Abstract and Applied Analysis

Volume 2013 (2013), Article ID 342682, 10 pages

http://dx.doi.org/10.1155/2013/342682

Research Article

Fine Spectra of Upper Triangular Triple-Band Matrices over the Sequence Space ()

Ali Karaisa¹ and Feyzi Başar²

¹Department of Mathematics, Necmettin Erbakan University, Karaciğan Mahallesi, Ankara Caddesi 74, 42060 Konya, Turkey

²Department of Mathematics, Fatih University, Hadımköy Campus, Büyükçekmece, 34500 Istanbul, Turkey

Received 27 September 2012; Revised 28 December 2012; Accepted 31 December 2012

Academic Editor: Simeon Reich

Copyright © 2013 Ali Karaisa and Feyzi Başar. 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 fine spectra of lower triangular triple-band matrices have been examined by several authors (e.g., Akhmedov (2006), Başar (2007), and Furken et al. (2010)). Here we determine the fine spectra of upper triangular triple-band matrices over the sequence space . The operator on sequence space on is defined by , where , with . In this paper we have obtained the results on the spectrum and point spectrum for the operator on the sequence space . Further, the results on continuous spectrum, residual spectrum, and fine spectrum of the operator on the sequence space are also derived. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator over the space and we give some applications.

1. Introduction

In functional analysis, the spectrum of an operator generalizes the notion of eigenvalues for matrices. The spectrum of an operator over a Banach space is partitioned into three parts, which are the point spectrum, the continuous spectrum, and the residual spectrum. The calculation of these three parts of the spectrum of an operator is called calculating the fine spectrum of the operator.

Over the years and different names the spectrum and fine spectra of linear operators defined by some triangle matrices over certain sequence spaces were studied.

By we denote the space of all complex-valued sequences. Any vector subspace of is called a sequence space. We write , , , and for the spaces of all bounded, convergent, null, and bounded variation sequences, respectively, which are the Banach spaces with the sup-norm and , respectively, where . Also by and we denote the spaces of all absolutely summable and -absolutely summable sequences, which are the Banach spaces with the norm , respectively, where .

Several authors studied the spectrum and fine spectrum of linear operators defined by some triangle matrices over some sequence spaces. We introduce knowledge in the existing literature concerning the spectrum and the fine spectrum. Cesàro operator of order one on the sequence space was studied by Gonzàlez [1], where . Also, weighted mean matrices of operators on have been investigated by Cartlidge [2]. The spectrum of the Cesàro operator of order one on the sequence spaces and were investigated by Okutoyi [3, 4]. The spectrum and fine spectrum of the Rally operators on the sequence space were examined by Yıldırım [5]. The fine spectrum of the difference operator over the sequence spaces and was studied by Altay and Başar [6]. The same authors also worked out the fine spectrum of the generalized difference operator over and , in [7]. Recently, the fine spectra of the difference operator over the sequence spaces and have been studied by Akhmedov and Başar [8, 9], where is the space consisting of the sequences such that and introduced by Başar and Altay [10] with . In the recent paper, Furkan et al. [11] have studied the fine spectrum of over the sequence spaces and with , where is a lower triangular triple-band matrix. Later, Karakaya and Altun have determined the fine spectra of upper triangular double-band matrices over the sequence spaces and , in [12]. Quite recently, Karaisa [13] has determined the fine spectrum of the generalized difference operator , defined as an upper triangular double-band matrix with the convergent sequences and having certain properties, over the sequence space , where .

In this paper, we study the fine spectrum of the generalized difference operator defined by a triple sequential band matrix acting on the sequence space (), with respect to Goldberg's classification. Additionally, we give the approximate point spectrum and defect spectrum and give some applications.

2. Preliminaries, Background, and Notation

Let and be two Banach spaces and be a bounded linear operator. By we denote range of , that is,

By we also denote the set of all bounded linear operators on into itself. If then the adjoint of is a bounded linear operator on the dual of defined by for all and .

Let be a complex normed space and be a linear operator with domain . With we associate the operator , where is a complex number and is the identity operator on . If has an inverse which is linear, we denote it by , that is,

and call it the resolvent operator of .

Many properties of and depend on , and spectral theory is concerned with those properties. For instance, we shall be interested in the set of all in the complex plane such that exists. The boundedness of is another property that will be essential. We shall also ask for what the domain of is dense in , to name just a few aspects For our investigation of , , and , we need some basic concepts in spectral theory which are given as follows (see [14, pp. 370-371]).

Let be a complex normed space and be a linear operator with domain . A regular value of is a complex number such that

(R1) exists,(R2) is bounded,(R3) is defined on a set which is dense in .

The resolvent set of is the set of all regular values of . Its complement in the complex plane is called the spectrum of . Furthermore, the spectrum is partitioned into three disjoint sets as follows. The point spectrum is the set such that does not exist. is called an eigenvalue of . The continuous spectrum is the set such that exists and satisfies (R3) but not (R2). The residual spectrum is the set such that exists but does not satisfy (R3).

In this section, following Appell et al. [15], we define the three more subdivisions of the spectrum called 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 (3) and (4) form a (not necessarily disjoint) subdivisions

of the spectrum. There is another subspectrum

which is often called compression spectrum in the literature.

By the definitions given above, we can illustrate the subdivisions of spectrum in Table 1.

Table 1: Subdivisions of spectrum of a linear operator.

1
2
3

exists and is bounded
exists and is unbounded
does not exist

A

—

B

C

From Goldberg [16] if , a Banach space, then there are three possibilities for :

(I),(II), (III),

and three possibilities for :

(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 , , , , , , , and . If is a complex number such that or , then is in the resolvent set of . The further classification gives rise to the fine spectrum of . If an operator is in state , for example, then and exists but is discontinuous and we write .

Let and be two sequence spaces and let be an infinite matrix of real or complex numbers , where . Then, we say that defines a matrix mapping from into and we denote it by writing if for every sequence the sequence , the transform of is in , where

By , we denote the class of all matrices such that . Thus, if and only if the series on the right side of (7) converges for each and every , and we have for all .

Proposition 1 (see [15, Proposition 1.3, 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 [15]).

Lemma 2 (see [16, p. 60]). The adjoint operator of    is onto if and only if has a bounded inverse.

Lemma 3 (see [16, p. 59]). has a dense range if and only if is one to one.

Our main focus in this paper is on the triple-band matrix , where

We assume here and after that and are complex parameters which do not simultaneously vanish. We introduce the introduce the operator from to itself by

3. Fine Spectra of Upper Triangular Triple-Band Matrices over the Sequence Space ()

In this section, we prove that the operator is a bounded linear operator and compute its norm. We essentially emphasize the fine spectrum of the operator in the case .

Theorem 4. The operator is a bounded linear operator and

Proof. Since the linearity of the operator is trivial, so it is omitted. Let us take . Then and observe that

which gives the fact that

Let , where . Then, since , and , it is easy to see by triangle inequality that

which leads us to the result that

Therefore, by combining the inequalities (12) and (14) we see that (10) holds which completes the proof.

If is a bounded matrix operator with the matrix , then it is known that the adjoint operator is defined by the transpose of the matrix . The dual space of is isomorphic to , where .

Before giving the main theorem of this section, we should note the following remark. In this work, here and in what follows, if is a complex number, then by we always mean the square root of with a nonnegative real part. If , then represents the square root of with . The same results are obtained if represents the other square root.

Theorem 5. Let be a complex number such that and define the set by

Then, .

Proof. Let . Then, by solving the equation for in terms of , we obtain

and if we denote , , and , we have

Now we must find . We have and if we use relation (18), we have

This implies that

In fact this sequence is obtained recursively by letting

The characteristic polynomial of the recurrence relation is . There are two cases.

Case 1. If whose roots are

elementary calculation on recurrent sequence gives that

In this case . Assume that . So we have

Since for any , we must have

It follows that . Now, for we can see that

for all . Taking limit on the inequality (26) as , we get

Thus for , is onto and by Lemma 2, has a bounded inverse. This means that

Case 2. If , a calculation on recurrent sequence gives that

Now, for we can see that

for all . Taking limit on the inequality (30) as , we obtain that

is convergent, since . Thus for , is onto and by Lemma 2, has a bounded inverse. This means that

Theorem 6. .

Proof. Consider with 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 which implies which contradicts the assumption . Hence, the equation has no solution .

Theorem 7. , where .

Proof. Let for . Then, by solving the system of linear equations

and we have

Assume that . Then, we choose and . We show that for all . Since are roots of the characteristic equation , we must have

Combining the fact with relation (35), we can see that

The same result may be obtained in case . Now , since . This shows that .

Now we assume that , that is, . We must show that . Therefore we obtain from the relation (35) that

Now we examine three cases.

Case 1 (). In this case we have and

Then, we have

Now, if , then we have which is not in . Otherwise,

Case 2 (). In this case we have and using the formula

we obtain that

which leads to

Case 3 (). In this case we have and so we have . Assume that . This implies that and . Thus we again derive (35)

Since , we must have . But this implies that , a contradiction which means that . Thus . This completes the proof.

Theorem 8. .

Proof. By Proposition 1, . Since by Theorem 6,

This completes the proof.

Theorem 9. Let be a complex number such that and define the set by

Then, .

Proof. By Theorem 7, we get

Since the spectrum of any bounded operator is closed, we have

and again from Theorems 5, 7, and 8,

Combining (49) and (50), we obtain that , where is defined by (47).

Theorem 10. , where .

Proof. Because the parts , , and are pairwise disjoint sets and the union of these sets is , the proof immediately follows from Theorems 7, 8, and 9.

Theorem 11. If , .

Proof. From Theorem 7, . Thus, does not exist. By Theorem 6 is one to one, so has a dense range in by Lemma 3.

Theorem 12. The following statements hold:

(i), (ii), (iii).

Proof. (i)  Since from Table 1,

we have by Theorem 8

Hence

(ii) Since the following equality

holds from Table 1, we derive by Theorems 8 and 11 that .

(iii) From Table 1, we have

By Theorem 6 it is immediate that .

4. Fine Spectra of Upper Triangular Triple-Band Matrices over the Sequence Space   ( )

In the present section, we determine the fine spectrum of the operator in case . We quote some lemmas which are needed in proving the theorems given in Section 4.

Lemma 13 (see [17, p. 253, Theorem 34.16]). 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 14 (see [17, p. 245, Theorem 34.3]). 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 15 (see [17, p. 254, Theorem 34.18]). Let and . Then, .

Theorem 16. The operator is a bounded linear operator and

Proof. Since the linearity of the operator is not hard, we omit the details. Now we prove that (56) holds for the operator on the space . It is trivial that for . Therefore, we have

which implies that

Let , where . Then, since , , and , it is easy to see by Minkowski's inequality that

which leads us to the result that

Therefore, by combining the inequalities in (58) and (59) we have (56), as desired.

If is a bounded matrix operator with the matrix , then it is known that the adjoint operator is defined by the transpose of the matrix . The dual space of is isomorphic to , where .

Theorem 17. Let be a complex number such that and define the set by

Then, .

Proof. We will show that is onto, for . Thus, for every , we find . is a triangle so it has an inverse. Also equation gives . It is sufficient to show that . We calculate that as follows:

where

It is known that from Theorem 5

Now, we show that , for . By Theorem 5, we know that if , . We assume that and ,

since and . This shows that . Similarly we can show that .

Now assume that . Then,

and simple calculation gives that if and only if ,

Hence, is onto. By Lemma 2, has a bounded inverse. This means that , where is defined by (61).

Theorem 18. .

Proof. Let with 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 which implies which contradicts the assumption . Hence, the equation has no solution .

In the case , since the proof of the theorems, in Section 4, determining the spectrum and fine spectrum of the matrix operator on the sequence space is similar to the case ; to avoid the repetition of similar statements, we give the results by the following theorem without proof.

Theorem 19. The following statements hold:

(i), (ii), (iii), (iv), (v), (vi), (vii).

5. Some Applications

In this section, we give two theorems related to Toeplitz matrix.

Theorem 20. Let be a polynomial that corresponds to the -tuple and let also be the roots of . Define as a Toeplitz matrix associated with , that is,

The resolvent operator over with , where the domain of the resolvent operator is the whole space , exists if and if only all the roots of the polynomial are outside the unit disc . That is if and if only , . In this case the resolvent operator is represented by

Proof. Suppose all the roots of the polynomial are outside the unit disc. The Toeplitz matrix associated with can be written as the product

Since multiplication of upper triangular Toeplitz matrices is commutative, we can see that

is left inverse of . Since all roots are outside the unit disc, then

Therefore each , for . Similarly we can say that . So we have .

Theorem 21. The resolvent operator of over with , where the domain of the resolvent operator is the space , exists if and only if . In this case, the resolvent operator is represented by the infinite banded Toeplitz matrix

Proof. By Theorem 20, we can see that is inverse of the matrix of . But this is not enough to say it is resolvent operator. By Lemmas 13, 14, and 15, , when . That is for , is a resolvent operator.

Acknowledgments

The authors would like to thank the referee for pointing out some mistakes and misprints in the earlier version of this paper. They would like to express their pleasure to Ms. Medine Yeşilkayagil, Department of Mathematics, Uşak University, 1 Eylül Campus, Uşak, Turkey, for many helpful suggestions and interesting comments on the revised form of the paper.

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