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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 476363, 9 pages
http://dx.doi.org/10.1155/2013/476363
Research Article

On the Domain of the Triangle on the Spaces of Null, Convergent, and Bounded Sequences

1Department of Mathematics and Computer Sciences, Avenue “Mother Teresa” Nr=5, 10000 Prishtinë, Kosovo
2Department of Mathematics, Faculty of Arts and Sciences, Fatih University, The Hadimköy Campus, Büyükçekmece, 34500 İstanbul, Turkey

Received 17 January 2013; Accepted 7 May 2013

Academic Editor: Douglas Anderson

Copyright © 2013 Naim L. Braha 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

We introduce the spaces of -null, -convergent, and -bounded sequences. We examine some topological properties of the spaces and give some inclusion relations concerning these sequence spaces. Furthermore, we compute -, -, and -duals of these spaces. Finally, we characterize some classes of matrix transformations from the spaces of -bounded and -convergent sequences to the spaces of bounded, almost convergent, almost null, and convergent sequences and present a Steinhaus type theorem.

1. Introduction

By , we denote the space of all complex sequences. If , then we simply write instead of . Also, we will use the conventions that , and is the sequence whose only nonzero term is in the place for each , where . Any vector subspace of is called a sequence space. We will write , and for the sequence spaces of all bounded, convergent, and null sequences, respectively. Further, by with , we denote the sequence space of all -absolutely convergent series, that is, . For simplicity in notation, here and in what follows, the summation without limits runs from to . Moreover, we write and for the spaces of all bounded and convergent series, respectively. A sequence space is called an -space if it is a complete linear metric space with continuous coordinates , where denotes the complex field and for all and every . A normed -space is called a -space, that is, a -space is a Banach space with continuous coordinates. The sequence spaces and are -spaces with the usual sup-norm given by . Also, the space is a -space with the usual norm defined by where . A sequence in a normed space is called a Schauder basis for if for every there is a unique sequence of scalars such that , that is, The alpha-, beta-, and gamma-duals , , and of a sequence space are, respectively, defined by

If is an infinite matrix with complex entries , where , then we write instead of . Also, we write for the sequence in the row of the matrix , that is, for every . Further, if then we define the -transform of as the sequence , where provided the series on the right hand side of (4) convergent for each .

Furthermore, the sequence is said to be -summable to if converges to which is called the -limit of . In addition, let and be sequence spaces. Then, we say that defines a matrix mapping from into if for every sequence the -transform of exists and is in . Moreover, we write for the class of all infinite matrices that map into . Thus, if and only if for all and for all . The matrix domain of an infinite matrix in a sequence space is defined by which is a sequence space. The approach constructing a new sequence space by means of the matrix domain of a triangle matrix was employed by several authors, see for instance [14]. In this paper, we introduce the spaces of -null, -convergent, and -bounded sequences which generalize the results given in [2]. Further, we define some related -spaces and construct their bases. Moreover, we establish some inclusion relations concerning those spaces and determine their alpha-, beta-, and gamma-duals. Finally, we characterize some classes of matrix transformations on these sequence spaces.

2. Notion of -Null, -Convergent, and -Bounded Sequences

Let be a strictly increasing sequence of positive real numbers tending to infinity, as and for each . From this last relation, it follows that . The first and second differences are defined as follows: and for all , where .

Let be a sequence of complex numbers, such that . We say that the sequence is -strongly convergent to a number if This generalizes the concept of -strong convergence (see [5]).

Lemma 1 (see [5]). A sequence of complex numbers -strongly converges to a number if and only if converges to in the ordinary sense and

Let us define the sequence by the -transform of a sequence , that is, for all . Throughout the text, we suppose that the terms of the sequences and    are connected with the relation (8).

Lemma 2 (see [5]). If a sequence converges to in the ordinary sense and condition (7) of Lemma 1 holds, then the sequence of complex numbers -strongly converges to .

Remark 3 (see [5]). From above results, we can conclude the following. The sequence of complex numbers -strongly converges to if and only if the following relation holds:

Now, we define the infinite matrix by for all . Then, -transform of a sequence is the sequence , where is given by the relation (8) for every . Thus, the sequence is -convergent if and only if is -summable. Further, if is -convergent then the -limit of exists and coincides with the ordinary limit of , that is, to say that the method is regular.

3. The Spaces of -Null, -Convergent, and -Bounded Sequences

We introduce the classes , , and of all -null, -convergent, and -bounded sequences of complex numbers, that is, Obviously, , , and are the linear spaces with respect to the usual operations coordinatewise addition and scalar multiplication of sequences. Here and after, by we denote any of the spaces , , and . It is not hard to see that the quantity is finite for every , and is a norm on .

Denote by the usual -norm, that is, to say that With the notation of (5), we can redefine the spaces , , and as follows:

Theorem 4. The sequence spaces , , and are -spaces with the norm given by

Proof. This follows from Theorem  4.3.12 given in [6] and the relations (14).

Theorem 5. The sequence spaces , , and are norm isomorphic to the spaces , , and , respectively.

Proof. Since the matrix is triangle, it has unique inverse which is also triangle matrix (see [6, ]). Therefore, the linear operator, defined by , for all , is bijective and norm preserving by relation (15).

As a consequence of Theorems 4 and 5, we get the following result.

Corollary 6. Define the sequence for every fixed bywhere . Then, one has the following.(1)The sequence is a Schauder basis for the space , and every has a unique representation: . (2)The sequence   is a Schauder basis for the space , and every has a unique representation: , where .

4. Some Inclusion Relations Related to the New Spaces

In this section, we give some inclusion relations concerning the spaces , , and .

Theorem 7. The inclusions strictly hold.

Proof. Let us suppose that , then it follows that and . In what follows we show that these inclusions are strict. The first inclusion follows from the fact that every sequence, which converges in ordinary sense, converges in -sense to the same limit. To prove the strictness of the inclusion , define the sequence by for all . Then, it follows that Therefore, it is trivial that .

Theorem 8. The equality holds.

Proof. First, we prove that . If a sequence converges in the ordinary sense to then it follows that converges in -sense, too. This gives the first inclusion. The converse inclusion follows from Lemma 1, in [5].

In what follows we describe some properties of the sequence in the space .

Theorem 9. For the sequence which is given in Section 2, the following relations are satisfied:(i)  if and only if ;(ii)  if and only if .

Proof. (i) Let us start with the expression After some calculations, we get
On the other hand, from the definition of the sequence we have
From the last relation, we have following two possibilities:(a) or (b).
Part (a) is satisfied if and only if is bounded. Part (b) is satisfied if and only if is unbounded.

Lemma 10. The inclusions and hold. Those spaces coincide if and only if for every , respectively, , where .

Lemma 11. The inclusion holds. Those spaces coincide if and only if for every .

Theorem 12. The inclusions , and strictly hold if and only if

Proof. Let us suppose that is strict. Then, from Lemma 11, it follows that there exists a sequence such that . Since , we have which leads us to the fact that . On the other hand, from relation (20), it follows that . The last relation is equivalent to by part (i) of Theorem 9. In a similar way we can conclude that the inclusions , are strict. In what follows we prove the sufficiency. Let Then, from, part (i) of Theorem 9, it follows that and . Let us define the sequence by for all . Then, we get the following estimation: Hence, which means that . If . Then, there exists a subsequence such that Now, let us define the sequence by for all . It follows from (28) that . On the other hand, Now, from the relations (27) and (29), we derive that . This completes the proof.

As an immediate result of Theorem 12, we have the following.

Corollary 13. The equalities , , and are satisfied if and only if

Proposition 14. The following statements hold.(i)Although and overlap, the space does not include the space . (ii) Although and overlap, the space does not include the space .

Proposition 15. If  , then the following statements hold.(i)Neither of the spaces and includes the other.(ii)Neither of the spaces and includes the other.(iii)Neither of the spaces and includes the other.

5. The -, -, and -Duals of the Spaces , , and

In this section, we determine the alpha-, beta-, and gamma-duals of the spaces , , and .

We need the following lemma due to Stieglitz and Tietz [3] in proving Theorem 17.

Lemma 16. if and only if Here and after, by one denotes the collection of all finite subsets of .

Theorem 17. The -dual of the spaces , , and is the set

Proof. Define the matrix with the aid of a sequence as follows: Then, , we have from Theorem 5 for all . From the relation (34), it follows that whenever if and only if whenever , that is, if and only if . By Lemma 16, this is possible if and only if Now, from definition of the sets and the matrix , it follows that (35) holds if and only if which gives that .
In a similar way, one can show that is the -dual of the spaces and . So, we omit the details.

Theorem 18. Define the sets , , , and as follows: Then, one has , and .

Proof. Since the proof is similar for the spaces and , we consider only the space . Let . Then, taking into account the relation (8) between the sequences and , we obtain that where and the matrix is defined by for all . Therefore, one can easily see from (38) that with if and only if with , where is defined by (40). That is, to say that if and only if is a matrix satisfying the conditions of Kojima-Schur's theorem (cf. Başar [7, Theorem , page 35]). This leads to the fact that .

Theorem 19. The -dual of the spaces , , and is the set .

Proof. This is similar to the proof of Theorem 18. So, we omit the details.

6. Some Matrix Transformation Related to Sequence Spaces , , and

In this section, we characterize the matrix transformations from the spaces and into the spaces , , , , and of bounded, almost convergent, almost null, convergent, and null sequences, respectively. We write throughout for brevity that for all , and we use these abbreviations with other letters, where .

Theorem 20. if and only if

Proof. Suppose that the conditions (42) and (43) hold, and take any . Then, the sequence for all , and this implies the existence of the -transform of .
Let us now consider the following equality derived by using the relation (8) from the partial sum of the series : for all . Therefore, we obtain from (44) with (42), as , that Now, by taking the sup-norm in (45), we derive that which shows the sufficiency of the conditions (42) and (43).
Conversely, suppose that . Then, since for all by the hypothesis, the necessity of (42) is trivial and (45) holds. Consider the continuous linear functionals defined on by the sequences as Since , and , it should follow with (45) that . This just says that the functionals defined by the rows of on are pointwise bounded. Hence, by Banach-Steinhaus theorem, 's are uniformly bounded which gives that there exists a constant such that for all . It therefore follows that holds for all which shows the necessity of the condition (43).
This step completes the proof.

Prior to characterizing the class of infinite matrices from the space into the space of almost convergent sequences, we give a short knowledge on the concept of almost convergence. The shift operator is defined on by for all . Banach limit is defined on , as a non-negative linear functional, such that and . sequence is said to be almost convergent to the generalized limit if all Banach limits of coincide and are equal to [8] and is denoted by . Let be the composition of with itself times and write for a sequence Lorentz [8] proved that if and only if , uniformly in . It is well known that a convergent sequence is almost convergent such that its ordinary and generalized limits are equal. By and , we denote the spaces of almost null and almost convergent sequences, respectively. Now, we can give the lemma characterizing the almost coercive matrices.

Lemma 21 (see [9, Theorem 1]). if and only if

Theorem 22. if and only if the conditions (42) and (43) hold, and

Proof. Let . Then, since , the necessity of (42) and (43) is immediately obtained from Theorem 20. To prove the necessity of (52), consider the sequence , defined by (16) for every fixed . Since exists and is in for every , one can easily see that for all , that is, the condition (52) is necessary.
Define the matrix by for all . Then, we derive from the equality (45) that . Since by the hypothesis, we have . Therefore, the matrix satisfies the condition (51) of Lemma 21 which is equivalent to the condition (53).
Conversely, suppose that the matrix satisfies the conditions (42), (43), (52), and (53), and . Reconsider the equality obtained from (45) with instead of . Then, the conditions (49), (50), and (51) are satisfied by the matrix . Hence, is almost coercive by Lemma 21 and this gives by passing to -limit in (45) that , that is, , as desired.
This concludes the proof.

As a direct consequence of Theorem 22, we have the following.

Corollary 23. if and only if the conditions (42) and (43) hold, and (52) and (53) hold with for all .

Theorem 24. if and only if the condition (42) holds, and the conditions

Corollary 25.   if and only if the conditions (42) and (54) hold, and (55) also holds with for all .

Now, we give the following lemma due to King [10] characterizing the class of almost conservative matrices.

Lemma 26. if and only if (49) and (50) hold, and

Theorem 27. if and only if the conditions (42), (43), and (52) hold, and the conditionalso holds.

Proof. This is obtained by a similar way used in proving Theorem 22 with Lemma 26 instead of Lemma 21. So, to avoid the repetition of the similar statements we omit the details.

Corollary 28. if and only if the conditions (42) and (43) hold, and the conditions (52) and (57) also hold with for all and , respectively; where by , we denote the class of infinite matrices such that for all .

Now, we give the following Steinhaus type theorem.

Theorem 29. The classes and are disjoint, where .

Proof. Suppose that the classes and are not disjoint. Then, there is at least one in the set . Therefore, one can derive by combining (53) and (52) with for all that which is contrary to the condition (57) with . This completes the proof.

Lemma 30 (see [11, Lemma 5.3]). Let , be any two sequence spaces, an infinite matrix, and a triangle matrix. Then, if and only if .

It is trivial that Lemma 30 has several consequences. Indeed, combining Lemma 30 with Theorems 20, 22, 24, and 27 and Corollaries 23, 25, and 28 by choosing as one of the special matrices , , , , , , or , one can easily obtain the following results.

Corollary 31. Let be an infinite matrix over the complex field. Then, the following statements hold.(i) if and only if (42) and (43) hold with instead of , where denotes the space of all sequences such that and was introduced by Başar and Altay [11]. (ii) if and only if (42) and (43) hold with instead of , where denotes the space of all sequences such that and was introduced by Altay et al. [12]. (iii) if and only if (42) and (43) hold with instead of , where denotes the space of all sequences such that and was introduced by Ng and Lee [13].(iv) if and only if (42) and (43) hold with instead of , where denotes the space of all sequences such that and was introduced by Altay and Başar [14].(v) if and only if (42) and (43) hold with instead of .

Corollary 32. Let be an infinite matrix over the complex field. Then, the following statements hold.(i) if and only if (42), (54), and (55) hold with instead of , where denotes the space of all sequences such that and was introduced by Kızmaz [15]. (ii) if and only if (42), (54), and (55) hold with instead of , where denotes the space of all sequences such that and was introduced by Altay and Başar [16].(iii) if and only if (42), (54), and (55) hold with instead of , where denotes the space of all sequences such that   and was introduced by Şengönül and Başar [17].(iv) if and only if (42), (54), and (55) hold with instead of , where denotes the space of all sequences such that and was introduced by Altay and Başar [18].(v) if and only if (42), (54) and (55) hold with instead of .

Corollary 33. Let be an infinite matrix over the complex field. Then, the following statements hold.(i) if and only if (42), (43), (52), and (53) hold with instead of , where denotes the space of all sequences such that and was introduced by Başar and Kirişçi [19].(ii) if and only if (42) and (43) hold, and (52) and (53) hold with for all and instead of , where denotes the space of all sequences such that and was introduced by Başar and Kirişçi [19].(iii) if and only if (42), (43), (52), and (53) hold with instead of .(iv) if and only if (42) and (43) hold, and (52) and (53) also hold with for all and , respectively, hold with for all and instead of .(v) if and only if the conditions of Corollary 28 hold with instead of .

Corollary 34. Let be an infinite matrix over the complex field. Then, the following statements hold. (i) if and only if (42), (43), (52), and (53) hold with instead of , where denotes the space of all sequences such that and was introduced by Kayaduman and Şengönül [20]. (ii) if and only if (42) and (43) hold, and (52) and (53) hold with for all and instead of , where denotes the space of all sequences such that and was introduced by Kayaduman and Şengönül [20]. (iii) if and only if (42), (43), (52), and (53) hold with instead of . (iv) if and only if (42) and (43) hold, and (52) and (53) also hold with for all and , respectively, with instead of . (v) if and only if the conditions of Corollary 28 hold with instead of .

Corollary 35. Let be an infinite matrix over the complex field. Then, the following statements hold.(i) if and only if (42), (43), (52), and (53) hold with instead of , where denotes the space of all sequences such that and was introduced by Sönmez [21]. (ii) if and only if (42) and (43) hold, and (52) and (53) hold with for all and instead of , where denotes the space of all sequences such that and was introduced by Sönmez [21]. (iii) if and only if the conditions (42), (43), (52), and (53) hold with instead of . (iv) if and only if the conditions (42) and (43) hold, and the conditions (52) and (53) also hold with for all and , respectively, hold with for all and instead of . (v) if and only if the conditions of Corollary 28 hold with instead of .

7. Conclusion

Mursaleen and Noman [2, 22, 23] have studied the domains , , , and of the matrix in the classical sequence spaces , , , and , respectively. Malkowsky and Rakočević [24] characterized some classes of matrix transformations and investigated related compact operators involving the spaces of -null, -convergent, and -bounded sequences. Quite recently, Sönmez and Başar [25] have introduced the spaces and of generalized difference sequences which generalize the paper due to Mursaleen and Noman [26]. Mursaleen and Noman [26] have derived some inclusion relations and determined the alpha-, beta-, and gamma-duals of those spaces and constructed their Schauder bases. Finally, Sönmez and Başar [25] have characterized some matrix classes from the spaces and to the spaces , , and . In the present paper, we emphasize the domains , , and of the matrix in the classical sequence spaces , , and . Our results are more general and comprehensive than the corresponding results of Mursaleen and Noman [2, 22, 23] derived with the matrix . We should note that, as a natural continuation of the present paper, one can study the domains and of the matrix in the classical sequence space and in the sequence space with and , where denotes the space of all sequences such that and introduced in the case by Başar and Altay [11] and in the case by Altay and Başar [27].

Acknowledgment

The authors would like to express their pleasure to the anonymous referees for constructive criticism of an earlier version of this paper which improved its readability.

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