Abstract

In this article, the necessary conditions on s-type Orlicz generalized difference sequence space to generate an operator ideal have been examined. Therefore, the s-type Orlicz generalized difference sequence space which fails to generate an operator ideal has been shown. We investigate the sufficient conditions on this sequence space to be premodular Banach special space of sequences, and the constructed pre-quasi operator ideal becomes small, simple, closed, Banach space and has eigenvalues identical with its s-numbers.

1. Introduction

The operator ideals have a wide field of mathematics in functional analysis, for instance, in eigenvalue distribution theorem, geometric structure of Banach spaces, and theory of fixed point. By , , , , and , we denote the spaces of all, convergent, bounded, r‐absolutely summable and null sequences of complex numbers, respectively. indicates the set of nonnegative integers. Tripathy et al. [1] introduced and studied the forward and backward generalized difference sequence spaces:where , , or , withrespectively. When , the generalized difference sequence spaces reduced to defined and investigated by Et and Çolak [2]. If , the generalized difference sequence spaces reduced to defined and investigated by Tripathy and Esi [3]. While if and , the generalized difference sequence spaces reduced to defined and studied by Kizmaz [4].

Definition 1. (see [5]). The backward generalized difference is called an absolute nondecreasing; if for all , then .
An Orlicz function [6] is a function , which is convex, continuous, and nondecreasing with , for and , as . An Orlicz function [7] is said to satisfy -condition for all values of , if there exists a constant , such that . The -condition is equivalent to , for all values of and for . Lindentrauss and Tzafriri [8] utilized the idea of an Orlicz function to define Orlicz sequence space:where is a Banach space with the Luxemburg norm:Every Orlicz sequence space contains a subspace that is isomorphic to or , for some .
Let , where indicates the space of sequences with positive real numbers, and we define the Orlicz backward generalized difference sequence space as follows:where for , , and , for all . It is a Banach space, withWhen , then studied in [5]. Mohiuddine et al. [9] investigated the applications of fractional-order difference operators by constructing Orlicz almost null and almost convergent sequence spaces. Yaying et al. [10] examined sequence spaces generated by the triple band generalized Fibonacci difference operator. By , we will indicate the set of every operators which are linear and bounded between Banach spaces and , and if , we write . The s-numbers [11] have many examples such as the r-th approximation number, denoted by , which is defined by , and the r-th Kolmogorov number, denoted by , which is defined by . 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 and . Pietsch [11] 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 , where is the closed class of all finite rank operators, and the class became simple Banach and small [12]. The strictly inclusion , whenever , and are infinite dimensional Banach spaces investigated through Makarov and Faried [13]. Faried and Bakery [14] gave a generalization of the class of quasi-operator ideal which is the pre-quasi operator ideal, and they examined several geometric and topological structures of and . Başarir and Kara [15] studied the compact operators on some Euler -difference sequence spaces. İlkhan et al. [16] investigated the multiplication operators on Cesáro second-order function spaces. The point of this article to explain some results of equipped with a pre-quasi norm . Firstly, we give the necessary conditions on any s-type to give an operator ideal. Secondly, some geometric and topological structures of have been studied, such as closed, small, simple Banach and . A strictly inclusion relation of has been determined for different Orlicz functions and .

2. Preliminaries and Definitions

Definition 2. (see [11]). An operator is called approximable if there are , for every and .
By , we will indicate the space of all approximable operators from to . The sequence with 1 in the -th coordinate, for every , will be used in the sequel.

Lemma 1. (see [11]). Let . If , then there are and such that , for all .

Definition 3. (see [11]). A Banach space is called simple if includes one and only one nontrivial closed ideal.

Theorem 1. (see [11]). If is Banach space with , then

Definition 4. (see[14]). The space of linear sequence spaces is called a special space of sequences (sss) if(1) with ,(2)let , , and , for every , then . This means be solid, and(3)if , then , wherever means the integral part of .

Definition 5. (see [5]). A subspace of the (sss) is called a premodular (sss) if there is a function verifying the following conditions:(i) for each and , where is the zero element of (ii)There exists such that , for all and (iii)For some , , for every (iv) with , which implies that (v)For some , (vi)If and , then there is with (vii)There is with , for any The (sss) is called pre-quasi normed (sss) if satisfies Parts (i)–(iii) of Definition 5 and when the space is complete under , then is called a pre-quasi Banach (sss).

Theorem 2. (see [5]). A pre-quasi norm (sss) , whenever it is premodular (sss).
By , we will denote the class of all bounded linear operators between any pair of Banach spaces.

Definition 6. (see [5]). A class is called an operator ideal if every satisfies the following conditions:(i)(ii)The space is linear over (iii)If , , and , then , where and are Banach spaces

Definition 7. (see [5]). A pre-quasi norm on the ideal is a function which satisfies the following conditions:(1)For all , and if and only if (2)There is such that , for all and (3)There is such that , for all (4)There is such that if , , and , then

Theorem 3. (see [14]). The class is an operator ideal, if is a (sss).

Theorem 4. (see [14]). The function forms a pre-quasi norm on , whenever be a premodular (sss).
The inequality [17] , where for all , and , will be used in the sequel.

3. Main Results

We give the necessary conditions on s-type under such that forms an operator ideal.

Theorem 5. For . If is an operator ideal, then the following conditions are satisfied:(1)The set contains , the space of all the sequences with finite nonzero numbers(2)If and , then (3)For all and , then (4)The sequence space is solid

Proof. let be an operator ideal.(i)We have . Hence, for all , we have . This gives that . Hence, .(ii)The space is linear over . Hence, for each and , we have . This implies that(iii)If , , and , then , where and are arbitrary Banach spaces. Therefore, if , , and , then . In addition, . By using condition 3, if , we have . This means that is solid.We explain that for any backward generalized difference , the space is not operator ideal.

Theorem 6. The space is not operator ideal, where is an Orlicz function satisfying -condition and .

Proof. if we choose , , , and for or , otherwise, for all . We have , for all , , and . Hence, the space is not solid. This finishes the proof.In this part, we give the conditions on Orlicz backward generalized difference sequence space to be premodular Banach (sss).

Theorem 7. If is an Orlicz function satisfying -condition and is an absolute nondecreasing, then the space is a premodular Banach (sss), where

Proof. (1)(i)Suppose . Since is nondecreasing, convex, and satisfying -condition and is an absolute nondecreasing, then there exists a number such thatfor some . Then, .(ii)Assume and . Since is satisfying -condition, we havewhere . Then, . Hence, from Parts (i) and (iii), the space is linear. Therefore, , for all and . Therefore, for all .(2)Suppose , for all and . is nondecreasing, and is an absolute nondecreasing. Hence, we haveso .(3)Assume that . We havethen .(i)Obviously, and (ii)There is where , for all and (iii)For some , we have , for all (iv)Clearly from the proof part (2)(v)From (3), we have that (vi)It is obvious that (vii)Since is satisfying -condition, there is with such that , for each and , if Hence, the space is premodular (sss). To explain that is a premodular Banach (sss), assume is a Cauchy sequence in , and then for each , there is such that for all , we haveSince is nondecreasing, hence, for and , we concludeTherefore, is a Cauchy sequence in for fixed , so for fixed . Hence, , for all . Finally, to show that , we haveTherefore, . This gives that is a premodular Banach (sss).
In view of Theorem 2, we conclude the following theorem.

Theorem 8. If is an Orlicz function satisfying -condition and is an absolute nondecreasing, then the space is pre-quasi Banach (sss), where

Corollary 1. If and is an absolute nondecreasing, then is a premodular Banach (sss), where , for all .

4. Pre-Quasi Banach Closed Ideal

We introduce the sufficient conditions on such that the class is Banach and closed.

Theorem 9. If is an Orlicz function satisfying -condition and is an absolute nondecreasing, then is a pre-quasi Banach operator ideal, with , for all and .

Proof. let the conditions be satisfied. Hence, from Theorems 3, 4, and 7, the function is a pre-quasi norm on the ideal . Let be a Cauchy sequence in . Since , we haveTherefore, is a Cauchy sequence in . Since is a Banach space, hence with and while , for each . From Parts (ii), (iii), and (iv) of Definition 5, we haveTherefore, . Hence, .

Theorem 10. If is an Orlicz function satisfying -condition and is an absolute nondecreasing, then is a pre-quasi closed operator ideal, with , for all and .

Proof. let the conditions be satisfied. Therefore, by using Theorems 3, 4, and 7, the function is a pre-quasi norm on the ideal . Assume for all and . Since , we haveHence, is a convergent sequence in . In addition, , for each . From Parts (ii), (iii), and (iv) of Definition 5, we getTherefore, . This gives that .

Corollary 2. is a pre-quasi closed and Banach, with , for all and , if and is an absolute nondecreasing.

5. Small and Simple Pre-Quasi Operator Ideal

We explain the sufficient conditions on such that the strictly inclusion relation of , for different and , has been happened.

Theorem 11. For any infinite dimensional Banach spaces and . Let and be two Orlicz functions satisfying -condition with for all and is an absolute nondecreasing, for all , then

Proof. Let the conditions be satisfied. If , we have . One can seeTherefore, . Next, if we choose such that and , for , we can find with and . Therefore, . Therefore, and . Clearly, . Choose such that , for . We have such that .

Corollary 3. For any infinite dimensional Banach spaces and , and absolute nondecreasing , for every , thenWe study the conditions such that the class is small.

Theorem 12. For any Banach spaces and with . Let be an Orlicz function satisfying -condition and be an absolute nondecreasing, then the class is small.

Proof. Let the conditions be verified. Therefore, be a pre-quasi Banach operator ideal, where . Let , so there is with for all . From Dvoretzky’s theorem [11] for , there are subspaces and quotient spaces of . By isomorphisms and will be mapped onto with and . Let be the natural embedding map from into and be the quotient map from onto . If we denote the Bernstein numbers [11] by , we havefor . Since is an Orlicz function satisfying -condition, we havefor some . Since is an arbitrary, we have a contradiction. So, and cannot be infinite dimensional while .
By the same manner, one can prove that the class is small.

Theorem 13. Hold any Banach spaces and with . Let be an Orlicz function satisfying -condition and is an absolute nondecreasing, then the class is small.
For which , is simple?

Theorem 14. For any infinite dimensional Banach spaces and . Let and be two Orlicz functions satisfying -condition with for all and is an absolute nondecreasing, for all , then

Proof. Assume that there is which is not approximable. By Lemma 1, we have and with . Therefore, for all , we getFrom Theorem 11, we obtain a contradiction. Hence, .

Corollary 4. For any infinite dimensional Banach spaces and . Let and be two Orlicz functions satisfying -condition with for all and is an absolute nondecreasing, for all , then

Proof. Clearly, since each approximable operator is compact.

Theorem 15. Pick up any Banach spaces and with . If is an Orlicz function satisfying -condition and is an absolute nondecreasing, then the class is simple.

Proof. Suppose that there is and . Therefore, from Lemma 1, one can find that with . This means that . Consequently, . Therefore, includes one and only one nontrivial closed ideal .

6. Eigenvalues of s-Type Orlicz Generalized Difference Sequence Space

We explain here the sufficient conditions on such that equals .

Theorem 16. Pick up any Banach spaces and with . If is an Orlicz function satisfying -condition and is an absolute nondecreasing, then

Proof. Suppose and then , we have . Since is continuous, so . Let be an invertible, for all , then exists and bounded, for each . Therefore, with . From the pre-quasi operator ideal of , one hasTherefore, . We have a contradiction, and then is not invertible, for all . Hence, is the eigenvalues of . Conversely, if , then and , for all . This gives that , for all , and then , for all . Therefore, , so . This completes the proof.

7. Conclusion

We have introduced the concept of the pre-quasi norm on the new sequence space generated by the domain of generalized backward difference operator in Orlicz sequence space. This space is not operator ideal since it is not solid. However, if the generalized backward difference operator is an absolute nondecreasing and Orlicz function satisfies -condition, then the operator ideal constructed by this sequence space and s-numbers will be Banach, closed, small, and simple. Finally, we have found the spectrum of all operators contained in this operator ideal.

Data Availability

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

Ethical Approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally to the writing of this paper and 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.