#### 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 [3]. 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 [3]. 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 [4] provided a generalization of the result for EP operators on Hilbert -modules. This has been studied by many other authors, see, e.g., [5–8] and references therein. More generally, is said to be a hypo-EP operator if [9]. 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 [12] and core partial order [13] 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 [16]. 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. [17–19]).

*Definition 1. *(see [20]). 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 [21], Lemma 2.2.4, and [22], 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 [17], Theorem 1, was proved for finite matrices.

Lemma 3 (see [23]). *Let , where , and . If and are closed, then if and only if and .*

*Proof. *The proof is similar to that in [22], Corollary 12, for Hilbert space operators.

*Definition 2. *(see [4]). 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 [4]). *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 [10], 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 [10], 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 [10]. In addition, these results originated from the research of the EP property of block matrices, according to Hartwig [24].

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 [10], and the alternative proofs of Theorem 4 and Corollary 9 can be, respectively, found in Theorem 3.8 and Theorem 3.9 of [10].

#### 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 [25]). *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 [22]). *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).