ON THE FINE SPECTRUM OF THE GENERALIZED DIFFERENCE OPERATOR B(r,s) OVER THE SEQUENCE SPACES c0 AND c

We determine the fine spectrum of the generalized difference operator B(r,s) defined by a band matrix over the sequence spaces c0 and c, and derive a Mercerian theorem. This generalizes our earlier work (2004) for the difference operator Δ, and includes as other special cases the right shift and the Zweier matrices.


Preliminaries, background, and notation
Let X and Y be the 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.Let X = {θ} be a nontrivial complex normed space and T : Ᏸ(T) → X a linear operator defined on a subspace Ᏸ(T) ⊆ X.We do not assume that D(T) is dense in X, or that T has closed graph {(x, Tx) : x ∈ D(T)} ⊆ X × X.We mean by the expression "T is invertible" that there exists a bounded linear operator S : R(T) → X for which ST = I on D(T) and R(T) = X; such that S = T −1 is necessarily uniquely determined, and linear; the boundedness of S means that T must be bounded below, in the sense that there is k > 0 for which Tx ≥ k x for all x ∈ D(T).Associated with each complex number, α is the perturbed operator defined on the same domain D(T) as T. The spectrum σ(T,X) consists of those α ∈ C for which T α is not invertible, and the resolvent is the mapping from the complement σ(T,X) of the spectrum into the algebra of bounded linear operators on X defined by α → T −1 α .The name resolvent is appropriate since T −1 α helps to solve the equation T α x = y.Thus, x = T −1 α y provided that T −1 α exists.More important, the investigation of properties of T −1 will be basic for an understanding of the operator T itself.Naturally, many properties of T α and T −1 α depend on α, and the spectral theory is concerned with those properties.For instance, we will be interested in the set of all α's in the complex plane such that T −1 α exists.Boundedness of T −1 α is another property that will be essential.We will also ask for what α's the domain of T −1 α is dense in X, to name just a few aspects.A regular value α of T is a complex number such that T −1 α exists and is bounded and whose domain is dense in X.For our investigation of T, T α , and T −1 α , we need some basic concepts in the spectral theory which are given as follows (see [8, pages 370-371]).
The resolvent set ρ(T,X) of T is the set of all regular values α of T. Furthermore, the spectrum σ(T,X) is partitioned into the following three disjoint sets.
The point (discrete) spectrum σ p (T,X) is the set such that T −1 α does not exist.A α ∈ σ p (T,X) is called an eigenvalue of T.
The continuous spectrum σ c (T,X) is the set such that T −1 α exists and is unbounded and the domain of T −1 α is dense in X.The residual spectrum σ r (T,X) is the set such that T −1 α exists (and may be bounded or not) but the domain of T −1 α is not dense in X.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 wellknown 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.
From Goldberg [6, pages 58-71], if X is a Banach space and T ∈ B(X), then there are three possibilities for R(T) and for T −1 : ( and (1) T −1 exists and is continuous, (2) T −1 exists but is discontinuous, (3) T −1 does not exist.Applying Golberg's classification to T α , we have three possibilities for T α and for and (1) T α is injective and T −1 α is continuous, (2) T α is injective and T −1 α is discontinuous, (3) T α is not injective.If these possibilities are combined in all possible ways, nine different states are created.These are labeled by I 1 , I 2 , I 3 , II 1 , II 2 , II 3 , III 1 , III 2 , and III 3 .If α is a complex number such that T α ∈ I 1 or T α ∈ II 1 , then α is in the resolvent set ρ(T,X) of T. The further classification gives rise to the fine spectrum of T. If an operator is in state II 2 for example, then R(T) = R(T) = X and T −1 exists but is discontinuous and we write α ∈ II 2 σ(T,X).
By a sequence space, we understand a linear subspace of the space w = C N of all complex sequences which contain φ, the set of all finitely nonzero sequences, where N ={0, 1,2,...}.
We write ∞ , c, c 0 , and bv for the spaces of all bounded, convergent, null, and bounded variation sequences, respectively.Also by p , we denote the space of all p-absolutely summable sequences, where 1 ≤ p < ∞.
Let A = (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.More generally, if µ is a normed sequence space, we can write D µ (A) for x ∈ w for which the sum in (1.3) converges in the norm of µ.We will write for the space of those matrices which send the whole of the sequence space λ into µ in this sense.Our main focus in this note is on the band matrix A = B(r,s), where We begin by determining when a matrix A induces a bounded operator from c to c.
We summarize the knowledge in the existing literature concerned with the spectrum of the linear operators defined by some particular limitation matrices over some sequence spaces.Wenger [13] examined the fine spectrum of the integer power of the Cesàro operator in c and Rhoades [12] generalized this result to the weighted mean methods.The fine spectrum of the Cesàro operator on the sequence space p has been studied by González [7], where 1 < p < ∞.The spectrum of the Cesàro operator on the sequence spaces c 0 and bv have also been investigated by Reade [11], Akhmedov and Bas ¸ar [1], and Okutoyi [10], respectively.The fine spectrum of the Rhaly operators on the sequence spaces c 0 and c has been examined by Yıldırım [15].Furthermore, Cos ¸kun [4] has studied the spectrum and fine spectrum for p-Cesàro operator acting on the space c 0 .More recently, de Malafosse [5] and Altay and Bas ¸ar [2] 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 In this work, our purpose is to determine the fine spectrum of the generalized difference operator B(r,s) on the sequence spaces c 0 and c, and to give a Mercerian theorem.The main results of the present work are more general than those of Altay and Bas ¸ar [2].

The spectrum of the operator B(r,s) on the sequence spaces c 0 and c
In this section, we examine the spectrum, the point spectrum, the continuous spectrum, the residual spectrum, and the fine spectrum of the operator B(r,s) on the sequence spaces c 0 and c.Finally, we also give a Mercerian theorem.

B. Altay and F. Bas ¸ar 3009
If α = r, then the operator B(r,s) − αI = B(0,s) is represented by the matrix Since R(B(0,s)) = c 0 , B(0,s) is not invertible.This completes the proof.
we find that if x n0 is the first nonzero entry of the sequence x = (x n ), then α = r and for all k ∈ N.This contradicts the fact that x n0 = 0, which completes the proof.
If T : c 0 → c 0 is a bounded linear operator with the matrix A, then it is known that the adjoint operator T * : c * 0 → c * 0 is defined by the transpose A t of the matrix A. It should be noted that the dual space c * 0 of c 0 is isometrically isomorphic to the Banach space 1 of absolutely summable sequences normed by Then, by solving the system of linear equations Proof.We show that the operator B(r,s) − αI has an inverse and R(B(r,s) − αI) = c 0 for α satisfying |α − r| < |s|.For α = r, the operator B(r,s) − αI is triangle, hence has an inverse.For α = r, the operator B(r,s) − αI is one to one, hence has an inverse.But B(r,s) * − αI is not one to one by Theorem 2.3.Now, Lemma 2.4 yields the fact that R(B(r,s) − αI) = c 0 and this step concludes the proof.
Proof.Since the operator B(r,s) − αI = B(0,s) for α = r, B(0,s) ∈ III 1 or ∈ III 2 by Theorem 2.5.To verify the fact that B(0,s) has a bounded inverse, it is enough to show that B(0,s) is bounded below.Indeed, one can easily see for all x ∈ c 0 that which means that B(0,s) is bounded below.This completes the proof.
Proof.For this, we prove that the operator B(r,s) − αI has an inverse and To verify that the operator B(r,s) − αI is not surjective, it is sufficient to show that there is no sequence x = (x n ) in c 0 such that (B(r,s) − αI)x = y for some y ∈ c 0 .Let us consider the sequence y = (1,0,0,...) ∈ c 0 .For this sequence, we obtain x n = {s/(α − r)} n /(r − α).This yields that x / ∈ c 0 , that is, B(r,s) − αI is not onto.This completes the proof.Proof.This is obtained in the similar way that is used in the proof of Theorem 2.1.
Proof.The proof may be obtained by proceeding as in proving Theorem 2.2.So, we omit the details.
If T : c → c is a bounded matrix operator with the matrix A, then T * : c * → c * acting on C ⊕ 1 has a matrix representation of the form where χ is the limit of the sequence of row sums of A minus the sum of the limit of the columns of A, and b is the column vector whose kth entry is the limit of the kth column of A for each k ∈ N.For B(r,s) : c → c, the matrix B(r,s) * ∈ B( 1) is of the form Proof. Suppose that B(r,s) * x = αx for x = θ in 1 .Then by solving the system of linear equations .13)we obtain that

.14)
If x 0 = 0, then α = r + s.So, α = r + s is an eigenvalue with the corresponding eigenvector x = (x 0 ,0,0,... Since the fine spectrum of the operator B(r,s) on c can be derived by analogy to that space c 0 , we omit the detail and give it without proof.Therefore, we have  Proof.It is known by Cartlidge [3] that if a matrix operator A is bounded on c, then σ(A,c) = σ(A, ∞ ).Now, the proof is immediate from Theorem 2.10 with A = B(r,s).
Subsequent to stating the concept of Mercerian theorem, we conclude this section by giving a Mercerian theorem.Let A be an infinite matrix and the set c A denotes the convergence field of that matrix A. A theorem which proves that c A = c is called a Mercerian theorem, after Mercer, who proved a significant theorem of this type [9, page 186].Now, we may give our final theorem.Proof.If α = 1, there is nothing to prove.Let us suppose that α = 1.Then, one can observe by Theorem 2.10 and the choice of α that B(r,s) − [α/(α − 1)]I has an inverse in B(c).That is to say, that

.16)
Since A is a triangle and is in B(c), A −1 is also conservative which implies that c A = c; see [14, page 12].

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This shows that x ∈ 1 if and only if |α − r| < |s|, as asserted.Now, we may give the following lemma required in the proof of theorems given in the present section.Lemma 2.4 [6, page 59].T has a dense range if and only if T * is one to one.Theorem 2.5.σ r (B(r,s),c 0 ) = {α ∈ C : |α − r| < |s|}.