# Map Ideal of Type the Domain ofr-Cesàro Matrix in the Variable Exponent ℓt. and Its Eigenvalue Distributions

Authors: Awad A. Bakery; OM Kalthum S. K. Mohamed
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, we define a new sequence space generated by the domain ofr-Cesàro matrix in Nakano sequence space. Some geometric and topological properties of this sequence space, the multiplication maps defined on it, and the eigenvalue distributions of map ideal with s-numbers that belong to this sequence space have been examined.


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## Body

## 1. Introduction

The vector spacesℓt. are contained in the variable exponent spaces Lt.. Regarding the 2nd half of the twentieth century, it used to be fulfilled that these variable exponent spaces constituted the proper framework for the mathematical components of numerous issues for which the classical Lebesgue spaces have been inadequate. The relevancy of these spaces and their homes made them a famous and environment friendly device in the remedy of a range of situations; these days, the region of Lt.Ω spaces is a prolific subject of lookup with ramifications achieving into very numerous mathematical specialties [1]. Learning about the variable exponent Lebesgue spaces Lt. obtained in addition impetus from the mathematical description of the hydrodynamics of non-Newtonian fluids [2, 3]. Applications of non-Newtonian fluids additionally known as electrorheological vary from their use in army science to civil engineering and orthopedics. By CN, ℓ∞, ℓr, and c0, we suggest the spaces of each, bounded, r-absolutely summable and null sequences of complex numbers N=0,1,2,⋯. We evidence the space of all, finite rank, approximable and compact bounded linear maps from a Banach space P into a Banach space Q by BP,Q, FP,Q, AP,Q, and KP,Q, and if P=Q, we mark BP, FP, AP, and KP, respectively (see [4, 5]). The ideal of all, finite rank, approximable and compact maps is denoted by B, F, A, and K. We designate el=0,0,⋯,1,0,0,⋯, as 1 presents at the lth coordinate, with l∈N.Lemma 1 [5].
Pick upU∈BP,Q. Assume U∉AP,Q; then, there are maps X∈BP and Y∈BQ so that YUXel=el, for every l∈N.Definition 2 [5].
A Banach spaceV is named simple if the algebra BV includes one and only one nontrivial closed ideal.Theorem 3 [5].
LetV be an infinite dimensional Banach space; then,
(1)FVÞAVÞKVÞBV.Definition 4 [6].
A mapU∈BV is entitled Fredholm if dimRangeUc<∞, dimkerU<∞, and RangeU is closed, where RangeUc mentions the complement of RangeU.Definition 5 [7].
A subclassW of B is named a map ideal if every component WP,Q=W∩BP,Q executes the next setup:
(i)
IΩ∈W, if Ω illustrates Banach space of one dimension(ii)
WP,Q is a linear space on C(iii)
SupposeX∈BP0,P, Y∈WP,Q, and Z∈BQ,Q0, then ZYX∈WP0,Q0, where P0 and Q0 are normed spacesFaried and Bakery [8] made current the notion of prequasi ideal which is added established than the quasi ideal.Definition 6.
A functionΨ:W⟶0,∞ is named a prequasi norm on the map ideal W if the next setting encompasses the following:
(1)
For eachX∈WP,Q, ΨX≥0 and ΨX=0⇔X=0(2)
We haveE0≥1 so as to ΨκX≤E0κΨX, for all X∈WP,Q and κ∈C(3)
We haveG0≥1 for ΨZ1+Z2≤G0ΨZ1+ΨZ2, for all Z1,Z2∈WP,Q(4)
We haveD0≥1 if X∈BP0,P, Y∈WP,Q, and Z∈BQ,Q0; then, ΨZYX≤D0ZΨYXTheorem 7 [8].
Ψ is prequasi norm on the map ideal W, whenever Ψ is a quasinorm on the map ideal W.Definition 8 [9].
Ans-number function is a map detailed on BP,Q which sort to every map X∈BP,Q a nonnegative scalar sequence slXl=0∞ overbearing that the next setting encompasses the following:
(a)
∥X∥=s0X≥s1X≥s2X≥⋯≥0, for every X∈BP,Q(b)
sl+a−1X1+X2≤slX1+saX2, for each X1,X2∈BP,Q and l, a∈N(c)
Ideal property:saZYX≤∥Z∥saY∥X∥, for all X∈BP0,P, Y∈BP,Q, and Z∈BQ,Q0, where P0 and Q0 are discretionary Banach spaces(d)
ForG∈BP,Q and γ∈C, one has saγG=∣γ∣saG(e)
Rank property: assumerankX≤a, then saX=0, for each X∈BP,Q(f)
Norming property:sl≥aIa=0 or sl<aIa=1, where Ia mirrors the unit map on the a-dimensional Hilbert space ℓ2a
In an assorted illustration ofs-numbers, we intimate the next setting:
(1)
Thea-th Kolmogorov number, demonstrated by daX, is indicated by(2)daX=infdimJ≤asupf≤1infg∈JXf−g.(2)
Thea-th approximation number, established by αaX, is denoted by(3)αaX=infX−Y:Y∈BP,QandrankY≤a.Notations 9 [8].
(4)BVs≔BVsP,Q;PandQareBanachSpaces,whereBVsP,Q≔X∈BP,Q:saXa=0∞∈V,BVα≔BVαP,Q;PandQareBanachSpaces,whereBVαP,Q≔X∈BP,Q:αaXa=0∞∈V,BVd≔BVdP,QPandQareBanachSpaces,whereBVdP,Q≔X∈BP,Q:daXa=0∞∈V.The ideals and multiplication mappings possess extensive grazing of mathematics in functional analysis, namely, in the theory of fixed point, eigenvalue distributions theorem, and geometric structure of Banach spaces. A few of map ideals in the class of Banach spaces or Hilbert spaces are evident by inconsistent scalar sequence spaces. For representative the ideal of compact maps is evident by the spacec0 and daX, for X∈BP,Q. Pietsch [5] approved the quasi-ideals Bℓbα, for 0<b<∞. He investigated that the ideals of nuclear maps and of Hilbert Schmidt maps between Hilbert spaces are explored by ℓ1 and ℓ2, respectively. He examined that Fℓb are dense in Bℓb, and the algebra Bℓb, where 1≤b<∞, constructed simple Banach space. Pietsch [10] approved that Bℓbα, for 0<b<∞, is small. Makarov and Faried [11] examined that for each infinite dimensional Banach space P, Q and r>b>0, then BℓbαP,QÞBℓrαP,QÜBP,Q. Yaying et al. [12] defined and examined the sequence space, χrt, whose r-Cesàro matrix is in ℓt, with r∈0,1 and 1≤t≤∞. They explored the quasi Banach ideal of type χrt, for r∈0,1 and 1<t<∞. They establish its Schauder basis, α−, β−, and γ− duals, and found certain matrix classes connected with this sequence space. Basarir and Kara investigated the compact mappings on some Euler Bm-difference sequence spaces [13], some difference sequence spaces of weighted means [14], the Riesz Bm-difference sequence space [15], the B-difference sequence space derived by weighted mean [16], and the mth order difference sequence space of generalized weighted mean [17]. Mursaleen and Noman [18, 19] introduced the compact mappings on some difference sequence spaces. The multiplication maps on Cesàro sequence spaces with the Luxemburg norm were studied by Komal et al. [20]. İlkhan et al. [21] examined the multiplication maps on Cesàro second-order function spaces. Recently, many authors in the literature have considered some nonabsolute-type sequence spaces and introduced recent high-quality papers, for example, Mursaleen and Noman [22] defined the sequence space ℓpλ and ℓ∞λ of nonabsolute type and proved that the spaces ℓpλ and ℓpλ are linearly isomorphic for 0<p≤∞, ℓpλ is a p-normed space and a BK-space in the cases for 0<p<1 and 1≤p≤∞, and formed the basis for the space ℓpλ for 1≤p<∞. In [23], they examined the α−, β−, and γ− duals of ℓpλ and ℓ∞λ of nonabsolute type, for 1≤p<∞. They characterized some related matrix classes and derived the characterizations of some other classes by means of a given basic lemma. On Cesàro summable sequences, Mursaleen and Başar [24] defined some spaces of double sequences whose Cesàro transforms are bounded, convergent in Pringsheim’s sense, null in Pringsheim’s sense, both convergent in Pringsheim’s sense, and bounded, regularly convergent, and absolutely q-summable, respectively, and examined some topological properties of those sequence spaces. The addicted inequality will be run down in the development [25]. If ra≥1 and xa,za∈C, with a∈N, and ℏ=supara, then
(5)xa+zara≤2ℏ−1xara+zara.Supposer∈0,1, tl∈R+N, where R+N is the space of all sequences of positive reals, and tl≥1, with l∈N, we define a new sequence space generated by the domain of r-Cesàro matrix in Nakano sequence space as
(6)cesrtυ=f=fk∈CN:υρf<∞,forsomeρ>0,where υf=∑l=0∞∑z=0lrzfz/l+1rtl and
(7)lr=1−rl1−r,r≠1,l,r=1.In casetl∈ℓ∞, we have
(8)cesrtυ=f=fk∈CN:υρf<∞,foranyρ>0.Remark 10.
(1)
Whenr=1 and tl=t, with l∈N, then cesrt is compressed to cest, introduced and studied by Ng and Lee [26]. Different types of Cesàro summable sequence spaces of nonabsolute type have been studied by many authors [27–31](2)
Iftl=t, with l∈N, cesrt is truncated to χrt studied by Yaying et al. [12]The goal of this paper is efficient like so in Section2 we offer the sufficient setting on any linear space of sequences V, and we mark it a private sequence space (pss), so as to the class BVs constructs a map ideal. We apply this theorem on cesrt. We define a subclass of the pss which we will call a premodular pss under the functional υ:V⟶0,∞. We explain the sufficient conditions on cesrt with definite functional υ to become premodular pss. Which implies that cesrt is a prequasi normed pss. In Section 3, we define a multiplication map on the prequasi normed pss, cesrtυ, and give the necessity and sufficient setup on this sequence space such that the multiplication map is bounded, approximable, invertible, Fredholm, and closed range. In Section 4, firstly, we introduce the sufficient settings (not necessary) on the premodular psscesrtυ so that F is dense in Bcesrtυs. This explains a negative answer of Rhoades [32] open problem about the linearity of s-type cesrtυ spaces. Secondly, we introduce the conditions on cesrtυ so that the components of prequasi ideal Bcesrts are complete and closed. Thirdly, we investigate the sufficient conditions on cesrtυ so as Bcesrtυα is precisely confined for altered powers. We explain the setup for which the prequasi ideal Bcesrtυα is minimum. Fourthly, we describe the setting for which the Banach prequasi ideal Bcesrtυs is simple. Fifthly, we expound the sufficient setting on cesrtυ so as to the class of all bounded linear maps which sequence of eigenvalues in cesrtυ equals Bcesrtυs.


## 2. Linear Problem

In this section, we offer the enough setting on any linear space of sequencesV, and we mark it private sequence space (pss), so as the class BVs creates a map ideal. We apply this setting on cesrt. We define a subclass of pss under the functional υ:V⟶0,∞, which we will call a premodular pss. We explain the enough setup on cesrt with definite functional υ to become premodular pss, which implies that cesrt is a prequasi normed pss.Definition 11.
The linear space of sequencesV is named a pss, if it satisfies the following:
(1)
eb∈V, with b∈N(2)
V is solid, i.e., for f=fb∈CN, g=gb∈V, and fb≤gb, over b∈N, then f∈V(3)
fb/2b=0∞∈V, while b/2 illustrates the integral part of b/2, if fbb=0∞∈VTheorem 12.
If the linear sequence spaceV is pss, then BVs is a map ideal.Proof.
Assume the linear sequence spaceV is pss.
(i)
SupposeX∈FP,Q and rankX=b, with b∈N. As eb∈V, with b∈N and by the linearity of V, one has sbXb=0∞=∑a=0b−1saXea∈V. Therefore, X∈BVsP,Q, this gives FP,Q⊆BVsP,Q(ii)
PresumeX1,X2∈BVsP,Q and ρ1,ρ2∈C, then sbX1b=0∞∈V and sbX2b=0∞∈V. As b≥2b/2, compared to the definition of s-numbers and sbX is a decreasing sequence, we get sbρ1X1+ρ2X2≤s2b/2ρ1X1+ρ2X2≤sb/2ρ1X1+sb/2ρ2X2=ρ1sb/2X1+ρ2sb/2X2, with b∈N. By using the linearity of V, conditions (24) and (25), one can see sbρ1X1+ρ2X2b=0∞∈V, so ρ1X1+ρ2X2∈BVsP,Q(iii)
LetX∈BP0,P, Y∈BVsP,Q, and Z∈BQ,Q0, one has sbYb=0∞∈V. As sbZYX≤ZsbYX. By using the linearity of V and condition (24), we have sbZYXb=0∞∈V, then ZYX∈BVsP0,Q0
Here and after, we will denote the space of all increasing sequences of real numbers byI.Theorem 13.
cesrt is pss, if tl∈I∩ℓ∞ with t0>1.Proof.
(1.i)
Assumef,g∈cesrt. As tl∈ℓ∞, one has(9)∑l=0∞∑z=0lrzfz+rzgzl+1rtl≤2ℏ−1∑l=0∞∑z=0lrzfzl+1rtl+∑l=0∞∑z=0lrzgzl+1rtl<∞,so f+g∈cesrt.
(1.ii)
Supposeρ∈C, f∈cesrt, and as tl∈ℓ∞, we obtain(10)∑l=0∞∑z=0lrzρfzl+1rtl≤suplρtl∑l=0∞∑z=0lrzfzl+1rtl<∞.
Hence,ρf∈cesrt. Relative to (1-i) and (1-ii), we have cesrt as a linear space.
Also astl∈I∩ℓ∞ with t0>1, one has
(11)∑l=0∞∑z=0lrzebzl+1rtl=∑l=b∞rbl+1rtl≤suplrbtl∑l=b∞1l+1rtl<∞.
Therefore,eb∈cesrt, with b∈N.
(1)
If∣fb∣≤∣gb∣, for each b∈N and ∣g∣∈cesrt. One can see(12)∑l=0∞∑z=0lrzfzl+1rtl≤∑l=0∞∑z=0lrzgzl+1rtl<∞.Hence, ∣f∣∈cesrt.
(2)
Assumefz∈cesrt, where tl∈I∩ℓ∞, we get(13)∑l=0∞∑z=0lrzfz/2l+1rtl=∑l=0∞∑z=02lrz∣fz/2∣2l+1rt2l+∑l=0∞∑z=02l+1rzfz/22l+2rt2l+1≤∑l=0∞12l+1rr2lfl+∑z=0lr2z+r2z+1fztl+∑l=0∞12l+2r∑z=0lr2z+r2z+1fztl≤2ℏ−1∑l=0∞1l+1r∑z=0lrzfztl+∑l=0∞2l+1r∑z=0lrzfztl+∑l=0∞2l+1r∑z=0lrzfztl≤22ℏ−1+2ℏ−1+2ℏ∑l=0∞1l+1r∑z=0lrzfztl<∞,so fz/2∈cesrt.By using Theorem12, we can get the next theorem.Theorem 14.
Pick uptl∈I∩ℓ∞ with t0>1, then Bcesrts is a map ideal.Definition 15.
A subclass ofpss is named a premodular pss, if there is a map υ:V⟶0,∞ with the following settings:
(i)
Whenf∈V, f=θ⇔υf=0, with υf≥0, where θ is the zero element of V(ii)
Iff∈V and ρ∈C, we have E0≥1 with υρf≤ρE0υf(iii)
υf+g≤G0υf+υg includes for some G0≥1, with f,g∈V(iv)
Forb∈N, ∣fb∣≤∣gb∣, we get υfb≤υgb(v)
The inequality,υfb≤υfb/2≤D0υfb includes, for D0≥1(vi)
IfF denotes the space of all sequences with finite nonzero coordinates, then F¯=Vυ(vii)
We haveϖ>0 so that υρ,0,0,0,⋯≥ϖρυ1,0,0,0,⋯, with ρ∈CDefinition 16.
ThepssVυ is named a prequasi normed pss, if υ supports the points (i)–(iii) of Definition 15. If V is complete equipped with υ, then Vυ is named a prequasi Banach pss.Theorem 17.
A prequasi normedpssVυ, whenever it is premodular pss.Theorem 18.
cesrtυ is a premodular pss, if tl∈I∩ℓ∞ with t0>1.Proof.
(i)
Easily,υf≥0 and υf=0⇔f=θ(ii)
We haveE0=max1,supl∣ρtl−1≥1 with υρf≤E0ρυf, for every f∈cesrt and ρ∈C(iii)
One hasυf+g≤2ℏ−1υf+υg, for each f,g∈cesrt(iv)
Definitely, from the proof part (24) of Theorem 13(v)
Indeed, from the proof part (25) of Theorem 13, D0≥22ℏ−1+2ℏ−1+2ℏ≥1(vi)
Obviously,F¯=cesrt(vii)
We have0<ϖ≤suplρtl−1 with υρ,0,0,0,⋯≥ϖρυ1,0,0,0,⋯, for each ρ≠0 and ϖ>0, if ρ=0By following Theorems17 and 18, we determine the next theorem.Theorem 19.
The spaceVυ is a prequasi normed pss, if tl∈I∩ℓ∞ with t0>1.


## 3. Multiplication Maps oncesrtυ

In this section, we define a multiplication map on the prequasi normedpsscesrtυ and investigate the necessity and sufficient setup on cesrtυ so as the multiplication map is bounded, invertible, approximable, Fredholm, and closed range map.Definition 20.
Ifω=ωk∈CN and Vυ is a prequasi normed pss, the map Hω:Vυ⟶Vυ is called a multiplication map on Vυ, when Hωf=ωbfb∈Vυ, with f∈Vυ. The multiplication map is named created by ω, if Hω∈BVυ.Theorem 21.
Pick upω∈CN and tl∈I∩ℓ∞ with t0>1, then ω∈ℓ∞, if and only if Hω∈Bcesrtυ.Proof.
Suppose the settings are confirmed. Letω∈ℓ∞. Hence, there is ν>0 so as to ωb≤ν, with b∈N. For f∈cesrtυ, one has
(14)υHωf=υωf=∑l=0∞∑z=0lrzωzfzl+1rtl≤∑l=0∞∑z=0lνrzfzl+1rtl≤suplνtl∑l=0∞∑z=0lrzfzl+1rtl=suplνtlυf.
Therefore,Hω∈Bcesrtυ.
On the contrary, letHω∈Bcesrtυ and ω∉ℓ∞. Hence, for all b∈N, there are xb∈N so as ωxb>b. We have
(15)υHωexb=υωexb=∑l=0∞∑z=0lrzωzexbzl+1rtl=∑l=xb∞rxbωxbl+1rtl>∑l=xb∞brxbl+1rtl>bt0υexb.
Hence,Hω∉Bcesrtυ. So ω∈ℓ∞.Theorem 22.
Assumeω∈CN and cesrtυ be a prequasi normed pss. Then, ωb=g, for every b∈N and g∈C with g=1, if and only if Hω is an isometry.Proof.
Let the sufficient condition be verified. One has(16)υHωf=υωf=∑l=0∞∑k=0lrkωkfkl+1rtl=∑l=0∞∑k=0lgrkfkl+1rtl=υf,with f∈cesrtυ. So Hω is an isometry.
Let the necessity condition be satisfied andωb<1, for some b=b0. We get
(17)υHωeb0=υωeb0=∑l=0∞∑k=0lrkωkeb0kl+1rtl=∑l=b0∞rb0ωb0l+1rtl<∑l=b0∞rb0l+1rtl=υeb0.
Also, whenωb0>1, it is easy to show that υHωeb0>υeb0, which is an inconsistency for the two cases. Therefore, ωb=1, for all b∈N.
ByF, we will denote the space of all sets with finite number of elements.Theorem 23.
Raise upω∈CN and tl∈I∩ℓ∞ with t0>1. Then, Hω∈Acesrtυ, if and only if ωbb=0∞∈c0.Proof.
LetHω∈Acesrtυ, so Hω∈Kcesrtυ. Suppose limb⟶∞ωb≠0. Therefore, we have ρ>0 so as the set Kρ=b∈N:ωb≥ρàF, if αbb∈N⊂Kρ. Hence, eαb:αb∈Kρ∈ℓ∞ is an infinite set in cesrtυ. Since
(18)υHωeαa−Hωeαb=υωeαa−ωeαb=∑l=0∞∑k=0lrkωkeαak−eαbkl+1rtl≥∑l=0∞∑k=0lrkρeαak−eαbkl+1rtl≥inflρtlυeαa−eαb,with αa,αb∈Kρ. Therefore, eαb:αb∈Kρ∈ℓ∞, which cannot have a convergent subsequence under Hω. Hence, Hω∉Kcesrtυ, which implies Hω∉Acesrtυ; this gives an inconsistency. So limb⟶∞ωb=0. On the other hand, let limb⟶∞ωb=0. Hence, for all ρ>0, one has Kρ=b∈N:ωb≥ρ⊂F. Hence, for each ρ>0, we have dimcesrtυKρ=dimCKρ<∞. So Hω∈FcesrtυKρ. Define ωa∈CN, for all a∈N, by
(19)ωab=ωb,b∈K1/a+1,0,otherwise.
It is clear thatHωa∈FcesrtυB1/a+1 as dimcesrtυB1/a+1<∞, for all a∈N. From tl∈I∩ℓ∞ with t0>1, one can see
(20)υHω−Hωaf=υωb−ωabfbb=0∞=∑l=0∞∑b=0lrbωb−ωabfbl+1rtl=∑l=0,l∈K1/a+1∞∑b=0lrbωb−ωabfbl+1rtl+∑l=0,l∉K1/a+1∞∑b=0lrbωb−ωabfbl+1rtl=∑l=0,l∉K1/a+1∞∑b=0lrbωbfbl+1rtl≤1a+1t0∑l=0,l∉K1/a+1∞∑b=0lrbfbl+1rtl<1a+1t0∑l=0∞∑b=0lrbfbl+1rtl=1a+1t0υf.
Hence,Hω−Hωa≤1/a+1t0. This gives that Hω is a limit of finite rank maps. Therefore, Hω∈Acesrtυ.Theorem 24.
Pick upω∈CN and tl∈I∩ℓ∞ with t0>1. Then, Hω∈Kcesrtυ, if and only if ωbb=0∞∈c0.Proof.
Obviously, sinceAcesrtυÞKcesrtυ.Corollary 25.
Iftl∈I∩ℓ∞ with t0>1, then KcesrtυÞBcesrtυ.Proof.
Asω=1,1,⋯ is created the multiplication map I on cesrtυ, I∉Kcesrtυ and I∈Bcesrtυ.Theorem 26.
Ifcesrtυ is a prequasi Banach pss and Hω∈Bcesrtυ, then there are α>0 and η>0 so as α<ωb<η, with b∈kerωc, if and only if RangeHω is closed.Proof.
Assume the sufficient setup is confirmed. Hence, there isρ>0 so as ωb≥ρ, with b∈kerωc. To show that RangeHω is closed, if g is a limit point of RangeHω, we have Hωfb∈cesrtυ, with b∈N so that limb⟶∞Hωfb=g. Obviously, the sequence Hωfb is a Cauchy sequence. As tl∈I∩ℓ∞ with t0>1, one has
(21)υHωfa−Hωfb=∑l=0∞∑k=0lrkωkfak−ωkfbkl+1rtl=∑l=0,l∈kerωc∞∑k=0lrkωkfak−ωkfbkl+1rtl+∑l=0,l∉kerωc∞∑k=0lrkωkfak−ωkfbkl+1rtl≥∑l=0,l∈kerωc∞∑k=0lrkωkfak−ωkfbkl+1rtl=∑l=0∞∑k=0lrkωkuak−ωkubkl+1rtl>∑l=0∞∑k=0lrkρuak−ubkl+1rtl≥inflρtlυua−ub,where
(22)uak=fak,k∈kerωc,0,k∉kerωc.
This implies thatua is a Cauchy sequence in cesrtυ. As cesrtυ is complete, there is f∈cesrtυ so that limb⟶∞ub=f. Since Hω∈Bcesrtυ, we have limb⟶∞Hωub=Hωf. But limb⟶∞Hωub=limb⟶∞Hωfb=g. So Hωf=g. Hence, g∈RangeHω. Therefore, RangeHω is closed. Next, suppose the necessity setup is satisfied. So there is ρ>0 so as υHωf≥ρυf, with f∈cesrtυkerωc. If K=b∈kerωc:ωb<ρ≠∅, then for a0∈K, one has
(23)υHωea0=υωbea0bb=0∞=∑l=0∞∑b=0lrbωbea0bl+1rtl<∑l=0∞∑b=0lrbea0bρl+1rtl≤suplρtlυea0.This gives an inconsistency. Therefore, K=ϕ, we have ∣ωb∣≥ρ, with b∈kerωc. This proves the theorem.Theorem 27.
Pick upω∈CN and cesrtυ be a prequasi Banach pss. Then, there is α>0 and η>0 so that α<ωb<η, with b∈N, if and only if Hω∈Bcesrtυ is invertible.Proof.
Let the setup be true. Assumeκ∈CN with κb=1/ωb. By using Theorem 21, the maps Hω and Hκ are bounded linear. We have Hω·Hκ=Hκ·Hω=I. Therefore, Hκ=Hω−1. Next, let Hω be invertible. So RangeHω=cesrtυN. Hence, RangeHω is closed. Therefore, by Theorem 26, there is α>0 so that ∣ωb∣≥α, for each b∈kerωc. We have kerω=∅, if ωb0=0, with b0∈N; this gives eb0∈kerHω which is an inconsistency, as kerHω is trivial. Therefore, ωb≥α, with b∈N. As Hω∈ℓ∞, from Theorem 21, there is η>0 so that ∣ωb∣≤η, with b∈N. Hence, one has α≤ωb≤η, with b∈N.Theorem 28.
Raise upcesrtυ to be a prequasi Banach pss and Hω∈Bcesrtυ. Then, Hω is a Fredholm map, if and only if (i) kerωÜN is finite and (ii) ωb≥ρ, with b∈kerωc.Proof.
Assume the sufficient condition is satisfied. LetkerωÜN be infinite; hence, eb∈kerHω, with b∈kerω. Since eb’s are linearly independent, this gives dimkerHω=∞; this implies an inconsistency. Hence, kerωÜN must be finite. The condition (ii) comes from Theorem 26. Next, let the setup (i) and (ii) be confirmed. From Theorem 26, the setup (ii) implies that RangeHω is closed. The setting (i) gives that dimkerHω<∞ and dimRangeHωc<∞. This implies that Hω is Fredholm.


## 4. Prequasi Ideal

In this section, firstly, we introduce the sufficient setting (not necessary) oncesrtυ such that F is dense in Bcesrtυs. This investigates a negative answer of Rhoades [32] open problem about the linearity of s-type cesrtυ spaces. Secondly, for which conditions on cesrtυ are Bcesrtυs complete and closed? Thirdly, we give the sufficient setup on cesrtυ such that Bcesrtυα is strictly contained for different powers. We explain the settings in order that Bcesrtυα is minimum. Fourthly, we explain the conditions so that the Banach prequasi ideal Bcesrtυs is simple. Fifthly, we give the sufficient conditions on cesrtυ such that the space of all bounded linear maps which sequence of eigenvalues in cesrtυ equals Bcesrtυs.


### 4.1. Finite Rank Prequasi Ideal

Theorem 29.
BcesrtυsP,Q=FP,Q¯, whenever tl∈I∩ℓ∞ with t0>1. But the converse is not necessarily true.Proof.
To show thatFP,Q¯⊆BcesrtυsP,Q, as el∈cesrtυ, with l∈N and cesrtυ is a linear space, let Z∈FP,Q, one has slZl=0∞∈F. To show that BcesrtυsP,Q⊆FP,Q¯, as t0>1 and tl∈I∩ℓ∞, one can see ∑l=0∞1/l+1rtl<∞. For Z∈BcesrtυsP,Q, we have slZl=0∞∈cesrtυ. As υslZl=0∞<∞, suppose ρ∈0,1, then there is l0∈N−0 with υslZl=l0∞<ρ/2ℏ+3ηd, for some d≥1, where η=max1,∑l=l0∞1/l+1rtl. As slZ is decreasing, one has
(24)∑l=l0+12l0∑j=0lrjs2l0Zl+1rtl≤∑l=l0+12l0∑j=0lrjsjZl+1rtl≤∑l=l0∞∑j=0lrjsjZl+1rtl<ρ2ℏ+3ηd.
Therefore, there isY∈F2l0P,Q so that rankY≤2l0 and
(25)∑l=2l0+13l0∑j=0lrjZ−Yl+1rtl≤∑l=l0+12l0∑j=0lrjZ−Yl+1rtl<ρ2ℏ+3ηd,since tl∈I∩ℓ∞, we have
(26)supl=l0∞∑j=0l0rjZ−Ytl<ρ22ℏ+2η.
Therefore, one has(27)∑l=0l0∑j=0lrjZ−Yl+1rtl<ρ2ℏ+3ηd.
Astl∈I∩ℓ∞, t0>1 and by using inequalities (5) and (24)–(27), one has
(28)dZ,Y=υslZ−Yl=0∞=∑l=03l0−1∑j=0lrjsjZ−Yl+1rtl+∑l=3l0∞∑j=0lrjsjZ−Yl+1rtl≤∑l=03l0∑j=0lrjZ−Yl+1rtl+∑l=l0∞∑j=0l+2l0rjsjZ−Yl+2l0+1rtl+2l0≤∑l=03l0∑j=0lrjZ−Yl+1rtl+∑l=l0∞∑j=0l+2l0rjsjZ−Yl+1rtl≤3∑l=0l0∑j=0irjZ−Yl+1rtl+∑l=l0∞∑j=02l0−1rjsjZ−Y+∑j=2l0l+2l0rjsjZ−Yl+1rtl≤3∑l=0l0∑j=0lrjZ−Yl+1rtl+2ℏ−1∑l=l0∞∑j=02l0−1rjsjZ−Yl+1rtl+∑l=l0∞∑j=2l0l+2l0rjsjZ−Yl+1rtl≤3∑l=0l0∑j=0lrjZ−Yl+1rtl+2ℏ−1∑l=l0∞∑j=02l0−1rljZ−Yl+1rtl+∑l=l0∞∑j=0lrj+2l0sj+2l0Z−Yl+1rtl≤3∑l=0l0∑j=0lrjZ−Yl+1rtl+22ℏ−1supl=l0∞∑j=0l0rjZ−Ytl∑l=l0∞l+1r−tl+2ℏ−1∑l=l0∞∑j=0lrjsjZl+1rtl<ρ.
Conversely, we give a counterexample asI4∈Bces0tυsP,Q but 0<r<1 is not verified. This confirms the proof.


### 4.2. Banach and Closed Prequasi Ideal

Theorem 30.
The functionΨ is a prequasi norm on BVυs, where ΨZ=υsbZb=0∞, for all Z∈BVυsP,Q, if Vυ is a premodular pss.Proof.
LetVυ be a premodular pss; hence, Ψ satisfies the following conditions:
(1)
For allX∈BVυsP,Q, ΨX=υsbXb=0∞≥0 and ΨX=υsbXb=0∞=0, if and only if sbX=0, for all b∈N, if and only if X=0(2)
There isE0≥1 such that ΨρX=υsbρXb=0∞≤E0ρΨX, for all X∈BVυsP,Q and ρ∈C(3)
There isD≥1 such that for all X1,X2∈BVυsP,Q, we have(29)ΨX1+X2=υsbX1+X2b=0∞≤G0υsb/2X1b=0∞+υsb/2X2b=0∞≤G0D0υsbX1b=0∞+υsbX2b=0∞≤DΨX1+ΨX2.(4)
There isρ≥1 such that if X∈BP0,P, Y∈BVυsP,Q, and Z∈BQ,Q0, then ΨZYX=υsbZYXb=0∞≤υXZsbYb=0∞≤ρXΨYZTheorem 31.
Pick uptl∈I∩ℓ∞ with t0>1, then Bcesrtυs,Ψ is a prequasi Banach ideal, where ΨX=υslXl=0∞.Proof.
Ascesrtυ is a premodular pss, hence from Theorem 30, Ψ is a prequasi norm on Bcesrtυs. Suppose Xbb∈N is a Cauchy sequence in BcesrtυsP,Q. As BP,Q⊇BcesrtυsP,Q, one has
(30)ΨXa−Xb=∑l=0∞∑z=0lrzszXa−Xbl+1rtl≥∑l=0∞Xa−Xbl+1rtl≥Xa−Xbt0,so Xbb∈N is a Cauchy sequence in BP,Q. Since BP,Q is a Banach space, there is X∈BP,Q with limb⟶∞Xb−X=0. Since slXbl=0∞∈cesrtυ, for all b∈N, from Definition 15 parts (ii), (iii), and (v), one can see
(31)ΨX=∑l=0∞∑z=0lrzszXl+1rtl≤2ℏ−1∑l=0∞∑z=0lrzsz/2X−Xbl+1rtl+2ℏ−1∑l=0∞∑z=0lrzsz/2Xbl+1rtl≤2ℏ−1∑l=0∞∑z=0lrzX−Xbl+1rtl+2ℏ−1D0∑l=0∞∑z=0lrzszXbl+1rtl<∞.
Hence,slXl=0∞∈cesrtυ, so X∈BcesrtυsP,Q.Theorem 32.
SupposeP, Q be normed spaces and tl∈I∩ℓ∞ with t0>1, then Bcesrtυs,Ψ is a prequasi closed ideal, where ΨX=υslXl=0∞.Proof.
Ascesrtυ is a premodular pss, from Theorem 30, Ψ is a prequasi norm on Bcesrtυs. Assume Xb∈BcesrtυsP,Q, for each b∈N and limb⟶∞ΨXb−X=0. As BP,Q⊇BcesrtυsP,Q, we have
(32)ΨX−Xb=∑l=0∞∑z=0lrzszX−Xbl+1rtl≥∑l=0∞X−Xbl+1rtl≥X−Xbt0.Hence, Xbb∈N is a convergent sequence in BP,Q. Since slXbl=0∞∈cesrtυ, for every b∈N, by using Definition 15 parts (ii), (iii), and (v), one can see
(33)ΨX=∑l=0∞∑z=0lrzszXl+1rtl≤2ℏ−1∑l=0∞∑z=0lrzsz/2X−Xbl+1rtl+2ℏ−1∑l=0∞∑z=0lrzsz/2Xbl+1rtl≤2ℏ−1∑l=0∞∑z=0lrzX−Xbl+1rtl+2ℏ−1D0∑l=0∞∑z=0lrzszXbl+1rtl<∞.
We getslXl=0∞∈cesrtυ, so X∈BcesrtυsP,Q.


### 4.3. Minimum Prequasi Ideal

Theorem 33.
For any infinite dimensional Banach spacesP, Q and tl1∈ℓ∞, tl2∈ℓ∞ with 1<tl1<tl2, for all l∈N, then
(34)Bcesrtn1υsP,QÞBcesrtn2υsP,QÜBP,Q.Proof.
SupposeZ∈Bcesrtn1υsP,Q, then slZ∈cesrtn1υ. One has
(35)∑l=0∞1l+1r∑z=0lrzszZtl2<∑l=0∞1l+1r∑z=0lrzszZtl1<∞.Then, Z∈Bcesrtn2υsP,Q. After, if we choose slZl=0∞ so as ∑z=0lrzszZ=l+1r/l+1tl1, we have Z∈BP,Q such that
(36)∑l=0∞1l+1r∑z=0lrzszZtl1=∑l=0∞1l+1=∞,∑l=0∞1l+1r∑z=0lrzszZtl2=∑l=0∞1l+1tl2/tl1<∞.
SoX∉Bcesrtn1υsP,Q and X∈Bcesrtn2υsP,Q. Clearly, Bcesrtn2υsP,Q⊂BP,Q. Next, if we take slXl=0∞ such that ∑z=0lrzszZ=l+1r/l+1tl2. We have X∈BP,Q so that X∉Bcesrtn2υsP,Q. This confirms the proof.Theorem 34.
Pick up any infinite dimensional Banach spacesP, Q and tl∈I∩ℓ∞ with t0>1; hence, Bcesrtυα is minimum.Proof.
Assume the setup is confirmed. SoBcesrtα,Ψ, where ΨZ=∑l=0∞1/l+1r∑z=0lrzαzZtl, is a prequasi Banach ideal. Let BcesrtαP,Q=BP,Q; hence, there is η>0 so as ΨZ≤ηZ, for each Z∈BP,Q. Then, by Dvoretzky’s theorem [33] with b∈N, one has quotient spaces P/Yb and subspaces Mb of Q which can be mapped onto ℓ2b by isomorphisms Vb and Xb with VbVb−1≤2 and XbXb−1≤2. If Ib is the identity map on ℓ2b, Tb is the quotient map from P onto P/Yb and Jb is the natural embedding map from Mb into Q. Assume mz be the Bernstein numbers [34]; hence,
(37)1=mzIb=mzXbXb−1IbVbVb−1≤XbmzXb−1IbVbVb−1=XbmzJbXb−1IbVbVb−1≤XbdzJbXb−1IbVbVb−1=XbdzJbXb−1IbVbTbVb−1≤XbαzJbXb−1IbVbTbVb−1,for 0≤l≤b. We have
(38)∑z=0lrz≤∑z=0lXbrzαzJbXb−1IbVbTbVb−1⇒1≤XbVb−1tl∑z=0lrzαzJbXb−1IbVbTbl+1rtl.
Hence, for someρ≥1, one has
(39)b+1≤ρXbVb−1∑l=0b∑z=0lrzαzJbXb−1IbVbTbl+1rtl⇒b+1≤ρXbVb−1ΨJbXb−1IbVbTb⇒b+1≤ρηXbVb−1JbXb−1IbVbTb⇒b+1≤ρηXbVb−1JbXb−1IbVbTb=ρηXbVb−1Xb−1IbVb≤4ρη.
We have an inconsistency, asb is an arbitrary. Then, P and Q both cannot be infinite dimensional when BcesrtαP,Q=BP,Q. This completes the proof.Theorem 35.
Upon any infinite dimensional Banach spacesP, Q and tl∈I∩ℓ∞ with t0>1, then Bcesrtd is minimum.


### 4.4. Simple Banach Prequasi Ideal

Theorem 36.
PresumeP, Q be infinite dimensional Banach spaces. Let tl1∈ℓ∞ and tl2∈ℓ∞ with 1<tl1<tl2, with l∈N, then
(40)BBcesrtn2υsP,Q,Bcesrtn1υsP,Q=ABcesrtn2υsP,Q,Bcesrtn1υsP,Q.Proof.
ForX∈BBcesrtn2υsP,Q,Bcesrtn1υsP,Q and X∉ABcesrtn2υsP,Q,Bcesrtn1υsP,Q. From Lemma 1, one has Y∈BBcesrtn2υsP,Q and Z∈BBcesrtn1υsP,Q with ZXYIb=Ib. Therefore, for each b∈N, we have
(41)IbBcesrtl1υsP,Q=∑l=0∞∑z=0lrzszIbl+1rtl1≤ZXYIbBcesrtl2υsP,Q≤∑l=0∞∑z=0irzszIjl+1rti2.
This defies Theorem33. Then, X∈ABcesrtn2υsP,Q,Bcesrtn1υsP,Q, which affects the proof.Corollary 37.
Upon any infinite dimensional Banach spacesP and Q, if tl1∈ℓ∞ and tl2∈ℓ∞ with 1<tl1<tl2, for every l∈N, then
(42)BBcesrtn2υsP,Q,Bcesrtn1υsP,Q=KBcesrtn2υsP,Q,Bcesrtn1υsP,Q.Proof.
Easily, asA⊂K.Theorem 38.
Raise uptl∈I∩ℓ∞ with t0>1, and then, Bcesrtυs be simple.Proof.
Let the closed idealKBcesrtυsP,Q include a map X∉ABcesrtυsP,Q. From Lemma 1, one has Y,Z∈BBcesrtυsP,Q with ZXYIb=Ib. This gives that IBcesrtυsP,Q∈KBcesrtυsP,Q. Accordingly, BBcesrtυsP,Q=KBcesrtυsP,Q. Hence, Bcesrtυs is a simple Banach space.


### 4.5. Eigenvalues ofs-Type Mappings

Notation 39.
(43)BVsρ≔BVsρP,Q;PandQareBanachSpaces,whereBVsρP,Q≔X∈BP,Q:ρlXn=0∞∈VandX−ρlXIisnotinvertible,foralll∈N.Theorem 40.
Pick up any infinite dimensional Banach spacesP and Q. Suppose tl∈I∩ℓ∞ with t0>1, then
(44)BcesrtυsρP,Q=BcesrtυsP,Q.Proof.
LetX∈BcesrtυsρP,Q; hence, ρlXl=0∞∈cesrtυ and X−ρlXI=0, for all l∈N. We have X=ρlXI, with l∈N, so slX=slρlXI=ρlX, with l∈N. Therefore, slXl=0∞∈cesrtυ, so X∈BcesrtυsP,Q.
Secondly, letX∈BcesrtυsP,Q. Therefore, slXl=0∞∈cesrtυ. Hence, we have
(45)∑l=0∞∑z=0lrzszX∑z=0lrztl≥∑l=0∞slXtl.
Soliml⟶∞slX=0. Assume X−slXI−1 exists, for every l∈N. Therefore, X−slXI−1 exists and bounded, for every l∈N. So liml⟶∞X−slXI−1=X−1 exists and bounded. From the prequasi operator ideal of Bcesrtυs,Ψ, we obtain
(46)I=XX−1∈BcesrtυsP,Q⇒slIl=0∞∈Ξr,t⇒liml⟶∞slI=0.
Butliml⟶∞slI=1. Therefore, X−slXI=0, for every l∈N. This gives X∈BcesrtυsρP,Q. This provides the proof.


## 4.1. Finite Rank Prequasi Ideal

Theorem 29.
BcesrtυsP,Q=FP,Q¯, whenever tl∈I∩ℓ∞ with t0>1. But the converse is not necessarily true.Proof.
To show thatFP,Q¯⊆BcesrtυsP,Q, as el∈cesrtυ, with l∈N and cesrtυ is a linear space, let Z∈FP,Q, one has slZl=0∞∈F. To show that BcesrtυsP,Q⊆FP,Q¯, as t0>1 and tl∈I∩ℓ∞, one can see ∑l=0∞1/l+1rtl<∞. For Z∈BcesrtυsP,Q, we have slZl=0∞∈cesrtυ. As υslZl=0∞<∞, suppose ρ∈0,1, then there is l0∈N−0 with υslZl=l0∞<ρ/2ℏ+3ηd, for some d≥1, where η=max1,∑l=l0∞1/l+1rtl. As slZ is decreasing, one has
(24)∑l=l0+12l0∑j=0lrjs2l0Zl+1rtl≤∑l=l0+12l0∑j=0lrjsjZl+1rtl≤∑l=l0∞∑j=0lrjsjZl+1rtl<ρ2ℏ+3ηd.
Therefore, there isY∈F2l0P,Q so that rankY≤2l0 and
(25)∑l=2l0+13l0∑j=0lrjZ−Yl+1rtl≤∑l=l0+12l0∑j=0lrjZ−Yl+1rtl<ρ2ℏ+3ηd,since tl∈I∩ℓ∞, we have
(26)supl=l0∞∑j=0l0rjZ−Ytl<ρ22ℏ+2η.
Therefore, one has(27)∑l=0l0∑j=0lrjZ−Yl+1rtl<ρ2ℏ+3ηd.
Astl∈I∩ℓ∞, t0>1 and by using inequalities (5) and (24)–(27), one has
(28)dZ,Y=υslZ−Yl=0∞=∑l=03l0−1∑j=0lrjsjZ−Yl+1rtl+∑l=3l0∞∑j=0lrjsjZ−Yl+1rtl≤∑l=03l0∑j=0lrjZ−Yl+1rtl+∑l=l0∞∑j=0l+2l0rjsjZ−Yl+2l0+1rtl+2l0≤∑l=03l0∑j=0lrjZ−Yl+1rtl+∑l=l0∞∑j=0l+2l0rjsjZ−Yl+1rtl≤3∑l=0l0∑j=0irjZ−Yl+1rtl+∑l=l0∞∑j=02l0−1rjsjZ−Y+∑j=2l0l+2l0rjsjZ−Yl+1rtl≤3∑l=0l0∑j=0lrjZ−Yl+1rtl+2ℏ−1∑l=l0∞∑j=02l0−1rjsjZ−Yl+1rtl+∑l=l0∞∑j=2l0l+2l0rjsjZ−Yl+1rtl≤3∑l=0l0∑j=0lrjZ−Yl+1rtl+2ℏ−1∑l=l0∞∑j=02l0−1rljZ−Yl+1rtl+∑l=l0∞∑j=0lrj+2l0sj+2l0Z−Yl+1rtl≤3∑l=0l0∑j=0lrjZ−Yl+1rtl+22ℏ−1supl=l0∞∑j=0l0rjZ−Ytl∑l=l0∞l+1r−tl+2ℏ−1∑l=l0∞∑j=0lrjsjZl+1rtl<ρ.
Conversely, we give a counterexample asI4∈Bces0tυsP,Q but 0<r<1 is not verified. This confirms the proof.


## 4.2. Banach and Closed Prequasi Ideal

Theorem 30.
The functionΨ is a prequasi norm on BVυs, where ΨZ=υsbZb=0∞, for all Z∈BVυsP,Q, if Vυ is a premodular pss.Proof.
LetVυ be a premodular pss; hence, Ψ satisfies the following conditions:
(1)
For allX∈BVυsP,Q, ΨX=υsbXb=0∞≥0 and ΨX=υsbXb=0∞=0, if and only if sbX=0, for all b∈N, if and only if X=0(2)
There isE0≥1 such that ΨρX=υsbρXb=0∞≤E0ρΨX, for all X∈BVυsP,Q and ρ∈C(3)
There isD≥1 such that for all X1,X2∈BVυsP,Q, we have(29)ΨX1+X2=υsbX1+X2b=0∞≤G0υsb/2X1b=0∞+υsb/2X2b=0∞≤G0D0υsbX1b=0∞+υsbX2b=0∞≤DΨX1+ΨX2.(4)
There isρ≥1 such that if X∈BP0,P, Y∈BVυsP,Q, and Z∈BQ,Q0, then ΨZYX=υsbZYXb=0∞≤υXZsbYb=0∞≤ρXΨYZTheorem 31.
Pick uptl∈I∩ℓ∞ with t0>1, then Bcesrtυs,Ψ is a prequasi Banach ideal, where ΨX=υslXl=0∞.Proof.
Ascesrtυ is a premodular pss, hence from Theorem 30, Ψ is a prequasi norm on Bcesrtυs. Suppose Xbb∈N is a Cauchy sequence in BcesrtυsP,Q. As BP,Q⊇BcesrtυsP,Q, one has
(30)ΨXa−Xb=∑l=0∞∑z=0lrzszXa−Xbl+1rtl≥∑l=0∞Xa−Xbl+1rtl≥Xa−Xbt0,so Xbb∈N is a Cauchy sequence in BP,Q. Since BP,Q is a Banach space, there is X∈BP,Q with limb⟶∞Xb−X=0. Since slXbl=0∞∈cesrtυ, for all b∈N, from Definition 15 parts (ii), (iii), and (v), one can see
(31)ΨX=∑l=0∞∑z=0lrzszXl+1rtl≤2ℏ−1∑l=0∞∑z=0lrzsz/2X−Xbl+1rtl+2ℏ−1∑l=0∞∑z=0lrzsz/2Xbl+1rtl≤2ℏ−1∑l=0∞∑z=0lrzX−Xbl+1rtl+2ℏ−1D0∑l=0∞∑z=0lrzszXbl+1rtl<∞.
Hence,slXl=0∞∈cesrtυ, so X∈BcesrtυsP,Q.Theorem 32.
SupposeP, Q be normed spaces and tl∈I∩ℓ∞ with t0>1, then Bcesrtυs,Ψ is a prequasi closed ideal, where ΨX=υslXl=0∞.Proof.
Ascesrtυ is a premodular pss, from Theorem 30, Ψ is a prequasi norm on Bcesrtυs. Assume Xb∈BcesrtυsP,Q, for each b∈N and limb⟶∞ΨXb−X=0. As BP,Q⊇BcesrtυsP,Q, we have
(32)ΨX−Xb=∑l=0∞∑z=0lrzszX−Xbl+1rtl≥∑l=0∞X−Xbl+1rtl≥X−Xbt0.Hence, Xbb∈N is a convergent sequence in BP,Q. Since slXbl=0∞∈cesrtυ, for every b∈N, by using Definition 15 parts (ii), (iii), and (v), one can see
(33)ΨX=∑l=0∞∑z=0lrzszXl+1rtl≤2ℏ−1∑l=0∞∑z=0lrzsz/2X−Xbl+1rtl+2ℏ−1∑l=0∞∑z=0lrzsz/2Xbl+1rtl≤2ℏ−1∑l=0∞∑z=0lrzX−Xbl+1rtl+2ℏ−1D0∑l=0∞∑z=0lrzszXbl+1rtl<∞.
We getslXl=0∞∈cesrtυ, so X∈BcesrtυsP,Q.


## 4.3. Minimum Prequasi Ideal

Theorem 33.
For any infinite dimensional Banach spacesP, Q and tl1∈ℓ∞, tl2∈ℓ∞ with 1<tl1<tl2, for all l∈N, then
(34)Bcesrtn1υsP,QÞBcesrtn2υsP,QÜBP,Q.Proof.
SupposeZ∈Bcesrtn1υsP,Q, then slZ∈cesrtn1υ. One has
(35)∑l=0∞1l+1r∑z=0lrzszZtl2<∑l=0∞1l+1r∑z=0lrzszZtl1<∞.Then, Z∈Bcesrtn2υsP,Q. After, if we choose slZl=0∞ so as ∑z=0lrzszZ=l+1r/l+1tl1, we have Z∈BP,Q such that
(36)∑l=0∞1l+1r∑z=0lrzszZtl1=∑l=0∞1l+1=∞,∑l=0∞1l+1r∑z=0lrzszZtl2=∑l=0∞1l+1tl2/tl1<∞.
SoX∉Bcesrtn1υsP,Q and X∈Bcesrtn2υsP,Q. Clearly, Bcesrtn2υsP,Q⊂BP,Q. Next, if we take slXl=0∞ such that ∑z=0lrzszZ=l+1r/l+1tl2. We have X∈BP,Q so that X∉Bcesrtn2υsP,Q. This confirms the proof.Theorem 34.
Pick up any infinite dimensional Banach spacesP, Q and tl∈I∩ℓ∞ with t0>1; hence, Bcesrtυα is minimum.Proof.
Assume the setup is confirmed. SoBcesrtα,Ψ, where ΨZ=∑l=0∞1/l+1r∑z=0lrzαzZtl, is a prequasi Banach ideal. Let BcesrtαP,Q=BP,Q; hence, there is η>0 so as ΨZ≤ηZ, for each Z∈BP,Q. Then, by Dvoretzky’s theorem [33] with b∈N, one has quotient spaces P/Yb and subspaces Mb of Q which can be mapped onto ℓ2b by isomorphisms Vb and Xb with VbVb−1≤2 and XbXb−1≤2. If Ib is the identity map on ℓ2b, Tb is the quotient map from P onto P/Yb and Jb is the natural embedding map from Mb into Q. Assume mz be the Bernstein numbers [34]; hence,
(37)1=mzIb=mzXbXb−1IbVbVb−1≤XbmzXb−1IbVbVb−1=XbmzJbXb−1IbVbVb−1≤XbdzJbXb−1IbVbVb−1=XbdzJbXb−1IbVbTbVb−1≤XbαzJbXb−1IbVbTbVb−1,for 0≤l≤b. We have
(38)∑z=0lrz≤∑z=0lXbrzαzJbXb−1IbVbTbVb−1⇒1≤XbVb−1tl∑z=0lrzαzJbXb−1IbVbTbl+1rtl.
Hence, for someρ≥1, one has
(39)b+1≤ρXbVb−1∑l=0b∑z=0lrzαzJbXb−1IbVbTbl+1rtl⇒b+1≤ρXbVb−1ΨJbXb−1IbVbTb⇒b+1≤ρηXbVb−1JbXb−1IbVbTb⇒b+1≤ρηXbVb−1JbXb−1IbVbTb=ρηXbVb−1Xb−1IbVb≤4ρη.
We have an inconsistency, asb is an arbitrary. Then, P and Q both cannot be infinite dimensional when BcesrtαP,Q=BP,Q. This completes the proof.Theorem 35.
Upon any infinite dimensional Banach spacesP, Q and tl∈I∩ℓ∞ with t0>1, then Bcesrtd is minimum.


## 4.4. Simple Banach Prequasi Ideal

Theorem 36.
PresumeP, Q be infinite dimensional Banach spaces. Let tl1∈ℓ∞ and tl2∈ℓ∞ with 1<tl1<tl2, with l∈N, then
(40)BBcesrtn2υsP,Q,Bcesrtn1υsP,Q=ABcesrtn2υsP,Q,Bcesrtn1υsP,Q.Proof.
ForX∈BBcesrtn2υsP,Q,Bcesrtn1υsP,Q and X∉ABcesrtn2υsP,Q,Bcesrtn1υsP,Q. From Lemma 1, one has Y∈BBcesrtn2υsP,Q and Z∈BBcesrtn1υsP,Q with ZXYIb=Ib. Therefore, for each b∈N, we have
(41)IbBcesrtl1υsP,Q=∑l=0∞∑z=0lrzszIbl+1rtl1≤ZXYIbBcesrtl2υsP,Q≤∑l=0∞∑z=0irzszIjl+1rti2.
This defies Theorem33. Then, X∈ABcesrtn2υsP,Q,Bcesrtn1υsP,Q, which affects the proof.Corollary 37.
Upon any infinite dimensional Banach spacesP and Q, if tl1∈ℓ∞ and tl2∈ℓ∞ with 1<tl1<tl2, for every l∈N, then
(42)BBcesrtn2υsP,Q,Bcesrtn1υsP,Q=KBcesrtn2υsP,Q,Bcesrtn1υsP,Q.Proof.
Easily, asA⊂K.Theorem 38.
Raise uptl∈I∩ℓ∞ with t0>1, and then, Bcesrtυs be simple.Proof.
Let the closed idealKBcesrtυsP,Q include a map X∉ABcesrtυsP,Q. From Lemma 1, one has Y,Z∈BBcesrtυsP,Q with ZXYIb=Ib. This gives that IBcesrtυsP,Q∈KBcesrtυsP,Q. Accordingly, BBcesrtυsP,Q=KBcesrtυsP,Q. Hence, Bcesrtυs is a simple Banach space.


## 4.5. Eigenvalues ofs-Type Mappings

Notation 39.
(43)BVsρ≔BVsρP,Q;PandQareBanachSpaces,whereBVsρP,Q≔X∈BP,Q:ρlXn=0∞∈VandX−ρlXIisnotinvertible,foralll∈N.Theorem 40.
Pick up any infinite dimensional Banach spacesP and Q. Suppose tl∈I∩ℓ∞ with t0>1, then
(44)BcesrtυsρP,Q=BcesrtυsP,Q.Proof.
LetX∈BcesrtυsρP,Q; hence, ρlXl=0∞∈cesrtυ and X−ρlXI=0, for all l∈N. We have X=ρlXI, with l∈N, so slX=slρlXI=ρlX, with l∈N. Therefore, slXl=0∞∈cesrtυ, so X∈BcesrtυsP,Q.
Secondly, letX∈BcesrtυsP,Q. Therefore, slXl=0∞∈cesrtυ. Hence, we have
(45)∑l=0∞∑z=0lrzszX∑z=0lrztl≥∑l=0∞slXtl.
Soliml⟶∞slX=0. Assume X−slXI−1 exists, for every l∈N. Therefore, X−slXI−1 exists and bounded, for every l∈N. So liml⟶∞X−slXI−1=X−1 exists and bounded. From the prequasi operator ideal of Bcesrtυs,Ψ, we obtain
(46)I=XX−1∈BcesrtυsP,Q⇒slIl=0∞∈Ξr,t⇒liml⟶∞slI=0.
Butliml⟶∞slI=1. Therefore, X−slXI=0, for every l∈N. This gives X∈BcesrtυsρP,Q. This provides the proof.


## 5. Conclusion

In this article, we explain some topological and geometric structure, of the multiplication maps defined oncesrtυ, of the class Bcesrtυs, and of the class Bcesrtυsρ. This article has a number of advantages for researchers such as studying the fixed points of any contraction maps on this prequasi normed sequence space which is a generalization of the quasi normed sequence spaces, a new general space of solutions for many difference equations, the spectrum of any bounded linear operators between any two Banach spaces with s-numbers in this sequence space and noting that the closed map ideals are certain to play an effective function in the principle of Banach lattices. We open the way for many authors to generalize the results by a sequence rl and generate cesrltlυ of nonabsolute type.



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*Source: 5567628-2021-04-28.xml*