Almost convergence and generalized weighted mean II
© Kirişci; licensee Springer. 2014
Received: 8 January 2013
Accepted: 28 January 2014
Published: 25 February 2014
In this paper, we investigate some new sequence spaces, which naturally emerge from the concepts of almost convergence and generalized weighted mean. The object of this paper is to introduce the new sequence spaces obtained as the matrix domain of generalized weighted mean in the spaces of almost null and almost convergent sequences. Furthermore, the beta and gamma dual spaces of the new spaces are determined and some classes of matrix transformations are characterized.
MSC:47A15, 46A32, 47D20.
Keywordsalmost convergence matrix domain sequence spaces generalized weighted mean β- and γ-duals matrix transformations
By a sequence space, we understand a linear subspace of the space of all complex sequences which contains ϕ, the set of all finitely non-zero sequences, where ℂ denotes the complex field and . We write , c, and for the classical spaces of all bounded, convergent and null sequences, respectively. Also by bs, cs, , and , we denote the spaces of all bounded, convergent, absolutely, and p-absolutely convergent series, respectively.
A sequence spaces μ with a linear topology is called a K-space if each of the maps defined by is continuous for all . A K-space is called an FK-space if μ is a complete linear metric space; a BK-space is a normed FK-space.
The sequence is the Schauder basis for and , and is the Schauder basis for the space c, while the space has no Schauder basis, where and e denote the sequences whose only non-zero entry is a 1 in the n th place for each and .
A subset M of a metric space is said to be dense in X if . A metric space is said to be separable if it contains a countable subset which is dense in X. Note that a nonseparable space has no Schauder basis.
for all ; here depends only on n and only on k.
for all .
The main purpose of present paper is to introduce the sequence spaces and derived as the domain of the generalized weighted mean in the spaces and f of almost null and almost convergent sequences, and to determine the β- and γ-duals of these spaces. Furthermore, some classes of matrix mappings on/in the space are characterized.
2 Spaces of almost null and almost convergent sequences
Lorentz  proved that if and only if uniformly in m. It is well known that a convergent sequence is almost convergent such that its ordinary and generalized limits are equal.
where is defined by (2.1). Also, by fs, we denote the space of all almost convergent series.
3 New sequence spaces and their duals
In this section, we introduce the sequence spaces and and give some results concerning them, and we determine their beta and gamma duals.
Şimşek et al.  have introduced a modular structure of the sequence spaces defined by Altay and Başar  and studied Kadec-Klee and uniform Opial properties of this sequence space on Köthe sequence spaces.
Theorem 3.2 The sequence spaces and strictly include the spaces and f, respectively.
Proof By the definition on the sequence spaces and f, it is immediate that and .
The sequence is not almost convergent but Gt is almost convergent to . This step completes the proof. □
Theorem 3.3 The inclusions and strictly hold.
Proof It is clear that and because of Theorem 3.2 and . Further, we show that these inclusions are strict.
Consider the sequence with the sequence y in the set given by Miller and Orhan  as , where the blocks of 0’s are increasing by factors of 100 and blocks of 1’s are increasing by factors of 10. Then the sequence x is not in but in the space . This shows that the inclusion is strict.
Since the inclusion strictly holds by Theorem 3.2, by combining this fact with the well-known strict inclusion , one can easily see that the inclusion also strictly holds, as was desired. □
It is known from Corollary 3.3 of Başar and Kirişçi  that the Banach space f has no Schauder basis. It is also known from Theorem 2.3 of Jarrah and Malkowsky  that the domain of a matrix A in a normed sequence space μ has a basis if and only if μ has a basis whenever is a triangle. Combining these two facts one can immediately conclude that neither the space nor the space have a Schauder basis.
Lemma 3.4 [, Theorem 2.1]
for all . Then the inclusion holds if and only if the matrix is in the classes , where is the diagonal matrix defined by for all .
Lemma 3.5 [, Theorem 3.1]
From Lemma 3.4 and Lemma 3.5, we may give the theorem determining the β- and γ-duals of the sequence space .
where is defined by (3.4). We therefore observe by (3.5) that or bs whenever if and only if or whenever . We obtain from Lemma 3.4 and Lemma 3.5 the result that or if and only if or , which is what we wished to prove. □
As a direct consequence of Theorem 3.6, we have the following.
4 Some matrix mappings related to the space
In this section, we give two theorems characterizing the classes of matrix transformations from the sequence space into any given sequence space μ and from any sequence space μ into the given sequence space .
for all .
for all and μ be any given sequence space. Then if and only if for all and .
Proof Let μ be any given sequence. Suppose that (4.9) holds between the infinite matrices and , and we take into account that the spaces and f are linearly isomorphic.
we obtain , which leads to the consequence .
as the result that and this shows that . This completes the proof. □
By changing the roles of the spaces and with μ, we have the following theorem.
Theorem 4.3 Suppose that the entries of the infinite matrices and are connected with the relation (4.1) and μ be any given sequence space. Then if and only if .
Equation (4.10) yields as the result that . Therefore, one can immediately observe from this that whenever if and only if whenever . This completes the proof. □
It is of course so that Theorem 4.2 and Theorem 4.3 have several consequences depending on the choice of sequence space μ and the sequences and . Therefore by Theorem 4.2 and Theorem 4.3, necessary and sufficient conditions for and may be derived by replacing the entries of E and by those of the entries of and , respectively, where the necessary and sufficient conditions on the matrices F and B are read from the concerning results in the existing literature.
If we get the space and the spaces , , and , which are isomorphic to instead of μ in Theorem 4.2, we obtain the following corollaries.
Then the necessary and sufficient conditions in order for A to belong to the class are obtained from Theorem 4.2 by replacing the entries of the matrix A by those of the matrix C; here as defined Altay et al. and Altay and Başar .
where for all . Then the necessary and sufficient conditions in order for A to belong to the class are obtained from in Theorem 4.2 by replacing the entries of the matrix A by those of the matrix D; here is the space of all sequences whose -transforms are in the space .
Remark 4.7 In the case in the space , this space reduces to the Cesàro sequence space of non-absolute type . Then Corollary 4.6 also includes the characterization of class , as a special case.
As in Corollaries 4.4-4.6 and Remark 4.7, the following corollaries are obtained for ; here the spaces , , are isomorphic to the space f.
defined by Kirişci .
defined by Başar and Kirişci .
defined by Kayaduman and Şengönul .
Prior to giving some consequences as an application of this idea, we give the following basic lemma, which is the collection of the characterization of matrix transformations related to almost convergence.
if and only if (4.17) and (4.18) hold.
if and only if (4.3) and (4.17)-(4.19) hold .
if and only if (4.6), (4.7), and (4.15) hold .
if and only if (4.6), (4.8), and (4.15) hold .
if and only if (4.6), (4.17),(4.18), and (4.21) hold .
if and only if (4.6), (4.8), (4.18), and (4.24) hold .
if and only if (4.6) and (4.17) hold .
if and only if (4.18) and (4.21)-(4.23) hold .
if and only if (4.17) and (4.22)-(4.24) hold .
if and only if (4.22) and (4.23) hold .
if and only if (4.25)-(4.28) hold .
Now, we can give the following results.
if and only if for all and (4.2)-(4.5) hold with instead of .
if and only if for all and (4.2) holds, (4.3) and (4.5) hold with and (4.4) holds with as instead of .
if and only if for all and (4.25) holds with instead of .
if and only if for all and (4.25)-(4.28) hold with instead of .
if and only if (4.2), (4.6), and (4.20) hold with instead of .
if and only if (4.2), (4.6), (4.7), and (4.8) hold with instead of .
if and only if (4.2), (4.6), and (4.7) hold with instead of .
if and only if (4.17), (4.18), (4.6), and (4.21) hold with instead of .
if and only if (4.18), (4.6), (4.8), and (4.21) hold with instead of .
if and only if (4.17) and (4.6) hold with instead of .
if and only if (4.18), (4.21), and (4.23) hold with instead of .
if and only if (4.21)-(4.24) hold with instead of .
if and only if (4.22) and (4.23) hold with instead of .
Here denotes the domain of the -generalized weighted mean in the sequence space fs.
As an essential work on the algebraic and topological properties of the spaces and f, Başar and Kirişçi  have recently introduced the sequence spaces and derived by the domain of the generalized difference matrix in the sequence spaces and f, respectively. Following Başar and Kirişçi , Kayaduman and Şengönül have studied the domain and of the Cesàro mean of order one in the spaces and f, in . They have determined the β- and γ-duals of the new spaces and , and they characterize some classes of matrix transformations on/in the new sequence spaces. They complete the paper by a nice section including some core theorems related to the matrix classes on/in the new sequence space . Quite recently, in , Sönmez has introduced the domain of the triple band matrix in the sequence space f. In this paper, the β- and γ-duals of the space are determined. Furthermore, the classes and of infinite matrices are characterized together with some other classes, where μ is any given sequence space. Finally, in  Candan has studied the sequence spaces and as the domain of the double sequential band matrix in the sequence spaces and f.
Since Kirişçi and Başar , Başar and Kirişçi , Kayaduman and Şengönül , Sönmez [26, 29], and Candan [27, 30] are recent works on the domain of certain triangle matrices in the spaces , f, and in the classical sequence spaces, the present paper is their natural continuation. Also these spaces are special cases of the notion of A-almost convergence and -convergence ([31, 32]) as well as analogous to the definition introduced in .
Finally, we should note that the investigation of the domain of some particular limitation matrices, namely Cesàro means of order m, Nörlund means, etc., in the spaces and f will lead to new results which are not comparable with the present results.
MK defined the new almost sequence spaces derived by generalized weighted mean and studied some properties. MK computed the duals of new spaces and characterized the matrix classes. In last section, it was summarized to studies in manuscripts and given some open problems by MK. The author read and approved the final manuscript.
Some of the results of this study presented in First International Conference on Analysis and Applied Mathematics (ICAAM 2012, Gumushane University, Turkey) .
I would like to express thanks to Professor Feyzi Başar, Department of Mathematics, Fatih University, Buyukcekmece, Istanbul-34500, Turkey, for his valuable help on some results and the useful comments which improved the presentation paper. I have benefited much from the constructive reports of the anonymous referees, and I am grateful for their valuable comments on the first draft on this paper, which improved the presentation and readability. This work was supported by Scientific Projects Coordination Unit of Istanbul University. Project number 27008.
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