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Jselfadjoint extensions for secondorder linear difference equations with complex coefficients
Advances in Difference Equations volume 2013, Article number: 3 (2013)
Abstract
This paper is concerned with secondorder linear difference equations with complex coefficients which are formally Jsymmetric. Both Jselfadjoint subspace extensions and Jselfadjoint operator extensions of the corresponding minimal subspace are completely characterized in terms of boundary conditions.
MSC:39A70, 47A06.
1 Introduction
In this paper, we consider the following secondorder linear difference equation with complex coefficients:
where I is the integer set {\{t\}}_{t=a}^{b}, a is a finite integer or −∞, and b is a finite integer or +∞ with ba\ge 3; Δ and ∇ are the forward and backward difference operators, respectively, i.e., \mathrm{\Delta}x(t)=x(t+1)x(t) and \mathrm{\nabla}x(t)=x(t)x(t1); p(t) and q(t) are complex with p(t)\ne 0 for t\in I, p(a1)\ne 0 if a is finite and p(b+1)\ne 0 if b is finite; w(t)>0 for t\in I; and λ is a spectral parameter.
Equation (1.1) is formally symmetric if and only if both p(t) and q(t) are real numbers. Therefore, if p(t) or q(t) are complex, then Eq. (1.1) is formally nonsymmetric. To study nonsymmetric operators, Glazman introduced the concept of Jsymmetric operators in [1] where J is a conjugation operator (see Definition 2.2). The minimal operators generated by SturmLiouville and some higherorder differential and difference expressions with complex coefficients are Jsymmetric operators in the related Hilbert spaces (e.g., [2–4]). Here, we remark that a bounded Jsymmetric operator is also called a complex symmetric operator (cf. [5, 6]). The operators generated by singular differential and difference expressions are not bounded in general.
It is well known that the study of spectra of symmetric (Jsymmetric) differential expressions is to consider the spectra of selfadjoint (Jselfadjoint) operators generated by such expressions. In general, under a certain definiteness condition, a formally differential expression can generate a minimal operator in a related Hilbert space and its adjoint is the corresponding maximal operator (see, e.g., [7, 8]). Generally, the selfadjoint (Jselfadjoint) operators are generated by extending the minimal operators. In addition, the eigenvalues of every selfadjoint (Jselfadjoint) extension of the corresponding minimal operator are different although the essential spectra of them are the same. Therefore, the characterization of selfadjoint (Jselfadjoint) extensions of a differential expression is a primary task in the study of its spectral problems; and the classical von Neumann selfadjoint extension theory and the GlazmanKreinNaimark (GKN) theory for symmetric operators were established [9, 10]. The related Jselfadjoint extension theory was also established (cf. [3, 11]). By using them, characterizations of selfadjoint (Jselfadjoint) extensions for differential expressions in terms of boundary conditions have been given (cf. [4, 7, 12, 13]). For other results for formally symmetric (Jsymmetric) differential expressions, the reader is referred to [14–22] and the references therein.
It has been found out that the minimal operators generated by some differential expressions may be nondensely defined and the maximal operators may be multivalued (e.g., see [[20], Example 2.2]). In particular, the maximal operator corresponding to Eq. (1.1) is multivalued, and the minimal operator is nondensely defined in the related Hilbert space (cf. [23]). Therefore, the selfadjoint extension theory for symmetric operators is not applicable in these cases. Coddington [24] extended the von Neumann selfadjoint extension theory for symmetric operators to Hermitian subspaces in 1973. Recently, Shi [25] extended the GKN theory for symmetric operators to Hermitian subspaces. Using GKN theory given in [25], Shi [23] first studied the selfadjoint extensions of (1.1) with real coefficients in the framework of subspaces in a product space. For Jsymmetric case, in order to study the Jselfadjoint extensions of Jsymmetric differential and difference expressions for which the minimal operators are nondensely defined or the maximal operators are multivalued, the theory for a JHermitian subspace was given in [26] which includes the GKN theorem for a JHermitian subspace. For the results for difference expressions, the reader is referred to [27–33].
The limit types of (1.1) which are directly related to how many boundary conditions should be added to get a Jselfadjoint extension have been investigated in [31, 32]. In the present paper, the Jselfadjoint subspace extensions and Jselfadjoint operator extensions of the minimal subspace corresponding to Eq. (1.1) with complex coefficients are studied. A complete characterization of them in terms of boundary conditions is given. These characterizations are basic in the study of spectral theory for Eq. (1.1).
The rest of this present paper is organized as follows. In Section 2, some basic concepts and fundamental results about subspaces and Eq. (1.1) are introduced. In Section 3, the maximal, preminimal, and minimal subspaces in the whole interval and the lefthand and righthand halfintervals are introduced and their properties are studied. The relationship among the defect indices of the minimal subspaces in the whole interval and the lefthand and righthand halfintervals is studied in Section 4. In Section 5, we pay our attention to Jselfadjoint subspace extensions of the minimal subspace in the whole interval. Finally, a complete characterization of Jselfadjoint operator extensions of the minimal operator in the whole interval is given in Section 6. Three examples are given in Section 7.
2 Preliminaries
In this section, we introduce some basic concepts and give some fundamental results about subspaces in a product space and present two results about Eq. (1.1).
By C denote the set of complex numbers, and by \overline{z} denote the complex conjugate of z\in \mathbf{C}. Let X be a complex Hilbert space with the inner product \u3008\cdot ,\cdot \u3009. The norm \parallel \cdot \parallel is defined by \parallel f\parallel ={\u3008f,f\u3009}^{1/2} for f\in X. Let {X}^{2} be the product space X\times X with the following induced inner product, denoted by \u3008\cdot ,\cdot \u3009 without any confusion:
Let T be a linear subspace in {X}^{2}. For briefness, a linear subspace is only called a subspace. For a subspace T in {X}^{2}, denote
Clearly, T(0)=\{0\} if and only if T can determine a unique linear operator from D(T) into X whose graph is T. Therefore, T is said to be an operator if T(0)=\{0\}.
Definition 2.1 [24]
Let T be a subspace in {X}^{2}.

(1)
Its adjoint, {T}^{\ast}, is defined by
{T}^{\ast}=\{(y,g)\in {X}^{2}:\u3008f,y\u3009=\u3008x,g\u3009\text{for all}(x,f)\in T\}. 
(2)
T is said to be a Hermitian subspace if T\subset {T}^{\ast}.

(3)
T is said to be a selfadjoint subspace if T={T}^{\ast}.
Lemma 2.1 [24]
Let T be a subspace in {X}^{2}. Then {T}^{\ast} is a closed subspace in {X}^{2}, {T}^{\ast}={(\overline{T})}^{\ast}, and {T}^{\ast \ast}=\overline{T}, where \overline{T} is the closure of T.
Definition 2.2 (see [[19], p.114] or [3])
An operator J defined on X is said to be a conjugation operator if for all x,y\in X,
It can be verified that J is a conjugate linear, normpreserving bijection on X and it holds that (see [[19], p.114])
The complex conjugation x\mapsto \overline{x} in any {l}^{2} space is a conjugation operator on {l}^{2}.
Definition 2.3 [26]
Let T be a subspace in {X}^{2} and J be a conjugation operator.

(1)
The Jadjoint of T, i.e., {T}_{J}^{\ast}, is defined by
{T}_{J}^{\ast}=\{(y,g)\in {X}^{2}:\u3008f,Jy\u3009=\u3008x,Jg\u3009\text{for all}(x,f)\in T\}. 
(2)
T is said to be a JHermitian subspace if T\subset {T}_{J}^{\ast}.

(3)
T is said to be a Jselfadjoint subspace if T={T}_{J}^{\ast}.

(4)
Let T be a JHermitian subspace. Then S is a Jselfadjoint subspace extension (briefly, JSSE) of T if T\subset S and S is a Jselfadjoint subspace.
Remark 2.1

(i)
It can be easily verified that {T}_{J}^{\ast} is a closed subspace. Consequently, a Jselfadjoint subspace T is a closed subspace since T={T}_{J}^{\ast}. In addition, {S}_{J}^{\ast}\subset {T}_{J}^{\ast} if T\subset S.

(ii)
From the definition, we have that \u3008f,Jy\u3009=\u3008x,Jg\u3009 holds for all (x,f)\in T and (y,g)\in {T}_{J}^{\ast}, and that T is a JHermitian subspace if and only if
\u3008f,Jy\u3009=\u3008x,Jg\u3009\phantom{\rule{1em}{0ex}}\text{for all}(x,f),(y,g)\in T. 
(iii)
Assume that T is not only Jsymmetric for some conjugation operator J but also symmetric, and that S is a JSSE of T. Then S is a selfadjoint subspace extension of T if and only if {S}_{J}^{\ast}={S}^{\ast}.
Lemma 2.2 [26]
Let T be a subspace in {X}^{2}. Then

(1)
{T}^{\ast}=\{(Jy,Jg):(y,g)\in {T}_{J}^{\ast}\};

(2)
{T}_{J}^{\ast}=\{(Jy,Jg):(y,g)\in {T}^{\ast}\}.
Lemma 2.3 [26]
Let T be a JHermitian subspace. Then (y,g)\in \overline{T} if and only if (y,g)\in {T}_{J}^{\ast} and \u3008f,Jy\u3009=\u3008x,Jg\u3009 for all (x,f)\in {T}_{J}^{\ast}.
Definition 2.4 [26]
Let T be a JHermitian subspace. Then d(T)=\frac{1}{2}dim{T}_{J}^{\ast}/\overline{T} is called to be the defect index of T.
Remark 2.2 By [[26], Remark 3.5], d(T) is a nonnegative integer or else infinite. Further, d(T)=d(\overline{T}). Then T and \overline{T} have the same JSSEs since every JSSE is closed.
Define the form [:] as
Then, for all {Y}_{j}=({x}_{j},{f}_{j})\in {T}_{J}^{\ast} (j=1,2,3) and \mu \in \mathbf{C}, it holds that
The following result which can be regarded as the GKN theorem for a JHermitian subspace was established in [26].
Theorem 2.1 Let T be a closed JHermitian subspace. Assume that d(T)=:d<+\mathrm{\infty}. Then a subspace S is a JSSE of T if and only if T\subset S\subset {T}_{J}^{\ast} and there exists {\{({x}_{j},{f}_{j})\}}_{j=1}^{d}\subset {T}_{J}^{\ast} such that

(i)
({x}_{1},{f}_{1}),({x}_{2},{f}_{2}),\dots ,({x}_{d},{f}_{d}) are linearly independent (modulo T);

(ii)
[({x}_{s},{f}_{s}):({x}_{j},{f}_{j})]=0 for s,j=1,2,\dots ,d;

(iii)
S=\{(y,g)\in {T}_{J}^{\ast}:[(y,g):({x}_{j},{f}_{j})]=0,j=1,2,\dots ,d\}.
Finally, we present two results for τ or Eq. (1.1). For briefness, introduce the conventions: for any given integer k, a+k=\mathrm{\infty} when a=\mathrm{\infty}, and b+k=+\mathrm{\infty} when b=+\mathrm{\infty}. Further, denote
In the case of a=\mathrm{\infty}, if {lim}_{t\to a}(x,y)(t) exists and is finite, then denote the limit by (x,y)(\mathrm{\infty}); and in the other case of b=+\mathrm{\infty}, if {lim}_{t\to b}(x,y)(t) exists and is finite, then denote the limit by (x,y)(\mathrm{\infty}).
We remark that the notation (x,y)(t) is also used in [23] where it is given by (x,y)(t)=p(t)[(\mathrm{\Delta}\overline{y}(t))x(t)\overline{y}(t)\mathrm{\Delta}x(t)]. So, the expression of (x,y)(t) in the present paper is different from that in [23].
It can be easily verified that the following result holds.
Lemma 2.4 For any x={\{x(t)\}}_{t=a1}^{b+1}, y={\{y(t)\}}_{t=a1}^{b+1}\subset \mathbf{C}, and for any m,n\in I with m\le n,
The following result is a direct consequence of Lemma 2.4.
Lemma 2.5 For each \lambda \in \mathbf{C}, let y and z be any solutions of (1.1). Then, for any given a1\le {t}_{0}\le b,
3 Maximal and minimal subspaces
In this section, we introduce the corresponding maximal, preminimal, and minimal subspaces to τ in the whole interval and the lefthand and righthand halfintervals and study their properties.
First, introduce the following space:
Then {l}_{w}^{2}(I) is a Hilbert space with the inner product
Clearly, x=y in {l}_{w}^{2}(I) if and only if x(t)=y(t), t\in I, i.e., \parallel xy\parallel =0, where \parallel x\parallel ={\u3008x,x\u3009}^{1/2}.
The formally adjoint operator of τ is
Now, introduce the maximal subspace H(\tau ) and the preminimal subspace {H}_{00}(\tau ) in {l}_{w}^{2}(I)\times {l}_{w}^{2}(I) corresponding to τ as follows.
The subspace {H}_{0}(\tau ):={\overline{H}}_{00}(\tau ) is called the minimal subspace corresponding to τ.
The endpoints a and b may be finite or infinite. In order to characterize the JSSEs of {H}_{0}(\tau ) in a unified form, we introduce the left and right maximal and minimal subspaces. Fix any integer a+1<{c}_{0}<b. Denote
and by \u3008\cdot ,\cdot \u3009, {\u3008\cdot ,\cdot \u3009}_{a}, {\u3008\cdot ,\cdot \u3009}_{b}, \parallel \cdot \parallel, {\parallel \cdot \parallel}_{a}, and {\parallel \cdot \parallel}_{b} denote the inner products and the norms of {l}_{w}^{2}(I), {l}_{w}^{2}({I}_{1}), and {l}_{w}^{2}({I}_{2}), respectively. For briefness, we still denote the inner products and norms of their product spaces {l}_{w}^{2}(I)\times {l}_{w}^{2}(I), {l}_{w}^{2}({I}_{1})\times {l}_{w}^{2}({I}_{1}), and {l}_{w}^{2}({I}_{2})\times {l}_{w}^{2}({I}_{2}) by the same notations as those for {l}_{w}^{2}(I), {l}_{w}^{2}({I}_{1}), and {l}_{w}^{2}({I}_{2}), respectively.
Let {H}_{a}(\tau ) and {H}_{a,00}(\tau ) be the left maximal and preminimal subspaces defined as in (3.1) with I replaced by {I}_{1}, respectively, and let {H}_{b}(\tau ) and {H}_{b,00}(\tau ) be the right maximal and preminimal subspaces defined as in (3.1) with I replaced by {I}_{2}, respectively. The subspaces {H}_{a,0}(\tau ):={\overline{H}}_{a,00}(\tau ) and {H}_{b,0}(\tau ):={\overline{H}}_{b,00}(\tau ) are called the left and right minimal subspaces corresponding to τ, respectively. By Lemma 2.1, one has
In the rest of the present paper, let J be the complex conjugate x\mapsto \overline{x}, i.e., Jx=\overline{x}. Then J is a conjugation operator on {l}_{w}^{2}(I) (or {l}_{w}^{2}({I}_{1}) or {l}_{w}^{2}({I}_{2})). By Lemma 2.2 and (3.2), one has that
The rest of this section is divided into three parts.
3.1 Properties of minimal subspaces and their adjoint and Jadjoint subspaces
In this subsection, we study the properties of minimal subspaces {H}_{0}(\tau ), {H}_{a,0}(\tau ), {H}_{b,0}(\tau ) and their adjoint and Jadjoint subspaces.
First, we have the following result.
Lemma 3.1 (see [[23], Lemma 3.1])
For each a+1\le {t}_{0}\le b1 (or a+1\le {t}_{0}\le {c}_{0}2 or {c}_{0}+1\le {t}_{0}\le b1) and for each \xi \in \mathbf{C}, there exists x\in D({H}_{00}(\tau )) (or D({H}_{a,00}(\tau )) or D({H}_{b,00}(\tau ))) such that x({t}_{0})=\xi and x(t)=0 for all t\ne {t}_{0}.
Theorem 3.1 H({\tau}^{+})\subset {H}_{00}^{\ast}(\tau ), {H}_{a}({\tau}^{+})\subset {H}_{a,00}^{\ast}(\tau ), {H}_{b}({\tau}^{+})\subset {H}_{b,00}^{\ast}(\tau ), and
Proof Since {H}_{a,00}(\tau ) and {H}_{b,00}(\tau ) are two special cases of {H}_{00}(\tau ), we only prove the results corresponding to {H}_{00}(\tau ).
For any given (x,f)\in {H}_{00}^{\ast}(\tau ), we have
which implies that
On the other hand, by using y\in D({H}_{00}(\tau )), it can be verified that
which, together with (3.6) and y(a)=y(b)=0 when a and b are finite, implies that
So, by Lemma 3.1 we get
Conversely, suppose that (x,f)\in {l}_{w}^{2}(I)\times {l}_{w}^{2}(I) satisfies (3.7). Then (3.5) holds for all (y,g)\in {H}_{00}(\tau ). Consequently, (x,f)\in {H}_{00}^{\ast}(\tau ). So, the first relation of (3.4) holds. In addition, the first relation of (3.4) directly yields that H({\tau}^{+})\subset {H}_{00}^{\ast}(\tau ). This completes the proof. □
Theorem 3.2 The subspaces {H}_{00}(\tau ), {H}_{a,00}(\tau ), and {H}_{b,00}(\tau ) are JHermitian subspaces in {l}_{w}^{2}(I)\times {l}_{w}^{2}(I), {l}_{w}^{2}({I}_{1})\times {l}_{w}^{2}({I}_{1}), and {l}_{w}^{2}({I}_{2})\times {l}_{w}^{2}({I}_{2}), respectively. Further, H(\tau )\subset {({H}_{00}(\tau ))}_{J}^{\ast}, {H}_{a}(\tau )\subset {({H}_{a,00}(\tau ))}_{J}^{\ast}, and {H}_{b}(\tau )\subset {({H}_{b,00}(\tau ))}_{J}^{\ast}, and
Proof It can be easily verified that {H}_{00}(\tau ), {H}_{a,00}(\tau ), and {H}_{b,00}(\tau ) are JHermitian subspaces in the corresponding Hilbert spaces by (ii) of Remark 2.1 and Lemma 2.4. Further, (3.8) can be concluded from Theorem 3.1 and Lemma 2.2. This completes the proof. □
Using Theorem 3.2 and with a similar argument to [[23], Corollary 3.1], we can get the following results.
Corollary 3.1 H(\tau )={({H}_{00}(\tau ))}_{J}^{\ast}={({H}_{0}(\tau ))}_{J}^{\ast}, {H}_{a}(\tau )={({H}_{a,00}(\tau ))}_{J}^{\ast}={({H}_{a,0}(\tau ))}_{J}^{\ast}, and {H}_{b}(\tau )={({H}_{b,00}(\tau ))}_{J}^{\ast}={({H}_{b,0}(\tau ))}_{J}^{\ast} in the sense of the norms \parallel \cdot \parallel, {\parallel \cdot \parallel}_{a}, and {\parallel \cdot \parallel}_{b}, respectively. Consequently, H(\tau ), {H}_{a}(\tau ), and {H}_{b}(\tau ) are closed subspaces in {l}_{w}^{2}(I)\times {l}_{w}^{2}(I), {l}_{w}^{2}({I}_{1})\times {l}_{w}^{2}({I}_{1}), and {l}_{w}^{2}({I}_{2})\times {l}_{w}^{2}({I}_{2}), respectively.
Remark 3.1 H(\tau )={({H}_{00}(\tau ))}_{J}^{\ast}={({H}_{0}(\tau ))}_{J}^{\ast} follows from (3.3) and the first relation of (3.8) in the special case that a=\mathrm{\infty} and b=+\mathrm{\infty}.
Now, we introduce the boundary forms on {l}_{w}^{2}(I)\times {l}_{w}^{2}(I), {l}_{w}^{2}({I}_{1})\times {l}_{w}^{2}({I}_{1}), and {l}_{w}^{2}({I}_{2})\times {l}_{w}^{2}({I}_{2}) as follows.
It can be easily shown that (2.3) holds for [:], {[:]}_{a}, and {[:]}_{b}, respectively.
Note that {H}_{0}(\tau ), {H}_{a,0}(\tau ), and {H}_{b,0}(\tau ) are closed. Then, by Lemma 2.3 and (3.3), {H}_{0}(\tau ), {H}_{a,0}(\tau ), and {H}_{b,0}(\tau ) can be expressed in terms of the boundary forms as follows.
Theorem 3.3 The subspaces {H}_{0}(\tau ), {H}_{a,0}(\tau ), and {H}_{b,0}(\tau ) are closed JHermitian operators in {l}_{w}^{2}(I), {l}_{w}^{2}({I}_{1}), and {l}_{w}^{2}({I}_{2}), respectively.
Proof We only prove the result for {H}_{0}(\tau ) since {H}_{a,0}(\tau ) and {H}_{b,0}(\tau ) can be regarded as two special cases of {H}_{0}(\tau ).
Since {H}_{0}(\tau ) is a JHermitian subspace by Theorem 3.2 and {H}_{0}(\tau )={\overline{H}}_{00}(\tau ), one has that {H}_{0}(\tau ) is a closed JHermitian subspace. So, it suffices to show that ({H}_{0}(\tau ))(0)=\{0\}. Suppose that (0,f)\in {H}_{0}(\tau ). Then, for all (y,g)\in H(\tau )\subset {({H}_{00}(\tau ))}_{J}^{\ast}, [(0,f):(y,g)]=\u3008f,Jy\u3009=0, that is,
In order to show f=0, the discussion is divided into three cases.
Case 1. The endpoints a and b are finite. For all (x,f)\in {({H}_{00}(\tau ))}_{J}^{\ast} with x(t)=0 for all t\in I, we get by Theorem 3.2 and (3.10) that
It can be easily shown that there exists (y,g)\in H(\tau ) such that y(a)=f(a) and y(t)=0 for all t\ne a. Inserting it into (3.11) yields f(a)=0. Similarly, f(b)=0. Hence, f=0.
Case 2. One of a and b is finite. With a similar argument to that for Case 1, one can show f(a)=0. Hence, f=0.
Case 3. a=\mathrm{\infty} and b=+\mathrm{\infty}. By Remark 3.1, H(\tau )={({H}_{00}(\tau ))}_{J}^{\ast}. So, by the first relation of (3.8), x(t)=0 for t\in I implies that f(t)=0 for t\in I. This completes the proof. □
Lemma 3.2 For every (x,f)\in {H}_{0}(\tau ), x(a)=0 in the case that a is finite and x(b)=0 in the case that b is finite.
Proof Fix any (x,f)\in {H}_{0}(\tau ). Then we have
If a is finite, then there exists ({y}_{0},{g}_{0})\in H(\tau ) such that {y}_{0}(a1)\ne 0 and {y}_{0}(t)=0 for all t\in I. Inserting ({y}_{0},{g}_{0}) into (3.12), we have that p(a1){y}_{0}(a1)x(a)=0. So, x(a)=0. One can get that x(b)=0 when b is finite similarly. This completes the proof. □
Theorem 3.4 The subspace {H}_{0}(\tau ) is a densely defined JHermitian operator in {l}_{w}^{2}(I) in the case that a=\mathrm{\infty} and b=+\mathrm{\infty} and a nondensely defined JHermitian operator in {l}_{w}^{2}(I) in the case that at least one of a and b is finite. Consequently, {H}_{a,0}(\tau ) and {H}_{b,0}(\tau ) are nondensely defined JHermitian operators in {l}_{w}^{2}({I}_{1}) and {l}_{w}^{2}({I}_{2}), respectively.
Proof By Theorem 3.3, Lemma 3.2, and a similar method to [[23], Theorem 3.3], this theorem can be proved. □
3.2 Characterizations of the three subspaces {\stackrel{\u02c6}{H}}_{0}(\tau ), {\stackrel{\u02c6}{H}}_{a,0}(\tau ), and {\stackrel{\u02c6}{H}}_{b,0}(\tau )
In this section, we introduce three subspaces {\stackrel{\u02c6}{H}}_{0}(\tau ), {\stackrel{\u02c6}{H}}_{a,0}(\tau ), and {\stackrel{\u02c6}{H}}_{b,0}(\tau ), and discuss their characterizations, which will play an important role in the study of JSSEs of {H}_{0}(\tau ).
First, define {\stackrel{\u02c6}{H}}_{0}(\tau ), {\stackrel{\u02c6}{H}}_{a,0}(\tau ), and {\stackrel{\u02c6}{H}}_{b,0}(\tau ) in {l}_{w}^{2}(I)\times {l}_{w}^{2}(I), {l}_{w}^{2}({I}_{1})\times {l}_{w}^{2}({I}_{1}), and {l}_{w}^{2}({I}_{2})\times {l}_{w}^{2}({I}_{2}) as follows:
Since [:], {[:]}_{a}, and {[:]}_{b} are defined in terms of the norms \parallel \cdot \parallel, {\parallel \cdot \parallel}_{a}, and {\parallel \cdot \parallel}_{b}, respectively, by Corollary 3.1 we get that {\stackrel{\u02c6}{H}}_{0}(\tau )={H}_{0}(\tau ), {\stackrel{\u02c6}{H}}_{a,0}(\tau )={H}_{a,0}(\tau ), and {\stackrel{\u02c6}{H}}_{b,0}(\tau )={H}_{b,0}(\tau ) in the sense of the norms \parallel \cdot \parallel, {\parallel \cdot \parallel}_{a}, and {\parallel \cdot \parallel}_{b}, respectively. So, {\stackrel{\u02c6}{H}}_{0}(\tau ), {\stackrel{\u02c6}{H}}_{a,0}(\tau ), and {\stackrel{\u02c6}{H}}_{b,0}(\tau ) are closed JHermitian operators in the corresponding spaces by Theorem 3.3.
In [23], the patching lemma [[23], Lemma 3.3] was used in the study of the selfadjoint subspace extensions for (1.1) with real coefficients. It also holds for (1.1) with complex coefficients here.
Lemma 3.3 [[23], Lemma 3.3]
For any given {\alpha}_{j},{\beta}_{j}\in \mathbf{C}, j=1,2, and any given {a}_{1},{b}_{1}\in I with {b}_{1}\ge {a}_{1}+1, there exists f={\{f(t)\}}_{t={a}_{1}}^{{b}_{1}}\subset \mathbf{C} such that the boundary value problem
has a solution x={\{x(t)\}}_{t={a}_{1}1}^{{b}_{1}+1}. Further, for any given ({x}_{1},{f}_{1}),({x}_{2},{f}_{2})\in H(\tau ), there exists (y,g)\in H(\tau ) such that
Remark 3.2 (see [[23], Remark 3.2])
Any two elements of {H}_{a}(\tau ) (or {H}_{b}(\tau )) can be patched together by some element of {H}_{a}(\tau ) (or {H}_{b}(\tau )) in a similar way as in Lemma 3.3. Further, any element of {H}_{a}(\tau ) and any element of {H}_{b}(\tau ) can be patched together by some element of H(\tau ) in a similar way as in Lemma 3.3.
The following result can be easily verified by Lemma 2.4, Theorem 3.2, and (3.3).
Lemma 3.4 For all x,y\in D({({H}_{0}(\tau ))}_{J}^{\ast}) or D({({H}_{a,0}(\tau ))}_{J}^{\ast}), {lim}_{t\to a1}(x,y)(t) exists and is finite in the case of a=\mathrm{\infty}, and for all x,y\in D({({H}_{0}(\tau ))}_{J}^{\ast}) or D({({H}_{b,0}(\tau ))}_{J}^{\ast}), {lim}_{t\to b}(x,y)(t) exists and is finite in the case of b=+\mathrm{\infty}. Moreover,
Using Lemma 3.3 and with a similar argument to [[23], Theorem 3.4], we have the other characterizations of three subspaces {\stackrel{\u02c6}{H}}_{0}(\tau ), {\stackrel{\u02c6}{H}}_{a,0}(\tau ), and {\stackrel{\u02c6}{H}}_{b,0}(\tau ).
Theorem 3.5
3.3 Characterizations of the left and right maximal subspaces
In this section, we characterize {H}_{a}(\tau ) and {H}_{b}(\tau ).
First, let d, {d}_{a}, and {d}_{b} be the defect indices of {H}_{0}(\tau ), {H}_{a,0}(\tau ), and {H}_{b,0}(\tau ), respectively. Then we have
Lemma 3.5 d=\frac{1}{2}dim\mathcal{D}, {d}_{b}=\frac{1}{2}dim{\mathcal{D}}_{b}, and {d}_{a}=\frac{1}{2}dim{\mathcal{D}}_{a}, where
Proof Since the proofs are similar, we only prove d=\frac{1}{2}dim\mathcal{D}.
First, it can be verified that
Next, we prove that
Let (y,g)\in H(\tau )\ominus {\stackrel{\u02c6}{H}}_{0}(\tau ), where ⊖ denotes the orthogonal complement of {\stackrel{\u02c6}{H}}_{0}(\tau ) in H(\tau ). Then
which yields that (g,y)\in {\stackrel{\u02c6}{H}}_{0}^{\ast}(\tau ). It can be easily verified that {\stackrel{\u02c6}{H}}_{0}^{\ast}(\tau )={H}_{0}^{\ast}(\tau ). So, (g,y)\in {H}_{0}^{\ast}(\tau ), and by Theorem 3.1, one has that
Since (y,g)\in H(\tau ) and w\ne 0, we get g=\frac{1}{w}\tau (y) on I. Inserting it into (3.16), we have
So, (y,g)\in \mathcal{D}. Conversely, suppose that (y,g)\in \mathcal{D}. Then (y,g)\in H(\tau ) and (3.17) holds, and then (3.16) holds. Then (g,y)\in {H}_{0}^{\ast}(\tau ) and hence (g,y)\in {\stackrel{\u02c6}{H}}_{0}^{\ast}(\tau ). So, (3.15) holds and hence (y,g)\in H(\tau )\ominus {\stackrel{\u02c6}{H}}_{0}(\tau ). So, (3.14) holds, which together with (3.13) implies that d=\frac{1}{2}dim\mathcal{D}. This completes the proof. □
Lemma 3.6 {d}_{b}=1 or 2 and {d}_{a}=1 or 2.
Proof By Lemma 3.5, dim{\mathcal{D}}_{b} is equal to the number of linearly independent solutions of
for which both y and \frac{1}{w}\tau y are in {l}_{w}^{2}({I}_{2}). Then {d}_{b}=\frac{1}{2}dim{\mathcal{D}}_{b}\le 2 since (3.18) has at most four linearly independent solutions. In addition, there exists ({z}_{j},{h}_{j})\in {H}_{b}(\tau ), j=1,2, such that
Note that ({z}_{1},{h}_{1}) and ({z}_{2},{h}_{2}) are linearly independent (modulo {\stackrel{\u02c6}{H}}_{b,0}(\tau )) and
Then {d}_{b}\ge 1 and hence 1\le {d}_{b}\le 2. Then {d}_{b}=1 or 2 since {d}_{b} is an integer.
The assertion {d}_{a}=1 or 2 can be proved similarly. This completes the proof. □
Lemma 3.7

(1)
If all the solutions of (1.1) restricted on {I}_{2} are in {l}_{w}^{2}({I}_{2}) for some {\lambda}_{0}\in \mathbf{C}, then the same is true for all \lambda \in \mathbf{C}.

(2)
If all the solutions of the equation
{\tau}^{+}(\frac{1}{w}\tau (y))(t)=\lambda w(t)y(t),\phantom{\rule{1em}{0ex}}{c}_{0}+1\le t<b1,(3.20)
are in {l}_{w}^{2}({I}_{2}) for some {\lambda}_{0}\in \mathbf{C}, then the same is true for all \lambda \in \mathbf{C}.
Proof The first result is [[31], Lemma 2.2]. Now, we prove the assertion (2). Clearly, this result holds if b is finite. So, we prove the case where b=+\mathrm{\infty}. By setting
Eq. (3.20) can be rewritten as the following discrete Hamiltonian system:
where
{I}_{2\times 2} is the 2\times 2 unit matrix, and W(t)=diag(w(t+1),0,0,0). It is evident that the assumptions ({A}_{1}) and ({A}_{2}) of [[27], Section 1] hold for (3.22). Let
with the inner product {\u3008Y,Z\u3009}_{W}={\sum}_{t={c}_{0}}^{\mathrm{\infty}}R{(Z)}^{\ast}(t)W(t)R(Y)(t), where {Y}^{\ast}(t) denotes the complex conjugate transpose of Y(t). We have from [[27], Theorem 5.5] that if there exists {\lambda}_{0}\in \mathbf{C} such that all the solutions of (3.22) are in {l}_{W}^{2}, then the same is true for all \lambda \in \mathbf{C}. Hence, the assertion (2) of this lemma follows. This completes the proof. □
Theorem 3.6 Let ({z}_{j},{h}_{j})\in {H}_{b}(\tau ) (j=1,2) be defined by (3.19). Then the following results hold:

(1)
In the case of {d}_{b}=1, for any given (x,f)\in {H}_{b}(\tau ), there exist uniquely ({y}_{0},{f}_{0})\in {\stackrel{\u02c6}{H}}_{b,0}(\tau ) and {c}_{1},{c}_{2}\in \mathbf{C} such that
x(t)={y}_{0}(t)+{c}_{1}{z}_{1}(t)+{c}_{2}{z}_{2}(t),\phantom{\rule{1em}{0ex}}{c}_{0}1\le t\le b+1.(3.23) 
(2)
In the case of {d}_{b}=2, let {\varphi}_{1} and {\varphi}_{2} be two linearly independent solutions of (1.1) restricted on {I}_{2}. Then {\varphi}_{1} and {\varphi}_{2} are in {l}_{w}^{2}({I}_{2}), and for any given (x,f)\in {H}_{b}(\tau ), there exist uniquely ({y}_{0},{f}_{0})\in {\stackrel{\u02c6}{H}}_{b,0}(\tau ) and {c}_{j},{d}_{j}\in \mathbf{C} (j=1,2) such that
x(t)={y}_{0}(t)+{c}_{1}{z}_{1}(t)+{c}_{2}{z}_{2}(t)+{d}_{1}{\varphi}_{1}(t)+{d}_{2}{\varphi}_{2}(t),\phantom{\rule{1em}{0ex}}{c}_{0}1\le t\le b+1.(3.24)
Proof Since dim{H}_{b}(\tau )/{\stackrel{\u02c6}{H}}_{b,0}(\tau )=2 in the case of {d}_{b}=1, one has that ({z}_{1},{h}_{1}) and ({z}_{2},{h}_{2}) defined by (3.19) form a basis of {H}_{b}(\tau )/{\stackrel{\u02c6}{H}}_{b,0}(\tau ). So, the first result holds.
In the case of {d}_{b}=2, one has that dim{H}_{b}(\tau )/{\stackrel{\u02c6}{H}}_{b,0}(\tau )=4. By Lemmas 3.5 and 3.7, all the solutions of (3.20) with \lambda =0 are in {l}_{w}^{2}({I}_{2}) and hence all the solutions of \tau (y)(t)=0 restricted on {I}_{2} are in {l}_{w}^{2}({I}_{2}). So, all the solutions of (1.1) restricted on {I}_{2} are in {l}_{w}^{2}({I}_{2}) by Lemma 3.7. Let {\varphi}_{1} and {\varphi}_{2} be two linearly independent solutions of (1.1). Then ({\varphi}_{1},\lambda {\varphi}_{1}), ({\varphi}_{2},\lambda {\varphi}_{2})\in {H}_{b}(\tau ). Set
Then it can be concluded that rank\mathrm{\Phi}=2. On the other hand, ({z}_{1},{h}_{1}), ({z}_{2},{h}_{2}), ({\varphi}_{1},\lambda {\varphi}_{1}), and ({\varphi}_{2},\lambda {\varphi}_{2}) are linearly independent (modulo {\stackrel{\u02c6}{H}}_{b,0}(\tau )). In fact, if
then by Theorem 3.5 and {\varphi}_{1},{\varphi}_{2}\in D({H}_{b}(\tau )),
This, together with Lemma 2.5 and rank\mathrm{\Phi}=2, implies that {c}_{j}=0 (1\le j\le 4). Then ({z}_{1},{h}_{1}), ({z}_{2},{h}_{2}), ({\varphi}_{1},\lambda {\varphi}_{1}), and ({\varphi}_{2},\lambda {\varphi}_{2}) form a basis of {H}_{b}(\tau )/{\stackrel{\u02c6}{H}}_{b,0}(\tau ). So, (3.24) holds. This completes the proof. □
Using a similar argument to Theorem 3.6, we can get the following result.
Theorem 3.7 Let ({\tilde{z}}_{j},{\tilde{h}}_{j})\in {H}_{a}(\tau ) (j=1,2) be defined by
Then the following results hold:

(1)
In the case of {d}_{a}=1, for any given (x,f)\in {H}_{a}(\tau ), there exist uniquely ({\tilde{y}}_{0},{\tilde{f}}_{0})\in {\stackrel{\u02c6}{H}}_{a,0}(\tau ) and {\tilde{c}}_{1},{\tilde{c}}_{2}\in \mathbf{C} such that
x(t)={\tilde{y}}_{0}(t)+{\tilde{c}}_{1}{\tilde{z}}_{1}(t)+{\tilde{c}}_{2}{\tilde{z}}_{2}(t),\phantom{\rule{1em}{0ex}}a1\le t\le {c}_{0}.(3.27) 
(2)
In the case of {d}_{a}=2, let {\tilde{\varphi}}_{1} and {\tilde{\varphi}}_{2} be two linearly independent solutions of equation (1.1) restricted on {I}_{1}. Then {\tilde{\varphi}}_{1} and {\tilde{\varphi}}_{2} are in {l}_{w}^{2}({I}_{1}), and for any given (x,f)\in {H}_{a}(\tau ), there exist uniquely ({\tilde{y}}_{0},{\tilde{f}}_{0})\in {\stackrel{\u02c6}{H}}_{a,0}(\tau ) and {\tilde{c}}_{j},{\tilde{d}}_{j}\in \mathbf{C} (j=1,2) such that
x(t)={\tilde{y}}_{0}(t)+{\tilde{c}}_{1}{\tilde{z}}_{1}(t)+{\tilde{c}}_{2}{\tilde{z}}_{2}(t)+{\tilde{d}}_{1}{\tilde{\varphi}}_{1}(t)+{\tilde{d}}_{2}{\tilde{\varphi}}_{2}(t),\phantom{\rule{1em}{0ex}}a1\le t\le {c}_{0}.(3.28)
4 Defect indices of {H}_{0}(\tau )
The following is the main result of this section.
Theorem 4.1 Let d, {d}_{a}, and {d}_{b} be the defect indices of {H}_{0}(\tau ), {H}_{a,0}(\tau ), and {H}_{b,0}(\tau ), respectively. Then d={d}_{a}+{d}_{b}2.
It is evident that Theorem 4.1 holds in the case that at least one of a and b is finite. So, it is only needed to consider the case that a=\mathrm{\infty} and b=+\mathrm{\infty}. Before proving Theorem 4.1, we prove three lemmas in this case.
Lemma 4.1 d=\frac{1}{2}dim\tilde{\mathcal{D}}, {d}_{b}=\frac{1}{2}dim{\tilde{\mathcal{D}}}_{b}, and {d}_{a}=\frac{1}{2}dim{\tilde{\mathcal{D}}}_{a}, where
Proof It can be easily verified that {({H}_{0}(\tau ))}_{J}^{\ast}={H}_{0}(\tau )\oplus \tilde{\mathcal{D}}. This gives that d=\frac{1}{2}dim\tilde{\mathcal{D}}. The other two relations are proved similarly. This completes the proof. □
For any given (x,f)\in {l}_{w}^{2}(I)\times {l}_{w}^{2}(I), denote
Then we have the following result.
Lemma 4.2 Let {\tilde{H}}_{0}(\tau ) be the restriction of {\stackrel{\u02c6}{H}}_{0}(\tau ) defined by
Then
Proof It can be easily verified by Theorem 3.5 that
Then it can be verified that
Relation (4.1) follows from (4.3) and Lemma 2.2. This completes the proof. □
Lemma 4.3 Let \tilde{d} be the defect index of {\tilde{H}}_{0}(\tau ). Then \tilde{d}={d}_{a}+{d}_{b}.
Proof It can be easily verified that {\tilde{H}}_{0}(\tau ) is a closed JHermitian operator in {l}_{w}^{2}(I) by the fact that {\stackrel{\u02c6}{H}}_{0}(\tau ) is a closed JHermitian operator in {l}_{w}^{2}(I). Set
in which {\tilde{\mathcal{D}}}_{a} and {\tilde{\mathcal{D}}}_{b} are given in Lemma 4.1. Now, we prove that {\mathcal{D}}_{a,b}={({\tilde{H}}_{0}(\tau ))}_{J}^{\ast}\ominus {\tilde{H}}_{0}(\tau ). Let (y,g)\in {({\tilde{H}}_{0}(\tau ))}_{J}^{\ast}\ominus {\tilde{H}}_{0}(\tau ). Then, for all (x,f)\in {\tilde{H}}_{0}(\tau ), (3.15) holds, which together with (4.2) implies that
Since {H}_{a,0}^{\ast}(\tau )={\stackrel{\u02c6}{H}}_{a,0}^{\ast}(\tau ) and {H}_{b,0}^{\ast}(\tau )={\stackrel{\u02c6}{H}}_{b,0}^{\ast}(\tau ), one has (y,g)\in {\mathcal{D}}_{a,b}. Conversely, suppose that (y,g)\in {\mathcal{D}}_{a,b}. It can be verified that (y,g)\in {({\tilde{H}}_{0}(\tau ))}_{J}^{\ast}\ominus {\tilde{H}}_{0}(\tau ) by (4.2). Hence, {\mathcal{D}}_{a,b}={({\tilde{H}}_{0}(\tau ))}_{J}^{\ast}\ominus {\tilde{H}}_{0}(\tau ). Therefore, \tilde{d}=\frac{1}{2}dim{\mathcal{D}}_{a,b}. It can be easily verified that dim{\mathcal{D}}_{a,b}=dim{\mathcal{D}}_{a}+dim{\mathcal{D}}_{b}. So, \tilde{d}={d}_{a}+{d}_{b} by Lemma 4.1. This completes the proof. □
Proof of Theorem 4.1 Set
There exist ({y}_{1},{g}_{1}),({y}_{2},{g}_{2})\in H(\tau ) such that
Then ({y}_{j},{g}_{j})\in {\stackrel{\u02c6}{H}}_{0}(\tau ) by Theorem 3.5, ({y}_{j},{g}_{j})\notin {\tilde{H}}_{0}(\tau ), and ({y}_{j},{g}_{j})\notin \tilde{H}(\tau ), j=1,2. We claim that
In fact, for each given (x,f)\in {\stackrel{\u02c6}{H}}_{0}(\tau ), the algebraic system
has a unique solution {({\tilde{c}}_{1},{\tilde{c}}_{2})}^{T}. Let \tilde{x}=x({\tilde{c}}_{1}{y}_{1}+{\tilde{c}}_{2}{y}_{2}) and \tilde{f}=f({\tilde{c}}_{1}{g}_{1}+{\tilde{c}}_{2}{g}_{2}). Then (\tilde{x},\tilde{f})\in {\tilde{H}}_{0}(\tau ). So, every (x,f)\in {\stackrel{\u02c6}{H}}_{0}(\tau ) can be uniquely expressed as a linear combination of some element of {\tilde{H}}_{0}(\tau ), ({y}_{1},{g}_{1}), and ({y}_{2},{g}_{2}). Therefore, (4.4) holds. Similarly, (4.5) can be proved.
Furthermore, there exists ({x}_{j},{f}_{j})\in {({\tilde{H}}_{0}(\tau ))}_{J}^{\ast} such that
where r(t):=p(t)+p(t1)+q(t). Suppose that there exists {c}_{j}\in \mathbf{C} such that {\sum}_{j=1}^{4}{c}_{j}({x}_{j},{f}_{j})\in \tilde{H}(\tau ). Then we get from w(t)\ne 0 for t\in I that
which implies that {\sum}_{j=1}^{4}{c}_{j}{x}_{j}(t)=0, a1\le t\le b+1. Therefore,
It can be obtained from (4.6) and (4.7) that {c}_{j}=0. So, ({x}_{1},{f}_{1}),\dots ,({x}_{4},{f}_{4}) are linearly independent (modulo \tilde{H}(\tau )). Further, we claim that
where U=span\{({x}_{1},{f}_{1}),({x}_{2},{f}_{2}),({x}_{3},{f}_{3}),({x}_{4},{f}_{4})\}. In fact, it is evident that \tilde{H}(\tau )\dot{+}U\subset {({\tilde{H}}_{0}(\tau ))}_{J}^{\ast}. Now, we show {({\tilde{H}}_{0}(\tau ))}_{J}^{\ast}\subset \tilde{H}(\tau )\dot{+}U. For each given (x,f)\in {({\tilde{H}}_{0}(\tau ))}_{J}^{\ast}, the algebraic system
has a unique solution {({\tilde{c}}_{1},{\tilde{c}}_{2},{\tilde{c}}_{3},{\tilde{c}}_{4})}^{T}. Let \tilde{x}=x{\sum}_{j=1}^{4}{\tilde{c}}_{j}{x}_{j} and \tilde{f}=f{\sum}_{j=1}^{4}{\tilde{c}}_{j}{f}_{j}. Then (\tilde{x},\tilde{f})\in \tilde{H}(\tau ). So, every (x,f)\in {({\tilde{H}}_{0}(\tau ))}_{J}^{\ast} can be uniquely expressed as a linear combination of some element of \tilde{H}(\tau ), ({x}_{1},{f}_{1}),\dots ,({x}_{4},{f}_{4}). Therefore, {({\tilde{H}}_{0}(\tau ))}_{J}^{\ast}\subset \tilde{H}(\tau )\dot{+}U and hence (4.8) holds.
Since ({y}_{1},{g}_{1}) and ({y}_{2},{g}_{2}) are linearly independent (modulo \tilde{H}(\tau )), it follows from (4.5) that dimH(\tau )/\tilde{H}(\tau )=2. Further, from (4.8),
Then \tilde{H}(\tau )\subset H(\tau )\subset {({\tilde{H}}_{0}(\tau ))}_{J}^{\ast} implies
Since
we get from (3.13), (4.4), and (4.9) that
which together with Lemma 4.3 implies that d={d}_{a}+{d}_{b}2. So, Theorem 4.1 holds. This completes the proof. □
5 Jselfadjoint subspace extensions of {H}_{0}(\tau )
By [[26], Theorem 4.3], {H}_{0}(\tau ) must have JSSEs since it is JHermitian. In this section, we give a complete characterization of all the JSSEs of {H}_{0}(\tau ) in terms of boundary conditions. This section consists of two subsections.
5.1 The general case
The discussion is divided into three cases: d=0, d=1, and d=2, which are equivalent to {d}_{a}={d}_{b}=1, {d}_{a}=1, {d}_{b}=2 or {d}_{a}=2, {d}_{b}=1, and {d}_{a}={d}_{b}=2, respectively, by Theorem 4.1.
The following result can be directly derived from Theorem 2.1 and Theorem 3.3.
Theorem 5.1 In the case of d=0, i.e., {d}_{a}={d}_{b}=1, {H}_{0}(\tau ) is a Jselfadjoint operator.
Theorem 5.2 In the case of d=1 with {d}_{a}=2 and {d}_{b}=1, let {\varphi}_{1} and {\varphi}_{2} be any two linearly independent solutions of (1.1). Then {H}_{1} is a JSSE of {H}_{0}(\tau ) (i.e., {H}_{00}(\tau )) if and only if there exists a matrix M\in {\mathbf{C}}^{1\times 2} such that M\ne 0 and
Proof Note that {\varphi}_{1}^{},{\varphi}_{2}^{}\in {l}_{w}^{2}({I}_{1}) by Theorem 3.7 and b=+\mathrm{\infty} in this case.
First, consider the sufficiency. Suppose that M=({m}_{1},{m}_{2})\ne 0. Let u={m}_{1}{\varphi}_{1}+{m}_{2}{\varphi}_{2}. It is evident that {u}^{}\in D({H}_{a}(\tau )). Fix any integers {a}_{1} and {b}_{1} with a<{a}_{1}+1<{c}_{0}<{b}_{1}1. By Remark 3.2, there exists \beta =(y,g)\in H(\tau ) such that
We claim that \beta \notin {H}_{0}(\tau ). Suppose on the contrary that \beta \in {H}_{0}(\tau ). Then \beta \in {\stackrel{\u02c6}{H}}_{0}(\tau ). Again by Remark 3.2, there exists ({y}_{j},{g}_{j})\in H(\tau ), j=1,2, such that
So, we get from Lemma 3.4 and \beta \in {\stackrel{\u02c6}{H}}_{0}(\tau ) that
which implies that M=0 since rank {(({\varphi}_{j},{\varphi}_{k})(a1))}_{2\times 2}=2 from Lemma 2.5 and the proof of Theorem 3.6. This contradicts M\ne 0. Hence, \beta \notin {H}_{0}(\tau ). Note that [\beta :\beta ]=0 and d=1. Then, by Theorem 2.1 and Corollary 3.1, the set
is a JSSE of {H}_{0}(\tau ). On the other hand, for any F=(x,f)\in H(\tau ), by Lemma 3.4 one has
which implies that {H}_{1}={H}_{2}. The sufficiency is shown.
Next, consider the necessity. Suppose that {H}_{2} is a JSSE of {H}_{0}(\tau ). By Theorem 2.1, Corollary 3.1, and d=1, there exists some element \beta =(y,g)\in H(\tau ) such that \beta \notin {H}_{0}(\tau ), [\beta :\beta ]=0, and (5.3) holds. By (1) in Theorem 3.6 and (2) in Theorem 3.7, there exist uniquely {y}_{b,0}\in D({\stackrel{\u02c6}{H}}_{b,0}(\tau )) and uniquely {y}_{a,0}\in D({\stackrel{\u02c6}{H}}_{a,0}(\tau )) such that
where {\tilde{c}}_{k}, {c}_{k}, {\tilde{d}}_{k}\in \mathbf{C} and {z}_{k}, {\tilde{z}}_{k}, k=1,2, are defined by (3.19) and (3.26). If {\tilde{d}}_{1}={\tilde{d}}_{2}=0, then it can be obtained from (5.4), (3.19), (3.26), Corollary 3.1, Lemma 3.4, and Theorem 3.5 that for all (x,f)\in {({H}_{0}(\tau ))}_{J}^{\ast} there exists (\stackrel{\u02c6}{x},\stackrel{\u02c6}{f})\in H(\tau ) such that
So, \beta \in {H}_{0}(\tau ), which contradicts \beta \notin {H}_{0}(\tau ). Therefore, {\tilde{d}}_{1}+{\tilde{d}}_{2}>0. Set
Then M\ne 0. Furthermore, for any (x,f)\in H(\tau ), by Lemma 3.4 one has
It follows from (5.4), (3.19), (3.26), and Theorem 3.5 that
So, {H}_{2} determined by (5.3) can be expressed as (5.1). The necessity is proved. The entire proof is complete. □
With a similar argument to Theorem 5.2, one can show the following result.
Theorem 5.3 In the case of d=1 with {d}_{a}=1 and {d}_{b}=2, let {\varphi}_{1} and {\varphi}_{2} be any two linearly independent solutions of (1.1). Then {H}_{1} is a JSSE of {H}_{0}(\tau ) (i.e., {H}_{00}(\tau )) if and only if there exists a matrix N\in {\mathbf{C}}^{1\times 2} such that N\ne 0 and
Theorem 5.4 In the case of d=2, let {\varphi}_{1} and {\varphi}_{2} be any two linearly independent solutions of (1.1). Then {H}_{1} is a JSSE of {H}_{0}(\tau ) (i.e., {H}_{00}(\tau )) if and only if there exist two matrices M,N\in {\mathbf{C}}^{2\times 2} such that
where Φ is defined by (3.25).
Proof Because d=2 is equivalent to {d}_{a}={d}_{b}=2, it follows that {\varphi}_{1}^{} and {\varphi}_{2}^{} are in {l}_{w}^{2}({I}_{1}), and {\varphi}_{1}^{+} and {\varphi}_{2}^{+} are in {l}_{w}^{2}({I}_{2}) and hence {\varphi}_{1} and {\varphi}_{2} are in {l}_{w}^{2}(I).
Step 1. Consider the sufficiency. Let M=({m}_{jk}), N=({n}_{jk}), and
It is evident that {\tilde{u}}_{j},{u}_{j}\in H(\tau ), j=1,2. Choose any integers {a}_{1} and {b}_{1} with a<{a}_{1}+1<{c}_{0}<{b}_{1}1<b. By Lemma 3.3 there exists {\beta}_{j}=({y}_{j},{g}_{j})\in H(\tau ) (j=1,2) such that
By Theorem 3.5, rank\mathrm{\Phi}=2, and rank(M,N)=2, it can be verified that {\beta}_{1} and {\beta}_{2} are linearly independent (modulo {H}_{0}(\tau )). Furthermore, by Lemmas 2.5 and 3.4, (5.6), and (5.8), we have
Therefore, by Theorem 2.1 and Corollary 3.1, it can be concluded that
is a JSSE of {H}_{0}(\tau ). For any F=(x,f)\in H(\tau ),
Lemma 3.4 and (5.10) yield that {H}_{1}={H}_{2}. The sufficiency is proved.
Step 2. Consider the necessity. Suppose that {H}_{2} is a JSSE of {H}_{0}(\tau ). By Theorem 2.1 and Corollary 3.1, there exist two linearly independent (modulo {H}_{0}(\tau )) elements {\beta}_{1} and {\beta}_{2} in H(\tau ) such that [{\beta}_{j}:{\beta}_{k}]=0, j,k=1,2, and (5.9) holds. Note that {\beta}_{j}=({y}_{j},{g}_{j})\in H(\tau ) and hence ({{y}_{j}}^{},{g}_{j}^{})\in {H}_{a}(\tau ) and ({{y}_{j}}^{+},{g}_{j}^{+})\in {H}_{b}(\tau ). By Theorems 3.7 and 3.6, there exist uniquely {\tilde{y}}_{j0}\in D({\stackrel{\u02c6}{H}}_{a,0}(\tau )), {y}_{j0}\in D({\stackrel{\u02c6}{H}}_{b,0}(\tau )), {\tilde{c}}_{jk},{\tilde{n}}_{jk},{c}_{jk},{n}_{jk}\in \mathbf{C} (j=1,2) such that
where {\tilde{z}}_{k} and {z}_{k} are defined by (3.19) and (3.26). Set
We will show that rank(M,N)=2. Otherwise, rank(M,N)<2. Then there exist {c}_{1},{c}_{2}\in \mathbf{C} with {c}_{1}+{c}_{2}>0 such that ({c}_{1},{c}_{2})(M,N)=0, i.e.,
Set \beta =({y}^{\prime},{g}^{\prime})={c}_{1}{\beta}_{1}+{c}_{2}{\beta}_{2}. Then \beta \in H(\tau ) and from (5.13) and Theorem 3.5,
By Theorems 3.6 and 3.7, for y\in D(H(\tau )), {y}^{+} can be uniquely expressed as (3.24) and {y}^{} can be uniquely expressed as (3.28). So, it follows from (5.14) and Theorem 3.5 that ({y}^{\prime},y)(a1)=({y}^{\prime},y)(b)=0 for all y\in D(H(\tau )). This, together with Corollary 3.1 and Lemma 3.4, implies that [\beta :{({H}_{00}(\tau ))}_{J}^{\ast}]=[\beta :H(\tau )]=0. Hence, \beta \in {H}_{0}(\tau ). Consequently, {\beta}_{1} and {\beta}_{2} are linearly dependent (modulo {H}_{0}(\tau )). This is a contradiction. So, rank(M,N)=2. Further, from [{\beta}_{j}:{\beta}_{k}]=0 and Lemmas 2.5 and 3.4, (5.11), and Theorem 3.5, we get that
So, M and N satisfy the second relation of (5.6).
Finally, for any (x,f)\in H(\tau ), it follows from (5.11) and Theorem 3.5 that (5.10) holds with M and N defined by (5.12). So, by Lemma 3.4, {H}_{2} determined by (5.9) can be expressed as (5.7). The necessity is proved. The entire proof is complete. □
5.2 The special cases
In this subsection, we characterize the JSSEs of {H}_{0}(\tau ) in the special cases that one of the two endpoints a and b is finite and that both a and b are finite.
First, consider the case that a is finite and b=+\mathrm{\infty}. By Lemma 3.5, {d}_{a}=2 in this case. Let {\varphi}_{1} and {\varphi}_{2} be two linearly independent solutions of (1.1) satisfying
Then {(({\varphi}_{j},{\varphi}_{k})(a1))}_{2\times 2}=\stackrel{\u02c6}{J} and hence by Lemma 2.5, \mathrm{\Phi}=\stackrel{\u02c6}{J}, where Φ is defined by (3.25) and \stackrel{\u02c6}{J}=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right). It can be obtained from (5.15) that