7_195297_akhmedov.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No. 1, 2012, 59-74 ISSN 1307-5543 – www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE - 02 JULY 2011, ISTANBUL TURKEY Some Spectral Properties of the Generalized Difference Operator ∆v Ali M. Akhmedov1,∗, Saad R. El-Shabrawy 2 1,2 Baku State University, Faculty of Mech. & Math., Z. Khalilov Str., 23, AZ 1148, Baku, Azerbaijan 2 Permanent Address: Mansoura University, Damietta Branch, Faculty of Science, Mathematics Department, New Damietta, Egypt. Abstract. In this paper we consider the generalized difference operator ∆v on the sequence spaces l1 and c0. The operator ∆v is represented by a lower triangular double band matrix whose nonzero entries are the elements of a sequence (vk) with certain conditions. We mainly review several recent results concerning the fine spectrum of the operator ∆v over the sequence spaces l1 and c0. Also, we provide some new results. Following that we give some illustrative examples which motivate the main results. Finally, we give notes on the fine spectrum of the operator ∆v . These notes attempt to present some ideas about changing the conditions on the sequence (vk) in the fine spectrum of the operator ∆v . The new results of this paper generalize and improve some recent results that appeared recently in the literature. Key Words and Phrases: Spectrum of an operator; Generalized difference operator; The sequence spaces l1 and c0. 1. Introduction Several authors have studied the spectrum and fine spectrum of linear operators defined by some particular limitation matrices over some sequence spaces. We summarize the knowl- edge in the existing literature concerning the spectrum and the fine spectrum. The fine spec- trum of the difference operator ∆ over the sequence spaces c0 and c has been studied by Altay and Başar [9]. Akhmedov and Başar [1, 2] have studied the fine spectrum of the difference operator ∆ over the sequence spaces lp and bvp, where 1 ≤ p <∞. Note that the sequence space bvp was studied by Başar and Altay [12] and Akhmedov and Başar [2]. Malafosse [22] ∗Corresponding author. Email addresses: akhmedovali�rambler.ru (A. Akhmedov), srshabrawy�yahoo. om (S. El-Shabrawy) http://www.ejpam.com 59 c© 2012 EJPAM All rights reserved. A. Akhmedov, S. El-Shabrawy / Eur. J. Pure Appl. Math, 5 (2012), 59-74 60 has studied the spectrum and the fine spectrum of the difference operator ∆ over the space sr , where sr denotes the Banach space of all sequences x = (xk) normed by ‖x‖sr = sup k∈N � �xk � � rk (r > 0). The fine spectrum of the Zweier matrix operator Z s over the sequence spaces l1 and bv has been examined by Altay and Karakuş [11]. The fine spectrum of the generalized difference operator B(r, s) over the sequence spaces c0 and c has been studied by Altay and Başar [10]. Also, the fine spectrum of the operator B(r, s) over the sequence spaces l1 and bv has been examined by Furkan et al. [18]. The fine spectrum of the operator B(r, s) over the sequence spaces lp and bvp, where 1 < p < ∞ has been determined by Bilgiç and Furkan [13]. The fine spectrum of the operator B(r, s, t) over the sequence spaces c0 and c has been studied by Furkan et al. [16]. Also, the fine spectrum of the operator B(r, s, t) over the sequence spaces lp and bvp, where 1 < p < ∞ has been determined by Furkan et al. [17]. Panigrahi and Srivastava [23] have studied the fine spectrum of the generalized second order difference operator ∆2 uv over the sequence space c0. The fine spectrum of the generalized difference operator ∆a,b over the sequence spaces c0 and c has been studied by Akhmedov and El- Shabrawy [3,5]. The fine spectrum of the upper triangular double-band matrices over the sequence spaces c0 and c has been determined by Karakaya and Altun [20]. The operator ∆v has been introduced firstly by Srivastava and Kumar [24]. The operator ∆v : (l1→ l1, c0 −→ c0) is defined as follows: ∆v x =∆v(xk) = (vk xk − vk−1 xk−1) ∞ k=0 with x−1 = v−1 = 0, (1) where the sequence (vk) is assumed to be either constant or strictly decreasing sequence of positive real numbers satisfying lim k→∞ vk = L > 0 and sup k vk ≤ 2L. (2) It is easy to verify that the operator ∆v is represented by a lower triangular double band matrix of the form ∆v =       v0 0 0 · · · −v0 v1 0 · · · 0 −v1 v2 · · · ... ... ... . . .       . (3) The fine spectrum of the generalized difference operator ∆v over the sequence spaces c0 and l1 was investigated by Srivastava and Kumar [24, 25]. In [7, 8], Akhmedov and El- Shabrawy have proved by counterexamples that some of the main results in [24, 25] are incorrect and the corresponding corrected results are provided. The fine spectrum of the operator ∆v over the sequence space c has been examined by Akhmedov and El-Shabrawy [6]. Recently, El-Shabrawy [15] has studied the fine spectrum of the operator ∆v over the sequence space lp, where 1 < p < ∞. Akhmedov and El-Shabrawy [4] have modified the A. Akhmedov, S. El-Shabrawy / Eur. J. Pure Appl. Math, 5 (2012), 59-74 61 definition of the operator ∆v and have determined the fine spectrum of the modified operator ∆v over the sequence spaces c and lp, where 1< p <∞. Note that, if (vk) is a constant sequence, say vk = L 6= 0 for all k ∈ N, then the operator∆v is reduced to the operator B(r, s) with r = L, s = −L and the results for the spectrum and the fine spectrum of the operator ∆v on the sequence spaces l1 and c0 follow immediately from the corresponding results in [10, 18]. Then, throughout this paper, the case when (vk) is a constant sequence is not considered. The rest of the paper is organized as follows. Section 2 presents some basic concepts of spectral theory concerning the spectrum and the fine spectrum of linear operators. Next, in Section 3, we mainly review several recent results concerning the fine spectrum of operator ∆v over the sequence spaces l1 and c0. Also, some new results are obtained. In Section 4 we give some illustrative examples to support the main results. In Section 5 we show some ideas about changing the conditions on the sequence (vk) in the fine spectrum of the operator ∆v. 2. Preliminaries By w, we shall denote the space of all real or complex valued sequences. Any vector subspace of w is called a sequence space. We shall write l∞, c, c0 and bv for the spaces of all bounded, convergent, null and bounded variation sequences, respectively. Also by l1, lp and bvp we denote the spaces of all absolutely summable sequences, p-absolutely summable sequences and p-bounded variation sequences, respectively. A triangle is a lower triangular matrix with all of the principal diagonal elements nonzero. Let λ and µ be two sequence spaces and A = (ank) be an infinite matrix of real or complex numbers ank, where n, k ∈ N = {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 = (xk) ∈ λ, the sequence Ax = {(Ax)n}, the A-transform of x , is in µ, where (Ax)n = ∑ k ank xk, (n ∈ N). (4) For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. By (λ,µ), we denote the class of all matrices A such that A : λ→ µ. Thus, A ∈ (λ,µ) if and only if the series on the right side of (4) converges for each n ∈ N and every x ∈ λ, and we have Ax = {(Ax)n}n∈N ∈ µ for all x ∈ λ. We use the convention that any term with negative subscript is equal to naught. We recall some basic concepts of spectral theory which are needed for our investigation [see 21, pp. 370-372]. Let X be a Banach space and T : X → X be a bounded linear operator. By R(T ), we denote the range of T , i.e., R(T ) = � y ∈ X : y = T x , x ∈ X . 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 . A. Akhmedov, S. El-Shabrawy / Eur. J. Pure Appl. Math, 5 (2012), 59-74 62 If T : l1→ l1 is a bounded linear operator with matrix A, then it is known that the adjoint operator T ∗ : l∗1 → l∗1 is defined by the transpose of the matrix A. It is well-known that the dual space l∗1 of l1 is isomorphic to l∞. Also, if T : c0 → c0 is a bounded linear operator with matrix A then the adjoint operator T ∗ : c∗0 → c∗0 is defined by the transpose of the matrix A. The dual space c∗0 of c0 is isomorphic to the Banach space l1. Let X 6= {θ} be a complex normed space and T : D(T ) → X be a linear operator with domain D(T )⊆ X . With T we associate the operator Tλ = T −λI , (5) where λ is a complex number and I is the identity operator on D(T ). If Tλ has an inverse which is linear, we denote it by T−1 λ , that is T−1 λ = (T −λI)−1, (6) and call it the resolvent operator of T . Many properties of Tλ and T−1 λ depend on λ, and spectral theory is concerned with those properties. For instance, we shall be interested in the set of all λ in the complex plane such that T−1 λ exists. The boundedness of T−1 λ is another property that will be essential. We shall also ask for what λ’s the domain of T−1 λ is dense in X , to name just a few aspects. Definition 1. Let X 6= {θ} be a complex normed space and T : D(T )→ X be a linear operator with domain D(T )⊆ X . A regular value λ of T is a complex number such that (R1) T−1 λ exists, (R2) T−1 λ is bounded, (R3) T−1 λ is defined on a set which is dense in X . The resolvent set of T , denoted by ρ(T, X ), is the set of all regular values λ of T . Its comple- ment 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 σp(T, X ) is the set such that T−1 λ does not exist. Any such λ ∈ σp(T, X ) is called an eigenvalue of T . The continuous spectrum σc(T, X ) is the set such that T−1 λ exists and satisfies (R3) but not (R2), that is, T−1 λ is unbounded. The residual spectrum σr(T, X ) is the set 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 . Hence if (T − λI)x = θ for some x 6= θ , then λ ∈ σp(T, X ), by definition, that is, λ is an eigenvalue of T . The vector x is then called an eigenvector of T corresponding to the eigenvalue λ. Now, we may give: Lemma 1 ([19, p. 59]). T has a dense range if and only if T ∗ is one to one. A. Akhmedov, S. El-Shabrawy / Eur. J. Pure Appl. Math, 5 (2012), 59-74 63 3. Recent and New Results on the Fine Spectrum of the Operator ∆ v on l1 and c0 In this section we mainly review several recent results concerning the fine spectrum of the operator ∆v on the sequence spaces l1 and c0. Also, we provide some new results. As we mentioned before, the case when (vk) is a constant sequence is not considered here. So, throughout this section, the sequence (vk) is assumed to be a strictly decreasing sequence of positive real numbers satisfying the conditions (2). 3.1. The Fine Spectrum of the Operator ∆v on l1 Srivastava and Kumar [25] investigated the fine spectrum of the operator ∆v on the se- quence space l1. But, incorrect results are obtained. Akhmedov and El-Shabrawy [8] have proved by a counterexample that the results concerning the point spectrum and the residual spectrum are incorrect. In this subsection we summarize the main results. Theorem 1. The operator ∆v : l1 −→ l1 is a bounded linear operator and (i) ∆v l1 = 2v0. (ii) σ(∆v, l1) = {λ ∈ C : |λ− L| ≤ L}. (iii) σp(∆v, l1) = φ. (iv) σp(∆ ∗ v, l∗1) = {λ ∈ C : |λ− L| ≤ L}. (v) σr(∆v, l1) = {λ ∈ C : |λ− L| ≤ L}. (vi) σc(∆v, l1) = φ. The results (i), (ii), (iv) and (vi) of Theorem 1 have been given by Srivastava and Kumar [25] and the results (iii) and (v) of Theorem 1 have been proved by Akhmedov and El- Shabrawy [8]. To help understanding we give the following example which disproves the statements of Srivastava and Kumar [25] concerning the point spectrum and the residual spectrum of the operator ∆v on l1. Example 1. Consider the sequence (vk), where vk = k+2 k+1 , k ∈ N. Clearly, (vk) is a strictly decreasing sequence of positive real numbers satisfying the conditions (2); where lim k→∞ vk = L = 1, sup k vk = v0 = 2≤ 2L. We can prove that v0 = 2 /∈ σp(∆v, l1). Indeed, suppose for contrary that there exists x = (xk) 6= θ in l1 such that ∆v x = v0 x. Then (v0 − v0)x0 = 0 and − vk xk + (vk+1− v0)xk+1 = 0, for all k ∈ N. If x0 = 0, then xk = 0, for all k ≥ 1, and so we have a contradiction since x 6= θ . Also, if x0 6= 0 then we can easily see that � �xk � �≥ � �x0 � � , for all k ≥ 1, A. Akhmedov, S. El-Shabrawy / Eur. J. Pure Appl. Math, 5 (2012), 59-74 64 and so we have a contradiction since x ∈ l1. Then v0 /∈ σp(∆v, l1). Similarly, we can prove that vk /∈ σp(∆v, l1) for all k ≥ 1. Thus σp(∆v, l1) = φ. Now, the operator ∆v − v0I on l1 is defined by (∆v − v0I)x = (0,−v0x0 + (v1− v0)x1,−v1 x1+ (v2− v0)x2, . . .), (7) where x = (xk) ∈ l1. The operator (∆v − v0 I)−1 exists since v0 /∈ σp(∆v, l1). But (∆v − v0 I)−1 does not satisfy (R3). Indeed, consider the sequence y = (1,0,0, . . .) in l1 and let y be the center of a small ball, say, of radius 1/3. Clearly, by (7), this ball does not intersect the range of the operator ∆v − v0 I . Then, the operator ∆v − v0 I does not have a dense range in l1. Hence, by definition, v0 ∈ σr(∆v, l1). 3.2. The Fine Spectrum of the Operator ∆ v on c0 Srivastava and Kumar [24] investigated the fine spectrum of the operator ∆v on the se- quence space c0. But, incorrect results are also obtained. Akhmedov and El-Shabrawy [7] have proved by a counterexample that the results concerning the point spectrum, the residual spectrum and the continuous spectrum are incorrect. In this subsection we summarize the main results. Theorem 2. The operator ∆v : c0 −→ c0 is a bounded linear operator and (i) ∆v c0 = v0 + v1. (ii) σ(∆v, c0) = {λ ∈ C : |λ− L| ≤ L}. (iii) σp(∆v, c0) = ∅. The bounded linearity of the operator ∆v on c0 has been given by the Srivastava and Kumar [24] and the norm of the operator ∆v on c0 has been revised by Akhmedov and El- Shabrawy [7]. Also, the result (ii) of Theorem 2 has been proved by Srivastava and Kumar [24] and the result (iii) of Theorem 2 has been proved by Akhmedov and El-Shabrawy [7]. The results concerning the point spectrum of the adjoint operator ∆∗v of ∆v are given by the following theorem. Theorem 3 ([7]). (i) {λ ∈ C : |λ− L| < L} ∪ � v0 ⊆ σp(∆ ∗ v, c∗0), (ii) ¨ λ ∈ C : sup k � � � λ−vk vk � � � < 1 « ⊆ σp(∆ ∗ v, c∗0), (iii) σp(∆ ∗ v, c∗0)⊆ § λ ∈ C : inf k � � � λ−vk vk � � � < 1 ª . We give the following example to support the results in Theorem 3. A. Akhmedov, S. El-Shabrawy / Eur. J. Pure Appl. Math, 5 (2012), 59-74 65 Example 2. Consider the sequence (vk), where vk = (k+3)2 (k+2)2+(k+3)2 , k ∈ N. We can show that 1 ∈ σp(∆ ∗ v, c∗0). But 1 /∈ {λ ∈ C : |λ− L| < L} ∪ � v0 and 1 /∈ � λ ∈ C : sup n � � � λ−vk vk � � �< 1 � . On the other hand if vk = k+3 2k+5 , k ∈ N, then 1 ∈ § λ ∈ C : inf k � � � λ−vk vk � � � < 1 ª . But 1 /∈ σp(∆ ∗ v, c∗0). The following theorem gives some results on the residual spectrum of the operator ∆v on c0. Theorem 4 ([7]). (i) {λ ∈ C : |λ− L| < L} ∪ � v0 ⊆ σr(∆v, c0), (ii) ¨ λ ∈ C : sup k � � � λ−vk vk � � � < 1 « ⊆ σr(∆v, c0), (iii) σr(∆v, c0)⊆ § λ ∈ C : inf k � � � λ−vk vk � � � < 1 ª . For the continuous spectrum of the operator ∆v on c0, we have the following theorem. Theorem 5 ([7]). (i) σc(∆v, c0)⊆ {λ ∈ C : |λ− L| = L} \ � v0 , (ii) {λ ∈ C : |λ− L| ≤ L} ∩ § λ ∈ C : inf k � � � λ−vk vk � � �≥ 1 ª ⊆ σc(∆v, c0). Also, we have the following theorem. Theorem 6 ([7]). (i) σr(∆v, c0) = σp(∆ ∗ v, c∗0). (ii) σc(∆v, c0) = σ(∆v, c0)\σp(∆ ∗ v, c∗0). Now, we give the following new results: Theorem 7. σp(∆ ∗ v, c∗0) = {λ ∈ C : |λ− L| < L} ∪H, where H = ( λ ∈ C : |λ− L| = L, ∞ ∑ k=0 � � � � � k ∏ i=0 λ− vi vi � � � � � <∞ ) . Proof. Suppose that ∆∗v f = λ f for f = ( f0, f1, f2, . . .) 6= θ in c∗0 ∼= l1. Then, by solving the system of equations v0 f0 − v0 f1 = λ f0, v1 f1 − v1 f2 = λ f1, ... vk fk − vk fk+1 = λ fk, ... A. Akhmedov, S. El-Shabrawy / Eur. J. Pure Appl. Math, 5 (2012), 59-74 66 we obtain fk+1 = vk −λ vk fk, for all k ∈ N. Then we must take f0 6= 0, since otherwise we would have f = θ . It is clear that, for all k ∈ N, the vector f = ( f0, f1, . . . , fk, 0,0, . . .) is an eigenvector of the operator ∆∗v corresponding to the eigenvalue λ = vk, where f0 6= 0 and fn = vn−1−λ vn−1 fn−1 for all n = 1,2,3, . . . , k. Thus � vk : k ∈ N ⊆ σp(∆ ∗ v, c∗0). On the other hand if λ 6= vk for all k ∈ N, then we can see that ∑ k � � fk � � <∞ if lim k→∞ � � � fk+1 fk � � �= � � � λ−L L � � � < 1. Also, it can be proved that H ⊆ σp(∆ ∗ v, c∗0). Thus {λ ∈ C : |λ− L| < L} ∪H ⊆ σp(∆ ∗ v, c∗0). The second inclusion can be proved analogously. Theorem 8. σr(∆v, c0) = {λ ∈ C : |λ− L| < L} ∪H. Proof. The proof follows immediately from Theorems 6(i) and 7. Theorem 9. σc(∆v, c0) = {λ ∈ C : |λ− L| = L} \H. Proof. The proof follows immediately from Theorems 2(ii), 2(iii) and 8. 4. Illustrative Examples In this section we give some illustrative examples on the fine spectrum of the operator ∆v on the sequence spaces l1 and c0. In the following example, we consider a strictly decreasing sequence (vk) of positive real numbers satisfying the conditions (2). It will be shown that the following equalities are not hold; σp(∆v, l1) = � v0, v1, v2, . . . and σr(∆v, l1) = {λ ∈ C : |λ− L| ≤ L}\ � v0, v1, v2, . . . . Example 3 ([8]). Consider the sequence (vk), where vk = k+3 2k+5 , k ∈ N. Clearly, (vk) is a strictly decreasing sequence of positive real numbers satisfying the conditions (2); where lim k→∞ vk = L = 1/2, sup k vk = 3/5 ≤ 1 = 2L. We can prove that v0 = 3/5 /∈ σp(∆v, l1). Indeed, suppose for contrary that there exists x = (xk) 6= θ in l1 such that ∆v x = v0 x. Then (v0 − v0)x0 = 0 and − vk xk + (vk+1− v0)xk+1 = 0, for all k ∈ N. If x0 = 0, then xk = 0, for all k ≥ 1, and so we have a contradiction since x 6= θ . Also, if x0 6= 0 then lim k→∞ � � � � xk+1 xk � � � � = � � � � L L − vo � � � � = 5> 1, A. Akhmedov, S. El-Shabrawy / Eur. J. Pure Appl. Math, 5 (2012), 59-74 67 and so we have a contradiction since x ∈ l1. Then v0 /∈ σp(∆v, l1). Similarly, we can prove that vk /∈ σp(∆v, l1) for all k ≥ 1, and so σp(∆v, l1) = φ. Now, the operator ∆v − v0I on l1 is defined by (∆v − v0I)x = (0,−v0 x0 + (v1− v0)x1, −v1 x1+ (v2 − v0)x2, . . .), (8) where x = (xk) ∈ l1. The operator (∆v − v0 I)−1 exists since v0 /∈ σp(∆v, l1). But (∆v − v0 I)−1 does not satisfy (R3). Indeed, consider the sequence y = (1,0,0, . . .) in l1 and let y be the center of a small ball, say, of radius 1/3. Clearly, by (8), this ball does not intersect the range of the operator ∆v − v0 I . Then, the operator ∆v − v0 I does not have a dense range in l1. Hence, by definition, v0 ∈ σr(∆v, l1). The following example disproves the statements of Srivastava and Kumar [24] concerning the point spectrum, the residual spectrum and the continuous spectrum of the operator ∆v on c0. More precisely, we consider a strictly decreasing sequence (vk) of positive real numbers satisfying the conditions (2) and it will be shown that the following equalities are not hold; σp(∆v, c0) = � v0, v1, v2, . . . , σr(∆v, c0) = {λ ∈ C : |λ− L| < L} \ � v0, v1, v2, . . . , σp(∆ ∗ v, c∗0) = {λ ∈ C : |λ− L| < L} , σc(∆v, c0) = {λ ∈ C : |λ− L| = L}\ � v0 . Example 4 ([7]). Consider the sequence (vk), where vk = (k+3)2 (k+2)2+(k+3)2 , k ∈ N. The sequence (vk) is a strictly decreasing sequence of positive real numbers satisfying the conditions (2); where lim k→∞ vk = L = 1/2, sup k vk = 9/13≤ 1= 2L. We can prove, as in Example 3, that vk /∈ σp(∆v, c0) for all k ∈ N. Also, we can prove that v0 ∈ σr(∆v, c0). On the other hand, for λ = 1, we have λ− vk vk = 1− vk vk = � k+ 2 k+ 3 �2 . If we suppose that ∆∗v f = (1) f for some f = ( f0, f1, f2, . . .) 6= θ in c∗0 ∼= l1, then we obtain that fk = vk−1−1 vk−1 fk−1, k ≥ 1. If we take f0 = 0, then f = θ and we have a contradiction since f 6= θ . If f0 6= 0, then ∞ ∑ k=0 � � fk � � = � � f0 � �+ � � f0 � � ∞ ∑ k=1 � � � � 1− v0 v0 � � � � � � � � 1− v1 v1 � � � � . . . � � � � 1− vk−1 vk−1 � � � � = � � f0 � �+ 4 � � f0 � � ∞ ∑ k=1 � 1 k+ 2 �2 <∞. Then 1 ∈ σp(∆ ∗ v, c∗0), and consequently 1 /∈ σc(∆v, c0). In the following example, we consider a sequence of positive real numbers (not necessarily strictly decreasing) satisfying the conditions (2) and we calculate the spectrum, the point spectrum, the residual spectrum and the continuous spectrum of the operator ∆v on c0. A. Akhmedov, S. El-Shabrawy / Eur. J. Pure Appl. Math, 5 (2012), 59-74 68 Example 5. Consider the sequence (vk), where vk = (k+2)2 (k+2)2+(k+3)2 , k ∈ N. We can prove that the operator ∆v : c0 −→ c0 is a bounded linear operator with the norm ∆v c0 = 1 and σ(∆v, c0) = ¨ λ ∈ C : � � � � λ− 1 2 � � � � ≤ 1 2 « , σp(∆v, c0) = ∅. σp(∆ ∗ v, c∗0) = ¨ λ ∈ C : � � � � λ− 1 2 � � � � < 1 2 « , σr(∆v, c0) = ¨ λ ∈ C : � � � � λ− 1 2 � � � � < 1 2 « , σc(∆v, c0) = ¨ λ ∈ C : � � � � λ− 1 2 � � � � = 1 2 « . In Example 5, we see that although the sequence (vk) is not strictly decreasing, the residual spectrum and the continuous spectrum in addition to the spectrum and the point spectrum of the operator ∆v are completely determined. In fact, If (vk) is assumed to be a sequence of positive real numbers (not necessarily strictly decreasing) satisfying the conditions (2), then we can have results similar to those in Section 3.2. This means that the condition that (vk) is a strictly decreasing is not an effective condition. In the next section we modify the definition of the operator ∆v in two ways by dropping the condition that (vk) is strictly decreasing sequence of positive real numbers and replacing the conditions (2) by another conditions. 5. Notes on the Fine Spectrum of the Operator ∆ v on c0 and l1 In this section we are going to show some ideas about changing the conditions on the sequence (vk) in the fine spectrum of the operator ∆v. We consider two modifications of the operator ∆v. More precisely, we modify the definition of the operator ∆v by changing the conditions on the sequence (vk) in two ways. First, we consider the sequence (vk) of nonzero real numbers such that lim k→∞ vk = L > 0 and sup k vk ≤ L, (9) and we study the fine spectrum of the modified operator ∆v on c0. Second, we consider the sequence (vk) of nonzero real numbers such that lim k→∞ vk = L > 0, vk ≥ L and vk 6= 2L, for all k ∈ N, (10) and we study the fine spectrum of the modified operator ∆v on l1. We should indicate the reader that we use the same symbol for the operator ∆v and its modifications here, since they have the same matrix representation and the difference between them lies in the conditions on the sequence (vk). A. Akhmedov, S. El-Shabrawy / Eur. J. Pure Appl. Math, 5 (2012), 59-74 69 5.1. The fine Spectrum of the Modified Operator ∆ v on c0 In this subsection we calculate the fine spectrum of the modified operator ∆v, which is represented by the matrix in (3) such that the conditions (9) are satisfied, on the sequence space c0. The modified operator ∆v of this form has been introduced and studied by Akhme- dov and El-Shabrawy [4] over the sequence spaces c and lp, where 1< p <∞. The results of this section improve the corresponding results in Section 3.2. We begin by determining when a matrix A induces a bounded linear operator from c0 to itself. Lemma 2 ([26, p. 129]). The matrix A = � ank � gives rise to a bounded linear operator T ∈ B � c0 � from c0 to itself if and only if (i) The rows of A are in l1 and their l1 norms are bounded, (ii) The columns of A are in c0. The operator norm of T is the supremum of the l1 norms of the rows. Corollary 1. The modified operator ∆v : c0 → c0 is a bounded linear operator with the norm ∆v c0 = sup k ( � �vk � �+ � �vk−1 � �). Theorem 10. Let D = {λ ∈ C : |λ− L| ≤ L} and E = ¦ vk : k ∈ N, � �vk − L � � > L © . Then σ(∆v, c0) = D ∪ E. Proof. First, we prove that (∆v − λI)−1 exists and is in B(c0) for λ /∈ D ∪ E and then the operator ∆v −λI is not invertible for λ ∈ D ∪ E. Let λ /∈ D ∪ E. Then, |λ− L| > L and λ 6= vk for all k ∈ N. So, ∆v − λI is triangle, and hence (∆v −λI)−1 exists. We can calculate that (∆v −λI)−1 =        1 (v0−λ) 0 0 · · · v0 (v0−λ)(v1−λ) 1 (v1−λ) 0 · · · v0v1 (v0−λ)(v1−λ)(v2−λ) v1 (v1−λ)(v2−λ) 1 (v2−λ) · · · ... ... ... . . .        . Then, the rows of (∆v − λI)−1 are in l1 and the supremum of the l1 norms of the rows of (∆v −λI)−1 is sup k Sk, where Sk =   1 � �vk −λ � � + � �vk−1 � � � �vk −λ � � � �vk−1−λ � � + . . . .+ � �vk−1 � � � �vk−2 � � . . . � �v0 � � � �vk −λ � � � �vk−1−λ � � . . . � �v0−λ � �   , k ∈ N. Then, we can easily prove that sup k Sk <∞. Also, it is clear that the columns of (∆v − λI)−1 are in c0. From Lemma 2, (∆v −λI)−1 ∈ (c0, c0). Thus σ(∆v, c0)⊆ D ∪ E. A. Akhmedov, S. El-Shabrawy / Eur. J. Pure Appl. Math, 5 (2012), 59-74 70 Conversely, suppose that λ /∈ σ(∆v, c0). Then (∆v − λI)−1 ∈ B(c0). Since (∆v − λI)−1- transform of the unit sequence e = (1,0,0, . . .) is in c0, we have lim k→∞ � � � vk vk+1−λ � � � = � � � L L−λ � � � ≤ 1 and λ 6= vk for all k ∈ N. Then {λ ∈ C : |λ− L| < L} ⊆ σ(∆v, c0) and � vk : k ∈ N ⊆ σ(∆v, c0). But σ(∆v, c0) is compact set, and so it is closed. Then D = {λ ∈ C : |λ− L| ≤ L} ⊆ σ(∆v, c0) and E = ¦ vk : k ∈ N, � �vk − L � � > L © ⊆ σ(∆v, c0). This completes the proof. The point spectrum of the operator ∆v on c0 is given by the following theorem. Theorem 11. σp(∆v, c0) = E. Proof. Suppose ∆v x = λx for x 6= θ = (0,0,0, . . .) in c0. Then by solving the system of equations v0 x0 = λx0 −v0 x0+ v1 x1 = λx1 −v1 x1+ v2 x2 = λx2 ...      we obtain (v0 −λ)x0 = 0 and − vk xk + (vk+1−λ)xk+1 = 0, for all k ∈ N. Hence, for all λ /∈ � vk : k ∈ N , we have xk = 0 for all k ∈ N, which contradicts our assump- tion. This shows that σp(∆v, c0) ⊆ � vk : k ∈ N . Also, if λ = L, then we can easily prove that λ /∈ σp(∆v, c0). Thus σp(∆v, c0)⊆ � vk : k ∈ N \{L}. Now, we will prove that λ ∈ σp(∆v, c0) if and only if λ ∈ E. If λ ∈ σp(∆v, c0), then λ = v j 6= L for some j ∈ N and there exists x ∈ c0, x 6= θ such that ∆v x = v j x . Then lim k→∞ � � � � xk+1 xk � � � � = � � � � � L L − v j � � � � � ≤ 1. But � � � L L−v j � � � 6= 1. Then λ= v j ∈ ¦ vk : k ∈ N, � �vk − L � � > L © = E. Thus σp(∆v, c0)⊆ E. Conversely, let λ ∈ E. Then there exists i ∈ N such that λ = vi 6= L and so we can take x 6= θ such that ∆v x = vi x and lim k→∞ � � � � xk+1 xk � � � � = � � � � L L − vi � � � � < 1, that is, x ∈ c0. Thus E ⊆ σp(∆v, c0). This completes the proof. We give the following lemma which is required in the proof of the next theorem. Lemma 3. Let λ ∈ {λ ∈ C : |λ− L| = L}. Then the series ∑ k � � � � (v0 −λ)(v1−λ) . . . (vk −λ) v0v1 . . . vk � � � � , is not a convergent series. A. Akhmedov, S. El-Shabrawy / Eur. J. Pure Appl. Math, 5 (2012), 59-74 71 Proof. Let λ= λ1 + iλ2 ∈ C such that |λ− L| = L. Then |λ|2 = λ2 1 +λ 2 2 = 2λ1 L. Also, � �vk −λ � � 2 = v2 k + (λ2 1 +λ 2 2)− 2λ1vk = v2 k − 2λ1(vk − L) ≥ v2 k . Therefore � � � � vk −λ vk � � � � ≥ 1, for all k ∈ N. This completes the proof. Theorem 12. σp(∆ ∗ v, c∗0) = {λ ∈ C : |λ− L| < L} ∪ E. Proof. Suppose that ∆∗v f = λ f for f = ( f0, f1, f2, . . .) 6= θ in c∗0 ∼= l1. Then, by solving the system of equations v0 f0 − v0 f1 = λ f0 v1 f1 − v1 f2 = λ f1 ... vk fk − vk fk+1 = λ fk, ... we obtain fk+1 = vk −λ vk fk, k ∈ N. Therefore, we must take f0 6= 0, since otherwise we would have f = θ . It is clear that, for all k ∈ N, the vector f = ( f0, f1, . . . , fk, 0,0, . . .) is an eigenvector of the operator ∆∗v corresponding to the eigenvalue λ = vk, where f0 6= 0 and fn = vn−1−λ vn−1 fn−1, for all n = 1,2, . . . , k. Thus � vk : k ∈ N ⊆ σp(∆ ∗ v, c∗0). Also, if λ 6= vk for all k ∈ N, then fk 6= 0 for all k ∈ N, and so, ∑ k � � fk � � <∞ if lim k→∞ � � � fk+1 fk � � � = � � � λ−L L � � �< 1. Thus {λ ∈ C : |λ− L| < L} ∪ E ⊆ σp(∆ ∗ v, c∗0). Conversely, if λ ∈ σp(∆ ∗ v, c∗0), then there exists f = ( f0, f1, f2, . . .) 6= θ in c∗0 ∼= l1, ∆∗v f = λ f . Then, fk+1 = vk−λ vk fk, k ∈ N and ∑ k � � fk � � <∞. Therefore lim k→∞ � � � fk+1 fk � � � = � � � λ−L L � � � < 1 or λ ∈ � vk : k ∈ N (note that |λ− L| = L contradicts with ∑ k � � fk � � <∞, by using Lemma 3. This completes the proof. Theorem 13. σr(∆v, c0) = σp(∆ ∗ v, c∗0)\σp(∆v, c0). A. Akhmedov, S. El-Shabrawy / Eur. J. Pure Appl. Math, 5 (2012), 59-74 72 Proof. The proof follows immediately from the definition of the residual spectrum and Lemma 1. Theorem 14. σr(∆v, c0) = {λ ∈ C : |λ− L| < L}. Proof. The proof follows immediately from Theorems 11, 12 and 13. Theorem 15. σc(∆v, c0) = σ(∆v, c0)\σp(∆ ∗ v, c∗0). Proof. The proof follows immediately from Theorems 11, 12 and 13. Theorem 16. σc(∆v, c0) = {λ ∈ C : |λ− L| = L}. Proof. The proof follows immediately from Theorems 10, 12 and 15. 5.2. The Fine Spectrum of the Modified Operator ∆v on l1 In this subsection we calculate the fine spectrum of the operator ∆v, which is represented by the matrix in (3) such that the conditions (10) are satisfied, on the sequence space l1. The results of this section generalize the corresponding results in Section 3.1. We begin by determining when a matrix A induces a bounded linear operator from l1 to itself. Lemma 4 ([14, p. 253, Theorem 34.16]). The matrix A= (ank) gives rise to a bounded linear operator T ∈ B(l1) from l1 to itself if and only if the supremum of l1 norms of the columns of A is bounded. Corollary 2. The operator ∆v : l1→ l1 is a bounded linear operator with the norm ∆v l1 = 2 sup k vk. By using arguments similar to those used in Section 5.1, we can prove the following main theorem. Theorem 17. (i) σ(∆v, l1) = D ∪ E. (ii) σp(∆v, l1) = E. (iii) σp(∆ ∗ v, l∗1) = D ∪ E. (iv) σr(∆v, l1) = D. (v) σc(∆v, l1) = φ. REFERENCES 73 6. Conclusion In this paper, the fine spectrum of the generalized difference operator ∆v on the sequence spaces c0 and l1 is commented on, and some new results are obtained. Illustrative examples are given as well. These examples are used not only to apply new results but also to dis- prove some recent results. Finally, two modifications of the operator ∆v are introduced. The new results of this paper generalize and improve some recent results that appeared in the literature. References [1] A Akhmedov and F Başar, On the fine spectra of the difference operator ∆ over the sequence space lp, (1≤ p <∞), Demonstratio Math. 39 (3). 585-595. 2006. [2] A Akhmedov and F Başar, The fine spectra of the difference operator∆ over the sequence space bvp, (1≤ p <∞), Acta Math. Sin. (Engl. Ser.) 23 (10). 1757-1768. 2007. [3] A Akhmedov and S El-Shabrawy, On the fine spectrum of the operator ∆a,b over the sequence space c, Comput. Math. Appl. 61. 2994-3002. 2011. [4] A Akhmedov and S El-Shabrawy, On the fine spectrum of the operator ∆v over the sequence spaces c and lp, (1< p <∞), Appl. Math. Inf. Sci. 5 (3). 635-654. 2011. [5] A Akhmedov and S El-Shabrawy, On the spectrum of the generalized difference operator ∆a,b over the sequence space c0, Baku Univ. News J., Phys. Math. Sci. Ser. 4. 12-21. 2010. [6] A Akhmedov and S El-Shabrawy, The spectrum of the generalized lower triangle double- band matrix ∆a over the sequence space c, Al-Azhar Univ. Eng. J., JAUES (special issue), 5 (9), 54-63. 2010. [7] A Akhmedov and S El-Shabrawy, Comments on “On the fine spectrum of the generalized difference operator ∆v over the sequence space c0”, submitted for publication. [8] A Akhmedov and S El-Shabrawy, Notes on the spectrum of lower triangular double-band matrices, submitted for publication. [9] B Altay and F Başar, On the fine spectrum of the difference operator ∆ on c0 and c, Inform. Sci. 168. 217-224. 2004. [10] B Altay and F Başar, On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces c0 and c, Int. J. Math. Math. Sci. 18. 3005-3013. 2005. [11] B Altay and M Karakuş, On the spectrum and the fine spectrum of the Zweier matrix as an operator on some sequence spaces, Thai J. Math. 3(2). 153-162. 2005. REFERENCES 74 [12] F Başar and B Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J. 55 (1). 136-147. 2003. [13] H Bilgiç and H Furkan, On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces lp and bvp, (1< p <∞), Nonlinear Anal. 68. 499-506. 2008. [14] B Choudhary and S Nanda, Functional Analysis with Applications, John Wiley & Sons Inc., New York. 1989. [15] S El-Shabrawy, On the spectrum of the operator ∆v over the space lp, (1 < p < ∞), Baku Univ. News J., Phys. Math. Sci. Ser., to appear. [16] H Furkan, H Bilgiç and B Altay, On the fine spectrum of the operator B(r, s, t) over c0 and c, Comput. Math. Appl. 53. 989-998. 2007. [17] H Furkan, H Bilgiç and F Başar, On the fine spectrum of the operator B(r, s, t) over the sequence spaces lp and bvp, (1< p <∞), Comput. Math. Appl. 60. 2141-2152. 2010. [18] H Furkan, H Bilgiç and K Kayaduman, On the fine spectrum of the generalized difference operator B(r, s) over the sequence spaces l1 and bv, Hokkaido Math. J. 35. 893-904. 2006. [19] S Goldberg, Unbounded Linear Operators: Theory and Applications, McGraw-Hill, Inc., New York, 1966. [20] V Karakaya and M Altun, Fine spectra of upper triangular double-band matrices, J. Comput. Appl. Math. 234. 1387-1394. 2010. [21] E Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons Inc., New York. 1978. [22] B de Malafosse, Properties of some sets of sequences and application to the spaces of bounded difference sequences of order µ, Hokkaido Math. J. 31. 283–299. 2002. [23] B Panigrahi and P Srivastava, Spectrum and fine spectrum of generalized second order difference operator ∆2 uv on sequence space c0, Thai J. Math. 9 (1). 57–74. 2011. [24] P Srivastava and S Kumar, On the fine spectrum of the generalized difference operator ∆v over the sequence space c0, Commun. Math. Anal. 6 (1). 8-21. 2009. [25] P Srivastava and S Kumar, Fine spectrum of the generalized difference operator ∆v on sequence space l1, Thai J. Math. 8 (2). 221–233. 2010. [26] A Wilansky, Summability Through Functional Analysis, in: North-Holland Mathematics Studies, vol. 85, North-Holland, Amsterdam, 1984.