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

Authors: Awad A. Bakery; OM Kalthum S. K. Mohamed
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this article, the necessary conditions ons-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.


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## Body

## 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. Byℂℕ, c, ℓ∞, ℓr, and c0, 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:(1)UΔnm=wk∈ℂℕ:Δnmwk∈U,UΔnm=wk∈ℂℕ:Δnmwk∈U,where m,n∈ℕ, U=ℓ∞, c or c0, with(2)Δnmwk=∑ν=0m−1νCνmwk+νn,Δnmwk=∑ν=0m−1νCνmwk−νn,respectively. When n=1, the generalized difference sequence spaces reduced to UΔm defined and investigated by Et and Çolak [2]. If m=1, the generalized difference sequence spaces reduced to UΔn defined and investigated by Tripathy and Esi [3]. While if n=1 and m=1, the generalized difference sequence spaces reduced to UΔ defined and studied by Kizmaz [4].Definition 1.
(see [5]). The backward generalized difference Δn+1m is called an absolute nondecreasing; if xi≤yi for all i∈ℕ, then Δn+1mxi≤Δn+1myi.
An Orlicz function [6] is a function ψ:0,∞⟶0,∞, which is convex, continuous, and nondecreasing with ψ0=0, ψu>0 for u>0 and ψu⟶∞, as u⟶∞. An Orlicz function ψ [7] is said to satisfy δ2-condition for all values of x≥0, if there exists a constant k>0, such that ψ2x≤kψx. The δ2-condition is equivalent to ψlx≤klψx, for all values of x and for l>1. Lindentrauss and Tzafriri [8] utilized the idea of an Orlicz function to define Orlicz sequence space:(3)ℓψ=u∈ℂℕ:ρβu<∞,for someβ>0,whereρu=∑k=0∞ψuk,where ℓψ,. is a Banach space with the Luxemburg norm:(4)u=infβ>0:ρuβ≤1.
Every Orlicz sequence space contains a subspace that is isomorphic toc0 or ℓq, for some 1≤q<∞.
Letr=rj∈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:(5)ℓψΔn+1mτ=w=wj∈ℂℕ:∃σ>0,withτσw<∞,where τw=∑j=0∞ψΔn+1mwj,wj=0 for j<0, Δn+1mwj=Δn+1m−1wj−Δn+1m−1wj−1, and Δ0wj=wj, for all j,n,m∈ℕ. It is a Banach space, with(6)w=infσ>0:τwσ≤1.
Whenψw=wr, then ℓψΔn+1m=ℓrΔn+1m 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 BW,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 BW. The s-numbers [11] have many examples such as the r-th approximation number, denoted by αrV, which is defined by αrV=infV−B:B∈BW,Zand rankB≤r, and the r-th Kolmogorov number, denoted by drV, which is defined by drV=infdimW≤rsupw≤1infv∈WVw−v. The following notations will be used in the sequel:(7)XS≔XSW,Z,whereXSW,Z≔V∈BW,Z:sjVj=0∞∈X,Xapp≔XappW,Z,whereXappW,Z≔V∈BW,Z:αjVj=0∞∈X,XKol≔XKolW,Z,whereXKolW,Z≔V∈BW,Z:djVj=0∞∈X,Xν≔XνW,Z,whereXνW,Z≔V∈BW,Z:νjVj=0∞∈X,V−νjVI=0,for allj∈ℕ.
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 operatorsBc formed by drV and c0. Pietsch [11] studied the quasi-ideals ℓrapp for r∈0,∞, the ideals of Hilbert Schmidt operators between Hilbert spaces constructed by ℓ2, and the ideals of nuclear operators generated by ℓ1. He explained that S¯=ℓrapp for r∈1,∞, where S¯ is the closed class of all finite rank operators, and the class ℓrapp became simple Banach and small [12]. The strictly inclusion ℓrappW,Z⊊ℓjappW,ZBW,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 ℓMS and cesrS. Başarir and Kara [15] studied the compact operators on some Euler Bm-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 ℓψΔn+1mτ equipped with a pre-quasi norm τ. Firstly, we give the necessary conditions on any s-type ℓψΔn+1mτ to give an operator ideal. Secondly, some geometric and topological structures of ℓψΔn+1mτS have been studied, such as closed, small, simple Banach and ℓψΔn+1mS=ℓψΔn+1mν. A strictly inclusion relation of ℓψΔn+1mS has been determined for different Orlicz functions and Δn+1m.


## 2. Preliminaries and Definitions

Definition 2.
(see [11]). An operator V∈BW is called approximable if there are Dr∈SW, for every r∈ℕ and limr⟶∞V−Dr=0.
ByΥW,Z, we will indicate the space of all approximable operators from W to Z. The sequence ej=0,0,…,1,0,0,… with 1 in the j-th coordinate, for every j∈ℕ, will be used in the sequel.Lemma 1.
(see [11]). Let V∈BW,Z. If V∉ΥW,Z, then there are G∈BW and B∈BZ such that BVGer=er, for all r∈ℕ.Definition 3.
(see [11]). A Banach space W is called simple if BW includes one and only one nontrivial closed ideal.Theorem 1.
(see [11]). If W is Banach space with dimW=∞, then(8)SW⊊ΥW⊊BcWBW.Definition 4.
(see[14]). The space of linear sequence spaces Y is called a special space of sequences (sss) if(1)
er∈Y with r∈ℕ,(2)
letu=ur∈ℂℕ, v=vr∈Y, and ur≤vr, for every r∈ℕ, then u∈Y. This means Y be solid, and(3)
ifurr=0∞∈Y, then ur/2r=0∞∈Y, wherever r/2 means the integral part of r/2.Definition 5.
(see [5]). A subspace of the (sss) Yτ is called a premodular (sss) if there is a function τ:Y⟶0,∞ verifying the following conditions:(i)
τy≥0 for each y∈Y and τy=0⇔y=θ, where θ is the zero element of Y(ii)
There existsa≥1 such that τηy≤aητy, for all y∈Y and η∈ℂ(iii)
For someb≥1, τy+z≤bτy+τz, for every y,z∈Y(iv)
yr≤zr with r∈ℕ, which implies that τyr≤τzr(v)
For someb0≥1, τyr≤τyr/2≤b0τyi(vi)
Ify=yrr=o∞∈Y and d>0, then there is r0∈ℕ with τyrr=r0∞<d(vii)
There ist>0 with τν,0,0,0,…≥tντ1,0,0,0,…, for any ν∈ℂ
The (sss)Yτ 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 Yτ is called a pre-quasi Banach (sss).Theorem 2.
(see [5]). A pre-quasi norm (sss) Yτ, whenever it is premodular (sss).
ByB, 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 RW,Z=R∩BW,Z satisfies the following conditions:(i)
R⊇S(ii)
The spaceRW,Z is linear over ℂ(iii)
IfV∈BW0,W, G∈RW,Z, and Q∈BZ,Z0, then QGV∈RW0,Z0, where W0 and Z0 are Banach spacesDefinition 7.
(see [5]). A pre-quasi norm on the ideal B is a function ζ:B⟶0,∞ which satisfies the following conditions:(1)
For allV∈BW,Z, ζV≥0 and ζV=0 if and only if V=0(2)
There isH≥1 such that ζηV≤HηζV, for all V∈BW,Z and η∈ℂ(3)
There isb≥1 such that ζV1+V2≤bζV1+ζV2, for all V1,V2∈BW,Z(4)
There isD≥1 such that if U∈BW0,W, T∈BW,Z, and V∈BZ,Z0, then ζVTU≤DVζTUTheorem 3.
(see [14]). The classXτS is an operator ideal, if Xτ is a (sss).Theorem 4.
(see [14]). The functionζV=τsrVr=0∞ forms a pre-quasi norm on XτS, whenever Xτ be a premodular (sss).
The inequality [17] ai+biqi≤Haiqi+biqi, where qi≥0 for all i∈ℕ, H=max1,2h−1 and h=supiqi, will be used in the sequel.


## 3. Main Results

We give the necessary conditions ons-type ℓψΔn+1m under τ:ℓψΔn+1m⟶0,∞ such that ℓψΔn+1mτS forms an operator ideal.Theorem 5.
ForℓψΔn+1mτ=x=snV∈ℂℕ:V∈BW,Zandτx<∞. If ℓψΔn+1mτS is an operator ideal, then the following conditions are satisfied:(1)
The setℓψΔn+1mτ contains F, the space of all the sequences with finite nonzero numbers(2)
IfsrV1r=0∞∈ℓψΔn+1mτ and srV2r=0∞∈ℓψΔn+1mτ, then srV1+V2r=0∞∈ℓψΔn+1mτ(3)
For allλ∈ℂ and srVr=0∞∈ℓψΔn+1mτ, then λsrVr=0∞∈ℓψΔn+1mτ(4)
The sequence spaceℓψΔn+1mτ is solidProof.
letℓψΔn+1mτS be an operator ideal.(i)
We haveSW,Z⊂ℓψΔn+1mτSW,Z. Hence, for all T∈SW,Z, we have srVr=0∞∈F. This gives that srVr=0∞∈ℓψΔn+1mτ. Hence, F⊂ℓψΔn+1mτ.(ii)
The spaceℓψΔn+1mτSW,Z is linear over ℂ. Hence, for each λ∈ℂ and V1,V2∈ℓψΔn+1mτSW,Z, we have V1+V2∈ℓψΔn+1mτSW,ZandλV1∈ℓψΔn+1mτSW,Z. This implies that(9)srV1r=0∞∈ℓψΔn+1mτ,srV2r=0∞∈ℓψΔn+1mτ⟹srV1+V2r=0∞∈ℓψΔn+1mτ,λ∈ℂ,srV1r=0∞∈ℓψΔn+1mτ⟹λsrV1r=0∞∈ℓψΔn+1mτ.(iii)
IfA∈BW0,W, B∈ℓψΔn+1mτSW,Z, and D∈BZ,Z0, then DBA∈ℓψΔn+1mτSW0,Z0, where W0 and Z0 are arbitrary Banach spaces. Therefore, if A∈BW0,W, srBr=0∞∈ℓψΔn+1mτ, and D∈BZ,Z0, then srDBAr=0∞∈ℓψΔn+1mτ. In addition, srDBA≤DsrBA. By using condition 3, if DAsrBr=0∞∈ℓψΔn+1mτ, we have srDBAr=0∞∈ℓψΔn+1mτ. This means that ℓψΔn+1mτ is solid.
We explain that for any backward generalized differenceΔn+1m, the space ℓψΔn+1mτS is not operator ideal.Theorem 6.
The spaceℓψΔn+1mτS is not operator ideal, where ψ is an Orlicz function satisfying δ2-condition and τw=∑j=0∞ψΔn+1mwj,for allw∈ℓψΔn+1m.Proof.
if we choosem=2, n=1, wk=1, and vk=wk for k=3s or vk=0, otherwise, for all s,k∈ℕ. We have vk≤wk, for all k∈ℕ, w∈ℓψΔ22τ, and v∉ℓψΔ22τ. Hence, the space ℓψΔn+1mτ 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.
Ifψ is an Orlicz function satisfying δ2-condition and Δn+1m is an absolute nondecreasing, then the space ℓψΔn+1mτ is a premodular Banach (sss), where(10)τw=∑j=0∞ψΔn+1mwj,for allw∈ℓψΔn+1m.Proof.
(1)
(i)
Supposev,w∈ℓψΔn+1m. Since ψ is nondecreasing, convex, and satisfying δ2-condition and Δn+1m is an absolute nondecreasing, then there exists a number b>0 such that
(11)τv+w=∑i=0∞ψΔn+1mvi+wi≤∑i=0∞ψΔn+1mvi+Δn+1mwi≤12∑i=0∞ψ2Δn+1mvi+∑i=0∞ψ2Δn+1mwi≤b2τv+τw≤Bτv+τw<∞,for someB=max1,b/2. Then, v+w∈ℓψΔn+1m.(ii)
Assumeλ∈ℂ and v∈ℓψΔn+1m. Since ψ is satisfying δ2-condition, we have(12)τλv=∑i=0∞ψΔn+1mλvr≤dλ∑i=0∞ψΔn+1mvr≤Dλτv<∞,whereD=max1,d. Then, λv∈ℓψΔn+1m. Hence, from Parts (i) and (iii), the space ℓψΔn+1m is linear. Therefore, er∈ℓq⊆ℓψ⊆ℓψΔn+1m, for all r∈ℕ and q≥1. Therefore, er∈ℓψΔn+1m for all r∈ℕ.(2)
Supposexi≤yi, for all i∈ℕ and y∈ℓψΔn+1m. ψ is nondecreasing, and Δn+1m is an absolute nondecreasing. Hence, we have(13)τx=∑i=0∞ψΔn+1mxi≤∑i=0∞ψΔn+1m|yi|=τy<∞,sox∈ℓψΔn+1m.(3)
Assume thatvr∈ℓψΔn+1m. We have(14)τvr2=∑i=0∞ψΔn+1mvr2≤2∑i=0∞ψΔn+1mvr=2τv,thenvr/2∈ℓψΔn+1m.(i)
Obviously,τw≥0 and τw=0⇔w=θ(ii)
There isD≥1 where τηw≤Dητw, for all w∈ℓψΔn+1m and η∈ℂ(iii)
For someB≥1, we have τv+w≤Bτv+τw, for all v,w∈ℓψΔn+1m(iv)
Clearly from the proof part (2)(v)
From (3), we have thatb0=2≥1(vi)
It is obvious thatF¯=ℓψΔn+1m(vii)
Sinceψ is satisfying δ2-condition, there is ζ with 0<ζ≤ψη/η such that τη,0,0,0,…≥ζητ1,0,0,0,…, for each η≠0 and ζ>0, if η=0
Hence, the spaceℓψΔn+1mτ is premodular (sss). To explain that ℓψΔn+1mτ is a premodular Banach (sss), assume xi=xkik=0∞ is a Cauchy sequence in ℓψΔn+1mτ, and then for each ε∈0,1, there is i0∈ℕ such that for all i,j≥i0, we have(15)τxi−xj=∑k=0∞ψΔn+1mxki−xkj<ψε.
Sinceψ is nondecreasing, hence, for i,j≥i0 and k∈ℕ, we conclude(16)Δn+1mxki−Δn+1mxkj<ε.
Therefore,Δn+1mxkj is a Cauchy sequence in ℂ for fixed k∈ℕ, so limj→∞Δn+1mxkj=Δn+1mxk0 for fixed k∈ℕ. Hence, τxi−x0<ψε, for all i≥i0. Finally, to show that x0∈ℓψΔn+1m, we have(17)τx0=τx0−xn+xn≤Bτxn−x0+τxn<∞.
Therefore,x0∈ℓψΔn+1m. This gives that ℓψΔn+1mτ is a premodular Banach (sss).
In view of Theorem2, we conclude the following theorem.Theorem 8.
Ifψ is an Orlicz function satisfying δ2-condition and Δn+1m is an absolute nondecreasing, then the space ℓψΔn+1mτ is pre-quasi Banach (sss), where(18)τx=∑j=0∞ψΔn+1mxj,for allw∈ℓψΔn+1m.,for allx∈ℓψΔn+1m.Corollary 1.
If0<p<∞ and Δn+1m is an absolute nondecreasing, then ℓpΔn+1mτ is a premodular Banach (sss), where τx=∑i=0∞Δn+1mxip, for all x∈ℓpΔn+1m.


## 4. Pre-Quasi Banach Closed Ideal

We introduce the sufficient conditions onℓψΔn+1m such that the class ℓψΔn+1mτS is Banach and closed.Theorem 9.
Ifψ is an Orlicz function satisfying δ2-condition and Δn+1m is an absolute nondecreasing, then ℓψΔn+1mτS,ζ is a pre-quasi Banach operator ideal, with τw=∑j=0∞ψΔn+1mwj, for all w∈ℓψΔn+1m and ζV=τsnVn=0∞.Proof.
let the conditions be satisfied. Hence, from Theorems3, 4, and 7, the function ζ is a pre-quasi norm on the ideal ℓψΔn+1mτS. Let Vj be a Cauchy sequence in ℓψΔn+1mτSW,Z. Since BW,Z⊇ℓψΔn+1mτSW,Z, we have(19)ζVi−Vj=∑k=0∞ψΔn+1mskVi−Vj≥ψΔn+1mVi−Vj.
Therefore,Vjj∈ℕ is a Cauchy sequence in BW,Z. Since BW,Z is a Banach space, hence T∈BW,Z with limj⟶∞Vj−V=0 and while snVin=0∞∈ℓψΔn+1mτ, for each i∈ℕ. From Parts (ii), (iii), and (iv) of Definition 5, we have(20)ζV=∑r=0∞ψΔn+1msrV−Vj+Vj≤B∑r=0∞ψΔn+1msr2V−Vj+∑r=0∞ψΔn+1msr2Vj≤B∑r=0∞ψΔn+1mV−Vj+Bb0∑r=0∞ψΔn+1msrVj<ε.
Therefore,srVr=0∞∈ℓψΔn+1mτ. Hence, V∈ℓψΔn+1mτSW,Z.Theorem 10.
Ifψ is an Orlicz function satisfying δ2-condition and Δn+1m is an absolute nondecreasing, then ℓψΔn+1mτS,ζ is a pre-quasi closed operator ideal, with τw=∑j=0∞ψΔn+1mwj, for all w∈ℓψΔn+1m and ζV=τsnVn=0∞.Proof.
let the conditions be satisfied. Therefore, by using Theorems3, 4, and 7, the function ζ is a pre-quasi norm on the ideal ℓψΔn+1mτS. Assume Vj∈ℓψΔn+1mτSW,Z for all j∈ℕ and limj⟶∞ζVj−V=0. Since BW,Z⊇ℓψΔn+1mτSW,Z, we have(21)ζV−Vj=∑k=0∞ψΔn+1mskV−Vj≥ψΔn+1mV−Vj.
Hence,Vjj∈ℕ is a convergent sequence in BW,Z. In addition, snVjn=0∞∈ℓψΔn+1mτ, for each j∈ℕ. From Parts (ii), (iii), and (iv) of Definition 5, we get(22)ζV=∑r=0∞ψΔn+1msrV−Vj+Vj≤B∑r=0∞ψΔn+1msr2V−Vj+∑r=0∞ψΔn+1msr2Vj≤B∑r=0∞ψΔn+1mV−Vj+Bb0∑r=0∞ψΔn+1msrVj<ε.
Therefore,srVr=0∞∈ℓψΔn+1mτ. This gives that V∈ℓψΔn+1mτSW,Z.Corollary 2.
ℓpΔn+1mτS,ζ is a pre-quasi closed and Banach, with τw=∑i=0∞Δn+1mwip, for all w∈ℓpΔn+1m and ζV=τsnVn=0∞, if 0<p<∞ and Δn+1m is an absolute nondecreasing.


## 5. Small and Simple Pre-Quasi Operator Ideal

We explain the sufficient conditions onℓψΔn+1m such that the strictly inclusion relation of ℓψΔn+1mS, for different ψ and Δn+1m, has been happened.Theorem 11.
For any infinite dimensional Banach spacesW and Z. Let ψ1 and ψ2 be two Orlicz functions satisfying δ2-condition with ψ2t<ψ1t for all t∈0,∞ and Δnm is an absolute nondecreasing, for all n,m∈ℕ, then(23)ℓr2Δn+1mSW,Z⊂≠ℓj2Δn+1m+1SW,Z⊂≠BW,Z.Proof.
Let the conditions be satisfied. IfV∈ℓψ1Δn+2mSW,Z, we have siV∈ℓψ1Δn+2m. One can see(24)∑j=0∞ψ2Δn+1m+1sjV<∑j=0∞ψ1Δn+2msjV<∞.
Therefore,V∈ℓψ2Δn+1m+1SW,Z. Next, if we choose sjVj=0∞ such that ψ1Δn+2msjV=j+1−1 and ψ2Δn+2msjV=j+1−2, for n,m∈ℕ, we can find V∈BW,Z with ∑j=0∞ψ1Δn+2msjV=∑j=0∞j+1−1=∞ and ∑j=0∞ψ2Δn+2msjV=∑j=0∞j+1−2<∞. Therefore, ℓψ2Δn+2m⊆ℓψ2Δn+1m+1. Therefore, V∉ℓψ1Δn+2mSW,Z and V∈ℓψ2Δn+1m+1SW,Z. Clearly, ℓψ2Δn+1m+1SW,Z⊂BW,Z. Choose sjVj=0∞ such that Δn+1m+1sjV=j+1−1, for n,m∈ℕ. We have V∈BW,Z such that V∉ℓψ2Δn+1m+1SW,Z.Corollary 3.
For any infinite dimensional Banach spacesW and Z, j≥r>0 and absolute nondecreasing Δnm, for every n,m∈ℕ, then(25)ℓψ2Δn+1mSW,Z⊂≠ℓjΔn+1m+1SW,Z⊂≠BW,Z.
We study the conditions such that the classℓψΔn+1mapp is small.Theorem 12.
For any Banach spacesW and Z with dimW=dimZ=∞. Let ψ be an Orlicz function satisfying δ2-condition and Δn+1m be an absolute nondecreasing, then the class ℓψΔn+1mapp is small.Proof.
Let the conditions be verified. Therefore,ℓψΔn+1mapp,ζ be a pre-quasi Banach operator ideal, where ζV=∑k=0∞ψΔn+1mαkV. Let ℓψΔn+1mappW,Z=BW,Z, so there is δ>0 with ζV≤δV for all V∈BW,Z. From Dvoretzky’s theorem [11] for j∈ℕ, there are subspaces Mj and quotient spaces W/Nj of Z. By isomorphisms Aj and Hj will be mapped Z onto ℓ2j with HjHj−1≤2 and AjAj−1≤2. Let Jj be the natural embedding map from Mj into Z and Qj be the quotient map from W onto W/Nj. If we denote the Bernstein numbers [11] by uj, we have(26)1=ukIj=ukAjAj−1IjHjHj−1≤AjukAj−1IjHjHj−1=AjukJjAj−1IjHjHj−1≤AjdkJjAj−1IjHjHj−1=AjdkJjAj−1IjHjQjHj−1≤AjαkJjAj−1IjHjQjHj−1,for 0≤k≤i. Since ψ is an Orlicz function satisfying δ2-condition, we have(27)i+1ψ1≤aAjHj−1∑k=0iψΔn+1mαkJjAj−1IjHjQj⇒i+1ψ1≤aAjHj−1gJjAj−1IjHjQj⇒i+1ψ1≤aδAjHj−1JjAj−1IjHjQj⇒i+1ψ1≤aδAjHj−1JjAj−1IjHjQj=LδAjHj−1Aj−1IjHj⇒i+1ψ1≤4aδ,for some a≥1. Since i is an arbitrary, we have a contradiction. So, W and Z cannot be infinite dimensional while ℓψΔn+1mappW,Z=BW,Z.
By the same manner, one can prove that the classℓψΔn+1mKol is small.Theorem 13.
Hold any Banach spacesW and Z with dimW=dimZ=∞. Let v be an Orlicz function satisfying δ2-condition and Δn+1m is an absolute nondecreasing, then the class ℓψΔn+1mKol is small.
For whichℓψΔn+1m, is ℓψΔn+1mS simple?Theorem 14.
For any infinite dimensional Banach spacesW and Z. Let ψ1 and ψ2 be two Orlicz functions satisfying δ2-condition with ψ2t<ψ1t for all t∈0,∞ and Δnm is an absolute nondecreasing, for all n,m∈ℕ, then(28)Bℓψ2Δn+1m+1S,ℓψ2Δn+1mS=Υℓψ2Δn+1m+1S,ℓψ2Δn+1mS.Proof.
Assume that there isV∈Bℓψ2Δn+1m+1S,ℓψ1Δn+2mS which is not approximable. By Lemma 1, we have G∈Bℓψ2Δn+1m+1S and B∈Bℓψ1Δn+2mS with BVGIk=Ik. Therefore, for all k∈ℕ, we get(29)Ikℓψ1Δn+2mS=∑n=0∞ψ1Δn+2msnIk≤BVGIkℓψ2Δn+1m+1S≤∑n=0∞ψ2Δn+1m+1snIk.
From Theorem11, we obtain a contradiction. Hence, V∈Υℓψ2Δn+1m+1S,ℓψ1Δn+2mS.Corollary 4.
For any infinite dimensional Banach spacesW and Z. Let ψ1 and ψ2 be two Orlicz functions satisfying δ2-condition with ψ2t<ψ1t for all t∈0,∞ and Δnm is an absolute nondecreasing, for all n,m∈ℕ, then(30)Bℓψ2Δn+1m+1S,ℓψ2Δn+1mS=Bcℓψ2Δn+1m+1S,ℓψ2Δn+1mS.Proof.
Clearly, since each approximable operator is compact.Theorem 15.
Pick up any Banach spacesW and Z with dimW=dimZ=∞. If ψ is an Orlicz function satisfying δ2-condition and Δnm is an absolute nondecreasing, then the class ℓψΔn+1mτS is simple.Proof.
Suppose that there isV∈BcℓψΔn+1mτS and V∉ΥℓψΔn+1mτS. Therefore, from Lemma 1, one can find that A,B∈BℓψΔn+1mτS with BVAIk=Ik. This means that IℓψΔn+1mτS∈BcℓψΔn+1mτS. Consequently, BℓψΔn+1mτS=BcℓψΔn+1mτS. Therefore, BℓψΔn+1mτS includes one and only one nontrivial closed ideal ΥℓψΔn+1mτS.


## 6. Eigenvalues ofs-Type Orlicz Generalized Difference Sequence Space

We explain here the sufficient conditions onℓψΔn+1m such that ℓψΔn+1mS equals ℓψΔn+1mν.Theorem 16.
Pick up any Banach spacesW and Z with dimW=dimZ=∞. If ψ is an Orlicz function satisfying δ2-condition and Δn+1m is an absolute nondecreasing, then(31)ℓψΔn+1mSW,Z=ℓψΔn+1mνW,Z.Proof.
SupposeV∈ℓψΔn+1mSW,Z and then srVr=0∞∈ℓψΔn+1m, we have ∑r=0∞ψΔn+1msrV<∞. Since Δn+1m is continuous, so limr⟶∞srV=0. Let V−srVI be an invertible, for all r∈ℕ, then V−srVI−1 exists and bounded, for each r∈ℕ. Therefore, limr⟶∞V−srVI−1=V−1 with V−1∈BZ,W. From the pre-quasi operator ideal of ℓψΔn+1mS,ζ, one has(32)I=VV−1∈ℓψΔn+1mSZ⟹srIr=0∞∈ℓψΔn+1m⇒limr⟶∞srI=0.
Therefore,limr⟶∞srI=1. We have a contradiction, and then V−srVI is not invertible, for all r∈ℕ. Hence, srVr=0∞ is the eigenvalues of V. Conversely, if V∈ℓψΔn+1mνW,Z, then νrVr=0∞∈ℓψΔn+1m and V−νrVI=0, for all n∈ℕ. This gives that V=νrVI, for all r∈ℕ, and then srV=srνrVI=νrV, for all r∈ℕ. Therefore, srVr=0∞∈ℓψΔn+1m, so V∈ℓψΔn+1mSW,Z. 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 satisfiesδ2-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.



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