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Defect index of the square of a formally selfadjoint secondorder difference expression
Advances in Difference Equations volume 2014, Article number: 48 (2014)
Abstract
This paper is concerned with the defect index of the square of a formally selfadjoint secondorder difference expression with real coefficients, which, in fact, is a class of formally selfadjoint fourthorder difference expressions. Sufficient and necessary conditions for such fourthorder difference expression to be a limit2 case, a limit3 case, and a limit4 case are given, with respect to the limit case of the secondorder difference expression. These results parallel the wellknown results of Everitt and Chaudhuri for differential expressions.
MSC:39A10, 34B20.
1 Introduction
In this paper we discuss properties of the defect index of a class of fourthorder formally selfadjoint difference expressions, which is derived from squaring the secondorder difference expression,
where
Δ and ∇ are 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); the discrete time interval ℐ is bounded from below; without loss generality, we denote \mathcal{I}={[0,+\mathrm{\infty})}_{\mathbb{Z}}; and the functions p(t), q(t) are all realvalued and p(t)\ne 0 for t\ge 1.
According to the classical von Neumann theory (cf. [1, 2]) and its generalization (cf. [3]), a symmetric operator or a nondensely defined Hermitian operator has a selfadjoint extension if and only if its positive and negative defect indices are equal and its selfadjoint extension domains have a close relationship with its defect index. So it is very important to determine the defect indices of both differential and difference expressions in the study of selfadjoint extensions.
The problem on the defect index of the secondorder formally selfadjoint linear differential expression with real coefficients
was first studied by Weyl [4]. It is well known that the defect index of ℒ is equal to the number of linearly independent square integrable solutions of equation \mathcal{L}y=\lambda y for each \lambda \in \mathbb{C}\setminus \mathbb{R}, where ℂ and ℝ denote the sets of the complex and real numbers. Later, some authors studied the defect index of {\mathcal{L}}^{2}, which, in fact, is a class of fourthorder formally selfadjoint linear differential expressions with real coefficient, and they have obtained a few excellent results [5–8].
The study of secondorder difference expressions L began with Atkinson’s work [9] and the properties of its defect index have been sufficiently discussed. It is well known that the defect index, say {d}_{L} is equal to the number of linearly independent solutions which are square summable of the difference equation
for any \lambda \in \mathbb{C}\setminus \mathbb{R}. The value range of defect index {d}_{L} is one or two. The secondorder difference expression L is called a limitpoint case at t=+\mathrm{\infty} if {d}_{L}=1, that is, (1.3) has just one solution which is square summable for any point \lambda \in \mathbb{C}\setminus \mathbb{R}; and it is a limitcircle case at t=+\mathrm{\infty} if {d}_{L}=2, that is, (1.3) has two linearly independent solutions which are square summable for any λ, real or complex. All difference expressions of the form (1.2) come within the limitpoint, limitcircle classification which depends only on the coefficients p and q and not on the parameter λ. Several criteria of the limitpoint and limitcircle cases have been established [10–13].
The general form of a formally selfadjoint fourthorder difference expression with real coefficients is
where {p}_{j} are all realvalued for 0\le j\le 2, and {p}_{2}(t)\ne 0 for all t\in \mathcal{I}.
Similarly as that of the secondorder difference expression L, the defect index, say {d}_{M}, of M is equal to the number of linear independent solutions which are square summable of the difference equation
for any \lambda \in \mathbb{C}\setminus \mathbb{R}. For such values of λ, the difference equation (1.5) may have two, three or four linearly independent solutions in {l}^{2} and for any particular M the number of such solutions is independent of λ. So all such difference expressions as M can be classified into three cases: limit2 (limitpoint), limit3 and limit4 (limitcircle) cases. In the limitcircle case all solutions are square summable whether ℑλ is zero or not.
In comparison with the secondorder difference expressions L, fewer criteria for the limit case of the difference expressions M have become known. Recently, it has been shown that all values of the defect index from 2 and 4 can be realized and some criteria for the limitpoint case were given in [14].
In this paper, we focus on the fourthorder difference expression {L}^{2} and discuss the relationship of the limit cases between L and {L}^{2}. It is worth noting that, different from the differential expression, the maximal operator generated by the difference expression L or M may be multivalued, and the minimal operator may be nondense [15, 16]. To solve this problem, we will apply the theory of subspaces to discuss the spectral theory of such a difference expression.
The rest of the paper is organized as follows. In Section 2, some preliminary work is given, including some basic concepts and useful results as regards subspaces, and the known result of the secondorder difference expression L and the fourthorder difference expression M. In Section 3, we pay attention to the defect index of the difference expression {L}^{2}. Sufficient and necessary conditions for {L}^{2} to be limit2 (limitpoint), limit3 and limit4 (limitcircle) cases are given, separately. These results parallel the Chaudhuri and Everitt’s result for differential expressions [5]. In the special case p(t)\equiv 1, which covers a number of examples which arise in practice, we establish a criterion for both L and {L}^{2} to be a limitpoint case. In the final section, i.e., Section 4, some examples are given to show that all the cases of the difference expressions L and {L}^{2} considered in Section 3 can be realized.
2 Preliminaries
In this section, we first recall some basic concepts and useful results about subspaces. The readers are referred to [3, 16].
Let X be a complex Hilbert space equipped with inner product \u3008\cdot ,\cdot \u3009, T be a linear subspace (briefly, subspace) in {X}^{2}:=X\times X, and \lambda \in \mathbb{C}. Denote
It can easily be verified that T(0)=\{0\} if and only if T can determine a unique linear operator from D(T) into X whose graph is just T. For convenience, we will identify a linear operator in X with a subspace in {X}^{2} via its graph.
Definition 2.1 [3]
Let T be a subspace in {X}^{2}.

(1)
The adjoint of T is defined by
{T}^{\ast}:=\{(y,g)\in {X}^{2}:\u3008y,f\u3009=\u3008g,x\u3009\text{for all}(x,f)\in T\}. 
(2)
T is said to be a Hermitian subspace if T\subset {T}^{\ast}. T is said to be a selfadjoint subspace if T={T}^{\ast}.

(3)
Let T be a Hermitian subspace. {T}_{1} is said to be a selfadjoint subspace extension (SSE) of T if T\subset {T}_{1} and {T}_{1} is a selfadjoint subspace.
Definition 2.2 [16]
Let T be a subspace in {X}^{2}. R{(T\lambda I)}^{\mathrm{\perp}} is called the defect space of T and λ, and dim(R{(T\lambda I)}^{\mathrm{\perp}}) is called the defect index of T and λ.
Lemma 2.1 A Hermitian subspace T in {X}^{2} with R(T)=X is a selfadjoint subspace.
Proof It suffices to show that {T}^{\ast}\subset T.
For any (y,g)\in {T}^{\ast}, since R(T)=X, there exists (h,g)\in T. Consequently, for any (x,f)\in T,
Again by R(T)=X, it follows that h=y, which implies that (y,g)\in T, and consequently, {T}^{\ast}\subset T. Hence, T is a selfadjoint subspace. □
Let T and S be a subspace in {X}^{2}. The product of T and S is defined by
Lemma 2.2 If T is a closed subspace in {X}^{2}. Then {T}^{\ast}T is a selfadjoint subspace in {X}^{2}.
Proof To show the result, we introduce an operator U in {X}^{2}, similarly to that given for a graph of an operator (see [[17], §51]), by putting
It is clear that U is a unitary operator and {U}^{2}=I, where I is the identity operator in {X}^{2}.
Since T is a closed subspace in {X}^{2}, so is UT. Therefore, the following formula is true:
Hence, by applying the operator U and {U}^{2}T=T, we have
Now, for any (h,0)\in {X}^{2}, it can be expressed uniquely in the form
or
where (x,f)\in T and (y,g)\in {T}^{\ast}. This yields
and, consequently,
where I is the identity operator in X. Thus, the range of {T}^{\ast}T+I coincides with the whole space X. Therefore, by Lemma 2.1, {T}^{\ast}T+I is a selfadjoint subspace in {X}^{2} and since {T}^{\ast}T=({T}^{\ast}T+I)I, one sees that {T}^{\ast}T also is selfadjoint. □
Corollary 2.1 If T is a selfadjoint subspace in {X}^{2}, then so is {T}^{n}, where n\ge 2 is any integer.
Next we introduce some notation for (1.1) and (1.2). Denote
Clearly {l}^{2} is a Hilbert space with the inner product
where y=z in {l}^{2} if and only if \parallel yz\parallel =0, i.e., y(t)=z(t), t\in \mathcal{I}, while \parallel \cdot \parallel is the induced norm.
The Green’s formula for L is
where
The corresponding maximal and preminimal subspaces to L were defined in some existing literature (e.g., [16]) by
and
respectively, and the minimal subspace was defined by {H}_{L,0}={\overline{H}}_{L,00}, where {\overline{H}}_{L,00} is the closure of {H}_{L,00}. It has been shown in [16] that {H}_{L} may be multivalued, and {H}_{L,00} and {H}_{L,0} are only nondensely defined Hermitian operators in {l}^{2}.
A sufficient condition for L to be a limitpoint case has been given.
Lemma 2.3 [10]
L is a limitpoint case at t=+\mathrm{\infty} if
A complete characterization of selfadjoint extension of {H}_{L,0} has been given in terms of boundary conditions. Here we recall one result which will be used.
Lemma 2.4 [16]
Assume that L is a limitpoint case at t=+\mathrm{\infty}. {H}_{L,1} is a SSE of {H}_{L,0} if and only if there exist real numbers a and b with {a}^{2}+{b}^{2}\ne 0 such that
Next, we consider the fourthorder formally selfadjoint difference expressions M. The Green formula for M is
where
The corresponding maximal and preminimal subspaces to M are defined as follows:
and the minimal subspace is defined by {H}_{M,0}={\overline{H}}_{M,00}. The following is a characterization of selfadjoint extension of {H}_{M,0} in the case when M is a limitpoint case at t=+\mathrm{\infty}.
Lemma 2.5 [18]
Assume that M is a limitpoint case at t=+\mathrm{\infty}. {H}_{M,1} is a SSE of {H}_{M,0} if and only if there exists a matrix {G}_{2\times 4} , satisfying rankG=2, GJ{G}^{\ast}=0, and
where
and {I}_{2} is the 2\times 2 identity matrix.
Finally, in this section, we make it clear that {L}^{2} is a special case of the fourthorder difference expressions M. In fact, from (1.2), one has
where
Further, applying L to (2.3), one has
where
Further, one has by formula (2.5) in [14] that
where
3 Main results
In this section, we first establish a sufficient and necessary condition for {L}^{2} to be limitcircle case.
Theorem 3.1 {L}^{2} is a limitcircle case at t=+\mathrm{\infty} if and only if L is a limitcircle case at t=+\mathrm{\infty}.
Proof We first consider the sufficiency. Suppose that L is a limitcircle case at t=+\mathrm{\infty}. Choose a complex number μ such that the roots of {\lambda}^{2}=\mu, say {\lambda}_{1} and {\lambda}_{2}, are distinct. For each j=1,2, let {x}_{1}^{j}, {x}_{2}^{j} be a fundamental set of solution of the difference equation
It can easily be verified that {x}_{k}^{j}, k=1,2, j=1,2, form a fundamental set of solutions of
Since L is a limitcircle case at t=+\mathrm{\infty}, {x}_{k}^{j}\in {l}^{2}, k=1,2, j=1,2. Hence, {L}^{2} is a limitcircle case at t=+\mathrm{\infty}.
Next we consider necessity. If {L}^{2} is a limitcircle case at t=+\mathrm{\infty}, we can choose μ and {\lambda}_{j} as above and conclude that all solutions of difference equation L[x](t)={\lambda}_{1}x(t) are in {l}^{2}. Therefore, L is a limitcircle case at t=+\mathrm{\infty}. The proof is complete. □
The following is a direct conclusion derived from Theorem 3.1.
Corollary 3.1 {L}^{2} is a limitpoint case or limit3 case at t=+\mathrm{\infty} if and only if L is a limitpoint case at t=+\mathrm{\infty}.
Theorem 3.1 completely describes the limit classification of {L}^{2} in the limitcircle case. So we will assume, for the rest of this paper, that L is a limitpoint case at t=+\mathrm{\infty}. In addition, we will assume, in the following of this paper, that μ is complex with \mathrm{\Im}\mu \ne 0, and that {\lambda}_{j}, j=1,2, are the two distinct complex roots of {\lambda}^{2}=\mu with \mathrm{\Im}{\lambda}_{j}\ne 0 for j=1,2. Such a choice of μ is always possible.
A sufficient and necessary condition for {L}^{2} to be a limitpoint case is obtained.
Theorem 3.2 Let L be a limitpoint case at t=+\mathrm{\infty}. {L}^{2} is a limitpoint case at t=+\mathrm{\infty} if and only if
Proof First, we consider the sufficiency. Suppose that (3.3) holds.
Consider then the two difference equations (3.1) for j=1,2. Since L is in the limitpoint case at t=+\mathrm{\infty} and each \mathrm{\Im}{\lambda}_{j}\ne 0, there will be two linearly independent solutions, {\varphi}_{j} and {\psi}_{j}, of (3.1) which satisfy {\varphi}_{j}\in {l}^{2} and {\psi}_{j}\notin {l}^{2}. In addition, by repeated application of L to (3.1), we see that {\varphi}_{j} and {\psi}_{j} are also solutions of (3.2).
Since {\lambda}_{i}\ne {\lambda}_{j} with i\ne j, it can easily be shown that these four solutions, {\{{\varphi}_{j},{\psi}_{j}\}}_{j=1}^{2}, of (3.2) are linearly independent on ℐ and so forms a basis of solutions for (3.2).
Suppose now that the result to be proved is not true, i.e. that {L}^{2} is not a limitpoint case at t=+\mathrm{\infty}. Then, from the TitchmarshWeyl theory of difference equations [19] or [20, 21], it follows that (3.2) must have exactly three linearly independent solutions in {l}^{2}. Since {\{{\varphi}_{j},{\psi}_{j}\}}_{j=1}^{2} is a basis of solutions for (3.2) and since {\varphi}_{j}\in {l}^{2} for j=1,2,\dots ,n, it follows that there must be at least one linearly independent solution, say ψ, of (3.2) which is of the form
such that not all the {c}_{j} are zero and \psi \in {l}^{2}.
Since ψ is a solution of (3.2), it follows that {L}^{2}\psi \in {l}^{2}. So by the assumption (3.3), L\psi \in {l}^{2}. So we have from this and the fact that \psi \in {l}^{2},
Since the matrix
is nonsingular, one can see from (3.4) that {c}_{j}{\psi}_{j}\in {l}^{2} for each j=1,2. Since {\psi}_{j}\notin {l}^{2} for all such j, it then follows that {c}_{1}={c}_{2}=0 and this is a contradiction with the assumption. Hence {L}^{2} is a limitpoint case at t=+\mathrm{\infty}.
Next we consider the necessity. Suppose that {L}^{2} is a limitpoint case at t=+\mathrm{\infty}. Since L is a limitpoint case at t=+\mathrm{\infty}, by Lemma 2.4, {H}_{L,1}, defined by (2.1), is a selfadjoint subspace extension of {H}_{L,0}.
Consider the square {H}_{L,1}^{2}, of the selfadjoint subspace {H}_{L,1}. It is known by Corollary 2.1 that {H}_{L,1}^{2} is also a selfadjoint subspace in {l}^{2}\times {l}^{2}, which can be characterized by
On the other hand, consider the subspace {H}_{{L}^{2},1}, generated by {L}^{2} and the boundary conditions
A calculation shows that {H}_{{L}^{2},1} is a selfadjoint subspace derived from {L}^{2}. In fact, by taking G={({g}_{ij})}_{2\times 4} with
one can verify that rankG=2, GJ{G}^{\ast}=0, and Gu(0)=0. So by Lemma 2.5, {H}_{{L}^{2},1} is a selfadjoint subspace derived from {L}^{2}.
Thus, {H}_{L,1}^{2} and {H}_{{L}^{2},1} are both selfadjoint subspaces derived from {L}^{2} with {H}_{L,1}^{2}\subset {H}_{{L}^{2},1}. Since no selfadjoint subspace can have a strict selfadjoint restriction, one must have {H}_{L,1}^{2}={H}_{{L}^{2},1}. If now we compare D({H}_{L,1}^{2}) and D({H}_{{L}^{2},1}), we see that x\in D({H}_{{L}^{2},1}) implies that Lx\in {l}^{2}.
For any x\in {l}^{2} satisfying {L}^{2}x\in {l}^{2}, we can redefine (if necessary) the values of x on {[2,2]}_{\mathbb{Z}}, which are finite points, so that the boundary conditions in (3.5) are satisfied. Let the new function be \tilde{x}. Then \tilde{x}\in D({H}_{{L}^{2},1}) and consequently, L\tilde{x}\in {l}^{2}. Since Lx and L\tilde{x} are different at finite points, it follows that Lx\in {l}^{2}. Hence, condition (3.3) holds.
The whole proof is complete. □
By the proof of Theorem 3.2, one can find some relationship of solutions of equations (3.1) and (3.2). Next, by exploiting this relationship, we prove a result which gives a necessary and sufficient condition for {L}^{2} to be a limit3 case at t=+\mathrm{\infty}.
Theorem 3.3 Assume that L is a limitpoint case at t=+\mathrm{\infty}. Let {\psi}_{j}\notin {l}^{2} be a solution of the equation Lx={\lambda}_{j}x for j=1,2. Then {L}^{2} is a limit3 case at t=+\mathrm{\infty} if and only if there exists a unique constant k\ne 0, such that
Proof Here it is worth noting that since {\psi}_{1}\notin {l}^{2} and {\psi}_{2}\notin {l}^{2} it follows that if (3.6) holds, then k is unique and not zero.
We first consider sufficiency. Assume that (3.6) holds. Define the function \psi (t) by
It can easily be verified that
It follows that L\psi \notin {l}^{2} since k is unique and not zero, while {L}^{2}\psi \in {l}^{2}. Now if {L}^{2} was in the limitpoint case, then it would follow from Theorem 3.2 that L\psi \in {l}^{2}, which is a contradiction. In addition, it follows from Theorem 3.1 that {L}^{2} is not a limitcircle case. So it must be a limit3 case at t=+\mathrm{\infty}.
Next, we consider the necessity. Suppose that {L}^{2} is a limit3 case at t=+\mathrm{\infty}. Let {\varphi}_{j}\in {l}^{2} be a solution of (3.1) for j=1,2. Thus we get four solutions, {\varphi}_{1}, {\varphi}_{2}, {\psi}_{1}, {\psi}_{2}, of (3.2). By the discussion in the proof of Theorem 3.2, it follows that these four solutions are linearly independent on ℐ and so they form a basis of solutions for (3.2). Since {L}^{2} is a limit3 case at t=+\mathrm{\infty}, there exists exactly one solution which belongs to {l}^{2} and is linearly independent of {\varphi}_{1} and {\varphi}_{2}, say ψ, of (3.2), which is a linear combination of {\psi}_{1} and {\psi}_{2}, i.e.,
It is evident that {c}_{1}\ne 0 since {\psi}_{2}\notin {l}^{2}. Define \tilde{\psi}={c}_{1}^{1}\psi. Then \tilde{\psi} is of form (3.6).
The whole proof is complete. □
Theorem 3.4 Assume that the equation {L}^{2}x=0 on ℐ has exactly three linearly independent solutions in {l}^{2}. Then {L}^{2} is a limit3 case at t=+\mathrm{\infty} and L is a limitpoint case at t=+\mathrm{\infty}.
Proof Since the equation {L}^{2}[x](t)=0 has exactly three linearly independent solutions in {l}^{2}, it follows that {d}_{{L}^{2}}\ge 3 by [[15], Corollary 5.2]. In addition, one has {d}_{{L}^{2}}\ne 4, otherwise, {L}^{2}[x](t)=0 would have four linearly independent solutions in {l}^{2}. Hence, {d}_{{L}^{2}}=3 and {L}^{2} is a limit3 case at t=+\mathrm{\infty}, and consequently L is a limitpoint case at t=+\mathrm{\infty} by Corollary 3.1. □
In the special case when p(t)\equiv 1, we have the following result.
Theorem 3.5 Assume that p(t)\equiv 1 for t\ge 1. If q(t)+q(t1) is bounded on ℐ, then both L and {L}^{2} are a limitpoint case.
Proof By (2.4) and (2.5), {L}^{2} takes the form
with
Firstly, L is a limitpoint case by Lemma 2.4, since p(t)\equiv 1 for t\ge 1. Secondly, q(t)+q(t1) is bounded on ℐ implies that condition (3.1) in [14] is satisfied (with respect to n=2 and \sigma (t)=1). Hence, {L}^{2} is a limitpoint case by [[14], Theorem 3.3]. □
4 Examples
We finally give some examples to show that all the cases of difference expressions L and {L}^{2} can be realized.

(1)
Both L and {L}^{2} are a limitpoint case at t=+\mathrm{\infty} if
p(t)=1,\phantom{\rule{2em}{0ex}}q(t)=0,\phantom{\rule{1em}{0ex}}t\ge 1
or
In fact, L is a limitpoint case by Lemma 2.3 and {L}^{2} is a limitpoint case by Theorem 3.5.

(2)
Both L and {L}^{2} are a limitcircle case at t=+\mathrm{\infty} if
p(t)=q(t)={4}^{t1},\phantom{\rule{1em}{0ex}}t\ge 0.
L is a limitcircle case by [[10], Example 3.2], and consequently {L}^{2} is a limitcircle case by Theorem 3.1.

(3)
L is a limitpoint case and {L}^{2} is a limit3 case at t=+\mathrm{\infty} if
p(t)={2}^{t},\phantom{\rule{2em}{0ex}}q(t)={2}^{t},\phantom{\rule{1em}{0ex}}t\ge 1.
We first show that L is a limitpoint case. The solutions of the difference equation Lx=0 are of the form {a}^{t} with a satisfying the equation 2{a}^{2}4a+1=0. The two roots of this equation are
and then {x}_{i}(t)={a}_{i}^{t} are two solutions of L[x]=0. It is evident that {x}_{1}\notin {l}^{2} and {x}_{2}\in {l}^{2}. Thus L is not a limitcircle case and so it must be a limitpoint at t=+\mathrm{\infty}.
Similarly, solutions of the equation {L}^{2}[x]=0 are also of the form {a}^{t} with a satisfying
The roots of (8{a}^{2}8a+1)=0 are
If {x}_{1} and {x}_{2} are defined as before and {x}_{j}(t)={a}_{j}^{t} for j=3,4, then we can see that {x}_{j}\in {l}^{2} for j=2,3,4, but {x}_{1} is not in {l}^{2}. Thus equation {L}^{2}[x]=0 has just three linearly independent solutions in {l}^{2}. This implies that {L}^{2} is not a limitpoint case at t=+\mathrm{\infty}.
On the other hand, a calculation shows that
Since {a}_{3}^{2}4{a}_{3}+1\ne 0 and 2{a}_{3}>1, it follows that L({x}_{3})\notin {l}^{2}. This shows that {L}^{2}[{x}_{3}]\in {l}^{2} cannot imply L[{x}_{3}]\notin {l}^{2}. So by Theorem 3.2, {L}^{2} is not a limitpoint case. Hence, {L}^{2} is a limit3 case at t=+\mathrm{\infty}.
Author’s contributions
The author worked on the results independently.
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This research was supported by the NNSF of China (Grants 11226160, 11301304, 11071143, 11101241).
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Ren, G. Defect index of the square of a formally selfadjoint secondorder difference expression. Adv Differ Equ 2014, 48 (2014). https://doi.org/10.1186/16871847201448
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DOI: https://doi.org/10.1186/16871847201448