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ISRN Mathematical Analysis

VolumeΒ 2012Β (2012), Article IDΒ 676835, 16 pages

http://dx.doi.org/10.5402/2012/676835

## The Theory for -Hermitian Subspaces in a Product Space

Department of Mathematics, Shandong University at Weihai, Weihai, Shandong 264209, China

Received 6 January 2012; Accepted 13 February 2012

Academic Editors: S.Β Deng and O.Β Miyagaki

Copyright Β© 2012 Huaqing Sun and Jiangang Qi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is concerned with the theory for -Hermitian subspaces. The defect index of a -Hermitian subspace is defined, and a formula for the defect index is established; the result that every -Hermitian subspace has a -self-adjoint subspace extension is obtained; all the -self-adjoint subspace extensions of a -Hermitian subspace are characterized. This theory will provide a fundamental basis for characterizations of -self-adjoint extensions for linear nonsymmetric expressions on general time scales in terms of boundary conditions, including both differential and difference cases.

#### 1. Introduction

The spectral theory for differential and difference has been investigated extensively. In general, under certain definiteness conditions, a formally symmetric differential expression can generate a minimal operator which is symmetric, that is, a densely defined Hermitian operator, in a related Hilbert space and its adjoint, is the corresponding maximal operator (see, e.g., [1β3]). There are many results on self-adjoint extensions of the minimal operators since self-adjoint extension problems are fundamental in the study of spectral theory for differential expressions [2β6]. However, for some formally symmetric differential expressions, their minimal operators may be nondensely defined, or their maximal operators may be multivalued (e.g., [7, Example 2.2]). Further, for a formally symmetric difference expression, even a second-order one, its minimal operator is nondensely defined, and its maximal operator is multivalued in the related Hilbert space in general [8]. Therefore, the classical von Neumann self-adjoint extension theory and the Glazman-Krein-Naimark (GKN) theory for symmetric operators are not applicable in these cases.

The appropriate framework is linear subspaces (linear relations in the terminology of [7, 9, 10]) in a Hilbert space to study the linear differential or difference expressions for which the corresponding operators are nondensely defined or multivalued. Lesch and Malamud studied formally symmetric Hamiltonian systems in the framework of linear subspaces [7]. Coddington studied self-adjoint extensions of Hermitian subspaces in a product space [11]. He had extended the von Neumann self-adjoint extension theory for symmetric operators to Hermitian subspaces. By applying the relevant results in [11], Shi established the GKN theory for Hermitian subspaces [12]. For more results about nondensely defined Hermitian operators or Hermitian subspaces, we refer to [13β15].

The study of spectral problems involving linear differential and difference expressions with complex-valued coefficients is becoming a well-established area of analysis, and many results have been obtained [1, 16β22]. Such expressions are not formally symmetric in general, and hence, the spectral theory of self-adjoint subspaces is not applicable. To study such problems, Glazman introduced a concept of -symmetric operators in [23], where is a conjugation operator given in Section 2. The minimal operators generated by certain differential and difference expressions with complex-valued coefficients are -symmetric operators in the related Hilbert spaces (e.g., [18, 24, 25]). -self-adjoint extension problems are also fundamental in the spectral theory for such expressions. Many results have been obtained on -self-adjoint extensions [24β27]. Knowles gave a complete solution to the problem of describing all the -self-adjoint extensions of any given -symmetric operator provided the regularity field of this operator is nonempty [26]. Given a differential or difference expression, it is in practice difficult however to determine whether the appropriate -symmetric operator has empty or nonempty regularity field. Therefore, Race established the theory for -self-adjoint extensions of -symmetric operators without the restrictions on the regularity fields [24]. However, the appropriate framework is also linear subspaces in a Hilbert space to study the linear nonsymmetric differential or difference expressions for which the corresponding minimal operators are nondensely defined, or the corresponding maximal operators are multivalued. So, the -self-adjoint extension theory mentioned the above needs to be extended to linear subspace when we consider the nonsymmetric Hamiltonian systems which induce the nondensely defined or multivalued operators.

In this present paper, the concept of the defect indices of -Hermitian subspaces is given and a formula for the defect indices is obtained. Further, the result that every -Hermitian subspace has a -self-adjoint subspace extension is given, and the characterizations for all the -self-adjoint subspace extensions of a -Hermitian subspace are established, which can be regarded as the GKN theorem for -Hermitian subspaces.

*Remark 1.1. *We will apply the results obtained in the present paper to characterizations of -self-adjoint extensions for linear Hamiltonian difference systems in terms of boundary conditions in the near future.

The rest of this present paper is organized as follows. In Section 2, some basic concepts and fundamental results about linear subspaces are introduced. In Section 3, the defect index of a -Hermitian subspace is defined, and a formula for the defect index is given. Section 4 pays attention to the existence of -self-adjoint subspace extensions and the GKN theorem for -Hermitian subspace.

#### 2. Preliminaries

In this section, we introduce some basic concepts and give some fundamental results about linear subspaces in a product space.

Let be a complex Hilbert space with the inner product . The norm is defined by for . Let be the product space . By definition, the elements of consist of all possible ordered pairs with and , and for arbitrary two elements , and , The null element of is . The inner product in is defined by and denotes the induced norm.

Let be a linear subspace in which is called to be a linear relation in [7, 9, 10]. For brevity, a linear subspace is only called a subspace. For subspace in , we shall use the following definitions and notations: Clearly, if and only if can determine a unique linear operator from into whose graph is , and is closed if and only if is closed. Since the graph of a linear operator in is a subspace in and a linear operator is identified with its graph, the concept of subspaces in generalizes that of linear operators in .

*Definition 2.1 (see [11]). *Let be a subspace in .(1)Its adjoint, , is defined by
(2) is said to be a Hermitian subspace if .(3) is said to be a self-adjoint subspace if .

Lemma 2.2 (see [11]). *Let be a subspace in , then is a closed subspace in , , and , where is the closure of .*

*Definition 2.3. *An operator defined on is said to be a conjugation operator if for all ,

It can be verified that is a conjugate linear, that is, and for and , and norm-preserving bijection on satisfying For example, the complex conjugation in any space is a conjugation operator on .

*Definition 2.4. *Let be a subspace in , and let be a conjugation operator.(1)Its -adjoint, , is defined by
(2) is said to be a -Hermitian subspace if .(3) is said to be a -self-adjoint subspace if .

*Remark 2.5. *(1) It can be easily verified that is a closed subspace. Consequently, a -self-adjoint subspace is a closed subspace since . In addition, if .

(2) From the definition, we have that for all and , and that is a -Hermitian subspace if and only if

(3) The concepts of -Hermitian and -self-adjoint subspaces generalize those of -symmetric and -self-adjoint operators, respectively (see, e.g., [1, 24] for the concepts of -symmetric and -self-adjoint operators).

Lemma 2.6. *Let be a subspace in , then*(1)*,*(2)*.*

*Proof. *Result (2) follows from result (1) and the second relation of (2.5). So, one needs only to prove result (1). Set . Let , then for all . So, the second relation of (2.5) yields that for all . Hence, . Then , which implies that . Conversely, let , then there exists such that . It follows from that for all , which implies that , that is, . So, . Consequently, , and result (1) holds.

*Remark 2.7. *Let be a subspace in , then from Lemmas 2.2 and 2.6, and the closedness of , one has that , and is -Hermitian if is -Hermitian.

Lemma 2.8. *Let be a closed -Hermitian subspace, then if and only if and for all .*

*Proof. *Let be a closed -Hermitian subspace. Clearly, the necessity holds by (2) of Remark 2.5. Now, consider the sufficiency. Suppose that and for all , then we get from (2.6) that for all . This, together with (1) of Lemma 2.6, implies that for all . So, , and hence, by Lemma 2.2. So, the sufficiency holds.

Lemma 2.9. *Let be a closed -Hermitian subspace in , then*(1)*,*(2)*.*

*Proof. *Since result (1) and the second relation of (2.5) imply that result (2) holds, it suffices to prove result (1). Set . Let , then there exists such that . By Lemma 2.8, implies that for all . Then , that is, . So, . Conversely, let , then for all , that is,
Clearly, (2.9) holds for all since . So, . This, together with (2.9) and Lemma 2.8, implies that . So, , that is, . Then, and then .

*Remark 2.10. *Since by Remark 2.7, Lemma 2.9 yields that and for a -Hermitian subspace which may not be closed.

#### 3. Defect Index of a -Hermitian Subspace

In this section, the definition of the defect index of a -Hermitian subspace is introduced, and a formula for the defect index is obtained.

Let be a closed -Hermitian subspace. It has been known that is a closed subspace by (1) of Remark 2.5. Then the closedness of and and gives that where denotes the orthogonal complement of in , that is, . Now, let be a closed -Hermitian subspace extension of , that is, and is -Hermitian. Then, it follows from the closedness of and that there exists a unique subspace such that Clearly, since and . Then by (3.1) and (3.2), can be expressed as Further, we have the following result.

Theorem 3.1. *Let be a closed -Hermitian subspace, and let be a -self-adjoint subspace extension (briefly, -SSE) of , that is, and is -self-adjoint, then
*

*Proof. *If is a -self-adjoint subspace, then and is the only -SSE of itself. So, (3.4) holds. Now, suppose that is -Hermitian but not -self-adjoint, that is, and . It follows that (3.2) holds with . Let , then (3.2) yields that . In the case of , let be a basis of , then
Define
Clearly, for and
since and is -self-adjoint. Further, () is a closed subspace since is closed. It holds that
Otherwise, for example, suppose that , then by Lemma 2.2, we have since and are closed. It contradicts . So, (3.8) holds. It follows from (3.8) and (2) of Lemma 2.6 that
We get from (3.7) and (3.9) that .

In the case of , we have the linear span of an infinite set in (3.5). So we can construct an infinite sequence of subspaces of the form (3.6) which satisfies the relations like those in (3.7) and (3.9). So, we have .

Next, we prove that . Since and are closed subspaces, there exists uniquely a closed subspace such that . Set . Then . If , let be a basis of , thenDefine
Clearly, it holds that for and
since and is -self-adjoint. Further, () is a closed subspace since is closed. Similarly, it holds that
We get from (3.12) and (3.13) that . It can be verified that
by Lemma 2.9. So, . Further, it can be verified that also holds for . Based on the above discussions, (3.4) holds.

*Remark 3.2. *(1) From Theorem 3.1 and its proof, one has that if one of the two dimensions in (3.4) is finite, so is the other and they are equal, and if one of the two dimensions is infinite, so is the other. Here, there is no distinction between degrees of infinity.

(2) The case for -symmetric operators was established in [24, Theorem 3.1].

*Remark 3.3. *Note that . We get from Theorem 3.1 that if is a -SSE of , which may not be closed, then it holds that

Now, we give the concept of the defect index of a -Hermitian subspace. The concept of the defect index of a -symmetric operator in was given by [24, Definition 3.2].

*Definition 3.4. *Let be a -Hermitian subspace, then is called to be the defect index of .

*Remark 3.5. *(1) It will be proved that every -Hermitian subspace has a -SSE in Theorem 4.3 in Section 4. So, by (3.15) we have that the defect index of every -Hermitian subspace is a nonnegative integer.

(2) Since by Remark 2.7 and every -SSE is closed, we have that a -symmetric subspace and its closure have the same defect index and the same -SSEs.

*Definition 3.6 (see [12]). *Let be a subspace in . The set
is called to be the regularity field of .

It is evident that for a subspace .

Lemma 3.7. *Let be a subspace in , then*(1)* for each ,*(2)*for each ,*(3)* for each .*

*Proof. *(1) Let . It is clear that
For every , we have from (3.18) that . So, for all , which implies that . Therefore, , and then . Conversely, for every , we have that for all . It follows that for all . So, , and hence by (3.18). So, . Consequently, , and result (1) is proved.

(2) By result (1) and Lemma 2.2, we have that . So, by the projection theorem, in order to prove (3.17), it suffices to show that with is closed in . It is evident that
Let . Since , one has that , that is, there exists a constant such that
Then we get from (3.19) and (3.20) that . So, determines a linear operator from to . Further, the closedness of and (3.20) imply that this operator is a closed and bounded operator. So, its domain is closed in . Therefore, (3.17) holds, and result (2) is proved.

(3) Let . We first show that there exists a constant such that
Assume the contrary, then there exists a sequence with () such that
Clearly, and imply that . So, for all by (1) of Remark 2.5, which, together with , implies that for ,
Define for . Then , , are linear functionals in . Since by (3.23) and , we have that is bounded for any given . Note that with is closed by the proof of result (2), and hence it is a Hilbert space with the inner product . So, by the resonance theorem, is bounded, that is, is bounded. Since , , we have a contradiction with (3.22). So, (3.21) holds.

Inserting (3.21) into , we get that
It can be easily verified from the closedness of and (3.24) that is closed in . So, result (1) implies that
By Remark 2.10, , while it can be verified that for . Therefore, (3.25) yields that result (3) holds.

If , we have the following results which give a formula for the defect index of a -Hermitian space.

Theorem 3.8. *Assume that is a -Hermitian subspace with . Let , then
**
where
*

*Proof. *We first prove (3.26). Since is -Hermitian, one has that is -Hermitian, and hence . Clearly, . So, . On the other hand, let . It follows from (3.17) and that there exist and such that , that is, . Let , then and . So, , and consequently, .

Now, let , then
Consequently, by , which can be obtained from (3.17). Since , one has by Definition 3.6 that there exists a constant such that . It follows that , which, together with , implies that . Then, if . So, (3.26) holds.

Next, we prove (3.28). Let . Set , then is closed since is closed. Let . We will show that
Let be linearly independent, then, by (3) of Lemma 3.7, there exists such that , . It follows from that for . In addition, if
then , that is, . So, for , and hence is linearly independent (modβ). Conversely, let be linearly independent (modβ), and let , then . If , then . So, (), and hence the set is linearly independent. Based on the above discussions, (3.30) holds. On the other hand, it is evident that
Further, by Lemma 2.6, we have that
It follows from (3.30)β(3.33) that , and hence (3.26) implies that
So, (3.28) holds by (1) of Lemma 3.7.

*Remark 3.9. *From Theorem 3.8, one has the following result: for a -Hermitian subspace , and are constants in which are equal to the defect index of . This result extends [24, Theorem 5.7] for -symmetric operators to -Hermitian subspaces. Similarly, there is no distinction between degrees of infinity.

#### 4. -Self-Adjoint Subspace Extensions of a -Hermitian Subspace

In this section, we consider the existence of -SSEs of a -Hermitian space and the characterizations of all the -SSEs.

Define the form as Then, if and only if . Further, for all and , Since the closure of a -Hermitian subspace is also a -Hermitian subspace by Remark 2.7, and and have the same defect indices and the same -SSEs by (2) of Remark 3.5, we shall assume that is closed in the rest of this section. Let be a closed subspace in , and let be a subspace, where is given in (3.1). Let be a restriction of to , that is,

then is called to be -Hermitian in if , and is called to be -self-adjoint in if .

*Remark 4.1. *From the definition, we have that for all and , and is -Hermitian in if and only if for all , .

Lemma 4.2. * Let be a closed -Hermitian subspace, and let be a subspace. Assume that , then*(1)* is -Hermitian if and only if is -Hermitian in ,*(2)* is -self-adjoint if and only if is -self-adjoint in .*

*Proof. *(1) Suppose that is -Hermitian. It can be easily verified that is -Hermitian in by (2) of Remark 2.5, , and Remark 4.1. So, the necessity holds. We now prove the sufficiency. Suppose that is -Hermitian in . For all , , we get from that
Since is -Hermitian and , it can be obtained from (2) of Remark 2.5 and Remark 4.1 that
So, is -Hermitian. The sufficiency holds, and result (1) is proved.

(2) First, consider the necessity. Suppose that is -self-adjoint, then is -Hermitian in , that is, , by result (1). It suffices to show that . For any given , there exist and such that the first relation of (4.4) holds. Let , then by Remark 4.1. Note that by . Then by (2) of Remark 2.5. Therefore, the first relation of (4.4) yields that
So, . Therefore, yields that , which, together with , , and , implies that . Hence, , and hence . The necessity holds.

Next, consider the sufficiency. Suppose that is -self-adjoint in . By result (1), one has that . It suffices to show that . Let , then since by . It follows from (3.1) that there exist and such that
We claim that . In fact, since , we have
Inserting the first relation of (4.4) and (4.7) into (4.8) and using (2) of Remark 2.5 and Remark 4.1, we get that for all . Then, . It follows from , , and (4.7) that . So, , and hence . The sufficiency holds.

Now, we give the following result on the existence of -SSEs.

Theorem 4.3. *Every closed -Hermitian subspace has a -SSE.*

*Proof. *Let be a closed -Hermitian subspace. If is -self-adjoint, then this conclusion holds. So, we assume that . To prove that has a -SSE, it suffices to prove that there exists a -self-adjoint subspace in by Lemma 4.2. The proof uses Zornβs lemma. Since , one has that . Choose and set . Then is -Hermitian in since . Let be the set of all the -Hermitian subspaces in which contain , then is not empty since . Further, let be ordered by extension, that is, if and only if , and let be an arbitrary totally ordered subset of . Set . Then, it can be verified that is -Hermitian in by Remark 4.1 and the fact that all the elements of are -Hermitian in . So, is an upper bound of in . Therefore, has a maximal element by Zornβs lemma. This means that has a maximal -Hermitian subspace extension, denoted by , in . We now prove . Suppose that on the contrary. Note that . Choose satisfying , and set . It can be verified by Remark 4.1 and the fact that is -Hermitian in that holds for all . Note that . Then, is -Hermitian in , which contradicts the maximality of . Hence, .

*Remark 4.4. *Since and its closure have the same -SSEs, we have that every -Hermitian subspace has a -SSE. In addition, Theorem 4.3 extends the relevant result, for example, [1, Chapter III, Theorem 5.8], for -symmetric operators to -Hermitian subspaces.

The following result will give a characterization of all the -SSEs.

Theorem 4.5. *Let be a closed -Hermitian subspace. Assume that , then a subspace is a -SSE of if and only if , and there exists such that*(1)* are linearly independent (mod),*(2)* for ,*(3)*.*

*Proof. *First, consider the necessity. Suppose that is a -SSE of , then it holds that since and . We also have that (3.2) holds and in (3.2) is -self-adjoint in by Lemma 4.2. Note that by Theorem 3.1. Then , and let be a basis of , then we get from (3.2) that (1) holds. In addition, since is -self-adjoint in , one has that (2) holds by Remark 4.1. For convenience, set
Now, we prove , that is, . Let , then by , and there exist and such that
Inserting (4.10) into and using (2) of Remark 2.5 and (2) of this theorem, we get that for . So, , and hence . Conversely, suppose that , then by (3.1), there exist and such that (4.7) holds. The definition of , (4.7), and (2) of Remark 2.5 implies that for ,
So, , which implies that since is -self-adjoint in . One has from (4.7) that , and consequently, . Hence, , that is, . The necessity holds.

Next, consider the sufficiency. Suppose that there exists such that conditions (1) and (2) hold and is given in condition (3). From (3.1), we have
It can be easily verified that the set satisfies conditions (1) and (2). Let , then is -Hermitian in since satisfies condition (2) and . By the proof of Theorem 4.3, there exists such that is -self-adjoint in and . Then, by Lemma 4.2, is a -SSE of , which, together with Theorem 3.1, yields that
Therefore, and dimβ imply that , and hence is -self-adjoint in . With a similar argument to the proof of , we have
where
On the other hand, it can be easily verified that . So, , and hence is -self-adjoint by (2) of Lemma 4.2. The sufficiency holds.

*Remark 4.6. *The case for -symmetric operators is given by [24, 27]. Theorem 4.5 can be regarded as the GKN theorem for -Hermitian subspaces, which will be used for characterizations of -self-adjoint extensions for linear Hamiltonian difference systems in terms of boundary conditions.

#### Acknowledgments

This research was supported by the NNSFs of China (Grants 11101241 and 11071143), the NNSF of Shandong Province (Grant ZR2011AQ002), and the independent innovation fund of Shandong University (Grant 2011ZRYQ003).

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