Fine Spectra of Upper Triangular Double-Band Matrices over the Sequence Space p , 1 < p < ∞

The operator A r̃, s̃ on sequence space on p is defined A r̃, s̃ x rkxk skxk 1 ∞ k 0, where x xk ∈ p, and r̃ and s̃ are two convergent sequences of nonzero real numbers satisfying certain conditions, where 1 < p < ∞ . The main purpose of this paper is to determine the fine spectrum with respect to the Goldberg’s classification of the operator A r̃, s̃ defined by a double sequential band matrix over the sequence space p. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator A r̃, s̃ over the space p.


Introduction
Let X and Y be Banach spaces, and let T : X → Y also be a bounded linear operator.By R T , we denote the range of T , that is, By B X , we also denote the set of all bounded linear operators on X into itself.If X is any Banach space and 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 Tx for all f ∈ X * and x ∈ X.Given an operator T ∈ B X , the set ρ T : λ ∈ C : T λ λI − T is a bijection 1.2 Discrete Dynamics in Nature and Society is called the resolvent set of T and its complement with respect to the complex plain is called the spectrum of T .By the closed graph theorem, the inverse operator is always bounded and is usually called resolvent operator of T at λ.

Subdivisions of the Spectrum
In this section, we give the definitions of the parts point spectrum, continuous spectrum, residual spectrum, approximate point spectrum, defect spectrum, and compression spectrum of the spectrum.There are many different ways to subdivide the spectrum of a bounded linear operator.Some of them are motivated by applications to physics, in particular, quantum mechanics.

The Point Spectrum, Continuous Spectrum, and Residual Spectrum
The name resolvent is appropriate, since T −1 λ helps to solve the equation T λ x y.Thus, x T −1 λ y provided T −1 λ exists.More important, the investigation of properties of T −1 λ will be basic for an understanding of the operator T itself.Naturally, many properties of T λ and T −1 λ depend on λ, and spectral theory is concerned with those properties.For instance, we will be interested in the set of all λ's in the complex plane such that T −1 λ exists.Boundedness of T −1 λ is another property that will be essential.We will also ask for what λ's the domain of T −1 λ is dense in X, to name just a few aspects.A regular value λ of T is a complex number such that T −1 λ exists and bounded and whose domain is dense in X.For our investigation of T , T λ , and T −1 λ , we need some basic concepts in spectral theory, which are given as follows see 1, pp. 370-371 : The resolvent set ρ T, X of T is the set of all regular values λ 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 such that T −1 λ does not exist.An λ ∈ σ p T, X is called an eigenvalue of T .
The continuous spectrum σ c T, X is the set such that T −1 λ exists and is unbounded and the domain of T −1 λ is dense in X.The residual spectrum σ r T, X is the set such that T −1 λ exists and may be bounded or not , but the domain of T −1 λ is not dense in X.Therefore, these three subspectra form a disjoint subdivisions σ T, X σ p T, X ∪ σ c T, X ∪ σ r T, X .

2.1
To avoid trivial misunderstandings, let us say that some of the sets defined above, may be empty.This is an existence problem, which we will have to discuss.Indeed, it is well known that σ c T, X σ r T, X ∅ and the spectrum σ T, X consists of only the set σ p T, X in the finite-dimensional case.

The Approximate Point Spectrum, Defect Spectrum, and Compression Spectrum
In this subsection, following Appell et al. 2 , 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 x k in X as a Weyl sequence for T if x k 1 and Tx k → 0, as k → ∞.In what follows, we call the set σ ap T, X : λ ∈ C : there exists a Weyl sequence for λI − T 2.2 the approximate point spectrum of T .Moreover, the subspectrum The two subspectra given by 2.2 and 2.3 form a not necessarily disjoint subdivision of the spectrum.There is another subspectrum which is often called compression spectrum in the literature.The compression spectrum gives rise to another not necessarily disjoint decomposition of the spectrum.Clearly, σ p T, X ⊆ σ ap T, X and σ co T, X ⊆ σ δ T, X .Moreover, comparing these subspectra with those in 2.1 we note that

2.7
Sometimes it is useful to relate the spectrum of a bounded linear operator to that of its adjoint.Building on classical existence and uniqueness results for linear operator equations in Banach spaces and their adjoints is also useful.Proposition 2.1 see 2, 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: 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 see 2 .

Goldberg's Classification of Spectrum
If X is a Banach space and T ∈ B X , then there are three possibilities for R T : If these possibilities are combined in all possible ways, nine different states are created.These are labelled by: A If an operator is in state C 2 , for example, then R T / X and T −1 exist but is discontinuous see 3 and Figure 1 .
If λ is a complex number such that T λ λI −T ∈ A 1 or T λ λI −T ∈ B 1 , then λ ∈ ρ T, X .All scalar values of λ not in ρ T, X comprise the spectrum of T .The further classification of σ T, X gives rise to the fine spectrum of T .That is, σ T, X can be divided into the subsets , and C 3 σ T, X .For example, if T λ λI − T is in a given state, C 2 say , then we write λ ∈ C 2 σ T, X .
By the definitions given above, we can illustrate the subdivisions 2.1 in Table 1.
Observe that the case in the first row and second column cannot occur in a Banach space X, by the closed graph theorem.If we are not in the third column, that is, if λ is not an eigenvalue of T , we may always consider the resolvent operator T −1 λ on a possibly "thin" domain of definition as "algebraic" inverse of λI − T .
By a sequence space, we understand a linear subspace of the space ω C N 1 of all complex sequences which contains φ, the set of all finitely nonzero sequences, where N 1 denotes the set of positive integers.We write ∞ , c, c 0 , and bv for the spaces of all bounded, convergent, null, and bounded variation sequences, which are the Banach spaces with the sup-norm , while φ is not a Banach space with respect to any norm, respectively, where N {0, 1, 2, . ..}.Also by p , we denote the Table 1: Subdivisions of spectrum of a linear operator.
space of all p-absolutely summable sequences, which is a Banach space with the norm a nk be an infinite matrix of complex numbers a nk , where n, k ∈ N, and write where D 00 A denotes the subspace of w consisting of x ∈ w for which the sum exists as a finite sum.For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞, and we will use the convention that any term with negative subscript is equal to naught.More generally if μ is a normed sequence space, we can write D μ A for the x ∈ w for which the sum in 2.8 converges in the norm of μ.We write for the space of those matrices which send the whole of the sequence space λ into μ in this sense.
We give a short survey concerning the spectrum and the fine spectrum of the linear operators defined by some particular triangle matrices over certain sequence spaces.The fine spectrum of the Cesàro operator of order one on the sequence space p studied by González 19 , where 1 < p < ∞.Also, weighted mean matrices of operators on p have been investigated by Cartlidge 20 .The spectrum of the Cesàro operator of order one on the sequence spaces bv 0 and bv investigated by Okutoyi 8,21 .The spectrum and fine spectrum of the Rhally operators on the sequence spaces c 0 , c, p , bv, and bv 0 were examined by Yıldırım 9,[22][23][24][25][26][27][28] The fine spectrum of the difference operator Δ over the sequence spaces c 0 and c was studied by Altay and Bas ¸ar 12 .The same authors also worked the fine spectrum of the generalized difference operator B r, s over c 0 and c, in 29 .The fine spectra of Δ over 1 and bv studied by Kayaduman and Furkan 30 .Recently, the fine spectra of the difference operator Δ over the sequence spaces p and bv p studied by Akhmedov and Bas ¸ar 31, 32 , where bv p is the space of p-bounded variation sequences and introduced by Bas ¸ar and Altay 33 with 1 p < ∞.Also, the fine spectrum of the generalized difference operator B r, s over the sequence spaces  2: Spectrum and fine spectrum of some triangle matrices in certain sequence spaces.In this paper, we study the fine spectrum of the generalized difference operator spectrum of the generalized difference operator defined by an upper double sequential band matrix acting on the sequence spaces p with respect to the Goldberg's classification.Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator A r, s over the spaces p .We quote some lemmas, which are needed in proving the theorems given in Section 3.

2.10
Then, we define the sequential generalized difference matrix A r, s by Therefore, we introduce the operator A r, s from p to itself by is not convergent.
Throughout the paper, by C and SD, we denote the set of constant sequences and the set of sequences of distinct real numbers, respectively.Theorem 3.3. 3.8 Proof.Let A r, s x αx for θ / x ∈ p Then, by solving linear equation . . .

3.9
x k α − r k /s k−1 x k−1 for all k 1 and Part 1. Assume that r, s ∈ C. Let r k r and s k s For all k ∈ N. We observe that x k α − r /s k x 0 .This shows that x ∈ p if and if only |α − r| < |s|, as asserted.
Part 2. Assume that r, s ∈ SD.We must take x 0 / 0, since x / 0. It is clear that, for all k ∈ N, the vector x x 0 , x 1 , . . ., x k , 0, 0, . . . is an eigenvector of the operator A r, s corresponding to the eigenvalue α r k , where x 0 / 0 and ∈ p .This completes the proof.

3.11
Proof.We prove the theorem by dividing into two parts.
Part 1. Assume that r, s ∈ C. Consider A r, s * f αf for f / θ 0, 0, 0, . . . in * p q .Then, by solving the system of linear equations the first nonzero entry of the sequence f f n and α r, then we get s n 0 f n 0 rf n 0 1 αf n 0 1 that implies f n 0 0, which contradicts the assumption f n 0 / 0. Hence, the equation A r, s * f αf has no solution f / θ.Part 2. Assume that r, s ∈ SD.Then, by solving the equation A r, s * f αf for f / θ 0, 0, 0, . . . in q , we obtain r 0 − α f 0 0 and r k 1 − α f k 1 s k f k 0 for all k ∈ N. Hence, for all α / ∈ {r k : k ∈ N}, we have f k 0 for all k ∈ N, which contradicts our assumption.So, α / ∈ σ p A r, s * , q .This shows that σ p A r, s * , q ⊆ {r k : k ∈ N} \ {r}.Now, we prove that α ∈ σ p A r, s * , q iff α ∈ B.

3.13
If α ∈ σ p A r, s * , q , then, by solving the equation A r, s * f αf for f / θ 0, 0, 0, . . . in q with α r 0 , which can expressed by the recursion relation 3.17 If we choose α r k / r for all k ∈ N 1 , then we get which can expressed by the recursion relation

3.19
Using ratio test,

3.22
That is, f ∈ q .So we have B ⊆ σ p A r, s * , q .This completes the proof.Proof.We will show that A α r, s * is onto, for |r − α| > |s|.Thus, for every y ∈ q , we find x ∈ q .A α r, s * is triangle so it has an inverse.Also equation A α r, s * x y gives A α r, s * −1 y x.It is sufficient to show that A α r, s * −1 ∈ q : q .We can calculate that A a nk A α r, s * −1 as follows: Therefore, the supremum of the 1 norms of the rows of A α r, s * −1 is S k , where

3.24
Now, we prove that

Conclusion
In the present work, as a natural continuation of Akhmedov and El-Shabrawy 15 and Srivastava and Kumar 18 , we have determined the spectrum and the fine spectrum of the double sequential band matrix A r, s on the space p .Many researchers determine the spectrum and fine spectrum of a matrix operator in some sequence spaces.In addition to this, we add the definition of some new divisions of spectrum called as approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator and give the related results for the matrix operator A r, s on the space p , which is a new development for this type works giving the fine spectrum of a matrix operator on a sequence space with respect to the Goldberg's classification.

and 1 T
−1 exists and is continuous, 2 T −1 exists but is discontinuous, 3 T −1 does not exist.
1 and bv determined by Furkan et al.34 .Recently, the fine spectrum of B r, s, t over the sequence spaces c 0 and c has been studied by Furkan et al.35 .Quite recently, de Malafosse 11 and Altay and Bas ¸ar 12 have, respectively, studied the spectrum and the fine spectrum of the difference operator on the sequence spaces s r and c 0 , c, where s r denotes the Banach space of all sequences x x k normed by x s r sup k∈N |x k |/r k , r > 0 .Altay and Karakus ¸ 36 have determined the fine spectrum of the Zweier matrix, which is a band matrix as an operator over the sequence spaces 1 and bv.Farés and de Malafosse 37 studied the spectra of the difference operator on the sequence spaces p α , where α n denotes the sequence of positive reals and p α is the Banach space of all sequences xx n normed by x p α∞ n 1 |x n |/α n p 1/p with p 1.Also the fine spectrum of the same operator over 1 and bv has been studied by Bilgic ¸and Furkan 13 .More recently the fine spectrum of the operator B r, s over p and bv p has been studied by Bilgic ¸and Furkan 38 .In 2010, Srivastava and Kumar 16 have determined the spectra and the fine spectra of generalized difference operator Δ ν on 1 , where Δ ν is defined by Δ ν nn ν n and Δ ν n 1,n −ν n for all n ∈ N, under certain conditions on the sequence ν ν n , and they have just generalized these results by the generalized difference operator Δ uv defined by Δ uv x u n x n v n−1 x n−1 n∈N for all n ∈ N, see 18 .Altun 39 has studied the fine spectra of the Toeplitz operators, which are represented by upper and lower triangular n-band infinite matrices, over the sequence spaces c 0 and c.Later, Karakaya and Altun have determined the fine spectra of upper triangular double-band matrices over the sequence spaces c 0 and c, in 40 .Quite recently, Akhmedov and El-Shabrawy 15 have obtained the fine spectrum of the generalized difference operator Δ a,b , defined as a double band matrix with the convergent sequences a a k and b b k having certain properties, over the sequence space c.Finally, the fine spectrum with respect to the Goldberg's classification of the operator B r, s, t defined by a triple band matrix over the sequence spaces p and bv p Table

Lemma 3 . 2
combining the inequalities in 3.3 and 3.5 we have 3.1 , as desired.see 42, p. 115, Lemma 3.1 .Let 1 < p < ∞.If α ∈ {α ∈ C : |r − α| |s|}, The matrix A a nk gives rise to a bounded linear operator T ∈ B 1 from 1 to itself if and only if the supremum of 1 norms of the columns of A is bounded.B ∞ from ∞ to itself if and only if the supremum of 1 norms of the rows of A is bounded.

Fine Spectra of Upper Triangular Double-Band Matrices over the Sequence Space
∈ p , where p > 1.Then, since s k x k 1 , r k x k ∈ p it is easy to see by Minkowski's inequality that where x x k ∈ p .2.12 3. p Theorem 3.1.The operator A r, s : p → p is a bounded linear operator and sup k∈N |r k | p |s k | p 1/p A r, s p sup k∈N |r k | sup k∈N |s k |. 3.1 Proof.Since the linearity of the operator A r, s is not difficult to prove, we omit the detail.Now we prove that 3.1 holds for the operator A r, s on the space p .It is trivial that A r, s e k 0, 0, . . ., s k−1 , r k , 0, . . ., 0, . . .for e k ∈ p .Therefore, we have p sup k∈N |r k | p |s k | p 1/p .
r, s , p .Conversely, let α ∈ σ p A r, s , p .Then, there exists x x 0 , x 1 , x 2 , . . . in p and we have x k α − r k /s k−1 x k−1 , for all k 1.Since x ∈ p , we can use ratio test.And so lim k Proof.The proof is obvious so is omitted.Let A {α ∈ C : |r − α| |s|} and B {r k : k ∈ N, |r − r k | > |s|}.
Lemma 3.5 see 3, p. 60 .The adjoint operator T * of T is onto if and only if T is a bounded operator.
5, A α r, s is bounded inverse.This means that σ c A r, s , p ⊆ {α ∈ C : |r − α| |s|}.3.30 Combining this with Theorem 3.3 and Theorem 3.7, we get σ A r, s , p ⊆ {α ∈ C : |r − α| |s|} ∪ B 3.31 and again from Theorem 3.3 {α ∈ C : |r − α| < |s|} ⊆ σ A r, s , p and B ⊆ σ A r, s , p .Since the spectrum of any bounded operator is closed, we have {α ∈ C : |r − α| |s|} ∪ B ⊆ σ A r, s , p .Proof.From Theorem 3.3, α ∈ σ p A r, s , p .Thus, A r, s − αI −1 does not exist.It is sufficient to show that the operator A r, s − αI is onto, that is, for given y y k ∈ p , we have to find x x k ∈ p such that A r, s − αI x y.Solving the linear equation