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:
{l}_{w}^{2}(I):=\{x={\{x(t)\}}_{t=a1}^{b+1}\subset \mathbf{C}:\sum _{t=a}^{b}w(t){x(t)}^{2}<+\mathrm{\infty}\}.
Then {l}_{w}^{2}(I) is a Hilbert space with the inner product
\u3008x,y\u3009:=\sum _{t=a}^{b}\overline{y}(t)w(t)x(t).
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
{\tau}^{+}(x)(t):=\mathrm{\nabla}(\overline{p}(t)\mathrm{\Delta}x(t))+\overline{q}(t)x(t),\phantom{\rule{1em}{0ex}}t\in I.
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.
\begin{array}{r}H(\tau )=\{(x,f)\in {l}_{w}^{2}(I)\times {l}_{w}^{2}(I):\tau (x)(t)=w(t)f(t),t\in I\},\\ {H}_{00}(\tau )=\{(x,f)\in H(\tau )\text{: there exist two integers}{\tilde{t}}_{0},{t}_{0}\in I\text{with}{\tilde{t}}_{0}{t}_{0}\\ \phantom{{H}_{00}(\tau )=\phantom{\rule{0.2em}{0ex}}}\text{such that}x(t)=0\text{for}t\le {\tilde{t}}_{0}\text{and}t\ge {t}_{0}\}.\end{array}
(3.1)
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
{I}_{1}:={\{t\}}_{t=a}^{t={c}_{0}1},\phantom{\rule{2em}{0ex}}{I}_{2}:={\{t\}}_{t={c}_{0}}^{b},
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
{H}_{0}^{\ast}(\tau )={H}_{00}^{\ast}(\tau ),\phantom{\rule{2em}{0ex}}{H}_{a,0}^{\ast}(\tau )={H}_{a,00}^{\ast}(\tau ),\phantom{\rule{2em}{0ex}}{H}_{b,0}^{\ast}(\tau )={H}_{b,00}^{\ast}(\tau ).
(3.2)
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
\begin{array}{r}{({H}_{0}(\tau ))}_{J}^{\ast}={({H}_{00}(\tau ))}_{J}^{\ast},\phantom{\rule{2em}{0ex}}{({H}_{a,0}(\tau ))}_{J}^{\ast}={({H}_{a,00}(\tau ))}_{J}^{\ast},\\ {({H}_{b,0}(\tau ))}_{J}^{\ast}={({H}_{b,00}(\tau ))}_{J}^{\ast}.\end{array}
(3.3)
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
\begin{array}{r}{H}_{00}^{\ast}(\tau )=\{(x,f)\in {l}_{w}^{2}(I)\times {l}_{w}^{2}(I):{\tau}^{+}(x)(t)=w(t)f(t),a+1\le t\le b1\},\\ {H}_{a,00}^{\ast}(\tau )=\{(x,f)\in {l}_{w}^{2}({I}_{1})\times {l}_{w}^{2}({I}_{1}):{\tau}^{+}(x)(t)=w(t)f(t),a+1\le t\le {c}_{0}2\},\\ {H}_{b,00}^{\ast}(\tau )=\{(x,f)\in {l}_{w}^{2}({I}_{2})\times {l}_{w}^{2}({I}_{2}):{\tau}^{+}(x)(t)=w(t)f(t),{c}_{0}+1\le t\le b1\}.\end{array}
(3.4)
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
\u3008f,y\u3009=\u3008x,g\u3009,\phantom{\rule{1em}{0ex}}\mathrm{\forall}(y,g)\in {H}_{00}(\tau ),
(3.5)
which implies that
\sum _{t=a}^{b}[\overline{y}(t)w(t)f(t){\tau}^{+}(\overline{y})(t)x(t)]=0.
(3.6)
On the other hand, by using y\in D({H}_{00}(\tau )), it can be verified that
\sum _{t=a}^{b}[\overline{y}(t){\tau}^{+}(x)(t){\tau}^{+}(\overline{y})(t)x(t)]=0,
which, together with (3.6) and y(a)=y(b)=0 when a and b are finite, implies that
\sum _{t=a+1}^{b1}\overline{y}(t)[w(t)f(t){\tau}^{+}(x)(t)]=0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in D({H}_{00}(\tau )).
So, by Lemma 3.1 we get
{\tau}^{+}(x)(t)=w(t)f(t),\phantom{\rule{1em}{0ex}}a+1\le t\le b1.
(3.7)
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
\begin{array}{r}{({H}_{00}(\tau ))}_{J}^{\ast}=\{(x,f)\in {l}_{w}^{2}(I)\times {l}_{w}^{2}(I):\tau (x)(t)=w(t)f(t),a+1\le t\le b1\},\\ {({H}_{a,00}(\tau ))}_{J}^{\ast}=\{(x,f)\in {l}_{w}^{2}({I}_{1})\times {l}_{w}^{2}({I}_{1}):\tau (x)(t)=w(t)f(t),a+1\le t\le {c}_{0}2\},\\ {({H}_{b,00}(\tau ))}_{J}^{\ast}=\{(x,f)\in {l}_{w}^{2}({I}_{2})\times {l}_{w}^{2}({I}_{2}):\tau (x)(t)=w(t)f(t),{c}_{0}+1\le t\le b1\}.\end{array}
(3.8)
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.
\begin{array}{r}[:]:{l}_{w}^{2}(I)\times {l}_{w}^{2}(I)\times {l}_{w}^{2}(I)\times {l}_{w}^{2}(I)\to \mathbf{C},\phantom{\rule{2em}{0ex}}((x,f),(y,g))\mapsto \u3008f,Jy\u3009\u3008x,Jg\u3009;\\ {[:]}_{a}:{l}_{w}^{2}({I}_{1})\times {l}_{w}^{2}({I}_{1})\times {l}_{w}^{2}({I}_{1})\times {l}_{w}^{2}({I}_{1})\to \mathbf{C},\phantom{\rule{2em}{0ex}}((x,f),(y,g))\mapsto {\u3008f,Jy\u3009}_{a}{\u3008x,Jg\u3009}_{a};\\ {[:]}_{b}:{l}_{w}^{2}({I}_{2})\times {l}_{w}^{2}({I}_{2})\times {l}_{w}^{2}({I}_{2})\times {l}_{w}^{2}(({I}_{2})\to \mathbf{C},\phantom{\rule{2em}{0ex}}((x,f),(y,g))\mapsto {\u3008f,Jy\u3009}_{b}{\u3008x,Jg\u3009}_{b}.\end{array}
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.
\begin{array}{r}{H}_{0}(\tau )=\{(x,f)\in {({H}_{00}(\tau ))}_{J}^{\ast}:[(x,f):{({H}_{00}(\tau ))}_{J}^{\ast}]=0\},\\ {H}_{a,0}(\tau )=\{(x,f)\in {({H}_{a,00}(\tau ))}_{J}^{\ast}:{[(x,f):{({H}_{a,00}(\tau ))}_{J}^{\ast}]}_{a}=0\},\\ {H}_{b,0}(\tau )=\{(x,f)\in {({H}_{b,00}(\tau ))}_{J}^{\ast}:{[(x,f):{({H}_{b,00}(\tau ))}_{J}^{\ast}]}_{b}=0\}.\end{array}
(3.9)
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,
\sum _{t=a}^{b}y(t)w(t)f(t)=0.
(3.10)
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
y(a)w(a)f(a)+y(b)w(b)f(b)=0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}(y,g)\in H(\tau ).
(3.11)
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
0=[(x,f):(y,g)]=\sum _{t=a}^{b}y(t)w(t)f(t)\sum _{t=a}^{b}g(t)w(t)x(t),\phantom{\rule{1em}{0ex}}\mathrm{\forall}(y,g)\in H(\tau ).
(3.12)
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:
\begin{array}{r}{\stackrel{\u02c6}{H}}_{0}(\tau ):=\{(x,f)\in H(\tau ):[(x,f):H(\tau )]=0\},\\ {\stackrel{\u02c6}{H}}_{a,0}(\tau ):=\{(x,f)\in {H}_{a}(\tau ):{[(x,f):{H}_{a}(\tau )]}_{a}=0\},\\ {\stackrel{\u02c6}{H}}_{b,0}(\tau ):=\{(x,f)\in {H}_{b}(\tau ):{[(x,f):{H}_{b}(\tau )]}_{b}=0\}.\end{array}
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
\begin{array}{r}\tau (x)(t)=w(t)f(t),\phantom{\rule{1em}{0ex}}{a}_{1}\le t\le {b}_{1},\\ x({a}_{1}1)={\alpha}_{1},\phantom{\rule{2em}{0ex}}x({a}_{1})={\alpha}_{2},\phantom{\rule{2em}{0ex}}x({b}_{1})={\beta}_{1},\phantom{\rule{2em}{0ex}}x({b}_{1}+1)={\beta}_{2}\end{array}
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
y(t)=\{\begin{array}{ll}{x}_{1}(t),& a1\le t\le {a}_{1},\\ {x}_{2}(t),& {b}_{1}\le t\le b+1,\end{array}\phantom{\rule{2em}{0ex}}g(t)=\{\begin{array}{ll}{f}_{1}(t),& a\le t\le {a}_{1}1,\\ {f}_{2}(t),& {b}_{1}+1\le t\le b.\end{array}
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,
\begin{array}{r}[(x,f):(y,g)]=(x,y)(b)(x,y)(a1),\phantom{\rule{1em}{0ex}}\mathrm{\forall}(x,f),(y,g)\in H(\tau ),\\ {[(x,f):(y,g)]}_{a}=(x,y)({c}_{0}1)(x,y)(a1),\phantom{\rule{1em}{0ex}}\mathrm{\forall}(x,f),(y,g)\in {H}_{a}(\tau ),\\ {[(x,f):(y,g)]}_{b}=(x,y)(b)(x,y)({c}_{0}1),\phantom{\rule{1em}{0ex}}\mathrm{\forall}(x,f),(y,g)\in {H}_{b}(\tau ).\end{array}
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
\begin{array}{r}{\stackrel{\u02c6}{H}}_{0}(\tau )=\{(x,f)\in H(\tau ):(x,y)(a1)=(x,y)(b)=0,\mathrm{\forall}y\in D(H(\tau ))\}.\\ {\stackrel{\u02c6}{H}}_{a,0}(\tau )=\{(x,f)\in {H}_{a}(\tau ):x({c}_{0}1)=x({c}_{0})=0\\ \phantom{{\stackrel{\u02c6}{H}}_{a,0}(\tau )=\phantom{\rule{0.2em}{0ex}}}\mathit{\text{and}}(x,y)(a1)=0,\mathrm{\forall}y\in D({H}_{a}(\tau ))\}.\\ {\stackrel{\u02c6}{H}}_{b,0}(\tau )=\{(x,f)\in {H}_{b}(\tau ):x({c}_{0}1)=x({c}_{0})=0\\ \phantom{{\stackrel{\u02c6}{H}}_{b,0}(\tau )=\phantom{\rule{0.2em}{0ex}}}\mathit{\text{and}}(x,y)(b)=0,\mathrm{\forall}y\in D({H}_{b}(\tau ))\}.\end{array}
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
\begin{array}{r}\mathcal{D}:=\{(y,g)\in H(\tau ):{\tau}^{+}(\frac{1}{w}\tau (y))(t)=w(t)y(t),\mathrm{\forall}a+1\le t\le b1\},\\ {\mathcal{D}}_{b}:=\{(y,g)\in {H}_{b}(\tau ):{\tau}^{+}(\frac{1}{w}\tau (y))(t)=w(t)y(t),\mathrm{\forall}{c}_{0}+1\le t\le b1\},\\ {\mathcal{D}}_{a}:=\{(y,g)\in {H}_{a}(\tau ):{\tau}^{+}(\frac{1}{w}\tau (y))(t)=w(t)y(t),\mathrm{\forall}a+1\le t\le {c}_{0}2\}.\end{array}
Proof Since the proofs are similar, we only prove d=\frac{1}{2}dim\mathcal{D}.
First, it can be verified that
dim{({H}_{0}(\tau ))}_{J}^{\ast}/{H}_{0}(\tau )=dimH(\tau )/{\stackrel{\u02c6}{H}}_{0}(\tau ).
(3.13)
Next, we prove that
H(\tau )={\stackrel{\u02c6}{H}}_{0}(\tau )\oplus \mathcal{D}\phantom{\rule{1em}{0ex}}\text{(orthogonal sum)}.
(3.14)
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
0=\u3008(y,g),(x,f)\u3009=\u3008y,x\u3009+\u3008g,f\u3009,\phantom{\rule{1em}{0ex}}\mathrm{\forall}(x,f)\in {\stackrel{\u02c6}{H}}_{0}(\tau ),
(3.15)
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
{\tau}^{+}(g)(t)=w(t)y(t),\phantom{\rule{1em}{0ex}}\mathrm{\forall}a+1\le t\le b1.
(3.16)
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
{\tau}^{+}(\frac{1}{w}\tau (y))(t)=w(t)y(t),\phantom{\rule{1em}{0ex}}\mathrm{\forall}a+1\le t\le b1.
(3.17)
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
{\tau}^{+}(\frac{1}{w}\tau (y))(t)=w(t)y(t),\phantom{\rule{1em}{0ex}}{c}_{0}+1\le t\le b1,
(3.18)
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
\begin{array}{r}{z}_{1}({c}_{0}1)=1,\phantom{\rule{2em}{0ex}}{z}_{1}({c}_{0})=0,\phantom{\rule{2em}{0ex}}{z}_{1}(t)=0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}{c}_{0}+1\le t\le b+1,\\ {z}_{2}({c}_{0}1)=0,\phantom{\rule{2em}{0ex}}{z}_{2}({c}_{0})=1,\phantom{\rule{2em}{0ex}}{z}_{2}(t)=0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}{c}_{0}+1\le t\le b+1.\end{array}
(3.19)
Note that ({z}_{1},{h}_{1}) and ({z}_{2},{h}_{2}) are linearly independent (modulo {\stackrel{\u02c6}{H}}_{b,0}(\tau )) and
{d}_{b}=\frac{1}{2}dim{H}_{b}(\tau )/{\stackrel{\u02c6}{H}}_{b,0}(\tau ).
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
\begin{array}{r}{x}_{1}(t)=y(t),\phantom{\rule{2em}{0ex}}{x}_{2}(t)=\frac{1}{w(t)}\tau (y)(t),\\ {x}_{3}(t)=\overline{p}(t)\mathrm{\Delta}(\frac{1}{w(t)}\tau (y)(t)),\phantom{\rule{2em}{0ex}}{x}_{4}(t)=p(t)\mathrm{\Delta}y(t),\end{array}
(3.21)
Eq. (3.20) can be rewritten as the following discrete Hamiltonian system:
\tilde{J}\mathrm{\Delta}Y(t)=(P(t)\lambda W(t))R(Y)(t),\phantom{\rule{1em}{0ex}}{c}_{0}\le t<\mathrm{\infty},
(3.22)
where
\begin{array}{c}Y(t)=\left(\begin{array}{c}{x}_{1}(t)\\ {x}_{2}(t)\\ {x}_{3}(t)\\ {x}_{4}(t)\end{array}\right),\phantom{\rule{2em}{0ex}}R(Y)(t)=\left(\begin{array}{c}{x}_{1}(t+1)\\ {x}_{2}(t+1)\\ {x}_{3}(t)\\ {x}_{4}(t)\end{array}\right),\phantom{\rule{2em}{0ex}}P(t)=\left(\begin{array}{cc}C(t)& 0\\ 0& B(t)\end{array}\right),\hfill \\ C(t)=\left(\begin{array}{cc}0& \overline{q}(t+1)\\ q(t+1)& w(t+1)\end{array}\right),\phantom{\rule{2em}{0ex}}B(t)=\left(\begin{array}{cc}0& 1/p(t)\\ 1/\overline{p}(t)& 0\end{array}\right),\phantom{\rule{2em}{0ex}}\tilde{J}=\left(\begin{array}{cc}0& {I}_{2\times 2}\\ {I}_{2\times 2}& 0\end{array}\right),\hfill \end{array}
{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
{l}_{W}^{2}:=\{Y={\{Y(t)\}}_{t={c}_{0}}^{\mathrm{\infty}}\subset {\mathbf{C}}^{4}:\sum _{t={c}_{0}}^{\mathrm{\infty}}R{(Y)}^{\ast}(t)W(t)R(Y)(t)<+\mathrm{\infty}\}
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
\mathrm{\Phi}:={(({\varphi}_{j},{\varphi}_{k})({c}_{0}1))}_{2\times 2}.
(3.25)
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
(\sum _{j=1}^{2}{c}_{j}{z}_{j}+\sum _{j=1}^{2}{c}_{j+1}{\varphi}_{j},\sum _{j=1}^{2}{c}_{j}{h}_{j}+\lambda \sum _{j=1}^{2}{c}_{j+1}{\varphi}_{j})\in {\stackrel{\u02c6}{H}}_{b,0}(\tau ),
then by Theorem 3.5 and {\varphi}_{1},{\varphi}_{2}\in D({H}_{b}(\tau )),
\{\begin{array}{l}{\sum}_{j=1}^{2}{c}_{j}{z}_{j}({c}_{0}1)+{\sum}_{j=1}^{2}{c}_{j+1}{\varphi}_{j}({c}_{0}1)=0,\\ {\sum}_{j=1}^{2}{c}_{j}{z}_{j}({c}_{0})+{\sum}_{j=1}^{2}{c}_{j+1}{\varphi}_{j}({c}_{0})=0,\\ {c}_{3}({\varphi}_{1},{\varphi}_{1})(b)+{c}_{4}({\varphi}_{2},{\varphi}_{1})(b)=0,\\ {c}_{3}({\varphi}_{1},{\varphi}_{2})(b)+{c}_{4}({\varphi}_{2},{\varphi}_{2})(b)=0.\end{array}
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
\begin{array}{r}{\tilde{z}}_{1}({c}_{0}1)=1,\phantom{\rule{2em}{0ex}}{\tilde{z}}_{1}({c}_{0})=0,\phantom{\rule{2em}{0ex}}{\tilde{z}}_{1}(t)=0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}a1\le t\le {c}_{0}2,\\ {\tilde{z}}_{2}({c}_{0}1)=0,\phantom{\rule{2em}{0ex}}{\tilde{z}}_{2}({c}_{0})=1,\phantom{\rule{2em}{0ex}}{\tilde{z}}_{2}(t)=0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}a1\le t\le {c}_{0}2.\end{array}
(3.26)
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)