Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 381069, 19 pages

http://dx.doi.org/10.1155/2012/381069

## Fine Spectra of Upper Triangular Double-Band Matrices over the Sequence Space ,

Department of Mathematics, Faculty of Sciences, Konya Necmettin Erbakan University, Karacian Mahallesi, Ankara Caddesi 74, 42060 Konya, Turkey

Received 8 May 2012; Accepted 9 July 2012

Academic Editor: Antonia Vecchio

Copyright © 2012 Ali Karaisa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The operator on sequence space on is defined , where , and and are two convergent sequences of nonzero real numbers satisfying certain conditions, where . The main purpose of this paper is to determine the fine spectrum with respect to the Goldberg's classification of the operator defined by a double sequential band matrix over the sequence space . Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator over the space .

#### 1. Introduction

Let and be Banach spaces, and let also be a bounded linear operator. By , we denote the range of , that is, By , we also denote the set of all bounded linear operators on into itself. If is any Banach space and , then the adjoint of is a bounded linear operator on the dual of defined by for all and .

Given an operator , the set
is called the *resolvent set* of and its complement with respect to the complex plain
is called the *spectrum* of . By the closed graph theorem, the inverse operator
is always bounded and is usually called *resolvent operator* of at .

#### 2. 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.

##### 2.1. The Point Spectrum, Continuous Spectrum, and Residual Spectrum

The name *resolvent* is appropriate, since helps to solve the equation . Thus, provided exists. More important, the investigation of properties of will be basic for an understanding of the operator itself. Naturally, many properties of and depend on , and spectral theory is concerned with those properties. For instance, we will be interested in the set of all 's in the complex plane such that exists. Boundedness of is another property that will be essential. We will also ask for what 's the domain of is dense in , to name just a few aspects. A *regular value * of is a complex number such that exists and bounded and whose domain is dense in . For our investigation of , , and , we need some basic concepts in spectral theory, which are given as follows (see [1, pp. 370-371]).

The *resolvent set * of is the set of all regular values of . Furthermore, the spectrum is partitioned into three disjoint sets as follows.

The *point (discrete) spectrum * is the set such that does not exist. An is called an *eigenvalue* of .

The *continuous spectrum * is the set such that exists and is unbounded and the domain of is dense in .

The *residual spectrum * is the set such that exists (and may be bounded or not), but the domain of is not dense in .

Therefore, these three subspectra form a disjoint subdivisions 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 and the spectrum consists of only the set in the finite-dimensional case.

##### 2.2. 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 in a Banach space , we call a sequence in as a *Weyl sequence* for if and , as .

In what follows, we call the set
the *approximate point spectrum of *. Moreover, the subspectrum
is called defect spectrum of .

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, and . Moreover, comparing these subspectra with those in (2.1) we note that

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 , p. 28]). *Spectra and subspectra of an operator and its adjoint are related by the following relations: *(a), (b), (c), (d), (e), (f), (g).

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 if is a Hilbert space and 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]).

##### 2.3. Goldberg's Classification of Spectrum

If is a Banach space and , then there are three possibilities for : (A), (B), (C), and (1) exists and is continuous, (2) exists but is discontinuous, (3) does not exist.

If these possibilities are combined in all possible ways, nine different states are created. These are labelled by: , , , , , , , , . If an operator is in state , for example, then and exist but is discontinuous (see [3] and Figure 1).

If is a complex number such that or , then . All scalar values of not in comprise the spectrum of . The further classification of gives rise to the fine spectrum of . That is, can be divided into the subsets , , , , , , and . For example, if is in a given state, (say), then we write .

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 , by the closed graph theorem. If we are not in the third column, that is, if is not an eigenvalue of , we may always consider the resolvent operator (on a possibly “thin” domain of definition) as “algebraic” inverse of .

By a *sequence space*, we understand a linear subspace of the space of all complex sequences which contains , the set of all finitely nonzero sequences, where denotes the set of positive integers. We write , , , and for the spaces of all bounded, convergent, null, and bounded variation sequences, which are the Banach spaces with the sup-norm and , while is not a Banach space with respect to any norm, respectively, where . Also by , we denote the space of all -absolutely summable sequences, which is a Banach space with the norm , where .

Let be an infinite matrix of complex numbers , where , and write where denotes the subspace of consisting of for which the sum exists as a finite sum. For simplicity in notation, here and in what follows, the summation without limits runs from to , and we will use the convention that any term with negative subscript is equal to naught. More generally if is a normed sequence space, we can write for the for which the sum in (2.8) converges in the norm of . We write for the space of those matrices which send the whole of the sequence space into 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 studied by González [19], where . Also, weighted mean matrices of operators on have been investigated by Cartlidge [20]. The spectrum of the Cesàro operator of order one on the sequence spaces and investigated by Okutoyi [8, 21]. The spectrum and fine spectrum of the Rhally operators on the sequence spaces , , , , and were examined by Yıldırım [9, 22–28]. The fine spectrum of the difference operator over the sequence spaces and was studied by Altay and Başar [12]. The same authors also worked the fine spectrum of the generalized difference operator over and , in [29]. The fine spectra of over and studied by Kayaduman and Furkan [30]. Recently, the fine spectra of the difference operator over the sequence spaces and studied by Akhmedov and Başar [31, 32], where is the space of -bounded variation sequences and introduced by Başar and Altay [33] with . Also, the fine spectrum of the generalized difference operator over the sequence spaces and determined by Furkan et al. [34]. Recently, the fine spectrum of over the sequence spaces and has been studied by Furkan et al. [35]. Quite recently, de Malafosse [11] and Altay and Başar [12] have, respectively, studied the spectrum and the fine spectrum of the difference operator on the sequence spaces and , , where denotes the Banach space of all sequences normed by , . Altay and Karakuş [36] have determined the fine spectrum of the Zweier matrix, which is a band matrix as an operator over the sequence spaces and . Farés and de Malafosse [37] studied the spectra of the difference operator on the sequence spaces , where denotes the sequence of positive reals and is the Banach space of all sequences normed by with . Also the fine spectrum of the same operator over and has been studied by Bilgiç and Furkan [13]. More recently the fine spectrum of the operator over and has been studied by Bilgiç and Furkan [38]. In 2010, Srivastava and Kumar [16] have determined the spectra and the fine spectra of generalized difference operator on , where is defined by and for all , under certain conditions on the sequence , and they have just generalized these results by the generalized difference operator defined by for all , (see [18]). Altun [39] has studied the fine spectra of the Toeplitz operators, which are represented by upper and lower triangular -band infinite matrices, over the sequence spaces and . Later, Karakaya and Altun have determined the fine spectra of upper triangular double-band matrices over the sequence spaces and , in [40]. Quite recently, Akhmedov and El-Shabrawy [15] have obtained the fine spectrum of the generalized difference operator , defined as a double band matrix with the convergent sequences and having certain properties, over the sequence space . Finally, the fine spectrum with respect to the Goldberg's classification of the operator defined by a triple band matrix over the sequence spaces and with has recently been studied by Furkan et al. [14]. At this stage, Table 2 may be useful.

Lemma 2.2 (see [41, p. 253, Theorem ]). *The matrix gives rise to a bounded linear operator from to itself if and only if the supremum of norms of the columns of is bounded. *

Lemma 2.3 (see [41, p. 245, Theorem ]). *The matrix gives rise to a bounded linear operator from to itself if and only if the supremum of norms of the rows of is bounded. *

Lemma 2.4 (see [41, p. 254, Theorem ]). * Let and . Then, . *

Let and be sequences whose entries either constants or distinct real numbers satisfying the following conditions:

Then, we define the sequential generalized difference matrix by

Therefore, we introduce the operator from to itself by

#### 3. Fine Spectra of Upper Triangular Double-Band Matrices over the Sequence Space

Theorem 3.1. *The operator is a bounded linear operator and
*

*Proof. *Since the linearity of the operator is not difficult to prove, we omit the detail. Now we prove that (3.1) holds for the operator on the space . It is trivial that for . Therefore, we have
which implies that
Let , where . Then, since it is easy to see by Minkowski's inequality that
which leads us to the result that
Therefore, by combining the inequalities in (3.3) and (3.5) we have (3.1), as desired.

Lemma 3.2 (see [42, p. 115, Lemma 3.1]). *Let . If
**
then the series
**
is not convergent. *

Throughout the paper, by and , we denote the set of constant sequences and the set of sequences of distinct real numbers, respectively.

Theorem 3.3.

*Proof. *Let for Then, by solving linear equation
for all and
*Part *. Assume that . Let and For all . We observe that . This shows that if and if only , as asserted.*Part *2. Assume that . We must take , since . It is clear that, for all , the vector is an eigenvector of the operator corresponding to the eigenvalue , where and , for . Thus . If , for all , then . If we take , since , . Hence . Conversely, let . Then, there exists in and we have , for all . Since , we can use ratio test. And so or . If , by Lemma 3.2 . This completes the proof.

Theorem 3.4.

*Proof. *We prove the theorem by dividing into two parts.*Part *1. Assume that . Consider for in . Then, by solving the system of linear equations
we find that if and if , which contradicts . If is the first nonzero entry of the sequence and , then we get that implies , which contradicts the assumption . Hence, the equation has no solution . *Part *2. Assume that . Then, by solving the equation for in , we obtain and for all . Hence, for all , we have for all , which contradicts our assumption. So, . This shows that . Now, we prove that
If , then, by solving the equation for in with ,
which can expressed by the recursion relation
Using ratio test,
But . Hence,
If we choose for all , then we get and
which can expressed by the recursion relation
Using ratio test,
But . So we have
Conversely, let . Then exist , , and
That is, . So we have . This completes the proof.

Lemma 3.5 (see [3, p. 60]). *The adjoint operator of is onto if and only if is a bounded operator. *

Theorem 3.6.

* Proof. *The proof is obvious so is omitted.

Theorem 3.7. *Let in and . .*

*Proof. *By Theorems 3.4 and 3.6, .

Theorem 3.8. *Let and . Then, the set is finite and . *

* Proof. *We will show that is onto, for . Thus, for every , we find . is triangle so it has an inverse. Also equation gives . It is sufficient to show that . We can calculate that as follows:
Therefore, the supremum of the norms of the rows of is , where
Now, we prove that . Since , then there exists such that with , for all ,
Therefore,
where
Then, and so
But there exist and a real number such that for all . Then, for all . Hence, . This shows that . Similarly, we can show that . By Lemma 2.4, we have
Hence, is onto. By Lemma 3.5, is bounded inverse. This means that
Combining this with Theorem 3.3 and Theorem 3.7, we get
and again from Theorem 3.3 and . Since the spectrum of any bounded operator is closed, we have

Combining (3.31) and (3.32), we get

Theorem 3.9. *Let in or . .*

* Proof. *The proof follows of immediately from Theorems 3.3, 3.7, and 3.8 because the parts , , and are pairwise disjoint sets and union of these sets is .

Theorem 3.10. *Let and . If , . *

* Proof. *From Theorem 3.3, . Thus, does not exist. It is sufficient to show that the operator is onto, that is, for given , we have to find such that . Solving the linear equation ,

let
Then, , where
for all . Then, since
we have
Since , is a convergent sequence of positive real numbers with limit . Hence, bounded and we have . Therefore,
This shows that . Thus is onto. So we have .

Theorem 3.11. *Let with , for all . Then, the following statements hold: *(i)*, *(ii)*, *(iii)*. *

* Proof. *(i) Since from Table 1,
we have by Theorem 3.7
Hence,

(ii) Since the following equality:
holds from Table 1, we derive by Theorems 3.8 and 3.10 that .

(iii) From Table 1, we have

By Theorem 3.4, it is immediate that .

Theorem 3.12. *Let . Then
*

*Proof. *We have by Theorem 3.4 and Part (e) of Proposition 2.1 that
By Theorems 3.7 and 3.4, we must have
Hence, . Additionally, since .

Therefore, we derive from Table 1, Theorems 3.8, and 3.10 that

#### 4. 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 on the space . 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 on the space , 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.

#### Acknowledgment

The authors would like to express their gratitude to Professor Feyzi Basar, Fatih University, Faculty of Art and Sciences, Department of Mathematics, The Hadımköy Campus, Büyükçekmece, Turkey, for his careful reading and for making some useful corrections, which improved the presentation of the paper.

#### References

- E. Kreyszig,
*Introductory Functional Analysis with Applications*, John Wiley & Sons, New York, NY, USA, 1978. - J. Appell, E. Pascale, and A. Vignoli,
*Nonlinear Spectral Theory*, vol. 10 of*de Gruyter Series in Nonlinear Analysis and Applications*, Walter de Gruyter, Berlin, Germany, 2004. View at Publisher · View at Google Scholar - S. Goldberg,
*Unbounded Linear Operators*, Dover Publications, New York, NY, USA, 1985. - R. B. Wenger, “The fine spectra of the Hölder summability operators,”
*Indian Journal of Pure and Applied Mathematics*, vol. 6, no. 6, pp. 695–712, 1975. View at Google Scholar · View at Zentralblatt MATH - B. E. Rhoades, “The fine spectra for weighted mean operators,”
*Pacific Journal of Mathematics*, vol. 104, no. 1, pp. 219–230, 1983. View at Google Scholar · View at Zentralblatt MATH - J. B. Reade, “On the spectrum of the Cesàro operator,”
*The Bulletin of the London Mathematical Society*, vol. 17, no. 3, pp. 263–267, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. M. Akhmedov and F. Başar, “On the fine spectrum of the Cesàro operator in ${c}_{0}$,”
*Mathematical Journal of Ibaraki University*, vol. 36, pp. 25–32, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. I. Okutoyi, “On the spectrum of ${C}_{1}$ as an operator on bv,”
*Communications de la Faculté des Sciences de l'Université d'Ankara A1*, vol. 41, no. 1-2, pp. 197–207, 1992. View at Google Scholar · View at Zentralblatt MATH - M. Yıldırım, “On the spectrum and fine spectrum of the compact Rhaly operators,”
*Indian Journal of Pure and Applied Mathematics*, vol. 27, no. 8, pp. 779–784, 1996. View at Google Scholar · View at Zentralblatt MATH - C. Coşkun, “The spectra and fine spectra for $p$-Cesàro operators,”
*Turkish Journal of Mathematics*, vol. 21, no. 2, pp. 207–212, 1997. View at Google Scholar - B. de Malafosse, “Properties of some sets of sequences and application to the spaces of bounded difference sequences of order $\mu $,”
*Hokkaido Mathematical Journal*, vol. 31, no. 2, pp. 283–299, 2002. View at Google Scholar · View at Zentralblatt MATH - B. Altay and F. Başar, “On the fine spectrum of the difference operator $\mathrm{\Delta}$ on ${c}_{0}$ and $c$,”
*Information Sciences*, vol. 168, no. 1–4, pp. 217–224, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Bilgiç and H. Furkan, “On the fine spectrum of the operator $B(r,s,t)$ over the sequence spaces ${l}_{1}$ and $bv$,”
*Mathematical and Computer Modelling*, vol. 45, no. 7-8, pp. 883–891, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Furkan, H. Bilgiç, and F. Başar, “On the fine spectrum of the operator $B(r,s,t)$ over the sequence spaces ${\ell}_{p}$ and $b{v}_{p}$, $(1<p<\infty )$,”
*Computers & Mathematics with Applications*, vol. 60, no. 7, pp. 2141–2152, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. M. Akhmedov and S. R. El-Shabrawy, “On the fine spectrum of the operator ${\mathrm{\Delta}}_{a,b}$ over the sequence space $c$,”
*Computers & Mathematics with Applications*, vol. 61, no. 10, pp. 2994–3002, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. D. Srivastava and S. Kumar, “Fine spectrum of the generalized difference operator ${\mathrm{\Delta}}_{v}$ on sequence space ${l}_{1}$,”
*Thai Journal of Mathematics*, vol. 8, no. 2, pp. 7–19, 2010. View at Google Scholar · View at Zentralblatt MATH - B. L. Panigrahi and P. D. Srivastava, “Spectrum and fine spectrum of generalized second order difference operator ${\mathrm{\Delta}}_{uv}^{2}$ on sequence space ${c}_{0}$,”
*Thai Journal of Mathematics*, vol. 9, no. 1, pp. 57–74, 2011. View at Google Scholar - P. D. Srivastava and S. Kumar, “Fine spectrum of the generalized difference operator ${\mathrm{\Delta}}_{uv}$ on sequence space ${l}_{1}$,”
*Applied Mathematics and Computation*, vol. 218, no. 11, pp. 6407–6414, 2012. View at Publisher · View at Google Scholar - M. González, “The fine spectrum of the Cesàro operator in ${\ell}_{p}$$(1<p<\infty )$,”
*Archiv der Mathematik*, vol. 44, no. 4, pp. 355–358, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. J. Cartlidge,
*Weighted mean matrices as operators on lp [Ph.D. thesis]*, Indiana University, 1978. - J. I. Okutoyi, “On the spectrum of ${C}_{1}$ as an operator on $b{v}_{0}$,”
*Journal of the Australian Mathematical Society A*, vol. 48, no. 1, pp. 79–86, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Yıldırım, “On the spectrum of the Rhaly operators on ${c}_{0}$ and $c$,”
*Indian Journal of Pure and Applied Mathematics*, vol. 29, no. 12, pp. 1301–1309, 1998. View at Google Scholar · View at Zentralblatt MATH - M. Yıldırım, “On the spectrum of the Rhaly operators on ${l}_{p}$,”
*Indian Journal of Pure and Applied Mathematics*, vol. 32, no. 2, pp. 191–198, 2001. View at Google Scholar · View at Zentralblatt MATH - M. Yıldırım, “The fine spectra of the Rhaly operators on ${c}_{0}$,”
*Turkish Journal of Mathematics*, vol. 26, no. 3, pp. 273–282, 2002. View at Google Scholar · View at Zentralblatt MATH - M. Yıldırım, “On the spectrum of the Rhaly operators on
*bv*,”*East Asian Mathematical Journal*, vol. 18, pp. 21–41, 2002. View at Google Scholar - M. Yıldırım, “On the spectrum and fine spectrum of the compact Rhaly operators,”
*Indian Journal of Pure and Applied Mathematics*, vol. 34, no. 10, pp. 1443–1452, 2003. View at Google Scholar · View at Zentralblatt MATH - M. Yildirim, “On the spectrum of the Rhaly operators on $b{v}_{0}$,”
*Communications of the Korean Mathematical Society*, vol. 18, no. 4, pp. 669–676, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Yildirim, “On the fine spectrum of the Rhaly operators on
*lp*,”*East Asian Mathematical Journal*, vol. 20, pp. 153–160, 2004. View at Google Scholar - B. Altay and F. Başar, “On the fine spectrum of the generalized difference operator $B(r,s)$ over the sequence spaces ${c}_{0}$ and $c$,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2005, no. 18, pp. 3005–3013, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Kayaduman and H. Furkan, “The fine spectra of the difference operator $\mathrm{\Delta}$ over the sequence spaces ${l}_{1}$ and $bv$,”
*International Mathematical Forum*, vol. 1, no. 24, pp. 1153–1160, 2006. View at Google Scholar · View at Zentralblatt MATH - A. M. Akhmedov and F. Başar, “On the fine spectra of the difference operator $\mathrm{\Delta}$ over the sequence space ${\ell}_{p}$,”
*Demonstratio Mathematica*, vol. 39, no. 3, pp. 585–595, 2006. View at Google Scholar · View at Zentralblatt MATH - A. M. Akhmedov and F. Başar, “On the fine spectra of the difference operator $\mathrm{\Delta}$ over the sequence space $b{v}_{p}$, $(1\le p<\infty )$,”
*Acta Mathematica Sinica*, vol. 23, no. 10, pp. 1757–1768, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Başar and B. Altay, “On the space of sequences of $p$-bounded variation and related matrix mappings,”
*Ukrainian Mathematical Journal*, vol. 55, no. 1, pp. 136–147, 2003. View at Publisher · View at Google Scholar - H. Furkan, H. Bilgiç, and K. Kayaduman, “On the fine spectrum of the generalized difference operator $B(r,s)$ over the sequence spaces ${l}_{1}$ and $bv$,”
*Hokkaido Mathematical Journal*, vol. 35, no. 4, pp. 897–908, 2006. View at Google Scholar - H. Furkan, H. Bilgiç, and B. Altay, “On the fine spectrum of the operator $B(r,s,t)$ over ${c}_{0}$ and $c$,”
*Computers & Mathematics with Applications*, vol. 53, no. 6, pp. 989–998, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Altay and M. Karakuş, “On the spectrum and the fine spectrum of the Zweier matrix as an operator on some sequence spaces,”
*Thai Journal of Mathematics*, vol. 3, no. 2, pp. 153–162, 2005. View at Google Scholar · View at Zentralblatt MATH - A. Farés and B. de Malafosse, “Spectra of the operator of the first difference in ${s}_{\alpha},{s}_{\alpha}^{0},{s}_{\alpha}^{(c)}$ and ${l}_{p}(\alpha )(1\le p<\infty )$
and application to matrix transformations,”
*Demonstratio Mathematica*, vol. 41, no. 3, pp. 661–676, 2008. View at Google Scholar · View at Zentralblatt MATH - H. Bilgiç and H. Furkan, “On the fine spectrum of the generalized difference operator $B(r,s)$ over the sequence spaces ${l}_{p}$ and $b{v}_{p}$$(1<p<\infty )$,”
*Nonlinear Analysis*, vol. 68, no. 3, pp. 499–506, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Altun, “On the fine spectra of triangular Toeplitz operators,”
*Applied Mathematics and Computation*, vol. 217, no. 20, pp. 8044–8051, 2011. View at Publisher · View at Google Scholar - V. Karakaya and M. Altun, “Fine spectra of upper triangular double-band matrices,”
*Journal of Computational and Applied Mathematics*, vol. 234, no. 5, pp. 1387–1394, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Choudhary and S. Nanda,
*Functional Analysis with Applications*, John Wiley & Sons, New York, NY, USA, 1989. - S. R. El-Shabrawy, “On the fine spectrum of the generalized difference operator ${\mathrm{\Delta}}_{a,b}$ over the sequence ${l}_{p}$, $(1<p<\infty )$,”
*Applied Mathematics & Information Sciences*, vol. 6, no. 1, supplement, pp. 111S–118S, 2012. View at Google Scholar