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

Guojing Ren ¹ and Huaqing Sun ²

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. [1–4]). 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 [5–8].

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 [11–15]. 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 [17–19, 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 , it follows from Lemma 11 and Theorem 18 that

On the other hand, by using Lemma 5 and Theorems 17 and 18, one has that

It is clear that

Combining (91)–(93), one has . This, together with Lemma 22, implies that . In addition, it has been shown in [21] that for any . So assertion (1) is true. The proof is complete.

Next, we discuss relationship among defect indices of , , and . For convenience, denote

for any . It is evident that , , and

On the other hand, for any and , we define by

Then (95) still holds, and consequently . Furthermore, it is clear that

The following result can be easily verified. So we omit its proof.

Lemma 29. Assume that , , and hold.

(1) If , then and . (2) If and with , then .

Let be the restriction of subspace , defined by

Lemma 30. Assume that , , and hold. Then

(1) if and only if and ; (2) if and only if and .

Proof. (1) We first consider the necessity. Fix any . Let , , , and be defined as (94). Then . By (1) of Lemma 29 one has that . In addition, for any , it follows from Remark 26 that there exits with

So one has

which implies that by the arbitrariness of and (3) of Theorem 27. With a similar argument, one can show .

Next, we consider the sufficiency. Fix any and . Let and be defined by (96). By (2) and (3) of Theorem 27 one has that . So . It follows from (2) of Lemma 29 that . For any , it follows from (1) of Lemma 29 that and . Thus one has by (2) and (3) of Theorem 27 that

This implies that by (1) of Theorem 27, and consequently, .

(2) We first consider the necessity. Fix any . Let , , , and be defined as (94). Then and .

Set for , where is defined by (19) with replaced by . It is clear that . For any , let and be defined by (96). Then by the above result (1) one has that . It follows that

By the arbitrariness of , one has that by Theorem 17. With a similar argument one can show .

Next, we consider the sufficiency. Fix any and . Let and be defined by (96). For any , it follows from the above result (1) that and . So one has by Theorem 17 that

This yields that . The whole proof is complete.

Lemma 31. Assume that , , and hold. Then and .

Proof. The first assertion holds because , and the second assertion can be proved by (1) of Lemma 30 and (95). The proof is complete.

In the following of the present paper, we assume that

.

By Definition 10, one has that if , then , and consequently . We do not consider this case in the present paper.

Theorem 32. Assume that , , , and hold. Then

Proof. The proof is divided into two steps.

Step  1. We show that

It follows from and Lemma 31 that . This, together with , , and Theorem 28, implies that for any , has just linearly independent solutions , , and has just linearly independent solutions , ; that is, for and for .

Set for . It is clear that for . Let , , be defined by (96). Then it follows from (2) of Lemma 30 that . Similarly, set for . Then one has for . It is evident that are linearly independent.

On the other hand, for any , it follows from (2) of Lemma 31 that and ; that is, is a solution of in , and is a solution of in . Therefore, there exist unique and such that

Noting the constructions of and by (96), one has that

This, together with Lemma 11, implies that (105) holds.

Step  2. We show that

It is evident that , , where are defined by (80). We claim that

In fact, for each , set . Let

Then

It can be easily verified that this decomposition is unique. Hence, (109) holds.

For any , we claim that

Since (109) holds, it suffices to show that

Suppose that and satisfy

that is,

Since and , it follows that . Since , it yields

which, together with (109), implies that for , and consequently . This yields that (113) holds.

Since and are closed -Hermitian subspaces, it follows that and are closed subspaces in , respectively. Hence, there exists a closed subspace in such that

In addition, again by the fact that , it follows that are linearly independent in . It follows from (113) that . Consequently,

This yields that (108) holds. It follows from (105) and (108) that (104) holds. The proof is complete.

Remark 33. Theorem 32 (formula (104)) generalizes the classical result for th order ordinary differential equations that go back to the classical work by Akhiezer and Glazman [1, Theorem 3 in Appendix 2]. To the case of symmetric Hamiltonian systems, formula (104) was extended in [15].

So it follows from Theorems 28 and 32 that if and only if , and if and only if . The following definition is obtained.

Definition 34. Assume that , , and hold. Then is said to be in the limit case at . In the special case that , is said to be in the limit point case at , and in the other special case that , is said to be in the limit circle case at .

The same definition can be given at provided that holds.

4.3. Characterizations of and

In this subsection, we characterize the maximal subspaces and . We first consider characterization of . Assume that holds and . Let . It follows from the proof of Lemma 11 that

where and , and are linearly independent (mod ). For convenience, denote

Clearly, , . So is finite for all , . Then the following result can be directly derived from (119) and (120).

Lemma 35. Assume that and hold, and . Then every can be expressed as

where and .

By Lemma 35, , defined by (78), can be uniquely expressed as

where and . Denote

Lemma 36. Assume that and hold, and . Then and . Furthermore, we can rearrange the order of such that

Proof. It follows from (122) that

which, together with (78), implies that

By Lemma 12, one has .

On the other hand, it follows from (122) that

for , , which, together with (78) and (2) of Theorem 27, implies that

Noting that , one has

We now want to show . By (3) of Lemma 14 one has

where . Since , we have that

which, together with (129), yields that

Because , one can rearrange the order of such that the first rows of are linearly independent; that is, (124) holds. The proof is complete.

Without loss of generality, we assume that (124) holds in the rest of this paper. Now, we can give a characterization of .

Theorem 37. Assume that and hold, , and are linearly independent solutions of in such that (124) holds. Then

is invertible, and any can be uniquely expressed as

where , are defined by (78), and .

Proof. Let , where and are and matrices, respectively. It follows from (124) that there exists an invertible matrix such that

So, it follows from (128) that , which is equivalent to . Since , it follows that is invertible. Multiplying (125) by from the right-hand side, we get

This implies that each of can be uniquely expressed as a linear combination of , , and . Therefore, (134) follows from Lemma 35.

Since for , can be uniquely expressed as

where , and . This, together with (78) and (2) of Theorem 27, implies that for

that is,

Therefore, is invertible from (124). This completes the proof.

With a similar argument, one can obtain the following characterization of .

Theorem 38. Assume that and hold, . Then, for some , system has linearly independent solutions in such that is invertible, where

and each can be uniquely expressed as

where , , and are defined as (79).

5. Characterizations of -SSEs of

In this section, we give a complete characterization of all the -SSEs of minimal subspace in terms of the square summable solutions of system . As a consequence, characterizations of all the -self-adjoint subspace extensions are obtained in the two special cases: the limit point and limit circle cases. The following discussion is divided into two parts based on the form of .

5.1. Both the Endpoints Are Infinite

Let , and assume that , , and hold, and in this subsection. It follows from Theorem 32 that

In addition, for some , let given in Theorems 37 and be given in Theorems 38.

Theorem 39. Assume that , , , and hold. Then a subspace is a -SSE of if and only if there exist two matrices and such that

(1) , (2) , and

where and are the same as those in Theorems 37 and 38.

Proof. We first show the sufficiency.

Suppose that there exist two matrices and such that conditions (1) and (2) hold and is defined by (143). We now prove that is a -self-adjoint subspace extension of by Lemma 9.

Denote

and set

Clearly, and for . By Remark 26, there exist such that

where and are specified by and , respectively.

Since and are liner subspaces, are linearly independent in (modulo ) if and only if are linearly independent in (modulo ). So it suffices to show that are linearly independent in (modulo ). Suppose that there exists such that

It follows from (145), (146), and (2) and (3) of Theorem 27 that

Since and are invertible, we get from (148) that . Then by condition (1). So, are linearly independent in (modulo ), and consequently are linearly independent in (modulo ).

Next, we show that for . It follows from (145) and (146) that

which, together with Lemma 24 and condition (2), implies that

Consequently, for , . Therefore, satisfy the conditions (1) and (2) of Lemma 9.

Note that for each , it follows that

where (145) and (146) have been used. Therefore, it follows from Lemma 24 that can be expressed as

Hence, is a -SSE of by Lemma 9. The sufficiency is proved.

We now show the necessity. Suppose that is a -SSE of . By Lemma 9 and Theorem 17, there exists a set of such that (152) holds. Write . Then and for . By Theorems 37 and 38, each and can be uniquely expressed as

where , , and . Set

First, we want to show that and satisfy condition (1). Otherwise, suppose that . Then there exists with such that

Then and . Let . We then have

For each , can be uniquely expressed as (134) by Theorem 37 and can be uniquely expressed as (141) by Theorem 38. So it follows from (156), (78), (79), and (2) and (3) of Theorem 27 that for any , , which yields that by (1) of Theorem 27. Then are linearly dependent in (modulo ), and consequently are linearly dependent in (modulo ). This is a contradiction. Hence, .

Next, we prove that and satisfy condition (2). It can be easily verified that

Hence, by Lemma 14 and for , , and satisfy condition (2).

In addition, it follows from (78), (79), (153), and (2) and (3) of Theorem 27 that

Hence, in (152) can be expressed as (143). The necessity is proved.

The entire proof is complete.

To end this subsection, we give characterizations of -SSEs of in four special cases of defect indices: ; ; ; .

In the case that , that is, by Theorem 32, the following result is derived from Lemma 11 and Theorem 17.

Theorem 40. Assume that , , , and hold. If , then is a -self-adjoint subspace.

In the case that , , it follows from Theorem 32 that . Let , , be linearly independent solutions in of satisfying . Then, by (3) of Lemma 14 one has that

The following result can be directly derived from Theorem 39.

Theorem 41. Assume that , , , , and hold. If is in l.p.c. at and in l.c.c. at . Let be linearly independent solutions in of satisfying . Then a subspace is a -SSE of if and only if there exists a matrix such that

In the case that , a similar result can be easily given. So we omit the details in this case.

In the case that , it follows from Theorem 32 that . Let be linearly independent solutions in of satisfying

Then, by Lemma 14, and , defined by (133) and (140), satisfy

The following result is a direct consequence of Theorem 39.

Theorem 42. Assume that , , , and hold. If is in l.c.c. at both and , then for any given , let be linearly independent solutions in of satisfying . Then a subspace is a -SSE of if and only if there exist two matrices and such that

Remark 43. As we have seen, there is no boundary condition at the endpoints at which system is in l.p.c., and the matrix or can be replaced by in the case that system is in l.c.c. at or .

5.2. At Least One of the Two Endpoints Is Finite

In this subsection, we characterize the -SSEs of in the special case that at least one of the two endpoints and is finite. We first consider the case that is finite, and is finite or infinite.

We point out that in this case, characterizations of all the -SSEs of can also be given by the proof of Theorem 39, provided that assumptions , , and hold. But, if there does not exist a such that both and are satisfied, then Theorem 39 fails. We will remark again that the division of is not necessary in the case that one of the two endpoints is finite, and characterizations of all the -SSEs of can still be given provided that and hold.

In the case that is finite, can be regarded as with , and is equivalent to . So all the characterizations for and given in Sections 3 and 4 are available to and , respectively, with replaced by . Assume that holds. Then for any given , as discussed in Section 4, let be linearly independent solutions in of such that is invertible, where is defined by (133) with is replaced by , . Then all the results of Theorem 37 hold with and replaced by and , respectively.

Theorem 44. Assume that the left endpoint is finite, and hold. Then a subspace is a -SSE of if and only if there exist two matrices and such that

(1) , (2) , and

Proof. The main idea of the proof is similar to that of Theorem 39.

We first show the sufficiency. Denote

and set

Clearly, for . By Remark 26, there exist () such that

where is specified by . It can be verified by the method used in the proof of Theorem 39 that the set satisfies the conditions (1) and (2) in Lemma 9. Note that for each , it follows that

Therefore, it follows from Lemma 24 that can be expressed as

Hence, is a -SSE of by Lemma 9. The sufficiency is proved.

We now show the necessity. Suppose that is a -SSE of . By Lemma 9 and Theorem 17, there exists a set of for such that conditions (1) and (2) in Lemma 9 hold, and can be expressed as (152). Write . Then for . By Theorem 37, each can be uniquely expressed as

where and . Set

With a similar argument to that used in the proof of Theorem 39, we can prove that and satisfy conditions (1) and (2). In addition, it is clear that in (170) can be expressed as (165). The necessity is proved, and then the entire proof is complete.

At the end of this subsection, we give the characterizations of -SSEs of in two special cases of defect indices.

In the special case that , Theorem 44 can be described in the following simpler form.

Theorem 45. Assume that the left endpoint is finite, and hold. If is in l.p.c. at , then a subspace is a -SSE of if and only if there exists a matrix satisfying the self-adjoint condition

such that can be defined by

In the other special case that , the following result is a direct consequence of Theorem 44.

Theorem 46. Assume that the left endpoint is finite, and hold. If is in l.c.c. at , let be linearly independent solutions in of satisfying . Then a subspace is a -SSE of if and only if there exist two matrices and such that

(1) , (2) , and

For the case that is finite and , it can be considered by a similar method. Here we only give the following basic result.

Theorem 47. Assume that the right endpoint is finite, and hold. Let , , be linearly independent solutions of in such that is invertible, where is defined as in Theorem 38 with replaced by . Then a subspace is a -SSE of if and only if there exist two matrices and such that

(1) , (2) , and

In the case that both the two endpoints and are finite, that is, , it is clear that by . The characterization of -SSEs given in Theorem 44 can be simplified as follows.

Theorem 48. Let . Assume that hold. Then a subspace is a -SSE of if and only if there exist two matrices and such that

(1) , (2) , and

Proof. Let , , be defined as those in Theorem 46. Then it follows that

where . It is evident . So by Lemma 14, one has that

Hence, the assertion follows from Theorem 46. This completes the proof.

Acknowledgment

This research was supported by the NNSF of China (Grants 11071143, 11101241, and 11226160).

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