Journal of Mathematics

Volume 2014 (2014), Article ID 876813, 13 pages

http://dx.doi.org/10.1155/2014/876813

Research Article

On Some New Generalized Difference Sequence Spaces of Nonabsolute Type

Osman Duyar,¹ Serkan Demiriz,² and Osman Özdemir²

¹Anatolian High School, 60200 Tokat, Turkey

²Department of Mathematics, Faculty of Arts and Science, Gaziosmanpaşa University, 60250 Tokat, Turkey

Received 30 April 2014; Accepted 21 August 2014; Published 9 September 2014

Academic Editor: Tepper L. Gill

Copyright © 2014 Osman Duyar et al. 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.

We define a new triangle matrix by the composition of the matrices and . Also, we introduce the sequence spaces , and by using matrix domain of the matrix on the classical sequence spaces , and , respectively, where . Moreover, we show that the space is norm isomorphic to for . Furthermore, we establish some inclusion relations concerning those spaces and determine -, -, and -duals of those spaces and construct the Schauder bases , and . Finally, we characterize the classes of infinite matrices where and .

1. Preliminaries, Background, and Notation

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. We write , and for the classical sequence spaces of all bounded, convergent, null, and absolutely -summable sequences, respectively, where . Also by and , we denote the spaces of all bounded and convergent series, respectively. We assume throughout unless stated otherwise that with and use the convention that any term with negative subscript is equal to zero. We denote throughout that the collection of all finite subsets of by .

Let be an infinite matrix and two sequence spaces. Then, defines a matrix mapping from to and is denoted by if for every sequence the sequence , the -transform of , is in , where

By , denote the class of all matrices such that . Thus, if and only if the series on the right hand side of (1) converges for each and , and we have for all . A sequence is said to be -summable to if converges to , which is called the -limit of .

A matrix is called a triangle if for and for all . It is trivial that holds for the triangle matrices and a sequence . Further, a triangle matrix has a unique inverse which is also a triangle matrix. Then, holds for all .

Let us give the definition of some triangle limitation matrices which are needed in the text. Let be a sequence of positive reals and write

Then the Cesàro mean of order one, Riesz mean with respect to the sequence , and Euler mean of order with are, respectively, defined by the matrices ,  , and , where

for all . We write for the set of all sequences such that for all . For , let . Let , and define the summation matrix , the difference matrix , and the generalized weighted mean or factorable matrix ,  ,   by

for all , where and depend only on and , respectively. Let and be nonzero real numbers, and define the generalized difference matrix by

for all . We note that if we choose and then the matrix is reduced to the backward difference.

For a sequence space , the matrix domain of an infinite matrix in the space is defined by

which is a sequence space. If is triangle, then one can easily observe that the sequence spaces and are linearly isomorphic; that is, .

Following Başar [1, page 51], we note that although in the most cases the new sequence space generated by the limitation matrix from a sequence space is the expansion or the contraction of the original space , it may be observed in some cases that those spaces overlap. Indeed, one can easily see that the inclusion, , strictly holds for . As this, one can deduce that the inclusion also strictly holds for . However, if we define with , that is, if and only if for some and some , and consider the matrix with the rows defined by for all , we have but which lead us to the consequences that and , where and is a sequence whose only nonzero term is a in th place for each . That is to say, the sequence spaces and overlap but neither contains the other.

The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by Wang [2], Ng and Lee [3], Malkowsky [4], Altay and Başar [5], Malkowsky and Savaş [6], Başarır [7], Aydın and Başar [8], Başar et al. [9], Şengönül and Başar [10], Altay [11], Polat and Başar [12], and Malkowsky et al. [13]. and are the transposes of the matrices , and , respectively, and are the spaces consisting of the sequences such that in the spaces and for , respectively, studied by Başarır [7]. More recently, the generalized difference matrix has been used by Kirişçi and Başar [14] for generalizing the difference spaces , and . Finally, the new technique for deducing certain topological properties, for example, -,  -, and  -properties, and determining the - and -duals of the domain of a triangle matrix in a sequence space has been given by Altay and Başar [15].

Let , and be nonzero real numbers, and define the generalized difference matrix by

for all . The inverse of is given by

We should record here that , , and . So, the results related to the matrix domain of the triple band matrix are more general and more comprehensive than the consequences on the matrix domain of ,   and and include them. We assume throughout that is a strictly increasing sequence of positive reals tending to ; that is

The main purpose of the present paper is to introduce the sequence spaces and to determine the -,  -, and -duals of this space, where denotes throughout any of the classical spaces ,  , or , and is the triple band matrix and the sequence is defined in (9). Furthermore, the Schauder bases for the spaces ,   and are given, and some topological properties of the spaces ,  , and are examined. Finally, some classes of matrix mappings on the spaces are characterized.

The paper is organized as follows. In Section 2, the -spaces ,  ,  , and of generalized difference sequence spaces are introduced and the Schauder bases of the spaces ,  , and are given. In Section 3, some inclusion relations concerning these spaces are examined. In Section 4, the -,  -, and -duals of the generalized difference sequence spaces of nonabsolute type are determined. In Section 5, the classes of infinite matrices are characterized, where and .

2. The Difference Sequence Spaces of Nonabsolute Type

The difference sequence spaces have been studied by several authors in different ways [16–34]. In the present section, we introduce the spaces ,  ,  , and and show that these spaces are -spaces of nonabsolute type which are norm isomorphic to the spaces ,  ,  , and , respectively. Furthermore, we give the bases of the spaces ,  , and .

We say that a sequence is -convergent to the number , called the -limit of , if as where

In particular, we say that is a -null sequence if as . Further, we say that is -bounded if , [35]. Recently, Mursaleen and Noman [35, 36] studied the sequence spaces ,  ,  , and of nonabsolute type as follows:

On the other hand, we define the matrix for all by

Then, the space can be restated with the notation of (6) that

More recently, Sönmez [37] has defined the sequence spaces , and as follows:

In fact, the sequence spaces , and can be considered as the set of all sequences whose -transforms are in the spaces , and , respectively. That is,

Now, we introduce the difference sequence spaces , and as follows:

Now, we define the triangle matrix ; that is,

for all . Further, for any sequence we define the sequence which will be used, as the -transform of ; that is,

In fact, the sequence spaces , and can be considered as the set of all sequences whose -transforms are in the spaces , and , respectively. That is,

Since the proof may also be obtained in a similar way as the other spaces, to avoid the repetition of the similar statements, we give the proof only for one of those spaces. Now, we may begin with the following theorem which is essential in the study.

Theorem 1. (i) The difference sequence spaces , , and are -spaces with the norm ; that is,

(ii) Let . Then is a -space with the norm ; that is,

Proof. Since (19) holds and , and are -spaces with respect to their natural norms (see [1, pages 16-17]) and the matrix is a triangle, Theorem 4.3.12 of Wilansky [38, page 63] gives the fact that , ,  , and are -spaces with the given norms. This completes the proof.

Remark 2. Let . Then the absolute property does not hold on the space ; that is, . This can be shown for at least one sequence in those spaces. Hence is the sequence space of nonabsolute type.

Theorem 3. The sequence spaces , , , and of nonabsolute type are norm isomorphic to the spaces , , , and , respectively; that is, , , , and .

Proof. We prove the theorem for the space . To prove our assertion we should show the existence of a linear bijection between the spaces and . Let be defined by (18). Then, for every and the linearity of is clear. Further, it is trivial that whenever and hence is injective.

Moreover, let and we define the sequence by

Then we obtain

Hence, we get for every that

This shows that and since , we conclude that . Thus, we deduce that and . Hence is surjective.

Moreover one can easily see for every that

which means that is norm preserving. Consequently is a linear bijection which shows that the spaces and are linearly isomorphic, as desired.

Let be a normed space. A sequence of points of is called a Schauder basis for if and only if, for each , there exists a unique sequence of scalars such that ; that is,

Because of the isomorphism , which is defined in the proof of Theorem 3, the inverse image of the basis of the spaces , and are the basis of new spaces , and , respectively. Therefore, we have the following.

Theorem 4. Let for all and . Define the sequence for every fixed by

Then, the following statements hold.

(i)The sequence is a basis for the spaces and ; any or has a unique representation of the form .(ii)The sequence is a basis for the space and any has a unique representation of the form , where .

3. The Inclusion Relations

In the present section, we prove some inclusion relations concerning the spaces , and .

Theorem 5. The inclusion strictly holds.

Proof. It is obvious that the inclusion holds. Further to show that this inclusion is strict, consider the sequence defined by

for all . Then

Then one can easily see that , where . Thus, this sequence is in but not in . Hence, the inclusion is strict and this completes the proof of the theorem.

Theorem 6. If then the inclusion strictly holds.

Proof. Suppose that and . Then and hence since the inclusion holds [35]. This shows that ; that is, . Further consider the sequence defined by for all . Then it is trivial that . On the other hand, it can easily be seen that . Hence, . Thus the sequence . Hence, the inclusion is strict. This completes the proof.

Lemma 7. if and only if  .

Theorem 8. Let be defined by

for all . Then the inclusion strictly holds if and only if .

Proof. Let be a subset of . Then we obtain that for every and the matrix is in the class . By using Lemma 7 it follows that

Now, by taking into account the definition of matrix given by (17), we have for every that

By Lemma 7, we have that

Now, we have for every that

and since by (33) and (34), we obtain that by (35)

which shows that where the sequence is defined by

for all .

Conversely, we suppose that . Then we have (37). Further, for every , we derive that

Then, (37) and (39) together imply that (35) holds. On the other hand, we have for every that

Therefore, it follows by (35)

Particularly if (i) , , , then we obtain , which shows that (33) holds. (ii) , , , and then we obtain , which shows that (34) holds. Thus, we deduce by relation (32) that (31) holds. This leads us with Lemma 7 to the consequence that . Hence, the inclusion holds. Finally, it is obvious that the sequence , defined in the proof of Theorem 6, is in but not in , so the inclusion is strict.

Theorem 9. The inclusion strictly holds.

Proof. To prove the validity of the inclusion , it suffices to show that, for every , there exists a positive real number such that . Let . Then, we have

so that and hence . Furthermore, we consider defined by for all . Then we have . Thus, we deduce that . Hence .

On the other hand, we know from Theorem 8 that the inclusion is strict. Since , the inclusion strictly holds.

4. The -, -, and -Duals of the Spaces of Nonabsolute Type

In this section, we determine the -, -, and -duals of the generalized difference sequence spaces , and of nonabsolute type.

We will firstly give the definition of -, -, and -duals of sequence spaces and secondly we give the lemmas due to [39] which are needed in proving the theorems given in Sections 4 and 5.

For the sequence spaces and , define the set by

With the notation of (43), the -, -, and -duals , and of a sequence space are defined by

Lemma 10. (i) if and only if

(ii) if and only if

Lemma 11. if and only if

Lemma 12. if and only if (47) and (48) hold, and

Lemma 13. if an only if (47) holds and

Lemma 14. if and only if (47) holds and

Lemma 15. if and only if (48) holds.

Lemma 16. if and only if (51) holds.

Now we consider the following sets:

where the matrices and are defined as follows:

for all and the is defined by

Theorem 17. (i) .

(ii) .

Proof. We prove the theorem for the space . Let . Then, we obtain the equality

by relation (22). Thus we observe by (55) that whenever if and only if whenever . This means that the sequence if and only if . Therefore we obtain by Lemma 10 with instead of that if and only if

which leads us to the consequence that . This completes the proof.

Theorem 18. (i)   .

(ii)   .

(iii) .

(iv) .

Proof. Consider the equality

Then we deduce by (57) that whenever if and only if whenever . This means that if and only if . Therefore, by using Lemma 11, we obtain

Hence, we conclude that .

Finally, we ended this section with the following theorem which determines the -duals of sequence spaces , and .

Theorem 19. (i)   .

(ii) .

5. Some Matrix Transformations Related to the Spaces of Nonabsolute Type for

In this final section, we state some results on the characterization of several classes of matrix mappings on the , and . We will write throughout for brevity that

for all . We assume that the series are convergent in the last equation.

We will begin with lemmas which are needed in the proof of our theorems.

Lemma 20. The matrix mappings between the -spaces are continuous.

Lemma 21. if and only if

Lemma 22. if and only if (48) holds and

Lemma 23. if and only if (48) and (61) hold.

Now, we may give our matrix transformations.

Theorem 24. Let be an infinite matrix and . Then, if and only if

Proof. If conditions (62)–(67) hold and is any sequence in the space then by using Theorem 18 we have that for all . Hence, -transform of exists. Furthermore, since the associated sequence is in the space , we may write for some suitable . Also, the matrix is in the class by Lemma 21 and condition (62), where .

Now, we may consider the th partial sum of the series which is derived by using relation (22):

Then, since and , exists and so the series converges for every . Also, it follows by (63) that we have . Therefore, if we pass to limit in (68) as , then we obtain by (65) that

which can be written as follows:

This yields by taking -norm that

Consequently, we have that ; that is, .

Conversely, if , we have for all which implies with Theorem 18 that conditions (63), (64), and (66) are necessary.

On the other hand, since and are -spaces, we have by Lemma 20 that there is a constant such that

holds for all . Let . Then, the sequence is in , where the sequences are defined by (27) for every fixed .

Since for each fixed , we have

Furthermore, for every , we obtain by (27) that

Hence, since inequality (72) is satisfied for the sequence , we have for any that

which shows the necessity of (62). Thus, it follows by Lemma 21 that .

Moreover, we consider the sequence defined by (22) for every and suppose that . Then, since such that by (18), the transforms and exist. Hence, the series and are convergent for every . So we infer that

Consequently, we obtain from (68) that

Hence, we deduce that

which leads us to the necessity of (65) and so the relation (68) holds, where .

Finally, since and , the necessity of (67) is immediate by (68). This completes the proof.

Theorem 25. if and only if (65) and (66) hold, and

Proof. This is an immediate consequence of Lemma 15 and Theorem 24.

Theorem 26. (i) if and only if (62),(63), and (64) hold and

(ii) if and only if (64), (79), and (81) hold.

Proof. This may be obtained by proceedings as in Theorem 24, above. So, we omit the detail.

Theorem 27. if and only if (65), (66), and (79) hold and

Proof. Suppose that satisfies conditions (65), (66), (79), (82), (83), and (84), and let be a sequence in the space ). Since (79) implies (64), we have by Theorem 18 that for all . Hence, exists. We also observe from (79) and (83) that

holds for every . So and hence the series converges, where is the sequence connected with by relation (18) such that . Also, it is obvious by combining Lemma 12 with conditions (79), (83), and (84) that the matrix is in the class .

Now, by following the similar way used in the proof of Theorem 24, we obtain that relation (69) holds, which can be written as follows:

If we pass the limit in (86) as we have that

which shows that ; that is, .

Conversely, suppose that . Since the inclusion holds, . Therefore, the necessity of conditions (65), (66), and (79) is obvious from Theorem 25. Furthermore, consider the sequence defined by (27) for every fixed . Then, one can see that and hence for every which shows the necessity of (83). Let . Then, since the linear transformation , defined as in the proof of Theorem 3 by analogy, is continuous and for each fixed , we obtain that

which shows that and hence . On the other hand, since and are the -spaces, Lemma 20 implies the continuity of the matrix mapping . Thus, we have for every that

This shows the necessity of (84).

Now, it follows by (79), (83), and (84) with Lemma 12 that . So by (65), (66) and relation (70) hold for all and .

Finally, the necessity of (82) is immediate by (70) since and . This completes the proof.

Theorem 28. if and only if (65), (66), and (79) hold and

Proof. This is obtained in the similar way used in the proof of Theorem 27 with Lemma 22 instead of Lemma 12 and so we omit the detail.

Theorem 29. if and only if (63), (79), (81), and (83) hold.

Proof. This is an immediate consequence of Lemma 11 and Theorems 18 and 26 (ii).

Theorem 30. if and only if (63), (79), and (83) hold.

Proof. This is an immediate consequence of Lemma 23 and Theorems 18 and 29.

Theorem 31. if and only if (64) and (81) hold and

Proof. Suppose that satisfies conditions (64), (81), and (91) and take any . Then, it is obvious that for each . Hence, is convergent for all ; that is, exists. Also, it is obvious by combining Lemma 16 with condition (91) that .

Now, we will show that for all . By using relation (68) we have that

By applying Hölder’s inequality to (92) we obtain that

which gives us by taking supremum over in this last inequality that

which implies that ; that is, .

Conversely, suppose that . Thus, exists and bounded for all . Also, for all which implies the necessity of (64) and (81). If we define the sequences such that then we have that

So, bearing in mind (81) one can easily obtain relation (92) by using relation (68). On the other hand, the sequences define the continuous linear functionals on the space as follows:

Since the spaces and are linear isomorphic, we have that

Also, since , it is obvious that

Therefore, by using the Banach-Steinhaus Theorem [38, see pages: 1-2] we obtain that

which shows the necessity of (91). This step completes the proof.

Theorem 32. if and only if (64), (81), (83), and (91) hold.

Proof. Suppose that conditions (64), (81), (83), and (91) hold. Then, for each . Hence, the series is convergent for each and for all which implies that exists. By using (83) we obtain that

Since

for all , in this situation, we see by passing to the limit in the last inequality as that

Because (102) holds for all positive integers , we have that

We remember that and are associated sequences by relation (22) where for .

Let be any positive number. Then, there exists a number such that

On the other hand, there is an integer such that

whenever . Therefore, we obtain

which shows that

Since

we have that which implies that .

The necessity part can be proved by using the similar way in the proof of Theorem 27, so we omit the detail.

Theorem 33. if and only if (64), (81), and (91) hold and

Proof. This result can be proved similarly as the proof of Theorem 32.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors wish to express their deep appreciation to the referees of this paper for their excellent remarks which helped them to present the paper in its final form.

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