#### Abstract

We study the solvability of the system of the adjointable operator equations , , and over Hilbert -modules. We give necessary and sufficient conditions for the existence of a solution and a positive solution of the system. We also derive representations for a general solution and a positive solution to this system. The above results generalize some recent results concerning the equations for operators with closed ranges.

#### 1. Introduction

Many results have been made on the study of solvability of equations for operators on Hilbert spaces and Hilbert -modules. In 1966, Douglas presented the famous Douglas theorem in [1]. He gave the conditions of the existence of the solution to the equation for operators on a Hilbert space. By using the generalized inverse of operators, Dajić and Koliha [2] got the existence of the common Hermitian and positive solution to the equations , for operators on a Hilbert space.

Hilbert -module is a natural generalization both of Hilbert space and -algebra and it has been an important tool in the theory of -algebra, especially in the study of -groups and induced representations (see [3–9]). Therefore it is meaningful to put forward a generalized version of the previous results about operator equations in the context of Hilbert -modules.

By using the generalized inverses of adjointable operators on a Hilbert -module, Wang and Dong recently obtained the necessary and sufficient conditions for the existence of a positive solution to the system of adjointable operator equations , , and for operators on Hilbert -modules in [10].

To use the generalized inverse, the authors mentioned above have to focus their attentions on those adjointable operators whose ranges are closed. However, closed range is a very strong condition in infinite dimensional case which general bounded (adjointable) linear operators may not satisfy. In fact an operator with closed range is also called a generalized Fredholm operator. In [6, 11], Fang et al. generalize the famous Douglas theorem from the case of the Hilbert spaces to the one of the Hilbert -modules and get some results about solutions to some equations and some systems of equations without the assumption of the closed ranges by use of some new approaches. They put the attention to the operators whose adjoint operators’ range closures are orthogonally complemented, which is automatically satisfied in Hilbert space case.

In this paper, along the same way as in [6, 11], we obtain the existence of the solution to the system of equations , , and which was studied in [10] and then two theorems about the existence of the positive solution to this system, which extend the main results in [10] from operators with closed range to operators whose adjoint operators’ range closures are orthogonally complemented.

#### 2. Preliminaries

First of all, we recall some knowledge about Hilbert -modules.

Throughout this paper, is a -algebra. An inner-product -module is a linear space which is a right -module, together with a map such that, for any , , and , the following conditions hold:(1); (2); (3); (4), and if and only if .

An inner-product -module which is complete with respect to the induced norm is called a (right) Hilbert -module.

Suppose that and are two Hilbert -modules; let be the set of all maps for which there is a map such that It is known that any element of must be a bounded linear operator, which is also -linear in the sense that for and . For any , the range and the null space of are denoted by and , respectively. We call the set of adjointable operators from to . We denote by the set of all bounded -linear maps, and therefore we have . In case , , to which we abbreviate , is a -algebra. Then for , is Hermitian (self-adjointable) if and only if for any , and positive if and only if for any , in which case, we denote by the unique positive element such that in the -algebra and then . Let be the sets of Hermitian and positive elements of , respectively. For any , we say if for any . For , the set of positive elements of the -algebra is a positive cone; we could easily verify that “≥” is a partial order on . For an operator , set Re, and is called real positive if .

We say that a closed submodule of is topologically complemented if there is a closed submodule of such that and and briefly denote the sum by , called the direct sum of and . Moreover, if where for all , we say is orthogonally complemented and briefly denote the sum by , called the orthogonal sum of and . In this case, and there exists unique orthogonal projection (i.e., idempotent and self-adjointable operator in ) onto . For two submodules and of , if , then .

Let ; then (1) and ; (2) if is closed, then so is , and in this case both and are orthogonally complemented and , (see [7], Theorem 3.2).

Any element of is called the inner inverse of and . is closed if and only if has a inner inverse. The Moore-Penrose inverse of is the unique inner inverse of which satisfies In this case, , and . Thus are the projections onto and , respectively.

Throughout this paper, , , , , and are Hilbert -modules. For an operator , if is orthogonally complemented, then and there exists an orthogonal decomposition . Let denote the orthogonal projection of onto and the projection ; then .

Lemma 1 (see [6, Theorem 1.1]). *Let and with being orthogonally complemented. The following statements are equivalent:*(1)* for some ;*(2)*there exists such that for all ;*(3)*there exists such that ; that is, has a solution;*(4)*. **Moreover there exists a unique operator which satisfies the conditions
**
In this case,
**
and is called the reduced solution of the equation .**The general solution to is of the form
**
where is arbitrary.*

Lemma 2 (see [6, Theorem 2.1]). *Let , , , and , suppose and are orthogonally complemented submodules in and , respectively, Then and have a common solution if and only if
**
In this case, the general solution is of the form:
**
where and are the reduced solutions of and , respectively, and is arbitrary.*

Lemma 3 (see [11, Theorem 3.1]). *Let , , and .*(1)* If the equation has a solution , then
*(2)* Suppose and are orthogonally complemented submodules of and , respectively. If
* *or
**then has a unique solution such that
**
which is called the reduced solution, and the general solution to is of the form
**
where .**Then, from and , one knows that . As in [11], one can obtain that .*

Lemma 4 (see [6, Theorem 1.3]). *Let such that is orthogonally complemented. Then has a positive solution if and only if .**In this case, , and the general positive solution is of the form
**
where is the positive reduced solution and is an arbitrary positive operator.*

Lemma 5 (see [6, Lemma 2.1]). *Let , and . Then if and only if , , and for any , , and any state .*

Lemma 6 (see [6, Proposition 2.2]). *Let and ,
**
such that is orthogonally complemented. Then and have a common positive solution , if and only if and .**In this case, the general positive solution can be expressed as , where is the positive reduced solution and is an arbitrary positive operator.*

Lemma 7 (see [11, Corollary 3.3(iii)]). *Let and such that and are orthogonally complemented, and or has the closed range; the equation has a positive solution if and only if and . In this case, the operator
**
is a positive solution for any and , where is the reduced solution.*

Let , and . Suppose and are orthogonally complemented submodules of ; the equation has the reduced solution . Set and Re, and assume that and are orthogonally complemented submodules of and , respectively. Set

It is clear that is a subset of the solution space to the equation .

Lemma 8 (see [11, Theorem 4.7]). *Let , , and . Suppose that and are orthogonally complemented submodules of , has the reduced solution , has the closed range, and is self-adjoint. Let with for some and . Then if and only if there exist and such that
**
in which case, .**As a consequence,
*

Lemma 9 (see [11, Theorem 4.12]). *Let , , and such that , and let and be orthogonally complemented submodules of . Suppose that has the reduced solution .*(1)*If has a positive solution , then and there exists a positive number such that
*(2)*Suppose that is orthogonally complemented in . If and
**for some positive number , then has a positive solution .*

#### 3. Main Results

Theorem 10. *Let () be Hilbert -modules, , , , , , , and .*(1)* If the system of operator equations , and has a solution , then
*(2)* Set . Suppose , , , and are orthogonally complemented submodules of . If
* *or
**and , then the system of operator equations , , and has a unique solution such that
**
In this case, the general solution is of the form
**
where , , and are the reduced solutions of , , and respectively. are arbitrary.*

*Proof. *(1) If the system of operator equations has a solution , it is easy to know that

(2) Since and are orthogonally complemented, , and . By Lemma 2, we know that have a common solution and it has the form:
where and are the reduced solutions of respectively.

Take into , then we can get

Since and (or and ), then
or
By Lemma 3(2), we know that the operator equation has a unique reduced solution and the general solution has the form
where and are arbitrary.

Hence the system of operator equations , and has a solution and it has the form

And it is easy to see that is the reduced solution to the system of operator equations , and ; and .

If and are orthogonally complemented submodules, then and are orthogonally complemented submodules.

*Remark 11. *Let () be Hilbert -modules, , , , , , , and . Set , , and as Lemma 6, and suppose , , and are orthogonally complemented submodules of . If , , , (or, , , , ), then the system of operator equations , , has a solution from Theorem 10. Let be the reduced solution to the system, and then we can obtain the next theorem about the positive solution to the system.

Theorem 12. *Let the notions and conditions as Remark 11. Set , , and .*(1)*If the system of operator equations , , and has a positive solution, then , , and there exists a positive number such that .*(2)*Suppose is orthogonally complemented submodule in . If , , and , for some , then the system of operator equations , , and has a positive solution.*

*Proof. *(1) If the system of operator equations , , and has a positive solution , then is the positive solution to the system of equations , . by Lemma 6, we know that and can be expressed as
where is the positive reduced solution to the system of equations , . Taking into yields that . And it has a positive solution. From Lemma 9, we know that , and there exists a positive number such that .

(2) If , we can get that the system of equations , has a positive solution , and it has the form
where is the positive reduced solution to the system of equations , . Take into ; then we can obtain that .

If is orthogonally complemented submodule in , and for some , by Lemma 9(2), we can easily get that the equation has a positive solution. Therefore the system of operator equations , , and has a positive solution.

Next, we give another theorem about the existence of the positive solution to the system of operator equations , , and . First we propose a lemma as follows.

Lemma 13. *Suppose that , , and and are orthogonally complemented submodules of . Let , and suppose has a closed range; then and both have positive solutions.**In this case, the general positive solutions are of the forms
**
where and are the reduced positive solutions, respectively, and are arbitrary. Furthermore, on has and .*

*Proof. *It is clear that , , so , and . Consider , , , , then it follows from Lemma 4 that , , and the general positive solutions are of the forms
where are arbitrary.

Consider , , ; then , , and , , , and . By Lemma 4, we know that , so . Similarly, we can obtain that ; then .

For simplicity, put

is the common reduced positive solution to , . Consider , ; is the reduced solution to . Consider . is the positive reduced solution to . is the positive reduced solution to . is the positive solution to . is the positive solution to . Take

By Lemma 6, we know that uniquely exists when is orthogonally complemented, , and . From Lemma 3, we know that uniquely exists when and , are orthogonally complemented, and . From Lemma 13, we know that uniquely exist when and are orthogonally complemented, has a closed range. By Lemmas 4 and 13, we know that and exist. So uniquely exists. In fact, if the conditions in the next theorem are satisfied, we can easily get that , , , , and uniquely exist.

Theorem 14. *Let () be Hilbert -modules, , , , , , , and . Suppose , , and are orthogonally complemented submodules of , , (or , ), and has closed range. The system of operator equations , , and has a positive solution if and only if
**
in which case the general common positive solution to , , and can be expressed as
**
where is the common positive reduced solution to , , is the positive reduced solution to , and are arbitrary positive solutions to , , respectively, such that is positive, is arbitrary, and is arbitrary positive operator in .*

*Proof. *Suppose is a positive solution to the system of the adjointable operator equations , , and ; then is a positive solution to the operator equations , . It follows from Lemma 6 that , , and has the form
where is the positive reduced solution of , , and is an arbitrary operator.

Take into ; we can get that
has a positive solution. Consider
By Lemma 13, we know that . Hence, if is positive, so is .

For all , ; that is, ; then since that . We can get ; then . Similarly, .

If and , then equations , have a positive solution by Lemma 6 and this positive solution can be expressed as
where is the positive reduced solution of the system of the adjointable operator equations , , and is an arbitrary operator.

Taking into , we can get

Now, we want to show that the equation has a positive solution. By Lemma 3, we know that the equation has a solution; then exists.

We rewrite , as , ; then , , and . Consider , ; the equations , both have positive solutions by Lemma 4 and they can be expressed as
where and are the positive reduced solutions to the equations , , respectively, and are arbitrary.

For operator equations , , we can obtain that , and , . By Lemma 4, we can get that and both have positive solutions. Let and be the positive reduced solutions, respectively. Hence

Let and . By Lemma 5, .

For the operator equation , by Lemma 7, has a positive solution and has the form
where is the positive reduced solution to the operator equation and and are arbitrary.

Then we claim that is the positive solution to , , and . In fact, we only need to prove that . Consider

Suppose that is a positive solution to the system of the operator equations , , and . It follows from Lemma 6 that can be expressed as
where is the positive reduced solution of the system of the operator equations , . Hence there is a positive operator such that

Let , ; it follows from
that and are positive solutions to the operator equations , , respectively. Consider
By Lemma 7, can be expressed as
where is the positive reduced solution to and and are arbitrary.

Take into ; we know that has the form that is expressed as (41).

Let , , and , and suppose that , are orthogonally complemented submodules of . If and (or , ) we can obtain that the operator equation has a solution by Lemma 3. Let be the reduced solution to and and suppose it has a closed range, where is the positive reduced solution to . is the positive reduced solution to . is the positive solution to . is the positive solution to .

Consider By Theorem 14, we can give necessary and sufficient conditions for the existence of a positive solution to operator equation .

Corollary 15. *Let the notions and conditions be as described above. Then the operator equation has a positive solution in if and only if
**
in this case the form of general positive solution to is
**
where is the positive reduced solution to , and are arbitrary positive solutions to , , respectively, such that is positive, and and are arbitrary.*

The results obtained in Lemma 9 give us the condition of the existence of the positive solution to , but they are restricted by the assumption that . The result in Corollary 15 does not have the constraint mentioned above. Clearly Theorem 14 extends Theorem 3.6 in [10], and Corollary 15 extends Corollary 4.1 in [10].

#### Conflict of Interests

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

#### Acknowledgments

This paper is supported by the National Natural Science Foundation of China (11071188) and “Chen Guang” Project (supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation) (10CGB25).