Journal of Mathematics
Volume 2020 (2020), Article ID 1608631, 7 pages
https://doi.org/10.1155/2020/1608631

Research Article

Multiplication Operators on Weighted Nakano (sss)

Awad A. Bakery,1,2 Afaf R. Abou Elmatty,² and OM Kalthum S. K. Mohamed1,3

¹University of Jeddah, College of Science and Arts at Khulis, Department of Mathematics, Jeddah, Saudi Arabia
²Department of Mathematics, Faculty of Science, Ain Shams University, P.O. Box 1156, Cairo 11566, Abbassia, Egypt
³Academy of Engineering and Medical Sciences, Department of Mathematics, Khartoum, Sudan

Correspondence should be addressed to OM Kalthum S. K. Mohamed; om_kalsoom2020@yahoo.com

Received 19 July 2020; Accepted 15 September 2020; Published 30 September 2020

Academic Editor: Kit C. Chan

Copyright © 2020 Awad A. Bakery 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.

Abstract.

In this article, we investigate the sufficient conditions on weighted Nakano sequence space to be premodular Banach (sss). We examine some topological and geometrical structures of the multiplication operators defined on weighted Nakano prequasi-normed (sss).

1. Introduction

Summability is very important in mathematical models and has numerous implementations, such as normal series theory, approximation theory, ideal transformations, and fixed point theory. For more details, see [1–4]. By , c, , , and co, we signify the spaces of each, convergent, bounded, r-absolutely summable, and convergent to zero sequences of complex numbers. indicates the set of nonnegative integers. Let  =  and be sequences of positive reals, and the weighted Nakano sequence space [5] is defined by

where . And is a Banach space, where

When , we have

(1)If , for all n ∈, then will reduce to (see [6, 7])(2)If , for all n ∈, then will reduce to which is studied by many authors (see [8–10])

By , we will indicate the set of every operators which are linear and bounded between Banach spaces W and Z, and if W = Z, we write . The multiplication operators have a wide field of mathematics in functional analysis, for instance, in eigenvalue distributions theorem, geometric structure of Banach spaces, and theory of fixed point. For more details, see [11–16]. The s-numbers [17] have many examples such as the rth approximation number, denoted by αr(V), is defined by αr(V) = inf {: B ∈ and rank(B) ≤ r}, and the rth Kolmogorov number, denoted by dr(V), is defined by dr(V) = infdimw ≤ r sup ≤1 inf ∈W. The following notations will be used in the sequel:

A few of operator ideals in the class of Hilbert spaces or Banach spaces are defined by distinct scalar sequence spaces, such as the ideal of compact operators formed by (dr (V)) and c₀. Pietsch [4] studied the quasi-ideals for , the ideals of Hilbert Schmidt operators between Hilbert spaces constructed by , and the ideals of nuclear operators generated by . He explained that for r ∈ [1, ∞), where is the closed class of all finite rank operators, and the class became simple Banach and small [1]. The strictly inclusion , whenever j > r > 0, W and Z are infinite-dimensional Banach spaces investigated through Makarov and Faried [2]. Faried and Bakery [18] gave a generalization of the class of quasi-operator ideal which is the prequasi-operator ideal, and they examined several geometric and topological structure of and (ces(r))S. On sequence spaces, Mursaleen and Noman [19, 20] investigated the compact operators on some difference sequence spaces. The multiplication operators on (ces(r), ||.||) with the Luxemburg norm ||.|| elaborated by Komal et al. [21] and Bakery and Abou Elmatty [5] gave the sufficient (not necessary) conditions on such that constructed a simple Banach prequi operator ideal. The prequasi-operator ideal became small and strictly contained for different weights and powers. The aim of this article to explain some results of equipped with the prequasi-norm . Firstly, we give the sufficient conditions on weighted Nakano sequence space to form premodular Banach (sss). Secondly, we give the necessity and sufficient conditions on weighted Nakano sequence space equipped with the prequasi-norm such that the multiplication operator defined on it is bounded, approximable, invertible, Fredholm, and closed range operator.

2. Preliminaries and Definitions

Definition 1 (see [4]). An operator is called approximable if there are Dr ∈, for every r ∈ and limr⟶∞ ||V—Dr|| = 0.
By and , we will indicate the space of all approximable and compact operators from W to Z, respectively.

Theorem 1 (see [4]). If W is Banach space with dim (W) = ∞, then .

Definition 2 (see [22]). An operator V∈ is called Fredholm if dim (R(V))c < ∞, dim (kerV) < ∞, and R(V) is closed, where (R(V))c denotes the complement of range V.
The sequence ej = (0, 0, 1, 0, 0, ...) with 1 in the jth coordinate, for every j ∈, will be used in the sequel.

Definition 3 (see [3]). The space of linear sequence spaces Y is called (sss)

(1)If er ∈ Y with r ∈,(2)Let u = (ur) ∈,  = ()∈ Y, and |ur|≤||, for every r ∈, then u ∈ Y. This means Y be solid,”(3)If (ur) , then ∈Y, wherever means the integral part of

Definition 4 (see [23]). A subspace of the (sss) is called a premodular (sss) if there is a function:  ⟶ [0, ∞) verifying the conditions:

(i)τ(y) ≥ 0, for each y ∈ Y and τ(y) = 0 ⇔ y = θ, where θ is the zero element of Y(ii)There exists a ≥ 1 such that τ(ηy) ≤ a|η| τ(y), for all y ∈ Y, and η ∈(iii)For some b ≥ 1, τ(y + z) ≤ b(τ(y) + τ(z)), for every y, z ∈ Y(iv)|yr| ≤ |zr| with r ∈, implies τ((yr)) ≤ τ((zr))(v)For some b0 ≥ 1, τ ((yr)) ≤ τ() ≤ b0τ ((yr))(vi)If y = ∈ Y, and d > 0, then there is r0 ∈ with τ() < d(vii)There is t > 0 with τ (ν, 0, 0, 0, ...) ≥ t|ν| τ (1, 0, 0, 0, ...), for any ? ∈

Definition 5 (see [23]). The (sss) is called prequasi-normed (sss) if τ satisfies the Parts (i)–(iii) of Definition 4, and when the space Y is complete under τ, then is called a prequasi-Banach (sss).

Theorem 2 (see [23]). A prequasi-norm (sss) , whenever it is premodular (sss).

The inequality [24]: where  ≥ 0, for all i ∈ and , will be used in the sequel.

3. Main Results

3.1. Prequasi-Norm on

We investigate the sufficient conditions on the sequence space equipped with prequasi-norm to form prequasi-Banach and closed (sss).

Theorem 3. , where , for all x ∈, is a premodular (sss), if the following conditions are satisfied:

(a1)The sequence ∈∩ is increasing(a2)Either (βn) is a monotonic decreasing or monotonic increasing such that there is C ≥ 1 for which β2n + 1 ≤ Cβn

Proof. Firstly, we have to prove is a (sss):

(1-i)Let x, y ∈. Since is bounded, we obtain

then x + y ∈.

(1-ii)Let λ ∈ and x ∈. Since is bounded, we have

(i)Then, λ x ∈ Therefore, by using Parts (1-i) and (1-ii) we have that the space is linear. Also, en ∈, for all n ∈, since

(2)Let , for all n ∈ and y ∈. Since βn > 0, for all n ∈, then τ(x) =  ≤  = τ(y) < ∞, we get x ∈.(3)Let ∈ and (βn) be an increasing sequence. There exists C ≥ 1 such that β2n + 1 ≤ Cβn and be an increasing; then, we have

then ∈. Secondly, we show that the functional τ on is a premodular:

(i)Evidently, τ (x) ≥ 0 and τ (x) = 0 ⇔ x = θ(ii)There is a steady a = max {1, supn } ≥ 1 such that τ (λx) ≤ a|λ| τ (x), for all x ∈and λ ∈(iii)There exists K ≥ 1 such that τ (x + y) ≤ K (τ(x) + τ(y)), for all x, y ∈(iv)Clearly, from the proof part (2) of Theorem 3, the condition is clear since the weighted Nakano sequence space is solid(v)It is obtained from (3) that  ≥ 1(vi)It is clear that  =  (vii)There exists a steady 0 <  ≤ , for λ  0 or  > 0, for λ = 0 such that τ (λ, 0, 0, 0, ...)   |λ| τ (1, 0, 0, 0, ...)

Theorem 4. If conditions (a1) and (a2) are satisfied, then , where for all x ∈, is a prequasi-Banach (sss).

Proof. Let the conditions be verified. By Theorem 3, the space is a premodular (sss). Therefore, from Theorem 2, the space is a prequasi-normed (sss). To prove that is a prequasi-Banach (sss), suppose xn =  be a Cauchy sequence in , then for every ε ∈ (0, 1), there exists a number ∈ such that, for all n, m ≥ , one has

Hence, for n, m ≥  and j ∈, we get  < ε.
So, is a Cauchy sequence in for fixed j ∈, and this gives for j ∈. Hence, τ (xn−x₀) < ε, for all n ≥ . Finally, to prove that x₀∈, we have τ (x₀) = τ (x₀ − xn + xn) ≤ τ(xn–x₀) + τ(xn) < ∞.
So, x₀∈. This means that is a prequasi-Banach (sss).

Theorem 5. If conditions (a1) and (a2) are satisfied, then , where , for all x ∈, is a prequasi-closed (sss).

Proof. Let the conditions be verified. By Theorem 3, the space is a premodular (sss). Therefore, from Theorem 2, the space is a prequasi-normed (sss). To prove that is a prequasi-closed (sss), suppose ∈ and limn ⟶ ∞ , then for every ε ∈ (0, 1), there exists a number ∈ such that for all n ≥ , one has

Hence, for n ≥  and j ∈, we get  < ε.
So, is a convergent sequence in for fixed j ∈, and this gives limm ⟶ ∞ for fixed j ∈. Finally, to prove that x₀∈, we have τ (x₀) = τ(x₀ − xn + xn) ≤ τ(xn − x₀) + τ(xn) < ∞.
So, x₀∈. This means that is a prequasi-closed (sss).

3.2. Bounded Multiplication Operator on

Here after, we investigate some topological and geometric structure of the multiplication operator acting on . In this section, we examine the sufficient conditions on the sequence space equipped with prequasi-norm τ such that the multiplication operator defined on is bounded and isometry.

Definition 6. Let κ ∈∩ ℓ∞ and Wτ be a prequasi-normed (sss). An operator Vκ: Wτ ⟶ Wτ is called multiplication operator if Vκ = κ  = ∈ W, for all ∈W. If Vκ ∈, we name it a multiplication operator generated by κ.

Theorem 6. Let κ ∈ and conditions (a1) and (a2) be satisfied; then, κ ∈ ℓ∞, if and only if , where , for all x ∈.

Proof. Let the conditions be satisfied. Assume κ ∈ ℓ∞. Therefore, there is ε > 0 with || ≤ ε, for every r ∈. For x ∈, since is bounded from above with  > 0, for all r ∈, then

This gives Vκ∈. Conversely, assume that Vκ∈. Let us assume κ ∉ ℓ∞; hence, for each j ∈, there is ∈ such that  > j. Since ∈∩ ℓ∞ is increasing, we have

This shows that Vκ∈. Therefore, κ ∈ ℓ∞.

Theorem 7. Pick up κ ∈ and be a prequasi-normed (sss), with τ(x) = , for all x ∈. Then,  = 1, for all r ∈, if and only if, Vκ is an isometry.

Proof. Suppose |κr| = 1, for all r ∈. Hence,

for all x ∈. Therefore, Vκ is an isometry. Conversely, Assume that |κi| < 1 for some i = i0. Since ∈∩ ℓ∞ is increasing, we obtain

When || > 1, we can prove that  > . Therefore, in both cases, we have a contradiction. So, |κr| = 1, for every r ∈.

3.3. Approximable Multiplication Operator on

In this section, we introduce the sufficient conditions on the sequence space equipped with prequasi-norm τ such that the multiplication operator defined on is an approximable and compact. By card (A), we indicate the cardinality of the set A.

Theorem 8. If κ ∈ and is a prequasi-normed (sss), where τ(x) = , for all x ∈, then, Vκ∈ if and only if ∈ c₀.

Proof. Assume that Vκ∈. Therefore, Vκ∈, to prove that the sequence belongs to c₀. Suppose ∉ c₀. Hence, there is δ > 0 such that the set  = {r ∈: |κr| ≥ δ} has card  = ∞. Assume ai ∈, for all i ∈. Hence, {: ai ∈ Aδ} is an infinite bounded set in . Let

for all ∈ Aδ. This shows {: ∈ Bδ} ∈ ℓ∞ which cannot have a convergent subsequence under Vκ. This proves that Vk ∉ . Then, Vk ∉, this gives a contradiction. So, limi ⟶ ∞ κi = 0. Conversely, let limi ⟶∞ κi = 0. Then, for each δ > 0, the set  = {i ∈: |κi| ≥ δ} has card (Aδ) < ∞. Hence, for every δ > 0, the space  = {x = (xi) ∈} is finite dimensional. Then, Vκ| is a finite rank operator. For every i ∈, dene κi∈ by (κ i)j = 
It is clear that has rank <∞ as dim  < ∞, for each i ∈. Therefore, since ∈∩ ℓ∞ is increasing, we obtain

This implies that and that Vκ is a limit of finite rank operators. Therefore, Vκ is an approximable operator.

Theorem 9. Let κ ∈ and be a prequasi-normed (sss), where τ(x) = , for all x ∈. Then, Vκ∈, if and only if, ∈ c₀.

Proof. It is simple so overlooked.

Corollary 1. If κ ∈, conditions (a1) and (a2) are satisfied, then , where τ(x) = , for all x ∈.

Proof. Since I is a multiplication operator on generated by κ = (1, 1, ...), therefore, I ∉ and I ∈.

3.4. Fredholm Multiplication Operator on

In this section, we give the sufficient conditions on the sequence space equipped with prequasi-norm such that the multiplication operator defined on has closed range, invertible, and Fredholm.

Theorem 10. If κ ∈, is prequasi-Banach (sss), where τ(x) = , for all x ∈, then κ is bounded away from zero on , if and only if, R(Vκ) is closed.

Proof. Let the sufficient condition be satisfied. Therefore, there is  > 0 with  ≥ , for all i ∈c, to show R(Vκ) is closed. Assume d be a limit point of R(Vκ). Therefore, there is Vκxi in , for all i ∈ such that lim i⟶∞ Vκxi = d. Obviously, (Vκxi) is a Cauchy sequence. Since ∈∩ ℓ∞ is increasing, one has

where

This shows that (yi) is a Cauchy sequence in . Since is complete, there is x such that limi ⟶ ∞yi = x. Since Vκ is continuous, then limi ⟶ ∞ Vκyi = Vκx. However, limi ⟶ ∞ Vκxi = lim i ⟶ ∞ Vκyi = d. Hence, Vκx = d. Therefore, d ∈ R(Vκ). This shows that R(Vκ) is closed. Conversely, let R(Vκ) be closed. Therefore, Vκ be bounded away from zero on . Hence, there exists  > 0 such that (Vκx) ≥  (x), for all x ∈.
Assume B = {r ∈: |κr|> }. If Bϕ, then for ∈ B, we obtain

and this gives contradiction. So, B = ϕ such that |κr| ≥ , for all r ∈. This completes the proof of the theorem.

Theorem 11. If κ ∈ and is a prequasi-Banach (sss), with τ =  , for all ∈, there are b > 0 and B > 0 such that b <|κr| < B, for all r ∈, if and only if, Vκ∈ is invertible.

Proof. Let the conditions be verified. Define γ ∈ CN by γr = . From Theorem 6, we have Vκ, Vγ∈ and Vκ.Vγ = Vγ. Vκ = I. Then, Vγ is the inverse of Vκ. Conversely, let Vκ be invertible. Hence, R (Vκ) = . This implies R (Vκ) is closed. From Theorem 10, there is b > 0 such that |κr| ≥ b, for all r ∈ (ker (κ))c. Now, ker (κ) = ϕ, else  = 0, for several ro∈, we get ∈ ker (Vκ). This gives a contradiction, since ker (Vκ) is trivial. So, |κr| ≥ b, for all r ∈. Since Vκ is bounded, hence from Theorem 6, there is B > 0 such that |κr| ≤ B, for all r ∈. Therefore, we have shown that b ≤ |κr| ≤ B, for all r ∈.

Theorem 12. If κ ∈ and is a prequasi-Banach (sss), where τ = , for all ∈, then Vκ ∈ is Fredholm operator, if and only if, (i) card (ker(κ)) < ∞ and (ii) |κr| ≥ , for all r ∈ (ker(κ))c.

Proof. Let Vκ be Fredholm. If card (ker(κ)) = ∞, hence en ∈ ker (Vκ), for all n ∈ ker(κ). Since en’s are linearly independent, this gives card (ker(Vκ)) = ∞. This is a contradiction. Therefore, card (ker(κ)) < ∞. By Theorem 10, condition (ii) is satisfied. Next, if the necessary conditions are verified, to show that Vκ is Fredholm. From Theorem 10, condition (ii) gives that R(Vκ) is closed. Condition (i) indicates that dim (ker(Vκ)) < ∞ and dim ((R(Vκ))c) < ∞. Therefore, Vκ is Fredholm.

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-02-054-DR. The authors, therefore, acknowledge with thanks the University technical and financial support.

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