Orlicz Generalized Difference Sequence Space and the Linked Pre-Quasi Operator Ideal

In this article, the necessary conditions on s-type Orlicz generalized difference sequence space to generate an operator ideal have been examined. +erefore, 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 snumbers.


Introduction
e 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 C N , c, ℓ ∞ , ℓ r , and c 0 , we denote the spaces of all, convergent, bounded, r-absolutely summable and null sequences of complex numbers, respectively. N indicates the set of nonnegative integers. Tripathy et al. [1] introduced and studied the forward and backward generalized difference sequence spaces: respectively. When n � 1, the generalized difference sequence spaces reduced to U( Δ (m) ) defined and investigated by Et and Çolak [2]. If m � 1, the generalized difference sequence spaces reduced to U(Δ n ) defined and investigated by Tripathy and Esi [3]. While if n � 1 and m � 1, the generalized difference sequence spaces reduced to U(Δ) defined and studied by Kizmaz [4].
Let r � (r j ) ∈ R +N , where R +N indicates the space of sequences with positive real numbers, and we define the Orlicz backward generalized difference sequence space as follows: where When ψ(w) � w r , then ℓ ψ (Δ m n+1 ) � ℓ r (Δ m n+1 ) 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 B(W, Z), 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 B(W). e s-numbers [11] have many examples such as the r-th approximation number, denoted by e 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 B c formed by (d r (V)) and c 0 . Pietsch [11] studied the quasi-ideals (ℓ r ) app for r ∈ (0, ∞), the ideals of Hilbert Schmidt operators between Hilbert spaces constructed by ℓ 2 , and the ideals of nuclear operators generated by ℓ 1 . He explained that S � (ℓ r ) app for r ∈ [ 1, ∞ ), where S is the closed class of all finite rank operators, and the class (ℓ r ) app became simple Banach and small [12]. e strictly inclusion (ℓ r ) app (W, Z)⊊ (ℓ j ) app (W, Z)B(W, Z), whenever j > r > 0, W and Z 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 (ℓ M ) S and (ces(r)) S . Başarir and Kara [15] studied the compact operators on some Euler B(m)-difference sequence spaces.İlkhan et al. [16] investigated the multiplication operators on Cesáro second-order function spaces. e point of this article to explain some results of ( ℓ ψ (Δ m n+1 ) ) τ equipped with a prequasi norm τ. Firstly, we give the necessary conditions on any s-type ( ℓ ψ (Δ m n+1 ) ) τ to give an operator ideal. Secondly, some geometric and topological structures of ( ℓ ψ (Δ m n+1 ) ) S τ have been studied, such as closed, small, simple Banach and ( ℓ ψ (Δ m n+1 ) ) S � ( ℓ ψ (Δ m n+1 ) ) ] . A strictly inclusion relation of ( ℓ ψ (Δ m n+1 ) ) S has been determined for different Orlicz functions and Δ m n+1 .

Preliminaries and Definitions
Definition 2 (see [11]). An operator V ∈ B(W) is called approximable if there are D r ∈ S(W), for every r ∈ N and lim r⟶∞ ‖V − D r ‖ � 0. By Υ(W, Z), we will indicate the space of all approximable operators from W to Z.
Definition 3 (see [11]). A Banach space W is called simple if B(W) includes one and only one nontrivial closed ideal.
Definition 4 (see [14]). e space of linear sequence spaces Y is called a special space of sequences (sss) if Definition 5 (see [5]). A subspace of the (sss) Y τ is called a premodular (sss) if there is a function τ: Y ⟶ [ 0, ∞ ) verifying the following conditions: . . .), for any ] ∈ C e (sss) Y τ is called pre-quasi normed (sss) if τ satisfies Parts (i)-(iii) of Definition 5 and when the space Y is complete under τ, then Y τ is called a pre-quasi Banach (sss).
By B, we will denote the class of all bounded linear operators between any pair of Banach spaces.
Definition 6 (see [5]). A class R⊆B is called an operator ideal if every R(W, Z) � R ∩ B(W, Z) satisfies the following conditions: where W 0 and Z 0 are Banach spaces Definition 7 (see [5]). A pre-quasi norm on the ideal B is a function ζ: B ⟶ [ 0, ∞ ) which satisfies the following conditions: Theorem 3 (see [14]). e class X S τ is an operator ideal, if X τ is a (sss).
forms a pre-quasi norm on X S τ , whenever X τ be a premodular (sss). e inequality [17]
(2) Suppose |x i | ≤ |y i |, for all i ∈ N and y ∈ ℓ ψ (Δ m n+1 ). ψ is nondecreasing, and Δ m n+1 is an absolute nondecreasing. Hence, we have k�0 is a Cauchy sequence in ( ℓ ψ (Δ m n+1 ) ) τ , and then for each ε ∈ (0, 1), there is i 0 ∈ N such that for all i, j ≥ i 0 , we have Since ψ is nondecreasing, hence, for i, j ≥ i 0 and k ∈ N, we conclude In view of eorem 2, we conclude the following theorem.

Theorem 9. If ψ is an Orlicz function satisfying δ 2 -condition and Δ m n+1 is an absolute nondecreasing, then
Proof. let the conditions be satisfied. Hence, from eorems 3, 4, and 7, the function ζ is a pre-quasi norm on the ideal Journal of Mathematics erefore, (V j ) j∈N is a Cauchy sequence in B(W, Z). Since B(W, Z) is a Banach space, hence T ∈ B(W, Z) with lim j⟶∞ ‖V j − V‖ � 0 and while (s n (V i )) ∞ n�0 ∈ ( ℓ ψ (Δ m n+1 ) ) τ , for each i ∈ N. From Parts (ii), (iii), and (iv) of Definition 5, we have erefore, □ Theorem 10. If ψ is an Orlicz function satisfying δ 2 -condition and Δ m n+1 is an absolute nondecreasing, then Proof. let the conditions be satisfied. erefore, by using eorems 3, 4, and 7, the function ζ is a pre-quasi norm on Hence, (V j ) j∈N is a convergent sequence in B(W, Z). In addition, (s n (V j )) ∞ n�0 ∈ ( ℓ ψ (Δ m n+1 ) ) τ , for each j ∈ N. From Parts (ii), (iii), and (iv) of Definition 5, we get is an absolute nondecreasing.

Small and Simple Pre-Quasi Operator Ideal
We explain the sufficient conditions on ℓ ψ (Δ m n+1 ) such that the strictly inclusion relation of ( ℓ ψ (Δ m n+1 ) ) S , for different ψ and Δ m n+1 , has been happened.

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. is 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 δ 2 -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
e data used to support the findings of the study are available from the corresponding author upon request.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.