Some Properties of Prequasi Normed Generalized de La Vallée Poussin’s Mean Sequence Space

Bakery, Awad A.; Mohammed, Mustafa M.

doi:https://doi.org/10.1155/2020/5406134

Journal of Function Spaces

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 5406134 | https://doi.org/10.1155/2020/5406134

Some Properties of Prequasi Normed Generalized de La Vallée Poussin’s Mean Sequence Space

Awad A. Bakery^1,2and Mustafa M. Mohammed^1,3

Academic Editor: Raúl E. Curto

Received09 Dec 2019

Revised24 Feb 2020

Accepted16 Apr 2020

Published11 May 2020

Abstract

In this article, we study some topological properties of the multiplication operator on generalized de La Vallée Poussin’s mean sequence space equipped with the prequasi norm and the prequasi operator ideal generated by -numbers and this sequence space.

1. Introduction

All through the article, by , we signify the space of all bounded linear transformations between arbitrary Banach spaces and ; if , we mark . We will denote , , and for the space of all finite rank, approximable, and compact transformations on , respectively. The set is composed of nonnegative integers () and real numbers (). Moreover, , , , , , and will denote the space of complex, bounded, null, r-th -numbers [1], r-th Kolmogorov, and approximation sequences, respectively, where . The multiplication operators and operator ideals have a wide field of mathematics in functional analysis, for instance, in eigenvalue distributions theorem, fixed point theorem, geometry of Banach spaces, and spectral theory. Singh and Kumar [2] investigated the connection between the composition operators and the multiplication operators on -spaces. They proved the equivalence between the composition operator and , where is the multiplication operator induced by . The roots of the multiplication operators are contained in the spectral theory. According to Hilbert space theorem, there is a unitary equivalence between a multiplication operator and each normal operator on a separable Hilbert space. For more subtleties on multiplication operators, see [2–7]. In view of sequence spaces, Mursaleen and Noman [8] investigated some difference sequence spaces under compact matrix transformation. The multiplication operators belong to , where is an Orlicz function, studied by Komal and Gupta [9]. Furthermore, Komal et al. [10] explained the multiplication operators in , for with the Luxemburg norm . In the operator ideal theory, the scalar sequence space is sometimes used to define operator ideals in the class of Banach spaces or Hilbert spaces, since the ideal is generated by and . A Banach space is called simple [11], if there is one and only one nontrivial closed ideal in . The quasi-ideals , where , is investigated by Pietsch [11]. Also, he proved that and generated the ideals of Hilbert Schmidt operators between Hilbert spaces and of nuclear operators, respectively. He also investigated the sufficient condition on so that and is simple Banach space. Pietsch [12] studied the smallness of . For any infinite dimensional Banach spaces and , Makarov and Faried [13] gave the sufficient conditions for which is strictly contained in , for every . Bakery [14] introduced some properties of , where is the generalized de La Vallée Poussin’s mean sequence space. Some properties of -type Orlicz-Lorentz sequence spaces are examined by Mohiuddine and Raj [15]. Faried and Bakery [16] introduced a generalization of the usual classes of operator ideal which is prequasi operator ideal. They explained some results of and . We introduce in this paper the concept of prequasi norm on , give the conditions on equipped with the prequasi norm to construct Banach space, and study the necessity and sufficient conditions on so that the multiplication operator defined on is bounded, invertible, approximable, closed range, and Fredholm operator. We investigate the sufficient conditions on the class to form small, simple, closed Banach prequasi operator ideal. Also, we examine the strict inclusion relation between and (the class of all bounded linear operators whose sequence of eigenvalues belongs to ).

2. Definitions and Preliminaries

We will denote , with 1 in the position for every .

Lemma 1 (see [11]). For , if , then there are operators and with for each .

Theorem 2 (see [11]). If is a Banach space with , then

Definition 3 (see [17]). An operator is called Fredholm if it satisfies , , and has closed range, where denotes the complement of the range .

Let , where , , , for all , and with , for all . Simsek et al. [18] defined the generalized de La Vallée Poussin’s mean sequence space as follows: and , for . The space , where is a Banach space. When , it is clear that

For more details on , see [19, 20].

Remark 4. (1)If , for every , then examined by Sanhan and Suantai [21](2)If and , for all , then . Some authors [22–24] investigated various sorts of Cesáro summable sequence spaces

Definition 5 (see [25]). A class of linear sequence spaces is called a special space of sequences (sss) if (1) for each (2)for , and for all , then (3)for , then , wherever means the integral part of

Theorem 6 (see [14]). is a (sss), if
(a1) is increasing and with
(a2) , where , , , for all , and are satisfied

Definition 7 (see [25]). A subclass of (sss) is called a premodular (sss) if there is a function verifying the conditions (i) for all and , here is the zero element of (ii)there is with for each , and (iii)there is , for all (iv)for for all , then (v)for some , (vi)for all and , there is with (vii)there is such that for each

Theorem 8 (see [14]). If conditions (a1) and (a2) are verified, then is a premodular (sss), with for each . The usual classes of operator ideal generalized to the prequasi operator ideal.

Definition 9 (see [25]). A function is called a prequasi norm on the ideal if it satisfies the following: (1)If , then and if and only if (2)There is with , for each and (3)There is such that , for every (4)There is , if , and then , where and are normed spaces

Notations 10 (see [16]).

Theorem 11 (see [16]). Let be a premodular (sss), then the function be a prequasi norm on .

During this paper, with and the inequality [26] , where , , and for all , will be used.

3. Main Results

We introduce a generalization of the usual norm, the concept of prequasi norm on . We investigate the sufficient conditions on equipped with a prequasi norm to form Banach space.

Definition 12. Assume is (sss). If there is a function satisfying the conditions (i) for all and (ii)there is with for every , and (iii)there is , we have for each

The space is called prequasi normed (sss). If is complete with , hence is called a prequasi Banach (sss). We express the following two theorems without verification, since they are clear.

Theorem 13. is a prequasi norm (sss), if it is quasi norm (sss).

Theorem 14. Every premodular (sss) is prequasi normed (sss).

Theorem 15. If conditions (a1) and (a2) are verified, then is a prequasi Banach (sss), with for all .

Proof. Let the conditions be satisfied, then from Theorem 8, the space is premodular (sss). From Theorem 14, we have which is a prequasi normed (sss). To show that is a prequasi Banach (sss), assume be a Cauchy sequence in . Therefore, for all , there is such that for all , one has Hence for and , we get So is a Cauchy sequence in for fixed ; this gives for fixed . Hence, , for all . Finally, to prove that , we have so . This completes the proof.

Corollary 16. Let , then be a prequasi Banach (sss), with for all .

4. Multiplication Operator on Prequasi Normed (sss)

We define here a multiplication operator on with a prequasi norm and investigate the sufficient and necessary conditions on the multiplication operator to become bounded, closed range operator, approximable, invertible, and Fredholm.

Definition 17. If , and is a prequasi normed (sss). An operator is called multiplication operator if , where , for all . When , is called the multiplication operator induced by .

Theorem 18. Let , the conditions (a1) and (a2) be satisfied, hence if and only if , with for every .

Proof. Let the conditions be satisfied and . Therefore, there is with , for each . For , we have where ; this implies that Conversely, let to show that . For, if , hence for all , there are with . We have This shows that . So, .

Theorem 19. If and is a prequasi normed (sss), with for each . Hence, for each if and only if is an isometry.

Proof. Suppose , for every . Therefore, for all . Hence, is an isometry. Conversely, let for some . Then, Also, if , then we get . Therefore, we have a contradiction in the two cases. So, , for each .

Theorem 20. If , the conditions (a1) and (a2) are satisfied and , where for all . Then, if and only if .

Proof. Let be an approximable operator. Hence, . To prove that , assume ; hence, there is with which is an infinite set. Suppose be in . Then, is an infinite bounded set in . We have for all . This proves which cannot have a convergent subsequence under . Therefore, this gives , so ; this implies a contradiction. Therefore, . Conversely, suppose . Hence, for all , the set is finite. We have which is a finite dimensional space for all . Hence, is a finite rank operator. Define as follows It is clear that since is a finite rank operator, then the space is finite dimensional for each . Therefore, by using conditions (a1) and (a2), we have This proves that . So, is a limit of finite rank operators. This finishes the proof.

Theorem 21. If , the conditions (a1) and (a2) are satisfied and , where for all . Then, if and only if .

Proof. It is clear so omitted.

Corollary 22. If conditions (a1) and (a2) are satisfied, we have with for each .

Proof. Since the multiplication operator induced by is , therefore .

Theorem 23. Let be a prequasi Banach (sss), with and for each . Hence, is bounded away from zero on if and only if has closed range.

Proof. Let the sufficient conditions be satisfied. Therefore, there is such that , for every . To show that is closed, suppose be a limit point of . Hence, there is in , for all such that . Clearly, the sequence is a Cauchy sequence. Therefore, since is nondecreasing, one can see where Hence, is a Cauchy sequence in since is complete. So, there is with . Since is continuous, hence . But . Therefore, . Hence, . This gives is closed. Let the necessary condition be satisfied; this means be closed. Therefore, is bounded away from zero on , i.e., there is such that , for all . Let . If , then for , since ϱ is nondecreasing, we have This gives a contradiction. Therefore, with for each . This completes the proof.

Theorem 24. If and is a prequasi Banach (sss), with for each . Hence, there is and such that ; for each if and only if is invertible.

Proof. Assume that the sufficient condition be verified. Define by . Then by Theorem 18, and are bounded linear operators. Also, . Hence, is the inverse of . Conversely, suppose the necessary condition be satisfied. Therefore, . Hence, is closed. From Theorem 23, there is so that , for all . Now, ; otherwise, , for some , we have . Since is trivial, this gives a contradiction. Therefore, , for each . From Theorem 18, since is bounded, there is so that , for every . Therefore, we have showed that , for all .

Theorem 25. If is a prequasi Banach (sss), with and for every . Hence, be Fredholm operator if and only if (i) be a finite subset of . (ii) , for each .

Proof. Let be Fredholm. Assume be an infinite subset of , then , for each . This gives ; hence, we have a contradiction. Therefore, be a finite subset of . From Theorem 23, condition (ii) follows. On the other hand, let the necessary condition be satisfied. From Theorem 23, condition (ii) shows that is closed and condition (i) gives that and . This finishes the proof.

5. Closed Banach Prequasi Ideal

We study the sufficient conditions on so that the prequasi ideal are Banach and closed.

Theorem 26. Let and be Banach spaces, (a1) and (a2) be satisfied, then is a prequasi Banach operator ideal, where and for each .

Proof. From Theorem 8, the space is a premodular (sss). Hence, by using Theorem 11, we have the function that is a prequasi norm on . Suppose be a Cauchy sequence in , then by part (vii) of Definition 7 there is and since , we get Therefore, be a Cauchy sequence in . Since is a Banach space, hence there is with . Since for each , from parts (ii), (iii), (iv), and (v) of Definition 7, we have Therefore, ; hence, .

Theorem 27. Let and be Banach spaces, (a1) and (a2) be satisfied, then is a prequasi closed operator ideal, where and for each .

Proof. From Theorem 8, the space is a premodular (sss). Hence, by using Theorem 11, we have the function that is a prequasi norm on . Suppose for each and . Since and from part (vii) of Definition 7, we have Hence, is convergent in . Since for each , from parts (ii), (iii), (iv), and (v) of Definition 7, we obtain Therefore, . Hence, .

6. Small and Simple Prequasi Banach Operator Ideal

We introduce some inclusion relations concerning prequasi operator ideal constructed by the sequence of -numbers and .

Theorem 28. If , are infinite dimensional Banach spaces, , with , and for every , then

Proof. Assume the sufficient conditions be verified and . Therefore, . We have Hence, . By taking such that , one can find so that Hence, and . Obviously, . By choosing such that . We have and . This finishes the proof.

Corollary 29. If , are infinite dimensional Banach spaces and , it is true that We investigate the sufficient conditions so that the prequasi Banach operator ideal is small.

Theorem 30. The prequasi Banach operator ideal is small, whenever , are infinite dimensional Banach spaces and the conditions (a1) and (a2) are satisfied.

Proof. If the sufficient conditions are verified, therefore the space is a prequasi Banach operator ideal, with and . Suppose that , so there is so that for each . From Dvoretzky’s theorem [27] for all , we obtain subspaces of and quotient spaces transform onto by isomorphisms and with and . If is the natural embedding map from into and is the quotient map from onto , suppose denotes the Bernstein numbers [28], we have for . Now Therefore, for some . Since is an arbitrary, this gives a contradiction. Therefore, and when . This finishes the proof.

We introduce the next theorem without verification; these can be set up utilizing standard procedure.

Theorem 31. The prequasi Banach operator ideal is small, whenever , are infinite dimensional Banach spaces and the conditions (a1) and (a2) are satisfied.

Corollary 32. The prequasi Banach operator ideal is small, if , are infinite dimensional Banach spaces and .

Corollary 33. The prequasi Banach operator ideal is small, if , are infinite dimensional Banach spaces and .

We examine the sufficient conditions so that the prequasi Banach operator ideal is simple.

Theorem 34. If , are infinite dimensional Banach spaces, , with , and for each , it is true that

Proof. Let the sufficient conditions be given, and . From Lemma 1, we have and with . For each , we obtain In view of Theorem 28, we get a contradiction. Therefore, .

Corollary 35. If , are infinite dimensional Banach spaces, , with , and for each , it is true that

Theorem 36. The prequasi Banach operator ideal is simple, if , are infinite dimensional Banach spaces and the conditions (a1) and (a2) are satisfied.

Proof. Let the sufficient conditions be given and there is with . From Lemma 1, we have so that . This means that . Consequently, . Therefore, is simple.

7. Eigenvalues of s-Type Operators

We investigate the strict inclusion relation between and (the class of all bounded linear operators whose sequence of eigenvalues belongs to ).

Notations 37. , where .

Theorem 38. If , are infinite dimensional Banach spaces and conditions (a1) and (a2) are satisfied, it is true that

Proof. Let the sufficient conditions be verified and . Therefore, . Hence, we have Hence, , so Assume exists for every . Therefore, exists and bounded for every . This gives exists and bounded. From the prequasi operator ideal of , we obtain We have a contradiction, since . Therefore, for every . Hence, is the eigenvalues of . Finally, if we choose such that , we find such that and if we take such that Hence, and . This finishes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was funded by the University of Jeddah, Saudi Arabia (under grant no. UJ-02-058-DR). The authors, therefore, acknowledge with thanks the university technical and financial support.

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Copyright © 2020 Awad A. Bakery and Mustafa M. Mohammed. 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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