A New RH-Regular Matrix Derived by Jordan’s Function and Its Domains on Some Double Sequence Spaces

Erdem, Sezer; Demiriz, Serkan

doi:https://doi.org/10.1155/2021/5594751

Journal of Function Spaces

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Abstract Introduction and Preliminaries Conclusions Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5594751 | https://doi.org/10.1155/2021/5594751

A New RH-Regular Matrix Derived by Jordan’s Function and Its Domains on Some Double Sequence Spaces

Sezer Erdem¹and Serkan Demiriz²

Academic Editor: Gen Qi Xu

Received08 Feb 2021

Revised26 Mar 2021

Accepted05 Apr 2021

Published19 May 2021

Abstract

In the present study, we introduce a new RH-regular 4D (4-dimensional) matrix derived by Jordan’s function and define double sequence spaces by using domains of 4D Jordan totient matrix on some classical double sequence spaces. Also, the -, -, and -duals of these spaces are determined. Finally, some classes of 4D matrices on these spaces are characterized.

1. Introduction and Preliminaries

The Jordans’s function , is described as the number of -tuples of positive integers all less than or equal to that form a coprime with -tuples together with , where and . is entitled as Jordan’s function after Camille Jordan, and it is one of many generalizations of the famous Euler-totient function . Since the equality holds for the coprime numbers , is multiplicative, and furthermore, it is an arithmetic function. If is the prime factorization of for , in that case,

It is also significant to say that the Euler-totient function is the special case of the Jordan’s function for . Other three important features of Jordan’s function areandwhere denotes the Möbius function (see [1]). Given any , the function is described as

If is the prime factorization of such that , in this fact, . For , the equality holds and , where are coprime.

A double sequence is a function such that , , where denotes any nonempty set. Any linear subspace of the linear space is named as double sequence space. Here, the symbol represents the complex field. is convergent in the Pringsheim’s sense provided that for all , there subsists with for . In that case, is named as the Pringsheim limit of and stated with . represents the space of all such which are called shortly as p-convergent. The space of all bounded double sequences is denoted by . Of particular interest is unlike single sequences, when , it might be . The bounded sequences which are also -convergent are indicated by , that is, . is entitled as regularly convergent if the limits , and , subsist, and the limits and subsist and are equivalent to the . The set of all regularly convergent double sequences is represented by , and also, , , and are Banach spaces with the norm . Moreover, and symbolize the spaces of all -convergent and bounded series, respectively. It should be noted that the double sequence space which was defined by Zeltser [2] is the special case of the space described byfor . It is also significant to advert that the space is Banach space with

The concept of almost convergence which is more general than the convergence of single sequences was acquainted by Lorentz [3], and later, present idea was conveyed and examined for double sequences in [4]. The spaces of all almost convergent and almost null double sequences denoted by and , respectively.

Consider that and describe as . In that case, the pair is entitled as double series. Here, the sequence is the sequence of partial sums of the double series.

Throughout this article, it will be used instead of , 4D instead of 4-dimensional, , , and . Double sequences , and described byand . If for or or both, is named as triangular matrix, and also, if , in that case, is called as triangle.

Let us consider and the 4D complex infinite matrix . If for all , subsists and is in for each , in that case, describes a matrix transformation from into , and it is written as . stands for the class of all 4D complex infinite matrices from into . In that case, if and only if and for all , where and .

The domain of a 4D complex infinite matrix in a double sequence space consists of the sequences whose -transforms are in ; that is,

The 2D Jordan matrix and its domain on the space of of absolutely -summable single sequences are described and examined by lkhan et al. [5]. The 4D Euler-totient matrix and domains of this matrix on double sequences , , , , and were described and examined by Demiriz and Erdem [6] and Erdem and Demiriz [7].

Suppose that , E(r,s), B(r,s,t,u), , , and denote the 4D Riesz, Euler, generalized difference, sequential band, Euler-totient, and binomial matrices, respectively. The domains of some 4D triangle matrices in a space are listed in Table 1.

2. The 4D Jordan Totient Matrix and Some Double Sequences

In this section, we define 4D Jordan totient matrix derived by Jordan’s function, and we show that this matrix is RH-regular. By using its domains on the spaces , , , , and for , we introduce the spaces , , , , and and also give inclusion relations related these newly described spaces. The reader can refer to the papers [15–30] and recent monograph [31] concerning with the domain of 4D triangles in the classical spaces of double sequences and related topics.

Now, we describe the 4D Jordan totient matrix as follows:for each . It should be noted that the 4D Euler-totient matrix defined by Demiriz and Erdem [6] is the special case of the matrix for . Moreover, it is clear that is a triangle, and the -transform of any is described as

The inverse of the triangle matrix is calculated as

In that case, is obtained by applying to (10) as

Definition 1 (see [32, 33]). For every , if and , then is called as RH-regular.

Lemma 2 (see [32, 33]). A 4D matrix is RH-regular if and only if

Theorem 3. The 4D Jordan totient matrix described by (9) is RH-regular.

Proof. Since () and () from the regularity of the 2D Jordan totient matrix [5], we obtain that (). Thus, RH-1 holds. From the equalitywe see that RH-2 holds. Sincethen for each and RH-3 holds. It can be shown similar with RH-3 that the 4D Jordan totient matrix satisfies RH-4. From the relation , we see that RH-5 holds. By using the relation and the positivity of the matrix , it is clear that the condition satisfies.

Now, we introduce the double sequence spaces by using the domains of 4D Jordan totient matrix as follows:

The sets which are newly defined above can be rewritten as , , and with the notation (8), where .

Theorem 4. The sets , , , , and are linear spaces, and the following statements hold:(i)The spaces , , and are Banach spaces withwhich are linearly norm isomorphic to , , and , respectively.(ii)The space is a complete seminormed space withwhich is linearly isomorphic to the space .(iii)For , is a complete -normed space withwhich is linearly -norm isomorphic to the space .(iv)For , is a Banach space withwhich is linearly norm isomorphic to the space .

Proof. Since the other items of the theorem can be proven with similar technique, we will pass the others except first one for the set .
It is left to the reader to show that is a normed linear space. To confirm the fact that is linearly norm isomorphic to , it is needed to sure the presence of any linear and norm keeping bijection from to . Let us describe the transformation such that for all . It is easy to see the linearity of . Furthermore, we can understand the injectivity of from the fact that . Let us suppose the sequences and whose terms areSo, we getwhich yields us that . Thus, it is obtained that , since . That is, is surjective and since the equality holds, keeps the norm. Moreover, is a Banach space, and we obtain the desired result from Section (b) of Corollary 6.3.41 in [34].
Now, we may give two theorems and a lemma related the inclusion relations.

Theorem 5. .

Proof. Consider . Since,it is obtained that , and thus, is subspace of .
Now, we give a lemma which is needed to prove our results.

Lemma 6 (see [35]). Suppose that is a 4D infinite matrix. Then, if and only if following conditions hold:

Theorem 7. is the subspace of , where .

Proof. To show the validity of the theorem, we just need to show that the 4D Jordan totient matrix satisfies the Lemma 6’s conditions.
From the equality , (24) holds. Since is RH-regular, then (25) holds. From the equality , (26) holds. For all , if we take , then for all and . Therefore, (27) holds. The condition (28) can be shown similarly with (27).

3. Duals

Duals of a are described by the following way:

Note that and if , then .

It is listed the spaces and their duals which are computed in this section in Table 2.

Theorem 8. The following statements hold:

Proof. (i)Since it can be proven by applying a similar way used in Theorem 3.1 in [7], we omit details(ii)For this part of the proof, it will be used contradiction method. Let us take a sequence . In that case, for all . Consider that . Thus, we obtain that , that is . Therefore, it must be and consequently the inclusion holdsConversely, let us take and . In this fact, we understand that from the equality (10), and therefore, . Thus, we obtain from the following inequalitythat and this completes the proof.

Let us state the following conditions which will be used to give the 4D martix classes (see [2, 8, 10, 22, 32, 35–37]).

It is listed necessary and sufficient conditions of with the following table.

Theorem 9. Consider the set defined byIf , then .

Proof. Let us describe the 4D matrix byIn this fact, by using the equality (12), we obtain thatThus, we conclude from the relation (35) that whenever if and only if whenever . This fact implies that if and only if . Therefore, by using the condition of the class for and in Table 3 with the matrix instead of , we obtain the desired consequence.

Theorem 10. Let us take , and the 4D matrix asIn that case,

Proof. (i)Let us take sequences and . In that case, with . So, from the equality (12),for all . Then, by considering the equality above, we deduce that whenever if and only if whenever . This leads us to the fact that if and only if , as claimed.(ii)This part can be easily proven similarly to the first part by using the description of the -dual. So, we omit the proof.Now, consider the sets which are defined bywhere the 4D matrix defined as in Theorem 10 and .

Theorem 11. The following statements hold:

Proof. From Theorem 10 and the conditions of the classes , , and in Table 3 with instead of , the accuracy of items (i), (ii), and (iii) can be easily seen, respectively.

Theorem 12. The following statements hold:(i)For , (ii)For ,

Proof. From Theorem 10 and the conditions of the classes , for , for and in Table 3 with instead of , the accuracy of items (i), (ii), (iii), and (iv) can be easily seen, respectively.

Theorem 13. The following statements hold:(i)For , (ii)For ,

Proof. From Theorem 10 and the conditions of the classes , , for , and for in Table 3 with instead of , the accuracy of items (i), (ii), (iii), and (iv) can be easily seen, respectively.

4. 4D Matrix Transformations

In the current part of the article, it will be given the classes and , where .

Theorem 14. Consider the 4D infinite matrices and whose elements are connected with the equalityIn that case; if and only if and , where .

Proof. Assume that . In that case, exists and is in for all such that , and it also implies that . By bearing in mind (12), we obtain thatfor all . In that case, if we take -limit on the equality (45) as , we have . Therefore, we obtain that whenever , that is .
Conversely, suppose that and . Let us take the sequence with . In that case, exists. By using the relation (12), one can derive from the relation (45) when passing -limit as thatwhich implies that . Therefore, , and this completes the proof.

Corollary 15. Consider the 4D infinite matrices and whose elements are connected with the relation (44). In that case, the necessary and sufficient conditions for the classes can be read from the Table 4.

Lemma 16 (see [8]). Let ,, be a 4D matrix and also be a 4D triangle. In that case, if and only if .
Now, let us define the 4D matrix byand give the following corollary.

Corollary 17. Consider the 4D infinite matrices and whose elements are connected with the relation (47). In that case, the necessary and sufficient conditions for the classes can be read from Table 5.

5. Conclusions

As we have stated some of them in Table 1, double sequence spaces which are obtained by using the domains of triangular matrices have been studied many times by many authors in the past. In the light of these and similar studies, in the current study, we described a new RH-regular and triangular 4D matrix derived by famous Jordan’s function which is an arithmetic function and introduced some double sequence spaces by using this matrix on some classical double sequence spaces. Moreover, we investigated their some properties and inclusion relations related them, computed duals, and characterized some matrix classes. We conclude that the results obtained from the 4D Jordan totient matrix is more general and extensive than the existent results of the authors [6, 7]. We expect that our results might be a reference for further studies in this field.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright

Copyright © 2021 Sezer Erdem and Serkan Demiriz. 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.

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