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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 904976, 19 pages
http://dx.doi.org/10.1155/2013/904976
Research Article

-Self-Adjoint Extensions for a Class of Discrete Linear Hamiltonian Systems

1School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, Shandong 250014, China
2Department of Mathematics, Shandong University at Weihai, Weihai, Shandong 264209, China

Received 15 January 2013; Accepted 18 March 2013

Academic Editor: Michiel Bertsch

Copyright © 2013 Guojing Ren and Huaqing Sun. 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 formally -self-adjoint discrete linear Hamiltonian systems on finite or infinite intervals. The minimal and maximal subspaces are characterized, and the defect indices of the minimal subspaces are discussed. All the -self-adjoint subspace extensions of the minimal subspace are completely characterized in terms of the square summable solutions and boundary conditions. As a consequence, characterizations of all the -self-adjoint subspace extensions are given in the limit point and limit circle cases.

1. Introduction

Consider the following discrete linear Hamiltonian system: where , is a finite integer or , is a finite integer or , and ; is the forward difference operator, that is, ; is the canonical symplectic matrix, that is, where is the unit matrix; the weighted function is a real symmetric matrix with for , and it is of the block diagonal form, where is a complex symmetric matrix, that is, . The partial right shift operator with and ; is a complex spectral parameter.

For briefness, denote in the case where and are finite integers; in the case where is finite and ; in the case where and is finite; in the case where and .

Since is symmetric, it can be blocked as where , , and are complex-valued matrices with and . Then, can be rewritten as

To ensure the existence and uniqueness of the solution of any initial value problem for , we always assume in the present paper that is invertible in .

It can be easily verified that contains the following complex coefficients vector difference equation of order : where are complex-valued matrices with , ; is invertible in ; is an real-valued with . In fact, by letting with , , and for , can be converted into , as well as , with It is obvious that is satisfied for .

The spectral theory of self-adjoint operators and self-adjoint extensions of symmetric operators (i.e., densely defined Hermitian operators) in Hilbert spaces has been well developed (cf. [14]). In general, under certain definiteness conditions, a formally self-adjoint differential expression can generate a minimal operator which is symmetric, and the defect index of the minimal operator is equal to the number of linearly independent square integrable solutions. All the characterizations of self-adjoint extensions of differential equation are obtained [58].

However, for difference equations, it was found in [9] that the minimal operator defined in [10] may be neither densely defined nor single-valued even if the definiteness condition is satisfied. This is an important difference between the differential and difference equations. In order to study the self-adjoint extensions of nondensely defined or multivalued Hermitian operators, some scholars tried to extend the concepts and theory for densely defined Hermitian operators to Hermitian subspaces [1115]. Recently, Shi extended the Glazman-Krein-Naimark (GKN) theory for symmetric operators to Hermitian subspaces [9]. Applying this GKN theory, the first author, with Shi and Sun, gave complete characterizations of self-adjoint extensions for second-order formally self-adjoint difference equations and general linear discrete Hamiltonian systems, separately [16, 17].

We note that when the coefficient in is not a Hermitian matrix, that is, , system is not formally self-adjoint, and the minimal subspace generated by is not Hermitian. Hence the spectral theory of self-adjoint operators or self-adjoint subspaces is not applicable. To solve this problem, Glazman introduced a concept of -symmetric operators in [3, 18] where is an operator. The minimal operators generated by certain differential expressions are -symmetric operators in the related Hilbert spaces [19, 20]. Monaquel and Schmidt [21] discussed the -functions of the following discrete Hamiltonian system: where is the backward difference operator, that is, , and weighted function . By letting , , can be converted into with

In [22], the result that every -Hermitian subspace has a -self-adjoint subspace extension has been given. Furthermore, a result about -self-adjoint subspace extension was obtained [22], which can be regarded as a GKN theorem for -Hermitian subspaces.

In the present paper, enlightened by the methods used in the study of self-adjoint subspace extensions of Hermitian subspaces, we will study the -self-adjoint subspace extensions of the minimal operator corresponding to system . A complete characterization of them in terms of boundary conditions is given by employing the GKN theorem for -Hermitian subspaces. The rest of this paper is organized as follows. In Section 2, some basic concepts and useful results about subspaces are briefly recalled. In Section 3, a conjugation operator is defined in the corresponding Hilbert space, and the maximal and minimal subspaces are discussed. In Section 4, the description of the minimal subspaces is given by the properties of their elements at the endpoints of the discussed intervals, the defect indices of minimal subspaces are discussed, and characterizations of the maximal subspaces are established. Section 5 pays attention to two characterizations of all the self-adjoint subspace extensions of the minimal subspace in terms of boundary conditions via linearly independent square summable solutions of . As a consequence, characterizations of all the self-adjoint subspace extensions are given in two special cases: the limit point and limit circle cases.

2. Fundamental Results on Subspaces

In this section, we recall some basic concepts and useful results about subspaces. For more results about nondensely defined -Hermitian operators or -Hermitian subspaces, we refer to [1719, 22] and some references cited therein. In addition, some properties of solutions of and a result about matrices are given at the end of this section.

By and we denote the sets of the real and the complex numbers, respectively. Let be a complex Hilbert space equipped with inner product , and two linear subspaces (briefly, subspace) in , and . Denote If , we write which is denoted by in the case that and are orthogonal.

Denote It can be easily verified that if and only if can determine a unique linear operator from into whose graph is just . For convenience, we will identify a linear operator in with a subspace in via its graph.

Definition 1 (see [11]). Let be a subspace in . (1) is said to be a Hermitian subspace if . Furthermore, is said to be a Hermitian operator if it is an operator, that is, . (2) is said to be a self-adjoint subspace if . Furthermore, is said to be a self-adjoint operator if it is an operator, that is, . (3) Let be a Hermitian subspace. is said to be a self-adjoint subspace extension (briefly, SSE) of if and is a self-adjoint subspace. (4) Let be a Hermitian operator. is said to be a self-adjoint operator extension (briefly, SOE) of if and is a self-adjoint operator.

Lemma 2 (see [11]). Let be a subspace in . Then (1) is a closed subspace in ; (2) and , where is the closure of ; (3) .

In [19], an operator defined in is said to be a conjugation operator if for all ,

Definition 3. Let be a subspace in and be a conjugation operator. (1)The -adjoint of is defined by (2) is said to be a -Hermitian subspace if . Furthermore, is said to be a -Hermitian operator if it is an operator, that is, . (3) is said to be a -self-adjoint subspace if . Furthermore, is said to be a -self-adjoint operator if it is an operator, that is, . (4)Let be a -Hermitian subspace. is said to be a -self-adjoint subspace extension (briefly, -SSE) of if and is a -self-adjoint subspace. (5)Let be a -Hermitian operator. is said to be a -self-adjoint operator extension (briefly, -SOE) of if and is a -self-adjoint operator.

Remark 4. (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 holds for all and , and that is a -Hermitian subspace if and only if for all .

Lemma 5 (see [22]). Let be a subspace in . Then (1) ; (2) .

It follows from Lemmas 2 and 5 that , and is -Hermitian if is -Hermitian.

Lemma 6 (see [22]). Every J-Hermitian subspace has a -SSE.

Definition 7. Let T be a -Hermitian subspace. Then is called to be the defect index of .

Next, we introduce a form on by

Lemma 8 (see [22]). Let be a -Hermitian subspace. Then

Lemma 9 (see [22]). Let be a closed -Hermitian subspace in and satisfy . Then a subspace is a -SSE of if and only if and there exists such that (1) are linearly independent in (modulo ); (2) , , ;(3) .

Lemma 9 can be regarded as a GKN theorem for -Hermitian subspaces. A set of which is satisfying (1) and (2) in Lemma 9 is called a GKN set of .

Definition 10. Let be a subspace in .(1) The set is called the resolvent set of .(2) The set is called the spectrum of . (3) The set is called to be the regularity field of .
It is evident that for any subspace in .

Lemma 11 (see [22]). Let be a -Hermitian subspace in with , and . Then

The following is a well-known result on the rank of matrices.

Lemma 12. Let be an matrix and an matrix. Then In particular, if , then

3. Relationship between the Maximal and Minimal Subspaces

This section is divided into three subsections. In the first subsection, we define a conjugation operator in a Hilbert space. In the second subsection, we define maximal and minimal subspaces generated by and discuss relationship between them. In the last subsection, we discuss the definiteness condition corresponding to .

3.1. Conjugation Operator

In this subsection, we define a conjugation operator in a Hilbert space and then discuss its properties.

Since and may be finite or infinite, we introduce the following conventions for briefness: means in the case of and means in the case of . Denote For any Hermitian matrix defined in , we define with the semiscalar product Furthermore, denote for . Since the weighted function may be singular in , is a seminorm. Introduce the quotient space Then is a Hilbert space with the inner product .

For a function , denote by the corresponding class in . And for any , denote by a representative of . It is evident that for any .

For any , denote by the conjugation of ; that is, It can be easily verified that if and only if . Here is the conjugation of matrix . Since each is an equivalent class, we define a operator defined on by The following result is obtained.

Lemma 13. defined by (24) is a conjugation operator defined on if and only if is real and symmetric in .

Proof. The sufficiency is evident. Next, we consider the necessity. Assume that defined by (24) is a conjugation operator in . Then for any , it follows from that By the arbitrariness of , one has that . This, together with , yields that is real. The proof is complete.

For any , we denote where is the canonical symplectic matrix given in Section 1. In the case of , if exists and is finite, then its limit is denoted by . In the case of , if exists and is finite, then its limit is denoted by .

Denote where and are called the natural difference operators corresponding to system . The following result can be easily verified, and so we omit the proof.

Lemma 14. Assume that holds. Let . (1) . (2) For any , (3) For any , , and any two solutions and of , it follows that
Moreover, let be a fundamental solution of , then

3.2. Relationship between the Maximal and Minimal Subspaces

In this subsection, we first introduce the maximal and minimal subspaces corresponding to and then show that the minimal subspace is -Hermitian, and its -adjoint subspace is just the maximal subspace.

Denote and define It can be easily verified that and are both linear subspaces in . Here, and are called the maximal and preminimal subspaces corresponding to or in , and is called the minimal subspace corresponding to or in .

Since the end points and may be finite or infinite, we need to divide into two subintervals in order to characterize the maximal and minimal subspaces in a unified form. Choose and fix it. Denote and denote by , and , the inner products and norms of , , respectively. Let and be defined by (31) with replaced by and , respectively. Furthermore, let and be the left maximal and preminimal subspaces defined by (32) with replaced by , respectively, and and the right maximal and preminimal subspaces defined by (32) with replaced by , respectively. The subspaces and are called the left and right minimal subspaces corresponding to system in and , respectively. Similarly, we can define , , and ; , , and ; , and .

The following result is directly derived from (1) of Lemma 14.

Lemma 15. Assume that holds. Then if and only if .

In order to study properties of the above subspaces, we first make some preparation.

Let be the fundamental solution matrix of with . For any finite subinterval with , denote It is evident that is a positive semidefinite matrix and dependent on . By the same method used in [23, Lemma 3.2], it follows that there exists a finite subinterval with such that for any finite subinterval with . In the present paper, we will always denote and define whenever is finite or infinite. In the case that is finite, can be taken as .

In the case that is finite, we define It is evident that is a bounded linear map and its range is a closed subset in .

In the case that is infinite, that is, or or , where , are finite integers, we introduce the following subspaces of , respectively: It can be easily shown that is dense in . In this case, we define By the method used in [23, Lemma 3.3], one has the following properties of .

Lemma 16. Assume that holds.(1) . (2) In the case that is finite, in the case that is infinite, let . Then there exist linearly independent elements , , such that (3) .

The following is the main result of this section.

Theorem 17. Assume that holds. Then , , and .

Proof. Since the method of the proofs is similar, we only show the first assertion. By , it suffices to show .
We first show that . Let . Then for any , there exists with such that in . So, it follows from (2) of Lemma 14 that This implies that .
Next, we show . Fix any . It suffices to show that there exists such that in . Let be a solution of on . For any , there exits with such that in . Thus, it follows from (2) of Lemma 14 that In addition, it is clear that Combining (45) and (46), one has that for all , By (2) and (3) of Lemma 16, we get that for any and any
The following discussion is divided into two parts.
Case 1. is finite. It is evident that . Then, from (2) of Lemma 16, there exists such that . This, together with (48), implies that . This is equivalent to . Let . Then and satisfies Hence, . Since is arbitrary, we have .
Case 2. is infinite. We only consider the case that . For the other two cases, it can be proved with similar arguments.
Let . With a similar argument as Case 2 of the proof of [23, Theorem 3.1], it can be shown that there exist linearly independent elements , and such that , Combining (48)–(50), one has that for any This implies that and consequently is a representative of such that . So . By the arbitrariness of one has .
The entire proof is complete.

The following result is directly derived from Lemmas 5 and 15, and Theorem 17.

Theorem 18. Assume that holds. Then , , and .

3.3. Definiteness Condition

In this subsection, we introduce the definiteness condition for , and give some important results on it. Since the proofs are similar to those given in [23], we omit the proofs.

The definiteness condition for or is given by the following. There exists a finite subinterval such that for any and for any nontrivial solution of , the following always holds: In particular, the definiteness condition for can be described as there exists a finite subinterval such that for any and for any nontrivial solution of , the following always holds:

Lemma 19. Assume that holds. Then holds if and only if there exists a finite subinterval such that one of the following holds: (1) ; (2) for some , every nontrivial solution of satisfies

By Lemma 19, if (52) (or (53)) holds for some , then it holds for every . In addition, if holds on some finite interval , then it holds on .

The following is another sufficient and necessary condition for the definiteness condition.

Lemma 20. Assume that holds. Then holds if and only if for any , there exists a unique such that for .

Remark 21. (1) It can be easily verified that the definiteness condition for holds if and only if that for holds.
(2) In the following of the present paper, we always assume that holds. In this case, we can write instead of in the rest of the present paper.
(3) Denote by and , the definiteness conditions for in and , and By the corresponding intervals, respectively. It is evident that one of and implies .
But cannot imply that there exists such that both and hold.
(4) Several sufficient conditions for the definiteness condition can be given. The reader is referred to [23, Section 4].
For convenience, denote

Lemma 22. Assume that holds. For any , if and only if holds.

4. Characterizations of Minimal and Maximal Subspaces and Defect Indices of Minimal Subspaces

This section is divided into three subsections. In the first subsection, we give all the characterizations of the minimal subspaces generated by in , , and . In the second subsection, we study the defect indices of the minimal subspaces. In the third subsection, characterizations of the maximal subspaces are established.

4.1. Characterizations of the Minimal Subspaces

In this subsection, we study characterizations of the minimal subspaces generated by in , , and .

The following result is a direct consequence of Theorem 17.

Theorem 23. Assume that holds. Then , , and are closed -Hermitian subspace in , , and , respectively.

Now, we introduce boundary forms on , , and by

Lemma 24. Assume that holds. (1) If holds, then for any , (2) If holds, then for any , (3) If holds, then for any ,

Proof. Since the proofs of (1)–(3) are similar, we only show that assertion (1) holds.
For any , we have from (2) of Lemma 14 that for any . This yields that exists and is finite for any . Similarly, it can be shown that exists and is finite for any . Hence, assertion (1) holds. The proof is complete.

Lemma 25. Assume that and hold. Then for any given finite subset with and for any given , there exists such that the following boundary value problem: has a solution .

Proof. Set Let , be the linearly independent solutions of system . Then we have In fact, the linear algebraic system where , can be written as which yields Since is a solution of system , it follows from that . Then ; that is, (65) has only a zero solution. Consequently, (64) holds.
Let , be any given vectors in . By (64), the linear algebraic system has a unique solution . Set . It follows from (68) that Let be a solution of the following initial value problem: Since for and , we get by (70) and (2) of Lemma 14 that Since are linearly independent in , we get from (69) and (71) that . So, is a solution of the following boundary value problem:
On the other hand, the linear algebraic system has a unique solution by (64). Set . Then, by (73) Let be a solution of the following initial value problem: Since for and , we get by (2) of Lemma 14 and (75) that which, together with (74), implies that . So, is a solution of the following boundary value problem: Set and . Then is a solution of the boundary value problem (62). The proof is complete.

Remark 26. Lemma 25 is called a patch lemma. Based on Lemma 25, any two elements of ( , , resp.) can be patched up to construct another new element of ( , , resp.). In particular, (1) if holds, we can take , , and , . Then there exist satisfying (2) if holds, we take , , and . Then there exist satisfying (3) if both and hold, then there exist satisfying The above auxiliary elements , , and will be very useful in the sequent discussions.

Theorem 27. Assume that holds.(1) If holds, then In particular, if , then (2) If holds, then (3) If holds, then

Proof. We first show that assertion (1) holds. By Lemmas 8 and 24, and Theorem 17, one has For convenience, denote Clearly, . We now show that . Fix any . It follows from (85) that for all , For any given , by Remark 26 there exists such that Thus, it follows from (87) that , and consequently for all .
In the case that , it is clear that So it remains to show that . It suffices to show that for any . Fix any , and let , , , and . Then by Lemma 25, there exist with and . Inserting these into (87) one has that . Similarly, one can show that . Thus . Therefore, assertion (1) has been shown.
With similar arguments, one can show that assertion (2) and (3) hold by using (78) and (79), separately. This completes the proof.

4.2. Defect Indices of Minimal Subspaces

In this subsection, we first give a valued range of the defect indices of and and then discuss the relationship among the defect indices of , , and .

For briefness, denote For any , let , , , and be defined as (56) with replaced by and , respectively.

The following results are obtained.

Theorem 28. Assume that holds.(1) If holds and , then for any , and . (2) If holds and , then for any , and .

Proof. Since the method of the proofs is the same, we only give the proof of assertion (1).
For any