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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 949282, 10 pages
http://dx.doi.org/10.1155/2013/949282
Research Article

Domain of the Double Sequential Band Matrix in the Sequence Space

Department of Mathematics, Faculty of Arts and Sciences, Fatih University, The Hadımköy Campus, Büyükçekmece, 34500 Istanbul, Turkey

Received 17 October 2012; Accepted 29 January 2013

Academic Editor: Ferhan M. Atici

Copyright © 2013 Havva Nergiz 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 sequence space was introduced by Maddox (1967). Quite recently, the domain of the generalized difference matrix in the sequence space has been investigated by Kirişçi and Başar (2010). In the present paper, the sequence space of nonabsolute type has been studied which is the domain of the generalized difference matrix in the sequence space . Furthermore, the alpha-, beta-, and gamma-duals of the space have been determined, and the Schauder basis has been given. The classes of matrix transformations from the space to the spaces , c and c0 have been characterized. Additionally, the characterizations of some other matrix transformations from the space to the Euler, Riesz, difference, and so forth sequence spaces have been obtained by means of a given lemma. The last section of the paper has been devoted to conclusion.

1. Preliminaries, Background, and Notation

By , we denote the space of all real valued sequences. Any vector subspace of is called a sequence space. We write , , and for the spaces of all bounded, convergent, and null sequences, respectively. Also by , , , and , we denote the spaces of all bounded, convergent, absolutely convergent and absolutely convergent series, respectively, where .

A linear topological space over the real field is said to be a paranormed space if there is a subadditive function such that and scalar multiplication is continuous; that is, and imply for all ’s in and all ’s in , where is the zero vector in the linear space .

Assume here and after that   is a bounded sequence of strictly positive real numbers with and . Then, the linear spaces were defined by Maddox [1] (see also Simons [2] and Nakano [3]) as follows: which is the complete space paranormed by For simplicity in notation, here and in what follows, the summation without limits runs from to . We assume throughout that and denote the collection of all finite subsets of by and use the convention that any term with negative subscript is equal to naught.

Let , be any 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 (3) converges for each and every , and we have for all . A sequence is said to be summable to if converges to which is called the limit of .

The shift operator is defined on by for all . A Banach limit is defined on , as a nonnegative linear functional, such that and , where . A sequence is said to be almost convergent to the generalized limit if all Banach limits of are and is denoted by . Lorentz [4] proved that It is well known that a convergent sequence is almost convergent such that its ordinary and generalized limits are equal. By , we denote the space of all almost convergent sequences; that is,

Define the double sequential band matrix by for all , where and are the convergent sequences. We should note that the double sequential band matrices were firstly used by Srivastava and Kumar [5, 6], Panigrahi and Srivastava [7], and Akhmedov and El-Shabrawy [8].

The main purpose of this paper, which is a continuation of Kirişçi and Başar [9], is to introduce the sequence space of nonabsolute type consisting of all sequences whose transforms are in the space . Furthermore, the basis is constructed and the alpha-, beta-, and gamma-duals are computed for the space . Moreover, the matrix transformations from the space to some sequence spaces are characterized. Finally, we note open problems and further suggestions.

It is clear that can be obtained as a special case of for and and it is also trivial that is reduced in the special case and to the generalized difference matrix . So, the results related to the matrix domain of the matrix are more general and more comprehensive than the corresponding consequences of the matrix domains of and .

The rest of this paper is organized as follows. In Section 2, the linear sequence space is defined and proved that it is a complete paranormed space with a Schauder basis. Section 3 is devoted to the determination of alpha-, beta-, and gamma-duals of the space . In Section 4, the classes , , , and of infinite matrices are characterized. Additionally, the characterizations of some other classes of matrix transformations from the space to the Euler, Riesz, difference, and so forth sequence spaces are obtained by means of a given lemma. In the final section of the paper, open problems and further suggestions are noted.

2. The Sequence Space of Nonabsolute Type

In this section, we introduce the complete paranormed linear sequence space .

The matrix domain of an infinite matrix in a sequence space is defined by Choudhary and Mishra [10] defined the sequence space which consists of all sequences such that transforms of them are in the space , where is defined by for all . Başar and Altay [11] have recently examined the space which is formerly defined by Başar in [12] as the set of all series whose sequences of partial sums are in . More recently, Aydn and Başar [13] have studied the space which is the domain of the matrix in the sequence space , where the matrix is defined by for all , such that for all and . Altay and Başar [14] have studied the sequence space which is derived from the sequence space of Maddox by the Riesz means . With the notation of (7), the spaces , and can be redefined by Following Choudhary and Mishra [10], Başar and Altay [11], Altay and Başar [1417], and Aydn and Başar [13, 18], we introduce the sequence space as the set of all sequences whose transforms are in the space ; that is It is trivial that in the case for all , the sequence space is reduced to the sequence space which is introduced by Kirişçi and Başar [9]. With the notation of (7), we can redefine the space as follows: Define the sequence , which will be frequently used, as the transform of a sequence ; that is, Since the spaces and are linearly isomorphic by Corollary 4, one can easily observe that if and only if , where the sequences and are connected with the relation (13).

Now, we may begin with the following theorem which is essential in the text.

Theorem 1. is a complete linear metric space paranormed by the paranorm

Proof. It is easy to see that the space is linear with respect to the coordinate-wise addition and scalar multiplication. Therefore, we first show that it is a paranormed space with the paranorm defined by (14).
It is clear that where and for all .
Let ; then by Minkowski’s inequality we have
Let be a sequence of scalars with , as , and let be a sequence of elements with , as . We observe that It follows from that for all sufficiently large ; hence Furthermore, we have
Also, we have Then, we obtain from (16), (17), (18), and (19) that , as . This shows that is a paranorm on .
Furthermore, if , then . Therefore for each . If we put , since and , we have . For , since we have . Continuing in this way, we obtain for all . That is, . This shows that is a total paranorm.
Now, we show that is complete. Let be any Cauchy sequence in where ,. Here and after, for short we write instead of . Then for a given , there exists a positive integer such that for all . Since for each fixed for every , is a Cauchy sequence of real numbers for every fixed . Since is complete, it converges, say as . Using these infinitely many limits we define the sequence . For each and By letting , we have for that This shows us . Since is a linear space, we conclude that ; It follows that , as in , thus we have shown that is complete.

Therefore, one can easily check that the absolute property does not hold on the space ; that is, , where . This says that is the sequence space of nonabsolute type.

Theorem 2. Convergence in is stronger than coordinate-wise convergence.

Proof. First we show that , as implies ; as for every . We fix , then we have Hence, we have for that which gives the fact that , as . Similarly, for each , we have , as .

A sequence space with a linear topology is called a space provided each of the maps defined by is continuous for all , where denotes the complex field. A space is called an space provided is complete linear metric space. An space whose topology is normable is called a space. Given a space , we denote the section of a sequence by , and we say that has the property if . If property holds for every , then we say that the space is called space (cf. [19]). Now, we may give the following.

Theorem 3. is the linear space under the coordinatewise addition and scalar multiplication which is the space with the norm

Proof. Because the first part of the theorem is a routine verification, we omit the detail. Since is the space with respect to its usual norm (see [20, pages 217-218]) and is a normal matrix, Theorem of Wilansky [21, page 61] gives the fact that is the space, where .

Let us suppose that for all . Then, it is known that which leads us to the immediate consequence that .

With the notation of (13), define the transformation from to by . Since is linear and bijection, we have the following.

Corollary 4. The sequence space of nonabsolute type is linearly paranorm isomorphic to the space , where for all .

Theorem 5. The space has AK.

Proof. For each , we put Let and be given. Then, there is such that Then we have for all , This shows that .
Now we have to show that this representation is unique. We assume that . Then for each , Hence, for each .
For , . Since , we have .
For , . Since , we also have .
Continuing in this way, we obtain for each . Therefore, the representation is unique.

We firstly define the concept of the Schauder basis for a paranormed sequence space and next give the basis of the sequence space .

Let be a paranormed space. A sequence of the elements of is called a basis for if and only if, for each , there exists a unique sequence of scalars such that The series which has the sum is then called the expansion of with respect to and written as . Since it is known that the matrix domain of a sequence space has a basis if and only if has a basis whenever is a triangle (cf. [22, Remark 2.4]), we have the following.

Corollary 6. Let and for all . Define the sequence of the elements of the space by for every fixed . Then, the sequence given by (31) is a basis for the space and any has a unique representation of the form .

3. The Alpha-, Beta-, and Gamma-Duals of the Space

In this section, we state and prove the theorems determining the alpha-, beta-, and gamma-duals of the sequence space of nonabsolute type.

For the sequence spaces and , the set defined by is called the multiplier space of the spaces and . With the notation of (32), the alpha-, beta-, and gamma-duals of a sequence space , which are, respectively, denoted by , , and , are defined by

Since the case may be established in similar way to the proof of the case , we omit the detail of that case and give the proof only for the case in Theorems 1012 below.

We begin with quoting three lemmas which are needed in proving Theorems 1012.

Lemma 7 ([23, (i) and (ii) of Theorem 1]). Let be an infinite matrix. Then, the following statements hold.(i)Let for all . Then, if and only if (ii)Let for all . Then, if and only if there exists an integer such that

Lemma 8 ([23, Corollary for Theorem 1]). Let for all . Then, if and only if (34) and (35) hold, and

Lemma 9 ([24, Theorem ]). Let be an infinite matrix. Then, the following statements hold(i)Let for all . Then, if and only if (ii)Let for all . Then, if and only if there exists an integer such that

Theorem 10. Define the sets and by Then,

Proof. Let us take any . By using (13) we obtain that holds for all which leads us to where is defined by for all . Thus, we observe by combining (42) with the condition (37) of Part (i) of Lemma 9 that whenever if and only if whenever . That means .

Theorem 11. Define the sets , , and by Then,

Proof. Take any and consider the equation obtained with (13) that where is defined by for all . Thus, we deduce from Lemma 8 with (46) that whenever if and only if whenever . Therefore, we derive from (35) and (36) that This shows that .

Theorem 12.

Proof. From Lemma 7 and (46), we obtain that whenever if and only if whenever , where is defined by (47). Therefore, we obtain from (34) and (35) that for , for .

4. Matrix Transformations on the Sequence Space

In this section, we characterize some matrix transformations on the space . Theorem 13 gives the exact conditions of the general case by combining the cases and . We consider only the case and leave the case to the reader because it can be proved in similar way.

Theorem 13. Let be an infinite matrix. Then, the following statements hold.(i)Let for all . Then, if and only if there exists an integer such that (ii)Let for all . Then, if and only if the condition (51) holds, and

Proof. Suppose that the conditions (50) and (51) hold, and . In this situation, since for every fixed , the transform of exists. Consider the following equality obtained by using the relation (13) that for all . Taking into account the hypothesis we derive from (53) as that Now, by combining (54) with the following inequality (see [23]) which holds for any and any where and , one can easily see that
Conversely, suppose that and for all . Then exists for every and this implies that for all . Now, the necessity of (51) is immediate. Besides, we have from (54) that the matrix defined by for all , is in the class . Then, satisfies the condition (35) which is equivalent to (50).
This completes the proof.

Lemma 14 ([25, Theorem 1]). if and only if (34) and (35) hold, and

Theorem 15. Let the entries of the matrices and be connected with the relation for all . Then, if and only if and for every fixed , where with for all .

Proof. Let and take . Then, we obtain the equality for all . Since exists, . Letting in the equality (61) we have . Since , then . That is .
Conversely, let , and , and take . Then, since and we have for all . So, exists. Therefore we obtain from equality (61) as that , that is .

Theorem 16. Let for all . Then, if and only if (50)–(52) hold and

Proof. Let and for all . Then, since the inclusion holds, the necessities of (50) and (51) are immediately obtained from part (i) of Theorem 13.
To prove the necessity of (62), consider the sequence defined by (31) which is in the space for every fixed . Because the transform of every exists and is in by the hypothesis, for every fixed which shows the necessity of (62).
Conversely suppose that conditions (50), (51), and (62) hold, and take any in the space . Then, exists. We observe for all that which gives the fact that by letting with (50) and (62) that This shows that and so which implies that the series converges for every .
Let us now consider the equality obtained from (54) with instead of where defined by for all . Therefore, we have at this stage from Lemma 8 that the matrix belongs to the class of infinite matrices. Thus, we see by (66) that Equation (67) means that whenever and this is what we wished to prove.

Therefore, we have the following

Corollary 17. Let for all . Then, if and only if (50)–(52) hold, and (62) also holds with for all .

Now, we give the following lemma given by Başar and Altay [26] which is useful for deriving the characterizations of the certain matrix classes via Theorems 13, 15, and 16 and Corollary 17.

Lemma 18 ([26, Lemma 5.3]). Let , be any two sequence spaces, let be an infinite matrix, and let also be a triangle matrix. Then, if and only if .

It is trivial that Lemma 18 has several consequences. Indeed, combining Lemma 18 with Theorems 13, 15, and 16 and Corollary 17, one can derive the following results.

Corollary 19. Let be an infinite matrix and define the matrix by Then, the necessary and sufficient conditions in order to belongs to anyone of the classes , and are obtained from the respective ones in Theorems 13, 16 and Corollary 17 by replacing the entries of the matrix by those of the matrix ; where , and , , respectively, denote the spaces of all sequences whose transforms are in the spaces and , and are recently studied by Altay et al. [27] and Altay and Başar [28], where denotes the Euler mean of order .

Corollary 20. Let be an infinite matrix and define the matrix by Then, the necessary and sufficient conditions in order to belongs to the class is obtained from Theorem 15 by replacing the entries of the matrix by those of the matrix ; where and denotes the space of all sequences whose transforms are in the space and is recently studied by Başar and Kirişçi [29].

Corollary 21. Let be an infinite matrix and define the matrix by Then, the necessary and sufficient conditions in order to belongs to the class is obtained from Theorem 15 by replacing the entries of the matrix by those of the matrix ; where and denotes the space of all sequences whose transforms are in the space and is recently studied by Sönmez [30].

Corollary 22. Let be an infinite matrix and define the matrix by Then, the necessary and sufficient conditions in order to belongs to the class is obtained from Theorem 15 by replacing the entries of the matrix by those of the matrix , where denotes the space of all sequences whose transforms are in the space and is recently studied by Kayaduman and Şengönül [31].

Corollary 23. Let be an infinite matrix and let be a sequence of positive numbers and define the matrix by where for all . Then, the necessary and sufficient conditions in order to belongs to anyone of the classes , and are obtained from the respective ones in Theorems 13, 16 and Corollary 17 by replacing the entries of the matrix by those of the matrix , where , , and are defined by Altay and Başar in [32] as the spaces of all sequences whose transforms are, respectively, in the spaces , , and , and are derived from the paranormed spaces , and in the case for all .

Since the spaces , , and reduce in the case to the Cesàro sequence spaces , , and of nonabsolute type, respectively, Corollary 23 also includes the characterizations of the classes , , and , as a special case, where and , are the Cesàro spaces of the sequences consisting of transforms are in the spaces and , and studied by Ng and Lee [33] and Şengönül and Başar [34], respectively, where denotes the Cesàro mean of order .

Corollary 24. Let be an infinite matrix and define the matrix by for all . Then, the necessary and sufficient conditions in order to belongs to anyone of the classes , and are obtained from the respective ones in Theorems 13 and 16 and Corollary 17 by replacing the entries of the matrix by those of the matrix , where , , denote the difference spaces of all bounded, convergent, and null sequences and are introduced by Kzmaz [35].

Corollary 25. Let be an infinite matrix and define the matrix by for all . Then the necessary and sufficient conditions in order to belongs to anyone of the classes , and are obtained from the respective ones in Theorems 13, 16 and Corollary 17 by replacing the entries of the matrix by those of the matrix , where denotes the set of those series converging to zero.

5. Conclusion

The difference spaces , , and were introduced by Kzmaz [35]. Since we essentially employ the infinite matrices which is more different than Kzmaz and the other authors following him, and use the technique of obtaining a new sequence space by the matrix domain of a triangle limitation method. Following this way, the domain of some triangle matrices in the sequence space was recently studied and were obtained certain topological and geometric results by Altay and Başar [14, 16], Choudhary and Mishra [10], Başar et al. [36], and Aydn and Başar [13]. Although is investigated, since , our results are more general than those of Başar et al. [36]. Also in case for all the results of the present study are reduced to the corresponding results of the recent paper of Kirişçi and Başar [9]. We should note that the difference spaces , and of Maddox’s spaces , , and   were studied by Ahmad and Mursaleen [37]. Of course, a natural continuation of the present paper is to study the sequence spaces , and to generalize the main results of Ahmad and Mursaleen [37] which fills up a gap in the existing literature.

It is clear that can be obtained as a special case of for and and it is also trivial that is reduced in the special case and to the generalized difference matrix . So, the results related to the domain of the matrix are much more general and more comprehensive than the corresponding consequences of the domain of the matrix . We should note from now that the main results of the present paper are given as an extended abstract without proof by Nergiz and Başar [38], and our next paper will be devoted to some geometric and topological properties of the space .

Acknowledgments

The authors would like to thank Professor Bilâl Altay, Department of Mathematical Education, Faculty of Education, İnönü University, 44280 Malatya, Turkey, for his careful reading and constructive criticism of an earlier version of this paper which improved the presentation and its readability. The main results of this paper were presented in part at the conference First International Conference on Analysis and Applied Mathematics (ICAAM 2012) to be held October 18–21, 2012, in Gümüşhane, Turkey, at the University of Gümüşhane.

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