equationswhere , , and .

Lemma 6 (see ). Let with closed range. Then, the operator is EP if and only if there exist Hilbert spaces and , unitary, and isomorphism such thatwhere .

Lemma 7 (see ). Let and . Suppose that and have closed ranges. Then, equation (33) has a common solution if and only if

In this case, the general common solution is given bywhere is arbitrary.

Now, we consider the EP solution to equation (33). By the Lemma 6, for the unitary operator , the solution has the following factorization:

Let , be closed, andwhere , , , , and , are closed. Then, equation (33) has an EP solution if and only if operator equationshave a common solution. By Lemma 7, we have the following theorem.

Theorem 5. Let and , and let , be closed. Suppose that is unitary such thatwhere , , , , and and are closed. Then, equation (33) has an EP solution if and only if

In this case, the general EP solution of (33) is given bywhere is arbitrary.

#### 4. Concluding Remarks

In this work, we have characterized hypo-EP and EP matrices of adjointable operators on Hilbert -modules, based on the generalized Schur complement, and an application of EP operator in operator equations is presented. In addition, the properties of hypo-EP and EP operators may have potential applications in some fields involving mathematics and its applications. In our opinion, it is worth establishing the hypo-EP and EP matrices of bounded linear operators on Krein space and of the adjointable operators on Krein -modules.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the NNSF of China (Nos. 11961052 and 11761029) and the NSF of Inner Mongolia (No. 2017MS0118).

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