The Theory for 𝐽-Hermitian Subspaces in a Product Space

Abstract

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

1. Introduction

The spectral theory for differential and difference has been investigated extensively. In general, under certain definiteness conditions, a formally symmetric differential expression can generate a minimal operator which is symmetric, that is, a densely defined Hermitian operator, in a related Hilbert space and its adjoint, is the corresponding maximal operator (see, e.g., [1–3]). There are many results on self-adjoint extensions of the minimal operators since self-adjoint extension problems are fundamental in the study of spectral theory for differential expressions [2–6]. However, for some formally symmetric differential expressions, their minimal operators may be nondensely defined, or their maximal operators may be multivalued (e.g., [7, Example 2.2]). Further, for a formally symmetric difference expression, even a second-order one, its minimal operator is nondensely defined, and its maximal operator is multivalued in the related Hilbert space in general [8]. Therefore, the classical von Neumann self-adjoint extension theory and the Glazman-Krein-Naimark (GKN) theory for symmetric operators are not applicable in these cases.

The appropriate framework is linear subspaces (linear relations in the terminology of [7, 9, 10]) in a Hilbert space to study the linear differential or difference expressions for which the corresponding operators are nondensely defined or multivalued. Lesch and Malamud studied formally symmetric Hamiltonian systems in the framework of linear subspaces [7]. Coddington studied self-adjoint extensions of Hermitian subspaces in a product space [11]. He had extended the von Neumann self-adjoint extension theory for symmetric operators to Hermitian subspaces. By applying the relevant results in [11], Shi established the GKN theory for Hermitian subspaces [12]. For more results about nondensely defined Hermitian operators or Hermitian subspaces, we refer to [13–15].

The study of spectral problems involving linear differential and difference expressions with complex-valued coefficients is becoming a well-established area of analysis, and many results have been obtained [1, 16–22]. Such expressions are not formally symmetric in general, and hence, the spectral theory of self-adjoint subspaces is not applicable. To study such problems, Glazman introduced a concept of 𝐽-symmetric operators in [23], where 𝐽 is a conjugation operator given in Section 2. The minimal operators generated by certain differential and difference expressions with complex-valued coefficients are 𝐽-symmetric operators in the related Hilbert spaces (e.g., [18, 24, 25]). 𝐽-self-adjoint extension problems are also fundamental in the spectral theory for such expressions. Many results have been obtained on 𝐽-self-adjoint extensions [24–27]. Knowles gave a complete solution to the problem of describing all the 𝐽-self-adjoint extensions of any given 𝐽-symmetric operator provided the regularity field of this operator is nonempty [26]. Given a differential or difference expression, it is in practice difficult however to determine whether the appropriate 𝐽-symmetric operator has empty or nonempty regularity field. Therefore, Race established the theory for 𝐽-self-adjoint extensions of 𝐽-symmetric operators without the restrictions on the regularity fields [24]. However, the appropriate framework is also linear subspaces in a Hilbert space to study the linear nonsymmetric differential or difference expressions for which the corresponding minimal operators are nondensely defined, or the corresponding maximal operators are multivalued. So, the 𝐽-self-adjoint extension theory mentioned the above needs to be extended to linear subspace when we consider the nonsymmetric Hamiltonian systems which induce the nondensely defined or multivalued operators.

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

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

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

2. Preliminaries

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

Let 𝑋 be a complex Hilbert space with the inner product ⟨⋅,⋅⟩. The norm ‖⋅‖ is defined by ‖𝑓‖=⟨𝑓,𝑓⟩^1/2 for 𝑓∈𝑋. Let 𝑋² be the product space 𝑋×𝑋. By definition, the elements of 𝑋² consist of all possible ordered pairs (𝑥,𝑓) with 𝑥∈𝑋 and 𝑓∈𝑋, and for arbitrary two elements (𝑥,𝑓), (𝑦,𝑔)∈𝑋² and 𝛼∈𝐶,𝛼(𝑥,𝑓)=(𝛼𝑥,𝛼𝑓),(𝑥,𝑓)+(𝑦,𝑔)=(𝑥+𝑦,𝑓+𝑔).(2.1) The null element of 𝑋² is (0,0). The inner product in 𝑋² is defined by⟨(𝑥,𝑓),(𝑦,𝑔)⟩^∗=⟨𝑥,𝑦⟩+⟨𝑓,𝑔⟩,(𝑥,𝑓),(𝑦,𝑔)∈𝑋²,(2.2) and ‖⋅‖^∗ denotes the induced norm.

Let 𝑇 be a linear subspace in 𝑋² which is called to be a linear relation in [7, 9, 10]. For brevity, a linear subspace is only called a subspace. For subspace 𝑇 in 𝑋², we shall use the following definitions and notations:𝐷𝑅((𝑇𝑇))=={{𝑓𝑥∈∈𝑋𝑋∶∶((𝑥𝑥,,𝑓𝑓))∈∈𝑇𝑇forsome𝑓∈𝑋},k𝑇e𝑇r(𝑥(𝑇))=={𝑓{𝑥∈∈𝑋𝑋∶∶(𝑥(,𝑥f𝑓,o0)r)∈s∈o𝑇m𝑇}e},𝑥,₋₁∈𝑋},𝑇−𝜆=={({𝑥(,𝑓𝑓,−𝑥)𝜆∶𝑥()𝑥∶,𝑓(𝑥),∈𝑓)𝑇∈},𝑇}.(2.3) Clearly, 𝑇(0)={0} if and only if 𝑇 can determine a unique linear operator from 𝐷(𝑇) into 𝑋 whose graph is 𝑇, and 𝑇⁻¹ is closed if and only if 𝑇 is closed. Since the graph of a linear operator in 𝑋 is a subspace in 𝑋² and a linear operator is identified with its graph, the concept of subspaces in 𝑋² generalizes that of linear operators in 𝑋.

Definition 2.1 (see [11]). Let 𝑇 be a subspace in 𝑋².(1)Its adjoint, 𝑇^∗, is defined by 𝑇^∗=(𝑦,𝑔)∈𝑋²∶⟨𝑓,𝑦⟩=⟨𝑥,𝑔⟩∀(𝑥,𝑓)∈𝑇.(2.4)(2)𝑇 is said to be a Hermitian subspace if 𝑇⊂𝑇^∗.(3)𝑇 is said to be a self-adjoint subspace if 𝑇=𝑇^∗.

Lemma 2.2 (see [11]). Let 𝑇 be a subspace in 𝑋², then 𝑇^∗ is a closed subspace in 𝑋², 𝑇^∗=(𝑇)^∗, and 𝑇^∗∗=𝑇, where 𝑇 is the closure of 𝑇.

Definition 2.3. An operator 𝐽 defined on 𝑋 is said to be a conjugation operator if for all 𝑥,𝑦∈𝑋, ⟨𝐽𝑥,𝐽𝑦⟩=⟨𝑦,𝑥⟩,𝐽²𝑥=𝑥.(2.5)

It can be verified that 𝐽 is a conjugate linear, that is, 𝐽(𝑥+𝑦)=𝐽𝑥+𝐽𝑦 and 𝐽(𝜆𝑥)=𝜆𝐽𝑥 for 𝑥,𝑦∈𝑋 and 𝜆∈𝐶, and norm-preserving bijection on 𝑋 satisfying⟨𝐽𝑥,𝑦⟩=⟨𝐽𝑦,𝑥⟩∀𝑥,𝑦∈𝑋.(2.6) For example, the complex conjugation 𝑥↦𝑥 in any 𝐿² space is a conjugation operator on 𝐿².

Definition 2.4. Let 𝑇 be a subspace in 𝑋², and let 𝐽 be a conjugation operator.(1)Its 𝐽-adjoint, 𝑇^∗𝐽, is defined by 𝑇^∗𝐽=(𝑦,𝑔)∈𝑋²∶⟨𝑓,𝐽𝑦⟩=⟨𝑥,𝐽𝑔⟩∀(𝑥,𝑓)∈𝑇.(2.7)(2)𝑇 is said to be a 𝐽-Hermitian subspace if 𝑇⊂𝑇^∗𝐽.(3)𝑇 is said to be a 𝐽-self-adjoint subspace if 𝑇=𝑇^∗𝐽.

Remark 2.5. (1) It can be easily verified that 𝑇^∗𝐽 is a closed subspace. Consequently, a 𝐽-self-adjoint subspace 𝑇 is a closed subspace since 𝑇=𝑇^∗𝐽. In addition, 𝑆^∗𝐽⊂𝑇^∗𝐽 if 𝑇⊂𝑆.
(2) From the definition, we have that ⟨𝑓,𝐽𝑦⟩=⟨𝑥,𝐽𝑔⟩ for all (𝑥,𝑓)∈𝑇 and (𝑦,𝑔)∈𝑇^∗𝐽, and that 𝑇 is a 𝐽-Hermitian subspace if and only if ⟨𝑓,𝐽𝑦⟩=⟨𝑥,𝐽𝑔⟩∀(𝑥,𝑓),(𝑦,𝑔)∈𝑇.(2.8)
(3) The concepts of 𝐽-Hermitian and 𝐽-self-adjoint subspaces generalize those of 𝐽-symmetric and 𝐽-self-adjoint operators, respectively (see, e.g., [1, 24] for the concepts of 𝐽-symmetric and 𝐽-self-adjoint operators).

Lemma 2.6. Let 𝑇 be a subspace in 𝑋², then(1)𝑇^∗={(𝐽𝑦,𝐽𝑔)∶(𝑦,𝑔)∈𝑇^∗𝐽},(2)𝑇^∗𝐽={(𝐽𝑦,𝐽𝑔)∶(𝑦,𝑔)∈𝑇^∗}.

Proof. Result (2) follows from result (1) and the second relation of (2.5). So, one needs only to prove result (1). Set 𝐷₁={(𝐽𝑦,𝐽𝑔)∶(𝑦,𝑔)∈𝑇^∗𝐽}. Let (𝑦,𝑔)∈𝑇^∗, then ⟨𝑓,𝑦⟩=⟨𝑥,𝑔⟩ for all (𝑥,𝑓)∈𝑇. So, the second relation of (2.5) yields that ⟨𝑓,𝐽(𝐽𝑦)⟩=⟨𝑥,𝐽(𝐽𝑔)⟩ for all (𝑥,𝑓)∈𝑇. Hence, (𝐽𝑦,𝐽𝑔)∈𝑇^∗𝐽. Then (𝑦,𝑔)=(𝐽²𝑦,𝐽²𝑔)∈𝐷₁, which implies that 𝑇^∗⊂𝐷₁. Conversely, let (𝑦,𝑔)∈𝐷₁, then there exists (𝑦,̃𝑔̃)∈𝑇^∗𝐽 such that (𝑦,𝑔)=(𝐽𝑦,̃𝐽𝑔̃). It follows from (𝑦,̃𝑔̃)∈𝑇^∗𝐽 that ⟨𝑓,𝐽𝑦⟩̃=⟨𝑥,𝐽𝑔̃⟩ for all (𝑥,𝑓)∈𝑇, which implies that (𝐽𝑦,̃𝐽𝑔̃)∈𝑇^∗, that is, (𝑦,𝑔)∈𝑇^∗. So, 𝐷₁⊂𝑇^∗. Consequently, 𝑇^∗=𝐷₁, and result (1) holds.

Remark 2.7. Let 𝑇 be a subspace in 𝑋², then from Lemmas 2.2 and 2.6, and the closedness of 𝑇^∗𝐽, one has that 𝑇^∗𝐽=(𝑇)^∗𝐽, and 𝑇 is 𝐽-Hermitian if 𝑇 is 𝐽-Hermitian.

Lemma 2.8. Let 𝑇 be a closed 𝐽-Hermitian subspace, then (𝑦,𝑔)∈𝑇 if and only if (𝑦,𝑔)∈𝑇^∗𝐽 and ⟨𝑓,𝐽𝑦⟩=⟨𝑥,𝐽𝑔⟩ for all (𝑥,𝑓)∈𝑇^∗𝐽.

Proof. Let 𝑇 be a closed 𝐽-Hermitian subspace. Clearly, the necessity holds by (2) of Remark 2.5. Now, consider the sufficiency. Suppose that (𝑦,𝑔)∈𝑇^∗𝐽 and ⟨𝑓,𝐽𝑦⟩=⟨𝑥,𝐽𝑔⟩ for all (𝑥,𝑓)∈𝑇^∗𝐽, then we get from (2.6) that ⟨𝑦,𝐽𝑓⟩=⟨𝑔,𝐽𝑥⟩ for all (𝑥,𝑓)∈𝑇^∗𝐽. This, together with (1) of Lemma 2.6, implies that ⟨𝑦,𝑓⟩=⟨𝑔,𝑥̃⟩ for all (𝑥̃,𝑓)∈𝑇^∗. So, (𝑦,𝑔)∈𝑇^∗∗, and hence, (𝑦,𝑔)∈𝑇 by Lemma 2.2. So, the sufficiency holds.

Lemma 2.9. Let 𝑇 be a closed 𝐽-Hermitian subspace in 𝑋², then(1)(𝑇^∗𝐽)^∗={(𝐽𝑦,𝐽𝑔)∶(𝑦,𝑔)∈𝑇},(2)𝑇={(𝐽𝑦,𝐽𝑔)∶(𝑦,𝑔)∈(𝑇^∗𝐽)^∗}.

Proof. Since result (1) and the second relation of (2.5) imply that result (2) holds, it suffices to prove result (1). Set 𝐷={(𝐽𝑦,𝐽𝑔)∶(𝑦,𝑔)∈𝑇}. Let (𝑦,𝑔)∈𝐷, then there exists (𝑦,̃𝑔̃)∈𝑇 such that (𝑦,𝑔)=(𝐽𝑦,̃𝐽𝑔̃). By Lemma 2.8, (𝑦,̃𝑔̃)∈𝑇 implies that ⟨𝑓,𝐽𝑦⟩̃=⟨𝑥,𝐽𝑔̃⟩ for all (𝑥,𝑓)∈𝑇^∗𝐽. Then (𝐽𝑦,̃𝐽𝑔̃)∈(𝑇^∗𝐽)^∗, that is, (𝑦,𝑔)∈(𝑇^∗𝐽)^∗. So, 𝐷⊂(𝑇^∗𝐽)^∗. Conversely, let (𝑦,𝑔)∈(𝑇^∗𝐽)^∗, then ⟨𝑓,𝑦⟩=⟨𝑥,𝑔⟩ for all (𝑥,𝑓)∈𝑇^∗𝐽, that is, ⟨𝑓,𝐽(𝐽𝑦)⟩=⟨𝑥,𝐽(𝐽𝑔)⟩∀(𝑥,𝑓)∈𝑇^∗𝐽.(2.9) Clearly, (2.9) holds for all (𝑥,𝑓)∈𝑇 since 𝑇⊂𝑇^∗𝐽. So, (𝐽𝑦,𝐽𝑔)∈𝑇^∗𝐽. This, together with (2.9) and Lemma 2.8, implies that (𝐽𝑦,𝐽𝑔)∈𝑇. So, (𝐽(𝐽𝑦),𝐽(𝐽𝑔))∈𝐷, that is, (𝑦,𝑔)∈𝐷. Then, (𝑇^∗𝐽)^∗⊂𝐷 and then (𝑇^∗𝐽)^∗=𝐷.

Remark 2.10. Since 𝑇^∗𝐽=(𝑇)^∗𝐽 by Remark 2.7, Lemma 2.9 yields that (𝑇^∗𝐽)^∗={(𝐽𝑦,𝐽𝑔)∶(𝑦,𝑔)∈𝑇} and 𝑇={(𝐽𝑦,𝐽𝑔)∶(𝑦,𝑔)∈(𝑇^∗𝐽)^∗} for a 𝐽-Hermitian subspace 𝑇 which may not be closed.

3. Defect Index of a 𝐽-Hermitian Subspace

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

Let 𝑇 be a closed 𝐽-Hermitian subspace. It has been known that 𝑇^∗𝐽 is a closed subspace by (1) of Remark 2.5. Then the closedness of 𝑇 and 𝑇^∗𝐽 and 𝑇⊂𝑇^∗𝐽 gives that𝑇^∗𝐽=𝑇⊕𝒯,(3.1) where 𝒯 denotes the orthogonal complement of 𝑇 in 𝑇^∗𝐽, that is, 𝒯=𝑇^∗𝐽⊝𝑇. Now, let 𝑆 be a closed 𝐽-Hermitian subspace extension of 𝑇, that is, 𝑇⊂𝑆 and 𝑆 is 𝐽-Hermitian. Then, it follows from the closedness of 𝑆 and 𝑇 that there exists a unique subspace 𝐾_𝑆,𝑇=𝑆⊝𝑇 such that𝑆=𝑇⊕𝐾_𝑆,𝑇.(3.2) Clearly, 𝑆⊂𝑇^∗𝐽 since 𝑆^∗𝐽⊂𝑇^∗𝐽 and 𝑆⊂𝑆^∗𝐽. Then by (3.1) and (3.2), 𝐾_𝑆,𝑇 can be expressed as𝐾_𝑆,𝑇=(𝑦,𝑔)∈𝒯∶thereexist𝑦₁,𝑔₁∈𝑇,𝑦₂,𝑔₂∈𝑆,suchthat𝑦₂,𝑔₂=𝑦₁,𝑔₁+(𝑦,𝑔).(3.3) Further, we have the following result.

Theorem 3.1. Let 𝑇 be a closed 𝐽-Hermitian subspace, and let 𝑆 be a 𝐽-self-adjoint subspace extension (briefly, 𝐽-SSE) of 𝑇, that is, 𝑇⊂𝑆 and 𝑆 is 𝐽-self-adjoint, then dim𝑇^∗𝐽𝑆=dim𝑆𝑇.(3.4)

Proof. If 𝑇 is a 𝐽-self-adjoint subspace, then 𝑇=𝑇^∗𝐽 and 𝑇 is the only 𝐽-SSE of itself. So, (3.4) holds. Now, suppose that 𝑇 is 𝐽-Hermitian but not 𝐽-self-adjoint, that is, 𝑇⊂𝑇^∗𝐽 and 𝑇≠𝑇^∗𝐽. It follows that (3.2) holds with 𝐾_𝑆,𝑇≠{0}. Let dim𝑆/𝑇=𝑚, then (3.2) yields that dim𝐾_𝑆,𝑇=𝑚. In the case of 𝑚<∞, let {(𝑥_𝑗,𝑓_𝑗)}^𝑚𝑗=1 be a basis of 𝐾_𝑆,𝑇, then 𝑆=𝑇⊕span𝑥₁,𝑓₁,𝑥₂,𝑓₂,…,𝑥_𝑚,𝑓_𝑚.(3.5) Define 𝑇_𝑗=𝑇⊕span𝑥₁,𝑓₁,𝑥₂,𝑓₂,…,𝑥_𝑗,𝑓_𝑗,𝑗=1,2,…,𝑚.(3.6) Clearly, 𝑇≠𝑇_𝑗≠𝑇_𝑗+1 for 𝑗=1,2,…,𝑚−1 and 𝑆=𝑆^∗𝐽=𝑇_𝑚^∗𝐽⊂𝑇_𝑚−1^∗𝐽⊂⋯⊂𝑇₂^∗𝐽⊂𝑇₁^∗𝐽⊂𝑇^∗𝐽,(3.7) since 𝑇⊂𝑇₁⊂𝑇₂⊂⋯⊂𝑇_𝑚=𝑆 and 𝑆 is 𝐽-self-adjoint. Further, 𝑇_𝑗 (𝑗=1,2,…,𝑚) is a closed subspace since 𝑇 is closed. It holds that 𝑇^∗≠𝑇^∗𝑗≠𝑇^∗𝑗+1,𝑗=1,2,…,𝑚−1.(3.8) Otherwise, for example, suppose that 𝑇^∗1=𝑇^∗2, then by Lemma 2.2, we have 𝑇₁=𝑇^1∗∗=𝑇^2∗∗=𝑇₂ since 𝑇₁ and 𝑇₂ are closed. It contradicts 𝑇₁≠𝑇₂. So, (3.8) holds. It follows from (3.8) and (2) of Lemma 2.6 that 𝑇^∗𝐽≠𝑇_𝑗^∗𝐽≠𝑇_𝑗+1^∗𝐽,𝑗=1,2,…,𝑚−1.(3.9) We get from (3.7) and (3.9) that dim𝑇^∗𝐽/𝑆≥𝑚=dim𝑆/𝑇.
In the case of 𝑚=+∞, we have the linear span of an infinite set in (3.5). So we can construct an infinite sequence of subspaces of the form (3.6) which satisfies the relations like those in (3.7) and (3.9). So, we have dim𝑇^∗𝐽/𝑆=+∞=dim𝑆/𝑇.
Next, we prove that dim𝑆/𝑇≥dim𝑇^∗𝐽/𝑆. Since 𝑇^∗𝐽 and 𝑆 are closed subspaces, there exists uniquely a closed subspace 𝐾_𝑇^∗𝐽_,𝑆=𝑇^∗𝐽⊝𝑆 such that 𝑇^∗𝐽=𝑆⊕𝐾_𝑇^∗𝐽_,𝑆. Set dim𝑇^∗𝐽/𝑆=𝑚′. Then dim𝐾_𝑇^∗𝐽_,𝑆=𝑚′. If 𝑚′<∞, let {(𝑦_𝑗,𝑔_𝑗)}^𝑚′_𝑗=1 be a basis of 𝐾_𝑇^∗𝐽_,𝑆, then𝑇^∗𝐽=𝑆⊕span𝑦₁,𝑔₁,𝑦₂,𝑔₂,…,𝑦_𝑚^′,𝑔_𝑚′.(3.10)Define 𝑆_𝑗=𝑆⊕span𝑦₁,𝑔₁,𝑦₂,𝑔₂,…,𝑦_𝑗,𝑔_𝑗,𝑗=1,2,…,𝑚′.(3.11) Clearly, it holds that 𝑆^∗𝐽=𝑆≠𝑆_𝑗≠𝑆_𝑗+1 for 𝑗=1,2,…,𝑚′−1 and 𝑇^∗𝐽^∗=𝑆^∗𝑚_′⊂𝑆^∗𝑚_′−1⊂⋯⊂𝑆^∗2⊂𝑆^∗1⊂𝑆^∗=𝑆^∗𝐽^∗,(3.12) since 𝑆⊂𝑆₁⊂𝑆₂⊂⋯⊂𝑆_𝑚′=𝑇^∗𝐽 and 𝑆 is 𝐽-self-adjoint. Further, 𝑆_𝑗 (𝑗=1,2,…,𝑚^) is a closed subspace since 𝑆 is closed. Similarly, it holds that 𝑆^∗𝐽^∗≠𝑆^∗𝑗≠𝑆^∗𝑗+1,𝑗=1,2,…,𝑚^−1.(3.13) We get from (3.12) and (3.13) that dim(𝑆^∗𝐽)^∗/(𝑇^∗𝐽)^∗≥𝑚′. It can be verified that dim𝑆^∗𝐽^∗𝑇^∗𝐽^∗=dim𝑆𝑇(3.14) by Lemma 2.9. So, dim𝑆/𝑇≥𝑚^=dim𝑇^∗𝐽/𝑆. Further, it can be verified that dim𝑆/𝑇≥dim𝑇^∗𝐽/𝑆 also holds for 𝑚^=+∞. Based on the above discussions, (3.4) holds.

Remark 3.2. (1) From Theorem 3.1 and its proof, one has that if one of the two dimensions in (3.4) is finite, so is the other and they are equal, and if one of the two dimensions is infinite, so is the other. Here, there is no distinction between degrees of infinity.
(2) The case for 𝐽-symmetric operators was established in [24, Theorem 3.1].

Remark 3.3. Note that 𝑇^∗𝐽=(𝑇)^∗𝐽. We get from Theorem 3.1 that if 𝑆 is a 𝐽-SSE of 𝑇, which may not be closed, then it holds that dim𝑇^∗𝐽𝑇=2dim𝑆𝑇.(3.15)
Now, we give the concept of the defect index of a 𝐽-Hermitian subspace. The concept of the defect index of a 𝐽-symmetric operator in 𝑋 was given by [24, Definition 3.2].

Definition 3.4. Let 𝑇 be a 𝐽-Hermitian subspace, then 𝑑(𝑇)=(1/2)dim𝑇^∗𝐽/𝑇 is called to be the defect index of 𝑇.

Remark 3.5. (1) It will be proved that every 𝐽-Hermitian subspace has a 𝐽-SSE in Theorem 4.3 in Section 4. So, by (3.15) we have that the defect index of every 𝐽-Hermitian subspace is a nonnegative integer.
(2) Since 𝑇^∗𝐽=(𝑇)^∗𝐽 by Remark 2.7 and every 𝐽-SSE is closed, we have that a 𝐽-symmetric subspace 𝑇 and its closure 𝑇 have the same defect index and the same 𝐽-SSEs.

Definition 3.6 (see [12]). Let 𝑇 be a subspace in 𝑋². The set Γ(𝑇)∶={𝜆∈𝐂∶thereexists𝑐(𝜆)>0suchthat‖𝑓−𝜆𝑥‖≥𝑐(𝜆)‖𝑥‖∀(𝑥,𝑓)∈𝑇}(3.16) is called to be the regularity field of 𝑇.
It is evident that Γ(𝑇)=Γ(𝑇) for a subspace 𝑇.

Lemma 3.7. Let 𝑇 be a subspace in 𝑋, then(1)𝑅(𝑇−𝜆)^⟂=ker(𝑇^∗−𝜆) for each 𝜆∈𝐂,(2)for each 𝜆∈Γ(𝑇),𝑋=𝑅𝑇−𝜆⊕ker𝑇^∗−𝜆(orthogonalsumin𝑋),(3.17)(3)𝑋=𝑅(𝑇^∗𝐽−𝜆) for each 𝜆∈Γ(𝑇).

Proof. (1) Let 𝜆∈𝐂. It is clear that ker𝑇^∗−𝜆=𝑥∈𝑋∶𝑥,𝜆𝑥∈𝑇^∗.(3.18) For every 𝑥∈ker(𝑇^∗−𝜆), we have from (3.18) that (𝑥,𝜆𝑥)∈𝑇^∗. So, ⟨𝑔,𝑥⟩=⟨𝑦,𝜆𝑥⟩ for all (𝑦,𝑔)∈𝑇, which implies that ⟨𝑔−𝜆𝑦,𝑥⟩=0. Therefore, 𝑥∈𝑅(𝑇−𝜆)^⟂, and then ker(𝑇^∗−𝜆)⊂𝑅(𝑇−𝜆)^⟂. Conversely, for every 𝑥∈𝑅(𝑇−𝜆)^⟂, we have that ⟨𝑔−𝜆𝑦,𝑥⟩=0 for all (𝑦,𝑔)∈𝑇. It follows that ⟨𝑔,𝑥⟩=⟨𝑦,𝜆𝑥⟩ for all (𝑦,𝑔)∈𝑇. So, (𝑥,𝜆𝑥)∈𝑇^∗, and hence 𝑥∈ker(𝑇^∗−𝜆) by (3.18). So, 𝑅(𝑇−𝜆)^⟂⊂ker(𝑇^∗−𝜆). Consequently, 𝑅(𝑇−𝜆)^⟂=ker(𝑇^∗−𝜆), and result (1) is proved.
(2) By result (1) and Lemma 2.2, we have that 𝑅(𝑇−𝜆)^⟂=ker((𝑇)^∗−𝜆)=ker(𝑇^∗−𝜆). So, by the projection theorem, in order to prove (3.17), it suffices to show that 𝑅(𝑇−𝜆) with 𝜆∈Γ(𝑇) is closed in 𝑋. It is evident that 𝑇−𝜆⁻¹=(𝑓−𝜆𝑥,𝑥)∶(𝑥,𝑓)∈𝑇.(3.19) Let 𝜆∈Γ(𝑇). Since Γ(𝑇)=Γ(𝑇), one has that 𝜆∈Γ(𝑇), that is, there exists a constant 𝑐(𝜆)>0 such that ‖𝑓−𝜆𝑥‖≥𝑐(𝜆)‖𝑥‖∀(𝑥,𝑓)∈𝑇.(3.20) Then we get from (3.19) and (3.20) that (𝑇−𝜆)⁻¹(0)={0}. So, (𝑇−𝜆)⁻¹ determines a linear operator from 𝑅(𝑇−𝜆) to 𝑋. Further, the closedness of 𝑇−𝜆 and (3.20) imply that this operator is a closed and bounded operator. So, its domain 𝑅(𝑇−𝜆) is closed in 𝑋. Therefore, (3.17) holds, and result (2) is proved.
(3) Let 𝜆∈Γ(𝑇). We first show that there exists a constant 𝑀>0 such that ‖𝑥‖≤𝑀‖𝑓−𝜆𝑥‖∀(𝑥,𝑓)∈𝑇^∗𝐽.(3.21) Assume the contrary, then there exists a sequence {𝑓_𝑘−𝜆𝑥_𝑘}^∞𝑘=1⊂𝑅(𝑇^∗𝐽−𝜆) with ‖𝑓_𝑘−𝜆𝑥_𝑘‖=1 (𝑘=1,2,…) such that ‖‖𝑥_𝑘‖‖>𝑘,𝑘=1,2,….(3.22) Clearly, (𝑥_𝑘,𝑓_𝑘)∈𝑇^∗𝐽 and (𝑇)^∗𝐽=𝑇^∗𝐽 imply that (𝑥_𝑘,𝑓_𝑘)∈(𝑇)^∗𝐽. So, ⟨𝑔,𝐽𝑥_𝑘⟩=⟨𝑦,𝐽𝑓_𝑘⟩ for all (𝑦,𝑔)∈𝑇 by (1) of Remark 2.5, which, together with ⟨𝜆𝑦,𝐽𝑥_𝑘⟩=⟨𝑦,𝐽(𝜆𝑥_𝑘)⟩, implies that for 𝑘=1,2,…, ⟨𝑔−𝜆𝑦,𝐽𝑥_𝑘⟩=𝑦,𝐽𝑓_𝑘−𝜆𝑥_𝑘∀(𝑦,𝑔)∈𝑇.(3.23) Define 𝜙_𝑘(𝑔−𝜆𝑦)=⟨𝑔−𝜆𝑦,𝐽𝑥_𝑘⟩ for (𝑦,𝑔)∈𝑇. Then 𝜙_𝑘, 𝑘=1,2…, are linear functionals in 𝑅(𝑇−𝜆). Since 𝜙_𝑘(𝑔−𝜆𝑦)=⟨𝑦,𝐽(𝑓_𝑘−𝜆𝑥_𝑘)⟩ by (3.23) and ‖𝐽(𝑓_𝑘−𝜆𝑥_𝑘)‖=‖𝑓_𝑘−𝜆𝑥_𝑘‖=1, we have that {𝜙_𝑘(𝑔−𝜆𝑦)}^∞𝑘=1 is bounded for any given 𝑔−𝜆𝑦∈𝑅(𝑇−𝜆). Note that 𝑅(𝑇−𝜆) with 𝜆∈Γ(𝑇) is closed by the proof of result (2), and hence it is a Hilbert space with the inner product ⟨⋅,⋅⟩. So, by the resonance theorem, {‖𝜙_𝑘‖}^∞𝑘=1 is bounded, that is, {‖𝐽𝑥_𝑘‖}^∞𝑘=1 is bounded. Since ‖𝐽𝑥_𝑘‖=‖𝑥_𝑘‖, 𝑘=1,2,…, we have a contradiction with (3.22). So, (3.21) holds.
Inserting (3.21) into ‖𝑥‖+‖𝑓‖≤‖𝑓−𝜆𝑥‖+(1+|𝜆|)‖𝑥‖, we get that ‖𝑥‖+‖𝑓‖≤1+1+||𝜆||𝑀‖𝑓−𝜆𝑥‖.(3.24) It can be easily verified from the closedness of 𝑇^∗𝐽 and (3.24) that 𝑅(𝑇^∗𝐽−𝜆) is closed in 𝑋. So, result (1) implies that 𝑋=𝑅𝑇^∗𝐽−𝜆⊕ker𝑇^∗𝐽^∗−𝜆.(3.25) By Remark 2.10, ker((𝑇^∗𝐽)^∗−𝜆)={𝐽𝑦∶𝑦∈ker(𝑇−𝜆)}, while it can be verified that ker(𝑇−𝜆)={0} for 𝜆∈Γ(𝑇)=Γ(𝑇). Therefore, (3.25) yields that result (3) holds.

If Γ(𝑇)≠∅, we have the following results which give a formula for the defect index of a 𝐽-Hermitian space.

Theorem 3.8. Assume that 𝑇 is a 𝐽-Hermitian subspace with Γ(𝑇)≠∅. Let 𝜆∈Γ(𝑇), then 𝑇^∗𝐽=𝑇+̇𝒩,(3.26) where 𝒩=(𝑦,𝑔)∈𝑇^∗𝐽∶𝑔−𝜆𝑦∈ker𝑇^∗−𝜆,(3.27)𝑑(𝑇)=dim𝑅(𝑇−𝜆)^⟂=dimker𝑇^∗−𝜆.(3.28)

Proof. We first prove (3.26). Since 𝑇 is 𝐽-Hermitian, one has that 𝑇 is 𝐽-Hermitian, and hence 𝑇⊂(𝑇)^∗𝐽=𝑇^∗𝐽. Clearly, 𝒩⊂𝑇^∗𝐽. So, 𝑇+𝒩⊂𝑇^∗𝐽. On the other hand, let (𝑥,𝑓)∈𝑇^∗𝐽. It follows from (3.17) and 𝜆∈Γ(𝑇) that there exist (𝑦,𝑔)∈𝑇 and 𝑤∈ker(𝑇^∗−𝜆) such that 𝑓−𝜆𝑥=𝑔−𝜆𝑦+𝑤, that is, (𝑓−𝑔)−𝜆(𝑥−𝑦)=𝑤. Let (𝑥̃,𝑓)=(𝑥−𝑦,𝑓−𝑔), then (𝑥,𝑓)=(𝑦,𝑔)+(𝑥̃,𝑓) and (𝑥̃,𝑓)∈𝒩. So, 𝑇^∗𝐽⊂𝑇+𝒩, and consequently, 𝑇^∗𝐽=𝑇+𝒩.
Now, let (𝑢,ℎ)∈𝑇∩𝒩, then ℎ−𝜆𝑢∈𝑅𝑇−𝜆,ℎ−𝜆𝑢∈ker𝑇^∗−𝜆.(3.29) Consequently, ℎ−𝜆𝑢=0 by 𝑅(𝑇−𝜆)^⟂=ker(𝑇^∗−𝜆), which can be obtained from (3.17). Since 𝜆∈Γ(𝑇)=Γ(𝑇), one has by Definition 3.6 that there exists a constant 𝑐(𝜆)>0 such that 0=‖ℎ−𝜆𝑢‖≥𝑐(𝜆)‖𝑢‖. It follows that 𝑢=0, which, together with ‖ℎ−𝜆𝑢‖=0, implies that ℎ=0. Then, (𝑢,ℎ)=(0,0) if (𝑢,ℎ)∈𝑇∩𝒩. So, (3.26) holds.
Next, we prove (3.28). Let 𝜆∈Γ(𝑇). Set 𝑈₁={(𝑥,𝜆𝑥)∈𝑇^∗𝐽}, then 𝑈₁ is closed since 𝑇^∗𝐽 is closed. Let 𝑈₂=𝒩/𝑈₁. We will show that dim𝑈₂=dimker𝑇^∗−𝜆.(3.30) Let {𝜓_𝑗}^𝑚𝑗=1⊂ker(𝑇^∗−𝜆) be linearly independent, then, by (3) of Lemma 3.7, there exists (𝑢_𝑗,ℎ_𝑗)∈𝑇^∗𝐽 such that 𝜓_𝑗=ℎ_𝑗−𝜆𝑢_𝑗, 1≤𝑗≤𝑚. It follows from 𝜓_𝑗∈ker(𝑇^∗−𝜆) that (𝑢_𝑗,ℎ_𝑗)∈𝑈₂ for 1≤𝑗≤𝑚. In addition, if ^𝑚_𝑗=1𝑐_𝑗𝑢_𝑗,ℎ_𝑗∈𝑈₁,𝑐_𝑗∈𝐂,(3.31) then ∑^𝑚𝑗=1𝑐_𝑗(ℎ_𝑗−𝜆𝑢_𝑗)=0, that is, ∑^𝑚𝑗=1𝑐_𝑗𝜓_𝑗=0. So, 𝑐_𝑗=0 for 1≤𝑗≤𝑚, and hence {(𝑢_𝑗,ℎ_𝑗)}^𝑚𝑗=1⊂𝑈₂ is linearly independent (mod 𝑈₁). Conversely, let {(𝑢_𝑗,ℎ_𝑗)}^𝑚𝑗=1⊂𝑈₂ be linearly independent (mod 𝑈₁), and let 𝜓_𝑗=ℎ_𝑗−𝜆𝑢_𝑗, then 𝜓_𝑗∈ker(𝑇^∗−𝜆). If ∑^𝑚𝑗=1𝑐_𝑗𝜓_𝑗=0, then ∑^𝑚𝑗=1𝑐_𝑗(𝑢_𝑗,ℎ_𝑗)∈𝑈₁. So, 𝑐_𝑗=0 (1≤𝑗≤𝑚), and hence the set {𝜓_𝑗}^𝑚𝑗=1 is linearly independent. Based on the above discussions, (3.30) holds. On the other hand, it is evident that dim𝑥,𝜆𝑥∈𝑇^∗=dimker𝑇^∗−𝜆.(3.32) Further, by Lemma 2.6, we have that dim𝑈₁=dim𝑥,𝜆𝑥∈𝑇^∗.(3.33) It follows from (3.30)–(3.33) that dim𝑈₁=dim𝑈₂, and hence (3.26) implies that 𝑑(𝑇)=12dim𝒩=dimker𝑇^∗−𝜆.(3.34) So, (3.28) holds by (1) of Lemma 3.7.

Remark 3.9. From Theorem 3.8, one has the following result: for a 𝐽-Hermitian subspace 𝑇, dim𝑅(𝑇−𝜆)^⟂ and dimker(𝑇^∗−𝜆) are constants in Γ(𝑇) which are equal to the defect index of 𝑇. This result extends [24, Theorem 5.7] for 𝐽-symmetric operators to 𝐽-Hermitian subspaces. Similarly, there is no distinction between degrees of infinity.

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

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

Define the form [∶] as[(𝑥,𝑓)∶(𝑦,𝑔)]=⟨𝑓,𝐽𝑦⟩−⟨𝑥,𝐽𝑔⟩,(𝑥,𝑓),(𝑦,𝑔)∈𝑇^∗𝐽.(4.1) Then, ⟨𝑓,𝐽𝑦⟩=⟨𝑥,𝐽𝑔⟩ if and only if [(𝑥,𝑓)∶(𝑦,𝑔)]=0. Further, for all 𝑌_𝑗=(𝑥_𝑗,𝑓_𝑗)∈𝑇^∗𝐽(𝑗=1,2,3) and 𝜇∈𝐂,𝑌₃∶𝜇𝑌𝑌₁₁+𝑌₂=𝑌₃∶𝑌₁+𝑌₃∶𝑌₂,𝑌₁+𝑌₂∶𝑌₃=𝑌₁∶𝑌₃+𝑌₂∶𝑌₃,∶𝑌₂=𝑌₁∶𝜇𝑌₂=𝜇𝑌₁∶𝑌₂,𝑌₁∶𝑌₂=−𝑌₂∶𝑌₁.(4.2) Since the closure 𝑇 of a 𝐽-Hermitian subspace 𝑇 is also a 𝐽-Hermitian subspace by Remark 2.7, and 𝑇 and 𝑇 have the same defect indices and the same 𝐽-SSEs by (2) of Remark 3.5, we shall assume that 𝑇 is closed in the rest of this section. Let 𝑇 be a closed subspace in 𝑋², and let 𝐾⊂𝒯 be a subspace, where 𝒯 is given in (3.1). Let 𝐾^∗𝐽|_𝒯 be a restriction of 𝐾^∗𝐽 to 𝒯, that is,𝐾^∗𝐽||_𝒯={(𝑦,𝑔)∈𝒯∶[(𝑥,𝑓)∶(𝑦,𝑔)]=0∀(𝑥,𝑓)∈𝐾},(4.3)

then 𝐾 is called to be 𝐽-Hermitian in 𝒯 if 𝐾⊂𝐾^∗𝐽|_𝒯, and 𝐾 is called to be 𝐽-self-adjoint in 𝒯 if 𝐾=𝐾^∗𝐽|_𝒯.

Remark 4.1. From the definition, we have that [(𝑥,𝑓)∶(𝑦,𝑔)]=0 for all (𝑥,𝑓)∈𝐾 and (𝑦,𝑔)∈𝐾^∗𝐽|_𝒯, and 𝐾 is 𝐽-Hermitian in 𝒯 if and only if [(𝑥,𝑓)∶(𝑦,𝑔)]=0 for all (𝑥,𝑓), (𝑦,𝑔)∈𝐾.

Lemma 4.2. Let 𝑇 be a closed 𝐽-Hermitian subspace, and let 𝐾⊂𝒯 be a subspace. Assume that 𝑆=𝑇⊕𝐾, then(1)𝑆 is 𝐽-Hermitian if and only if 𝐾 is 𝐽-Hermitian in 𝒯,(2)𝑆 is 𝐽-self-adjoint if and only if 𝐾 is 𝐽-self-adjoint in 𝒯.

Proof. (1) Suppose that 𝑆 is 𝐽-Hermitian. It can be easily verified that 𝐾 is 𝐽-Hermitian in 𝒯 by (2) of Remark 2.5, 𝐾⊂𝑆, and Remark 4.1. So, the necessity holds. We now prove the sufficiency. Suppose that 𝐾 is 𝐽-Hermitian in 𝒯. For all (𝑥,𝑓), (𝑦,𝑔)∈𝑆, we get from 𝑆=𝑇⊕𝐾 that ((𝑥𝑦,,𝑓𝑔))==𝑥𝑦₁₁,𝑓₁+𝑥₂,𝑓₂,𝑥₁,𝑓₁∈𝑇,𝑥₂,𝑓₂∈𝐾,,𝑔₁+𝑦₂,𝑔₂,𝑦₁,𝑔₁∈𝑇,𝑦₂,𝑔₂∈𝐾.(4.4) Since 𝑇 is 𝐽-Hermitian and 𝐾⊂𝐾^∗𝐽|_𝒯⊂𝑇^∗𝐽, it can be obtained from (2) of Remark 2.5 and Remark 4.1 that [(𝑥,𝑓)∶(𝑦,𝑔)]=+𝑥₁𝑥,₂𝑓₁∶𝑦₁,𝑔₁+𝑥₁,𝑓₁∶𝑦₂,𝑔₂,𝑓₂∶𝑦₁,𝑔₁+𝑥₂,𝑓₂∶𝑦₂,𝑔₂=0.(4.5) So, 𝑆 is 𝐽-Hermitian. The sufficiency holds, and result (1) is proved.
(2) First, consider the necessity. Suppose that 𝑆 is 𝐽-self-adjoint, then 𝐾 is 𝐽-Hermitian in 𝒯, that is, 𝐾⊂𝐾^∗𝐽|_𝒯, by result (1). It suffices to show that 𝐾^∗𝐽|_𝒯⊂𝐾. For any given (𝑥,𝑓)∈𝑆, there exist (𝑥₁,𝑓₁)∈𝑇 and (𝑥₂,𝑓₂)∈𝐾 such that the first relation of (4.4) holds. Let (𝑦,𝑔)∈𝐾^∗𝐽|_𝒯, then [(𝑥₂,𝑓₂)∶(𝑦,𝑔)]=0 by Remark 4.1. Note that (𝑦,𝑔)∈𝑇^∗𝐽 by 𝐾^∗𝐽|_𝒯⊂𝑇^∗𝐽. Then [(𝑥₁,𝑓₁)∶(𝑦,𝑔)]=0 by (2) of Remark 2.5. Therefore, the first relation of (4.4) yields that [(𝑥,𝑓)∶(𝑦,𝑔)]=𝑥₁,𝑓₁∶(𝑦,𝑔)+𝑥₂,𝑓₂∶(𝑦,𝑔)=0∀(𝑥,𝑓)∈𝑆.(4.6) So, (𝑦,𝑔)∈𝑆^∗𝐽. Therefore, 𝑆=𝑆^∗𝐽 yields that (𝑦,𝑔)∈𝑆, which, together with (𝑦,𝑔)∈𝐾^∗𝐽|_𝒯, 𝐾^∗𝐽|_𝒯∩𝑇={0}, and 𝑆=𝑇⊕𝐾, implies that (𝑦,𝑔)∈𝐾. Hence, 𝐾^∗𝐽|_𝒯⊂𝐾, and hence 𝐾=𝐾^∗𝐽|_𝒯. The necessity holds.
Next, consider the sufficiency. Suppose that 𝐾 is 𝐽-self-adjoint in 𝒯. By result (1), one has that 𝑆⊂𝑆^∗𝐽. It suffices to show that 𝑆^∗𝐽⊂𝑆. Let (𝑦,𝑔)∈𝑆^∗𝐽, then (𝑦,𝑔)∈𝑇^∗𝐽 since 𝑆^∗𝐽⊂𝑇^∗𝐽 by 𝑇⊂𝑆. It follows from (3.1) that there exist (𝑦₁,𝑔₁)∈𝑇 and (𝑦₂,𝑔₂)∈𝒯 such that (𝑦,𝑔)=𝑦₁,𝑔₁+𝑦₂,𝑔₂.(4.7) We claim that (𝑦₂,𝑔₂)∈𝐾. In fact, since (𝑦,𝑔)∈𝑆^∗𝐽, we have [(𝑥,𝑓)∶(𝑦,𝑔)]=0∀(𝑥,𝑓)∈𝑆.(4.8) Inserting the first relation of (4.4) and (4.7) into (4.8) and using (2) of Remark 2.5 and Remark 4.1, we get that [(𝑥₂,𝑓₂)∶(𝑦₂,𝑔₂)]=0 for all (𝑥₂,𝑓₂)∈𝐾. Then, (𝑦₂,𝑔₂)∈𝐾^∗𝐽|_𝒯=𝐾. It follows from (𝑦₂,𝑔₂)∈𝐾, 𝑆=𝑇⊕𝐾, and (4.7) that (𝑦,𝑔)∈𝑆. So, 𝑆^∗𝐽⊂𝑆, and hence 𝑆=𝑆^∗𝐽. The sufficiency holds.

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

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

Proof. Let 𝑇 be a closed 𝐽-Hermitian subspace. If 𝑇 is 𝐽-self-adjoint, then this conclusion holds. So, we assume that 𝑇≠𝑇^∗𝐽. To prove that 𝑇 has a 𝐽-SSE, it suffices to prove that there exists a 𝐽-self-adjoint subspace 𝐾 in 𝒯 by Lemma 4.2. The proof uses Zorn’s lemma. Since 𝑇≠𝑇^∗𝐽, one has that 𝒯≠{0}. Choose 0≠(𝑥₀,𝑓₀)∈𝒯 and set 𝐾₀=span{(𝑥₀,𝑓₀)}. Then 𝐾₀ is 𝐽-Hermitian in 𝒯 since [(𝑥₀,𝑓₀)∶(𝑥₀,𝑓₀)]=0. Let 𝒦 be the set of all the 𝐽-Hermitian subspaces in 𝒯 which contain 𝐾₀, then 𝒦 is not empty since 𝐾₀∈𝒦. Further, let 𝒦 be ordered by extension, that is, 𝐴<𝐵 if and only if 𝐴⊂𝐵, and let ℳ={𝐾_𝛼} be an arbitrary totally ordered subset of 𝒦. Set 𝐾=⋃_𝛼𝐾_𝛼. Then, it can be verified that 𝐾 is 𝐽-Hermitian in 𝒯 by Remark 4.1 and the fact that all the elements of ℳ are 𝐽-Hermitian in 𝒯. So, 𝐾 is an upper bound of ℳ in 𝒦. Therefore, 𝒦 has a maximal element by Zorn’s lemma. This means that 𝐾₀ has a maximal 𝐽-Hermitian subspace extension, denoted by 𝐾, in 𝒯. We now prove 𝐾=𝐾^∗𝐽|_𝒯. Suppose that 𝐾≠𝐾^∗𝐽|_𝒯 on the contrary. Note that 𝐾⊂𝐾^∗𝐽|_𝒯. Choose (𝑥̃₀,𝑓₀)∈𝐾^∗𝐽|_𝒯 satisfying (𝑥̃₀,𝑓₀)∉𝐾, and set 𝐾=𝐾+̇span{(𝑥̃₀,𝑓₀)}. It can be verified by Remark 4.1 and the fact that 𝐾 is 𝐽-Hermitian in 𝒯 that [(𝑥,𝑓)∶(𝑦,𝑔)]=0 holds for all (𝑥,𝑓),(𝑦,𝑔)∈𝐾. Note that 𝐾⊂𝒯. Then, 𝐾 is 𝐽-Hermitian in 𝒯, which contradicts the maximality of 𝐾. Hence, 𝐾=𝐾^∗𝐽|_𝒯.

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

Theorem 4.5. Let 𝑇 be a closed 𝐽-Hermitian subspace. Assume that 𝑑(𝑇)=𝑑<+∞, then a subspace 𝑆 is a 𝐽-SSE of 𝑇 if and only if 𝑇⊂𝑆⊂𝑇^∗𝐽, and there exists {(𝑥_𝑗,𝑓_𝑗)}^𝑑𝑗=1⊂𝑇^∗𝐽 such that(1)(𝑥₁,𝑓₁),(𝑥₂,𝑓₂),…,(𝑥_𝑑,𝑓_𝑑) are linearly independent (mod𝑇),(2)[(𝑥_𝑠,𝑓_𝑠)∶(𝑥_𝑗,𝑓_𝑗)]=0 for 𝑠,𝑗=1,2,…,𝑑,(3)𝑆={(𝑦,𝑔)∈𝑇^∗𝐽∶[(𝑦,𝑔)∶(𝑥_𝑗,𝑓_𝑗)]=0,𝑗=1,2,…,𝑑}.

Proof. First, consider the necessity. Suppose that 𝑆 is a 𝐽-SSE of 𝑇, then it holds that 𝑇⊂𝑆⊂𝑇^∗𝐽 since 𝑆^∗𝐽⊂𝑇^∗𝐽 and 𝑆=𝑆^∗𝐽. We also have that (3.2) holds and 𝐾_𝑆,𝑇 in (3.2) is 𝐽-self-adjoint in 𝒯 by Lemma 4.2. Note that dim𝑆/𝑇=𝑑 by Theorem 3.1. Then dim𝐾_𝑆,𝑇=𝑑, and let {(𝑥_𝑗,𝑓_𝑗)}^𝑑𝑗=1 be a basis of 𝐾_𝑆,𝑇, then we get from (3.2) that (1) holds. In addition, since 𝐾_𝑆,𝑇 is 𝐽-self-adjoint in 𝒯, one has that (2) holds by Remark 4.1. For convenience, set𝐷=(𝑦,𝑔)∈𝑇^∗𝐽∶(𝑦,𝑔)∶𝑥_𝑗,𝑓_𝑗=0,𝑗=1,2,…,𝑑.(4.9) Now, we prove 𝑇⊕𝐾_𝑆,𝑇=𝐷, that is, 𝑆=𝐷. Let (𝑦,𝑔)∈𝑇⊕𝐾_𝑆,𝑇, then (𝑦,𝑔)∈𝑇^∗𝐽 by 𝑆⊂𝑇^∗𝐽, and there exist (𝑦,̃𝑔̃)∈𝑇 and 𝑐_𝑗∈𝐶 such that (𝑦,𝑔)=(𝑦,̃𝑔̃)+^𝑑_𝑗=1𝑐_𝑗𝑥_𝑗,𝑓_𝑗.(4.10) Inserting (4.10) into [(𝑦,𝑔)∶(𝑥_𝑠,𝑓_𝑠)] and using (2) of Remark 2.5 and (2) of this theorem, we get that [(𝑦,𝑔)∶(𝑥_𝑠,𝑓_𝑠)]=0 for 𝑠=1,2,…,𝑑. So, (𝑦,𝑔)∈𝐷, and hence 𝑇⊕𝐾_𝑆,𝑇⊂𝐷. Conversely, suppose that (𝑦,𝑔)∈𝐷, then by (3.1), there exist (𝑦₁,𝑔₁)∈𝑇 and (𝑦₂,𝑔₂)∈𝒯 such that (4.7) holds. The definition of 𝐷, (4.7), and (2) of Remark 2.5 implies that for 𝑠=1,2,…,𝑑, 𝑦₂,𝑔₂∶𝑥_𝑠,𝑓_𝑠=(𝑦,𝑔)∶𝑥_𝑠,𝑓_𝑠−𝑦₁,𝑔₁∶𝑥_𝑠,𝑓_𝑠=0.(4.11) So, (𝑦₂,𝑔₂)∈(𝐾_𝑆,𝑇)^∗𝐽|_𝒯, which implies that (𝑦₂,𝑔₂)∈𝐾_𝑆,𝑇 since 𝐾_𝑆,𝑇 is 𝐽-self-adjoint in 𝒯. One has from (4.7) that (𝑦,𝑔)∈𝑇⊕𝐾_𝑆,𝑇, and consequently, 𝐷⊂𝑇⊕𝐾_𝑆,𝑇. Hence, 𝑇⊕𝐾_𝑆,𝑇=𝐷, that is, 𝑆=𝐷. The necessity holds.
Next, consider the sufficiency. Suppose that there exists {(𝑥_𝑗,𝑓_𝑗)}^𝑑𝑗=1⊂𝑇^∗𝐽 such that conditions (1) and (2) hold and 𝑆 is given in condition (3). From (3.1), we have 𝑥_𝑗,𝑓_𝑗=𝑥_𝑗0,𝑓_𝑗0+𝑥̃_𝑗,𝑓_𝑗,𝑥^𝑗0,𝑓_𝑗0∈𝑇,𝑥̃_𝑗,𝑓_𝑗∈𝒯,𝑗=1,2,…,𝑑.(4.12) It can be easily verified that the set {(𝑥̃_𝑗,𝑓_𝑗)}^𝑑𝑗=1 satisfies conditions (1) and (2). Let 𝐾=span{(𝑥̃₁,𝑓₁),(𝑥̃₂,𝑓₂),…,(𝑥̃_𝑑,𝑓_𝑑)}, then 𝐾 is 𝐽-Hermitian in 𝒯 since {(𝑥̃_𝑗,𝑓_𝑗)}^𝑑𝑗=1 satisfies condition (2) and 𝐾⊂𝒯. By the proof of Theorem 4.3, there exists 𝐾₁⊂𝒯 such that 𝐾₁ is 𝐽-self-adjoint in 𝒯 and 𝐾⊂𝐾₁. Then, by Lemma 4.2, 𝑇⊕𝐾₁ is a 𝐽-SSE of 𝑇, which, together with Theorem 3.1, yields that 𝑑=dim𝑇⊕𝐾¹𝑇=dim𝐾¹≥𝑑.(4.13) Therefore, 𝐾⊂𝐾₁ and dim 𝐾=𝑑 imply that 𝐾=𝐾₁, and hence 𝐾 is 𝐽-self-adjoint in 𝒯. With a similar argument to the proof of 𝑇⊕𝐾_𝑆,𝑇=𝐷, we have 𝑇⊕𝐾=𝐷,(4.14) where 𝐷=(𝑦,𝑔)∈𝑇^∗𝐽∶(𝑦,𝑔)∶𝑥̃_𝑗,𝑓_𝑗=0,𝑗=1,2,…,𝑑.(4.15) On the other hand, it can be easily verified that 𝐷=𝑆. So, 𝑆=𝑇⊕𝐾, and hence 𝑆 is 𝐽-self-adjoint by (2) of Lemma 4.2. The sufficiency holds.

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

Acknowledgments

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