Journal of Mathematics

Volume 2018, Article ID 1826485, 7 pages

https://doi.org/10.1155/2018/1826485

## On the Characterization of Some Classes of Four-Dimensional Matrices and Almost -Summable Double Sequences

Department of Mathematics Education, Ishik University, Ishik Campus, 100 Meter Street, Erbil, Iraq

Correspondence should be addressed to Orhan Tug; qi.ude.kihsi@gut.nahro

Received 13 January 2018; Accepted 12 March 2018; Published 19 April 2018

Academic Editor: Alfred Peris

Copyright © 2018 Orhan Tug. 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 firstly summarize the related literature about -summability of double sequence spaces and almost -summable double sequence spaces. Then we characterize some new matrix classes of , , and of four-dimensional matrices in both cases of and , and we complete this work with some significant results.

#### 1. Preliminaries, Background, and Notations

We denote the set of all complex valued double sequence by which is a vector space with coordinatewise addition and scalar multiplication. Any subspace of is called a double sequence space. A double sequence of complex numbers is called bounded if , where . The space of all bounded double sequences is denoted by which is a Banach space with the norm . Consider the double sequence . If for every there exists a natural number and such that for all , then the double sequence is said to be convergent in Pringsheim’s sense to the limit point ; say that , where indicates the complex field. The space denotes the set of all convergent double sequences in Pringsheim’s sense. Although every convergent single sequence is bounded, this is not hold for double sequences in general. That is, there are such double sequences which are convergent in Pringsheim’s sense but not bounded. Actually, Boos [1, p. 16] defined the sequence by Then it is clearly seen that but , so . The set denotes the space of both bounded and convergent double sequences; that is, . Hardy [2] showed that a double sequence is said to converge regularly to if and the limits , and , exist, and the limits and exist and are equal to the -limit of . Moreover, by and , we may denote the spaces of all null double sequences contained in the sequence spaces and , respectively. Móricz [3] proved that the double sequence spaces , , , and are Banach spaces with the norm . The space of all absolutely summable double sequences corresponding to the space of summable single sequences was defined by Başar and Sever [4]; that is, which is a Banach space with the norm . Then, the space which is a special case of the space with is introduced by Zeltser [5].

Let be a double sequence space and converging with respect to some linear convergence rule is . Then, the sum of a double series relating to this rule is defined by . Throughout the paper, the summations from 0 to without limits, that is, , mean that .

Here and after, unless otherwise stated, we consider that denotes any of the symbols , or .

The dual , the dual with respect to the convergence, and the dual of double sequence space are, respectively, defined by It is easy to see for any two spaces and of double sequences that whenever and . Additionally, it is known that the inclusion holds while the inclusion does not hold, since the convergence of the double sequence of partial sum of a double series does not guarantee its boundedness.

Here, we shall be concerned with four-dimensional matrix transformation from any double sequence space to any double sequence space . Given any four-dimensional infinite matrix , where , any double sequence , we write , the transform of , exists for every sequence and it is in , whereThe four-dimensional matrix domain has fundamental importance for this article. Therefore, this concept is presented in this paragraph. The summability domain of in a space of double sequences is described as Notation (4) says that maps the space into the space if and we denote the set of all four-dimensional matrices, transforming the space into the space , by . Thus, if and only if the double series on the right side of (4) converges in the sense of for each ; that is, for all and we have for all , where for all . Moreover, the following definitions are significant in order to classify the four-dimensional matrices. A four-dimensional matrix is called if and is called if it is and

One can be obtained by using the notations of Zeltser [6] that we may define the double sequences , and by , , and for all and we may write the set by .

In order to establish a new sequence space, special triangular matrices were previously used. These new spaces derived by the domain of matrices are expansion or the contraction of the original space, in general. Adams [7] defined that the four-dimensional infinite matrix is called a triangular matrix if for or or both. We also say by [7] that an infinite matrix is said to be a triangular if for all . Moreover, Cooke [8] showed that every triangular matrix has a unique inverse which is also a triangular matrix.

The four-dimensional generalized difference matrix and matrix domain of it on some double sequence spaces were recently defined by Tuğ and Başar [9] and studied by Tuğ [10–12]. The matrix was defined by Tuğ and Başar [9] as for and for all . Therefore, the transform of a double sequence was defined byfor all . Moreover, the matrix which is the inverse of the matrix was calculated as for all . Furthermore, Tuğ and Başar [9] obtained the relation between and by

Tu and Başar [9] and Tuǧ [10] have introduced and studied the new double sequence spaces , , , , and as the domain of four-dimensional generalized difference matrix in the spaces , , , , and , respectively. That is,

Lorentz [13] introduced the concept of almost convergence for single sequence and Moricz and Rhoades [14] extended and studied this concept for double sequence. A double sequence of complex numbers is said to be almost convergent to a generalized limit if In this case, is called the limit of the double sequence . Throughout the paper, denotes the space of all almost convergent double sequences; that is, It is known that a convergent double sequence need not be almost convergent. But it is well known that every bounded convergent double sequence is also almost convergent and every almost convergent double sequence is bounded. That is, the inclusion holds, and each inclusion is proper. A double sequence is called almost Cauchy which was introduced by Čunjalo [15] if for every there exists a positive integer such that for all and . Mursaleen and Mohiuddine [16] proved that every double sequence is almost convergent if and only if it is almost Cauchy.

Moricz and Rhoades [14] considered that four-dimensional matrices transforming every almost convergent double sequence into a convergent double sequence with the same limit. Almost conservative and almost regular matrices for single sequences were characterized by King [17] and almost conservative and almost regular four-dimensional matrices for double sequences were defined and characterized by Zeltser et al. [18]. Mursaleen [19] introduced the almost strongly regularity for double sequences. A four-dimensional matrix is called almost strongly regular if it transforms every almost convergent double sequence into an almost convergent double sequence with the same limit.

*Definition 1 (see [18]). *A four-dimensional matrix is said to be almost conservative matrix if it transforms every convergent double sequence into an almost convergent double sequence space; that is, .

*Definition 2 (see [18]). *A four-dimensional matrix is said to be almost regular if it is conservative and for all .

Tuğ [11, 12] has recently studied new almost convergent double sequence spaces and as the domain of four-dimensional generalized difference matrix in the spaces and , respectively. That is,

In this paper, as natural continuation of [10, 11], we characterize some new matrix classes , , and of four-dimensional matrices in both cases and . Throughout the paper, we suppose that the terms of double sequence and are connected with relation (11) and the four-dimensional generalized difference matrix will be presented with .

#### 2. Main Results

In this section, we characterize some new four-dimensional matrix classes , , in both cases of and , and . Then we complete this section with some significant results of four-dimensional matrix mapping via the dual summability methods for double sequences which has been introduced and studied by Başar [20] and Yeşilkayagil and Başar [21] and has been recently applied in [10, 12].

Theorem 3. *Let be a four-dimensional infinite matrix. Then the following statements hold.**(a) Let . Then, if and only if**(b) Let . Then, if and only if condition (18) holds and*

*Proof. *(a) Let and . Then, exists and is in for all . Since the inclusion holds, then we may say that condition (17) is necessary. Moreover, since exists and is in for all , it is also provided for that which gives us condition (18) which is also necessary.

Conversely, suppose that conditions (17) and (18) hold and be any double sequence in . Since by corollary 3.2 of [22] for each , exists. By using conditions (17) and (18) for each , we have the following inequality:then we may say that . Thus, the series is convergent for all . Moreover, by condition (18) we may say that for every there exists a positive such that for all . Thus It gives us that exists; that is, .

This completes the proof of .

(b) Let . The necessity of conditions (18) and (19) may be easily shown in the similar method which is used in (a) with the Hölder inequality. So we pass the repetition.

Conversely, suppose that conditions (18) and (19) hold and be a double sequence in the space . It can be written with (18) for all thatsay that . Therefore, the double series is convergent for every . Moreover, for any given , there exist such that for every . So, by using Hölder’s inequality with relations (23) and (24) we have power to write that for all . After passing -limit as we can say that . This last step completes the proof.

Theorem 4. *Let be a four-dimensional infinite matrix. Then the following statements hold.**(a) Let . Then, if and only if(b) . Then, if and only if condition (27) holds and*

*Proof. *Suppose that the four-dimensional matrix . Then, exist and is in for all . Since then we can say that . Moreover, we may have the following equality:holds for all . Therefore, we have by taking limit as in (29). Now we can define the new four-dimensional matrix as So we can write here that whenever if and only if whenever which says that . Thus conditions (17) and (18) hold for , and conditions (18) and (19) hold for with instead of . This gives us the results as desired.

Theorem 5. *Let be a four-dimensional infinite matrix. Then the following statements hold.**(a) Let . Then, if and only if**(b) Let . Then, if and only if condition (32) holds and*

*Proof. *Suppose that the four-dimensional matrix . Then, as the method used in the proof of Theorem 4, exist and is in for all . Since then we can say that and the equalityholds for all . Therefore, we have by taking limit as in (34). Now we may define the following four-dimensional matrix as So we can write here that whenever if and only if whenever which says that . Thus conditions (17) and (18) hold for , and conditions (18) and (19) hold for with instead of . This gives us the expected results and completes the proof.

Corollary 6. *Suppose that the elements of four-dimensional infinite matrices and are connected with relation Then the following statements hold.**(i) Let . Then iff (17) and (18) hold with instead of .**(ii) Let . Then iff (18) and (19) hold with instead of .*

#### 3. Conclusion

The concept of almost convergence of single sequence was introduced by Lorentz [13]. In 2010, Mursaleen [23] investigated the certain properties of the space of almost convergent sequences denoted by . Then many of the mathematicians have studied the matrix domain on almost null and almost convergent sequences spaces (see [24–27]).

The almost convergence for double sequence was introduced by Moricz and Rhoades [14] and studied on by many researchers (see [16, 28–36]). Yeşilkayagil and Basar [37] recently studied the topological properties of the spaces of almost null and almost convergent double sequences.

The domain of four-dimensional generalized difference matrix on some double sequence spaces , , , and has been introduced and studied by Tu and Başar [9] and by Tu [10]. They examined some topological properties. Furthermore, they determined the , , and duals of some new double sequence spaces and characterized some classes of four-dimensional matrix transformations related to the new double sequence spaces.

The domain of the four-dimensional generalized difference matrix in the spaces of almost null and almost convergent double sequences has been recently introduced and studied by Tu [11, 12]. He examined some topological properties and determined , , and duals of the space . Moreover, he characterized some new classes of four-dimensional matrix mappings related to the sequence space .

In this work, we tried to fill one of the gaps in the four-dimensional matrix characterization of the matrix classes , , and in both cases of and . The characterization of the matrix classes , , , and and calculating the dual of the space where are still open problems for readers.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The author declares here that there are no conflicts of interest.

#### Authors’ Contributions

The author characterized some new four-dimensional matrix classes and proved the necessary and sufficient conditions. In the last section, some studies were summarized and some open problems were given by the author. The author read and approved the final manuscript.

#### Acknowledgments

The author would like to thank Professor Dr. Feyzi Başar for his careful reading and valuable suggestions on the first version of this paper which improved the presentation and readability. The author also would like thank to the audience of seminar sessions which was held by Mathematics Education Department of Ishik University for their valuable suggestions in order to improve the quality of this study. This work was supported by the Research Center of Ishik University, Erbil-IRAQ.

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