This paper is concerned with formally J-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 J-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 J-self-adjoint subspace extensions are given in the limit point and limit circle cases.

1. Introduction

Consider the following discrete linear Hamiltonian system:

where

, a is a finite integer or a = −∞, b is a finite integer or b = +∞, and b − a ≥ 1; Δ is the forward difference operator, that is, Δy(t) = y(t + 1) − y(t); 𝒥 is the canonical symplectic matrix, that is,

()

where I_n is the n × n unit matrix; the weighted function W(t) is a 2n × 2n real symmetric matrix with W(t) ≥ 0 for t ∈ ℐ, and it is of the block diagonal form,

()

where P(t) is a 2n × 2n complex symmetric matrix, that is, P^T(t) = P(t). The partial right shift operator R(y)(t) = (u^T(t + 1), v^T(t)) ^T with y(t) = (u^T(t), v^T(t)) ^T and u(t), v(t) ∈ ℂⁿ; λ is a complex spectral parameter.

For briefness, denote ℐ = [a, b] in the case where a and b are finite integers; ℐ = [a, +∞) in the case where a is finite and b = +∞; ℐ = (−∞, b] in the case where a = −∞ and b is finite; ℐ = (−∞, +∞) in the case where a = −∞ and b = +∞.

Since P(t) is symmetric, it can be blocked as

()

where A, B, and C are n × n complex-valued matrices with C^T = C and B^T = B. Then, (1λ) can be rewritten as

To ensure the existence and uniqueness of the solution of any initial value problem for (1λ), we always assume in the present paper that

(A₁) I_n − A(t) is invertible in ℐ.

It can be easily verified that (1λ) contains the following complex coefficients vector difference equation of order 2m:

where p_j(t) are l × l complex-valued matrices with

, 0 ≤ j ≤ m; p_m(t) is invertible in ℐ; w(t) is an l × l real-valued with w(t) ≥ 0. In fact, by letting y = (u^T, v^T) ^T with

, and

()

for 1 ≤ j ≤ m, (3λ) can be converted into (1λ), as well as (2λ), with

()

It is obvious that (A₁) is satisfied for (3λ).

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 P(t) in (1λ) is not a Hermitian matrix, that is, P^*(t) ≠ P(t), system (1λ) is not formally self-adjoint, and the minimal subspace generated by (1λ) 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 J-symmetric operators in [3, 18] where J is an operator. The minimal operators generated by certain differential expressions are J-symmetric operators in the related Hilbert spaces [19, 20]. Monaquel and Schmidt [21] discussed the M-functions of the following discrete Hamiltonian system:

where ∇ is the backward difference operator, that is, ∇z(t) = z(t) − z(t − 1), and weighted function H(t) = diag {H₁(t), H₂(t)}. By letting z₁(t) = v(t), z₂(t) = u(t + 1), (1λ) can be converted into (4_−λ) with

()

In [22], the result that every J-Hermitian subspace has a J-self-adjoint subspace extension has been given. Furthermore, a result about J-self-adjoint subspace extension was obtained [22], which can be regarded as a GKN theorem for J-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 J-self-adjoint subspace extensions of the minimal operator corresponding to system (1λ). A complete characterization of them in terms of boundary conditions is given by employing the GKN theorem for J-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 J 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 (1λ). 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 J-Hermitian operators or J-Hermitian subspaces, we refer to [17–19, 22] and some references cited therein. In addition, some properties of solutions of (1λ) and a result about matrices are given at the end of this section.

By ℝ and ℂ we denote the sets of the real and the complex numbers, respectively. Let X be a complex Hilbert space equipped with inner product 〈·, ·〉, T and S two linear subspaces (briefly, subspace) in X² : = X × X, and λ ∈ ℂ. Denote

()

If T∩S = {0}, we write

()

which is denoted by T ⊕ S in the case that T and S are orthogonal.

Denote

()

It can be easily verified that T(0) = {0} if and only if T can determine a unique linear operator from Dom T into X whose graph is just T. For convenience, we will identify a linear operator in X with a subspace in X² via its graph.

Definition 1 (see [11].)Let T be a subspace in X².

(1)
T is said to be a Hermitian subspace if T ⊂ T^*. Furthermore, T is said to be a Hermitian operator if it is an operator, that is, T(0) = {0}.
(2)
T is said to be a self-adjoint subspace if T = T^*. Furthermore, T is said to be a self-adjoint operator if it is an operator, that is, T(0) = {0}.
(3)
Let T be a Hermitian subspace. T₁ is said to be a self-adjoint subspace extension (briefly, SSE) of T if T ⊂ T₁ and T₁ is a self-adjoint subspace.
(4)
Let T be a Hermitian operator. T₁ is said to be a self-adjoint operator extension (briefly, SOE) of T if T ⊂ T₁ and T₁ is a self-adjoint operator.

Lemma 2 (see [11].)Let T be a subspace in X². Then

(1)
T^* is a closed subspace in X²;
(2)
and , where is the closure of T;
(3)
.

In [19], an operator J defined in X is said to be a conjugation operator if for all x, y ∈ X,

()

Definition 3. Let T be a subspace in X² and J be a conjugation operator.

(1)
The J-adjoint of T is defined by
()
(2)
T is said to be a J-Hermitian subspace if . Furthermore, T is said to be a J-Hermitian operator if it is an operator, that is, T(0) = {0}.
(3)
T is said to be a J-self-adjoint subspace if . Furthermore, T is said to be a J-self-adjoint operator if it is an operator, that is, T(0) = {0}.
(4)
Let T be a J-Hermitian subspace. T₁ is said to be a J-self-adjoint subspace extension (briefly, J-SSE) of T if T ⊂ T₁ and T₁ is a J-self-adjoint subspace.
(5)
Let T be a J-Hermitian operator. T₁ is said to be a J-self-adjoint operator extension (briefly, J-SOE) of T if T ⊂ T₁ and T₁ is a J-self-adjoint operator.

Remark 4. (1) It can be easily verified that is a closed subspace. Consequently, a J-self-adjoint subspace T is a closed subspace since . In addition, if T ⊂ S.

(2) From the definition, we have that 〈f, Jy〉 = 〈x, Jg〉 holds for all (x, f) ∈ T and , and that T is a J-Hermitian subspace if and only if 〈f, Jy〉 = 〈x, Jg〉 for all (x, f), (y, g) ∈ T.

Lemma 5 (see [22].)Let T be a subspace in X². Then

(1)
;
(2)
.

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

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

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

Next, we introduce a form on X² × X² by

()

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

()

Lemma 9 (see [22].)Let T be a closed J-Hermitian subspace in X² and satisfy d = d(T)<+∞. Then a subspace T₁ is a J-SSE of T if and only if and there exists such that

(1)
(x₁, f₁), (x₂, f₂), …, (x_d, f_d) are linearly independent in (modulo T);
(2)
[(x_j, f_j):(x_k, f_k)] = 0, 1 ≤ j, k ≤ d;
(3)
.

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

Definition 10. Let T be a subspace in X².

(1)
The set
()
is called the resolvent set of T.
(2)
The set σ(T): = ℂ∖ρ(T) is called the spectrum of T.
(3)
The set
()
is called to be the regularity field of T.

It is evident that ρ(T) ⊂ Γ(T) for any subspace T in X².

Lemma 11 (see [22].)Let T be a J-Hermitian subspace in X² with Γ(T) ≠ ∅, and λ ∈ Γ(T). Then

()

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

Lemma 12. Let A be an m × l matrix and B an l × n matrix. Then

()

In particular, if AB = 0, 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 (1λ) and discuss relationship between them. In the last subsection, we discuss the definiteness condition corresponding to (1λ).

3.1. Conjugation Operator

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

Since b and a may be finite or infinite, we introduce the following conventions for briefness: b + 1 means +∞ in the case of b = +∞ and a − 1 means −∞ in the case of a = −∞. Denote

()

For any 2n × 2n Hermitian matrix W(t) ≥ 0 defined in ℐ, we define

()

with the semiscalar product

()

Furthermore, denote ∥y∥ : = (〈y,y〉)^1/2 for

. Since the weighted function W(t) may be singular in ℐ, ∥·∥ is a seminorm. Introduce the quotient space

()

Then

is a Hilbert space with the inner product 〈·, ·〉.

For a function , denote by y^π the corresponding class in . And for any , denote by a representative of y^π. It is evident that 〈y^π, z^π〉 = 〈y, z〉 for any .

For any

, denote by J₀y the conjugation of y; that is,

()

It can be easily verified that

if and only if

. Here

is the conjugation of matrix W. Since each

is an equivalent class, we define a operator J defined on

()

The following result is obtained.

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

Proof. The sufficiency is evident. Next, we consider the necessity. Assume that J defined by (24) is a conjugation operator in . Then for any , it follows from 〈Jx^π, Jy^π〉 = 〈y^π, x^π〉 that

()

By the arbitrariness of x^π, y^π, one has that W(t) = W^T(t). This, together with W(t) = W^*(t), yields that W(t) is real. The proof is complete.

For any x, y ∈ l(ℐ), we denote

()

where 𝒥 is the canonical symplectic matrix given in Section 1. In the case of b = +∞, if lim _t→b (x, y)(t) exists and is finite, then its limit is denoted by (x, y)(+∞). In the case of a = −∞, if lim _t→a (x, y)(t) exists and is finite, then its limit is denoted by (x, y)(−∞).

Denote

()

where τ and δ are called the natural difference operators corresponding to system (1λ). The following result can be easily verified, and so we omit the proof.

Lemma 14. Assume that (A₁) holds. Let x, y ∈ l(ℐ).

(1)
δ(J₀y) = J₀τ(y), τ(J₀y) = J₀δ(y).
(2)
For any s, k ∈ ℐ,
()
(3)
For any λ ∈ ℂ, c₀ ∈ ℐ, and any two solutions x(t) and y(t) of (1λ), it follows that
()

Moreover, let Y(t, λ) be a fundamental solution of (1λ), then

()

3.2. Relationship between the Maximal and Minimal Subspaces

In this subsection, we first introduce the maximal and minimal subspaces corresponding to (1λ) and then show that the minimal subspace is J-Hermitian, and its J-adjoint subspace is just the maximal subspace.

Denote

()

and define

()

It can be easily verified that H(τ) and H₀₀(τ) are both linear subspaces in

. Here, H(τ) and H₀₀(τ) are called the maximal and preminimal subspaces corresponding to τ or (1λ) in

, and

is called the minimal subspace corresponding to τ or (1λ) in

Since the end points a and b may be finite or infinite, we need to divide ℐ into two subintervals in order to characterize the maximal and minimal subspaces in a unified form. Choose a < c₀ < b and fix it. Denote

()

and denote by 〈·, ·〉 _a, 〈·, ·〉 _b and ∥·∥_a, ∥·∥_b the inner products and norms of

, respectively. Let

and

be defined by (31) with ℐ replaced by ℐ_a and ℐ_b, respectively. Furthermore, let H_a(τ) and H_a,00(τ) be the left maximal and preminimal subspaces defined by (32) with ℐ replaced by ℐ_a, respectively, and H_b(τ) and H_b,00(τ) the right maximal and preminimal subspaces defined by (32) with ℐ replaced by ℐ_b, respectively. The subspaces

and

are called the left and right minimal subspaces corresponding to system (1λ) in ℐ_a and ℐ_b, respectively. Similarly, we can define H(δ), H₀₀(δ), and H₀(δ); H_b(δ), H_b,00(δ), and H_b,0(δ); H_a(δ), H_a,00(δ) and H_a,0(δ).

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

Lemma 15. Assume that (A₁) holds. Then (y^π, g^π) ∈ H(τ) if and only if (Jy^π, Jg^π) ∈ H(δ).

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

Let Y(t) be the fundamental solution matrix of (1₀) with Y(c₀) = I_2n. For any finite subinterval ℐ^′ with c₀ ∈ ℐ^′ ⊂ ℐ, denote

()

It is evident that Φ_ℐ′ is a 2n × 2n positive semidefinite matrix and dependent on ℐ^′. By the same method used in [23, Lemma 3.2], it follows that there exists a finite subinterval ℐ₀ with c₀ ∈ ℐ₀ ⊂ ℐ such that

()

for any finite subinterval ℐ^′ with ℐ₀ ⊂ ℐ^′ ⊂ ℐ. In the present paper, we will always denote ℐ₀ : = [s₀, t₀] and define

()

whenever ℐ is finite or infinite. In the case that ℐ is finite, ℐ₀ can be taken as ℐ.

In the case that ℐ is finite, we define

()

It is evident that ϕ_ℐ is a bounded linear map and its range is a closed subset in ℂ²ⁿ.

In the case that ℐ is infinite, that is, ℐ = [a, +∞) or ℐ = (−∞, b] or ℐ = (−∞, +∞), where a, b 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 ϕ_ℐ.

Lemma 16. Assume that (A₁) holds.

(1)
Ran ϕ_ℐ = Ran Φ_ℐ.
(2)
In the case that ℐ is finite,
()
in the case that ℐ is infinite, let l = rank Φ_ℐ. Then there exist linearly independent elements , 1 ≤ j ≤ l, such that
()
(3)
Ker ϕ_ℐ ⊂ Ran (H₀₀(τ)).

The following is the main result of this section.

Theorem 17. Assume that (A₁) 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 (y^π, g^π) ∈ H(τ). Then for any (x^π, f^π) ∈ H₀₀(τ), there exists x ∈ x^π with such that τ(x)(t) = W(t)R(f)(t) in ℐ. So, it follows from (2) of Lemma 14 that

()

This implies that

Next, we show . Fix any . It suffices to show that there exists y₀ ∈ y^π such that τ(y₀)(t) = W(t)R(g)(t) in ℐ. Let z be a solution of τ(z)(t) = W(t)R(g)(t) on ℐ. For any (x^π, f^π) ∈ H₀₀(τ), there exits x ∈ x^π with such that τ(x)(t) = W(t)R(f)(t) in ℐ. Thus, it follows from (2) of Lemma 14 that

()

In addition, it is clear that

()

Combining (45) and (46), one has that for all (x^π, f^π) ∈ H₀₀(τ),

()

By (2) and (3) of Lemma 16, we get that for any h^π ∈ Ker ϕ_ℐ and any ξ ∈ ℂ²ⁿ

()

The following discussion is divided into two parts.

Case 1. ℐ is finite. It is evident that . Then, from (2) of Lemma 16, there exists ξ₀ ∈ Ran Φ_ℐ such that (J₀(y − z − Yξ₀)) ^π ∈ Ker ϕ_ℐ. This, together with (48), implies that (J₀(y − z − Yξ₀)) ^π = 0. This is equivalent to (y − z − Yξ₀) ^π = 0. Let y₀(t): = z(t) + Y(t)ξ₀. Then y₀ ∈ y^π and satisfies

()

Hence, (y^π, g^π) ∈ H(τ). Since

is arbitrary, we have

Case 2. ℐ is infinite. We only consider the case that ℐ = [a, +∞). For the other two cases, it can be proved with similar arguments.

Let rank Φ_ℐ = l. 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 ξ₀ ∈ Ran Φ_ℐ such that (y − z − Yξ₀) ^π ∈ Ker ϕ_ℐ,

()

Combining (48)–(50), one has that for any

()

This implies that

and consequently y₀ : = z + Yξ₀ is a representative of y^π such that τ(y₀)(t) = W(t)R(g)(t). So (y^π, g^π) ∈ H(τ). By the arbitrariness of (y^π, g^π) 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 (A₁) holds. Then , , and .

3.3. Definiteness Condition

In this subsection, we introduce the definiteness condition for (1λ), 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 (1λ) or H(τ) is given by the following.

(A₂) There exists a finite subinterval ℐ₁ ⊂ ℐ such that for any λ ∈ ℂ and for any nontrivial solution y(t) of (1λ), the following always holds:
()

In particular, the definiteness condition for (3λ) can be described as there exists a finite subinterval ℐ₁ ⊂ ℐ such that for any λ ∈ ℂ and for any nontrivial solution z(t) of (3λ), the following always holds:

()

Lemma 19. Assume that (A₁) holds. Then (A₂) holds if and only if there exists a finite subinterval ℐ₁ ⊂ ℐ such that one of the following holds:

(1)
;
(2)
for some λ ∈ ℂ, every nontrivial solution y(t) of (1λ) satisfies
()

By Lemma 19, if (52) (or (53)) holds for some λ ∈ ℂ, then it holds for every λ ∈ ℂ. In addition, if (A₂) holds on some finite interval ℐ₁, then it holds on ℐ₀ = [s₀, t₀].

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

Lemma 20. Assume that (A₁) holds. Then (A₂) holds if and only if for any (y^π, g^π) ∈ H(τ), there exists a unique y ∈ y^π such that τ(y)(t) = W(t)R(g)(t) for t ∈ ℐ.

Remark 21. (1) It can be easily verified that the definiteness condition for H(τ) holds if and only if that for H(δ) holds.

(2) In the following of the present paper, we always assume that (A₂) holds. In this case, we can write (y, g^π) ∈ H(τ) instead of (y^π, g^π) ∈ H(τ) in the rest of the present paper.

(3) Denote by (A_a,2) and (A_b,2), the definiteness conditions for (1λ) in ℐ_a and ℐ_b, and

()

By the corresponding intervals, respectively. It is evident that one of (A_a,2) and (A_b,2) implies (A₂).

But (A₂) cannot imply that there exists c₀ ∈ ℐ such that both (A_a,2) and (A_b,2) 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 (A₁) holds. For any λ ∈ ℂ, dim ℳ_λ = dim M_λ if and only if (A₂) 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 (1λ) in ℐ, ℐ_a, and ℐ_b. 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 (1λ) in ℐ, ℐ_a, and ℐ_b.

The following result is a direct consequence of Theorem 17.

Theorem 23. Assume that (A₁) holds. Then H₀(τ), H_b,0(τ), and H_a,0(τ) are closed J-Hermitian subspace in , , and , respectively.

Now, we introduce boundary forms on

, and

()

Lemma 24. Assume that (A₁) holds.

(1)
If (A₂) holds, then for any (x, f^π), (y, g^π) ∈ H(τ),
()
(2)
If (A_a,2) holds, then for any (x, f^π), (y, g^π) ∈ H_a(τ),
()
(3)
If (A_b,2) holds, then for any (x, f^π), (y, g^π) ∈ H_b(τ),
()

Proof. Since the proofs of (1)–(3) are similar, we only show that assertion (1) holds.

For any (x, f^π), (y, g^π) ∈ H(τ), we have from (2) of Lemma 14 that

()

for any s < k ∈ ℐ. This yields that lim _t→b (x, y)(t) exists and is finite for any (x, f^π), (y, g^π) ∈ H(τ). Similarly, it can be shown that lim _t→a (x, y)(a) exists and is finite for any (x, f^π), (y, g^π) ∈ H(τ). Hence, assertion (1) holds. The proof is complete.

Lemma 25. Assume that (A₁) and (A₂) hold. Then for any given finite subset ℐ₁ = [s, k] with ℐ₀ ⊂ ℐ₁ ⊂ ℐ and for any given α, β ∈ ℂ²ⁿ, there exists such that the following boundary value problem:

()

has a solution

Proof. Set

()

Let ϕ_j, 1 ≤ j ≤ 2n, be the linearly independent solutions of system (1₀). Then we have

()

In fact, the linear algebraic system

()

where C = (c₁, c₂, …, c_2n) ^T ∈ ℂ²ⁿ, can be written as

()

which yields

()

Since

is a solution of system (1₀), it follows from (A₂) that

. Then C = 0; that is, (65) has only a zero solution. Consequently, (64) holds.

Let α, β be any given vectors in ℂ²ⁿ. By (64), the linear algebraic system

()

has a unique solution C₁ ∈ ℂ²ⁿ. Set ψ₁ = (ϕ₁, ϕ₂, …, ϕ_2n)C₁. It follows from (68) that

()

Let u(t) be a solution of the following initial value problem:

()

Since τ(ϕ_i)(t) = 0 for t ∈ ℐ and 1 ≤ i ≤ 2n, we get by (70) and (2) of Lemma 14 that

()

Since ϕ₁, ϕ₂, …, ϕ_2n are linearly independent in l(ℐ), we get from (69) and (71) that u(k + 1) = β. So, u(t) is a solution of the following boundary value problem:

()

On the other hand, the linear algebraic system

()

has a unique solution C₂ ∈ ℂ²ⁿ by (64). Set ψ₂ = (ϕ₁, ϕ₂, …, ϕ_2n)C₂. Then, by (73)

()

Let v(t) be a solution of the following initial value problem:

()

Since τ(ϕ_i)(t) = 0 for t ∈ ℐ and 1 ≤ i ≤ 2n, we get by (2) of Lemma 14 and (75) that

()

which, together with (74), implies that v(s) = α. So, v(t) is a solution of the following boundary value problem:

()

Set ψ = ψ₁ − ψ₂ and ϕ = u + v. Then ϕ 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 H(τ) (H_b(τ), H_a(τ), resp.) can be patched up to construct another new element of H(τ) (H_b(τ), H_a(τ), resp.). In particular,

(1)
if (A_b,2) holds, we can take , , and β = 0, 1 ≤ i ≤ 2n. Then there exist satisfying
()
(2)
if (A_a,2) holds, we take , α = 0, and β = e_i. Then there exist satisfying
()
(3)
if both (A_b,2) and (A_a,2) hold, then there exist satisfying
()

The above auxiliary elements

, and

will be very useful in the sequent discussions.

Theorem 27. Assume that (A₁) holds.

(1)
If (A₂) holds, then
()
In particular, if ℐ = [a, b], then
()
(2)
If (A_b,2) holds, then
()
(3)
If (A_a,2) holds, then
()

Proof. We first show that assertion (1) holds. By Lemmas 8 and 24, and Theorem 17, one has

()

For convenience, denote

()

Clearly, H⁰(τ) ⊂ H₀(τ). We now show that H₀(τ) ⊂ H⁰(τ). Fix any (x, f^π) ∈ H₀(τ). It follows from (85) that for all (y, g^π) ∈ H(τ),

()

For any given (y, g^π) ∈ H(τ), by Remark 26 there exists (z, h^π) ∈ H(τ) such that

()

Thus, it follows from (87) that (x, y)(b + 1) = (x, z)(b + 1) = (x, z)(a) = 0, and consequently (x, y)(a) = 0 for all (y, g^π) ∈ H(τ).

In the case that ℐ = [a, b], it is clear that

()

So it remains to show that H₀(τ) = H₀₀(τ). It suffices to show that x(a) = x(b + 1) = 0 for any (x, f^π) ∈ H₀(τ). Fix any (x, f^π) ∈ H₀(τ), and let ℐ₀ = ℐ, α = e_i, 1 ≤ i ≤ 2n, and β = 0. Then by Lemma 25, there exist

with y_i(a) = e_i and y_i(b + 1) = 0. Inserting these y_i into (87) one has that x(a) = 0. Similarly, one can show that x(b + 1) = 0. Thus H₀(τ) = H₀₀(τ). 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 H_b,0(τ) and H_a,0(τ) and then discuss the relationship among the defect indices of H_b,0(τ), H_a,0(τ), and H₀(τ).

For briefness, denote

()

For any λ ∈ ℂ, let ℳ_b,λ, M_b,λ, ℳ_a,λ, and M_a,λ be defined as (56) with ℐ replaced by ℐ_b and ℐ_a, respectively.

The following results are obtained.

Theorem 28. Assume that (A₁) holds.

(1)
If (A_b,2) holds and Γ(H_b,0(τ)) ≠ ∅, then d_b = dim ℳ_b,λ for any λ ∈ Γ(H_b,0(τ)), and n ≤ d_b ≤ 2n.
(2)
If (A_a,2) holds and Γ(H_a,0(τ)) ≠ ∅, then d_a = dim ℳ_a,λ for any λ ∈ Γ(H_a,0(τ)), and n ≤ d_a ≤ 2n.

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

For any λ ∈ Γ(H_b,0(τ)), 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 d_b = dim M_b,λ. This, together with Lemma 22, implies that d_b = dim ℳ_b,λ. In addition, it has been shown in [21] that n ≤ dim ℳ_b,λ ≤ 2n for any λ ∈ ℂ. So assertion (1) is true. The proof is complete.

Next, we discuss relationship among defect indices of H₀(τ), H_a,0(τ), and H_b,0(τ). For convenience, denote

()

for any

. It is evident that

, and

()

On the other hand, for any

and

, we define y 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 (A₁), (A_a,2), and (A_b,2) hold.

(1)
If (y, g^π) ∈ H(τ), then and .
(2)
If and with y_a(c₀) = y_b(c₀), then (y, g^π) ∈ H(τ).

Let

be the restriction of subspace H₀(τ), defined by

()

Lemma 30. Assume that (A₁), (A_a,2), and (A_b,2) hold. Then

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

Proof. (1) We first consider the necessity. Fix any . Let y_a, y_b, g_a, and g_b be defined as (94). Then y_a(c₀) = y_b(c₀) = 0. By (1) of Lemma 29 one has that . In addition, for any , it follows from Remark 26 that there exits (z, h^π) ∈ H(τ) 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 y and g be defined by (96). By (2) and (3) of Theorem 27 one has that y_a(c₀) = y_b(c₀) = 0. So y(c₀) = 0. It follows from (2) of Lemma 29 that (y, g^π) ∈ H(τ). For any (x, f^π) ∈ H(τ), it follows from (1) of Lemma 29 that and . Thus one has by (2) and (3) of Theorem 27 that

()

This implies that (y, g^π) ∈ H₀(τ) by (1) of Theorem 27, and consequently,

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

Set x_b(t) = f_b(t) = 0 for , where is defined by (19) with ℐ replaced by ℐ_b. It is clear that . For any , let x and f 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 y and g 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 (A₁), (A_a,2), and (A_b,2) 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

(A₃) Γ(H₀(τ)) ≠ ∅.

By Definition 10, one has that if Γ(H₀(τ)) = ∅, then ρ(H₀(τ)) = ∅, and consequently σ(H₀(τ)) = ℂ. We do not consider this case in the present paper.

Theorem 32. Assume that (A₁), (A_a,2), (A_b,2), and (A₃) hold. Then

()

Proof. The proof is divided into two steps.

Step 1. We show that

()

It follows from Γ(H₀(τ)) ≠ ∅ and Lemma 31 that Γ(H_b,0(τ))∩Γ(H_a,0(τ)) ≠ ∅. This, together with (A_a,2), (A_b,2), and Theorem 28, implies that for any λ ∈ Γ(H_b,0(τ))∩Γ(H_a,0(τ)), (1λ) has just d_b linearly independent solutions , 1 ≤ j ≤ d_b, and (1λ) has just d_a linearly independent solutions , 1 ≤ j ≤ d_a; that is, for 1 ≤ j ≤ d_b and for 1 ≤ j ≤ d_a.

Set for . It is clear that for 1 ≤ j ≤ d_a. Let x_j, 1 ≤ j ≤ d_a, be defined by (96). Then it follows from (2) of Lemma 30 that . Similarly, set for . Then one has for 1 ≤ j ≤ d_b. 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 (1λ) in ℐ_a, and is a solution of (1λ) in ℐ_b. Therefore, there exist unique c_j ∈ ℂ (1 ≤ j ≤ d_a) and d_j ∈ ℂ (1 ≤ j ≤ d_b) such that

()

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

()

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

Step 2. We show that

()

It is evident that , 1 ≤ i ≤ 2n, where are defined by (80). We claim that

()

In fact, for each (y, g^π) ∈ H₀(τ), set C₀ = (c₁, c₂, …, c_2n) ^T = y(c₀). Let

()

Then

()

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

For any λ ∈ Γ(H₀(τ)), we claim that

()

Since (109) holds, it suffices to show that

()

Suppose that

and c₁, c₂, …, c_2n ∈ ℂ satisfy

()

that is,

()

Since

and

, it follows that

. Since λ ∈ Γ(H₀(τ)), it yields

()

which, together with (109), implies that c_j = 0 for 1 ≤ j ≤ 2n, and consequently g^π = y^π = 0. This yields that (113) holds.

Since H₀(τ) and are closed J-Hermitian subspaces, it follows that Ran (H₀(τ) − λ) and are closed subspaces in , respectively. Hence, there exists a closed subspace Q in such that

()

In addition, again by the fact that λ ∈ Γ(H_b,0(τ))∩Γ(H_a,0(τ)), it follows that

are linearly independent in

. It follows from (113) that dim Q = 2n. 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 2nth 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 d = 0 if and only if d_a = d_b = n, and d = 2n if and only if d_a = d_b = 2n. The following definition is obtained.

Definition 34. Assume that (A₁), (A₂), and (A_b,2) hold. Then (1λ) is said to be in the limit d_b case at t = b. In the special case that d_b = n, (1λ) is said to be in the limit point case (l.p.c.) at t = b, and in the other special case that d_b = 2n, (1λ) is said to be in the limit circle case (l.c.c.) at t = b.

The same definition can be given at t = a provided that (A_a,2) holds.

4.3. Characterizations of H_a(τ) and H_b(τ)

In this subsection, we characterize the maximal subspaces H_a(τ) and H_b(τ). We first consider characterization of H_b(τ). Assume that (A_b,2) holds and Γ(H_b,0(τ)) ≠ ∅. Let λ ∈ Γ(H_b,0(τ)). It follows from the proof of Lemma 11 that

()

where

and

, and

are linearly independent (mod U₁). For convenience, denote

()

Clearly,

, 1 ≤ i ≤ 2d_b. So

is finite for all 1 ≤ i, j ≤ 2d_b. Then the following result can be directly derived from (119) and (120).

Lemma 35. Assume that (A₁) and (A_b,2) hold, and Γ(H_b,0(τ)) ≠ ∅. Then every y ∈ Dom H_b(τ) can be expressed as

()

where

and a_i ∈ ℂ.

By Lemma 35,

, defined by (78), can be uniquely expressed as

()

where

and a_ij ∈ ℂ. Denote

()

Lemma 36. Assume that (A₁) and (A_b,2) hold, and Γ(H_b,0(τ)) ≠ ∅. Then rank E = 2n and rank F₁ = 2d_b − 2n. 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 rank E = 2n.

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

()

for 1 ≤ i ≤ 2n, 1 ≤ s ≤ 2d_b, which, together with (78) and (2) of Theorem 27, implies that

()

Noting that rank E = 2n, one has

()

We now want to show rank F₁ ≥ 2d_b − 2n. By (3) of Lemma 14 one has

()

where

. Since rank X₀ = d_b, we have that

()

which, together with (129), yields that

()

Because rank F₂ = 2d_b − 2n ≤ d_b, one can rearrange the order of

such that the first 2d_b − 2n rows of F₂ 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 H_b(τ).

Theorem 37. Assume that (A₁) and (A_b,2) hold, λ ∈ Γ(H_b,0(τ)) ≠ ∅, and are linearly independent solutions of (1λ) in such that (124) holds. Then

()

is invertible, and any y ∈ Dom H_b(τ) can be uniquely expressed as

()

where

are defined by (78), and c_i, d_j ∈ ℂ.

Proof. Let E = (E₁, E₂), where E₁ and E₂ are 2n × (2d_b − 2n) and 2n × 2n matrices, respectively. It follows from (124) that there exists an invertible matrix L such that

()

So, it follows from (128) that E₁ + E₂F₃ = 0, which is equivalent to E₁ = −E₂F₃. Since rank E = 2n, it follows that E₂ 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 1 ≤ j ≤ d_b, can be uniquely expressed as

()

where

, and b_jl, c_js ∈ ℂ. This, together with (78) and (2) of Theorem 27, implies that for 1 ≤ i ≤ 2d_b − 2n, 1 ≤ j ≤ d_b

()

that is,

()

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

With a similar argument, one can obtain the following characterization of H_a(τ).

Theorem 38. Assume that (A₁) and (A_a,2) hold, Γ(H_a,0(τ)) ≠ ∅. Then, for some λ ∈ Γ(H_a,0(τ)), system (1λ) has 2d_a − 2n linearly independent solutions in such that G_a is invertible, where

()

and each y ∈ Dom H_a(τ) can be uniquely expressed as

()

where

, c_i, d_j ∈ ℂ, and

are defined as (79).

5. Characterizations of J-SSEs of H₀(τ)

In this section, we give a complete characterization of all the J-SSEs of minimal subspace H₀(τ) in terms of the square summable solutions of system (1λ). As a consequence, characterizations of all the J-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 ℐ.

5.1. Both the Endpoints Are Infinite

Let ℐ = (−∞, +∞), and assume that (A₁), (A_a,2), and (A_b,2) hold, and Γ(H₀(τ)) ≠ ∅ in this subsection. It follows from Theorem 32 that

()

In addition, for some λ ∈ Γ(H₀(τ), let

given in Theorems 37 and

be given in Theorems 38.

Theorem 39. Assume that (A₁), (A_a,2), (A_b,2), and (A₃) hold. Then a subspace is a J-SSE of H₀(τ) if and only if there exist two matrices and such that

(1)
rank (M, N) = d,
(2)
NG_bN^T − MG_aM^T = 0, and
()

where G_b and G_a 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 T is defined by (143). We now prove that T is a J-self-adjoint subspace extension of H₀(τ) by Lemma 9.

Denote

()

and set

()

Clearly,

and

for 1 ≤ i ≤ d. By Remark 26, there exist

such that

()

where

and

are specified by (A_a,2) and (A_b,2), respectively.

Since H(τ) and H₀(τ) are liner subspaces, β₁, β₂, …, β_d are linearly independent in H(τ) (modulo H₀(τ)) if and only if ω₁, ω₂, …, ω_d are linearly independent in Dom H(τ) (modulo Dom H₀(τ)). So it suffices to show that ω₁, ω₂, …, ω_d are linearly independent in Dom H(τ) (modulo Dom H₀(τ)). Suppose that there exists C = (c₁, c₂, …, c_d) ∈ ℂ^d such that

()

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

()

Since G_b and G_a are invertible, we get from (148) that CM = CN = 0. Then C = 0 by condition (1). So, ω₁, ω₂, …, ω_d are linearly independent in Dom H(τ) (modulo Dom H₀(τ)), and consequently β₁, β₂, …, β_d are linearly independent in H(τ) (modulo H₀(τ)).

Next, we show that [β_i : β_j] = 0 for 1 ≤ i, j ≤ d. It follows from (145) and (146) that

()

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

()

Consequently, [β_i : β_j] = 0 for 1 ≤ i, j ≤ d. Therefore,

satisfy the conditions (1) and (2) of Lemma 9.

Note that for each y ∈ Dom H(τ), it follows that

()

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

()

Hence, T is a J-SSE of H₀(τ) by Lemma 9. The sufficiency is proved.

We now show the necessity. Suppose that T is a J-SSE of H₀(τ). By Lemma 9 and Theorem 17, there exists a set of such that (152) holds. Write . Then and for 1 ≤ j ≤ d. By Theorems 37 and 38, each and can be uniquely expressed as

()

where

, and d_ij, c_ij, n_js, m_js ∈ ℂ. Set

()

First, we want to show that M and N satisfy condition (1). Otherwise, suppose that rank (M, N) < d. Then there exists C = (c₁, c₂, …, c_d) ∈ ℂ^d with C ≠ 0 such that

()

Then CM = 0 and CN = 0. Let

. We then have

()

For each y ∈ Dom H(τ), y_b can be uniquely expressed as (134) by Theorem 37 and y_a 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 y ∈ Dom H(τ), (ω, y)(+∞) = (ω, y)(−∞) = 0, which yields that ω ∈ Dom H₀(τ) by (1) of Theorem 27. Then ω₁, ω₂, …, ω_d are linearly dependent in Dom H(τ) (modulo Dom H₀(τ)), and consequently β₁, β₂, …, β_d are linearly dependent in H(τ) (modulo H₀(τ)). This is a contradiction. Hence, rank (M, N) = d.

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

()

Hence, by Lemma 14 and [β_i : β_j] = 0 for 1 ≤ i, j ≤ d, M and N satisfy condition (2).

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

()

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

The entire proof is complete.

To end this subsection, we give characterizations of J-SSEs of H₀(τ) in four special cases of defect indices: d_a = d_b = n; d_a = n, d_b = 2n; d_a = 2n, d_b = n; d_a = d_b = 2n.

In the case that d_a = d_b = n, that is, d = 0 by Theorem 32, the following result is derived from Lemma 11 and Theorem 17.

Theorem 40. Assume that (A₁), (A_a,2), (A_b,2), and (A₃) hold. If d = 0, then H₀(τ) is a J-self-adjoint subspace.

In the case that d_a = n, d_b = 2n, it follows from Theorem 32 that d = n. Let

, (1 ≤ i ≤ 2n), be 2n linearly independent solutions in

of (1λ) satisfying

. Then, by (3) of Lemma 14 one has that

()

The following result can be directly derived from Theorem 39.

Theorem 41. Assume that (A₁), (A₂), (A_a,2), (A_b,2), and (A₃) hold. If (1λ) is in l.p.c. at t = −∞ and in l.c.c. at t = +∞. Let be 2n linearly independent solutions in of (1λ) satisfying . Then a subspace is a J-SSE of H₀(τ) if and only if there exists a matrix N_n×2n such that

()

In the case that d_a = 2n, d_b = n, a similar result can be easily given. So we omit the details in this case.

In the case that d_a = d_b = 2n, it follows from Theorem 32 that d = 2n. Let χ_i (1 ≤ i ≤ 2n) be 2n linearly independent solutions in

of (1λ) satisfying

()

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

()

The following result is a direct consequence of Theorem 39.

Theorem 42. Assume that (A₁), (A_a,2), (A_b,2), and (A₃) hold. If (1λ) is in l.c.c. at both t = +∞ and t = −∞, then for any given λ ∈ Γ(H₀(τ)), let χ_i (1 ≤ i ≤ 2n) be 2n linearly independent solutions in of (1λ) satisfying (χ₁(c₀), χ₂(c₀), …, ϕ_2n(c₀)) = I_2n. Then a subspace is a J-SSE of H₀(τ) if and only if there exist two matrices M_2n×2n and N_2n×2n such that

()

Remark 43. As we have seen, there is no boundary condition at the endpoints at which system (1λ) is in l.p.c., and the matrix G_a or G_b can be replaced by 𝒥 in the case that system (1λ) is in l.c.c. at t = a or t = b.

5.2. At Least One of the Two Endpoints Is Finite

In this subsection, we characterize the J-SSEs of H₀(τ) in the special case that at least one of the two endpoints a and b is finite. We first consider the case that a is finite, and b is finite or infinite.

We point out that in this case, characterizations of all the J-SSEs of H₀(τ) can also be given by the proof of Theorem 39, provided that assumptions (A_b,2), (A_a,2), and (A₃) hold. But, if there does not exist a c₀ ∈ ℐ such that both (A_a,2) and (A_b,2) are satisfied, then Theorem 39 fails. We will remark again that the division of ℐ is not necessary in the case that one of the two endpoints is finite, and characterizations of all the J-SSEs of H₀(τ) can still be given provided that (A₃) and (A₃) hold.

In the case that a is finite, ℐ can be regarded as ℐ_b with a = c₀, and (A₂) is equivalent to (A_b,2). So all the characterizations for H_b,0(τ) and H_b(τ) given in Sections 3 and 4 are available to H₀(τ) and H(τ), respectively, with c₀ replaced by a. Assume that (A₂) holds. Then for any given λ ∈ Γ(H₀(τ)), as discussed in Section 4, let χ₁, …, χ_2d−2n be 2d − 2n linearly independent solutions in of (1λ) such that G is invertible, where G is defined by (133) with is replaced by χ_j, 1 ≤ j ≤ 2d − 2n. Then all the results of Theorem 37 hold with G_b and replaced by G and χ_j, respectively.

Theorem 44. Assume that the left endpoint a is finite, and (A₁)–(A₃) hold. Then a subspace is a J-SSE of H₀(τ) if and only if there exist two matrices M_d×2n and N_d×(2d−2n) such that

(1)
rank (M, N) = d,
(2)
M𝒥M^T − NGN^T = 0, and
()

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

We first show the sufficiency. Denote

()

and set

()

Clearly, w_i ∈ Dom H(τ) for 1 ≤ i ≤ d. By Remark 26, there exist β_i : = (ω_i, ρ^π) ∈ H(τ) (1 ≤ i ≤ d) such that

()

where t₀ is specified by (A₂). 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 y ∈ Dom H(τ), it follows that

()

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

()

Hence, T is a J-SSE of H₀(τ) by Lemma 9. The sufficiency is proved.

We now show the necessity. Suppose that T is a J-SSE of H₀(τ). By Lemma 9 and Theorem 17, there exists a set of for {H₀(τ), H(τ)} such that conditions (1) and (2) in Lemma 9 hold, and T can be expressed as (152). Write β_i : = (ω_i, ρ^π). Then ω_j ∈ Dom H(τ) for 1 ≤ j ≤ d. By Theorem 37, each ω_i can be uniquely expressed as

()

where

and a_ij, n_is ∈ ℂ. Set

()

With a similar argument to that used in the proof of Theorem 39, we can prove that M and N satisfy conditions (1) and (2). In addition, it is clear that T 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 J-SSEs of H₀(τ) in two special cases of defect indices.

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

Theorem 45. Assume that the left endpoint a is finite, and (A₁)–(A₃) hold. If (1λ) is in l.p.c. at t = b, then a subspace is a J-SSE of H₀(τ) if and only if there exists a matrix M_n×2n satisfying the self-adjoint condition

()

such that T can be defined by

()

In the other special case that d = 2n, the following result is a direct consequence of Theorem 44.

Theorem 46. Assume that the left endpoint a is finite, and (A₁)–(A₃) hold. If (1λ) is in l.c.c. at t = b, let χ_i (1 ≤ i ≤ 2n) be 2n linearly independent solutions in of (1λ) satisfying (χ₁(c₀), χ₂(c₀), …, χ_2n(c₀)) = I_2n. Then a subspace is a J-SSE of H₀(τ) if and only if there exist two 2n × 2n matrices M and N such that

(1)
rank (M, N) = 2n,
(2)
M𝒥M^T = N𝒥N^T, and
()

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

Theorem 47. Assume that the right endpoint b is finite, and (A₁)–(A₃) hold. Let χ_j, 1 ≤ j ≤ 2d − 2n, be linearly independent solutions of (1λ) in such that G is invertible, where G is defined as G_a in Theorem 38 with replaced by χ_j. Then a subspace is a J-SSE of H₀(τ) if and only if there exist two matrices M_n×2n and N_d×(2d−2n) such that

(1)
rank (M, N) = d,
(2)
MGM^T − N𝒥N^T = 0, and
()

In the case that both the two endpoints a and b are finite, that is, ℐ = [a, b], it is clear that d = 2n by (A₂). The characterization of J-SSEs given in Theorem 44 can be simplified as follows.

Theorem 48. Let ℐ = [a, b]. Assume that (A₁)–(A₃) hold. Then a subspace is a J-SSE of H₀(τ) if and only if there exist two 2n × 2n matrices M and N such that

(1)
rank (M, N) = 2n,
(2)
N𝒥N^T = M𝒥M^T, and
()

Proof. Let , 1 ≤ j ≤ 2n, be defined as those in Theorem 46. Then it follows that

()

where N₁ = N(χ₁, …, χ_2n) ^T(b + 1)𝒥. It is evident (χ₁, …, χ_2n)(b + 1) = I_2n. 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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All articles

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

Abstract

1. Introduction

2. Fundamental Results on Subspaces

3. Relationship between the Maximal and Minimal Subspaces

3.1. Conjugation Operator

3.2. Relationship between the Maximal and Minimal Subspaces

3.3. Definiteness Condition

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

4.1. Characterizations of the Minimal Subspaces

4.2. Defect Indices of Minimal Subspaces

4.3. Characterizations of H_a(τ) and H_b(τ)

5. Characterizations of J-SSEs of H₀(τ)

5.1. Both the Endpoints Are Infinite

5.2. At Least One of the Two Endpoints Is Finite

Acknowledgment

References

References

Information

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

Abstract

1. Introduction

2. Fundamental Results on Subspaces

3. Relationship between the Maximal and Minimal Subspaces

3.1. Conjugation Operator

3.2. Relationship between the Maximal and Minimal Subspaces

3.3. Definiteness Condition

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

4.1. Characterizations of the Minimal Subspaces

4.2. Defect Indices of Minimal Subspaces

4.3. Characterizations of Ha(τ) and Hb(τ)

5. Characterizations of J-SSEs of H0(τ)

5.1. Both the Endpoints Are Infinite

5.2. At Least One of the Two Endpoints Is Finite

Acknowledgment

References

References

Related

Information

4.3. Characterizations of H_a(τ) and H_b(τ)

5. Characterizations of J-SSEs of H₀(τ)