Spectrum and fine spectrum of the Zweier matrix over the sequence space cs

By w, we denote the space of all real or complex valued sequences. Throughout the paper c, c0, bv, cs, bs, l1, l∞ represent the spaces of all convergent, null, bounded variation, convergent series, bounded series, absolutely summable and bounded sequences respectively. Also bv0 denotes the sequence space bv ∩ c0. Fine spectra of various matrix operators on different sequence spaces have been examined by several authors. Fine spectrum of the operator ∆a,b on the sequence space c was determined by Akhmedov and El-Shabrawy [1]. The fine spectra of the Cesàro operator C1 over the sequence space bvp, (1 ≤ p < ∞) was determined by Akhmedov and Başar [2]. Altay and Başar [3,4] determined the fine spectrum of the difference operator ∆ and the generalized difference operator B(r, s) on the sequence spaces c0 and c. The spectrum and fine spectrum of the Zweier Matrix on the sequence spaces l1 and bv were studied by Altay and Karakuş [5]. Altun [6,7] determined the fine spectra of triangular Toeplitz operators and tridiagonal symmetric matrices over some sequence spaces. Furkan, Bilgiç and Kayaduman [14] have determined the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces l1 and bv. Fine spectra of operator B(r, s, t) over the sequence spaces l1 and bv and generalized difference operator B(r, s) over the sequence spaces lp and bvp, (1 ≤ p < ∞) were studied by Bilgiç and Furkan [11,12]. Fine spectrum of the generalized difference operator ∆v on the sequence space l1


Introduction
By w, we denote the space of all real or complex valued sequences.Throughout the paper c, c 0 , bv, cs, bs, ℓ 1 , ℓ ∞ represent the spaces of all convergent, null, bounded variation, convergent series, bounded series, absolutely summable and bounded sequences respectively.Also bv 0 denotes the sequence space bv ∩ c 0 .
Fine spectra of various matrix operators on different sequence spaces have been examined by several authors.Fine spectrum of the operator ∆ a,b on the sequence space c was determined by Akhmedov and El-Shabrawy [1].The fine spectra of the Cesàro operator C 1 over the sequence space bv p , (1 ≤ p < ∞) was determined by Akhmedov and Başar [2].Altay and Başar [3,4] determined the fine spectrum of the difference operator ∆ and the generalized difference operator B(r, s) on the sequence spaces c 0 and c.The spectrum and fine spectrum of the Zweier Matrix on the sequence spaces ℓ 1 and bv were studied by Altay and Karakuş [5].Altun [6,7] determined the fine spectra of triangular Toeplitz operators and tridiagonal symmetric matrices over some sequence spaces.Furkan, Bilgiç and Kayaduman [14] have determined the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces ℓ 1 and bv.Fine spectra of operator B(r, s, t) over the sequence spaces ℓ 1 and bv and generalized difference operator B(r, s) over the sequence spaces ℓ p and bv p , (1 ≤ p < ∞) were studied by Bilgiç and Furkan [11,12].Fine spectrum of the generalized difference operator ∆ v on the sequence space ℓ 1 2000 Mathematics Subject Classification: 47A10, 47B37.

Rituparna Das
was investigated by Srivastava and Kumar [28].Panigrahi and Srivastava [24,25] studied the spectrum and fine spectrum of the second order difference operator ∆ 2 uv on the sequence space c 0 and generalized second order forward difference operator ∆ 2 uvw on the sequence space ℓ 1 .Fine spectra of upper triangular double-band matrix U (r, s) over the sequence spaces c 0 and c were studied by Karakaya and Altun [20].Karaisa and Başar [19] have determined the spectrum and fine spectrum of the upper traiangular matrix A(r, s, t) over the sequence space ℓ p , (0 < p < ∞).In a further development, they have also determined the approximate point spectrum, defect spectrum and compression spectrum of the operator A(r, s, t) on the sequence space ℓ p , (0 < p < ∞).
In this paper, we shall determine the spectrum and fine spectrum of the lower triangular matrix Z s on the sequence space cs .Also,we determine the approximate point spectrum, the defect spectrum and the compression spectrum of the operator Z s on the sequence space cs.Clearly, cs = {x = (x n ) ∈ w : lim n→∞ n i=0 x i exists} is a Banach space with respect to the norm

Preliminaries and Background
Let X and Y be Banach spaces and T : X → Y be a bounded linear operator.By R(T ), we denote the range of T , i.e.

R(T ) = {y ∈
By B(X) ,we denote the set of all bounded linear operators on X into itself.If T ∈ B(X), then the adjoint T * of T is a bounded linear operator on the dual X * of X defined by (T * f )(x) = f (T x), for all f ∈ X * and x ∈ X.Let X = {θ} be a complex normed linear space, where θ is the zero element and T : D(T ) → X be a linear operator with domain D(T ) ⊆ X.With T , we associate the operator where λ is a complex number and I is the identity operator on D(T ).If T λ has an inverse which is linear, we denote it by T −1 λ , that is and call it the resolvent operator of T .
A regular value λ of T is a complex number such that The resolvent set of T , denoted by ρ(T, X), is the set of all regular values λ of T .Its complement σ(T, X) = C − ρ(T, X) in the complex plane C is called the spectrum of T .Furthermore, the spectrum σ(T, X) is partitioned into three disjoint sets as follows: The point(discrete) spectrum σ p (T, X) is the set of all λ ∈ C such that T −1 λ does not exist.Any such λ ∈ σ p (T, X) is called an eigenvalue of T .
The continuous spectrum σ c (T, X) is the set of all λ ∈ C such that T −1 λ exists and satisfies (R3), but not (R2), that is, T −1 λ is unbounded.The residual spectrum σ r (T, X) is the set of all λ ∈ C such that T −1 λ exists (and may be bounded or not), but does not satisfy (R3), that is, the domain of T −1 λ is not dense in X.
From Goldberg [17], if X is a Banach space and T ∈ B(X) , then there are three possibilities for R(T ) and T −1 : If these possibilities are combined in all possible ways, nine different states are created which may be shown as in the Table 1.
Table 1: Subdivisions of spectrum of a linear operator These are labeled by: Again, following Appell et al. [8], we define the three more subdivisions of the spectrum called as the approximate point spectrum, defect spectrum and compression spectrum.
Given a bounded linear operator T in a Banach space X, we call a sequence The approximate point spectrum of T , denoted by σ ap (T, X) , is defined as the set σ ap (T, X) = {λ ∈ C : there exists a Weyl sequence for T − λI} (2.1) The defect spectrum of T , denoted by σ δ (T, X) , is defined as the set The two subspectra given by equations (2.1) and (2.2) form a (not necessarily disjoint) subdivisions of the spectrum.There is another subspectrum which is often called the compression spectrum of T .The compression spectrum gives rise to another (not necessarily disjoint) decomposition Clearly, σ p (T, X) ⊆ σ ap (T, X) and σ co (T, X) ⊆ σ δ (T, X) .Moreover, it is easy to verify that σ r (T, X) = σ co (T, X) \ σ p (T, X) and σ c (T, By the definitions given above, we can illustrate the subdivisions of spectrum of a bounded linear operator in the Table 2. The relations (c)-(f) show that the approximate point spectrum is in a certain sense dual to defect spectrum, and the point spectrum dual to the compression spectrum.The equality (g) implies, in particular, that σ(T, X) = σ ap (T, X) if X is a Hilbert space and T is normal.Roughly speaking, this shows that normal (in particular, self-adjoint) operators on Hilbert spaces are most similar to matrices in finite dimensional spaces (Appell et al. [8]).
Let λ and µ be two sequence spaces and A = (a nk ) be an infinite matrix of real or complex numbers a nk , where n, k ∈ N 0 = {0, 1, 2, ...}.Then, we say that A defines a matrix mapping from λ into µ, and we denote it by A : λ → µ , if for every sequence x = (x k ) ∈ λ, the sequence Ax = {(Ax) n }, the A-transform of x, is in µ, where (2.5) By (λ : µ), we denote the class of all matrices such that A : λ → µ.Thus, A ∈ (λ : µ) if and only if the series on the right hand side of equation (2.5) converges for each n ∈ N 0 and every x ∈ λ and we have Ax = {(Ax) n } n∈N0 ∈ µ for all x ∈ λ.
The Zweier matrix Z s is an infinite lower triangular matrix of the form where s = 0, 1.
The following results will be used in order to establish the results of this article.
Lemma 2.1.[Wilansky [35] Example 6B, Page 130] The matrix A = (a nk ) gives rise to a bounded linear operator T ∈ B(cs) from cs to itself if and only if: and hence, Z s (cs:cs) ≤ |s| + |1 − s|.Hence the result.✷ From Theorem 2.1 in [4], we get the spectrum of the operator B(r, s) on the sequence space c 0 is σ(B(r, s), c 0 ) = {α ∈ C : |α − r| ≤ |s|}, where the operator B(r, s) is given by the lower triangular matrix where s = 0.The lower triangular matrix Z s is a special case of B(r, s).Also the sequence space cs is a subspace of c 0 .Therefore we can expect that In the following theorem we give an independent proof of our expectation.Since |α − s| > |1 − s|, so for all k, the series is also convergent.So, by Lemma 2.1, (Z s − αI) −1 is in (cs : cs).This shows that σ(Z s , cs) is not convergent and hence, x = (x n ) / ∈ cs.Therefore, (Z s − αI) −1 is not in (cs : cs) and so α ∈ σ(Z s , cs).If α = s, then the operator Z s − αI is represented by the matrix Since, the range of Z s − αI is not dense, so α ∈ σ(Z s , cs).Hence, This completes the proof.✷ Theorem 3.3.The point spectrum of the operator Z s over cs is given by Proof: Let α be an eigenvalue of the operator Z s .Then there exists x = θ = (0, 0, 0, ...) in cs such that Z s x = αx.Then, we have If T : cs → cs is a bounded linear operator represented by a matrix A, then it is known that the adjoint operator T * : cs * → cs * is defined by the transpose A t of the matrix A. It should be noted that the dual space cs * of cs is isometrically isomorphic to the Banach space bv of all bounded variation sequences normed by Proof: Let α be an eigenvalue of the operator Z * s .Then there exists x = θ = (0, 0, 0, ...) in bv such that Z * s x = αx.Then, we have Then, we have Proof: If α = s, the range of Z s − αI is not dense.So, from Table 2 and Theorem 3.3, we have α ∈ σ r (Z s , cs).
To prove the result, it is enough to show that the operator Z s − αI is bounded below.It is easy to verify that for all x ∈ cs , we have which shows that the operator Z s − αI is bounded below and so Z s − αI has a bounded inverse.This completes the theorem.✷ Theorem 3.8.If α = s and α ∈ σ r (Z s , cs) , then α ∈ III 2 σ(Z s , cs).

Table 2 :
Subdivisions of spectrum of a linear operator Proposition 2.1.[Appell et al. [8], Proposition 1.3, p. 28] Spectra and subspectra of an operator T ∈ B(X) and its adjoint T * ∈ B(X * ) are related by the following relations: