# Small Pre-Quasi Banach Operator Ideals of Type Orlicz-Cesáro Mean Sequence Spaces

Authors: Awad A. Bakery; Mustafa M. Mohammed
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this paper, we give the sufficient conditions on Orlicz-Cesáro mean sequence spacescesφ, where φ is an Orlicz function such that the class Scesφ of all bounded linear operators between arbitrary Banach spaces with its sequence of s-numbers which belong to cesφ forms an operator ideal. The completeness and denseness of its ideal components are specified and Scesφ constructs a pre-quasi Banach operator ideal. Some inclusion relations between the pre-quasi operator ideals and the inclusion relations for their duals are explained. Moreover, we have presented the sufficient conditions on cesφ such that the pre-quasi Banach operator ideal generated by approximation number is small. The above results coincide with that known for cesp  (1<p<∞).


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## Body

## 1. Introduction

Throughout the paper, byw, we mean the space of all real sequences, R the real numbers, and N={0,1,2,…} and L(X,Y) the space of all bounded linear operators from a normed space X into a normed space Y. The operator ideals theory takes an importance in functional analysis, since it has numerous applications in fixed point theorem, geometry of Banach spaces, spectral theory, eigenvalue distributions theorem, etc. Some of the operator ideals in the class of normed spaces or Banach spaces in functional analysis are characterized by various scalar sequence spaces. For example the ideal of compact operators is defined by kolmogorov numbers and the space c0 of convergent to zero sequences. Pietsch [1] inspected the operator ideals framed by the approximation numbers and the classical sequence space lp(0<p<∞). He proved that the ideals of Hilbert Schmidt operators and nuclear operators between Hilbert spaces are defined by l2 and l1, respectively, and the sequence of approximation numbers. In [2], Faried and Bakery examined the operator ideals developed by generalized Cesáro, Orlicz sequence spaces lM, and the approximation numbers. In [3], Faried and Bakery studied the operator ideals constructed by s- numbers, generalized Cesáro and Orlicz sequence spaces lM and show that the operator ideal formed by the previous sequence spaces and approximation numbers is small under certain conditions. Also summation process and sequences spaces applications are closely related to Korovkin type approximation theorems and linear positive operators studied by Costarelli and Vinti [4] and Altomare [5]. The idea of this paper is to examine a generalized class Scesφ by using Orlicz-Cesáro mean sequence spaces cesφ and the sequence of s-numbers, for which Scesφ constructs an operator ideal. The components of Scesφ as a pre-quasi Banach operator ideal containing finite dimensional operators as a dense subset and its completeness are proved. The inclusion relations between the pre-quasi operator ideals and the inclusion relations for their duals are determined. Finally, we show that the pre-quasi Banach operator ideal formed by the approximation numbers and cesφ is small under certain conditions. These results coincide with that known for cesp, (1<p<∞) in [3]. Furthermore we give some examples which support our main results.


## 2. Definitions and Preliminaries

Definition 1 (see [6]).
The sequence(sn(T))n=0∞, for all T∈L(X,Y) is named an s-function and the number sn(T) is called the nths- number of T if the following are satisfied: (a)
monotonicity:‖T‖=s0(T)≥s1(T)≥s2(T)≥⋯≥0 for all T∈L(X,Y);(b)
additivity:sm+n-1(T1+T2)≤sm(T1)+sn(T2) for all T1,T2∈L(X,Y), m, n∈N;(c)
property of ideal:sn(RTP)≤‖R‖sn(T)‖P‖ for all P∈L(X0,X), T∈L(X,Y), and R∈L(Y,Y0), where X0 and Y0 are normed spaces;(d)
sn(βT)=βsn(T) for every T∈L(X,Y), β∈R;(e)
rank property: ifrank(T)≤n then sn(T)=0 for every T∈L(X,Y);(f)
property of norming:(1)siIj=1,ifi<j;0,ifi≥j,whereIj is the identity operator on Rj.There are a few instances ofs-numbers; we notice the accompanying conditions:(1)
The n-th approximation number, denoted byαn(T), is defined by αn(T)=inf{‖T-B‖:B∈L(X,Y)andrank(B)≤n}.(2)
The n-th Hilbert number, denoted byhn(T), is defined by(2)hnT=sup⁡αnATB:A:Y→l2≤1andB:l2→X≤1.(3)
The n-th Weyl number, denoted byxn(T), is defined by(3)xnT=inf⁡αnTB:B:l2→X≤1.(4)
The n-th Kolmogorov number, denoted bydn(T), is defined by(4)dnT=infdimY≤nsupx≤1infy∈YTx-y.(5)
The n-th Gel’fand number, denoted bycn(T), is defined by cn(T)=αn(JYT), where JY is a metric injection from the space Y to a higher space l∞(Ψ) for an adequate index set Ψ. This number is independent of the choice of the higher space l∞(Ψ).(6)
The n-th Chang number, denoted byyn(T), is defined by(5)ynT=inf⁡αnAT:A:Y→l2≤1.Remark 2 (see [6]).
Among all thes-number sequences characterized above, it is easy to check that the approximation number, αn(T), is the largest and the Hilbert number, hn(T), is the smallest s-number sequence, i.e., hn(T)≤sn(T)≤αn(T) for any bounded linear operator T. If T is defined on a Hilbert space and compact, then all the s-numbers correspond with the eigenvalues of |T|, where |T|=(T∗T)1/2.Theorem 3 ([6], p.115).
LetT∈L(X,Y). Then(6)hnT≤xnT≤cnT≤αnT,hnT≤ynT≤dnT≤αnT.Theorem 4 ([6], p.90).
Ans-number sequence is called injective if, for any metric injection K∈L(Y,Y0), sn(T)=sn(KT) for all T∈L(X,Y).Theorem 5 ([6], p.95).
Ans-number sequence is called surjective if, for any metric surjection P∈L(X0,X), sn(T)=sn(TP) for all T∈L(X,Y).Theorem 6 ([6], pp.90-94).
The Weyl and Gel’fand numbers are injective.Theorem 7 ([6], pp.95).
The Chang and Kolmogorov numbers are surjective.Definition 8.
A finite rank operator is a bounded linear operator whose dimension of the range space is finite.Definition 9 ((duals-numbers) [7]).
For eachs-number sequence s=(sn), a dual s-number function sd=(snd) is defined by(7)sndT=snT′forallT∈LX,Y,where T′ is the dual of T.Definition 10 ([8], p.152)).
Ans-number sequence is called symmetric if sn(T)≥sn(T′) for all T∈L(X,Y). If sn(T)=sn(T′), then the s-number sequence is said to be completely symmetric.Presently we express some known results of dual of ans-number sequence.Theorem 11 ([8], p.152).
The approximation numbers are symmetric, i.e.,αn(T′)≤αn(T) for T∈L(X,Y).Remark 12 (see [9]).
α n ( T ) = α n ( T ′ ) for every compact operator T.Theorem 13 ([8], p.153).
LetT∈L(X,Y). Then(8)cnT′≤dnT,cnT=dnT′.In addition, if T is a compact operator then dn(T)=cn(T′).Theorem 14 ([6], p.96).
LetT∈L(X,Y). Then(9)ynT′≤xnT,xnT=ynT′,i.e., Chang numbers and Weyl numbers are dual to each other.Theorem 15 ([8], p.153).
The Hilbert numbers are completely symmetric, i.e.,hn(T′)=hn(T) for all T∈L(X,Y).Definition 16 (see [10, 11]).
The operator idealU≔{U(X,Y);XandYareBanachSpaces} is a subclass of linear bounded operators such that its components U(X,Y) which are subsets of L(X,Y) fulfill the accompanying conditions: (i)
IA∈U where A indicates one dimensional Banach space, where U⊂L.(ii)
ForT1,T2∈U(X,Y), then β1T1+β2T2∈U(X,Y) for any scalars β1,β2.(iii)
IfT∈L(X0,X), R∈U(X,Y), and P∈L(Y,Y0), then PRT∈U(X0,Y0).Definition 17 (see [12, 13]).
An Orlicz function is a functionφ:[0,∞)→[0,∞), which is nondecreasing, convex, and continuous with φ(0)=0 and φ(x)>0 for x>0 and limx→∞φ(x)=∞.Definition 18.
An Orlicz functionφ is said to satisfy Δ2-condition for every values of x≥0, if there is a>0, such that φ(2x)≤aφ(x). The Δ2-condition is corresponding to φ(mx)≤amφ(x) for every values of m>1 and x.Lindenstrauss and Tzafriri [14] utilized the idea of an Olicz function to define Orlicz sequence space: (10)lφ=x∈ω:ρλx<∞forsomeλ>0whereρx=∑k=0∞φxk,(lφ,‖.‖) is a Banach space with the Luxemburg norm: (11)xlφ=inf⁡λ>0:ρλ-1x≤1.Every Orlicz sequence space contains a subspace that is isomorphic to lp, for some 1≤p<∞ or c0 ([15], Theorem 4.a.9).In the recent past lot of work has been done on sequence spaces defined by Orlicz functions by Altin et al. [16], Et et al. ([17, 18]), Tripathy et al. ([19–21]), and Mohiuddine et al. ([22–25]).Given an Orlicz functionφ, the Orlicz-Cesáro mean sequence spaces is defined by (12)cesφ=u=ui∈ω:ρβu<∞forsomeβ>0,ρu=∑i=0∞ϕ∑j=0iuji+1.(cesφ,‖.‖) is a Banach space with the Luxemburg norm given by (13)ucesφ=inf⁡β>0:ρβ-1u≤1.It seems that Orlicz-Cesáro mean sequence spaces cesφ appeared for the first time in 1988, when Lim and Yee found their dual spaces [26]. Recently Cui, Hudzik, Petrot, Suantai, and Szymaszkiewicz obtained important properties of spaces cesφ [27]. In 2007 Maligranda, Petrot, and Suantai showed that cesφ is not B-convex, if φ∈Δ2 and cesφ≠0 [28]. The extreme points and strong X-points of cesφ have been characterized by Foralewski, Hudzik, and Szymaszkiewicz in [29]. In the case when φ(u)=up, 1≤p<∞, the space cesφ is just a Cesáro sequence space cesp, with the norm given by(14)ucesp=∑i=0∞∑j=0iuji+1p1/p.It is well known that ces1={0} [30].Definition 19 (see [31]).
The Matuszewska Orlicz lower indexαφ of an Orlicz function φ is defined as follows: (15)αφ=sup⁡p>0:∃K>0∀0<λ,t≤1φλt≤Ktpφλ.Theorem 20 (see [31]).
For any Orlicz functionφ, we have αφ>1 if and only if lφ⊂cesφ. In particular, if αφ>1 then cesφ≠{0}.Theorem 21 (see [31]).
Letφ1 and φ2 be Orlicz functions. If there exist b,t0>0 such that φ2(t0)>0 and φ2(t)≤φ1(bt) for all t∈[0,t0], then cesφ1⊂cesφ2.Theorem 22 (see [31]).
Letφ1 and φ2 be Orlicz functions and αφ1>1, then cesφ1⊂cesφ2 if and only if there exist b,t0>0 such that φ2(t0)>0 and φ2(t)≤φ1(bt) for all t∈[0,t0].Definition 23 (see [2]).
A class of linear sequence spacesE is called a special space of sequences (sss) having three properties: (1)
ei∈E for all i∈N,(2)
ifx=(xi)∈w, y=(yi)∈E and |xi|≤|yi| for every i∈N, then x∈E, “i.e., E is solid”,(3)
if(xi)i=0∞∈E, then (x[i/2])i=0∞∈E, wherever [i/2] means the integral part of i/2.Definition 24 (see [2]).
A subclass of the special space of sequences is called a premodular (sss) if there is a functionϱ:E→[0,∞[ fulfilling the accompanying conditions: (i)
ϱ(x)≥0 for each x∈E and ϱ(x)=0⇔x=θ, where θ is the zero element of E,(ii)
there existsL≥1 such that ϱ(λx)≤L|λ|ϱ(x) for all x∈E, and for any scalar λ,(iii)
for someK≥1, we have ϱ(x+y)≤K(ϱ(x)+ϱ(y)) for every x,y∈E,(iv)
if|xi|≤|yi| for all i∈N, then ϱ((xi))≤ϱ((yi)),(v)
for someK0≥1, we have(16)ϱxi≤ϱxi/2≤K0ϱxi,(vi)
the set of all finite sequences isϱ-dense in E. This means for each x=(xi)i=o∞∈E and for each ε>0 there exists m∈N such that ϱ((xi)i=m∞)<ε,(vii)
there exists a constantξ>0 such that ϱ(λ,0,0,0,…)≥ξ|λ|ϱ(1,0,0,0,…) for any λ∈R.We denote(Eϱ,ϱ) for the linear space E equipped with the metrizable topology generated by ϱ.Theorem 25 (see [32]).
IfX, Y are infinite dimensional Banach spaces and λi is a monotonic decreasing sequence to zero, then there exists a bounded linear operator T such that(17)116λ3i≤αiT≤8λi+1.Notations 26 (see [3]).
S E ≔ { S E ( X , Y ) ;  X   a n d   Y   a r e   B a n a c h   S p a c e s }, where S E ( X , Y ) ≔ { T ∈ L ( X , Y ) : ( ( s i ( T ) ) i = 0 ∞ ∈ E }. Also S E a p p ≔ { S E a p p ( X , Y ) ;  X   a n d   Y   a r e   B a n a c h   S p a c e s }, where S E a p p ( X , Y ) ≔ { T ∈ L ( X , Y ) : ( ( α i ( T ) ) i = 0 ∞ ∈ E }.Theorem 27 (see [3]).
IfE is a (sss), then SE is an operator ideal.The concept of pre-quasi operator ideal which is more general than the usual classes of operator ideal.Definition 28 (see [3]).
A functiong:Ω→[0,∞) is said to be a pre-quasi norm on the ideal Ω fulfilling the accompanying conditions: (1)
for allT∈Ω(X,Y), g(T)≥0 and g(T)=0 if and only if T=0,(2)
there exists a constantL≥1 such that g(βT)≤L|β|g(T), for all T∈Ω(X,Y) and β∈R,(3)
there exists a constantK≥1 such that g(T1+T2)≤K[g(T1)+g(T2)], for all T1,T2∈Ω(X,Y),(4)
there exists a constantC≥1 such that if P∈L(X0,X), R∈Ω(X,Y), and T∈L(Y,Y0), then g(TRP)≤C‖T‖g(R)‖P‖, where X0 and Y0 are normed spaces.Theorem 29 (see [3]).
Every quasi norm on the idealΩ is a pre-quasi norm on the ideal Ω.Here and after, we defineei={0,0,…,1,0,0,…} where 1 appears at the ith place for all i∈N.


## 3. Main Results

We give here the conditions on Orlicz-Cesáro mean sequence spacescesφ such that the class Scesφ of all bounded linear operators between arbitrary Banach spaces with its sequence of s-numbers which belong to cesφ forms an operator ideal.Theorem 30.
Ifφ is an Orlicz function satisfying Δ2-condition and αφ>1, then Scesφ is an operator ideal.Proof.
(1-i) Letx,y∈cesφ. Since φ is nondecreasing, convex, and satisfying Δ2-condition, we get for some k>0 that(18)∑n=0∞φ∑i=0nxi+yin+1≤k∑n=0∞φ∑i=0nxin+1+∑n=0∞φ∑i=0nyin+1<∞,then x+y∈cesφ.
(1-ii) Letλ∈R and x∈cesφ, and since φ is convex and satisfying Δ2-condition, we get for some k>0 that(19)∑n=0∞φ∑i=0nλxin+1≤λk∑n=0∞φ∑i=0nxin+1<∞,then λx∈cesφ; from (1-i) and (1-ii) cesφ is a linear space. Since en∈lφ, for all n∈N and αφ>1, then from Theorem 20, we get en∈cesφ, for all n∈N.
( 2 ) Let |xn|≤|yn| for all n∈N and y∈cesφ; since φ is nondecreasing, then we have(20)∑n=0∞φ∑i=0nxin+1≤∑n=0∞φ∑i=0nyin+1<∞,and we get x∈cesφ.
( 3 ) Let (xn)∈cesφ. Since φ is satisfying Δ2-condition, we get for some k>0 that(21)∑n=0∞φ∑i=0nxi/2n+1≤k+1∑n=0∞φ∑i=0nxin+1<∞,then xn/2∈cesφ. Then cesφ is a (sss); hence by Theorem 27, Scesφ is an operator ideal.Corollary 31.
S c e s q is an operator ideal, if 1<q<∞.We give the conditions on Orlicz-Cesáro mean sequence spacescesφ such that the ideal of the finite rank operators is dense in Scesφ(X,Y).Theorem 32.
S c e s φ ( X , Y ) = F ( X , Y ) ¯, if φ is an Orlicz function satisfying Δ2-condition and αφ>1.Proof.
Let us defineϱ(u)=∑i=0∞φ∑j=0i|uj|/i+1 on cesφ. First, we have to show that F(X,Y)¯⊆Scesφ(X,Y). Since αφ>1, we have ei∈cesφ for each i∈N and φ is an Orlicz function satisfying Δ2-condition, so for each finite operator P∈F(X,Y), i.e., we obtain (si(P))i=0∞ which contains only finitely many terms different from zero; hence P∈Scesφ(X,Y). Currently we prove that Scesφ(X,Y)⊆F(X,Y)¯; let P∈Scesφ(X,Y); we have (si(P))i=0∞∈cesφ; and hence ϱ(si(P))i=0∞<∞. By taking ε∈(0,1), hence there exists a i0∈N-{0} such that ϱ((si(P))i=i0∞)<ε/9δC2 for some c≥1, where δ=max⁡{1,∑i=i0∞φ1/i+1}. As si(P) is decreasing for every i∈N and φ is nondecreasing, we have(22)i0φs2i0P≤∑i=i0+12i0φ∑j=0isjPi+1≤∑i=i0∞φ∑j=0isjPi+1<ε9δC2.Hence, there exists B∈F2i0(X,Y) such that rank B≤2i0 and(23)i0φP-B≤∑i=i0+12i0φ∑j=0iP-Bi+1<ε9δC2.Since φ is right continuous at 0 and nondecreasing, then on considering this(24)P-B<ε6C2i0δ.Let k1>0, k2>0 and C=max⁡{1,k1,k2}, since φ is Orlicz function and by using (22), (23), and (24), we have (25)
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S c e s p ( X , Y ) = F ( X , Y ) ¯, if 1<p<∞.We express the accompanying theorem without verification; these can be set up utilizing standard procedure.Theorem 34.
The functiong(P)=∑i=0∞φ∑j=0isjP/i+1 is a pre-quasi norm on Scesφ, if φ is an Orlicz function satisfying Δ2-condition and αφ>1.We give the sufficient conditions on Orlicz-Cesáro mean sequence spacescesφ such that the components of the pre-quasi operator ideal Scesφ are complete.Theorem 35.
IfX and Y are Banach spaces, φ is an Orlicz function satisfying Δ2-condition and αφ>1, then (Scesφ(X,Y),g) is a pre-quasi Banach operator ideal.Proof.
Sinceφ is an Orlicz function satisfying Δ2-condition, then the function g(P)=ϱ((sn(P))n=0∞)=∑n=0∞φ∑m=0nsmP/n+1 is a pre-quasi norm on Scesφ. Let (Pm) be a Cauchy sequence in Scesφ(X,Y). Since L(X,Y)⊇Scesφ(X,Y) and αφ>1, we can find a constant ξ>0 such that (26)gPi-Pj=ϱsnPi-Pjn=0∞≥ϱs0Pi-Pj,0,0,0,…=ϱPi-Pj,0,0,0,…≥ξPi-Pjϱ1,0,0,0,…,then (Pm)m∈N is also a Cauchy sequence in L(X,Y). While the space L(X,Y) is a Banach space, there exists P∈L(X,Y) such that limm→∞Pm-P=0, while (sn(Pm))n=0∞∈cesφ for every m∈N. Since ϱ is continuous at θ and for some K≥1, we obtain (27)gP=ϱsnPn=0∞=ϱsnP-Pm+Pmn=0∞≤Kϱsn/2P-Pmn=0∞+Kϱαn/2Pmn=0∞≤KϱPm-Pn=0∞+KϱsnPmn=0∞<∞,we have (sn(P))n=0∞∈cesφ, and then P∈Scesφ(X,Y).Corollary 36.
IfX and Y are Banach spaces and 1<q<∞, then (Scesq(X,Y),g) is quasi Banach operator ideal, where g(P)=ϱ((sn(P))n=0∞)=∑n=0∞∑m=0nsmP/n+1q1/q.Theorem 37.
Letφ1, φ2 be Orlicz functions and αφ1>1. For any infinite dimensional Banach spaces X, Y and if there exist b,t0>0 such that φ2(t0)>0 and φ2(t)≤φ1(bt) for all t∈[0,t0], it is true that (28)Scesφ1appX,YScesφ2appX,Y⫋LX,Y.Proof.
LetX and Y be infinite dimensional Banach spaces and there exist b,t0>0 such that φ2(t0)>0 and φ2(t)≤φ1(bt) for all t∈[0,t0]; if P∈Scesφ1app(X,Y), then (αn(P))∈cesφ1. From Theorems 21, 22, and 25, we have cesφ1⊂cesφ2; hence P∈Scesφ2app(X,Y). It is easy to see that Scesφ2app(X,Y)⊂L(X,Y).Corollary 38.
For any infinite dimensional Banach spacesX, Y, and 1<p<q<∞, then ScespappX,YScesqapp(X,Y)⫋L(X,Y).We now study some properties of the pre-quasi Banach operator idealScesφ.Theorem 39.
The pre-quasi Banach operator ideal (Scesφ, g) is injective, if the s-number sequence is injective.Proof.
LetT∈L(X,Y) and P∈L(Y,Y0) be any metric injection. Assume that PT∈Scesφ(X,Y0), then ϱ(sn(PT))<∞. Since the s-number sequence is injective, we have sn(PT)=sn(T), for all T∈L(X,Y), n∈N. So ϱ(sn(T))=ϱ(sn(PT))<∞. Hence T∈Scesφ(X,Y) and clearly g(T)=g(PT) is verified.Remark 40.
The pre-quasi Banach operator ideal (ScesφWeyl, g) and the pre-quasi Banach operator ideal (ScesφGel, g) are injective pre-quasi Banach operator ideal.Theorem 41.
The pre-quasi Banach operator ideal (Scesφ, g) is surjective, if the s-number sequence is surjective.Proof.
LetT∈L(X,Y) and P∈L(X0,X) be any metric surjection. Suppose that TP∈Scesφ(X0,Y), then ϱ(sn(TP))<∞. Since the s-number sequence is surjective, we have sn(TP)=sn(T), for all T∈L(X,Y), n∈N. So ϱ(sn(T))=ϱ(sn(TP))<∞. Hence T∈Scesφ(X,Y) and clearly g(T)=g(TP) is verified.Remark 42.
The pre-quasi Banach operator ideal (ScesφChang, g) and the pre-quasi Banach operator ideal (ScesφKol, g) are surjective pre-quasi Banach operator ideal.Likewise, we have the accompanying inclusion relations between the pre-quasi Banach operator ideals.Theorem 43.
( 1 )   S c e s φ a p p ⊆ S c e s φ K o l ⊆ S c e s φ C h a n g ⊆ S c e s φ H i l b.
( 2 )   S c e s φ a p p ⊆ S c e s φ G e l ⊆ S c e s φ W e y l ⊆ S c e s φ H i l b.Proof.
Sincehn(T)≤yn(T)≤dn(T)≤αn(T) and hn(T)≤xn(T)≤cn(T)≤αn(T) for every n∈N and ϱ is nondecreasing, we obtain (29)ϱhnT≤ϱynT≤ϱdnT≤ϱαnT,ϱhnT≤ϱxnT≤ϱcnT≤ϱαnT.Hence the result is as follows.We presently express the dual of the pre-quasi operator ideal formed by differents- number sequences.Theorem 44.
The pre-quasi operator idealScesφHilb is completely symmetric and the pre-quasi operator ideal Scesφapp is symmetric.Proof.
Sincehn(T′)=hn(T) and αn(T′)≤αn(T), for all T∈L(X,Y), we have ScesφHilb=(ScesφHilb)′ and Scesφapp⊆(Scesφapp)′.In perspective on Theorem13, we express the following result without proof.Theorem 45.
The pre-quasi operator idealScesφKol⊆(ScesφGel)′ and ScesφGel=(ScesφKol)′. In addition if T is a compact operator from X to Y, then ScesφKol=(ScesφGel)′.In perspective on Theorem14, we express the following result without proof.Theorem 46.
The pre-quasi operator idealScesφChang=(ScesφWeyl)′ and ScesφWeyl=(ScesφChang)′.Theorem 47.
Ifφ is an Orlicz function satisfying Δ2-condition and αφ>1, then the pre-quasi Banach operator ideal Scesφapp is small.Proof.
Sinceφ is an Orlicz function and αφ>1, take β=∑i=0∞φ(1/i+1). Then (Scesφapp,g), where g(T)=ϱ((αn(T))n=0∞)=1/β∑n=0∞φ∑m=0nαm(T)/n+1 is a pre-quasi Banach operator ideal. Let X and Y be any two Banach spaces. Assume that Scesφapp(X,Y)=L(X,Y), then there exists a constant C>0 such that g(T)≤C‖T‖ for all T∈L(X,Y). Suppose that X and Y are infinite dimensional Banach spaces. Then by Dvoretzky’s theorem [8] for m∈N, we have quotient spaces X/Mm and subspaces Nm of Y which can be mapped onto l2m by isomorphisms Vm and Bm such that ‖Vm‖‖Vm-1‖≤2 and ‖Bm‖‖Bm-1‖≤2. Consider Im be the identity map on l2m, Pm be the quotient map from X onto X/Mm, and Qm be the natural embedding map from Nm into Y. Let vn be the Bernstein numbers [7], then(30)1=vnIm=vnBmBm-1ImVmVm-1≤BmvnBm-1ImVmVm-1=BmvnQmBm-1ImVmVm-1≤BmdnQmBm-1ImVmVm-1=BmdnQmBm-1ImVmQmVm-1≤BmαnQmBm-1ImVmQmVm-1,for 1≤i≤m. Now since φ is nondecreasing and having Δ2-condition, we have (31)∑j=0i1≤∑j=0iBmαjQmBm-1ImVmPmVm-1⇒1i+1i+1≤Bm1i+1∑j=0iαjQmBm-1ImVmPmVm-1⇒φ1≤LBmVm-1φ1i+1∑j=0iαjQmBm-1ImVmPm.Therefore (32)∑i=0mφ1≤LBmVm-1∑i=0mφ1i+1∑j=0iαjQmBm-1ImVmPm⇒φ1βm+1≤LBmVm-11β∑i=0mφ1i+1∑j=0iαjQmBm-1ImVmPm⇒φ1βm+1≤LBmVm-1gQmBm-1ImVmPm⇒φ1βm+1≤LCBmVm-1QmBm-1ImVmPm⇒φ1βm+1≤LCBmVm-1QmBm-1ImVmPm=LCBmVm-1Bm-1ImVm⇒φ1βm+1≤4LC,for some L≥1. Thus we arrive at a contradiction since m is arbitrary. Hence X and Y both cannot be infinite dimensional when Scesφapp(X,Y)=L(X,Y).Theorem 48.
Ifφ is an Orlicz function satisfying Δ2-condition and αφ>1, then the pre-quasi Banach operator ideal ScesφKol is small.Corollary 49.
Ifp∈(1,∞), then the quasi Banach operator ideal Scespapp is small.Corollary 50.
Ifp∈(1,∞), then the quasi Banach operator ideal ScespKol is small.


## 4. Examples

We give some examples which support our main results.Example 1.
Letφ be an Orlicz function; the subspace cesφh of all order continuous elements of cesφ is defined as [27] (33)cesφh=x∈cesφ:∀k>0∃nk∈N∑n=nk∞φkn∑i=1nxi<∞.If φ is an Orlicz function satisfying Δ2-condition and αφ>1, then the following conditions are satisfied: (1)
Scesφh is an operator ideal.(2)
Scesφh(X,Y)=F(X,Y)¯.(3)
IfX and Y are Banach spaces, then (Scesφh(X,Y),g) is pre-quasi Banach operator ideal.(4)
The pre-quasi Banach operator idealScesφhapp is small.Proof.
Sinceφ is an Orlicz function satisfying Δ2-condition and αφ>1, then from Theorem (5) in [31] we have cesφh=cesφ which completes the proof.Example 2.
Letφ be defined as (34)φt=altl+al-1tl-1+…+a1t,whereai>0forall1≤i≤l,l∈N,l>1andt≥0.It is clear that φ is an Orlicz function and αφ=l>1. Also φ is satisfying Δ2-condition since(35)limsupt→0+⁡φ2tφt≤2l<∞.Then the following conditions are satisfied: (1)
Scesφ is an operator ideal.(2)
Scesφ(X,Y)=F(X,Y)¯.(3)
IfX and Y are Banach spaces, then (Scesφ(X,Y),g) is pre-quasi Banach operator ideal.(4)
The pre-quasi Banach operator idealScesφapp is small.In the following two examples we will explain the importance of the sufficient conditions.Example 3.
Letφ be defined as (36)φt=0ift=0,-tln⁡tift∈0,1e,32et2-t+12eift∈1e,∞.It is clear that φ is an Orlicz function. Since ∑n=1∞φ1/n=∑n=1∞1/nln⁡n=∞, hence cesφ={0}. The space Scesφ is not operator ideal since IK∉Scesφ. Also since φ is convex function and for p>1, we have(37)limt→0+⁡φλtφλtp=limt→0+⁡t1-pln⁡λln⁡λt=limt→0+⁡1-pt1-pln⁡λ=∞,for all λ∈(0,1], then αφ=1. Although φ is satisfying Δ2-condition since(38)limsupt→0+⁡φ2tφt=limsupt→0+⁡2ln⁡tln⁡2t≤2<∞.Example 4.
Letφ(u)=∫0uf(t)dt, where f(t) is defined as (39)ft=0ift=0,1n!ift∈1n+1!,1n!forn=1,2,3…,tift∈1,∞.It is clear that φ is an Orlicz function. Let T∈Scesφ with sn(T)=1/n! for all n∈N. We have for n>2 that (40)φsn2T=∫02/n!ftdt>∫1/n!2/n!ftdt>∫1/n!1/n-1!ftdt>1n!n-1!,nφsnT=n∫01/n!ftdt<nsup0≤t≤1/n!ft∫01/n!1dt<1n!n-1!.Hence 2T∉Scesφ, so the space Scesφ is not operator ideal and φ∉Δ2. Also since φ is convex function and for p>1, we have (41)limt→0+⁡φλtφλtp=limt→0+⁡t-p=∞,for all λ∈(0,1], then αφ=1.



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