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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 435076, 20 pages
Generalized Difference Spaces of Non-Absolute Type of Convergent and Null Sequences
1Department of Mathematics, Faculty of Sciences, Erciyes University, Melikgazi, 38039 Kayseri, Turkey
2Department of Mathematics, Faculty of Art and Sciences, Fatih University, The Hadımköy Campus, Büyükçekmece, 34500 İstanbul, Turkey
Received 17 February 2012; Accepted 10 October 2012
Academic Editor: Stevo Stevic
Copyright © 2012 Abdulcabbar Sönmez 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.
The aim of the present paper is to introduce the spaces and of generalized difference sequences which generalize the paper due to Mursaleen and Noman (2010). These spaces are the BK-spaces of non-absolute type and norm isomorphic to the spaces and , respectively. Furthermore, we derive some inclusion relations determine the , , and duals of those spaces, and construct their Schauder bases. Finally, we characterize some matrix classes from the spaces , and to the spaces , , and c.
By , we denote the space of all complex valued sequences. Any vector subspace of is called a sequence space. 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 and . A K-space is called an -space provided is a complete linear metric space. An -space whose topology is normable is called a -space (see [1, pages 272-273]) which contains , the set of all finitely nonzero sequences. We write , and for the classical sequence spaces of all bounded, convergent, and null sequences, respectively, which are -spaces with the usual sup-norm defined by , where, here and in the sequel, the supremum is taken over all . Also by and , we denote the spaces of all absolutely and -absolutely convergent series, respectively, which are -spaces with the usual norm defined by , where . For simplicity in notation, here and in what follows, the summation without limits runs from to . Also by and , we denote the spaces of all bounded and convergent series, respectively.
Let and be two sequence spaces, and let be an infinite matrix of 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 , -transform of , exists and is in , where By , we denote the class of all infinite matrices such that . Thus if and only if the series on the right side of (1.1) converges for each and every , and for all . A sequence is said to be -summable to if converges to , which is called the -limit of .
The domain of an infinite matrix in a sequence space is defined by We denote the collection of all finite subsets of by . Also, we write for the sequence whose only nonzero term is a in the th place for each .
The approach of constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by several authors, for example, [2–14]. They introduced the sequence spaces ( and in , ( and ( in , in , (, and in , ( in , and in , and in , ( and in , and in , and in  and and in , in , and in ; where, , , , and denote the Nörlund, Cesáro, Riesz, and Euler means, respectively, , , and are, respectively, defined in [5, 9, 12], , and . Also and denote the sequence spaces generated from the Maddox’s spaces and by Başarir . In the present paper, following [2–14], we introduce the difference sequence spaces and of non-absolute type and derive some related results. We also establish some inclusion relations. Furthermore, we determine the -, -, and -duals of those spaces and construct their bases. Finally, we characterize some classes of infinite matrices concerning the spaces and .
The rest of this paper is organized, as follows.
In Section 2, the -spaces and of generalized difference sequences are introduced. Section 3 is devoted to inclusion relations concerning with the spaces and . In Sections 4 and 5, the Schauder bases of the spaces and are given and the -, -, and -duals of the generalized difference sequence spaces and of non-absolute type are determined, respectively. In Section 6, the classes , , , , , and of matrix transformations are characterized, where . Also, by means of a given basic lemma, the characterizations of some other classes involving the Euler, difference, Riesz, and Cesàro sequence spaces are derived. In the final section of the paper, we note the significance of the present results in the literature related with difference sequence spaces and record some further suggestions.
2. The Difference Sequence Spaces and of Non-Absolute Type
The difference sequence spaces have been studied by several authors in different ways (see e.g. [12, 16–21]). In the present section, we introduce the spaces , and , and show that these spaces are -spaces of non-absolute type which are norm isomorphic to the spaces and , respectively.
We assume throughout that is a strictly increasing sequence of positive reals tending to , that is,
Recently, Mursaleen and Noman  studied the sequence spaces and of non-absolute type, and later they introduced the difference sequence spaces and in  of non-absolute type as follows: Here and after, we use the convention that any term with a negative subscript is equal to zero, for example, and . With the notation of (1.2) we can redefine the spaces and by where denotes the band matrix representing the difference operator, that is, for .
Let and be nonzero real numbers and define the generalized difference matrix by for all . The -transform of a sequence is We note that the matrix can be reduced to the difference matrices in case and . So, the results related to the matrix domain of the matrix are more general and more comprehensive than the consequences of the matrices domain of and include them.
Now, following Başar and Altay  and Aydın and Başar , we proceed slightly differently to Kızmaz  and the other authors following him and employ a technique of obtaining a new sequence space by means of the matrix domain of a triangle limitation method.
We thus introduce the difference sequence spaces and , which are the generalization of the spaces and introduced by Mursaleen and Noman , as follows: With the notation of (1.2), we can redefine the spaces and as where denotes the generalized difference matrix defined by (2.4).
It is immediate by (2.7) that the sets and are linear spaces with coordinatewise addition and scalar multiplication, that is, and are the sequence spaces of generalized differences.
On the other hand, we define the triangle matrix by for all . With a direct calculation we derive the equality and every which leads us together with (1.2) to the fact that
Further, for any sequence we define the sequence which will be frequently used as the -transform of , that is, and so we have Where, here and in what follows, the summation running from to is equal to zero when .
Now, we may begin with the following theorem which is essential in the text.
Theorem 2.1. The difference sequence spaces and are BK-spaces with the norm , that is,
Proof. Since (2.10) holds and and are -spaces with respect to their natural norms (see [23, pages 217-218]) and the matrix is a triangle, Theorem of Wilansky [24, page 63] gives the fact that and are -spaces with the given norms. This completes the proof.
Remark 2.2. One can easily check that the absolute property does not hold on the spaces and , that is, and for at least one sequence in the spaces and , and this shows that and are the sequence spaces of non-absolute type, where .
Now, we give the final theorem of this section.
Theorem 2.3. The sequence spaces and of non-absolute type are norm isomorphic to the spaces and , respectively, that is, and .
Proof. To prove this, we should show the existence of a linear bijection between the spaces and . Consider the transformation defined, with the notation of (2.11), from to by . Then, for every and the linearity of is clear. Further, it is trivial that whenever and hence is injective.
Furthermore, let and define the sequence by Then, we obtain Hence, for every , we get by (2.9) 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 show that the spaces and are linearly isomorphic.
It is clear that if the spaces and are replaced by the spaces and , respectively, then we obtain the fact that . This completes the proof.
3. The Inclusion Relations
In the present section, we establish some inclusion relations concerning with the spaces and . We may begin with the following theorem.
Theorem 3.1. 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, we obtain by (2.9) that which shows that and hence , where . Thus, the sequence is in but not in . Hence, the inclusion is strict and this completes the proof.
Theorem 3.2. If , then the inclusion strictly holds.
Proof. Suppose that and . Then and hence , since the inclusion . This shows that . Consequently, the inclusion holds. Further consider the sequence defined by for all . Then, it is trivial that . On the other hand, it can easily seen that . Hence, which means that . Thus, the sequence is in but not in . We therefore deduce that the inclusion is strict. This completes the proof.
On the other hand, we recall that if and , then , namely, is stronger than the ordinary convergence, hence we have the following
Corollary 3.3. The inclusions and strictly hold.
Further, it is obvious that the sequence , defined in the proof of Theorem 3.2, is in but not in . This leads us to the following result.
Corollary 3.4. Although the spaces and overlap, the space does not include the space .
Now, to prove the next theorem, we need the following lemma [24, page 4].
Lemma 3.5. if and only if .
Theorem 3.6. The inclusion strictly holds if and only if , where the sequence is defined by
Proof. Suppose that the inclusion holds. Then we obtain that for every and hence the matrix is in the class . Thus it follows by Lemma 3.5 that
Now, by taking into account the definition of the matrix given by (2.8), we have for every that
Thus, the condition (3.3) implies both
Now we have for every that
and since by (3.5), we obtain by (3.6) that
which shows that .
Conversely, suppose that . Then we have (3.8). Further, for every , we derive that Then, (3.9) and (3.8) together imply that (3.6) holds. On the other hand, we have for every that Therefore, it follows by (3.6) that . Particularly, if we take and , then we have which shows that (3.5) holds. Thus, we deduce by the relation (3.4) that (3.3) holds. This leads us with Lemma 3.5 to the consequence that . Hence, the inclusion holds and is strict by Corollary 3.4. This completes the proof.
4. The Bases for the Spaces and
In the present section, we give two sequences of the points of the spaces and which form the bases for those spaces.
If a normed sequence space contains a sequence with the property that for every there is a unique sequence of scalars such that then is called a Schauder basis (or briefly basis) for . The series which has the sum is then called the expansion of with respect to and is written as .
Now, since the transformation defined from to in the proof of Theorem 2.3 is an isomorphism, the inverse image of the basis of the space is the basis for the new space . Therefore, we have the following.
Theorem 4.1. Let for all and . Define the sequence for every fixed by Then, the following statements hold. (a)The sequence is a basis for the space and any has a unique representation of the form . (b)The sequence is a basis for the space and any has a unique representation of the form , where .
Corollary 4.2. The difference sequence spaces and are seprable.
5. The -, -, and -Duals of the Spaces and
In this section, we state and prove the theorems determining the -, -, and -duals of the generalized difference sequence spaces and of non-absolute type.
For arbitrary sequence spaces and , the set defined by is called the multiplier space of and . One can easily observe for a sequence space with and that the inclusions and hold, respectively.
With the notation of (5.1), the -, -, and -duals of a sequence space , which are respectively, denoted by , , and , are defined by It is clear that . Also it can be obviously seen that the inclusions , , and hold whenever .
Lemma 5.1. if and only if
Lemma 5.2. if and only if
Lemma 5.4. if and only if (5.5) holds.
Now, we prove the following result.
Theorem 5.5. The -dual of the spaces and is the set where the matrix is defined via the sequence by for all .
Proof. Let . Then, by bearing in mind the relations (2.11) and (2.14), it is immediate that the equality holds for all . Thus, we observe by (5.9) that whenever or if and only if whenever or . This means that the sequence or if and only if . We therefore obtain by Lemma 5.4 with instead of that if and only if which leads us to the consequence that . This completes the proof.
Theorem 5.6. Define the sets , , , and , as follows: Then and .
Proof. Consider the equality
where the matrix is defined by
for all . Then, we deduce by (5.12) that whenever if and only if whenever . This means that if and only if . Therefore, by using Lemma 5.2, we derive from (5.4) and (5.5) that
Therefore, we conclude that .
Similarly, we deduce from Lemma 5.3 with (5.12) that if and only if . Therefore, we derive from (5.4) and (5.5) that (5.14), (5.15) hold.
Further, with a simple calculation one can easily see that the equality holds for all . Consequently, from (5.6) we obtain that Hence, we deduce that . This completes the proof.
Finally, we close this section with the following theorem which determines the -dual of the spaces and :
Theorem 5.8. The -duals of the spaces and are the set .
6. Certain Matrix Mappings Related to the Spaces and
In this final section, we characterize the matrix classes , , , , , and , where . Also, by means of a given basic lemma, we derive the characterizations of some other classes involving the Euler, difference, Riesz, and Cesàro sequence spaces.
For an infinite matrix , we write for brevity that for all provided the convergence of the series.
The following lemmas will be needed in proving our main results.
Lemma 6.1 (see [24, page 57]). The matrix mappings between the BK-spaces are continuous.
Lemma 6.2 (see [25, pages 7-8]). if and only if
Now, we give the following results on the matrix transformations.
Proof. Suppose that the conditions (6.4)–(6.9) hold and take any . Then, we have by Theorem 5.6 that for all and this implies that the -transform of exists. Also, it is clear that the associated sequence is in the space and hence as for some suitable . Further, it follows by combining Lemma 6.2 with (6.5) that the matrix is in the class , where .
Let us now consider the following equality derived by using the relation (2.11) from the th partial sum of the series : Then, since and , the series converges for every . Furthermore, it follows by (6.4) that the series converges for all and hence as . Therefore, if we pass to limit in (6.12) as then we obtain by (6.8) that which can be written as follows: This yields by taking -norm that which leads us to the consequence that . Hence, .
Conversely, suppose that , where . Then for all which implies with Theorem 5.6 that the conditions (6.6) and (6.7) are necessary.
On the other hand, since and are -spaces, we have by Lemma 6.1 that there is a constant such that holds for all . Now, . Then, the sequence is in , where the sequence is defined by (4.2) for every fixed .
Since for each fixed , we have
Furthermore, for every , we obtain by (4.2) that Hence, since the inequality (6.16) is satisfied for the sequence , we have for any that which shows the necessity of (6.5). Thus, it follows by Lemma 6.2 that .
Now, let and consider the sequence defined by (2.14) for every . Then, such that , that is, the sequences and are connected by the relation (2.11). Therefore, and exist. This leads us to the convergence of the series and for every . We thus deduce that Consequently, we obtain from (6.12) as that and since , we conclude that which shows the necessity of (6.8). Then relation (6.14) holds.
Finally, since and , the necessity of (6.9) is immediate by (6.14). This completes the proof of Part (i) of the theorem.
Since Part (ii) can be proved by using the similar way that used in the proof of Part (i) with Lemma 5.4 instead of Lemma 6.2, we leave the details to the reader.
Theorem 6.7. Let be an infinite matrix over the complex field. Then, the following statements hold. (i) Let . Then, if and only if (6.5) and (6.6) hold, and (ii) if and only if (6.10) and (6.24) hold.
Proof. It is natural that the present theorem can be proved by the same technique used in the proof of Theorem 6.5, above, and so we omit the proof.
Proof. Suppose that satisfies the conditions (6.7), (6.8), (6.10), (6.25), (6.26), and (6.27) and take any . Then, since (6.10) implies (6.6), we have by Theorem 5.6 that for all and hence exists. We also observe from (6.10) and (6.26) that
holds for every . This implies that and hence the series converges, where is the sequence connected with by the relation (2.11) such that as . Further it is obvious by combining Lemma 5.3 with the condition (6.10), (6.26), and (6.27) that the matrix is in the class .
Now reasoning as in the proof of Theorem 6.5, we obtain that the relation (6.13) holds which can be written as follows: In this situation, we see by passing to the limit in (6.29) as that the first term on the right tends to by (6.10) and (6.26). The second term tends to by (6.27) and the last term to by (6.25). Consequently, we have which shows that , that is to say that .
Conversely, Suppose that . Then, since the inclusion holds, it is trivial that . Therefore, the necessity of the conditions (6.7), (6.8), and (6.10) are obvious from Theorem 6.5. Further, consider the sequence defined by (4.2) for every fixed . Then, it is easily seen that and hence for every which shows the necessity of (6.26). Let . Then, since the linear transformation , defined as in the proof of Theorem 2.3 by analogy, is continuous and for each fixed , we obtain that which shows that and hence . On the other hand, since and are the BK-spaces, Lemma 6.1 implies the continuity of the matrix mapping . Thus, we have for every that This shows the necessity of (6.27).
Now, it follows by (6.10), (6.26), and (6.27) with Lemma 5.3 that . So by (6.7), and (6.8), relation (6.14) holds for all and , and and are connected by relation (2.11), where ().
Lastly, since and ; the necessity of (6.25) is immediate by (6.14). This step concludes the proof.
Let denotes any of the classical sequence spaces , , or . Then, consisting of the sequences such that is called as the difference sequence space which was introduced by Kızmaz . Kızmaz  proved that is a Banach space with the norm , where and the inclusion relation strictly holds. He also determined the -, -, and -duals of the difference spaces and characterized the classes and of infinite matrices, where . Following Kızmaz , Sarıgöl  extended the difference space to the space defined by and computed the -, -, and -duals of the space , where . It is easily seen that , if and , if . Recently, the difference spaces consisting of the sequences such that have been studied in the case by Altay and Başar  and in the case by Başar and Altay , Çolak et al. , and Malkowsky et al. . Quite recently, Mursaleen and Noman have introduced the spaces and of -convergent and -null sequences and nextly studied the difference spaces and in [21, 22], respectively. Of course, there is a wide literature concerning the difference sequence spaces. By the domain of the generalized difference matrix in the spaces of -convergent and -null sequences we have generalized the difference spaces and defined by Mursaleen and Noman . Since the generalized difference matrix reduces, in the special case , , to the usual difference matrix ; our results are more general and more comprehensive than the corresponding results of Mursaleen and Noman .
Since the difference spaces of -bounded and absolutely -summable sequences are not studied, the domain of both the difference matrix , and the generalized difference matrix in those spaces are still open. So, it is meaningful to fill this gap.
The authors have benefited a lot from discussion on the main results of the earlier version of the paper with Professor Bilâl Altay, Department of Mathematical Education, Faculty of Education, nönü University, Turkey. So, they wish to express their gratitude for his many helpful suggestions and interesting comments.
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