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ISRN Mathematical Analysis
VolumeΒ 2012Β (2012), Article IDΒ 676835, 16 pages
http://dx.doi.org/10.5402/2012/676835
Research Article

The Theory for 𝐽-Hermitian Subspaces in a Product Space

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

Received 6 January 2012; Accepted 13 February 2012

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

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

Abstract

This paper is concerned with the theory for 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 βŸ¨β‹…,β‹…βŸ©. The norm β€–β‹…β€– is defined by ‖𝑓‖=βŸ¨π‘“,π‘“βŸ©1/2 for π‘“βˆˆπ‘‹. Let 𝑋2 be the product space 𝑋×𝑋. By definition, the elements of 𝑋2 consist of all possible ordered pairs (π‘₯,𝑓) with π‘₯βˆˆπ‘‹ and π‘“βˆˆπ‘‹, and for arbitrary two elements (π‘₯,𝑓), (𝑦,𝑔)βˆˆπ‘‹2 and π›ΌβˆˆπΆ,𝛼(π‘₯,𝑓)=(𝛼π‘₯,𝛼𝑓),(π‘₯,𝑓)+(𝑦,𝑔)=(π‘₯+𝑦,𝑓+𝑔).(2.1) The null element of 𝑋2 is (0,0). The inner product in 𝑋2 is defined by⟨(π‘₯,𝑓),(𝑦,𝑔)βŸ©βˆ—=⟨π‘₯,π‘¦βŸ©+βŸ¨π‘“,π‘”βŸ©,(π‘₯,𝑓),(𝑦,𝑔)βˆˆπ‘‹2,(2.2) and β€–β‹…β€–βˆ— 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:𝐷(𝑇)={π‘₯βˆˆπ‘‹βˆΆ(π‘₯,𝑓)βˆˆπ‘‡forsomeπ‘…π‘“βˆˆπ‘‹},(𝑇)={π‘“βˆˆπ‘‹βˆΆ(π‘₯,𝑓)βˆˆπ‘‡forsome𝑇π‘₯βˆˆπ‘‹},𝑇(π‘₯)={π‘“βˆˆπ‘‹βˆΆ(π‘₯,𝑓)βˆˆπ‘‡},ker(𝑇)={π‘₯βˆˆπ‘‹βˆΆ(π‘₯,0)βˆˆπ‘‡},βˆ’1={(𝑓,π‘₯)∢(π‘₯,𝑓)βˆˆπ‘‡},π‘‡βˆ’πœ†={(π‘₯,π‘“βˆ’πœ†π‘₯)∢(π‘₯,𝑓)βˆˆπ‘‡}.(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 π‘‡βˆ’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, π‘‡βˆ—, is defined by π‘‡βˆ—=ξ€½(𝑦,𝑔)βˆˆπ‘‹2ξ€Ύ.βˆΆβŸ¨π‘“,π‘¦βŸ©=⟨π‘₯,π‘”βŸ©βˆ€(π‘₯,𝑓)βˆˆπ‘‡(2.4)(2)𝑇 is said to be a Hermitian subspace if π‘‡βŠ‚π‘‡βˆ—.(3)𝑇 is said to be a self-adjoint subspace if 𝑇=π‘‡βˆ—.

Lemma 2.2 (see [11]). Let 𝑇 be a subspace in 𝑋2, then π‘‡βˆ— is a closed subspace in 𝑋2, π‘‡βˆ—=(𝑇)βˆ—, and π‘‡βˆ—βˆ—=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 π‘₯,π‘¦βˆˆπ‘‹, ⟨𝐽π‘₯,π½π‘¦βŸ©=βŸ¨π‘¦,π‘₯⟩,𝐽2π‘₯=π‘₯.(2.5)

It can be verified that 𝐽 is a conjugate linear, that is, 𝐽(π‘₯+𝑦)=𝐽π‘₯+𝐽𝑦 and 𝐽(πœ†π‘₯)=πœ†π½π‘₯ for π‘₯,π‘¦βˆˆπ‘‹ and πœ†βˆˆπΆ, and norm-preserving bijection on 𝑋 satisfying⟨𝐽π‘₯,π‘¦βŸ©=βŸ¨π½π‘¦,π‘₯βŸ©βˆ€π‘₯,π‘¦βˆˆπ‘‹.(2.6) For example, the complex conjugation π‘₯↦π‘₯ 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, π‘‡βˆ—π½, is defined by π‘‡βˆ—π½=ξ€½(𝑦,𝑔)βˆˆπ‘‹2ξ€Ύ.βˆΆβŸ¨π‘“,π½π‘¦βŸ©=⟨π‘₯,π½π‘”βŸ©βˆ€(π‘₯,𝑓)βˆˆπ‘‡(2.7)(2)𝑇 is said to be a 𝐽-Hermitian subspace if π‘‡βŠ‚π‘‡βˆ—π½.(3)𝑇 is said to be a 𝐽-self-adjoint subspace if 𝑇=π‘‡βˆ—π½.

Remark 2.5. (1) It can be easily verified that π‘‡βˆ—π½ is a closed subspace. Consequently, a 𝐽-self-adjoint subspace 𝑇 is a closed subspace since 𝑇=π‘‡βˆ—π½. In addition, π‘†βˆ—π½βŠ‚π‘‡βˆ—π½ if π‘‡βŠ‚π‘†.
(2) From the definition, we have that βŸ¨π‘“,π½π‘¦βŸ©=⟨π‘₯,π½π‘”βŸ© for all (π‘₯,𝑓)βˆˆπ‘‡ and (𝑦,𝑔)βˆˆπ‘‡βˆ—π½, and that 𝑇 is a 𝐽-Hermitian subspace if and only if βŸ¨π‘“,π½π‘¦βŸ©=⟨π‘₯,π½π‘”βŸ©βˆ€(π‘₯,𝑓),(𝑦,𝑔)βˆˆπ‘‡.(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)π‘‡βˆ—={(𝐽𝑦,𝐽𝑔)∢(𝑦,𝑔)βˆˆπ‘‡βˆ—π½},(2)π‘‡βˆ—π½={(𝐽𝑦,𝐽𝑔)∢(𝑦,𝑔)βˆˆπ‘‡βˆ—}.

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)βˆˆπ‘‡βˆ—π½}. Let (𝑦,𝑔)βˆˆπ‘‡βˆ—, then βŸ¨π‘“,π‘¦βŸ©=⟨π‘₯,π‘”βŸ© for all (π‘₯,𝑓)βˆˆπ‘‡. So, the second relation of (2.5) yields that βŸ¨π‘“,𝐽(𝐽𝑦)⟩=⟨π‘₯,𝐽(𝐽𝑔)⟩ for all (π‘₯,𝑓)βˆˆπ‘‡. Hence, (𝐽𝑦,𝐽𝑔)βˆˆπ‘‡βˆ—π½. Then (𝑦,𝑔)=(𝐽2𝑦,𝐽2𝑔)∈𝐷1, which implies that π‘‡βˆ—βŠ‚π·1. Conversely, let (𝑦,𝑔)∈𝐷1, then there exists (̃𝑦,̃𝑔)βˆˆπ‘‡βˆ—π½ such that (𝑦,𝑔)=(𝐽̃𝑦,𝐽̃𝑔). It follows from (̃𝑦,̃𝑔)βˆˆπ‘‡βˆ—π½ that βŸ¨π‘“,π½Μƒπ‘¦βŸ©=⟨π‘₯,π½Μƒπ‘”βŸ© for all (π‘₯,𝑓)βˆˆπ‘‡, which implies that (𝐽̃𝑦,𝐽̃𝑔)βˆˆπ‘‡βˆ—, that is, (𝑦,𝑔)βˆˆπ‘‡βˆ—. So, 𝐷1βŠ‚π‘‡βˆ—. Consequently, π‘‡βˆ—=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 π‘‡βˆ—π½, one has that π‘‡βˆ—π½=(𝑇)βˆ—π½, and 𝑇 is 𝐽-Hermitian if 𝑇 is 𝐽-Hermitian.

Lemma 2.8. Let 𝑇 be a closed 𝐽-Hermitian subspace, then (𝑦,𝑔)βˆˆπ‘‡ if and only if (𝑦,𝑔)βˆˆπ‘‡βˆ—π½ and βŸ¨π‘“,π½π‘¦βŸ©=⟨π‘₯,π½π‘”βŸ© for all (π‘₯,𝑓)βˆˆπ‘‡βˆ—π½.

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)βˆˆπ‘‡βˆ—π½ and βŸ¨π‘“,π½π‘¦βŸ©=⟨π‘₯,π½π‘”βŸ© for all (π‘₯,𝑓)βˆˆπ‘‡βˆ—π½, then we get from (2.6) that βŸ¨π‘¦,π½π‘“βŸ©=βŸ¨π‘”,𝐽π‘₯⟩ for all (π‘₯,𝑓)βˆˆπ‘‡βˆ—π½. This, together with (1) of Lemma 2.6, implies that ξ‚βŸ¨π‘¦,π‘“βŸ©=βŸ¨π‘”,Μƒπ‘₯⟩ for all (Μƒπ‘₯,𝑓)βˆˆπ‘‡βˆ—. So, (𝑦,𝑔)βˆˆπ‘‡βˆ—βˆ—, and hence, (𝑦,𝑔)βˆˆπ‘‡ by Lemma 2.2. So, the sufficiency holds.

Lemma 2.9. Let 𝑇 be a closed 𝐽-Hermitian subspace in 𝑋2, then(1)(π‘‡βˆ—π½)βˆ—={(𝐽𝑦,𝐽𝑔)∢(𝑦,𝑔)βˆˆπ‘‡},(2)𝑇={(𝐽𝑦,𝐽𝑔)∢(𝑦,𝑔)∈(π‘‡βˆ—π½)βˆ—}.

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)βˆˆπ‘‡}. Let (𝑦,𝑔)∈𝐷, then there exists (̃𝑦,̃𝑔)βˆˆπ‘‡ such that (𝑦,𝑔)=(𝐽̃𝑦,𝐽̃𝑔). By Lemma 2.8, (̃𝑦,̃𝑔)βˆˆπ‘‡ implies that βŸ¨π‘“,π½Μƒπ‘¦βŸ©=⟨π‘₯,π½Μƒπ‘”βŸ© for all (π‘₯,𝑓)βˆˆπ‘‡βˆ—π½. Then (𝐽̃𝑦,𝐽̃𝑔)∈(π‘‡βˆ—π½)βˆ—, that is, (𝑦,𝑔)∈(π‘‡βˆ—π½)βˆ—. So, π·βŠ‚(π‘‡βˆ—π½)βˆ—. Conversely, let (𝑦,𝑔)∈(π‘‡βˆ—π½)βˆ—, then βŸ¨π‘“,π‘¦βŸ©=⟨π‘₯,π‘”βŸ© for all (π‘₯,𝑓)βˆˆπ‘‡βˆ—π½, that is, βŸ¨π‘“,𝐽(𝐽𝑦)⟩=⟨π‘₯,𝐽(𝐽𝑔)βŸ©βˆ€(π‘₯,𝑓)βˆˆπ‘‡βˆ—π½.(2.9) Clearly, (2.9) holds for all (π‘₯,𝑓)βˆˆπ‘‡ since π‘‡βŠ‚π‘‡βˆ—π½. So, (𝐽𝑦,𝐽𝑔)βˆˆπ‘‡βˆ—π½. This, together with (2.9) and Lemma 2.8, implies that (𝐽𝑦,𝐽𝑔)βˆˆπ‘‡. So, (𝐽(𝐽𝑦),𝐽(𝐽𝑔))∈𝐷, that is, (𝑦,𝑔)∈𝐷. Then, (π‘‡βˆ—π½)βˆ—βŠ‚π· and then (π‘‡βˆ—π½)βˆ—=𝐷.

Remark 2.10. Since π‘‡βˆ—π½=(𝑇)βˆ—π½ by Remark 2.7, Lemma 2.9 yields that (π‘‡βˆ—π½)βˆ—={(𝐽𝑦,𝐽𝑔)∢(𝑦,𝑔)βˆˆπ‘‡} and 𝑇={(𝐽𝑦,𝐽𝑔)∢(𝑦,𝑔)∈(π‘‡βˆ—π½)βˆ—} 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 π‘‡βˆ—π½ is a closed subspace by (1) of Remark 2.5. Then the closedness of 𝑇 and π‘‡βˆ—π½ and π‘‡βŠ‚π‘‡βˆ—π½ gives thatπ‘‡βˆ—π½=π‘‡βŠ•π’―,(3.1) where 𝒯 denotes the orthogonal complement of 𝑇 in π‘‡βˆ—π½, that is, 𝒯=π‘‡βˆ—π½βŠπ‘‡. Now, let 𝑆 be a closed 𝐽-Hermitian subspace extension of 𝑇, that is, π‘‡βŠ‚π‘† and 𝑆 is 𝐽-Hermitian. Then, it follows from the closedness of 𝑆 and 𝑇 that there exists a unique subspace 𝐾𝑆,𝑇=π‘†βŠπ‘‡ such that𝑆=π‘‡βŠ•πΎπ‘†,𝑇.(3.2) Clearly, π‘†βŠ‚π‘‡βˆ—π½ since π‘†βˆ—π½βŠ‚π‘‡βˆ—π½ and π‘†βŠ‚π‘†βˆ—π½. Then by (3.1) and (3.2), 𝐾𝑆,𝑇 can be expressed as𝐾𝑆,𝑇=ξ€½(𝑦,𝑔)βˆˆπ’―βˆΆthereexist𝑦1,𝑔1ξ€Έξ€·π‘¦βˆˆπ‘‡,2,𝑔2ξ€Έβˆˆπ‘†,suchthat𝑦2,𝑔2ξ€Έ=𝑦1,𝑔1ξ€Έξ€Ύ.+(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, π‘‡βŠ‚π‘† and 𝑆 is 𝐽-self-adjoint, then 𝑇dimβˆ—π½π‘†π‘†=dim𝑇.(3.4)

Proof. If 𝑇 is a 𝐽-self-adjoint subspace, then 𝑇=π‘‡βˆ—π½ 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, π‘‡βŠ‚π‘‡βˆ—π½ and π‘‡β‰ π‘‡βˆ—π½. It follows that (3.2) holds with 𝐾𝑆,𝑇≠{0}. Let dim𝑆/𝑇=π‘š, then (3.2) yields that dim𝐾𝑆,𝑇=π‘š. In the case of π‘š<∞, let {(π‘₯𝑗,𝑓𝑗)}π‘šπ‘—=1 be a basis of 𝐾𝑆,𝑇, then π‘₯𝑆=π‘‡βŠ•spanξ€½ξ€·1,𝑓1ξ€Έ,ξ€·π‘₯2,𝑓2ξ€Έξ€·π‘₯,…,π‘š,π‘“π‘š.ξ€Έξ€Ύ(3.5) Define 𝑇𝑗π‘₯=π‘‡βŠ•spanξ€½ξ€·1,𝑓1ξ€Έ,ξ€·π‘₯2,𝑓2ξ€Έξ€·π‘₯,…,𝑗,𝑓𝑗,𝑗=1,2,…,π‘š.(3.6) Clearly, 𝑇≠𝑇𝑗≠𝑇𝑗+1 for 𝑗=1,2,…,π‘šβˆ’1 and 𝑆=π‘†βˆ—π½=ξ€·π‘‡π‘šξ€Έβˆ—π½βŠ‚ξ€·π‘‡π‘šβˆ’1ξ€Έβˆ—π½ξ€·π‘‡βŠ‚β‹―βŠ‚2ξ€Έβˆ—π½βŠ‚ξ€·π‘‡1ξ€Έβˆ—π½βŠ‚π‘‡βˆ—π½,(3.7) since π‘‡βŠ‚π‘‡1βŠ‚π‘‡2βŠ‚β‹―βŠ‚π‘‡π‘š=𝑆 and 𝑆 is 𝐽-self-adjoint. Further, 𝑇𝑗 (𝑗=1,2,…,π‘š) is a closed subspace since 𝑇 is closed. It holds that π‘‡βˆ—β‰ π‘‡βˆ—π‘—β‰ π‘‡βˆ—π‘—+1,𝑗=1,2,…,π‘šβˆ’1.(3.8) Otherwise, for example, suppose that π‘‡βˆ—1=π‘‡βˆ—2, then by Lemma 2.2, we have 𝑇1=𝑇1βˆ—βˆ—=𝑇2βˆ—βˆ—=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 π‘‡βˆ—π½β‰ ξ€·π‘‡π‘—ξ€Έβˆ—π½β‰ ξ€·π‘‡π‘—+1ξ€Έβˆ—π½,𝑗=1,2,…,π‘šβˆ’1.(3.9) We get from (3.7) and (3.9) that dimπ‘‡βˆ—π½/𝑆β‰₯π‘š=dim𝑆/𝑇.
In the case of π‘š=+∞, 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 dimπ‘‡βˆ—π½/𝑆=+∞=dim𝑆/𝑇.
Next, we prove that dim𝑆/𝑇β‰₯dimπ‘‡βˆ—π½/𝑆. Since π‘‡βˆ—π½ and 𝑆 are closed subspaces, there exists uniquely a closed subspace πΎπ‘‡βˆ—π½,𝑆=π‘‡βˆ—π½βŠπ‘† such that π‘‡βˆ—π½=π‘†βŠ•πΎπ‘‡βˆ—π½,𝑆. Set dimπ‘‡βˆ—π½/𝑆=π‘šβ€². Then dimπΎπ‘‡βˆ—π½,𝑆=π‘šβ€². If π‘šβ€²<∞, let {(𝑦𝑗,𝑔𝑗)}π‘šβ€²π‘—=1 be a basis of πΎπ‘‡βˆ—π½,𝑆, thenπ‘‡βˆ—π½π‘¦=π‘†βŠ•spanξ€½ξ€·1,𝑔1ξ€Έ,𝑦2,𝑔2𝑦,…,π‘šβ€²,π‘”π‘šβ€².ξ€Έξ€Ύ(3.10)Define 𝑆𝑗𝑦=π‘†βŠ•spanξ€½ξ€·1,𝑔1ξ€Έ,𝑦2,𝑔2𝑦,…,𝑗,𝑔𝑗,𝑗=1,2,…,π‘šβ€².(3.11) Clearly, it holds that π‘†βˆ—π½=𝑆≠𝑆𝑗≠𝑆𝑗+1 for 𝑗=1,2,…,π‘šβ€²βˆ’1 and ξ€·π‘‡βˆ—π½ξ€Έβˆ—=π‘†βˆ—π‘šβ€²βŠ‚π‘†βˆ—π‘šβ€²βˆ’1βŠ‚β‹―βŠ‚π‘†βˆ—2βŠ‚π‘†βˆ—1βŠ‚π‘†βˆ—=ξ€·π‘†βˆ—π½ξ€Έβˆ—,(3.12) since π‘†βŠ‚π‘†1βŠ‚π‘†2βŠ‚β‹―βŠ‚π‘†π‘šβ€²=π‘‡βˆ—π½ and 𝑆 is 𝐽-self-adjoint. Further, 𝑆𝑗 (𝑗=1,2,…,π‘šξ…ž) is a closed subspace since 𝑆 is closed. Similarly, it holds that ξ€·π‘†βˆ—π½ξ€Έβˆ—β‰ π‘†βˆ—π‘—β‰ π‘†βˆ—π‘—+1,𝑗=1,2,…,π‘šξ…žβˆ’1.(3.13) We get from (3.12) and (3.13) that dim(π‘†βˆ—π½)βˆ—/(π‘‡βˆ—π½)βˆ—β‰₯π‘šβ€². It can be verified that 𝑆dimβˆ—π½ξ€Έβˆ—ξ€·π‘‡βˆ—π½ξ€Έβˆ—π‘†=dim𝑇(3.14) by Lemma 2.9. So, dim𝑆/𝑇β‰₯π‘šξ…ž=dimπ‘‡βˆ—π½/𝑆. Further, it can be verified that dim𝑆/𝑇β‰₯dimπ‘‡βˆ—π½/𝑆 also holds for π‘šξ…ž=+∞. 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 π‘‡βˆ—π½=(𝑇)βˆ—π½. We get from Theorem 3.1 that if 𝑆 is a 𝐽-SSE of 𝑇, which may not be closed, then it holds that 𝑇dimβˆ—π½π‘‡π‘†=2dim𝑇.(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)dimπ‘‡βˆ—π½/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 π‘‡βˆ—π½=(𝑇)βˆ—π½ 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 Ξ“(𝑇)∢={πœ†βˆˆπ‚βˆΆthereexists𝑐(πœ†)>0suchthatβ€–π‘“βˆ’πœ†π‘₯β€–β‰₯𝑐(πœ†)β€–π‘₯β€–βˆ€(π‘₯,𝑓)βˆˆπ‘‡}(3.16) is called to be the regularity field of 𝑇.
It is evident that Ξ“(𝑇)=Ξ“(𝑇) for a subspace 𝑇.

Lemma 3.7. Let 𝑇 be a subspace in 𝑋, then(1)𝑅(π‘‡βˆ’πœ†)βŸ‚=ker(π‘‡βˆ—βˆ’πœ†) for each πœ†βˆˆπ‚,(2)for each πœ†βˆˆΞ“(𝑇),𝑋=π‘…ξ‚ξ‚€π‘‡π‘‡βˆ’πœ†βŠ•kerβˆ—βˆ’πœ†ξ‚(orthogonalsumin𝑋),(3.17)(3)𝑋=𝑅(π‘‡βˆ—π½βˆ’πœ†) for each πœ†βˆˆΞ“(𝑇).

Proof. (1) Let πœ†βˆˆπ‚. It is clear that 𝑇kerβˆ—βˆ’πœ†ξ‚=π‘₯βˆˆπ‘‹βˆΆπ‘₯,ξ‚πœ†π‘₯βˆˆπ‘‡βˆ—ξ‚‡.(3.18) For every π‘₯∈ker(π‘‡βˆ—βˆ’πœ†), we have from (3.18) that (π‘₯,πœ†π‘₯)βˆˆπ‘‡βˆ—. So, βŸ¨π‘”,π‘₯⟩=βŸ¨π‘¦,πœ†π‘₯⟩ for all (𝑦,𝑔)βˆˆπ‘‡, which implies that βŸ¨π‘”βˆ’πœ†π‘¦,π‘₯⟩=0. Therefore, π‘₯βˆˆπ‘…(π‘‡βˆ’πœ†)βŸ‚, and then ker(π‘‡βˆ—βˆ’πœ†)βŠ‚π‘…(π‘‡βˆ’πœ†)βŸ‚. Conversely, for every π‘₯βˆˆπ‘…(π‘‡βˆ’πœ†)βŸ‚, we have that βŸ¨π‘”βˆ’πœ†π‘¦,π‘₯⟩=0 for all (𝑦,𝑔)βˆˆπ‘‡. It follows that βŸ¨π‘”,π‘₯⟩=βŸ¨π‘¦,πœ†π‘₯⟩ for all (𝑦,𝑔)βˆˆπ‘‡. So, (π‘₯,πœ†π‘₯)βˆˆπ‘‡βˆ—, and hence π‘₯∈ker(π‘‡βˆ—βˆ’πœ†) by (3.18). So, 𝑅(π‘‡βˆ’πœ†)βŸ‚βŠ‚ker(π‘‡βˆ—βˆ’πœ†). Consequently, 𝑅(π‘‡βˆ’πœ†)βŸ‚=ker(π‘‡βˆ—βˆ’πœ†), and result (1) is proved.
(2) By result (1) and Lemma 2.2, we have that 𝑅(π‘‡βˆ’πœ†)βŸ‚=ker((𝑇)βˆ—βˆ’πœ†)=ker(π‘‡βˆ—βˆ’πœ†). So, by the projection theorem, in order to prove (3.17), it suffices to show that 𝑅(π‘‡βˆ’πœ†) with πœ†βˆˆΞ“(𝑇) is closed in 𝑋. It is evident that ξ‚€ξ‚π‘‡βˆ’πœ†βˆ’1=(π‘“βˆ’πœ†π‘₯,π‘₯)∢(π‘₯,𝑓)βˆˆπ‘‡ξ‚‡.(3.19) Let πœ†βˆˆΞ“(𝑇). Since Ξ“(𝑇)=Ξ“(𝑇), one has that πœ†βˆˆΞ“(𝑇), that is, there exists a constant 𝑐(πœ†)>0 such that (β€–π‘“βˆ’πœ†π‘₯β€–β‰₯π‘πœ†)β€–π‘₯β€–βˆ€(π‘₯,𝑓)βˆˆπ‘‡.(3.20) Then we get from (3.19) and (3.20) that (π‘‡βˆ’πœ†)βˆ’1(0)={0}. So, (π‘‡βˆ’πœ†)βˆ’1 determines a linear operator from 𝑅(π‘‡βˆ’πœ†) to 𝑋. Further, the closedness of π‘‡βˆ’πœ† and (3.20) imply that this operator is a closed and bounded operator. So, its domain 𝑅(π‘‡βˆ’πœ†) is closed in 𝑋. Therefore, (3.17) holds, and result (2) is proved.
(3) Let πœ†βˆˆΞ“(𝑇). We first show that there exists a constant 𝑀>0 such that β€–π‘₯β€–β‰€π‘€β€–π‘“βˆ’πœ†π‘₯β€–βˆ€(π‘₯,𝑓)βˆˆπ‘‡βˆ—π½.(3.21) Assume the contrary, then there exists a sequence {π‘“π‘˜βˆ’πœ†π‘₯π‘˜}βˆžπ‘˜=1βŠ‚π‘…(π‘‡βˆ—π½βˆ’πœ†) with β€–π‘“π‘˜βˆ’πœ†π‘₯π‘˜β€–=1 (π‘˜=1,2,…) such that β€–β€–π‘₯π‘˜β€–β€–>π‘˜,π‘˜=1,2,….(3.22) Clearly, (π‘₯π‘˜,π‘“π‘˜)βˆˆπ‘‡βˆ—π½ and (𝑇)βˆ—π½=π‘‡βˆ—π½ imply that (π‘₯π‘˜,π‘“π‘˜)∈(𝑇)βˆ—π½. So, βŸ¨π‘”,𝐽π‘₯π‘˜βŸ©=βŸ¨π‘¦,π½π‘“π‘˜βŸ© for all (𝑦,𝑔)βˆˆπ‘‡ by (1) of Remark 2.5, which, together with βŸ¨πœ†π‘¦,𝐽π‘₯π‘˜βŸ©=βŸ¨π‘¦,𝐽(πœ†π‘₯π‘˜)⟩, implies that for π‘˜=1,2,…, βŸ¨π‘”βˆ’πœ†π‘¦,𝐽π‘₯π‘˜ξ«ξ€·π‘“βŸ©=𝑦,π½π‘˜βˆ’πœ†π‘₯π‘˜ξ€Έξ¬βˆ€(𝑦,𝑔)βˆˆπ‘‡.(3.23) Define πœ™π‘˜(π‘”βˆ’πœ†π‘¦)=βŸ¨π‘”βˆ’πœ†π‘¦,𝐽π‘₯π‘˜βŸ© for (𝑦,𝑔)βˆˆπ‘‡. Then πœ™π‘˜, π‘˜=1,2…, are linear functionals in 𝑅(π‘‡βˆ’πœ†). Since πœ™π‘˜(π‘”βˆ’πœ†π‘¦)=βŸ¨π‘¦,𝐽(π‘“π‘˜βˆ’πœ†π‘₯π‘˜)⟩ by (3.23) and ‖𝐽(π‘“π‘˜βˆ’πœ†π‘₯π‘˜)β€–=β€–π‘“π‘˜βˆ’πœ†π‘₯π‘˜β€–=1, we have that {πœ™π‘˜(π‘”βˆ’πœ†π‘¦)}βˆžπ‘˜=1 is bounded for any given π‘”βˆ’πœ†π‘¦βˆˆπ‘…(π‘‡βˆ’πœ†). Note that 𝑅(π‘‡βˆ’πœ†) with πœ†βˆˆΞ“(𝑇) is closed by the proof of result (2), and hence it is a Hilbert space with the inner product βŸ¨β‹…,β‹…βŸ©. So, by the resonance theorem, {β€–πœ™π‘˜β€–}βˆžπ‘˜=1 is bounded, that is, {‖𝐽π‘₯π‘˜β€–}βˆžπ‘˜=1 is bounded. Since ‖𝐽π‘₯π‘˜β€–=β€–π‘₯π‘˜β€–, π‘˜=1,2,…, we have a contradiction with (3.22). So, (3.21) holds.
Inserting (3.21) into β€–π‘₯β€–+β€–π‘“β€–β‰€β€–π‘“βˆ’πœ†π‘₯β€–+(1+|πœ†|)β€–π‘₯β€–, we get that ξ€Ίξ€·||πœ†||𝑀‖π‘₯β€–+‖𝑓‖≀1+1+β€–π‘“βˆ’πœ†π‘₯β€–.(3.24) It can be easily verified from the closedness of π‘‡βˆ—π½ and (3.24) that 𝑅(π‘‡βˆ—π½βˆ’πœ†) is closed in 𝑋. So, result (1) implies that 𝑇𝑋=π‘…βˆ—π½ξ€Έξ‚€ξ€·π‘‡βˆ’πœ†βŠ•kerβˆ—π½ξ€Έβˆ—βˆ’πœ†ξ‚.(3.25) By Remark 2.10, ker((π‘‡βˆ—π½)βˆ—βˆ’πœ†)={π½π‘¦βˆΆπ‘¦βˆˆker(π‘‡βˆ’πœ†)}, while it can be verified that ker(π‘‡βˆ’πœ†)={0} for πœ†βˆˆΞ“(𝑇)=Ξ“(𝑇). Therefore, (3.25) yields that result (3) holds.

If Ξ“(𝑇)β‰ βˆ…, 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 Ξ“(𝑇)β‰ βˆ…. Let πœ†βˆˆΞ“(𝑇), then π‘‡βˆ—π½=𝑇̇+𝒩,(3.26) where 𝒩=(𝑦,𝑔)βˆˆπ‘‡βˆ—π½ξ‚€π‘‡βˆΆπ‘”βˆ’πœ†π‘¦βˆˆkerβˆ—βˆ’πœ†,(3.27)𝑑(𝑇)=dim𝑅(π‘‡βˆ’πœ†)βŸ‚ξ‚€π‘‡=dimkerβˆ—βˆ’πœ†ξ‚.(3.28)

Proof. We first prove (3.26). Since 𝑇 is 𝐽-Hermitian, one has that 𝑇 is 𝐽-Hermitian, and hence π‘‡βŠ‚(𝑇)βˆ—π½=π‘‡βˆ—π½. Clearly, π’©βŠ‚π‘‡βˆ—π½. So, 𝑇+π’©βŠ‚π‘‡βˆ—π½. On the other hand, let (π‘₯,𝑓)βˆˆπ‘‡βˆ—π½. It follows from (3.17) and πœ†βˆˆΞ“(𝑇) that there exist (𝑦,𝑔)βˆˆπ‘‡ and π‘€βˆˆker(π‘‡βˆ—βˆ’πœ†) such that π‘“βˆ’πœ†π‘₯=π‘”βˆ’πœ†π‘¦+𝑀, that is, (π‘“βˆ’π‘”)βˆ’πœ†(π‘₯βˆ’π‘¦)=𝑀. Let (Μƒπ‘₯,𝑓)=(π‘₯βˆ’π‘¦,π‘“βˆ’π‘”), then (π‘₯,𝑓)=(𝑦,𝑔)+(Μƒπ‘₯,𝑓) and (Μƒπ‘₯,𝑓)βˆˆπ’©. So, π‘‡βˆ—π½βŠ‚π‘‡+𝒩, and consequently, π‘‡βˆ—π½=𝑇+𝒩.
Now, let (𝑒,β„Ž)βˆˆπ‘‡βˆ©π’©, then ξ‚€β„Žβˆ’πœ†π‘’βˆˆπ‘…ξ‚ξ‚€π‘‡π‘‡βˆ’πœ†,β„Žβˆ’πœ†π‘’βˆˆkerβˆ—βˆ’πœ†ξ‚.(3.29) Consequently, β„Žβˆ’πœ†π‘’=0 by 𝑅(π‘‡βˆ’πœ†)βŸ‚=ker(π‘‡βˆ—βˆ’πœ†), which can be obtained from (3.17). Since πœ†βˆˆΞ“(𝑇)=Ξ“(𝑇), one has by Definition 3.6 that there exists a constant 𝑐(πœ†)>0 such that 0=β€–β„Žβˆ’πœ†π‘’β€–β‰₯𝑐(πœ†)‖𝑒‖. It follows that 𝑒=0, which, together with β€–β„Žβˆ’πœ†π‘’β€–=0, implies that β„Ž=0. Then, (𝑒,β„Ž)=(0,0) if (𝑒,β„Ž)βˆˆπ‘‡βˆ©π’©. So, (3.26) holds.
Next, we prove (3.28). Let πœ†βˆˆΞ“(𝑇). Set π‘ˆ1={(π‘₯,πœ†π‘₯)βˆˆπ‘‡βˆ—π½}, then π‘ˆ1 is closed since π‘‡βˆ—π½ is closed. Let π‘ˆ2=𝒩/π‘ˆ1. We will show that dimπ‘ˆ2𝑇=dimkerβˆ—βˆ’πœ†ξ‚.(3.30) Let {πœ“π‘—}π‘šπ‘—=1βŠ‚ker(π‘‡βˆ—βˆ’πœ†) be linearly independent, then, by (3) of Lemma 3.7, there exists (𝑒𝑗,β„Žπ‘—)βˆˆπ‘‡βˆ—π½ such that πœ“π‘—=β„Žπ‘—βˆ’πœ†π‘’π‘—, 1β‰€π‘—β‰€π‘š. It follows from πœ“π‘—βˆˆker(π‘‡βˆ—βˆ’πœ†) that (𝑒𝑗,β„Žπ‘—)βˆˆπ‘ˆ2 for 1β‰€π‘—β‰€π‘š. In addition, if π‘šξ“π‘—=1𝑐𝑗𝑒𝑗,β„Žπ‘—ξ€Έβˆˆπ‘ˆ1,π‘π‘—βˆˆπ‚,(3.31) then βˆ‘π‘šπ‘—=1𝑐𝑗(β„Žπ‘—βˆ’πœ†π‘’π‘—)=0, that is, βˆ‘π‘šπ‘—=1π‘π‘—πœ“π‘—=0. So, 𝑐𝑗=0 for 1β‰€π‘—β‰€π‘š, and hence {(𝑒𝑗,β„Žπ‘—)}π‘šπ‘—=1βŠ‚π‘ˆ2 is linearly independent (modβ€‰π‘ˆ1). Conversely, let {(𝑒𝑗,β„Žπ‘—)}π‘šπ‘—=1βŠ‚π‘ˆ2 be linearly independent (modβ€‰π‘ˆ1), and let πœ“π‘—=β„Žπ‘—βˆ’πœ†π‘’π‘—, then πœ“π‘—βˆˆker(π‘‡βˆ—βˆ’πœ†). If βˆ‘π‘šπ‘—=1π‘π‘—πœ“π‘—=0, then βˆ‘π‘šπ‘—=1𝑐𝑗(𝑒𝑗,β„Žπ‘—)βˆˆπ‘ˆ1. So, 𝑐𝑗=0 (1β‰€π‘—β‰€π‘š), and hence the set {πœ“π‘—}π‘šπ‘—=1 is linearly independent. Based on the above discussions, (3.30) holds. On the other hand, it is evident that dimπ‘₯,ξ‚πœ†π‘₯βˆˆπ‘‡βˆ—ξ‚‡ξ‚€π‘‡=dimkerβˆ—βˆ’πœ†ξ‚.(3.32) Further, by Lemma 2.6, we have that dimπ‘ˆ1=dimπ‘₯,ξ‚πœ†π‘₯βˆˆπ‘‡βˆ—ξ‚‡.(3.33) It follows from (3.30)–(3.33) that dimπ‘ˆ1=dimπ‘ˆ2, and hence (3.26) implies that 1𝑑(𝑇)=2𝑇dim𝒩=dimkerβˆ—βˆ’πœ†ξ‚.(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(π‘‡βˆ’πœ†)βŸ‚ and dimker(π‘‡βˆ—βˆ’πœ†) are constants in Ξ“(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[](π‘₯,𝑓)∢(𝑦,𝑔)=βŸ¨π‘“,π½π‘¦βŸ©βˆ’βŸ¨π‘₯,π½π‘”βŸ©,(π‘₯,𝑓),(𝑦,𝑔)βˆˆπ‘‡βˆ—π½.(4.1) Then, βŸ¨π‘“,π½π‘¦βŸ©=⟨π‘₯,π½π‘”βŸ© if and only if [(π‘₯,𝑓)∢(𝑦,𝑔)]=0. Further, for all π‘Œπ‘—=(π‘₯𝑗,𝑓𝑗)βˆˆπ‘‡βˆ—π½(𝑗=1,2,3) and πœ‡βˆˆπ‚,ξ€Ίπ‘Œ3βˆΆπ‘Œ1+π‘Œ2ξ€»=ξ€Ίπ‘Œ3βˆΆπ‘Œ1ξ€»+ξ€Ίπ‘Œ3βˆΆπ‘Œ2ξ€»,ξ€Ίπ‘Œ1+π‘Œ2βˆΆπ‘Œ3ξ€»=ξ€Ίπ‘Œ1βˆΆπ‘Œ3ξ€»+ξ€Ίπ‘Œ2βˆΆπ‘Œ3ξ€»,ξ€Ίπœ‡π‘Œ1βˆΆπ‘Œ2ξ€»=ξ€Ίπ‘Œ1βˆΆπœ‡π‘Œ2ξ€»ξ€Ίπ‘Œ=πœ‡1βˆΆπ‘Œ2ξ€»,ξ€Ίπ‘Œ1βˆΆπ‘Œ2ξ€»ξ€Ίπ‘Œ=βˆ’2βˆΆπ‘Œ1ξ€».(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 πΎβŠ‚π’― be a subspace, where 𝒯 is given in (3.1). Let πΎβˆ—π½|𝒯 be a restriction of πΎβˆ—π½ to 𝒯, that is,πΎβˆ—π½||𝒯[]={(𝑦,𝑔)βˆˆπ’―βˆΆ(π‘₯,𝑓)∢(𝑦,𝑔)=0βˆ€(π‘₯,𝑓)∈𝐾},(4.3)

then 𝐾 is called to be 𝐽-Hermitian in 𝒯 if πΎβŠ‚πΎβˆ—π½|𝒯, and 𝐾 is called to be 𝐽-self-adjoint in 𝒯 if 𝐾=πΎβˆ—π½|𝒯.

Remark 4.1. From the definition, we have that [(π‘₯,𝑓)∢(𝑦,𝑔)]=0 for all (π‘₯,𝑓)∈𝐾 and (𝑦,𝑔)βˆˆπΎβˆ—π½|𝒯, and 𝐾 is 𝐽-Hermitian in 𝒯 if and only if [(π‘₯,𝑓)∢(𝑦,𝑔)]=0 for all (π‘₯,𝑓), (𝑦,𝑔)∈𝐾.

Lemma 4.2. Let 𝑇 be a closed 𝐽-Hermitian subspace, and let πΎβŠ‚π’― be a subspace. Assume that 𝑆=π‘‡βŠ•πΎ, 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, πΎβŠ‚π‘†, and Remark 4.1. So, the necessity holds. We now prove the sufficiency. Suppose that 𝐾 is 𝐽-Hermitian in 𝒯. For all (π‘₯,𝑓), (𝑦,𝑔)βˆˆπ‘†, we get from 𝑆=π‘‡βŠ•πΎ that ξ€·π‘₯(π‘₯,𝑓)=1,𝑓1ξ€Έ+ξ€·π‘₯2,𝑓2ξ€Έ,ξ€·π‘₯1,𝑓1ξ€Έξ€·π‘₯βˆˆπ‘‡,2,𝑓2ξ€Έ(ξ€·π‘¦βˆˆπΎ,𝑦,𝑔)=1,𝑔1ξ€Έ+𝑦2,𝑔2ξ€Έ,𝑦1,𝑔1ξ€Έξ€·π‘¦βˆˆπ‘‡,2,𝑔2ξ€ΈβˆˆπΎ.(4.4) Since 𝑇 is 𝐽-Hermitian and πΎβŠ‚πΎβˆ—π½|π’―βŠ‚π‘‡βˆ—π½, it can be obtained from (2) of Remark 2.5 and Remark 4.1 that []=π‘₯(π‘₯,𝑓)∢(𝑦,𝑔)ξ€Ίξ€·1,𝑓1ξ€ΈβˆΆξ€·π‘¦1,𝑔1+π‘₯ξ€Έξ€»ξ€Ίξ€·1,𝑓1ξ€ΈβˆΆξ€·π‘¦2,𝑔2+π‘₯ξ€Έξ€»ξ€Ίξ€·2,𝑓2ξ€ΈβˆΆξ€·π‘¦1,𝑔1+π‘₯ξ€Έξ€»ξ€Ίξ€·2,𝑓2ξ€ΈβˆΆξ€·π‘¦2,𝑔2ξ€Έξ€»=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, πΎβŠ‚πΎβˆ—π½|𝒯, by result (1). It suffices to show that πΎβˆ—π½|π’―βŠ‚πΎ. For any given (π‘₯,𝑓)βˆˆπ‘†, there exist (π‘₯1,𝑓1)βˆˆπ‘‡ and (π‘₯2,𝑓2)∈𝐾 such that the first relation of (4.4) holds. Let (𝑦,𝑔)βˆˆπΎβˆ—π½|𝒯, then [(π‘₯2,𝑓2)∢(𝑦,𝑔)]=0 by Remark 4.1. Note that (𝑦,𝑔)βˆˆπ‘‡βˆ—π½ by πΎβˆ—π½|π’―βŠ‚π‘‡βˆ—π½. Then [(π‘₯1,𝑓1)∢(𝑦,𝑔)]=0 by (2) of Remark 2.5. Therefore, the first relation of (4.4) yields that []=π‘₯(π‘₯,𝑓)∢(𝑦,𝑔)ξ€Ίξ€·1,𝑓1ξ€ΈβˆΆξ€»+π‘₯(𝑦,𝑔)ξ€Ίξ€·2,𝑓2ξ€ΈβˆΆξ€»(𝑦,𝑔)=0βˆ€(π‘₯,𝑓)βˆˆπ‘†.(4.6) So, (𝑦,𝑔)βˆˆπ‘†βˆ—π½. Therefore, 𝑆=π‘†βˆ—π½ yields that (𝑦,𝑔)βˆˆπ‘†, which, together with (𝑦,𝑔)βˆˆπΎβˆ—π½|𝒯, πΎβˆ—π½|π’―βˆ©π‘‡={0}, and 𝑆=π‘‡βŠ•πΎ, implies that (𝑦,𝑔)∈𝐾. Hence, πΎβˆ—π½|π’―βŠ‚πΎ, and hence 𝐾=πΎβˆ—π½|𝒯. The necessity holds.
Next, consider the sufficiency. Suppose that 𝐾 is 𝐽-self-adjoint in 𝒯. By result (1), one has that π‘†βŠ‚π‘†βˆ—π½. It suffices to show that π‘†βˆ—π½βŠ‚π‘†. Let (𝑦,𝑔)βˆˆπ‘†βˆ—π½, then (𝑦,𝑔)βˆˆπ‘‡βˆ—π½ since π‘†βˆ—π½βŠ‚π‘‡βˆ—π½ by π‘‡βŠ‚π‘†. It follows from (3.1) that there exist (𝑦1,𝑔1)βˆˆπ‘‡ and (𝑦2,𝑔2)βˆˆπ’― such that 𝑦(𝑦,𝑔)=1,𝑔1ξ€Έ+𝑦2,𝑔2ξ€Έ.(4.7) We claim that (𝑦2,𝑔2)∈𝐾. In fact, since (𝑦,𝑔)βˆˆπ‘†βˆ—π½, we have [](π‘₯,𝑓)∢(𝑦,𝑔)=0βˆ€(π‘₯,𝑓)βˆˆπ‘†.(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 [(π‘₯2,𝑓2)∢(𝑦2,𝑔2)]=0 for all (π‘₯2,𝑓2)∈𝐾. Then, (𝑦2,𝑔2)βˆˆπΎβˆ—π½|𝒯=𝐾. It follows from (𝑦2,𝑔2)∈𝐾, 𝑆=π‘‡βŠ•πΎ, and (4.7) that (𝑦,𝑔)βˆˆπ‘†. So, π‘†βˆ—π½βŠ‚π‘†, and hence 𝑆=π‘†βˆ—π½. 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 π‘‡β‰ π‘‡βˆ—π½. 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 π‘‡β‰ π‘‡βˆ—π½, one has that 𝒯≠{0}. Choose 0β‰ (π‘₯0,𝑓0)βˆˆπ’― and set 𝐾0=span{(π‘₯0,𝑓0)}. Then 𝐾0 is 𝐽-Hermitian in 𝒯 since [(π‘₯0,𝑓0)∢(π‘₯0,𝑓0)]=0. Let 𝒦 be the set of all the 𝐽-Hermitian subspaces in 𝒯 which contain 𝐾0, then 𝒦 is not empty since 𝐾0βˆˆπ’¦. Further, let 𝒦 be ordered by extension, that is, 𝐴<𝐡 if and only if π΄βŠ‚π΅, and let β„³={𝐾𝛼} be an arbitrary totally ordered subset of 𝒦. Set ⋃𝐾=𝛼𝐾𝛼. Then, it can be verified that 𝐾 is 𝐽-Hermitian in 𝒯 by Remark 4.1 and the fact that all the elements of β„³ are 𝐽-Hermitian in 𝒯. So, 𝐾 is an upper bound of β„³ 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=πΎβˆ—π½|𝒯. Suppose that πΎβ‰ πΎβˆ—π½|𝒯 on the contrary. Note that πΎβŠ‚πΎβˆ—π½|𝒯. Choose (Μƒπ‘₯0,𝑓0)βˆˆπΎβˆ—π½|𝒯 satisfying (Μƒπ‘₯0,𝑓0)βˆ‰πΎ, and set ̇𝐾=𝐾+span{(Μƒπ‘₯0,𝑓0)}. It can be verified by Remark 4.1 and the fact that 𝐾 is 𝐽-Hermitian in 𝒯 that [(π‘₯,𝑓)∢(𝑦,𝑔)]=0 holds for all 𝐾(π‘₯,𝑓),(𝑦,𝑔)∈. Note that ξ‚πΎβŠ‚π’―. Then, 𝐾 is 𝐽-Hermitian in 𝒯, which contradicts the maximality of 𝐾. Hence, 𝐾=πΎβˆ—π½|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 π‘‡βŠ‚π‘†βŠ‚π‘‡βˆ—π½, and there exists {(π‘₯𝑗,𝑓𝑗)}𝑑𝑗=1βŠ‚π‘‡βˆ—π½ such that(1)(π‘₯1,𝑓1),(π‘₯2,𝑓2),…,(π‘₯𝑑,𝑓𝑑) are linearly independent (mod𝑇),(2)[(π‘₯𝑠,𝑓𝑠)∢(π‘₯𝑗,𝑓𝑗)]=0 for 𝑠,𝑗=1,2,…,𝑑,(3)𝑆={(𝑦,𝑔)βˆˆπ‘‡βˆ—π½βˆΆ[(𝑦,𝑔)∢(π‘₯𝑗,𝑓𝑗)]=0,𝑗=1,2,…,𝑑}.

Proof. First, consider the necessity. Suppose that 𝑆 is a 𝐽-SSE of 𝑇, then it holds that π‘‡βŠ‚π‘†βŠ‚π‘‡βˆ—π½ since π‘†βˆ—π½βŠ‚π‘‡βˆ—π½ and 𝑆=π‘†βˆ—π½. We also have that (3.2) holds and 𝐾𝑆,𝑇 in (3.2) is 𝐽-self-adjoint in 𝒯 by Lemma 4.2. Note that dim𝑆/𝑇=𝑑 by Theorem 3.1. Then dim𝐾𝑆,𝑇=𝑑, and let {(π‘₯𝑗,𝑓𝑗)}𝑑𝑗=1 be a basis of 𝐾𝑆,𝑇, then we get from (3.2) that (1) holds. In addition, since 𝐾𝑆,𝑇 is 𝐽-self-adjoint in 𝒯, one has that (2) holds by Remark 4.1. For convenience, set𝐷=(𝑦,𝑔)βˆˆπ‘‡βˆ—π½βˆΆξ€Ίξ€·π‘₯(𝑦,𝑔)βˆΆπ‘—,𝑓𝑗.ξ€Έξ€»=0,𝑗=1,2,…,𝑑(4.9) Now, we prove π‘‡βŠ•πΎπ‘†,𝑇=𝐷, that is, 𝑆=𝐷. Let (𝑦,𝑔)βˆˆπ‘‡βŠ•πΎπ‘†,𝑇, then (𝑦,𝑔)βˆˆπ‘‡βˆ—π½ by π‘†βŠ‚π‘‡βˆ—π½, and there exist (̃𝑦,̃𝑔)βˆˆπ‘‡ and π‘π‘—βˆˆπΆ such that (𝑦,𝑔)=(̃𝑦,̃𝑔)+𝑑𝑗=1𝑐𝑗π‘₯𝑗,𝑓𝑗.(4.10) Inserting (4.10) into [(𝑦,𝑔)∢(π‘₯𝑠,𝑓𝑠)] and using (2) of Remark 2.5 and (2) of this theorem, we get that [(𝑦,𝑔)∢(π‘₯𝑠,𝑓𝑠)]=0 for 𝑠=1,2,…,𝑑. So, (𝑦,𝑔)∈𝐷, and hence π‘‡βŠ•πΎπ‘†,π‘‡βŠ‚π·. Conversely, suppose that (𝑦,𝑔)∈𝐷, then by (3.1), there exist (𝑦1,𝑔1)βˆˆπ‘‡ and (𝑦2,𝑔2)βˆˆπ’― such that (4.7) holds. The definition of 𝐷, (4.7), and (2) of Remark 2.5 implies that for 𝑠=1,2,…,𝑑, 𝑦2,𝑔2ξ€ΈβˆΆξ€·π‘₯𝑠,𝑓𝑠=ξ€Ίξ€·π‘₯ξ€Έξ€»(𝑦,𝑔)βˆΆπ‘ ,π‘“π‘ βˆ’π‘¦ξ€Έξ€»ξ€Ίξ€·1,𝑔1ξ€ΈβˆΆξ€·π‘₯𝑠,𝑓𝑠=0.(4.11) So, (𝑦2,𝑔2)∈(𝐾𝑆,𝑇)βˆ—π½|𝒯, which implies that (𝑦2,𝑔2)βˆˆπΎπ‘†,𝑇 since 𝐾𝑆,𝑇 is 𝐽-self-adjoint in 𝒯. One has from (4.7) that (𝑦,𝑔)βˆˆπ‘‡βŠ•πΎπ‘†,𝑇, and consequently, π·βŠ‚π‘‡βŠ•πΎπ‘†,𝑇. Hence, π‘‡βŠ•πΎπ‘†,𝑇=𝐷, that is, 𝑆=𝐷. The necessity holds.
Next, consider the sufficiency. Suppose that there exists {(π‘₯𝑗,𝑓𝑗)}𝑑𝑗=1βŠ‚π‘‡βˆ—π½ such that conditions (1) and (2) hold and 𝑆 is given in condition (3). From (3.1), we have ξ€·π‘₯𝑗,𝑓𝑗=ξ€·π‘₯𝑗0,𝑓𝑗0ξ€Έ+ξ‚€Μƒπ‘₯𝑗,𝑓𝑗,ξ€·π‘₯𝑗0,𝑓𝑗0ξ€Έξ‚€βˆˆπ‘‡,Μƒπ‘₯𝑗,ξ‚π‘“π‘—ξ‚βˆˆπ’―,𝑗=1,2,…,𝑑.(4.12) It can be easily verified that the set {(Μƒπ‘₯𝑗,𝑓𝑗)}𝑑𝑗=1 satisfies conditions (1) and (2). Let 𝐾=span{(Μƒπ‘₯1,𝑓1),(Μƒπ‘₯2,𝑓2),…,(Μƒπ‘₯𝑑,𝑓𝑑)}, then 𝐾 is 𝐽-Hermitian in 𝒯 since {(Μƒπ‘₯𝑗,𝑓𝑗)}𝑑𝑗=1 satisfies condition (2) and ξ‚πΎβŠ‚π’―. By the proof of Theorem 4.3, there exists 𝐾1βŠ‚π’― such that 𝐾1 is 𝐽-self-adjoint in 𝒯 and ξ‚πΎβŠ‚πΎ1. Then, by Lemma 4.2, π‘‡βŠ•πΎ1 is a 𝐽-SSE of 𝑇, which, together with Theorem 3.1, yields that 𝑑=dimπ‘‡βŠ•πΎ1𝑇=dim𝐾1β‰₯𝑑.(4.13) Therefore, ξ‚πΎβŠ‚πΎ1 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 π‘‡βŠ•πΎπ‘†,𝑇=𝐷, we have ξ‚ξ‚π‘‡βŠ•πΎ=𝐷,(4.14) where 𝐷=(𝑦,𝑔)βˆˆπ‘‡βˆ—π½βˆΆξ‚ƒξ‚€(𝑦,𝑔)βˆΆΜƒπ‘₯𝑗,𝑓𝑗.=0,𝑗=1,2,…,𝑑(4.15) On the other hand, it can be easily verified that 𝐷=𝑆. So, 𝐾𝑆=π‘‡βŠ•, 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).

References

  1. D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, The Clarendon Press Oxford University Press, New York, NY, USA, 1987.
  2. W. N. Everitt and L. Markus, Boundary Value Problems and Symplectic Algebra for Ordinary Differential and Quasi-Differential Operators, vol. 61, American Mathematical Society, Providence, RI, USA, 1999.
  3. J. Weidmann, Spectral Theory of Ordinary Differential Operators, vol. 1258 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1987.
  4. S. Z. Fu, β€œOn the self-adjoint extensions of symmetric ordinary differential operators in direct sum spaces,” Journal of Differential Equations, vol. 100, no. 2, pp. 269–291, 1992. View at Publisher Β· View at Google Scholar
  5. J. Sun, β€œOn the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices,” Acta Mathematica Sinica, vol. 2, no. 2, pp. 152–167, 1986. View at Publisher Β· View at Google Scholar
  6. A. Wang, J. Sun, and A. Zettl, β€œCharacterization of domains of self-adjoint ordinary differential operators,” Journal of Differential Equations, vol. 246, no. 4, pp. 1600–1622, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. M. Lesch and M. Malamud, β€œOn the deficiency indices and self-adjointness of symmetric Hamiltonian systems,” Journal of Differential Equations, vol. 189, no. 2, pp. 556–615, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. Y. Shi and H. Sun, β€œSelf-adjoint extensions for second-order symmetric linear difference equations,” Linear Algebra and its Applications, vol. 434, no. 4, pp. 903–930, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. R. Arens, β€œOperational calculus of linear relations,” Pacific Journal of Mathematics, vol. 11, pp. 9–23, 1961. View at Zentralblatt MATH
  10. H. Langer and B. Textorius, β€œOn generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space,” Pacific Journal of Mathematics, vol. 72, no. 1, pp. 135–165, 1977.
  11. E. A. Coddington, Extension Theory of Formally Normal and Symmetric Subspaces, Memoirs of the American Mathematical Society, No. 134, American Mathematical Society, Providence, RI, USA, 1973.
  12. Y. Shi, β€œThe Glazman-Krein-Naimark theory for Hermitian subspaces,” The Journal of Operator Theory. In press.
  13. E. A. Coddington, β€œSelf-adjoint subspace extensions of nondensely defined symmetric operators,” Advances in Mathematics, vol. 14, pp. 309–332, 1974. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  14. E. A. Coddington and A. Dijksma, β€œSelf-adjoint subspaces and eigenfunction expansions for ordinary differential subspaces,” Journal of Differential Equations, vol. 20, no. 2, pp. 473–526, 1976. View at Publisher Β· View at Google Scholar
  15. A. Dijksma and H. S. V. de Snoo, β€œSelf-adjoint extensions of symmetric subspaces,” Pacific Journal of Mathematics, vol. 54, pp. 71–100, 1974. View at Zentralblatt MATH
  16. B. M. Brown, D. K. R. McCormack, W. D. Evans, and M. Plum, β€œOn the spectrum of second-order differential operators with complex coefficients,” Proceedings of The Royal Society of London Series A, vol. 455, no. 1984, pp. 1235–1257, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  17. B. M. Brown and M. Marletta, β€œSpectral inclusion and spectral exactness for singular non-self-adjoint Sturm-Liouville problems,” Proceedings of The Royal Society of London Series A, vol. 457, no. 2005, pp. 117–139, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  18. J. Qi, Z. Zheng, and H. Sun, β€œClassification of Sturm-Liouville differential equations with complex coefficients and operator realizations,” Proceedings of The Royal Society of London Series A, vol. 467, no. 2131, pp. 1835–1850, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  19. A. R. Sims, β€œSecondary conditions for linear differential operators of the second order,” Journal of Mathematics and Mechanics, vol. 6, pp. 247–285, 1957. View at Zentralblatt MATH
  20. H. Sun and J. Qi, β€œOn classification of second-order differential equations with complex coefficients,” Journal of Mathematical Analysis and Applications, vol. 372, no. 2, pp. 585–597, 2010. View at Publisher Β· View at Google Scholar
  21. H. Sun, J. Qi, and H. Jing, β€œClassification of non-self-adjoint singular Sturm-Liouville difference equations,” Applied Mathematics and Computation, vol. 217, no. 20, pp. 8020–8030, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  22. R. H. Wilson, β€œNon-self-adjoint difference operators and their spectrum,” Proceedings of The Royal Society of London Series A, vol. 461, no. 2057, pp. 1505–1531, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  23. I. M. Glazman, β€œAn analogue of the extension theory of Hermitian operators and a non-symmetric one-dimensional boundary problem on a half-axis,” Doklady Akademii Nauk SSSR, vol. 115, pp. 214–216, 1957 (Russian). View at Zentralblatt MATH
  24. D. Race, β€œThe theory of J-self-adjoint extensions of J-symmetric operators,” Journal of Differential Equations, vol. 57, pp. 258–274, 1985.
  25. Z. J. Shang, β€œOn J-self-adjoint extensions of J-symmetric ordinary differential operators,” Journal of Differential Equations, vol. 73, no. 1, pp. 153–177, 1988. View at Publisher Β· View at Google Scholar
  26. I. Knowles, β€œOn the boundary conditions characterizing J-self-adjoint extensions of J-symmetric operators,” Journal of Differential Equations, vol. 40, no. 2, pp. 193–216, 1981. View at Publisher Β· View at Google Scholar
  27. J. L. Liu, β€œJ self-adjoint extensions of J symmetric operators,” Acta Scientiarum Naturalium Universitatis Intramongolicae, vol. 23, no. 3, pp. 312–316, 1992.