# The Theory forJ-Hermitian Subspaces in  a Product Space

Authors: Huaqing Sun; Jiangang Qi
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

This paper is concerned with the theory forJ-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.


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## Body

## 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 ofJ-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 ofJ-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 Section2, 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.LetX 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,(2.1)α(x,f)=(αx,αf),(x,f)+(y,g)=(x+y,f+g).
The null element of X2 is (0,0). The inner product in X2 is defined by(2.2)〈(x,f),(y,g)〉*=〈x,y〉+〈f,g〉,(x,f),(y,g)∈X2,
and ∥·∥* denotes the induced norm.LetT 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:(2.3)D(T)={x∈X:(x,f)∈Tforsomef∈X},R(T)={f∈X:(x,f)∈Tforsomex∈X},T(x)={f∈X:(x,f)∈T},ker⁡(T)={x∈X:(x,0)∈T},T-1={(f,x):(x,f)∈T},T-λ={(x,f-λx):(x,f)∈T}.
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]).
LetT be a subspace in X2.(1)
Its adjoint,T*, is defined by
(2.4)T*={(y,g)∈X2:〈f,y〉=〈x,g〉∀(x,f)∈T}.(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]).
LetT 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 operatorJ defined on X is said to be a conjugation operator if for all x,y∈X,
(2.5)〈Jx,Jy〉=〈y,x〉,J2x=x.It can be verified thatJ is a conjugate linear, that is, J(x+y)=Jx+Jy and J(λx)=λ¯Jx for x,y∈X and λ∈C, and norm-preserving bijection on X satisfying(2.6)〈Jx,y〉=〈Jy,x〉∀x,y∈X.
For example, the complex conjugation x↦x¯ in any L2 space is a conjugation operator on L2.Definition 2.4.
LetT be a subspace in X2, and let J be a conjugation operator.(1)
ItsJ-adjoint, TJ*, is defined by
(2.7)TJ*={(y,g)∈X2:〈f,Jy〉=〈x,Jg〉∀(x,f)∈T}.(2)
T is said to be a J-Hermitian subspace if T⊂TJ*.(3)
T is said to be a J-self-adjoint subspace if T=TJ*.Remark 2.5.
(1)  It can be easily verified thatTJ* is a closed subspace. Consequently, a J-self-adjoint subspace T is a closed subspace since T=TJ*. In addition, SJ*⊂TJ* if T⊂S.
(2) From the definition, we have that〈f,Jy〉=〈x,Jg〉 for all (x,f)∈T and (y,g)∈TJ*, and that T is a J-Hermitian subspace if and only if
(2.8)〈f,Jy〉=〈x,Jg〉∀(x,f),(y,g)∈T.
(3) The concepts ofJ-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.
LetT be a subspace in X2, then(1)
T*={(Jy,Jg):(y,g)∈TJ*},(2)
TJ*={(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)∈TJ*}. 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)∈TJ*. Then (y,g)=(J2y,J2g)∈D1, which implies that T*⊂D1. Conversely, let (y,g)∈D1, then there exists (ỹ,g̃)∈TJ* such that (y,g)=(Jỹ,Jg̃). It follows from (ỹ,g̃)∈TJ* that 〈f,Jỹ〉=〈x,Jg̃〉 for all (x,f)∈T, which implies that (Jỹ,Jg̃)∈T*, that is, (y,g)∈T*. So, D1⊂T*. Consequently, T*=D1, and result (1) holds.Remark 2.7.
LetT be a subspace in X2, then from Lemmas 2.2 and 2.6, and the closedness of TJ*, one has that TJ*=(T¯)J*, and T¯ is J-Hermitian if T is J-Hermitian.Lemma 2.8.
LetT be a closed J-Hermitian subspace, then (y,g)∈T if and only if (y,g)∈TJ* and 〈f,Jy〉=〈x,Jg〉 for all (x,f)∈TJ*.Proof.
LetT be a closed J-Hermitian subspace. Clearly, the necessity holds by (2) of Remark 2.5. Now, consider the sufficiency. Suppose that (y,g)∈TJ* and 〈f,Jy〉=〈x,Jg〉 for all (x,f)∈TJ*, then we get from (2.6) that 〈y,Jf〉=〈g,Jx〉 for all (x,f)∈TJ*. This, together with (1) of Lemma 2.6, implies that 〈y,f̃〉=〈g,x̃〉 for all (x̃,f̃)∈T*. So, (y,g)∈T**, and hence, (y,g)∈T by Lemma 2.2. So, the sufficiency holds.Lemma 2.9.
LetT be a closed J-Hermitian subspace in X2, then(1)
(TJ*)*={(Jy,Jg):(y,g)∈T},(2)
T={(Jy,Jg):(y,g)∈(TJ*)*}.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 (ỹ,g̃)∈T such that (y,g)=(Jỹ,Jg̃). By Lemma 2.8, (ỹ,g̃)∈T implies that 〈f,Jỹ〉=〈x,Jg̃〉 for all (x,f)∈TJ*. Then (Jỹ,Jg̃)∈(TJ*)*, that is, (y,g)∈(TJ*)*. So, D⊂(TJ*)*. Conversely, let (y,g)∈(TJ*)*, then 〈f,y〉=〈x,g〉 for all (x,f)∈TJ*, that is,
(2.9)〈f,J(Jy)〉=〈x,J(Jg)〉∀(x,f)∈TJ*.
Clearly, (2.9) holds for all (x,f)∈T since T⊂TJ*. So, (Jy,Jg)∈TJ*. 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, (TJ*)*⊂D and then (TJ*)*=D.Remark 2.10.
SinceTJ*=(T¯)J* by Remark 2.7, Lemma 2.9 yields that (TJ*)*={(Jy,Jg):(y,g)∈T¯} and T¯={(Jy,Jg):(y,g)∈(TJ*)*} for a J-Hermitian subspace T which may not be closed.


## 3. Defect Index of aJ-Hermitian Subspace

In this section, the definition of the defect index of aJ-Hermitian subspace is introduced, and a formula for the defect index is obtained.LetT be a closed J-Hermitian subspace. It has been known that TJ* is a closed subspace by (1) of Remark 2.5. Then the closedness of T and TJ* and T⊂TJ* gives that(3.1)TJ*=T⊕T,
where 𝒯 denotes the orthogonal complement of T in TJ*, that is, 𝒯=TJ*⊝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 KS,T=S⊝T such that(3.2)S=T⊕KS,T.
Clearly, S⊂TJ* since SJ*⊂TJ* and S⊂SJ*. Then by (3.1) and (3.2), KS,T can be expressed as(3.3)KS,T={(y,g)∈T:thereexist(y1,g1)∈T,(y2,g2)∈S,suchthat(y2,g2)=(y1,g1)+(y,g)}.
Further, we have the following result.Theorem 3.1.
LetT 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
(3.4)dim⁡TJ*S=dim⁡ST.Proof.
IfT is a J-self-adjoint subspace, then T=TJ* 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⊂TJ* and T≠TJ*. It follows that (3.2) holds with KS,T≠{0}. Let dimS/T=m, then (3.2) yields that dimKS,T=m. In the case of m<∞, let {(xj,fj)}j=1m be a basis of KS,T, then
(3.5)S=T⊕span⁡{(x1,f1),(x2,f2),…,(xm,fm)}.
Define
(3.6)Tj=T⊕span⁡{(x1,f1),(x2,f2),…,(xj,fj)},j=1,2,…,m.
Clearly, T≠Tj≠Tj+1 for j=1,2,…,m-1 and
(3.7)S=SJ*=(Tm)J*⊂(Tm-1)J*⊂⋯⊂(T2)J*⊂(T1)J*⊂TJ*,
since T⊂T1⊂T2⊂⋯⊂Tm=S and S is J-self-adjoint. Further, Tj (j=1,2,…,m) is a closed subspace since T is closed. It holds that
(3.8)T*≠Tj*≠Tj+1*,j=1,2,…,m-1.
Otherwise, for example, suppose that T1*=T2*, then by Lemma 2.2, we have T1=T1**=T2**=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
(3.9)TJ*≠(Tj)J*≠(Tj+1)J*,j=1,2,…,m-1.
We get from (3.7) and (3.9) that dimTJ*/S≥m=dimS/T.
In the case ofm=+∞, 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 dimTJ*/S=+∞=dimS/T.
Next, we prove thatdimS/T≥dimTJ*/S. Since TJ* and S are closed subspaces, there exists uniquely a closed subspace KTJ*,S=TJ*⊝S such that TJ*=S⊕KTJ*,S. Set dimTJ*/S=m′. Then dimKTJ*,S=m′. If m′<∞, let {(yj,gj)}j=1m′ be a basis of KTJ*,S, then(3.10)TJ*=S⊕span⁡{(y1,g1),(y2,g2),…,(ym′,gm′)}.Define
(3.11)Sj=S⊕span⁡{(y1,g1),(y2,g2),…,(yj,gj)},j=1,2,…,m′.
Clearly, it holds that SJ*=S≠Sj≠Sj+1 for j=1,2,…,m′-1 and
(3.12)(TJ*)*=Sm′*⊂Sm′-1*⊂⋯⊂S2*⊂S1*⊂S*=(SJ*)*,
since S⊂S1⊂S2⊂⋯⊂Sm′=TJ* and S is J-self-adjoint. Further, Sj (j=1,2,…,m′) is a closed subspace since S is closed. Similarly, it holds that
(3.13)(SJ*)*≠Sj*≠Sj+1*,j=1,2,…,m′-1.
We get from (3.12) and (3.13) that dim(SJ*)*/(TJ*)*≥m′. It can be verified that
(3.14)dim⁡(SJ*)*(TJ*)*=dim⁡ST
by Lemma 2.9. So, dimS/T≥m′=dimTJ*/S. Further, it can be verified that dimS/T≥dimTJ*/S also holds for m′=+∞. Based on the above discussions, (3.4) holds.Remark 3.2.
(1) From Theorem3.1 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 forJ-symmetric operators was established in [24, Theorem 3.1].Remark 3.3.
Note thatTJ*=(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
(3.15)dim⁡TJ*T¯=2dim⁡ST¯.
Now, we give the concept of the defect index of aJ-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.
LetT be a J-Hermitian subspace, then d(T)=(1/2)dimTJ*/T¯ is called to be the defect index of T.Remark 3.5.
(1) It will be proved that everyJ-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) SinceTJ*=(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]).
LetT be a subspace in X2. The set
(3.16)Γ(T):={λ∈C:thereexistsc(λ)>0suchthat‖f-λx‖≥c(λ)‖x‖∀(x,f)∈T}
is called to be the regularity field of T.
It is evident thatΓ(T)=Γ(T¯) for a subspace T.Lemma 3.7.
LetT be a subspace in X, then(1)
R(T-λ)⊥=ker(T*-λ¯) for each λ∈C,(2)
for eachλ∈Γ(T),(3.17)X=R(T¯-λ)⊕ker⁡(T*-λ¯)(orthogonal sum inX),(3)
X=R(TJ*-λ) for each λ∈Γ(T).Proof.
(1) Letλ∈C. It is clear that
(3.18)ker⁡(T*-λ¯)={x∈X:(x,λ¯x)∈T*}.
For every x∈ker(T*-λ¯), we have from (3.18) that (x,λ¯x)∈T*. So, 〈g,x〉=〈y,λ¯x〉 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,λ¯x〉 for all (y,g)∈T. So, (x,λ¯x)∈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 Lemma2.2, 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
(3.19)(T¯-λ)-1={(f-λx,x):(x,f)∈T¯}.
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
(3.20)‖f-λx‖≥c(λ)‖x‖∀(x,f)∈T¯.
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) Letλ∈Γ(T). We first show that there exists a constant M>0 such that
(3.21)‖x‖≤M‖f-λx‖∀(x,f)∈TJ*.
Assume the contrary, then there exists a sequence {fk-λxk}k=1∞⊂R(TJ*-λ) with ∥fk-λxk∥=1 (k=1,2,…) such that
(3.22)‖xk‖>k,k=1,2,….
Clearly, (xk,fk)∈TJ* and (T¯)J*=TJ* imply that (xk,fk)∈(T¯)J*. So, 〈g,Jxk〉=〈y,Jfk〉 for all (y,g)∈T¯ by (1) of Remark 2.5, which, together with 〈λy,Jxk〉=〈y,J(λxk)〉, implies that for k=1,2,…,
(3.23)〈g-λy,Jxk〉=〈y,J(fk-λxk)〉∀(y,g)∈T¯.
Define ϕk(g-λy)=〈g-λy,Jxk〉 for (y,g)∈T¯. Then ϕk, k=1,2…, are linear functionals in R(T¯-λ). Since ϕk(g-λy)=〈y,J(fk-λxk)〉 by (3.23) and ∥J(fk-λxk)∥=∥fk-λxk∥=1, we have that {ϕk(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, {∥ϕk∥}k=1∞ is bounded, that is, {∥Jxk∥}k=1∞ is bounded. Since ∥Jxk∥=∥xk∥, 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
(3.24)‖x‖+‖f‖≤[1+(1+|λ|)M]‖f-λx‖.
It can be easily verified from the closedness of TJ* and (3.24) that R(TJ*-λ) is closed in X. So, result (1) implies that
(3.25)X=R(TJ*-λ)⊕ker⁡((TJ*)*-λ¯).
By Remark 2.10, ker((TJ*)*-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.IfΓ(T)≠∅, we have the following results which give a formula for the defect index of a J-Hermitian space.Theorem 3.8.
Assume thatT is a J-Hermitian subspace with Γ(T)≠∅. Let λ∈Γ(T), then
(3.26)TJ*=T¯+̇N,
where
(3.27)N={(y,g)∈TJ*:g-λy∈ker⁡(T*-λ¯)},(3.28)d(T)=dim⁡R(T-λ)⊥=dim⁡ker⁡(T*-λ¯).Proof.
We first prove (3.26). Since T is J-Hermitian, one has that T¯ is J-Hermitian, and hence T¯⊂(T¯)J*=TJ*. Clearly, 𝒩⊂TJ*. So, T¯+𝒩⊂TJ*. On the other hand, let (x,f)∈TJ*. 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,f̃)=(x-y,f-g), then (x,f)=(y,g)+(x̃,f̃) and (x̃,f̃)∈𝒩. So, TJ*⊂T¯+𝒩, and consequently, TJ*=T¯+𝒩.
Now, let(u,h)∈T¯∩𝒩, then
(3.29)h-λu∈R(T¯-λ),h-λu∈ker⁡(T*-λ¯).
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)∈TJ*}, then U1 is closed since TJ* is closed. Let U2=𝒩/U1. We will show that
(3.30)dim⁡U2=dim⁡ker⁡(T*-λ¯).
Let {ψj}j=1m⊂ker(T*-λ¯) be linearly independent, then, by (3) of Lemma 3.7, there exists (uj,hj)∈TJ* such that ψj=hj-λuj,  1≤j≤m. It follows from ψj∈ker(T*-λ¯) that (uj,hj)∈U2 for 1≤j≤m. In addition, if
(3.31)∑j=1mcj(uj,hj)∈U1,cj∈C,
then ∑j=1mcj(hj-λuj)=0, that is, ∑j=1mcjψj=0. So, cj=0 for 1≤j≤m, and hence {(uj,hj)}j=1m⊂U2 is linearly independent (mod U1). Conversely, let {(uj,hj)}j=1m⊂U2 be linearly independent (mod U1), and let ψj=hj-λuj, then ψj∈ker(T*-λ¯). If ∑j=1mcjψj=0, then ∑j=1mcj(uj,hj)∈U1. So, cj=0 (1≤j≤m), and hence the set {ψj}j=1m is linearly independent. Based on the above discussions, (3.30) holds. On the other hand, it is evident that
(3.32)dim⁡{(x,λ¯x)∈T*}=dim⁡ker⁡(T*-λ¯).
Further, by Lemma 2.6, we have that
(3.33)dim⁡U1=dim⁡{(x,λ¯x)∈T*}.
It follows from (3.30)–(3.33) that dimU1=dimU2, and hence (3.26) implies that
(3.34)d(T)=12dim⁡N=dim⁡ker⁡(T*-λ¯).
So, (3.28) holds by (1) of Lemma 3.7.Remark 3.9.
From Theorem3.8, 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 ofJ-SSEs of a J-Hermitian space and the characterizations of all the J-SSEs.Define the form[:] as(4.1)[(x,f):(y,g)]=〈f,Jy〉-〈x,Jg〉,(x,f),(y,g)∈TJ*.
Then, 〈f,Jy〉=〈x,Jg〉 if and only if [(x,f):(y,g)]=0. Further, for all Yj=(xj,fj)∈TJ*(j=1,2,3) and μ∈C,(4.2)[Y3:Y1+Y2]=[Y3:Y1]+[Y3:Y2],[Y1+Y2:Y3]=[Y1:Y3]+[Y2:Y3],[μY1:Y2]=[Y1:μY2]=μ[Y1:Y2],[Y1:Y2]=-[Y2:Y1].
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 KJ*|𝒯 be a restriction of KJ* to 𝒯, that is,(4.3)KJ*|T={(y,g)∈T:[(x,f):(y,g)]=0∀(x,f)∈K},thenK is called to be J-Hermitian in 𝒯 if K⊂KJ*|𝒯, and K is called to be J-self-adjoint in 𝒯 if K=KJ*|𝒯.Remark 4.1.
From the definition, we have that[(x,f):(y,g)]=0 for all (x,f)∈K and (y,g)∈KJ*|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.
LetT 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 thatS 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
(4.4)(x,f)=(x1,f1)+(x2,f2),(x1,f1)∈T,(x2,f2)∈K,(y,g)=(y1,g1)+(y2,g2),(y1,g1)∈T,(y2,g2)∈K.
Since T is J-Hermitian and K⊂KJ*|𝒯⊂TJ*, it can be obtained from (2) of Remark 2.5 and Remark 4.1 that
(4.5)[(x,f):(y,g)]=[(x1,f1):(y1,g1)]+[(x1,f1):(y2,g2)]+[(x2,f2):(y1,g1)]+[(x2,f2):(y2,g2)]=0.
So, S is J-Hermitian. The sufficiency holds, and result (1) is proved.
(2) First, consider the necessity. Suppose thatS is J-self-adjoint, then K is J-Hermitian in 𝒯, that is, K⊂KJ*|𝒯, by result (1). It suffices to show that KJ*|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)∈KJ*|𝒯, then [(x2,f2):(y,g)]=0 by Remark 4.1. Note that (y,g)∈TJ* by KJ*|𝒯⊂TJ*. Then [(x1,f1):(y,g)]=0 by (2) of Remark 2.5. Therefore, the first relation of (4.4) yields that
(4.6)[(x,f):(y,g)]=[(x1,f1):(y,g)]+[(x2,f2):(y,g)]=0∀(x,f)∈S.
So, (y,g)∈SJ*. Therefore, S=SJ* yields that (y,g)∈S, which, together with (y,g)∈KJ*|𝒯, KJ*|𝒯∩T={0}, and S=T⊕K, implies that (y,g)∈K. Hence, KJ*|𝒯⊂K, and hence K=KJ*|𝒯. The necessity holds.
Next, consider the sufficiency. Suppose thatK is J-self-adjoint in 𝒯. By result (1), one has that S⊂SJ*. It suffices to show that SJ*⊂S. Let (y,g)∈SJ*, then (y,g)∈TJ* since SJ*⊂TJ* by T⊂S. It follows from (3.1) that there exist (y1,g1)∈T and (y2,g2)∈𝒯 such that
(4.7)(y,g)=(y1,g1)+(y2,g2).
We claim that (y2,g2)∈K. In fact, since (y,g)∈SJ*, we have
(4.8)[(x,f):(y,g)]=0∀(x,f)∈S.
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)∈KJ*|𝒯=K. It follows from (y2,g2)∈K, S=T⊕K, and (4.7) that (y,g)∈S. So, SJ*⊂S, and hence S=SJ*. The sufficiency holds.Now, we give the following result on the existence ofJ-SSEs.Theorem 4.3.
Every closedJ-Hermitian subspace has a J-SSE.Proof.
LetT be a closed J-Hermitian subspace. If T is J-self-adjoint, then this conclusion holds. So, we assume that T≠TJ*. 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≠TJ*, 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={Kα} be an arbitrary totally ordered subset of 𝒦. Set K̂=⋃αKα. Then, it can be verified that K̂ is J-Hermitian in 𝒯 by Remark 4.1 and the fact that all the elements of ℳ are J-Hermitian in 𝒯. So, K̂ 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=KJ*|𝒯. Suppose that K≠KJ*|𝒯 on the contrary. Note that K⊂KJ*|𝒯. Choose (x̃0,f̃0)∈KJ*|𝒯 satisfying (x̃0,f̃0)∉K, and set K̃=K+̇span{(x̃0,f̃0)}. 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)∈K̃. Note that K̃⊂𝒯. Then, K̃ is J-Hermitian in 𝒯, which contradicts the maximality of K. Hence, K=KJ*|𝒯.Remark 4.4.
SinceT 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 theJ-SSEs.Theorem 4.5.
LetT 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⊂TJ*, and there exists {(xj,fj)}j=1d⊂TJ* such that(1)
(x1,f1),(x2,f2),…,(xd,fd) are linearly independent (modT),(2)
[(xs,fs):(xj,fj)]=0 for s,j=1,2,…,d,(3)
S={(y,g)∈TJ*:[(y,g):(xj,fj)]=0,j=1,2,…,d}.Proof.
First, consider the necessity. Suppose thatS is a J-SSE of T, then it holds that T⊂S⊂TJ* since SJ*⊂TJ* and S=SJ*. 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 {(xj,fj)}j=1d 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(4.9)D={(y,g)∈TJ*:[(y,g):(xj,fj)]=0,j=1,2,…,d}.
Now, we prove T⊕KS,T=D, that is, S=D. Let (y,g)∈T⊕KS,T, then (y,g)∈TJ* by S⊂TJ*, and there exist (ỹ,g̃)∈T and cj∈C such that
(4.10)(y,g)=(ỹ,g̃)+∑j=1dcj(xj,fj).
Inserting (4.10) into [(y,g):(xs,fs)] and using (2) of Remark 2.5 and (2) of this theorem, we get that [(y,g):(xs,fs)]=0 for s=1,2,…,d. So, (y,g)∈D, and hence T⊕KS,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,
(4.11)[(y2,g2):(xs,fs)]=[(y,g):(xs,fs)]-[(y1,g1):(xs,fs)]=0.
So, (y2,g2)∈(KS,T)J*|𝒯, which implies that (y2,g2)∈KS,T since KS,T is J-self-adjoint in 𝒯. One has from (4.7) that (y,g)∈T⊕KS,T, and consequently, D⊂T⊕KS,T. Hence, T⊕KS,T=D, that is, S=D. The necessity holds.
Next, consider the sufficiency. Suppose that there exists{(xj,fj)}j=1d⊂TJ* such that conditions (1) and (2) hold and S is given in condition (3). From (3.1), we have
(4.12)(xj,fj)=(xj0,fj0)+(x̃j,f̃j),(xj0,fj0)∈T,(x̃j,f̃j)∈T,j=1,2,…,d.
It can be easily verified that the set {(x̃j,f̃j)}j=1d satisfies conditions (1) and (2). Let K̃=span{(x̃1,f̃1),(x̃2,f̃2),…,(x̃d,f̃d)}, then K̃ is J-Hermitian in 𝒯 since {(x̃j,f̃j)}j=1d satisfies condition (2) and K̃⊂𝒯. By the proof of Theorem 4.3, there exists K1⊂𝒯 such that K1 is J-self-adjoint in 𝒯 and K̃⊂K1. Then, by Lemma 4.2, T⊕K1 is a J-SSE of T, which, together with Theorem 3.1, yields that
(4.13)d=dim⁡(T⊕K1)T=dim⁡K1≥d.
Therefore, K̃⊂K1 and dim K̃=d imply that K̃=K1, and hence K̃ is J-self-adjoint in 𝒯. With a similar argument to the proof of T⊕KS,T=D, we have
(4.14)T⊕K̃=D̃,
where
(4.15)D̃={(y,g)∈TJ*:[(y,g):(x̃j,f̃j)]=0,j=1,2,…,d}.
On the other hand, it can be easily verified that D̃=S. So, S=T⊕K̃, and hence S is J-self-adjoint by (2) of Lemma 4.2. The sufficiency holds.Remark 4.6.
The case forJ-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.



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*Source: 676835-2012-04-24.xml*