Partition of the spectra for the lower triangular double band matrix as generalized difference operator Δv over the sequence spaces c and lp (1 < p < ∞)

Let the sequence (vk) is assumed to be either constant or strictly decreasing sequence of positive real numbers satisfying limk?? vk = L > 0 and supk vk ? 2L. Then the generalized difference operator ?v is ?v x = ?v (xn) = (vnxn ? vn?1xn?1)? n=0 with x?1 = v?1 = 0. The aim of this paper is to obtain the approximate point spectrum, the defect spectrum and the compression spectrum of the operator ?v and modified of the operator ?v on the sequence spaces c and ????p (1 < p < ?).


Introduction
We know that there exists strictly the relationship between matrices and operators. The eigenvalues of matrices have been contained spectrum of an operator. The spectral theory is one of the most useful tools in science. There exist many its applications in mathematics and physics which contain matrix theory, control theory, function theory, differential and integral equations, complex analysis, and quantum physics. For example, atomic energy levels are determined and therefore the frequency of a laser or the spectral signature of a star are obtained by it in quantum mechanics. The resolvent set of the band operators is important for solving in above explanations problems. Band matrices emerge in many areas of mathematics and its applications. Tridiagonal, or more general, banded matrices are used in telecommunication system analysis, finite difference methods for solving partial differential equations, linear recurrence systems with non-constant coefficients, etc, (see [27]).
Quite recently, many authors have studied several types of spectra which have important applications; for example, the approximate point spectrum, defect spectrum, compression spectrum, essential spectrum, etc.
Let L : X → Y be a bounded linear operator where X and Y are Banach spaces. Denote the range of L, R(L) = {y ∈ Y : y = Lx, x ∈ X} and B(X)={L : X → X : L is bounded linear operator} .
Assume that X be a Banach space and L ∈ B(X). The adjoint operator L * ∈ B(X * ) of L is defined by (L * f ) (x) = f (Lx) for all f ∈ X * and x ∈ X where X * is the dual space X.
Let X is a complex normed linear space and D(L) ⊂ X be domain of L where L : D (L) → X is a linear operator. For L ∈ B(X) we determine a complex number λ by the operator (λI − L) denoted by L λ which has the same domain D(L), such that I is the identity operator. Recall that the resolvent operator of L λ is L −1 λ := (λI − L) −1 . Let λ ∈ C. If L −1 λ exists, is bounded and, is defined on a set which is dense in X then λ is called a regular value of L.
The set ρ(L, X) of all regular values of L is called the resolvent set of L. σ(L, X) := C\ρ(L; X) is called the spectrum of L where C is complex plane. Hence those values λ ∈ C for which L λ is not invertible are contained in the spectrum σ(L, X).
The spectrum σ(L, X) is union of three disjoint sets as follows: The point (discrete) spectrum σ p (L, X) is the set such that L −1 λ does not exist. Further λ ∈ σ p (L, X) is called the eigen value of L. We say that λ ∈ C belongs to the continuous spectrum σ c (L, X) of L if the resolvent operator L −1 λ is defined on a dense subspace of X and is unbounded. Furthermore, we say that λ ∈ C belongs to the residual spectrum σ r (L, X) of L if the resolvent operator L −1 λ exists, but its domain of definition (i.e. the range R(λI − L) of (λI − L) is not dense in X; in this case L −1 λ may be bounded or unbounded. Together with the point spectrum, these two subspectra form a disjoint subdivision Also the spectrum σ(L, X) is partitioned into three sets which are not necessarily disjoint as follows: If there exists a sequence (x k ) in X shuch that kx k k = 1 and kLx k k → 0 as k → ∞ then (x k ) is called Weyl sequence for L.
We call the set σ ap (L, X) := {λ ∈ C : there exists aWeyl sequence for λI − L} (1.2) the approximate point spectrum of L. Moreover, the subspectrum is called defect spectrum of L. There exists another subspectrum, which is often called compression spectrum in the literature. Clearly, σ p (L, X) ⊆ σ ap (L, X) and σ co (L, X) ⊆ σ δ (L, X).
The following Proposition is quitly useful for calculating the separation of the spectrum of linear operator in Banach spaces.
If these possibilities are combined in all possible ways, nine different states are created. These are labelled by: I 1 , I 2 , I 3 , II 1 , II 2 , II 3 , III 1 , III 2 , III 3 . If an operator is in state III 2 for example, then R(T ) 6 = X and T −1 exists but is discontinuous (see [16]).
By the definitions given above, we can write following table and is bounded and is unbounded does not exists λ ∈ σ p (L, X) λ ∈ σ co (L, X) λ ∈ σ co (L, X) Table 1 Let us denote the set of all sequences; the space of all null sequences; space of all convergent sequences; space of all sequences such that P k | x k | p < ∞ by w; c 0 ; c; p ; respectively. Lemma 1 ([16], Theorem II 3.11). The adjoint operator T * is onto if and only if T has a bounded inverse.
Lemma 2 ([16], Theorem II 3.7). A linear operator T has a dense range if and only if the adjoint operator T * is one to one.
Lemma 3 ([17], Sections 28 Theorem 2). The sequence of the factors in a convergent infinite product always tends 1.

Results and discussion
The matrices which are the infinite element or finite difference problems are frequently banded in numerical analysis. We define the relationship between the problem variables helping by these matrices. The bandedness is confirmed with variables which are not conjugate in arbitrarily large distances. We can furthermore divide these matrices. For example, there are banded matrices with every element in the band is nonzero. We generally encounter these matrices while we are separating one-dimensional problems.
In addition, there are also band matrices in higher dimensional problems. Herein the bands are thiner. For example, the matrix which its bandwidth is the square root of the matrix dimension, correspond to partial differential equation defined in a square domain where the five diagonals are not zero in the band. Unfortunately, if we apply Gaussian elimination to this matrix, we obtain matrix which has the band with many non-zero elements. Therefore the resolvent set of the band operators is important for solving such problems (see [20]).
In the last years, several authors have investigated spectral divisions of generalized difference matrices. For example, Akhmedov and El-Shabrawy, [1,2] have investigated the spectrum and fine spectrum of the generalized lower triangle double-band matrix ∆ v over the sequence spaces c 0 , c and p , where 1 < p < ∞. The fine spectrum of the difference operator ∆ over the sequence spaces c 0 and c, has investigated by Altay and Başar [3] etc.
The above-mentioned articles are concerned with the decomposition of the spectrum which defined by Goldberg. However, in [9] Durna and Yildirim have investigated subdivision of the spectra for factorable matrices on c 0 and in [5] Başar, Durna and Yildirim have investigated subdivisions of the spectra for generalized difference operator over certain sequence spaces. In [22], the norm and spectrum of the Cesàro matrix considered as a bounded operator on bv 0 ∩ ∞ were studied by Tripathy and Saikia. In [23], Tripathy and Paul examined the spectra of the operator D(r, 0, 0, s) on sequence spaces c 0 and c. In [24], the spectra of the Rhaly operator on the class of bounded statistically null bounded variation sequence space was determined by Tripathy and Das. In [19], Paul and Tripathy investigated the fine spectrum of the operator D(r, 0, 0, s) over a sequence space bv 0 . In [25], Tripathy and Das determined the spectrum and subdivisions of the spectrum of the upper triangular matrix U (r, s) on the sequence space cs. In [6], the spectrum and fine spectrum of the lower triangular matrix B (r, s, t) on the sequence space cs were studied by Das and Tripathy. In [8], the fine spectrum of the lower triangular matrix B(r, s) over the Hahn sequence space was investigated by Das. In [10], [11] Durna has studied subdivision of the spectra for the generalized difference operators over the sequence spaces c 0 , c and p , (1 < p < ∞). In [18], Paul and Tripathy studied the spectrum of the operator D (r, 0, 0, s) over the sequence spaces p and bv p . In [7], Das has calculated the spectrum and fine spectrum of the upper triangular matrix U (r 1 , r 2 ; s 1 , s 2 ) over the sequence space c 0 . In [15], El-Shabrawy and Abu-Janah determined spectra and the fine spectra of generalized difference operator B (r, s) on the sequence spaces bv 0 and h. In [28], Yildirim and Durna examined the spectrum and some subdivisions of the spectrum of discrete generalized Cesaro operators on p , (1 < p < ∞).
In [26], the fine spectrum of the upper triangular matrix U (r, 0, 0, s) over the squence spaces c 0 and c was studied by Tripathy and Das. In [12], Durna et al. studied partition of the spectra for the generalized difference operator B(r, s) on the sequence space cs, in [13], Durna studied subdivision of spectra for some lower triangular doule-band matrices as operators on c 0 .
2.1. The fine spectrum of the operator ∆ v on c and p , 1 < p < ∞ In [21] Srivastava and Kumar have defined the generalized difference operator ∆ v as follows: Let the sequence (v k ) is assumed to be either constant or strictly decreasing sequence of positive real numbers satisfying If v k = L 6 = 0 for all k ∈ N is a constant sequence, then the operator ∆ v is the operator B (r, s) with r = L, s = −L and the results for the subdivisions of the spectra for generalized difference operator ∆ v over c 0 , c, p and bv p have been studied in [5].
The fine spectrum of the operator ∆ v has been investigated by Akhmedov and El-Shabrawy [1] and [2] on the sequence space c. In this study, let us assume that v 0 6 = 2L. Herein we mention the main results.
The following lemma is useful for finding the adjoint of a linear transformation on the sequence space c.
From Lemma 4 the adjoint of ∆ v : c → c is the matrix where (a k ) and (b nk ) are nonnegative real numbers and p ≥ 2r. Proof.
Let us investigate whether the operator (λI − ∆ v ) * = λI − ∆ * v is surjective or not. Does there exist x ∈ 1 for all y ∈ 1 such that (λI − ∆ * v ) x = y? Firstly, we assume that λ = 0. In this case, there is no x ∈ 1 for y = e 0 = (1, 0, 0, . . .) ∈ 1 such that (−∆ * v ) x = y. Therefore ∆ * v is not surjective. Hence from Lemma 1, we have 0 / ∈ III 1 σ (∆ v , c). Now, we assume that λ 6 = 0. In this case, if (λI − ∆ * v ) x = y for all y ∈ 1 , then we obtain that (2.4) Now, we must show that x ∈ 1 . That is, is the series Since for all n ∈ N, Hence for λ ∈ σ r (∆ v , c), the series P 1 and P 2 are convergent if and only if λ ∈ H ∪ {v k : k ∈ N}. Now, let us investigate the series P 3 to be convergent. If λ ∈ {v k : k ∈ N}, then it is clear that the series P 3 is convergent. Let λ ∈ H. Then, we get Thus, if we take p = 4, r = 2, a k = |y k | and b nk =¯n we obtain that  Table 1. Also, since from Table 1

Proof. (a) It is clear from Theorem 1 and Theorem 6 since
, the proof is finished from Theorem 1. (c) It is clear from Theorem 4 since from Table 1 Proof. It is clear from Theorem 7 and Proposition 1 (c) and ( Let T : p → p (1 < p < ∞) be a bounded linear operator and A be its matrix representation. We know that the adjoint operator T * : * p → * p is a bounded linear operator and A t is its matrix representation. We notice that the dual space * p of p is isomorphic to q with p −1 + q −1 = 1. Proof.
Let us investigate whether the operator (λI − ∆ v ) * = λI − ∆ * v is surjective or not. Does there exist x ∈ q for all y ∈ q such that . Therefore Thus Now, we must show that x ∈ q . That is, is the series ∞ P n=0 |x n | q convergent? From Minkowski inequality, we have Since for all n ∈ N, from Lemma 3, the product Thus for λ ∈ σ r (∆ v , c), the series P 5 is convergent if and only if λ ∈ H 1 ∪ {v k : k ∈ N}. Now, let us investigate the series P 6 to be convergent. If λ ∈ {v k : k ∈ N}, then it is clear that the series P 6 is convergent. Let λ ∈ H 1 . We get Therefore, if we take p = 2, r = 1, a k = |y k | q and b nk =¯n Proof. It is clear from Theorem 11 and Theorem 13, since

Proof.
(a) It is clear from Theorem 8 and Theorem 13. (b) From Theorem 9, . Also, the proof is finished from Theorem 8, since from Table 1 It is clear from Theorem 11, since from Table 1, Proof. It is clear from Theorem 14 and Proposition 1 (c) and (d). 2 2.2. The fine spectrum of the modified operator ∆ v on c and p , 1 < p < ∞ Akhmedov and El-Shabrawy [2] have modified the generalized difference operator ∆ v which is represented by the matrix They have eliminated the condition: the sequence (v k ) is strictly decreasing sequence of positive real numbers. Also they have put another condition instead of condition (2.2). That is throughout this section, the sequence (v k ) is assumed to be a sequence of nonzero real numbers which is either constant or satisfying the conditions lim k→∞ v k = L > 0 (2.6) and sup k v k ≤ L. (2.7) Hereafter the sequence (v k ) satisfies these properties adopted by Akhmedov and El-Shabrawy in [2].
2.2.1. Partition of the spectrum of the modified operator ∆ v on p , 1 < p < ∞ Akhmedov and El-Shabrawy [2] have examined the spectrum, the point spectrum, the residual spectrum and the continuous spectrum of the modified operator ∆ v over the sequence space p , (1 < p < ∞). Herein we mention the main results.

Proof.
Let us investigate whether the operator (λI Now, we must show that x ∈ q . That is, is the series from Theorem 18, we get¯L − λ L¯< 1. Thus from Lemma 3 , the infinite This means that lim n→∞ |x n | q 6 = 0. Hence λ ∈ σ r (∆ v , p ) and for all k ∈ N, λ 6 = v k implies x / ∈ q . In this case, λI−∆ * v is not surjective and from Lemma 1, λI −∆ v does not have bounded inverse. Now, we assume that λ ∈ σ r (∆ v , p ) and for some k 0 ∈ N, λ = v k . In this case, since the products are zero on the right hand of (2.5), x n = y n−1 v n−1 . Let us take elements v k such that |v k − L| < |L|, for k ∈ N. If L > 0, then all v k are positive in circle, if L < 0, then all v k are negative in circle. Hence there are two cases for elements of the set {v k : k ∈ N, |v k − L| < |L|}. 1. case: If L > 0, then there exists M > 0 such that M < v k < L. Hence That is, the operator (λI − ∆ v ) * is surjective if and only if {λ ∈ v k : k ∈ N, |v k − L| < |L|}. Hence from Lemma 1, λI − ∆ v has a bounded inverse. 2 Corollary 5.
Proof. It is clear from Theorem 18 and Theorem 20, since Thus we have

Proof.
It is clear from Theorem 21, since from Table 1, Proof.