Stability of deficiency indices for discrete Hamiltonian systems under bounded perturbations

This paper is concerned with stability of deficiency indices for discrete Hamiltonian systems under perturbations. By applying the perturbation theory of Hermitian linear relations we establish the invariance of deficiency indices for discrete Hamiltonian systems under bounded perturbations. As a consequence, we obtain the invariance of limit types for the systems under bounded perturbations. In particular, we build several criteria of the invariance of the limit circle and limit point cases for the systems. Some of these results improve and extend some previous results.


Introduction
In this paper, we consider the following discrete linear Hamiltonian systems with one singular endpoint: J y(t) -P(t)R(y)(t) = λW (t)R(y)(t), t ∈ I, ( 1 . 1 λ ) where I is the integer set {t} +∞ t=0 , J is the canonical symplectic matrix, that is, J = 0 -I n I n 0 , I n is the n × n unit matrix, is the forward difference operator, that is, y(t) = y(t + 1)y(t); the weight function W (t) = diag{W 1 (t), W 2 (t)}, where W 1 (t) and W 2 (t) are n × n nonnegative Hermitian matrices, the matrix P(t) can be written as where A(t), B(t), and C(t) are n × n complex-valued matrices, B(t) and C(t) are Hermitian matrices, A * (t) is the complex conjugate transpose of A(t), the partial right shift operator R(y)(t) = (u T (t + 1), v T (t)) T with y(t) = (u T (t), v T (t)) T and u(t), v(t) ∈ C n , and λ is a complex spectral parameter.
Discrete Hamiltonian systems are of growing interest in recent years because of their wide applications (see [3][4][5][6][7][8][9][10][11][12] and references therein). Although discrete Hamiltonian systems originate from the discretization of continuous Hamiltonian systems, there is an important difference between them. It is well known that under certain condition, the minimal and maximal operators generated by continuous Hamiltonian systems are densely defined and single-valued, respectively [13,14]. However, the minimal and maximal operators generated by the general discrete Hamiltonian systems may be neither densely defined nor single-valued in general even though the definiteness condition is satisfied [5-7, 15, 16]. This fact was ignored in some existing literature including [3,17]. This is an essential difficulty that we would encounter in the study of the stability of deficiency indices for discrete Hamiltonian systems under perturbations because the classical operator theory is not applicable in this case.
To overcome this difficulty, we will apply the theory of linear relations to study system (1.1 λ ). In 1961, Arens [18] initiated the study of linear relations, and his work was followed by many scholars [19][20][21][22][23][24][25][26][27][28][29][30]. In particular, perturbation theory of linear relations has received lots of attention, and some excellent results have been obtained, including stability of closedness, boundedness, self-adjointness, and spectra of linear relations (see [25,[27][28][29][30]). Recently, we studied the stability of deficiency indices of Hermitian relations and obtained several criteria of invariance of deficiency indices of Hermitian relations under relatively bounded perturbations [31]. Then, using our perturbation results, we obtained the invariance of deficiency indices of second-order symmetric linear difference equations under perturbations [32], which can be seen as the simplest example of system (1.1 λ ). In this paper, we apply the results given in [31] to study the stability of deficiency indices for (1.1 λ ).
It is well known that the deficiency indices of symmetric operators or Hermitian relations play a decisive role in their self-adjoint extensions. By the generalized von Neumann theory [19] and the GKN theory [26] a symmetric operator or Hermitian relation has a self-adjoint extension if and only if its positive and negative deficiency indices are equal; moreover, the numbers and types of boundary conditions of its self-adjoint extensions are determined by its deficiency indices. So it is necessary to pay attention to the stability of their deficiency indices under perturbations.
To the best of our knowledge, there seems to be a few results about stability of deficiency indices for discrete Hamiltonian systems under perturbations. In 2013, by using the generalized von Neumann theory Zheng [33] obtained the invariance of the minimal and maximal deficiency indices for (1.1 λ ) with P(t) under bounded perturbations. In the present paper, we apply the perturbation theory of Hermitian relations obtained in [31] to establish several criteria of stability of deficiency indices for (1.1 λ ) with both of P(t) and W (t) under bounded perturbations. Our technique is obviously different from that in [33]. By using it we could obtain the invariance of any deficiency index for (1.1 λ ) with P(t) under bounded perturbations. These results not only cover the results obtained in [33], but also some of them improve or weaken the conditions of the existing results. In addition, we note that almost all criteria for limit types of (1.1 λ ) were established only for the limit point and limit circle cases. However, there are seldom criteria of the intermediate cases for (1.1 λ ). We remark that the results given in the present paper provide an alternate way to determine the limit types of system (1.1 λ ).
The rest of this paper is organized as follows. In Sect. 2, we introduce some notations, basic concepts, and useful fundamental results about linear relations and recall some fundamental results about system (1.1 λ ). In Sect. 3, we establish several criteria of stability of deficiency indices for system (1.1 λ ) under bounded perturbations by using the perturbation theory of Hermitian relations obtained in [31]. As a consequence, we obtain the invariance of limit types for the systems under bounded perturbations. In particular, we build several criteria of the invariance of the limit circle and limit point cases for the systems. Finally, we present an example to illustrate the perturbation results obtained in Sect. 4.

Preliminaries
This section is divided into two parts. In Sect. 2.1, we introduce some notations, basic concepts, and fundamental results about linear relations. In Sect. 2.2, we first recall the maximal, preminimal, and minimal relations corresponding to system (1.1 λ ). Then we list some useful results about (1.1 λ ), which will be used in the sequent sections.

Some notations, concepts, and results about linear relations
By C and R we denote the sets of complex numbers and real numbers, respectively. Let X be a complex Hilbert space with inner product ·, · , and let T be a linear relation in the product space X 2 with the following induced inner product, still denoted by ·, · without any confusion: The domain D(T) and range R(T) of T are respectively defined by Its adjoint is defined by Further, denote It is evident that T(0) = {0} if and only if T can uniquely determine a linear operator from D(T) into X whose graph is T.
A linear relation T is called closed if it is a closed subspace in X 2 , Hermitian if T ⊂ T * , and self-adjoint if T = T * .
Let T and S be two linear relations in X 2 , and let λ ∈ C. Define In the following, we recall concepts of the norm of a linear relation and relatively boundedness of two linear relations.
Let T be a linear relation in X 2 . The quotient space X/T(0) is a Hilbert space [34] with the inner product Now define the natural quotient map Further, definẽ If T < ∞, then T is said to be bounded [30].
Next, we recall a criterion of stability of defect indices of Hermitian relations under relatively bounded perturbations, which will take a key role in the study of stability of deficiency indices for (1.1 λ ) under perturbations.

Some fundamental results about system (1.1 λ )
In this subsection, we first introduce the concepts of maximal, preminimal, and minimal relations and then list some useful results about system (1.1 λ ).
We denote with the semiscalar product Further, we define y := ( y, y ) 1/2 for y ∈ L 2 W (I). Since the weight function W (t) may be singular in I, · is a seminorm. We denote Then L 2 W (I) is a Hilbert space with the inner product ·, · (see [3,Lemma 2.5]). For a function y ∈ L 2 W (I), we denote by [y] the corresponding class in L 2 W (I). Set The natural difference operator corresponding to system (1.1 λ ) is Set where H is called the maximal relation, and H 00 is called the preminimal relation corresponding to L or system (1.1 λ ); H 0 := H 00 is called the minimal relation corresponding to L or to system (1. In particular, L is said to be in the limit point case (l.p.c.) at t = +∞ if d + (H 0 ) = d -(H 0 ) = n, and in the limit circle case (l.c.c.) at t = +∞ if d + (H 0 ) = d -(H 0 ) = 2n. We refer to the cases n < d ± (H 0 ) < 2n as L in the intermediate cases at t = +∞.
By n λ (H 0 ) we denote the number of linearly independent solutions of (1.1 λ ) in L 2 W (I). By [5, Corollary 5.1] we know that n λ (H 0 ) = d λ (H 0 ) if and only if the following condition is satisfied: (A 2 ) There exists a finite subset I 0 := [s 0 , t 0 ] ⊂ I such that for some λ ∈ C and any nontrivial solution y(t) of (1.1 λ ),  1 λ ). It was shown in [5] that if (2.1) holds for some λ ∈ C, then it holds for any λ ∈ C. As pointed out in Sect. 1, the definiteness condition (A 2 ) cannot guarantee H 0 to be densely defined or H to be single-valued. In fact, if there exists t 0 ∈ I such that W (t 0 ) = 0, then H 0 is Hermitian and nondensely defined in . So the classical perturbation theory of symmetric operators is not available in the study of stability of deficiency indices of H 0 under perturbations. We will apply the result about the perturbation of Hermitian relations, that is, Lemma 2.1, to study this problem in the present paper.
where w j (t) is the minimal eigenvalue of W j (t) for j = 1, 2.
Remark 2.2 Theorem 2.1 was significantly generalized in [35]. In the smallest deficiency index case, that is, holds. In addition, if (A 2 ) holds, then n λ (H 0 ) ≡ n for all λ ∈ C.

Main results
In this section, we study the stability of deficiency indices for system (1.1 λ ) with coefficient matrices under bounded perturbations with respect to the weight functions. We establish several criteria of stability of deficiency indices for (1.1 λ ) under bounded perturbations by using the perturbation theory of Hermitian relations obtained in [31]. As a consequence, we obtain the invariance of limit types of (1.1 λ ) under bounded perturbations. In particular, we build several criteria of the invariance of the limit circle and limit point cases for the systems. Consider the perturbed discrete Hamiltonian system with P(t) and W (t) perturbed bỹ P(t) andW (t), respectively, that is, where the weight functionW (t) = diag{W 1 (t),W 2 (t)} with n × n matricesW j (t) ≥ 0, j = 1, 2;P(t) can be written as The rest of this section is divided into two parts based on whether the weight matrix is perturbed.

Stability of deficient indices for (1.1 λ ) in the case ofW(t) = W(t)
In this subsection, we pay our attention on stability of deficiency indices for system (1.1 λ ) in the case ofW (t) = W (t) for t ∈ I. By applying the perturbation theory of Hermitian relations, we establish several criteria of stability of deficiency indices for (1.1 λ ) under bounded perturbations. As a consequence, we obtain the invariance of limit types of (1.1 λ ) under bounded perturbations. Theorem 3.1 Assume that (A 1 ) holds. LetW (t) = W (t) for t ∈ I. If there exist nonnegative constants c j and real-valued functions c j (t) with |c j (t)| ≤ c j for j = 1, 2 such that Proof It follows from (3.2) thatÃ(t) = A(t) for t ∈ I. So (Ã 1 ) holds since (A 1 ) holds. Since the deficient indices of H 0 andH 0 are equal to those of H 00 andH 00 , respectively, it suffices to show that By Remark 2.1 and the assumption thatW (t) = W (t) for t ∈ I it follows that H 00 andH 00 are both Hermitian relations in L 2 W (I) × L 2 W (I). Next, we will prove (3.3) by Lemma 2.1. The proof is divided into three steps.
Based on the discussions in Step 1, for any [y] ∈ D(H 00 ) = D(H 00 ) = D(H 00 -H 00 ), there exist g,g ∈ L 2 W (I) and y ∈ [y] with y ∈ L 2 W ,0 (I) such that  (1) By using the generalized von Neumann theory, Zheng [33] showed the invariance of the minimal and maximal deficiency indices of (1.1 λ ) under assumptions (A 1 ) and (A 2 ),W (t) = W (t) for t ∈ I, and the condition that where B 0 and C 0 are constants. Theorem 3.1 extends the results in [33]. By applying the perturbation theory of Hermitian relations we do not need that condition (A 2 ) holds in Theorem 3.1. Furthermore, we obtain the invariance for any deficient index of (1.1 λ ) under assumption (A 1 ),W (t) = W (t) for t ∈ I, and condition (3.2) in Theorem 3.1. (2) Note that (3.2) implies that c 1 (t) and c 2 (t) are both real-valued functions. In fact, P(t) -P(t) is Hermitian sinceP(t) and P(t) are Hermitian matrices. This, together with W (t) ≥ 0 and (3.2), yields that c 1 (t) and c 2 (t) are real-valued. With a similar argument, B 0 and C 0 in (3.6) also must be real numbers. This fact was ignored in the proof of Corollaries 3.1 and 3.2 in [33], where the author regarded the perturbation as (λλ 0 )W (t) for any λ, λ 0 ∈ C.
Let E and F be two Hermitian matrices. In this paper, we write E ≥ F if E -F ≥ 0. By comparing with Theorem 3.1 the following result imposes a weaker restriction oñ P(t) -P(t) when W (t) =W (t) > 0.
Proof The main idea of the proof is similar to that of Theorem 3.1. It suffices to show that (3.3) holds by Lemma 2.1. The proof is divided into three steps.
In the particular case that W 1 (t) > 0 and W 2 (t) = 0, we obtain the following result.
Proof The main idea of the proof is similar to that of Theorem 3.2 with only Step 1 being replaced by the following: Step 1. We prove that D(H 00 ) = D(H 00 ). It is evident that L 2W ,0 (I) = L 2 W ,0 (I) sinceW (t) = W (t) for t ∈ I. For any [y] ∈ D(H 00 ), there exist y ∈ [y] and g ∈ L 2 W (I) such that y ∈ L 2 W ,0 (I) and (3.4) holds. Next, we will show that [y] ∈ D(H 00 ). To prove [y] ∈ D(H 00 ), it suffices to prove that there existsg ∈ L 2 W (I) such that (3.8) holds. By (3.7) and the assumptions that W 1 (t) =W 1 (t) > 0 and W 2 (t) =W 2 (t) ≡ 0 for t ∈ I, we have that Inserting it into (3.8), we get that for t ∈ I, where y(t) = (y T 1 (t), y T 2 (t)) T , g(t) = (g T 1 (t), g T 2 (t)) T , andg(t) = (g T 1 (t),g T 2 (t)) T with y j (t), g j (t),g j (t) ∈ C n , j = 1, 2. In addition, noting that y ∈ L 2 W ,0 (I) and g ∈ L 2 W (I), we have that t∈I R(g) * (t)W (t)R(g)(t) < +∞. So the corresponding class Remark 3.2 We remark that (3.7) implies that c 1 and c 2 are real numbers since P(t),P(t), and W (t) are Hermitian matrices.
The following result is a direct consequence of Theorems 3.1-3.3.

Corollary 3.1
If any of the conditions of Theorems 3.1, 3.2, and 3.3 hold, then L is in the limit (d + (H 0 ), d -(H 0 )) case at +∞ if and only ifL is in the limit (d + (H 0 ), d -(H 0 )) case at +∞. In particular, L is in l.c.c. at +∞ if and only ifL is in l.c.c. at +∞; L is in l.p.c. at +∞ if and only ifL is in l.p.c. at +∞.

Stability of deficient indices for (1.1 λ ) in the case ofW(t) = W(t)
In this subsection, we study the stability of deficiency indices for system (1.1 λ ) wheñ W (t) = W (t) for t ∈ I. Note that H 0 andH 0 are defined in (L 2 W (I)) 2 and (L 2W (I)) 2 , respectively, and it is difficult to study the stability of deficiency indices of H 0 andH 0 since (L 2 W (I)) 2 and (L 2W (I)) 2 are different spaces. So we turn to study the invariance of the limit circle and limit point cases.  (3.11) then L is in l.c.c. at +∞ if and only ifL is in l.c.c. at +∞.
Proof First, consider the necessity. Suppose that L is in l.c.c. at +∞. Then d ± (H 0 ) = 2n. By L is in l.c.c. at +∞.
With a similar argument, the sufficiency can be shown by noting that (3.11) implies This completes the proof.
Introduce the following new system: If we regard (3.12 λ ) as the perturbation of (1.1 λ ) and (3.1 λ ) as the perturbation of (3.12 λ ), then the following two results can be directly derived by Theorem 3.4 and Corollary 3.1.   Remark 3.4 Note that the number of linearly independent square summable solutions of (1.1 λ ) is invariant if its coefficient matrices P(t) and W (t) vary at finite points. Therefore we remark that the results obtained in the present paper still hold if there exists t 0 ∈ I such that the assumptions of our results are satisfied for t ≥ t 0 .

Examples
In this section, we give an example illustrating the perturbation results obtained in this paper. It is well known that there are a lot of limit circle and limit point criteria for system (1.1 λ ). The criteria of stability of deficiency indices of (1.1 λ ) obtained in this paper provide an alternate way to determine the extreme limit type. However, to the best of our knowledge, there are seldom criteria of the intermediate cases for (1.1 λ ). Therefore it is very difficult to determine if (1. for t ∈ I. It is obvious that assumptions (A 1 ) and (A 2 ) hold. So, d λ (H 0 ) = n λ (H 0 ). Next, we will show that (1.1 λ ) with those matrix functions is in the limit-3 case at t = +∞. In fact, by Lemma 2.1 in [37], n λ (H 0 ) is equal to the number of linearly independent solutions of the following fourth-order difference equation: in l 2 for any λ ∈ C. The solutions of (4.2 λ ) with λ = 0 are of the form α t with α satisfying the equation 6α 2 -5α + 1 3α 2 -5α + 2 = 0.
nor in l.c.c. at t = +∞ by Theorem 2.1 and Remark 2.2. So (1.1 λ ) is in the limit-3 case at t = +∞. Now we consider system (3.1 λ ) with n = 2, A(t) = 0 1 0 0 ,B(t) = diag 0, 3 -t /2 , C(t) = diag sin t + e -t -2, -3 t ,W (t) = diag{1, 0, 0, 0} for t ∈ I. It is evident that assumptions (Ã 1 ) and (Ã 2 ) still hold. But it is difficult to determine the limit type of this system as discussed before. As pointed out in the first paragraph of this section, almost all criteria for limit types were established only for the limit point and limit circle cases. However, there are seldom criteria of the intermediate cases.
In addition, we note that the existing criteria for limit types are not applicable to (3.1 λ ) with matrix functions satisfying (4.3). At this point, we find that we may regard (4.3) as the perturbations of (4.1). Moreover, it is easy to verify that the conditions of Theorem 3.1 are satisfied with c 1 (t) = 2 -sin te -t , c 2 (t) = c 2 = 0, and c 1 = 4. Therefore (3.1 λ ) with (4.3) is also in the limit-3 case at t = +∞ by Theorem 3.1.