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Tuğ, Orhan; Malkowsky, Eberhard; Hazarika, Bipan; Yaying, Taja

doi:https://doi.org/10.1155/2022/6832559

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2022 | Article ID 6832559 | https://doi.org/10.1155/2022/6832559

On the New Generalized Hahn Sequence Space

Orhan Tuğ,¹Eberhard Malkowsky,²Bipan Hazarika,³and Taja Yaying⁴

Academic Editor: Jaume Giné

Received30 Jun 2022

Accepted25 Aug 2022

Published16 Sept 2022

Abstract

In this article, we define the new generalized Hahn sequence space , where is monotonically increasing sequence with for all , and . Then, we prove some topological properties and calculate the , , and duals of . Furthermore, we characterize the new matrix classes , where , and , where . In the last section, we prove the necessary and sufficient conditions of the matrix transformations from into , and from into .

1. Introduction and Basic Notations

The set of all complex valued sequences is denoted by and each vector subspace of is called a sequence space. The sets , , and are bounded, convergent and null sequence spaces, respectively. Moreover, , , , , and are the spaces of all absolutely summable, absolutely summable, convergent series, null series, bounded series and sequeences of bounded variation, respectively, where .

The alpha-dual , beta-dual and gamma-dual of a sequence space are defined by

Let be an infinite matrix and . We write and then we say that defines a matrix transformation from into as if for every . We denote the set of all infinite matrices that map the sequence space into the sequence space by . Thus, if and only if the right side of (2) converges for every , that is, for all and we have for all . The set is called the domain of the matrix in It is known that the matrix domain is also a sequence space. The readers may refer to these nice papers [1–3] and the textbooks [4–8] concerning domain of special matrices in classical sequence spaces and the theory of summability.

If a normed sequence space contains a sequence with the following property that for every there is a unique sequence of scalars such that then is called a Schauder basis for . The series which has the sum is then called the expansion of with respect to and written as .

If is an -space, and is a basis for then is said to have property, where is a sequence whose only term in place is the others are zero for each and . If is dense in , then is called -space, thus implies .

The sequence space defined by is called Hahn sequence space, named after its introducer H. Hahn [9]. The space is a space with the norm

Rao [10] proved that the space is a space with with respect to the norm

Later on Goes [11] introduced a generalised Hahn sequence space with for all defined by

Quiet recently a scientific study of a more generalized Hahn sequence space is carried out by Malkowsky et al. [12], defined as follows:

The authors proved that the space is a space with with respect to the norm where is an unbounded and monotonic increasing sequence of positive reals. Besides, the authors stated and proved various significant results concerning characterization of matrix transformations between the space and classical spaces, and characterization of compact operators on the space using Hausdorff measure of non-compactness. We refer to [2, 10, 13–23] and the survey paper [22] for more studies and results related to Hahn sequence space.

2. The New Generalized Hahn Sequence Space

In this section, we introduce the new generalized Hahn sequence space as follow where is an unbounded monotone increasing sequence of positive real numbers with for all , , and .

Remark 1. It is clear that if we consider , then becomes .
If we consider where

Moreover, if we write the terms of starting from to as follow and then when we get sum . Then we have whenever Equivalently, the sequence may also represented by where the matrix is defined by

Clearly

Theorem 2. The new Hahn sequence space is a linear complete normed space space with the norm defined as

Proof. The linearity is clear. Now, we observe that the is a Banach space.
(N1) if since for each , , that is, for all , and implies for all , that is, .
(N2) For every and , (N3) For every , Now we should show that the space is complete. Suppose that be a Cauchy sequence in the space . Therefore, for every , there exists a natural number such that Since the Cauchy sequence is in the space , we obtain Now we can write the following inequality for fixed , by considering the inequality for that Then we have the following result since for all Therefore, is a Cauchy sequence of Complex numbers for each . Since is complete, then converges to an arbitrary sequence as , that is, for every there exists such that for each . Therefore, we have the following with an arbitrary that hence, by letting limit as over (24) we obtain Moreover, we obtain by (23) for any and which says that . Furthermore, So, it shows that .

Corollary 3. The Hahn sequence space is a space.

Theorem 4. The space does not have property.

Proof. If the sequence space has property, then every sequence is the limit of its section , i.e., Let us consider and for all . Then obviously and, since , that is, . But, if we write for every as the section of the sequence , then we obtain for all Hence, as . It completes the proof.

Theorem 5. The space is linearly isomorphic to the space

Proof. Let us define the mapping by Linearity of is obvious. Since the triangle is the matrix representation of therefore is invertible too. Now, let and consider the following equality: This implies that is onto and preserves the norm. Consequently defines an isomorphism from to This completes the proof.

Theorem 6. Let us define a sequence of elements of the space for every fixed by Then the sequence is a basis for the space and any has unique representation of the form where for all and .

3. , And Duals of

In this section, we first calculate the dual of the space and then we give some basic lemmas to prove the , and duals of the space . The following calculations on the dual of the space has been stated by Goes ([11], 4.1 Theorem, p. 485) and it has recently been proved by Malkowsky et al. [12], Proposition 2.3., p.5) as

Moreover, Goes ([11], 4.2 Corolary (b)) showed the dual of the space as follows:

Now we can calculate the dual fo the space .

Lemma 7.

Proof. Suppose that and be given. Then we have the partial sum of the series as where the infinite matrix is defined as It gives that , that is, whenever if and only if whenever . Therefore, if and only if . By Lemma 10(d) and since we have that This completes the proof.

Now we have the following useful Lemmas for the further computations.

Lemma 8 (see [12], Theorem 11., p.8). We have (a) if and only if(b) if and only if (39) holds, and the limits

Lemma 9 (see [12], Theorem 13., p.9). We have if and only if

Lemma 10 (see [4], Theorem 9.7.3, p.356). Let , . Then we have (a) if and only if(b) if and only if (40) holds and(c) if and only if (43) holds(d) if and only if

Theorem 11. Let , . Then

Proof. Suppose that be given and consider the following equation that where the matrix is defined as for all . It gives us that whenever if and only if whenever . Therefore, whenever if and only if . Therefore, the condition in (42) of Lemma 10(a) holds with instead of , that is, It gives the dual of the space .

Theorem 12.

Proof. Suppose that and are given. Let us consider the following equality. where the matrix is defined as in (37) for all . Then we can say that whenever if and only if whenever . Therefore, if and only if . Then the condition in Lemma 10(b) holds with instead of , that is Thus, we can obtain that . It is clearly seen that and then as we desired.

Theorem 13.

Proof. Since we have similar pattern with Theorem 12, we can conclude the proof with the following computation only.
Let us say that whenever if and only if whenever . Therefore, if and only if . Then the condition in Lemma 10(c) holds with instead of , that is Thus, we can obtain that as we desired.

4. Some Matrix Transformations from and into the Space

In this section, we characterize the classes , where , and , where .

Remark 14. If for every , then characterization of Theorem 17-Corollary 23 yields the characterisation of , where

Definition 15 (see [8], Definition 7.4.2). Let be a space. A subset of the set called a determining set for if is the absolutely convex hull of .

Proposition 16 (see [12], Proposition 3.2, p.8). Let Then is a determining set for .

Theorem 17. The infinite matrix if and only if

Proof. Suppose that is a determining set for the sequence space . The sequence space is space with . Let . Then Hence, which gives the condition in (55).

Proposition 18 (see [1], Lemma 5.3). Let be any two sequence spaces, be an infinite matrix and a triangle matrix. Then if and only if .

Corollary 19. The infinite matrix if and only if

Corollary 20. The infinite matrix if and only if

Theorem 21. The infinite matrix if and only if

Proof. The proof is similar to that of Theorem 17 with the norm replaced by the norm . Therefore, we have only the following norm

If we take supremum from both sides of above equality, then we can easily see the condition in (60).

Theorem 22. The infinite matrix if and only if the condition in (60) holds and

Proof. Suppose that . Then exists and is in for each . Thus, the partial sum of the series converges, that is the condition in (62) is immediate. The rest of the proof is similar to that of Theorem 17 by replacing the norm with the norm . Hence, we obtain the necessity of the condition in (4.4).

Corollary 23. The infinite matrix if and only if the conditions in (60) and (62) hold with for every .

Theorem 24. The infinite matrix if and only if

Proof. Suppose that be a determining set of and for each . Then the column of where Then,

Hence, the sequence is bounded in if and only if which shows the necessity of the condition in (64). Moreover, for all , then the condition in (63) is also immediate.

Corollary 25. The infinite matrix if and only if the conditions in (63) and (64) hold, and

Theorem 26. The infinite matrix if and only if where the matrix is defined as .

Proof. Suppose that and exists and is in for every . Thus, . Let be a sequence such that . If and , then . By ([8], Lemma 8.5.3(i)), we can say that . It says that . So the necessity of the conditions in (68) and (69) are immediate.

Theorem 27. The infinite matrix if and only if the condition in (68) holds and where the matrix is defined as .

Proof. The proof is similar to that of Theorem 26 with and by ([8], Lemma 8.5.3(i)), we can say that . So the necessity of the conditions in (68) and (70) are immediate.

Corollary 28. The infinite matrix if and only if the conditions in (67), (68), (70) hold.

Remark 29. If we consider for every , then the characterization of Theorem 17, Corollary 19-Corollary 23 yields the characterisation of , where , and the characterization of Theorem 24-Corollary 28 yields the characterisation of, where (see [18], p.13-16).

5. Matrix Transformations on the Hahn Sequence Space

In this section, we characterize matrix and where and , respectively. The following lemma is significant for our investigation:

Lemma 30 (see [1]). Matrix transformation between -spaces are continuous.

Theorem 31. if and only if

Proof. Let and This means that exists and is contained in the space for each Apparently This establishes the necessity of condition in (71).
We recall that or equivalently for each This leads us to the following equality for each Since and are spaces, Lemma 30 ascertains that there exists a positive number such that for all Therefore, by the aid of Hölder’s inequality together with (73) and Theorem 5, we get for all This establishes the necessity of condition in (72).
Conversely, assume that conditions in (71) and (72) hold and take any Then which implies that exists. Again applying Hölder’s inequality and the fact that we deduce that Consequently, This completes the proof.

Theorem 32. if and only if (71) and (72) hold, and for each there exists such that

Proof. Let Then exists and is contained in the space for all Since Thus the necessity of conditions in (71) and (72) follows from Theorem 31. Let us consider and define the matrix by for all Then by virtue of equality (73), we get that Since so This proves the necessity of the condition in (77).
Conversely, assume that the conditions in (71), (72) and (77) hold. Then for each which implies that exists for all Therefore we again obtained equality (73). Then, one can notice that the conditions in (71) and (77) correspond to the conditions in (43) and (40), respectively, with instead of This concludes that Hence

Replacing by in Theorem 32, we obtain the following corollary:

Corollary 33. if and only if (71) and (72) hold, and (77) also holds with for all

Theorem 34. if and only if (71) holds, and

Proof. This is similar to the proof of Theorem 31. Hence details are omitted.

Theorem 35. if and only if the condition in (71) holds and

Proof. Let and suppose that such that exists and is in the space for each . Thus, we clerly see that which proves the necessity of condition in (71).
By using the relation or equivalently for each between the terms of the sequences and , we reach the equality in (73) which can be written as in the following where the infinite matrix is defined as for each . Since and are spaces, we revisit Lemma 30 that there exists a positive real number such that for all . Therefore, by using Hölder’s inequality together with (73) and Theorem 5, we get for all . This shows the necessity of condition in (82). Moreover, since for every the necessity of condition in (81) is also seen.
Conversely, suppose that the conditions in (71), (81), and (82) hold. Let us take any . Then which implies that exists. The rest of proof follows the similar path to that of Theorem 31 with the following inequality Thus, . This completes the proof.

Since the proof of the following results can be done similary to that of Theorem 35, we give them without their proofs.

Corollary 36. The followings hold. (i)The infinite matrix if and only if the condition in (71) holds and(ii)The infinite matrix if and only if the condition in (71) holds and(iii)The infinite matrix if and only if the conditions in (71) and (88) hold, and

Theorem 37. if and only if

Proof. Since is a space by Corollary 3, we apply ([8], Example 8.4.1) to obtain that if and only if the columns of form a bounded set in .
We have Hence the set is bounded in if and only if , which gives the condition in (91). Moreover for all , which gives the condition in (87). This completes the proof.

Now we give the following corollaries without their proofs since the proofs are followed similarly to that of Theorem 37.

Corollary 38. The followings hold. (i)The infinite matrix if and only if the condition in (63) holds, and(ii)The infinite matrix if and only ifwhere the matrix is defined as . (iii)The infinite matrix if and only if the condition in (94) holds andwhere the matrix is defined as .

6. Conclusion

Most recently, Malkowsky at al [12] defined the generalized Hahn sequence space , where is an unbounded monotone increasing sequence of positive real numbers, and characterized several classes of bounded linear operators or matrix transformations from into , and also from into . Moreover, the norms of the corresponding bounded linear operators and the Hausdorff measure of non-compactness for the operators in the above classes, and one application given by a tridiagonal matrix to present a Fredholm operator from into itself were studied in [12].

Malkowsky [14] established the characterisations of the classes of bounded linear operators from the generalised Hahn sequence space into the spaces , and . Moreover, he proved estimates for the Hausdorff measure of noncompactness of bounded linear operators from into , and identities for the Hausdorff measure of noncompactness of bounded linear operators from to , and then he used these results to characterise the classes of compact operators from to and .

Dolianin-Deki and Gili [20] established the characterisations of the classes of bounded linear operators from the generalised Hahn sequence space into the spaces , and . Then they proved estimates for the Hausdorff measure of noncompactness of bounded linear operators from into , and identities for the Hausdorff measure of noncompactness of bounded linear operators from into , and then they used these results to characterise the classes of compact operators from to and .

In this article, we defined new generalized Hahn sequence space , where is an unbounded monotone increasing sequence of positive real numbers with for all , and . Then, we proved some topological properties and showed some inclusion relations. Moreover, we calculate the , , and duals of . Furthermore, we characterize the new matrix classes , where , and , where . In the last section, we prove the necessary and sufficient conditions of the matrix transformations from into , and from into .

Data Availability

No data were used to support this study.

Conflicts of Interest

The author(s) declare(s) that they have no conflicts of interest.

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Copyright © 2022 Orhan Tuğ et al. 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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