# Pre-Quasi Simple Banach Operator Ideal Generated bys−Numbers

Authors: Awad A. Bakery; Afaf R. Abou Elmatty
Journal: Journal of Function Spaces
 (2020)

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## Abstract

LetE be a weighted Nakano sequence space or generalized Cesáro sequence space defined by weighted mean and by using s−numbers of operators from a Banach space X into a Banach space Y. We give the sufficient (not necessary) conditions on E such that the components SEX,Y≔T∈LX,Y:snTn=0∞∈E of the class SE form pre-quasi operator ideal, the class of all finite rank operators are dense in the Banach pre-quasi ideal SE, the pre-quasi operator ideal formed by the sequence of approximation numbers is strictly contained for different weights and powers, the pre-quasi Banach Operator ideal formed by the sequence of approximation numbers is small, and finally, the pre-quasi Banach operator ideal constructed by s−numbers is simple Banach space.


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## Body

## 1. Introduction

All through the paper,(1)LX,Y=T:X⟶Y;Tis a bounded linear operator;XandYare Banach spaces,and if X=Y, we write LX; by w, we denote the space of all real sequences and θ is the zero vector of E. Due to the immense applications in geometry of Banach spaces, spectral theory, geometry of Banach spaces, theory of eigenvalue distributions etc., the theory of operator ideals goals possesses an uncommon essentialness in useful examination. Some of operator ideals in the class of Banach spaces or Hilbert spaces are defined by different scalar sequence spaces. For example, the ideal of compact operators is defined by the space c0 of null sequences and Kolmogorov numbers. Pietsch [1] examined the quasi-ideals formed by the approximation numbers and classical sequence space ℓp0<p<∞. He showed that the ideals of nuclear operators and of Hilbert Schmidt operators between Hilbert spaces are defined by ℓ1 and ℓ2, respectively. Also, he proved that the class of all finite rank operators is dense in the Banach quasi ideal and the algebra Lℓp, where 1≤p<∞ contains one and only one nontrivial closed ideal. Pietsch [2] showed that the quasi Banach operator ideal formed by the sequence of approximation numbers is small. Makarov and Faried [3] proved that the quasi-operator ideal formed by the sequence of approximation numbers is strictly contained for different powers, i.e., for any infinite dimensional Banach spaces X and Y and for any q>p>0, it is true that SℓpappX,Y⫋SℓqappX,Y⫋LX,Y. In [4], Faried and Bakery studied the operator ideals constructed by approximation numbers and generalized Cesáro and Orlicz sequence spaces ℓM. In [5], Faried and Bakery introduced the concept of pre-quasi operator ideal which is more general than the usual classes of operator ideal; they studied the operator ideals constructed by s− numbers, generalized Cesáro and Orlicz sequence spaces ℓM, and showed that the operator ideal formed by the previous sequence spaces and approximation numbers is small under certain conditions. The idea of this paper is to study a generalized class SE by using the sequence of s-numbers and E (weighted Nakano sequence space or generalized Cesáro sequence space); we give sufficient (not necessary) conditions on E such that SE constructs a pre-quasi operator ideal, which gives a negative answer of Rhoades [6] open problem about the linearity of E−type spaces SE. The components of SE as a pre-quasi Banach operator ideal containing finite dimensional operators as a dense subset and its completeness are proved. The pre-quasi operator ideal formed by the sequence of approximation numbers is strictly contained for different weights and powers are determined. Finally, we show that the pre-quasi Banach operator ideal formed by E and approximation numbers is small under certain conditions. Furthermore, the sufficient conditions for which the pre-quasi Banach operator ideal constructed by s−numbers is a simple Banach space.


## 2. Definitions and Preliminaries

Definition 1.
(see[7]). An s-number function is a map defined on LX,Y which associates to each operator T∈LX,Y, a nonnegative scaler sequence snTn=0∞, assuming that the the following states are verified:(a)
T=s0T≥s1T≥s2T≥⋯≥0, for T∈LX,Y(b)
sm+n−1T1+T2≤smT1+snT2 for all T1,T2∈LX,Y, m, n∈ℕ(c)
Ideal property:snRVT≤RsnVT for all T∈LX0,X, V∈LX,Y and R∈LY,Y0, where X0 and Y0 are arbitrary Banach spaces(d)
IfG∈LX,Y and λ∈ℝ, we obtain snλG=λsnG(e)
Rank property: ifrankT≤n then snT=0 for each T∈LX,Y(f)
Norming property:sr≥nIn=0 or sr<nIn=1, where In represents the unit operator on the n-dimensional Hilbert space ℓ2n
There are several examples ofs-numbers, we mention the following:(1)
The n-th approximation number, denoted byαnT, is defined by(2)αnT=infT−B:B∈LX,Yand rankB≤n.(2)
Then-th Gel’fand number, denoted by cnT, is defined by cnT=αnJYT, where JY is a metric injection from the normed space Y to a higher space l∞Λ for an adequate index set Λ. This number is independent of the choice of the higher space l∞Λ.(3)
Then-th Kolmogorov number, denoted by dnT, is defined by(3)dnT=infdimY≤nsupx≤1infy∈YTx−y.(4)
Then-th Weyl number, denoted by xnT, is defined by(4)xnT=infαnTB:B:ℓ2⟶X≤1.(5)
Then-th Chang number, denoted by ynT, is defined by(5)ynT=infαnBT:B:Y⟶ℓ2≤1.(6)
Then-th Hilbert number, denoted by hnT, is defined by(6)hnT=supαnBTA:B:Y⟶ℓ2≤1andA:ℓ2⟶X≤1.Remark.
(see[7]). Among all the s-number sequences defined above, it is easy to verify that the approximation number, αnT, is the largest and the Hilbert number, hnT, is the smallest s-number sequence, i.e., hnT≤snT≤αnT for any bounded linear operator T. If T is compact and defined on a Hilbert space, then all the s-numbers coincide with the eigenvalues of T, where T=T∗T1/2.Theorem 1.
(see [7], p.115). If T∈LX,Y, then(7)hnT≤xnT≤cnT≤αnT,hnT≤ynT≤dnT≤αnT.Definition 2.
(see [1]). A finite rank operator is a bounded linear operator whose dimension of the range space is finite. The space of all finite rank operators on E is denoted by FE.Definition 3.
(see[1]). A bounded linear operator A:E⟶E (where E is a Banach space) is called approximable if there are Sn∈FE, for all n∈ℕ such that limn⟶∞A−Sn=0. The space of all approximable operators on E is denoted by ΛE.Lemma 1 (see [1]).
LetT∈LX,Y. If T is not approximable, then there are operators G∈LX,X and B∈LY,Y, such that BTGek=ek for all k∈ℕ.Definition 4.
(see [1]). A Banach space X is called simple if the algebra LX contains one and only one nontrivial closed ideal.Definition 5.
(see [1]). A bounded linear operator A:E⟶E (where E is a Banach space) is called compact if AB1 has compact closure, where B1 denotes the closed unit ball of E. The space of all compact operators on E is denoted by LcE.Theorem 2 (see [1]).
IfE is infinite dimensional Banach space, we have(8)FE⫋ΛE⫋LcE⫋LE.Definition 6.
(see [1]). Let L be the class of all bounded linear operators between any arbitrary Banach spaces. A subclass U of L is called an operator ideal if each element UX,Y=U∩LX,Y fulfills the following conditions:(i)
IϜ∈U where Ϝ represents Banach space of one dimension(ii)
The spaceUX,Y is linear over ℝ(iii)
IfT∈LX0,X, V∈UX,Y and R∈LY,Y0; then, RVT∈UX0,Y0 (see [8, 9])
The concept of pre-quasi operator ideal is more general than the usual classes of operator ideal.Definition 7.
(see [5]). A function g:Ω⟶0,∞ is said to be a pre-quasi norm on the ideal Ω if the following conditions holds:(1)
For allT∈ΩX,Y, gT≥0 and gT=0, if and only if T=0(2)
There exists a constantM≥1 such that gλT≤MλgT, for all T∈ΩX,Y and λ∈ℝ(3)
There exists a constantK≥1 such that gT1+T2≤KgT1+gT2, for all T1,T2∈ΩX,Y(4)
There exists a constantC≥1 such that if T∈LX0,X, P∈ΩX,Y and R∈LY,Y0; then gRPT≤CRgPT, where X0 and Y0 are normed spaces.Theorem 3 (see [5]).
Every quasi norm on the idealΩ is a pre-quasi norm on the ideal Ω.
Letp=pn be a positive real and βnn∈ℕ be a sequence of positive real; the weighted Nakano sequence space is defined by(9)ℓβpn=x=xk∈ω:ρλx<∞for someλ>0whereρx=∑k=0∞βkxkpk.And ℓβpn,⋅ is a Banach space, where(10)x=infη>0:ρxη≤1.
Whenpn is bounded, we mark(11)ℓβpn=x=xk∈ω:∑k=0∞βkxkpk<∞.
Ifβn=1 for all n∈ℕ, then ℓβpn will reduce to ℓpn studied in [10, 11].
In [12], Şengönül defined the sequence space as(12)cesan,pn=x=xk∈ω:∃λ>0withρλx<∞,ρx=∑n=0∞an∑k=0nxkpk,where an and pn are the sequences of positive real and pn≥1 for all n∈ℕ. With the norm,(13)x=infη>0:ρxη≤1.Note.
(1)
Takingan=1/n+1 for all n∈ℕ, then cesan,pn is reduced to cespn studied by Sanhan and Suantai [13]
(2)
Takingan=1/n+1 and pn=p for all n∈ℕ, then cesan,pn is reduced to cesp studied by many authors (see [14–16])Definition 8.
(see [5]). Let E be a linear space of sequences, then E is called a (sss) if(1)
Forn∈ℕ, en∈E(2)
E is solid; i.e., assumingx=xn∈w, y=yn∈E and xn≤yn for all n∈ℕ, then x∈E(3)
xn/2n=0∞∈E, where n/2 indicates the integral part of n/2, whenever xnn=0∞∈EDefinition 9.
(see [5]). A subclass of the (sss) is called a premodular (sss) assuming that we have a map ρ:E⟶0,∞ with the following:(i)
Forx∈E, x=θ⟺ρx=0 with ρx≥0(ii)
For eachx∈E and scalar λ, we get a real number L≥1 for which ρλx≤λLρx(iii)
ρx+y≤Kρx+ρy for each x,y∈E, holds for a few numbers K≥1(iv)
Forn∈ℕ, xn≤yn, we obtain ρxn≤ρyn(v)
The inequality,ρxn≤ρxn/2≤K0ρxn holds, for some numbers K0≥1(vi)
F¯=Eρ, where F is the space of finite sequences(vii)
There is a steadyξ>0 such that ρλ,0,0,0,…≥ξλρ1,0,0,0,… for any λ∈ℝ
Condition (ii) gives the continuity ofρx at θ. The linear space E enriched with the metric topology formed by the premodular ρ will be indicated by Eρ. Moreover, condition (16) in Definition 8 and condition (vi) in Definition 9 explain that enn∈ℕ is a Schauder basis of Eρ.Notations.
The setsSE, SEX,Y, SEapp, and SEappX,Y (cf. [5]) as follows:(14)SE≔SEX,Y;XandYare Banach spaces,whereSEX,Y≔T∈LX,Y:siTi=0∞∈E.Also,SEapp≔SEappX,Y;XandYare Banach spaces,whereSEappX,Y≔T∈LX,Y:αiTi=0∞∈E.Theorem 4 (see [5]).
IfE is a (sss), then SE is an operator ideal.Theorem 5 (see [3]).
IfX and Y are infinite dimensional Banach spaces and μi is a monotonic decreasing sequence to zero, then there exists a bounded linear operator T such that(15)116μ3i≤αiT≤8μi+1.
Now and after, defineen=0,0,…,1,0,0,… where 1 appears at the nth place for all n∈ℕ and the given inequality will be used in the sequel:(16)an+bnpn≤Hanpn+bnpn,where H=max1,2h−1, h=supnpn, and pn≥1 for all n∈ℕ (see [17]).


## 3. Main Results

### 3.1. Linear Problem

We examine here the operator ideals created bys−numbers and also weighted Nakano sequence space or generalized Cesáro sequence space defined by weighted mean such that those classes of all bounded linear operators T between arbitrary Banach spaces with αnT in these sequence spaces type an ideal operator.Theorem 6.
ℓ
β
p
n is a (sss), if the following conditions are satisfied:(a1)
The sequencepn is increasing and bounded from above with pn>0 for all n∈ℕ(a2)
Eitherβn is monotonic decreasing or monotonic increasing such that there exists a constant C≥1, for which β2n+1≤CβnProof.
(1) Letx,y∈ℓβpn. Since pn is bounded, we get
(17)
∑
n
=
0
∞
β
n
x
n
+
y
n
p
n
≤
H
∑
n
=
0
∞
β
n
x
n
p
n
+
∑
n
=
0
∞
β
n
y
n
p
n
<
∞
,then x+y∈ℓβpn(2) Letλ∈ℝ and x∈ℓβpn. Since pn is bounded, we have(18)∑n=0∞βnλxnpn≤supnλpn∑n=0∞βnxnpn<∞
Then,λx∈ℓβpn. Therefore, by using Parts (1) and (2), we have that the space ℓβpn is linear. Also, en∈ℓβpn for all n∈ℕ, since(19)∑i=0∞βienipi=βn.(2)
Letxn≤yn for all n∈ℕ and y∈ℓβpn. Since βn>0 for all n∈ℕ, then(20)∑n=0∞βnxnpn≤∑n=0∞βnynpn<∞,and we get x∈ℓβpn.(3)
Letxn∈ℓβpn, βn be an increasing sequence. There exists C>0 such that β2n+1≤Cβn and pn be increasing; then, we have(21)∑n=0∞βnxn/2pn=∑n=0∞β2nxnp2n+∑n=0∞β2n+1xnp2n+1≤∑n=0∞β2nxnpn+∑n=0∞β2n+1xnpn≤2C∑n=0∞βnxnpn,and then xn/2∈ℓβpn.Theorem 7.
ces
a
n
,
p
n is a (sss), if the following conditions are satisfied:(b1)
The sequencepn is increasing and bounded with p0>1(b2)
∑n=0∞anpn<∞Proof.
(1)
Given thatx,y∈cesan,pn and λ1,λ2∈ℝ. Since pn is bounded, we have
(22)
∑
n
=
0
∞
a
n
∑
k
=
0
n
λ
1
x
k
+
λ
2
y
k
p
n
≤
H
sup
n
λ
1
p
n
∑
n
=
0
∞
a
n
∑
k
=
0
n
x
k
p
n
+
sup
n
λ
2
p
n
∑
n
=
0
∞
a
n
∑
k
=
0
n
y
k
p
n
<
∞
.
Hence,λ1x+λ2y∈cesan,pn; then, the space cesan,pn is linear. Also prove that em∈cesan,pn for all m∈ℕ, since ∑n=0∞anpn<∞. So we get(23)∑n=0∞an∑k=0nemkpn=∑n=m∞anpn≤∑n=0∞anpn<∞.
Hence,em∈cesan,pn.(2)
Letxn≤yn for all n∈ℕ and y∈cesan,pn. Since an>0 for all n∈ℕ, then(24)∑n=0∞an∑k=0nxkpn≤∑n=0∞an∑k=0nykpn<∞,and we get x∈cesan,pn.(3)
Letxn∈cesan,pn. Since pn is increasing and the sequence an with ∑n=0∞anpn<∞ is decreasing, then we have(25)∑n=0∞an∑k=0nxk/2pn=∑n=0∞a2n∑k=02nxk/2p2n+∑n=0∞a2n+1∑k=02n+1xk/2p2n+1≤∑n=0∞a2n∑k=02n2xk+xnpn+∑n=0∞a2n+1∑k=02n+12xkpn≤2h−1∑n=0∞2an∑k=0nxkpn+∑n=0∞an∑k=0nxkpn+∑n=0∞2an∑k=0nxk/2pn≤2h−12h+1∑n=0∞an∑k=0nxkpn+2h∑n=0∞an∑k=0nxkpn≤22h−1+2h−1+2h∑n=0∞an∑k=0nxkpn<∞.Hence, xn/2∈cesan,pn.
By using Theorem4, we can get the following corollaries:Corollary 1.
Let conditions (a1) and (a2) be satisfied; then,Sℓβpnapp be an operator ideal.Corollary 2.
S
ℓ
p
n
app is an operator ideal, if the sequence pn is increasing and bounded from above with pn>0 for all n∈ℕ.Corollary 3.
Ifp∈0,∞, then Sℓpapp is an operator ideal.Corollary 4.
Conditions (b1) and (b2) are satisfied; hence,Scesan,pnapp is an operator ideal.Corollary 5.
Assumepn is increasing with p0>1 and bounded, so Scespnapp is an operator ideal.Corollary 6.
Ifp∈1,∞, then Scespapp is an operator ideal.


## 3.1. Linear Problem

We examine here the operator ideals created bys−numbers and also weighted Nakano sequence space or generalized Cesáro sequence space defined by weighted mean such that those classes of all bounded linear operators T between arbitrary Banach spaces with αnT in these sequence spaces type an ideal operator.Theorem 6.
ℓ
β
p
n is a (sss), if the following conditions are satisfied:(a1)
The sequencepn is increasing and bounded from above with pn>0 for all n∈ℕ(a2)
Eitherβn is monotonic decreasing or monotonic increasing such that there exists a constant C≥1, for which β2n+1≤CβnProof.
(1) Letx,y∈ℓβpn. Since pn is bounded, we get
(17)
∑
n
=
0
∞
β
n
x
n
+
y
n
p
n
≤
H
∑
n
=
0
∞
β
n
x
n
p
n
+
∑
n
=
0
∞
β
n
y
n
p
n
<
∞
,then x+y∈ℓβpn(2) Letλ∈ℝ and x∈ℓβpn. Since pn is bounded, we have(18)∑n=0∞βnλxnpn≤supnλpn∑n=0∞βnxnpn<∞
Then,λx∈ℓβpn. Therefore, by using Parts (1) and (2), we have that the space ℓβpn is linear. Also, en∈ℓβpn for all n∈ℕ, since(19)∑i=0∞βienipi=βn.(2)
Letxn≤yn for all n∈ℕ and y∈ℓβpn. Since βn>0 for all n∈ℕ, then(20)∑n=0∞βnxnpn≤∑n=0∞βnynpn<∞,and we get x∈ℓβpn.(3)
Letxn∈ℓβpn, βn be an increasing sequence. There exists C>0 such that β2n+1≤Cβn and pn be increasing; then, we have(21)∑n=0∞βnxn/2pn=∑n=0∞β2nxnp2n+∑n=0∞β2n+1xnp2n+1≤∑n=0∞β2nxnpn+∑n=0∞β2n+1xnpn≤2C∑n=0∞βnxnpn,and then xn/2∈ℓβpn.Theorem 7.
ces
a
n
,
p
n is a (sss), if the following conditions are satisfied:(b1)
The sequencepn is increasing and bounded with p0>1(b2)
∑n=0∞anpn<∞Proof.
(1)
Given thatx,y∈cesan,pn and λ1,λ2∈ℝ. Since pn is bounded, we have
(22)
∑
n
=
0
∞
a
n
∑
k
=
0
n
λ
1
x
k
+
λ
2
y
k
p
n
≤
H
sup
n
λ
1
p
n
∑
n
=
0
∞
a
n
∑
k
=
0
n
x
k
p
n
+
sup
n
λ
2
p
n
∑
n
=
0
∞
a
n
∑
k
=
0
n
y
k
p
n
<
∞
.
Hence,λ1x+λ2y∈cesan,pn; then, the space cesan,pn is linear. Also prove that em∈cesan,pn for all m∈ℕ, since ∑n=0∞anpn<∞. So we get(23)∑n=0∞an∑k=0nemkpn=∑n=m∞anpn≤∑n=0∞anpn<∞.
Hence,em∈cesan,pn.(2)
Letxn≤yn for all n∈ℕ and y∈cesan,pn. Since an>0 for all n∈ℕ, then(24)∑n=0∞an∑k=0nxkpn≤∑n=0∞an∑k=0nykpn<∞,and we get x∈cesan,pn.(3)
Letxn∈cesan,pn. Since pn is increasing and the sequence an with ∑n=0∞anpn<∞ is decreasing, then we have(25)∑n=0∞an∑k=0nxk/2pn=∑n=0∞a2n∑k=02nxk/2p2n+∑n=0∞a2n+1∑k=02n+1xk/2p2n+1≤∑n=0∞a2n∑k=02n2xk+xnpn+∑n=0∞a2n+1∑k=02n+12xkpn≤2h−1∑n=0∞2an∑k=0nxkpn+∑n=0∞an∑k=0nxkpn+∑n=0∞2an∑k=0nxk/2pn≤2h−12h+1∑n=0∞an∑k=0nxkpn+2h∑n=0∞an∑k=0nxkpn≤22h−1+2h−1+2h∑n=0∞an∑k=0nxkpn<∞.Hence, xn/2∈cesan,pn.
By using Theorem4, we can get the following corollaries:Corollary 1.
Let conditions (a1) and (a2) be satisfied; then,Sℓβpnapp be an operator ideal.Corollary 2.
S
ℓ
p
n
app is an operator ideal, if the sequence pn is increasing and bounded from above with pn>0 for all n∈ℕ.Corollary 3.
Ifp∈0,∞, then Sℓpapp is an operator ideal.Corollary 4.
Conditions (b1) and (b2) are satisfied; hence,Scesan,pnapp is an operator ideal.Corollary 5.
Assumepn is increasing with p0>1 and bounded, so Scespnapp is an operator ideal.Corollary 6.
Ifp∈1,∞, then Scespapp is an operator ideal.


## 4. Topological Problem

The following question arises naturally: which sufficient conditions (not necessary) on the sequence spaceE (weighted Nakano sequence space and generalized Cesáro sequence space defined by weighted mean) are the ideal of the finite rank operators in the class of Banach spaces dense in SE? This gives a negative answer of Rhoades [6] open problem about the linearity of E−type spaces SE.Theorem 8.
F
X
,
Y
¯
=
S
ℓ
β
p
n
X
,
Y, whenever conditions (a1) and (a2) are satisfied.Proof.
First, we substantiate that each finite operatorT∈FX,Y belongs to SℓβpnX,Y. Given that enn∈ℕ⊂ℓβpn and the space ℓβpn is linear, then for all finite operators T∈FX,Y, i.e., the sequence snTn∈ℕ contains only finitely many numbers different from zero. Currently, we substantiate that SℓβpnX,Y⊆FX,Y¯. On taking T∈SℓβpnX,Y,we obtain snTn=0∞∈ℓβpn; hence, ρsnTn=0∞<∞; let ε∈0,1; at that point, there exists a m∈ℕ−0 such that ρsnTn=m∞<ε/4C2 for some C≥1. While snTn∈ℕ is decreasing, we get(26)∑n=m+12mβns2mTpn≤∑n=m+12mβnsnTpn≤∑n=m∞βnsnTpn<ε4C2.
Hence, there existsA∈F2mX,Y; rank A≤2m and(27)∑n=2m+13mβnT−Apn≤∑n=m+12mβnT−Apn<ε4C2.
Sincepn is a bounded, consider(28)∑n=0mβnT−Apn<ε4C2.
Letβn be monotonic increasing such that there exists a constant C≥1 for which β2n+1≤Cβn. Then, we have for n≥m that(29)β2m+n≤β2m+2n+1≤Cβm+n≤Cβ2n≤Cβ2n+1≤C2βn.
Sincepn is increasing, inequalities (26)–(29) give(30)dT,A=ρsnT−An=0∞=∑n=03m−1βnsnT−Apn+∑n=3m∞βnsnT−Apn≤∑n=03mβnT−Apn+∑n=m∞βn+2msn+2mT−Apn+2m≤3∑n=0mβnT−Apn+C2∑n=m∞βnsnTpn<ε.
SinceI3∈Sℓ1n, condition (a1) is not satisfied which gives a counter example of the converse statement. This finishes the proof.
From Theorem8, we can say that if (a1) and (a2) are satisfied, then every compact operator would be approximated by finite rank operators and the converse is not always true.Theorem 9.
F
X
,
Y
¯
=
S
ces
a
n
,
p
n
X
,
Y; assume that states (b1) and (b2) are fulfilled and the converse is not always true.Proof.
Primary sinceen∈cesan,pn, for each n∈ℕ and the space cesan,pn is linear, then for every finite mapping T∈FX,Y, i.e., the sequence snTn∈ℕ contains only fnitely many numbers different from zero. Hence, FX,Y¯⊆Scesan,pnX,Y. By letting T∈Scesan,pnX,Y, we obtain snTn∈ℕ∈cesan,pn, while ραnTn∈ℕ<∞. Let ε∈0,1. Then, there exists a number m∈ℕ−0 such that ρsnTn=m∞<ε/2h+2δC for some C≥1, where δ=max1,∑n=m∞anpn. As snT is decreasing for every n∈ℕ, we obtain(31)∑n=m+12man∑k=0ns2mTpn≤∑n=m+12man∑k=0nskTpn≤∑n=m∞an∑k=0nskTpn<ε2h+2δC.Then, there exists A∈F2mX,Y and rank A≤2m and(32)∑n=2m+13man∑k=0nT−Apn≤∑n=m+12man∑k=0nT−Apn<ε2h+2δC,and since pn is bounded, consider(33)supn=m∞∑k=0mT−Apn<ε2hδ.Hence, set(34)∑n=0man∑k=0nT−Apn<ε2h+3δC.
However,pn is increasing and an is decreasing for each n∈ℕ; by using inequalities (31)–(34), we have(35)dT,A=ρsnT−An=0∞=∑n=03m−1an∑k=0nskT−Apn+∑n=3m∞an∑k=0nskT−Apn≤∑n=03man∑k=0nT−Apn+∑n=m∞an+2m∑k=0n+2mskT−Apn+2m≤3∑n=0man∑k=0nT−Apn+∑n=m∞an∑k=02m−1skT−A+an∑k=2mn+2mskT−Apn≤3∑n=0man∑k=0nT−Apn+2h−1∑n=m∞an∑k=02m−1skT−Apn+∑n=m∞an∑k=2mn+2mskT−Apn≤3∑n=0man∑k=0nT−Apn+2h−1∑n=m∞an∑k=0mT−Apn+∑n=m∞an∑k=0nsk+2mT−Apn≤3∑n=0man∑k=0nT−Apn+2h−1supn=m∞∑k=0mT−Apn∑n=m∞anpn+2h−1∑n=m∞an∑k=0nskTpn<ε.
SinceI3∈Sces1,1, condition (b2) is not satisfied which gives a counter example of the converse statement. This finishes the proof.
From Theorem9, we can say that if conditions (b1) and (b2) are satisfied, then every compact operators would be approximated by finite rank operators and the converse is not always true.Corollary 7.
Ifpn is an increasing with pn>0 for all n∈ℕ and bounded from above, then SlpX,Y=FX,Y¯.Corollary 8.
If0<p<∞, then SlpX,Y=FX,Y¯.Corollary 9.
S
ces
p
n
X
,
Y
=
F
X
,
Y
¯, if pn is increasing with p0>1 and limn⟶∞suppn<∞.Corollary 10.
S
ces
p
X
,
Y
=
F
X
,
Y
¯, if 1<p<∞.


## 5. Completeness of the Pre-Quasi Ideal Components

For which sequence spaceE are the components of pre-quasi operator ideal SE complete?Theorem 10.
ℓ
β
p
n is a premodular (sss), if conditions (a1) and (a2) are satisfied.Proof.
We define the functionalρ on ℓβpn as ρx=∑n=0∞βnxnpn:(i)
Evidently,ρx≥0 and ρx=0⟺x=θ(ii)
There is a steadyL=max1,supnλpn≥1 such that ρλx≤Lλρx for all x∈ℓβpn and λ∈ℝ(iii)
We have the inequalityρx+y≤Hρx+ρy for all x,y∈ℓβpn(iv)
Clearly follows from inequality (20) of Theorem 6(v)
It obtained from (27) Theorem 6 that K0≥2(vi)
It is clear thatF¯=ℓβpn(vii)
There exists a steady0<ξ≤λp0−1 such that(36)ρλ,0,0,0,…≥ξλρ1,0,0,0,…for anyλ∈ℝTheorem 11.
ces
a
n
,
p
n is a premodular (sss), if conditions (b1) and (b2) are satisfied.Proof.
We define the functionalρ on cesan,pn as ρx=∑n=0∞an∑k=0nxkpn:(i)
Clearly,ρx≥0 and ρx=0⟺x=θ.(ii)
There is a numberL=max1,supnλpn≥1 with ρλx≤Lλρx for all x∈cesan,pn and λ∈ℝ.(iii)
We have the inequalityρx+y≤Hρx+ρy for all x,y∈cesan,pn.(iv)
It clearly follows from inequality (24) of Theorem 7.(v)
It is clear from (27) Theorem 7, that K0≥22h−1+2h−1+2h≥1.(vi)
It is clear thatF¯=cesan,pn.(vii)
There exists a steady0ξ≤λ|p0−1 such that ρλ,0,0,0,…≥ξλρ1,0,0,0,… for any λ∈ℝ. We state the following theorem without proof, and this can be established using standard technique.Theorem 12.
The functiongP=ϱsiPi=0∞ is a pre-quasi norm on SEϱ, where Eϱ is a premodular (sss).Theorem 13.
IfX and Y are Banach spaces and Eρ is a premodular (sss), then SEρ,g, where gT=ρsnTn=0∞ is a pre-quasi Banach operator ideal.Proof.
SinceEρ is a premodular (sss), then the function gT=ρsnTn=0∞ is a pre-quasi norm on SEρ. Let Tm be a Cauchy sequence in SEρX,Y, then by utilizing Part (vii) of Definition 9 and since LX,Y⊇SEρX,Y, we get(37)gTi−Tj=ρsnTi−Tjn=0∞≥ρs0Ti−Tj,0,0,0,…=ρTi−Tj,0,0,0,…≥ξTm−Tjρ1,0,0,0,….Then, Tmm∈ℕ is a Cauchy sequence in LX,Y. While the space LX,Y is a Banach space, so there exists T∈LX,Y with limm⟶∞Tm−T=0, and while snTmn=0∞∈Eρ for each m∈ℕ; hence, using Parts (iii) and (iv) of Definition 9 and as ρ is continuous at θ, we obtain(38)gT=ρsnTn=0∞=ρsnT−Tm+Tmn=0∞≤Kρsn/2T−Tmn=0∞+Kρsn/2Tmn=0∞≤KρTm−Tn=0∞+KρsnTmn=0∞<ε,and we have snTn=0∞∈Eρ, then T∈SEρX,Y.Corollary 11.
IfX and Y are Banach spaces and conditions (a1) and (a2) are satisfied, then Sℓβpn is a pre-quasi Banach operator ideal.Corollary 12.
IfX and Y are Banach spaces, pn is increasing with pn>0 for all n∈ℕ and bounded from above, then Sℓpn is a pre-quasi Banach operator ideal.Corollary 13.
IfX and Y are Banach spaces and 0<p<∞, then Sℓp is a pre-quasi Banach operator ideal.Corollary 14.
IfX and Y are Banach spaces and the conditions (b1) and (b2) are satisfied, then Scesan,pn is a pre-quasi Banach operator ideal.Corollary 15.
IfX and Y are Banach spaces and pn is increasing with p0>1 and limn⟶∞suppn<∞, then Scespn is a pre-quasi Banach operator ideal.Corollary 16.
IfX and Y are Banach spaces and p∈1,∞, then UcespappX,Y is complete.


## 6. Smallness of the Pre-Quasi Banach Operator Ideal

We give here the sufficient conditions on the weighted Nakano sequence space such that the pre-quasi operator ideal formed by the sequence of approximation numbers and this sequence space is strictly contained for different weights and powers.Theorem 14.
For any infinite dimensional Banach spacesX and Y and for any 0<pn<qn and 0<an<bn for all n∈ℕ, it is true that(39)SℓbnpnappX,Y⫋SℓanqnappX,Y⫋LX,Y.Proof.
LetX and Y be infinite dimensional Banach spaces and for any 0<pn<qn and 0<an<bn for all n∈ℕ, if T∈SℓbnpnappX,Y, then αnT∈ℓbnpn. One can see that(40)∑n=0∞anαnTqn<∑n=0∞bnαnTpn<∞.Hence, T∈SℓanqnappX,Y. Next, if we take an with supan<∞, bn−1∈ℓqn/pnand μn=1/bnpn. So by using Theorem 5, one can find T∈LX,Y with(41)116b3np3n≤αnT≤8bn+1pn+1,such that T does not belong to SℓbnpnappX,Y and T∈SℓanqnappX,Y.
It is easy to see thatSℓanqnappX,Y⊂LX,Y. Next, if we take μn=1/anqn. So by using Theorem 5, one can find T∈LX,Y with(42)116a3nq3n≤αnT≤8an+1qn+1,such that T does not belong to SℓanqnappX,Y. This finishes the proof.Corollary 17 (see [3]).
For any infinite dimensional Banach spacesX and Y and for any q>p>0, it is true that SℓpappX,Y⫋SℓqappX,Y⫋LX,Y.Lemma 2.
Ifan is a sequence of positive real with an≤1 for all n∈ℕ, pn is monotonic increasing bounded sequence with pn≥1 for all n∈ℕ or an>1 for all n∈ℕ, and pn is monotonic decreasing bounded sequence with pn≥1 for all n∈ℕ, one has the following inequality:(43)∑k=0nakpn<2nsuppn−1∑k=0nakpk.Proof.
By using inequality (16) and the sufficient conditions, one has(44)∑k=0nakpn=a0+∑k=1nakpn<2suppn−1a0pn+∑k=1nakpn=2suppn−1a0pn+a1+∑k=2nakpn<2suppn−1a0pn+2suppn−1a1pn+∑k=2nakpn<2suppn−12suppn−1a0pn+2suppn−1a1pn+∑k=2nakpn<22suppn−1a0pn+a1pn+∑k=2nakpn<23suppn−1a0pn+a1pn+a2pn+∑k=3nakpn⋮<2n−1ssuppn−1a0pn+a1pn+a2pn+⋯+an−2pn+∑k=n−1nakpn<2nsuppn−1∑k=0nakpn<2nsuppn−1∑k=0nakpk.
We give here the sufficient conditions on the generalized Cesáro sequence space defined by weighted mean such that the pre-quasi operator ideal formed by the sequence of approximation numbers, and this sequence space is strictly contained for different weights and powers.Theorem 15.
For any infinite dimensional Banach spacesX and Y and for any 1<pn<qn for all n∈ℕ, it is true that Scesbn,pnappX,Y⫋Scesan,qnappX,Y⫋LX,Y, where pn and qn are the monotonic increasing bounded sequences.Proof.
LetX and Y be infinite dimensional Banach spaces and for any 0<pn<qn and 0<an<bn for all n∈ℕ; if T∈Scesbn,pnappX,Y, then αnT∈cesbn,pn. One has(45)∑n=0∞an∑k=0nαkTqn<∑n=0∞bn∑k=0nαkTpn<∞.Hence, T∈Scesan,qnappX,Y. Next, if we take an with sup2supqn−1nanqn<∞, bn−1∈ℓqn/pn and μn=1/bnpn. So by using Theorem 5 and Lemma 2, one can find T∈LX,Y with(46)116b3np3n≤αnT≤8bn+1pn+1,such that T does not belong to Scesbn,pnappX,Y and T∈Scesan,qnappX,Y.
It is easy to see thatScesan,qnappX,Y⊂LX,Y. Next, we take μn=1/anqn. So by using Theorem 5, one can find T∈LX,Y with(47)116a3nq3n≤αnT≤8an+1qn+1,such that T does not belong to Scesan,qnappX,Y. This finishes the proof.Corollary 18.
For any infinite dimensional Banach spacesX and Y and 1<p<q<∞, then(48)ScespappX,Y⫋ScesqappX,Y⫋LX,Y.
In this part, we give the conditions for which the pre-quasi Banach Operator idealScesai,qiapp is small.Theorem 16.
If conditions (b1), (b2), andnan∉ℓpn are satisfied, then the pre-quasi Banach operator ideal Scesai,piapp is small.Proof.
Sincepi is an increasing sequence with p0>1 and ai∈ℓpi, take λ=∑i=0∞aipi1/h. Then, Scesai,piapp,g, where gT=1/λ∑i=0∞ai∑j=0iαjTpi1/h is a pre-quasi Banach operator ideal. Let X and Y be any two Banach spaces. Suppose that Scesai,piappX,Y=LX,Y; then, there exists a constant C>0 such that gT≤CT for all T∈LX,Y. Assume that X and Y be infinite dimensional Banach spaces. Hence, by Dvoretzky’s Theorem [18] for m∈ℕ, we have quotient spaces X/Nm and subspaces Mm of Y which can be mapped onto ℓ2m by isomorphisms Hm and Am such that HmHm−1≤2 and AmAm−1≤2. Let Im be the identity map on ℓ2m, Qm be the quotient map from X onto X/Nm, and Jm be the natural embedding map from Mm into Y. Let un be the Bernstein numbers [19]; then,(49)1=unIm=unAmAm−1ImHmHm−1≤AmunAm−1ImHmHm−1=AmunJmAm−1ImHmHm−1≤AmdnJmAm−1ImHmHm−1=AmdnJmAm−1ImHmQmHm−1≤AmαnJmAm−1ImHmQmHm−1,for 1≤i≤m. Now,(50)∑j=0iai≤∑j=0iAmaiαjJmAm−1ImHmQmHm−1⟹i+1ai≤Amai∑j=0iαjJmAm−1ImHmQmHm−1⟹i+1aipi≤AmHm−1piai∑j=0iαjJmAm−1ImHmQmpi.
Therefore,(51)∑i=0mi+1aipi1/h≤LAmHm−1∑i=0mai∑j=0iαjJmAm−1ImHmQmpi1/h⟹1λ∑i=0mi+1aipi1/h≤LAmHm−11λ∑i=0mai∑j=0iαjJmAm−1ImHmQmpi1/h⟹1λ∑i=0mi+1aipi1/h≤LAmHm−1gJmAm−1ImHmQm⟹1λ∑i=0mi+1aipi1/h≤LCAmHm−1JmAm−1ImHmQm⟹1λ∑i=0mi+1aipi1/h≤LCAmHm−1JmAm−1ImHmQm=LCAmHm−1Am−1ImHm⟹1λ∑i=0mi+1aipi1/h≤4LC,for some L≥1. Thus, we arrive at a contradiction since m is an arbitrary and nan∉ℓpn. Thus, X and Y both cannot be infinite dimensional when Scesai,piappX,Y=LX,Y and hence, the result.Theorem 17.
Ifpi is an increasing and p0>1, then the pre-quasi Banach operator ideal ScespiKol is small.Corollary 19.
If1<p<∞, then the quasi Banach operator ideal Scespapp is small.Corollary 20.
If1<p<∞, then the quasi Banach operator ideal ScespKol is small.
In this part, we give the conditions for which the pre-quasi Banach operator idealSℓβpnapp is small.Theorem 18.
If conditions (a1), (a2), andβn∉ℓ1 are satisfied, then the pre-quasi Banach operator ideal Sℓβpnapp is small.Proof.
Since conditions (a1), (a2), andβn∉ℓ1 are satisfied, then Sℓβpnapp,g, where gT=∑i=0∞βiαiTpi, is a pre-quasi Banach operator ideal. Let X and Y be any two Banach spaces. Suppose that SℓβpnappX,Y=LX,Y. Then, there exists a constant C>0 such that gT≤CT for all T∈LX,Y. Assume that X and Y are infinite dimensional Banach spaces. By using inequality (49) and βn∉ℓ1, one obtains(52)1≤AmHm−1piαiJmAm−1ImHmQmpi⟹βi≤LAmβiαiJmAm−1ImHmQmpiHm−1⟹∑i=0mβi≤LAmHm−1∑i=0mβiαiJmAm−1ImHmQmpi⟹∑i=0mβi≤LAmHm−1gJmAm−1ImHmQm⟹∑i=0mβi≤LCAmHm−1JmAm−1ImHmQm⟹∑i=0mβi≤LCAmHm−1JmAm−1ImHmQm=LCAmHm−1Am−1ImHm⟹∑i=0mβi≤4LC,for some L≥1. Thus, we arrive at a contradiction since m is an arbitrary. Thus, X and Y both cannot be infinite dimensional when SℓβpnappX,Y=LX,Y and hence, the result.Corollary 21 (see [2]).
If0<p<∞, then the quasi Banach operator ideal Sℓpapp is small.Corollary 22.
If0<p<∞, then the quasi Banach operator ideal SℓpKol is small.


## 7. Pre-Quasi Simple Banach Operator Ideal

The following question arises naturally: for which weighted Nakano sequence space or generalized Cesáro sequence space defined by weighted mean is the pre-quasi Banach ideal simple?Theorem 19.
Ifpn and qn are bounded sequences with 1≤pn<qn and 0<an<bn for all n∈ℕ, then(53)LSℓanqn,Sℓbnpn=ΛSℓanqn,Sℓbnpn.Proof.
Suppose that there existsT∈LSℓanqn,Sℓbnpn which is not approximable. According to Lemma 1, we can find X∈LSℓanqn,Sℓanqn and B∈LSℓbnpn,Sℓbnpn with BTXIk=Ik. Then, it follows for all k∈ℕ that(54)IkSℓbnpn=∑n=0∞bnαnIkpn1/suppn=∑n=0k−1bn1/suppn≤BTXIkSℓanqn≤∑n=0∞anαnIkqn1/supqn=∑n=0k−1an1/supqn.
But this is impossible.Corollary 23.
Ifpn and qn are bounded sequences with 1≤pn<qn and 0<an<bn for all n∈ℕ, then(55)LSℓanqn,Sℓbnpn=LCSℓanqn,Sℓbnpn.Proof.
Every approximable operator is compact.Theorem 20.
Ifpn and qn are bounded sequences with 1<pn<qn and 0<an<bn for all n∈ℕ, then(56)LScesan,qn,Scesbn,pn=ΛScesan,qn,Scesbn,pn.Proof.
Suppose that there existsT∈LScesan,qn,Scesbn,pn which is not approximable. According to Lemma 1, we can find X∈LScesan,qn,Scesan,qn and B∈LScesbn,pn,Scesbn,pn with BTXIk=Ik. Then, it follows for all k∈ℕ that(57)IkScesbn,pn=∑n=0∞bn∑i=0nαiIkpn1/suppn=∑n=0∞bn∑i=0k−1αiIkpn1/suppn=∑n=0∞kbnpn1/suppn≤BTXIkScesan,qn≤∑n=0∞an∑i=0nαiIkqn1/supqn=∑n=0∞an∑i=0k−1αiIkqn1/supqn=∑n=0∞kanqn1/supqn.
But this is impossible.Corollary 24.
Ifpn and qn are bounded sequences with 1<pn<qn and 0<an<bn for all n∈ℕ, then(58)LScesan,qn,Scesbn,pn=LCScesan,qn,Scesbn,pn.Theorem 21.
For a bounded sequencepn with 1≤pn<∞ and bn>0 for all n∈ℕ, the pre-quasi Banach space Sℓbnpn is simple.Proof.
Suppose that the closed idealLCSℓbnpn contains an operator T which is not approximable. According to Lemma 1, we can find X,B∈LSℓbnpn with BTXIk=Ik. This means that ISℓbnpn∈LCSℓbnpn. Consequently, LSℓbnpn=LCSℓbnpn. Therefore, ΛSℓbnpn is the only nontrivial closed ideal in LSℓbnpn.Theorem 22.
For a bounded sequencepn with 1<pn<∞ and bn>0 for all n∈ℕ, the pre-quasi Banach space Scesbn,pn is simple.Proof.
Suppose that the closed idealLCScesbn,pn contains an operator T which is not approximable. According to Lemma 1, we can find X,B∈LScesbn,pn with BTXIk=Ik. This means that IScesbn,pn∈LCScesbn,pn. Consequently, LScesbn,pn=LCScesbn,pn. Therefore, ΛScesbn,pn is the only nontrivial closed ideal in LScesbn,pn.



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