- Research
- Open access
- Published:
Two new sequence spaces generated by the composition of m th order generalized difference matrix and lambda matrix
Journal of Inequalities and Applications volume 2014, Article number: 274 (2014)
Abstract
In this work, we introduce two sequence spaces and generated by the composition of m th order generalized difference matrix and lambda matrix and define an isomorphism between new sequence spaces and classical sequence spaces. Afterward, we investigate inclusion relations and obtain the Schauder basis of those spaces. Furthermore, we determine their α-, β- and γ-duals. Lastly, we characterize some matrix classes related to those spaces.
MSC:40C05, 40H05, 46B45.
1 Introduction
The family of all real (or complex) valued sequences is denoted by w. w is a vector space under point-wise addition and scalar multiplication. Any vector subspace of w is called a sequence space. In the literature, the classical sequence spaces are symbolized with , , c and which are called all bounded, null, convergent and absolutely p-summable sequence spaces, respectively, where .
A sequence space X with a linear topology is called a K-space provided each of the maps defined by is continuous for all . It is assumed that w is always endowed with its locally convex topology generated by the sequence of semi-norms on w, where , . A K-space X is called an FK-space provided X is a complete linear metric space. An FK-space whose topology is normable is called a BK-space [1].
The classical sequence spaces , and c are BK-spaces with their usual defined by and is a BK-space with its defined by
where [2].
Given an infinite matrix with , for all and a sequence , the A-transform of x is defined by
and is assumed to be convergent for all [3]. For using simple notations here and in what follows, the summation without limits runs from 0 to ∞. If implies that , then we say that A defines a matrix mapping from X into Y and denote it by . By using the notation , we mean the class of all infinite matrices A such that .
For an arbitrary sequence space X, is called the domain of an infinite matrix A and is defined by
which is also a sequence space. By bs and cs we denote the spaces of all bounded and convergent series, and define them by means of the matrix domain of the summation matrix such that and , respectively, where is defined by
which is a triangle matrix too. A matrix A is called a triangle if for and for all . Moreover, a triangle matrix A uniquely has an inverse which is also a triangle matrix.
Unless stated otherwise, any term with negative subscript is assumed to be zero. The theory of matrix transformation was prompted by summability theory which was obtained by Cesàro, Riesz and others. The Cesàro mean of order one and the Riesz mean according to the sequence are defined by using the matrices and such that
respectively, where , () and .
Moreover, the theory of matrix transformation has been continued until nowadays. Many authors have constructed new sequence spaces by using matrix domain of infinite matrices. For example, and in [4], and in [5], and in [6], in [7], and in [8], and in [9], , and in [10], in [11], and in [12], and in [13]. Also, many authors introduced new sequence spaces by using especially difference matrices. For instance, , and in [14], , and in [15], and in [16], , and in [17], , and in [18], , and in [19], , and in [20], , and in [21], , and in [22], in [23], , , and in [24], , , and in [25], , and in [26], in [27].
In this work, we introduce two sequence spaces and generated by the composition of m th order generalized difference matrix and lambda matrix and define an isomorphism between new sequence spaces and classical sequence spaces. Afterward, we investigate inclusion relations and obtain the Schauder basis of those spaces. Furthermore, we determine their α-, β- and γ-duals. Lastly, we characterize some matrix classes related to those spaces.
2 Two new sequence spaces
In this section, we give some historical information and define the sequence spaces and generated by the composition of m th order generalized difference matrix and lambda matrix. Moreover, we speak of some inclusion relations.
The idea of using the notion of λ-convergent was first motivated by Mursaleen and Noman in [28]. They defined the sequence spaces , and by means of the Lambda matrix such that
and
where consists of positive reals such that
and the lambda matrix is defined by
for all . Here, we would like to touch on a point, if we take and , for all , we obtain the Cesàro mean of order one and the Riesz mean matrix, respectively. So, the matrix generalizes the and matrices.
Also, they improved their work by constructing the spaces and in [29]. The sequence spaces and are defined by
and
where Δ is a difference matrix.
Afterward, Sönmez and Başar defined the sequence spaces and in [30] and improved Mursaleen and Noman’s work as follows:
and
where is called a double band (generalized difference) matrix defined by
Let r and s be non-zero real numbers, then the m th order generalized difference matrix is defined by
for all and . We want to recall that , , , , … where , , , , … are double band (that is, the generalized difference matrix), triple band, quadruple band, quinary band, … matrix, respectively. Moreover, , , . So, our results obtained from the matrix domain of the m th order generalized difference matrix are more general and more extensive than the results on the matrix domain of , , , , … , , and Δ.
By considering the definition of m th order generalized difference matrix , we define the sequence spaces and as follows:
and
If we recall the notation of (1.2), the sequence spaces and can be redefined by the matrix domain of as follows:
Also, by constructing a triangle matrix so that
for all and , we rearrange the sequence spaces and by means of the matrix as follows:
So, for a given arbitrary sequence , the -transform of x is denoted by
for all , or, by using another representation, we can rewrite the sequence as follows:
for all .
Theorem 2.1 The sequence spaces and are BK-spaces according to their norms defined by
Proof It is known that c and are BK-spaces with their [2]. Also (2.2) holds and is a triangle matrix. If we consider these three facts and Theorem 4.3.12 of Wilansky [3], we conclude that and are BK-spaces. This step completes the proof. □
Theorem 2.2 The sequence spaces and are linearly isomorphic to the sequence spaces and c, respectively.
Proof To avoid the repetition of similar statements, we give the proof of the theorem for only the sequence space . For the proof, the existence of a linear bijection between the spaces and c should be shown. Let us define a transformation L such that , . Then it is clear that for all , . Also, it is trivial that L is a linear transformation and whenever . On account of this, L is injective.
Furthermore, for a given sequence , we define the sequence as follows:
for all . Then, for every , we obtain
If we consider the equality above, for all , we conclude that
So, and since , we bring to a conclusion that . Hence, we conclude that and . Thus L is surjective.
Furthermore, we have for every that
So, L is norm preserving. As a consequence, L is a linear bijection. This step shows that the spaces and c are linearly isomorphic, namely . □
Lemma 2.3 [28]
The inclusions and hold.
Theorem 2.4 The inclusion strictly holds.
Proof It is well known that every null sequence is also convergent. So, the inclusion holds. Now we define a sequence such that
for all . Then we obtain by (2.3) that
for all , which gives us that , where . Then , namely . This shows that the inclusion strictly holds. This step completes the proof. □
Theorem 2.5 If , the inclusion is strict.
Proof It is clear that whenever . Assume that and . Then and because of , . If we consider this fact and Lemma 2.3, we deduce that . This shows that . As a consequence, holds. Now we define a sequence such that for all and . Then it is obvious that but . Because of , we conclude that and thereby . This shows that strictly holds if . This step completes the proof. □
If we combine Theorem 2.4 and Theorem 2.5, we give the following result.
Corollary 2.6 If , the inclusions and are strict.
Now we define two sequences and such that and for all and . Then we can see that and , namely . Also, it is clear that . These two facts give us the following corollary.
Corollary 2.7 The spaces and overlap but the space does not include the space .
Now we give the following lemma which is needed in the next theorem.
Lemma 2.8 [31]
.
Theorem 2.9 Let a sequence be as follows:
for all . Then the inclusion is strict if and only if .
Proof We assume that the inclusion holds. Then it is obvious that for every , , namely . Thus . If we consider the last result and Lemma 2.8, we deduce that
Also, by using the definition of the matrix , we obtain by the equality above that
By considering equalities (2.5), (2.6), …, (2.3+m) and (2.4+m), we conclude that
For all , we write the equality as follows:
By combining , (2.5) and (2.5+m), we deduce that
This means that .
On the contrary, assume that . Then we have (2.6+m). Also, for all , we write the inequality as follows:
By combining the last inequality and (2.6+m), we conclude that
Specially, if we take (, and ), (, and ), … then we obtain , , …, respectively. These equalities show that (2.4+m), (2.3+m), …, (2.6) and (2.5) hold, respectively. If we take into account the last result and Lemma 2.8, we conclude that . Thus, the inclusion holds and is strict by Corollary 2.7. This step completes the proof. □
3 The Schauder basis and α-, β- and γ-duals
In the present section, we give the Schauder basis and determine α-, β- and γ-duals of the sequence spaces and .
Let be a normed space. A set is called a Schauder basis for X if for every there exist unique scalars , , such that ; i.e.,
as .
Note that the Hamel basis is free from topology, whereas the Schauder basis involves convergence and hence topology (see [1]).
For example, let be a sequence with 1 in the k th place and zeros elsewhere, and let . Then the sequence is a Schauder basis for . Moreover, is a Schauder basis for c.
Due to the transformation, L defined in the proof of Theorem 2.2 is an isomorphism; the inverse image of is a Schauder basis for .
Now we give the following results.
Theorem 3.1 Let for all . For every fixed , we define the sequences and such that
Then
-
(a)
The sequence is a Schauder basis for the space , and every has a unique representation of the form
-
(b)
The sequence is a Schauder basis for the space , and every has a unique representation of the form
where .
If we consider the results of Theorem 2.1 and Theorem 3.1, we give the following result.
Corollary 3.2 The sequence spaces and are separable.
Given arbitrary sequence spaces X and Y, the set defined by
is called the multiplier space of X and Y. For a sequence space Z with , one can easily observe that and hold, in turn.
By using the sequence spaces , cs and bs and notation (3.1), the α-, β- and γ-duals of a sequence space X are defined by
respectively.
Now we give some properties which are needed in the next lemma
where ℱ is the collection of all finite subsets of ℕ and .
Lemma 3.3 [31]
Let be an infinite matrix, then the following hold:
-
(i)
;
-
(ii)
;
-
(iii)
;
-
(iv)
;
-
(v)
;
-
(vi)
, and ;
-
(vii)
, .
Theorem 3.4 Define the set by
where the matrix is defined by means of the sequence by
Then .
Proof For given , by taking into account the sequence that is defined in the proof of Theorem 2.2, we obtain
for all . If we consider the equality above, we conclude that whenever or if and only if whenever or c. This means that if and only if . If we consider this and Lemma 3.3(i), we write
and conclude . This step completes the proof. □
Theorem 3.5 Given the sets , , and as follows:
and
where
Then and .
Proof Given , by considering the sequence that is defined in the proof of Theorem 2.2, we obtain
, where the matrix is defined as follows:
for all . Then whenever if and only if whenever . This shows that if and only if . If we consider this and Lemma 3.3(ii), we obtain
and
These results show that .
By using a similar way, we obtain if and only if . If we consider this and Lemma 3.3(iii), we conclude that (3.6), (3.7) and (3.8) hold.
Moreover, one can easily see that
As a consequence, we derive from (3.5) that
Since condition (3.6) is weaker, it can be omitted.
Therefore we conclude that . This step completes the proof. □
Theorem 3.6 .
Proof It can be proved by combining the proof method of Theorem 3.5 and Lemma 3.3(iv). □
4 Matrix transformations
In the present section, we determine some matrix classes related to the sequence spaces and . Let us begin with two lemmas which are needed in the proof of theorems.
Lemma 4.1 [3]
Any matrix map between BK-spaces is continuous.
Lemma 4.2 [32]
Let X, Y be any two sequence spaces, A be an infinite matrix and U be a triangle matrix. Then .
For simplicity of notation, in what follows, we use the following equalities.
and
for all provided the convergence of the series. Also, unless stated otherwise, we assume throughout Section 4 that the sequence is connected with the sequence as follows:
for all .
Theorem 4.3 Given an infinite matrix of complex numbers, the following statements hold.
-
(1)
Let . Then if and only if
(4.1)(4.2)(4.3)(4.4)(4.5) -
(2)
if and only if (4.3) and (4.4) hold, and
(4.6)(4.7)
Proof For a given sequence , we assume that conditions (4.1)-(4.5) hold. Then, by remembering Theorem 3.5, we deduce that for all . Therefore the A-transform of x exists. Moreover, it is trivial that , namely . Furthermore, if we consider Lemmas 3.3 and 4.1, we conclude that the matrix , where .
Now, we consider the following equality:
Then exists and the series converges for all . Moreover, we derive from (4.3) that the series converges for all ; and therefore . Hence, if we take limit (4.8) side by side as , we obtain by (4.4) that
for all . Then we write the equality above as follows:
for all . Also, we know and . Then we have and . By taking -norm (4.10) side by side, we obtain that
Therefore and so .
On the contrary, assume that , where . This leads us to for all . Then, if we consider Theorem 3.5, conditions (4.2) and (4.3) hold.
We know that and are BK-spaces. If we combine this fact and Lemma 4.1, we conclude that there is a constant such that
holds for all . Let us define a sequence such that for every fixed , where the sequence and .
We know from Theorem 3.1 that and , . Then we obtain
and
for all . Since inequality (4.11) holds for every , the inequality is satisfied also for . Then we have
for all . Therefore (4.1) holds. If we consider this and Lemma 3.3(v), we conclude that . Given . Then so that . Hence Ax and exist. So one can easily see that the series and are convergent for all . Thus, we conclude that
or all . As a consequence, if we pass to the limit in (4.8) as , we obtain
for all . Because of , this leads us
for all . Therefore, (4.4) holds. If we take , also relation (4.10) holds. Because of , we conclude . The last result is the necessity of (4.5). This step completes the proof of part (1).
If we take Lemma 3.3(iv) instead of Lemma 3.3(v), the second part of theorem can be proved similarly. □
Moreover, from (4.12) we derive
If we combine (4.13) and (4.6), we conclude that
exists for each . So, condition (4.6) implies condition (4.2).
Theorem 4.4 Given an infinite matrix of complex numbers, the following statements hold.
-
(1)
Let . Then if and only if (4.1) and (4.2) hold, and
(4.14)(4.15) -
(2)
if and only if (4.6), (4.14) and (4.15) hold.
Proof From Lemma 3.3(iv) and (v), we know and . Therefore, the theorem can be proved similarly. □
Theorem 4.5 if and only if (4.3), (4.4) and (4.6) hold, and the conditions
hold.
Proof Given arbitrary , we assume that conditions (4.3), (4.4), (4.6), (4.16), (4.17) and (4.18) hold for an infinite matrix . We consider Theorem 3.5, and condition (4.6) implies condition (4.2). Then we conclude that for all , and so Ax exists. From (4.6) and (4.17) we have
for all . This leads us to , and therefore the series converges, where and so . If we combine Lemma 3.3(iii) with conditions (4.6), (4.17) and (4.18), we deduce that . Also from condition (4.9) we have
With a basic calculation, we obtain
for all . If we pass to the limit in (4.19), we write
This shows that and so .
On the contrary, we assume that . Since every convergent sequence is also bounded, we deduce that . If we consider this fact and Theorem 4.3, we conclude that conditions (4.3), (4.4) and (4.6) hold. Let us take the sequences and defined in Theorem 3.1 and the proof of Theorem 4.3, respectively. Then it is clear that for every . Hence condition (4.17) holds. Moreover, from Theorem 2.2 we know that the transformation , is continuous. So, we write
for all , where
This leads us to and so .
It is well known that c is a BK-space. If we combine Theorem 2.1 and Lemma 4.1, we conclude that the matrix transformation is continuous. Therefore the equality
holds for all . This shows that (4.18) holds.
By considering conditions (4.6), (4.17), (4.18) and Lemma 3.3(iii), we deduce that . Hence, (4.3), (4.4) and the last result give us that condition (4.10) holds for all and . Finally, if we consider and (4.10), we conclude that condition (4.16) holds. This step completes the proof. □
Theorem 4.6 if and only if (4.3), (4.4), (4.6) and the following conditions hold:
Proof In Theorem 4.5, if we take Lemma 3.3(vi) instead of Lemma 3.3(iii), the present theorem can be proved by using a similar way. □
Theorem 4.7 if and only if (4.6), (4.14), (4.15) and (4.17) hold.
Proof If we combine Lemma 3.3(ii) Theorem 3.5 and Theorem 4.4(2), the present theorem can be proved by using a similar way. □
Theorem 4.8 if and only if (4.6), (4.14), (4.15), (4.20), (4.21) and (4.22) hold.
Proof If we combine Lemma 3.3(vii), Theorem 3.5 and Theorem 4.7, the present theorem can be proved by using a similar way. □
Now, by using Lemma 4.2, we give one more result.
Corollary 4.9 Given an infinite matrix of complex numbers, we define a matrix as follows:
for all . Then A belongs to matrix classes , , , , and if and only if E belongs to matrix classes , , , , and .
Finally, we put a period to our work by mentioning as of now that the sequence space of almost convergent sequences derived by the domain of m th order generalized difference matrix will be defined and studied analogously in the next paper.
References
Choudhary B, Nanda S: Functional Analysis with Applications. Wiley, New Delhi; 1989.
Musayev B, Alp M: Fonksiyonel Analiz. Balcı Yayınları, Ankara; 2000.
Wilansky A North-Holland Mathematics Studies 85. In Summability Through Functional Analysis. Elsevier, Amsterdam; 1984.
Wang C-S: On Nörlund sequence spaces. Tamkang J. Math. 1978, 9: 269-274.
Aydın C, Başar F:Some new sequence spaces which include the spaces and . Demonstratio Math. 2005,38(3):641-656.
Aydın C, Başar F: On the new sequence spaces which include the spaces and c . Hokkaido Math. J. 2004,33(2):383-398. 10.14492/hokmj/1285766172
Malkowsky E, Savaş E: Matrix transformations between sequence spaces of generalized weighted means. Appl. Math. Comput. 2004, 147: 333-345. 10.1016/S0096-3003(02)00670-7
Ng P-N, Lee P-Y: Cesàro sequence spaces of non-absolute type. Comment. Math. (Prace Mat.) 1978,20(2):429-433.
Altay B, Başar F, Mursaleen M:On the Euler sequence spaces which include the spaces and . I. Inform. Sci. 2006,176(10):1450-1462. 10.1016/j.ins.2005.05.008
Malkowsky E: Recent results in the theory of matrix transformations in sequence spaces. Mat. Vesnik 1997, 49: 187-196.
Altay B, Başar F: On the paranormed Riesz sequence spaces of non-absolute type. Southeast Asian Bull. Math. 2002,26(5):701-715.
Altay B, Başar F: Some Euler sequence spaces of non-absolute type. Ukrainian Math. J. 2005,57(1):1-17. 10.1007/s11253-005-0168-9
Şengönül M, Başar F: Some new Cesàro sequence spaces of non-absolute type which include the spaces and c . Soochow J. Math. 2005,31(1):107-119.
Kızmaz H: On certain sequence spaces. Canad. Math. Bull. 1981,24(2):169-176. 10.4153/CMB-1981-027-5
Et M: On some difference sequence spaces. Turkish J. Math. 1993, 17: 18-24.
Aydın C, Başar F: Some new difference sequence spaces. Appl. Math. Comput. 2004,157(3):677-693. 10.1016/j.amc.2003.08.055
Et M, Çolak R: On some generalized difference sequence spaces. Soochow J. Math. 1995,21(4):377-386.
Ahmad ZU, Mursaleen : Köthe-Toeplitz duals of some new sequence spaces and their matrix maps. Publ. Inst. Math. (Beograd) 1987, 42: 57-61.
Mursaleen M: Generalized spaces of difference sequences. J. Math. Anal. Appl. 1996,203(3):738-745. 10.1006/jmaa.1996.0409
Asma Ç, Çolak R: On the Köthe-Toeplitz duals of some generalized sets of difference sequences. Demonstratio Math. 2000, 33: 797-803.
Bektaş Ç: On some new generalized sequence spaces. J. Math. Anal. Appl. 2003, 277: 681-688. 10.1016/S0022-247X(02)00619-4
Et M, Başarır M: On some new generalized difference sequence spaces. Period. Math. Hungar. 1997,35(3):169-175. 10.1023/A:1004597132128
Başarır M, Kayıkçı M: On generalized -Riesz difference sequence space and β -property. J. Inequal. Appl. 2009. Article ID 385029, 2009: Article ID 385029
Kirişçi M, Başar F: Some new sequence spaces derived by the domain of generalized difference matrix. Comput. Math. Appl. 2010,60(5):1299-1309. 10.1016/j.camwa.2010.06.010
Sönmez A: Some new sequence spaces derived by the domain of the triple band matrix. Comput. Math. Appl. 2011,62(2):641-650. 10.1016/j.camwa.2011.05.045
Malkowsky E, Parashar SD: Matrix transformations in space of bounded and convergent sequences of order m . Analysis. 1997, 17: 87-97.
Malkowsky E, Mursaleen M, Suantai S: The dual spaces of sets of difference sequences of order m and matrix transformations. Acta Math. Sinica (Engl. Ser.) 2007,23(3):521-532. 10.1007/s10114-005-0719-x
Mursaleen M, Noman AK: On the spaces of λ -convergent and bounded sequences. Thai J. Math. 2010,8(2):311-329.
Mursaleen M, Noman AK: On some new difference sequence spaces of non-absolute type. Math. Comput. Modelling. 2010, 52: 603-617. 10.1016/j.mcm.2010.04.006
Sönmez A, Başar F: Generalized difference spaces of non-absolute type of convergent and null sequences. Abstr. Appl. Anal. 2012. Article ID 435076, 2012: Article ID 435076
Stieglitz M, Tietz H: Matrix transformationen von folgenräumen eine ergebnisübersicht. Math. Z. 1977, 154: 1-16. 10.1007/BF01215107
Başar F, Altay B: On the space of sequences of p -bounded variation and related matrix mappings. Ukrainian Math. J. 2003,55(1):136-147. 10.1023/A:1025080820961
Acknowledgements
We would like to express our thanks to the anonymous reviewers for their valuable comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Bişgin, M.C., Sönmez, A. Two new sequence spaces generated by the composition of m th order generalized difference matrix and lambda matrix. J Inequal Appl 2014, 274 (2014). https://doi.org/10.1186/1029-242X-2014-274
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-274