# Some Properties of Prequasi Normed Generalized de La Vallée Poussin’s Mean Sequence Space

Authors: Awad A. Bakery; Mustafa M. Mohammed
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this article, we study some topological properties of the multiplication operator on generalized de La Vallée Poussin’s mean sequence space equipped with the prequasi norm and the prequasi operator ideal generated bys-numbers and this sequence space.


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## Body

## 1. Introduction

All through the article, byLU,V, we signify the space of all bounded linear transformations between arbitrary Banach spaces U and V; if U=V, we mark LU. We will denote FV, ΨV, and LcV for the space of all finite rank, approximable, and compact transformations on V, respectively. The set is composed of nonnegative integers (ℕ) and real numbers (ℝ). Moreover, ℂℕ, ℓ∞, C0, srA, drA, and αrA will denote the space of complex, bounded, null, r-th s-numbers [1], r-th Kolmogorov, and approximation sequences, respectively, where A∈LU,V. The multiplication operators and operator ideals have a wide field of mathematics in functional analysis, for instance, in eigenvalue distributions theorem, fixed point theorem, geometry of Banach spaces, and spectral theory. Singh and Kumar [2] investigated the connection between the composition operators and the multiplication operators on Lq-spaces. They proved the equivalence between the composition operator A∈LcLqU;ℂ and Bβ∈LcLqU;ℂ, where Bβ is the multiplication operator induced by β=dνB−1/dν. The roots of the multiplication operators are contained in the spectral theory. According to Hilbert space theorem, there is a unitary equivalence between a multiplication operator and each normal operator on a separable Hilbert space. For more subtleties on multiplication operators, see [2–7]. In view of sequence spaces, Mursaleen and Noman [8] investigated some difference sequence spaces under compact matrix transformation. The multiplication operators belong to Lℓφ, where φ is an Orlicz function, studied by Komal and Gupta [9]. Furthermore, Komal et al. [10] explained the multiplication operators in Lcesq., for q∈1,∞ with the Luxemburg norm .. In the operator ideal theory, the scalar sequence space is sometimes used to define operator ideals in the class of Banach spaces or Hilbert spaces, since the ideal LcU is generated by C0 and drA. A Banach space U is called simple [11], if there is one and only one nontrivial closed ideal in LU. The quasi-ideals Sℓqapp, where q∈0,∞, is investigated by Pietsch [11]. Also, he proved that ℓ2 and ℓ1 generated the ideals of Hilbert Schmidt operators between Hilbert spaces and of nuclear operators, respectively. He also investigated the sufficient condition on ℓq so that FU,V¯=SℓqappU,V and Lℓq is simple Banach space. Pietsch [12] studied the smallness of Sℓqapp. For any infinite dimensional Banach spaces U and V, Makarov and Faried [13] gave the sufficient conditions for which Sℓrapp is strictly contained in Sℓmapp, for every m>r>0. Bakery [14] introduced some properties of SVγ,papp, where Vγ,p is the generalized de La Vallée Poussin’s mean sequence space. Some properties of s-type Orlicz-Lorentz sequence spaces are examined by Mohiuddine and Raj [15]. Faried and Bakery [16] introduced a generalization of the usual classes of operator ideal which is prequasi operator ideal. They explained some results of Scesq and Sℓφ. We introduce in this paper the concept of prequasi norm on Vγ,p, give the conditions on Vγ,p equipped with the prequasi norm to construct Banach space, and study the necessity and sufficient conditions on Vγ,p so that the multiplication operator defined on Vγ,p is bounded, invertible, approximable, closed range, and Fredholm operator. We investigate the sufficient conditions on the class SVγ,p to form small, simple, closed Banach prequasi operator ideal. Also, we examine the strict inclusion relation between SVγ,p and SEλ (the class of all bounded linear operators whose sequence of eigenvalues belongs to Vγ,p).


## 2. Definitions and Preliminaries

We will denoteej=0,0,⋯,1,0,0,⋯, with 1 in the jth position for every j∈ℕ.Lemma 1 (see [11]).
ForT∈LV,W, if T∉ΨV,W, then there are operators B∈LW and G∈LV with BTGei=ei for each i∈ℕ.Theorem 2 (see [11]).
IfW is a Banach space with dimW=∞, then
(1)FW⫋ΨW⫋LcW⫋LW.Definition 3 (see [17]).
An operatorD∈LV is called Fredholm if it satisfies dimkerD<∞, dimRDc<∞, and D has closed range, where RDc denotes the complement of the range D.Letγk∈ℝ+ℕ, where γ0=1, γk+1≥γk, γk+1≤γk+1, for all k∈ℕ, limk→∞γk=∞ and pi∈ℝ+ℕ with pi≥1, for all i∈ℕ. Simsek et al. [18] defined the generalized de La Vallée Poussin’s mean sequence space as follows:
(2)Vγ,p=x=xi∈ℂℕ:∃η>0withσηx<∞whereσx=∑i=0∞∑k∈Gixkγipi,and Gi=i−γi+1,i, for i∈ℕ. The space Vγ,p,., where
(3)x=infη>0:σxη≤1,is a Banach space. When pn∈ℓ∞, it is clear that
(4)Vγ,p=xi∈ℂℕ:∑i=0∞∑k∈Gixkγipi<∞.For more details onVγ,p, see [19, 20].Remark 4.
(1)
Ifγi=i+1, for every i∈ℕ, then Vγ,p=cespn examined by Sanhan and Suantai [21]
(2)
Ifγi=i+1 and pi=p, for all i∈ℕ, then Vγ,p=cesp. Some authors [22–24] investigated various sorts of Cesáro summable sequence spacesDefinition 5 (see [25]).
A class of linear sequence spacesX is called a special space of sequences (sss) if
(1)
ej∈X for each j∈ℕ
(2)
foru=uj∈ℂℕ, v=vj∈X and uj≤vj for all j∈ℕ, then u∈X
(3)
forujj=0∞∈X, then uj/2j=0∞∈X, wherever j/2 means the integral part of j/2Theorem 6 (see [14]).
Vγ,p is a (sss), if
(a1)pk is increasing and pk∈ℓ∞ with p0>1
(a2)γk∈ℝ+ℕ, where γ0=1, γk+1≥γk, γk+1≤γk+1, for all k∈ℕ, limk→∞γk=∞ and γk−1∈ℓpk are satisfiedDefinition 7 (see [25]).
A subclass of (sss) is called a premodular (sss) if there is a functionσ:X⟶0,∞ verifying the conditions
(i)
σv≥0 for all v∈X and σv=0⇔v=θ, here θ is the zero element of X
(ii)
there isL≥1 with σηv≤L∣η∣σv for each v∈X, and η∈ℂ
(iii)
there isK≥1, σu+v≤Kσu+σv for all u,v∈X
(iv)
forui≤vi for all i∈ℕ, then σui≤σvi
(v)
for someK0≥1, σui≤σui/2≤K0σui
(vi)
for allu=ujj=o∞∈X and ε>0, there is s∈ℕ with σujj=s∞<ε
(vii)
there isξ>0 such that ση,0,0,0,⋯≥ξησ1,0,0,0,⋯ for each η∈ℂTheorem 8 (see [14]).
If conditions (a1) and (a2) are verified, thenVγ,pσ is a premodular (sss), with σx=∑j=0∞∑k∈Gjxk/γjpj for each x∈Vγ,p. The usual classes of operator ideal generalized to the prequasi operator ideal.Definition 9 (see [25]).
A functionf:Φ⟶0,∞ is called a prequasi norm on the ideal Φ if it satisfies the following:
(1)
IfA∈ΦU,V, then fA≥0 and fA=0 if and only if A=0
(2)
There isM≥1 with fηA≤M∣η∣fA, for each A∈ΦU,V and η∈ℂ
(3)
There isK≥1 such that fA1+A2≤KfA1+fA2, for every A1,A2∈ΦU,V
(4)
There isC≥1, if A∈LU0,U, P∈ΦU,V and R∈LV,V0 then fRPA≤CRfPA, where U0 and V0 are normed spacesNotations 10 (see [16]).
(5)SE≔SEU,V;UandVareBanachSpaces,withSEU,V≔A∈LU,V:siAi=0∞∈E.SEapp≔SEappU,V;UandVareBanachSpaces,withSEappU,V≔A∈LU,V:αiAi=0∞∈E.SEKol≔SEKolU,V;UandVareBanachSpaces,withSEKolU,V≔T∈LU,V:diAi=0∞∈E.Theorem 11 (see [16]).
LetEϱ be a premodular (sss), then the function gT=ϱsiTi=0∞ be a prequasi norm on SEϱ.During this paper,pi∈ℝ+ℕ with pi∈ℓ∞ and the inequality [26] aj+bjpj≤Hajpj+bjpj, where H=2h−1, h=supjpj, and pj≥1 for all j∈ℕ, will be used.


## 3. Main Results

We introduce a generalization of the usual norm, the concept of prequasi norm onVγ,p. We investigate the sufficient conditions on Vγ,p equipped with a prequasi norm to form Banach space.Definition 12.
AssumeX is (sss). If there is a function σ:X⟶0,∞ satisfying the conditions
(i)
σv≥0 for all v∈X and σv=0⇔v=θ
(ii)
there isL≥1 with σηv≤Lησv for every v∈X, and η∈ℂ
(iii)
there isK≥1, we have σu+v≤Kσu+σv for each u,v∈XThe spaceXσ is called prequasi normed (sss). If X is complete with σ, hence Xσ is called a prequasi Banach (sss). We express the following two theorems without verification, since they are clear.Theorem 13.
X is a prequasi norm (sss), if it is quasi norm (sss).Theorem 14.
Every premodular (sss) is prequasi normed (sss).Theorem 15.
If conditions (a1) and (a2) are verified, thenVγ,pϱ is a prequasi Banach (sss), with ϱx=∑i=0∞∑j∈Gixj/γipi for all x∈Vγ,p.Proof.
Let the conditions be satisfied, then from Theorem8, the space Vγ,pϱ is premodular (sss). From Theorem 14, we have Vγ,pϱ which is a prequasi normed (sss). To show that Vγ,pϱ is a prequasi Banach (sss), assume xn=xknk=0∞ be a Cauchy sequence in Vγ,pϱ. Therefore, for all ε∈0,1, there is n0∈ℕ such that for all n,m≥n0, one has
(6)ϱxn−xm=∑i=0∞∑j∈Gixjn−xjmγipi<εh.
Hence forn,m≥n0 and j∈ℕ, we get
(7)xjn−xjm<ε.
Soxjm is a Cauchy sequence in ℂ for fixed j∈ℕ; this gives limm→∞xjm=xj0 for fixed j∈ℕ. Hence, ϱxn−x0<εh, for all n≥n0. Finally, to prove that x0∈Vγ,p, we have
(8)ϱx0=ϱx0−xn+xn≤Hϱxn−x0+ϱxn<∞,so x0∈Vγ,p. This completes the proof.Corollary 16.
Let1<p<∞, then cespϱ be a prequasi Banach (sss), with ϱx=∑i=0∞∑j=0ixj/i+1p for all x∈cesp.


## 4. Multiplication Operator on Prequasi Normed (sss)

We define here a multiplication operator onVγ,p with a prequasi norm and investigate the sufficient and necessary conditions on the multiplication operator to become bounded, closed range operator, approximable, invertible, and Fredholm.Definition 17.
Ifη∈ℂℕ, η∈ℓ∞ and Xϱ is a prequasi normed (sss). An operator Tη is called multiplication operator if Tη:X⟶X, where Tηy=ηy=ηkykk=0∞, for all y∈X. When Tη∈LX, Tη is called the multiplication operator induced by η.Theorem 18.
Letα∈ℂℕ, the conditions (a1) and (a2) be satisfied, hence α∈ℓ∞ if and only if Tα∈LVγ,pϱ, with ϱx=∑i=0∞∑j∈Gixj/γipi for every x∈Vγ,pϱ.Proof.
Let the conditions be satisfied andα∈ℓ∞. Therefore, there is D>0 with ∣αk∣≤D, for each k∈ℕ. For x∈Vγ,pϱ, we have
(9)ϱTαx=ϱαx=ϱαkxkk=0∞=ϱαkxkk=0∞=∑i=0∞∑j∈Giαjxjγipi≤∑i=0∞∑j∈GiCxjγipi≤Lϱx,where L=supiDpi; this implies that Tα∈LVγ,pϱ. Conversely, let Tα∈LVγ,pϱ to show that α∈ℓ∞. For, if α∉ℓ∞, hence for all n∈ℕ, there are in∈ℕ with αin>n. We have
(10)ϱTαein=ϱαein=ϱαjeinjj=0∞=∑i=0∞∑j∈Giαjeinjγipi=∑i=0∞αinγipi>∑i=0∞nγipi≥infinpiϱein=np0ϱein.
This shows thatTα∉LVγ,pϱ. So, α∈ℓ∞.Theorem 19.
Ifα∈ℂℕ and Vγ,pϱ is a prequasi normed (sss), with ϱx=∑i=0∞∑j∈Gixj/γipi for each x∈Vγ,pϱ. Hence, αj=1 for each j∈ℕ if and only if Tα is an isometry.Proof.
Supposeαj=1, for every j∈ℕ. Therefore,
(11)ϱTαx=ϱαx=ϱαjxjj=0∞=∑i=0∞∑j∈Giαjxjγipi=∑i=0∞∑j∈Gixjγipi=ϱx,for all x∈Vγ,pϱ. Hence, Tα is an isometry. Conversely, let ∣αn∣<1 for some n=n0. Then,
(12)ϱTαen0=ϱαen0=ϱαjen0jj=0∞=∑i=0∞∑j∈Giαjen0jγipi=∑i=0∞αn0γipi<∑i=0∞1γipi<ϱen0.
Also, if∣αn0∣>1, then we get ϱTαen0>ϱen0. Therefore, we have a contradiction in the two cases. So, ∣αj∣=1, for each j∈ℕ.Theorem 20.
Ifα∈ℂℕ, the conditions (a1) and (a2) are satisfied and Tα∈LVγ,pϱ, where ρx=∑r=0∞∑j∈Grxj/γrpr for all x∈Vγ,pϱ. Then, Tα∈ΨVγ,pϱ if and only if αnn=0∞∈C0.Proof.
LetTα be an approximable operator. Hence, Tα∈LcVγ,pϱ. To prove that αrr=0∞∈C0, assume αrr=0∞∉C0; hence, there is δ>0 with Bδ=r∈ℕ:αr≥δ which is an infinite set. Suppose d1,⋯d2,⋯,dn,⋯ be in Bδ. Then, edn:dn∈Bδ is an infinite bounded set in Vγ,pϱ. We have
(13)ϱTαedn−Tαedm=ϱαedn−αedm=ϱαkednk−epmkk=0∞=∑i=0∞∑j∈Giαjednj−epmjγipi≥∑i=0∞∑j∈Giδednj−epmjγipi≥infnδpn∑i=0∞∑j∈Giednj−epmjγipi=infnδpnρedn−edm,for all dn,dm∈Bδ. This proves edn:dn∈Bδ∈ℓ∞ which cannot have a convergent subsequence under Tα. Therefore, this gives Tα∉LcVγ,pϱ, so Tα∉ΨVγ,pϱ; this implies a contradiction. Therefore, limn→∞αn=0. Conversely, suppose limn→∞αn=0. Hence, for all δ>0, the set Bδ=n∈ℕ:αn≥δ is finite. We have
(14)Vγ,pϱBδ=x=xn∈Vγ,pϱ:n∈Bδ,which is a finite dimensional space for all δ>0. Hence, TαVγ,pϱBδ is a finite rank operator. Define αi∈ℂℕ as follows
(15)αij=αj,j∈B1/i,0,otherwise.
It is clear that sinceTαi is a finite rank operator, then the space Vγ,pϱB1/i is finite dimensional for each i∈ℕ. Therefore, by using conditions (a1) and (a2), we have
(16)ϱTα−Tαnx=ϱαj−αnjxjj=0∞=∑i=0,i∈B1/n∞∑j∈Giαj−αnjxjγipi+∑i=0,i∉B1/n∞∑j∈Giαj−αnjxjγipi=∑i=0,i∈B1/n∞∑j∈Gi,j∉B1/nαjxjγipi+∑i=0,i∉B1/n∞∑j∈Gi,j∉B1/nαjxjγipi<1n∑i=0∞∑j∈Gixjγipi=1nϱx.
This proves thatTα−Tαn≤1/n. So, Tα is a limit of finite rank operators. This finishes the proof.Theorem 21.
Ifα∈ℂℕ, the conditions (a1) and (a2) are satisfied and Tα∈LVγ,pϱ, where ρx=∑i=0∞∑j∈Gixj/γipi for all x∈Vγ,pϱ. Then, Tα∈LcVγ,pϱ if and only if αnn=0∞∈C0.Proof.
It is clear so omitted.Corollary 22.
If conditions (a1) and (a2) are satisfied, we have(17)LcVγ,pϱ⫋LVγ,pϱ,with ϱx=∑i=0∞∑j∈Gixj/γipi for each x∈Vγ,p.Proof.
Since the multiplication operator induced byα=1,1, is I∈LVγ,pϱ, therefore I∉LcVγ,pϱ.Theorem 23.
LetVγ,pϱ be a prequasi Banach (sss), with Tα∈LVγ,pϱ) and ϱx=∑i=0∞∑j∈Gixj/γipi for each x∈Vγ,p. Hence, α is bounded away from zero on kerαc if and only if Tα has closed range.Proof.
Let the sufficient conditions be satisfied. Therefore, there isε>0 such that αj≥ε, for every j∈kerαc. To show that RTα is closed, suppose q be a limit point of RTα. Hence, there is Tαxj in Vγ,pϱ, for all j∈ℕ such that limj→∞Tαxj=q. Clearly, the sequence Tαxj is a Cauchy sequence. Therefore, since ρ is nondecreasing, one can see
(18)ϱTαxn−Tαxm=ϱαjxnj−αjxmjj=0∞=∑i=0∞∑j∈Giαjxnj−αjxmjγipi=∑i=0,i∈kerαc∞∑j∈Giαjxnj−αjxmjγipi+∑i=0,i∉kerαc∞∑j∈Giαjxnj−αjxmjγipi≥∑i=0,i∈kerαc∞∑j∈Giαjxnj−αjxmjγipi=∑i=0∞∑j∈Giαjynj−αjymjγipi>∑i=0∞∑j∈Giεynj−εymjγipi=ϱεyn−ym,where
(19)ynj=xnj,j∈kerαc,0,j∉kerαc.
Hence,yn is a Cauchy sequence in Vγ,pϱ since Vγ,pϱ is complete. So, there is x∈Vγ,pϱ with limn→∞yn=x. Since Tα is continuous, hence limn→∞Tαyn=Tαx. But limn→∞Tαxn=limn→∞Tαyn=q. Therefore, Tαx=q. Hence, q∈RTα. This gives RTα is closed. Let the necessary condition be satisfied; this means RTα be closed. Therefore, Tα is bounded away from zero on Vγ,pρkerαc, i.e., there is ε>0 such that ϱTαx≥ϱεx, for all x∈Vγ,pϱkerαc. Let D=j∈kerαc:αj<ε. If D≠ϕ, then for n0∈D, since ϱ is nondecreasing, we have
(20)ϱTαen0=ϱαjen0jj=0∞=ϱαjen0jj=0∞=∑i=0∞∑j∈Giαjen0jγipi<∑i=0∞∑j∈Giεen0ji+1pi=ϱεen0.
This gives a contradiction. Therefore,D=ϕ with ∣αj∣≥ε, for each j∈kerαc. This completes the proof.Theorem 24.
Ifα∈ℂℕ and Vγ,pϱ is a prequasi Banach (sss), with ϱx=∑i=0∞∑j∈Gixj/γipi for each x∈Vγ,p. Hence, there is b>0 and B>0 such that b<αj<B; for each j∈ℕ if and only if Tα∈LVγ,pϱ is invertible.Proof.
Assume that the sufficient condition be verified. Defineη∈ℂℕ by ηj=1/αj. Then by Theorem 18, Tα and Tη are bounded linear operators. Also, Tα·Tη=Tη·Tα=I. Hence, Tη is the inverse of Tα. Conversely, suppose the necessary condition be satisfied. Therefore, RTα=Vγ,pρℕ. Hence, RTα is closed. From Theorem 23, there is b>0 so that αj≥b, for all j∈kerαc. Now, kerα=ϕ; otherwise, αn0=0, for some n0∈ℕ, we have en0∈kerTα. Since kerTα is trivial, this gives a contradiction. Therefore, αj≥b, for each j∈ℕ. From Theorem 18, since Tα is bounded, there is B>0 so that αj≤B, for every j∈ℕ. Therefore, we have showed that b≤αj≤B, for all j∈ℕ.Theorem 25.
IfVγ,pϱ is a prequasi Banach (sss), with Tα∈LVγ,pϱ and ρy=∑i=0∞∑j∈Giyj/γipi for every y∈Vγ,p. Hence, Tα be Fredholm operator if and only if (i) kerα be a finite subset of ℕ. (ii) αj≥ε, for each j∈kerαc.Proof.
LetTα be Fredholm. Assume kerα be an infinite subset of ℕ, then ej∈kerTα, for each j∈kerα. This gives dimkerTα=∞; hence, we have a contradiction. Therefore, kerα be a finite subset of ℕ. From Theorem 23, condition (ii) follows. On the other hand, let the necessary condition be satisfied. From Theorem 23, condition (ii) shows that RTα is closed and condition (i) gives that dimkerTα<∞ and dimRTαc<∞. This finishes the proof.


## 5. Closed Banach Prequasi Ideal

We study the sufficient conditions onVγ,p so that the prequasi ideal SVγ,p are Banach and closed.Theorem 26.
LetX and Y be Banach spaces, (a1) and (a2) be satisfied, then SVγ,pρ,g is a prequasi Banach operator ideal, where gT=ρsnTn=0∞ and ρx=∑i=0∞∑j∈Gixj/γipi for each x∈Vγ,p.Proof.
From Theorem8, the space Vγ,pρ is a premodular (sss). Hence, by using Theorem 11, we have the function gT=ρsnTn=0∞ that is a prequasi norm on SVγ,pϱ. Suppose Tj be a Cauchy sequence in SVγ,pϱX,Y, then by part (vii) of Definition 7 there is ξ>0 and since SVγ,pϱX,Y⊆LX,Y, we get
(21)gTi−Tj=∑r=0∞1γr∑k∈GrskTi−Tjpr≥∑r=0∞1γrTi−Tjpr≥ξTi−Tj∑r=0∞1γrpr.
Therefore,Tjj∈ℕ be a Cauchy sequence in LX,Y. Since LX,Y is a Banach space, hence there is T∈LX,Y with limi→∞Ti−T=0. Since snTin=0∞∈Vγ,pϱ for each i∈ℕ, from parts (ii), (iii), (iv), and (v) of Definition 7, we have
(22)gT=∑r=0∞1γr∑k∈GrskTpr=∑r=0∞1γr∑k∈GrskT−Tm+Tmpr≤K∑r=0∞1γr∑k∈Grsk2T−Tmpr+K∑r=0∞1γr∑k∈Grsk2Tmpr≤K∑r=0∞1γr∑k∈GrTm−Tpr+KK0∑r=0∞1γr∑k∈GrskTmpr<ε.
Therefore,sjTj=0∞∈Vγ,pρ; hence, T∈SVγ,pϱX,Y.Theorem 27.
LetX and Y be Banach spaces, (a1) and (a2) be satisfied, then SVγ,pϱ,g is a prequasi closed operator ideal, where gT=ρsnTn=0∞ and ρx=∑i=0∞∑j∈Gixj/γipi for each x∈Vγ,p.Proof.
From Theorem8, the space Vγ,pρ is a premodular (sss). Hence, by using Theorem 11, we have the function gT=ρsnTn=0∞ that is a prequasi norm on SVγ,pϱ. Suppose Tj∈SVγ,pϱX,Y for each j∈ℕ and limm→∞gTm−T=0. Since SVγ,pϱX,Y⊆LX,Y and from part (vii) of Definition 7, we have
(23)gT−Tm=∑r=0∞1γr∑k∈GrskT−Tmpr≥∑r=0∞1γrT−Tmpr≥ξT−Tm∑r=0∞1γrpr.
Hence,Tmm∈ℕ is convergent in LX,Y. Since snTin=0∞∈Vγ,pϱ for each i∈ℕ, from parts (ii), (iii), (iv), and (v) of Definition 7, we obtain
(24)gT=∑r=0∞1γr∑k∈GrskTpr=∑r=0∞1γr∑k∈GrskT−Tm+Tmpr≤K∑r=0∞1γr∑k∈Grsk/2T−Tmpr+K∑r=0∞1γr∑k∈Grsk/2Tmpr≤K∑r=0∞1γr∑k∈GrTm−Tpr+KK0∑r=0∞1γr∑k∈GrskTmpr<ε.
Therefore,snTn=0∞∈Vγ,pρ. Hence, T∈SVγ,pϱX,Y.


## 6. Small and Simple Prequasi Banach Operator Ideal

We introduce some inclusion relations concerning prequasi operator ideal constructed by the sequence ofs-numbers and Vγ,p.Theorem 28.
IfX, Y are infinite dimensional Banach spaces, pr∈ℓ∞, qr∈ℓ∞ with 1<pr<qr, and 1≤γr≤νr for every r∈ℕ, then
(25)SVγ,pX,Y⫋SVν,qX,Y⫋LX,Y.Proof.
Assume the sufficient conditions be verified andT∈SVγ,pX,Y. Therefore, skT∈Vγ,p. We have
(26)∑r=0∞1νr∑k∈GrskTqr<∑r=0∞1γr∑k∈GrskTpr<∞.Hence, T∈SVν,qX,Y. By taking srTr=0∞ such that ∑k∈GrskT=γrr+1−1/pr, one can find T∈LX,Y so that
(27)∑r=0∞1γr∑k∈GrskTpr=∑r=0∞1r+1=∞,∑r=0∞1νr∑k∈GrskTqr≤∑r=0∞1γr∑k∈GrskTqr=∑r=0∞1r+1qr/pr<∞.
Hence,T∈SVν,qX,Y and T∉SVγ,pX,Y. Obviously, SVγ,pX,Y⊂LX,Y. By choosing srTr=0∞ such that ∑k∈GrskT=νrr+1−1/qr. We have T∈LX,Y and T∉SVν,qX,Y. This finishes the proof.Corollary 29.
IfX, Y are infinite dimensional Banach spaces and 1<p<q<∞, it is true that
(28)ScespX,Y⫋ScesqX,Y⫋LX,Y.We investigate the sufficient conditions so that the prequasi Banach operator ideal SVγ,papp is small.Theorem 30.
The prequasi Banach operator idealSVγ,papp is small, whenever X, Y are infinite dimensional Banach spaces and the conditions (a1) and (a2) are satisfied.Proof.
If the sufficient conditions are verified, therefore the spaceSVγ,papp,g is a prequasi Banach operator ideal, with gT=1/η∑i=0∞γi−1∑j∈GiαjTpi1/h and η=∑i=0∞γi−pi1/h. Suppose that SVγ,pappX,Y=LX,Y, so there is C>0 so that gT≤CT for each T∈LX,Y. From Dvoretzky’s theorem [27] for all m∈ℕ, we obtain subspaces Mm of Y and quotient spaces X/Nm transform onto ℓ2m by isomorphisms Am and Hm with AmAm−1≤2 and HmHm−1≤2. If Jm is the natural embedding map from Mm into Y and Qm is the quotient map from X onto X/Nm, suppose vn denotes the Bernstein numbers [28], we have
(29)1=unIm=vnAmAm−1ImHmHm−1≤AmvnAm−1ImHmHm−1=AmvnJmAm−1ImHmHm−1≤AmdnJmAm−1ImHmHm−1=AmdnJmAm−1ImHmQmHm−1≤AmαnJmAm−1ImHmQmHm−1,for 0≤i≤m. Now
(30)∑j∈Giγi−1≤∑j∈GiAmγi−1αjJmAm−1ImHmQmHm−1⇒1≤Amγi−1∑j∈GiαjJmAm−1ImHmQmHm−1⇒1≤AmHm−1piγi−1∑j∈GiαjJmAm−1ImHmQmpi.
Therefore,(31)m+11/h≤LAmHm−1∑i=0mγi−1∑j∈GiαjJmAm−1ImHmQmpi1/h⇒1ηm+11/h≤LAmHm−11η∑i=0mγi−1∑j∈GiαjJmAm−1ImHmQmpi1/h⇒1ηm+11/h≤LAmHm−1gJmAm−1ImHmQm⇒1ηm+11/h≤LCAmHm−1JmAm−1ImHmQm⇒1ηm+11/h≤LCAmHm−1JmAm−1ImHmQm=LCAmHm−1Am−1ImHm⇒1ηm+11/h≤4LC,for some L≥1. Since m is an arbitrary, this gives a contradiction. Therefore, dimX<∞ and dimY<∞ when SVγ,pappX,Y=LX,Y. This finishes the proof.We introduce the next theorem without verification; these can be set up utilizing standard procedure.Theorem 31.
The prequasi Banach operator idealSVγ,pKol is small, whenever X, Y are infinite dimensional Banach spaces and the conditions (a1) and (a2) are satisfied.Corollary 32.
The prequasi Banach operator idealScespapp is small, if X, Y are infinite dimensional Banach spaces and p∈1,∞.Corollary 33.
The prequasi Banach operator idealScespKol is small, if X, Y are infinite dimensional Banach spaces and p∈1,∞.We examine the sufficient conditions so that the prequasi Banach operator idealSVγ,p is simple.Theorem 34.
IfX, Y are infinite dimensional Banach spaces, pj∈ℓ∞, qj∈ℓ∞ with 1<pj<qj, and 1≤γj≤νj for each j∈ℕ, it is true that
(32)ΨSVν,q,SVγ,p=LSVν,q,SVγ,p.Proof.
Let the sufficient conditions be given,T∈LSVν,q,SVγ,p and T∉ΨSVν,q,SVγ,p. From Lemma 1, we have A∈LSVν,q and B∈LSVγ,p with BTAIk=Ik. For each k∈ℕ, we obtain
(33)IkSVγ,p=∑n=0∞1γn∑i∈GnsiIkpn≤BTAIkSVν,q≤∑n=0∞1νn∑i∈GnsiIkqn.
In view of Theorem28, we get a contradiction. Therefore, T∈ΨSVν,q,SVγ,p.Corollary 35.
IfX, Y are infinite dimensional Banach spaces, pj∈ℓ∞, qj∈ℓ∞ with 1<pj<qj, and 1≤γj≤νj for each j∈ℕ, it is true that
(34)LCScesqn,Scespn=LSVν,q,SVγ,p.Theorem 36.
The prequasi Banach operator idealSVγ,p is simple, if X, Y are infinite dimensional Banach spaces and the conditions (a1) and (a2) are satisfied.Proof.
Let the sufficient conditions be given and there isT∈LCSVγ,p with T∉ΨSVγ,p. From Lemma 1, we have A,B∈LSVγ,p so that BTAIk=Ik. This means that ISVγ,p∈LCSVγ,p. Consequently, LSVγ,p=LCSVγ,p. Therefore, SVγ,p is simple.


## 7. Eigenvalues of s-Type Operators

We investigate the strict inclusion relation betweenSVγ,p and SEλ (the class of all bounded linear operators whose sequence of eigenvalues belongs to Vγ,p).Notations 37.
SEλ≔SEλX,Y;XandYareBanachSpaces, where SEλX,Y≔T∈LX,Y:λiTn=0∞∈EandT−λnTI−1;doesnotexistforeveryn∈ℕ.Theorem 38.
IfX, Y are infinite dimensional Banach spaces and conditions (a1) and (a2) are satisfied, it is true that
(35)SVγ,pX,Y⫋SVγ,pλX,Y.Proof.
Let the sufficient conditions be verified andT∈SVγ,pX,Y. Therefore, snTn=0∞∈Vγ,p. Hence, we have
(36)∑r=0∞1γr∑k∈GrskTpr≥∑r=0∞srTpr.
Hence,srTr=0∞∈ℓpr, so limr→∞srT=0. Assume T−srTI−1 exists for every r∈ℕ. Therefore, T−srTI−1 exists and bounded for every r∈ℕ. This gives limr→∞T−srTI−1=T−1 exists and bounded. From the prequasi operator ideal of SVγ,p,g, we obtain
(37)I=TT−1∈SVγ,pX,Y⇒srIr=0∞∈Vγ,p⇒limr→∞srI=0.
We have a contradiction, sincelimr→∞srI=1. Therefore, T−srTI=0 for every r∈ℕ. Hence, srTr=0∞ is the eigenvalues of T. Finally, if we choose srTr=0∞ such that ∑k∈GrskT=γrr+1−1/pr, we find T∈LX,Y such that ∑r=0∞1/γr∑k∈GrskTpr=∑r=0∞1/r+1=∞ and if we take λrTr=0∞ such that ∑k∈GrλkT=γr/r+1. Hence, T∉SVγ,pX,Y and T∈SVγ,pλX,Y. This finishes the proof.



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