Research ArticlePre-Quasi Simple Banach Operator IdealGenerated by sminusNumbers

Awad A Bakery 12 and Afaf R Abou Elmatty2

1University of Jeddah College of Science and Arts at Khulis Department of Mathematics Jeddah Saudi Arabia2Department of Mathematics Faculty of Science Ain Shams University PO Box 1156 Abbassia Cairo 11566 Egypt

Correspondence should be addressed to Awad A Bakery awad_bakeryyahoocom

Received 10 September 2019 Accepted 11 February 2020 Published 15 May 2020

Academic Editor Shanhe Wu

Copyright copy 2020 Awad A Bakery and Afaf R Abou Elmatty )is is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

Let E be a weighted Nakano sequence space or generalized Cesaro sequence space defined by weighted mean and by usingsminusnumbers of operators from a Banach space X into a Banach space Y We give the sufficient (not necessary) conditions on E suchthat the components SE(X Y) ≔ T isin L(X Y) (sn(T))infinn0 isin E1113864 1113865 of the class SE form pre-quasi operator ideal the class of all finiterank operators are dense in the Banach pre-quasi ideal SE the pre-quasi operator ideal formed by the sequence of approximationnumbers is strictly contained for different weights and powers the pre-quasi Banach Operator ideal formed by the sequence ofapproximation numbers is small and finally the pre-quasi Banach operator ideal constructed by sminusnumbers is simpleBanach space

1 Introduction

All through the paper

L(X Y) T X⟶ Y T is a bounded linear operator X andY are Banach spaces1113864 1113865 (1)

and if X Y we write L(X) by w we denote the space of allreal sequences and θ is the zero vector of E Due to theimmense applications in geometry of Banach spaces spectraltheory geometry of Banach spaces theory of eigenvaluedistributions etc the theory of operator ideals goals pos-sesses an uncommon essentialness in useful examinationSome of operator ideals in the class of Banach spaces orHilbert spaces are defined by different scalar sequencespaces For example the ideal of compact operators is de-fined by the space c0 of null sequences and Kolmogorovnumbers Pietsch [1] examined the quasi-ideals formed bythe approximation numbers and classical sequence spaceℓp(0ltpltinfin) He showed that the ideals of nuclear

operators and of Hilbert Schmidt operators between Hilbertspaces are defined by ℓ1 and ℓ2 respectively Also he provedthat the class of all finite rank operators is dense in theBanach quasi ideal and the algebra L(ℓp) where (1lepltinfin)

contains one and only one nontrivial closed ideal Pietsch [2]showed that the quasi Banach operator ideal formed by thesequence of approximation numbers is small Makarov andFaried [3] proved that the quasi-operator ideal formed by thesequence of approximation numbers is strictly contained fordifferent powers ie for any infinite dimensional Banachspaces X and Y and for any qgtpgt 0 it is true thatSappℓp (X Y)⫋ S

appℓq (X Y)⫋L(X Y) In [4] Faried and Bakery

studied the operator ideals constructed by approximation

HindawiJournal of Function SpacesVolume 2020 Article ID 9164781 11 pageshttpsdoiorg10115520209164781

numbers and generalized Cesaro and Orlicz sequence spacesℓM In [5] Faried and Bakery introduced the concept of pre-quasi operator ideal which is more general than the usualclasses of operator ideal they studied the operator idealsconstructed by sminus numbers generalized Cesaro and Orliczsequence spaces ℓM and showed that the operator idealformed by the previous sequence spaces and approximationnumbers is small under certain conditions )e idea of thispaper is to study a generalized class SE by using the sequenceof s-numbers and E (weighted Nakano sequence space orgeneralized Cesaro sequence space) we give sufficient (notnecessary) conditions on E such that SE constructs a pre-quasi operator ideal which gives a negative answer ofRhoades [6] open problem about the linearity of Eminustypespaces SE )e components of SE as a pre-quasi Banachoperator ideal containing finite dimensional operators as adense subset and its completeness are proved )e pre-quasioperator ideal formed by the sequence of approximationnumbers is strictly contained for different weights and powersare determined Finally we show that the pre-quasi Banachoperator ideal formed by E and approximation numbers is smallunder certain conditions Furthermore the sufficient conditionsfor which the pre-quasi Banach operator ideal constructed bysminusnumbers is a simple Banach space

2 Definitions and Preliminaries

Definition 1 (see[7]) An s-number function is a map de-fined on L(X Y) which associates to each operatorT isin L(X Y) a nonnegative scaler sequence (sn(T))infinn0assuming that the the following states are verified

(a) T s0(T)ge s1(T)ge s2(T)ge middot middot middot ge 0 for T isin L(X

Y)

(b) sm+nminus1(T1 + T2)le sm(T1) + sn(T2) for all T1 T2 isinL(X Y) m n isin N

(c) Ideal property sn(RVT)le Rsn(V)T for allT isin L(X0 X) V isin L(X Y) and R isin L(Y Y0) whereX0 and Y0 are arbitrary Banach spaces

(d) If G isin L(X Y) and λ isin R we obtainsn(λG) |λ|sn(G)

(e) Rank property if rank(T)le n then sn(T) 0 foreach T isin L(X Y)

(f ) Norming property srgen(In) 0 or srltn(In) 1where In represents the unit operator on the n-di-mensional Hilbert space ℓn

2

)ere are several examples of s-numbers wemention thefollowing

(1) )e n-th approximation number denoted by αn(T)is defined by

αn(T) inf T minus B B isin L(X Y) and rank(B)le n

(2)

(2) )e n-th Gelrsquofand number denoted by cn(T) isdefined by cn(T) αn(JYT) where JY is a metric

injection from the normed space Y to a higher spacelinfin(Λ) for an adequate index set Λ )is number isindependent of the choice of the higher space linfin(Λ)

(3) )e n-th Kolmogorov number denoted by dn(T) isdefined by

dn(T) infdimYlen

supxle1

infyisinY

Tx minus y (3)

(4) )e n-th Weyl number denoted by xn(T) is definedby

xn(T) inf αn(TB) B ℓ2⟶ X

le 11113966 1113967 (4)

(5) )e n-th Chang number denoted by yn(T) is de-fined by

yn(T) inf αn(BT) B Y⟶ ℓ2

le 11113966 1113967 (5)

(6) )e n-th Hilbert number denoted by hn(T) is de-fined by

hn(T) sup αn(BTA) B Y⟶ ℓ2

1113966

le 1 and A ℓ2⟶ X

le 11113967(6)

Remark (see[7]) Among all the s-number sequences de-fined above it is easy to verify that the approximationnumber αn(T) is the largest and the Hilbert number hn(T)is the smallest s-number sequence ie hn(T)le sn(T)leαn(T) for any bounded linear operator T If T is compact anddefined on a Hilbert space then all the s-numbers coincidewith the eigenvalues of |T| where |T| (TlowastT)12

Theorem 1 (see [7] p115) If T isin L(X Y) then

hn(T)le xn(T)le cn(T)le αn(T)

hn(T)leyn(T)ledn(T)le αn(T)(7)

Definition 2 (see [1]) A finite rank operator is a boundedlinear operator whose dimension of the range space is finite)e space of all finite rank operators on E is denoted byF(E)

Definition 3 (see[1]) A bounded linear operator A E⟶ E

(where E is a Banach space) is called approximable if thereare Sn isin F(E) for all n isin N such that limn⟶infinA minus Sn 0)e space of all approximable operators on E is denoted byΛ(E)

Lemma 1 (see [1]) Let T isin L(X Y) If T is not approxim-able then there are operators G isin L(X X) and B isin L(Y Y)such that BTGek ek for all k isin N

Definition 4 (see [1]) A Banach space X is called simple ifthe algebra L(X) contains one and only one nontrivialclosed ideal

2 Journal of Function Spaces

Definition 5 (see [1]) A bounded linear operatorA E⟶ E (where E is a Banach space) is called compact ifA(B1) has compact closure where B1 denotes the closed unitball of E )e space of all compact operators on E is denotedby Lc(E)

Theorem 2 (see [1]) If E is infinite dimensional Banachspace we have

F(E)⫋Λ(E)⫋Lc(E)⫋ L(E) (8)

Definition 6 (see [1]) Let L be the class of all bounded linearoperators between any arbitrary Banach spaces A subclass Uof L is called an operator ideal if each element U(X Y)

U cap L(X Y) fulfills the following conditions

(i) I5 isin U where 5 represents Banach space of onedimension

(ii) )e space U(X Y) is linear over R(iii) If T isin L(X0 X) V isin U(X Y) and R isin L(Y Y0)

then RVT isin U(X0 Y0) (see [8 9])

)e concept of pre-quasi operator ideal is more generalthan the usual classes of operator ideal

Definition 7 (see [5]) A function g Ω⟶ [0infin) is said tobe a pre-quasi norm on the ideal Ω if the following con-ditions holds

(1) For all T isin Ω(X Y) g(T)ge 0 and g(T) 0 if andonly if T 0

(2) )ere exists a constant Mge 1 such that g(λT) leM|λ|g(T) for all T isin Ω(X Y) and λ isin R

(3) )ere exists a constant Kge 1 such thatg(T1 + T2)leK[g(T1) + g(T2)] for all T1 T2 isin Ω(X Y)

(4) )ere exists a constant Cge 1 such that ifT isin L(X0 X) P isin Ω(X Y) and R isin L(Y Y0) theng(RPT) leCRg(P)T where X0 and Y0 arenormed spaces

Theorem 3 (see [5]) Every quasi norm on the ideal Ω is apre-quasi norm on the ideal Ω

Let p (pn) be a positive real and (βn)nisinN be a sequenceof positive real the weighted Nakano sequence space isdefined by

ℓ pn( )β x xk( 1113857 isin ω ρ(λx)ltinfin for some λgt 01113864 1113865 where ρ(x)

1113944infin

k0βk xk

111386811138681113868111386811138681113868111386811138681113872 1113873

pk

(9)

And (ℓ(pn)

β middot) is a Banach space where

x inf ηgt 0 ρx

η1113888 1113889le 11113896 1113897 (10)

When (pn) is bounded we mark

ℓ pn( )β x xk( 1113857 isin ω 1113944

infin

k0βk xk

11138681113868111386811138681113868111386811138681113868pk ltinfin

⎧⎨

⎩

⎫⎬

⎭ (11)

If βn 1 for all n isin N then ℓ(pn)

β will reduce to ℓ(pn)

studied in [10 11]In [12] Sengonul defined the sequence space as

ces an( 1113857 pn( 1113857( 1113857 x xk( 1113857 isin ω existλgt 0with ρ(λx)ltinfin1113864 1113865

ρ(x) 1113944infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pk

(12)

where (an) and (pn) are the sequences of positive real andpn ge 1 for all n isin N With the norm

x inf ηgt 0 ρx

η1113888 1113889le 11113896 1113897 (13)

Note

(1) Taking an (1(n + 1)) for all n isin N thences((an) (pn)) is reduced to ces(pn) studied bySanhan and Suantai [13]

(2) Taking an (1(n + 1)) and pn p for all n isin Nthen ces((an) (pn)) is reduced to cesp studied bymany authors (see [14ndash16])

Definition 8 (see [5]) Let E be a linear space of sequencesthen E is called a (sss) if

(1) For n isin N en isin E

(2) E is solid ie assuming x (xn) isin w y (yn) isin E

and |xn|le |yn| for all n isin N then x isin E

(3) (x[n2])infinn0 isin E where [n2] indicates the integral

part of [n2] whenever (xn)infinn0 isin E

Definition 9 (see [5]) A subclass of the (sss) is called apremodular (sss) assuming that we have a mapρ E⟶ [0infin[ with the following

(i) For x isin E x θ⟺ ρ(x) 0 with ρ(x)ge 0(ii) For each x isin E and scalar λ we get a real number

Lge 1 for which ρ(λx)le |λ|Lρ(x)

(iii) ρ(x + y)leK(ρ(x) + ρ(y)) for each x y isin E holdsfor a few numbers Kge 1

(iv) For n isin N |xn|le |yn| we obtain ρ(xn)le ρ(yn)

(v) )e inequality ρ(xn)le ρ(x[n2])leK0ρ(xn) holdsfor some numbers K0 ge 1

(vi) F Eρ where F is the space of finite sequences(vii) )ere is a steady ξ gt 0 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin R

Condition (ii) gives the continuity of ρ(x) at θ)e linearspace E enriched with the metric topology formed by thepremodular ρ will be indicated by Eρ Moreover condition(16) in Definition 8 and condition (vi) in Definition 9 explainthat (en)nisinN is a Schauder basis of Eρ

Journal of Function Spaces 3

Notations )e sets SE SE(X Y) SappE and S

appE (X Y) (cf

[5]) as follows

SE ≔ SE(X Y) X andY are Banach spaces1113864 1113865 where

SE(X Y) ≔ T isin L(X Y) si(T)( 1113857(infini0 isin E1113864 1113865Also

SappE ≔ S

appE (X Y) X andY are Banach spaces1113864 1113865 where

SappE (X Y) ≔ T isin L(X Y) αi(T)( 1113857(

infini0 isin E1113864 1113865

(14)

Theorem 4 (see [5]) If E is a (sss) then SE is an operatorideal

Theorem 5 (see [3]) If X and Y are infinite dimensionalBanach spaces and (μi) is a monotonic decreasing sequenceto zero then there exists a bounded linear operator T suchthat

116μ3i le αi(T)le 8μi+1 (15)

Now and after define en 0 0 1 0 0 where 1appears at the nth place for all n isin N and the given inequalitywill be used in the sequel

an + bn

11138681113868111386811138681113868111386811138681113868pn leH an

11138681113868111386811138681113868111386811138681113868pn + bn

11138681113868111386811138681113868111386811138681113868pn1113872 1113873 (16)

whereH max 1 2hminus 11113864 1113865 h supnpn and pn ge 1 for all n isin N(see [17])

3 Main Results

31 Linear Problem We examine here the operator idealscreated by sminusnumbers and also weighted Nakano sequencespace or generalized Cesaro sequence space defined byweighted mean such that those classes of all bounded linearoperators T between arbitrary Banach spaces with (αn(T))

in these sequence spaces type an ideal operator

Theorem 6 ℓ(pn)

β is a (sss) if the following conditions aresatisfied

(a1) Ie sequence (pn) is increasing and bounded fromabove with pn gt 0 for all n isin N

(a2) Either (βn) is monotonic decreasing or monotonicincreasing such that there exists a constant Cge 1 forwhich β2n+1 leCβn

Proof (1) Let x y isin ℓ(pn)

β Since (pn) is bounded we get

1113944

infin

n0βn xn + yn

11138681113868111386811138681113868111386811138681113868pn leH 1113944

infin

n0βn xn

11138681113868111386811138681113868111386811138681113868pn + 1113944infin

n0βn yn

11138681113868111386811138681113868111386811138681113868pn⎛⎝ ⎞⎠ltinfin

(17)

then x + y isin ℓ(pn)

β

(2) Let λ isin R and x isin ℓ(pn)

β Since (pn) is bounded wehave

1113944

infin

n0βn λxn

11138681113868111386811138681113868111386811138681113868pn le sup

n

|λ|pn1113944

infin

n0βn xn

11138681113868111386811138681113868111386811138681113868pn ltinfin (18)

)en λx isin ℓ(pn)

β )erefore by using Parts (1) and (2) wehave that the space ℓ(pn)

β is linear Also en isin ℓ(pn)

β for alln isin N since

1113944

infin

i0βi en(i)

11138681113868111386811138681113868111386811138681113868pi βn (19)

(2) Let |xn|le |yn| for all n isin N and y isin ℓ(pn)

β Since βn gt 0for all n isin N then

1113944

infin

n0βn xn

11138681113868111386811138681113868111386811138681113868pn le 1113944infin

n0βn yn

11138681113868111386811138681113868111386811138681113868pn ltinfin (20)

and we get x isin ℓ(pn)

β

(3) Let (xn) isin ℓ(pn)

β (βn) be an increasing sequence)ere exists Cgt 0 such that β2n+1 leCβn and (pn) beincreasing then we have

1113944

infin

n0βn x[n2]

11138681113868111386811138681113868111386811138681113868pn 1113944infin

n0β2n xn

11138681113868111386811138681113868111386811138681113868p2n + 1113944infin

n0β2n+1 xn

11138681113868111386811138681113868111386811138681113868p2n+1

le 1113944

infin

n0β2n xn

11138681113868111386811138681113868111386811138681113868pn + 1113944

infin

n0β2n+1 xn

11138681113868111386811138681113868111386811138681113868pn le 2C 1113944

infin

n0βn xn

11138681113868111386811138681113868111386811138681113868pn

(21)

and then (x[n2]) isin ℓ(pn)

β

Theorem 7 ces((an) (pn)) is a (sss) if the following con-ditions are satisfied

(b1) Ie sequence (pn) is increasing and bounded withp0 gt 1

(b2) 1113936infinn0 (an)pn ltinfin

Proof (1) Given that x y isin ces((an) (pn)) and λ1 λ2 isin RSince (pn) is bounded we have

1113944

infin

n0an 1113944

n

k0λ1xk + λ2yk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

leH supn

λ11113868111386811138681113868

1113868111386811138681113868pn 1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

⎛⎝

+ supn

λ21113868111386811138681113868

1113868111386811138681113868pn 1113944

infin

n0an 1113944

n

k0yk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

⎞⎠ltinfin

(22)

Hence λ1x + λ2y isin ces((an) (pn)) then the spaceces((an) (pn)) is linear Also prove that em isin ces((an) (pn))

for all m isin N since 1113936infinn0 (an)pn ltinfin So we get

1113944

infin

n0an 1113944

n

k0em(k)

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

1113944infin

nm

an( 1113857pn le 1113944infin

n0an( 1113857

pn ltinfin

(23)

Hence em isin ces((an) (pn))

4 Journal of Function Spaces

(2) Let |xn|le |yn| for all n isin N and y isin ces((an) (pn))Since an gt 0 for all n isin N then

1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 1113944

infin

n0an 1113944

n

k0yk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

ltinfin (24)

and we get x isin ces((an) (pn))

(3) Let (xn) isin ces((an) (pn)) Since (pn) is increasingand the sequence (an) with 1113936

infinn0 (an)pn ltinfin is de-

creasing then we have

1113944

infin

n0an 1113944

n

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

1113944infin

n0a2n 1113944

2n

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

p2n

+ 1113944infin

n0a2n+1 1113944

2n+1

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

p2n+1

le 1113944infin

n0a2n 1113944

2n

k02 xk

11138681113868111386811138681113868111386811138681113868 + xn

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠⎛⎝ ⎞⎠

pn

+ 1113944infin

n0a2n+1 1113944

2n+1

k02 xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 2hminus 11113944

infin

n02an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

+ 1113944infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠ + 1113944infin

n02an 1113944

n

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 2hminus 1 2h+ 11113872 1113873 1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

+ 2h1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 22hminus 1+ 2hminus 1

+ 2h1113872 1113873 1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

ltinfin

(25)

Hence (x[n2]) isin ces((an) (pn))By using)eorem 4 we can get the following corollaries

Corollary 1 Let conditions (a1) and (a2) be satisfied thenSappℓ(pn )

β

be an operator ideal

Corollary 2 Sappℓ(pn ) is an operator ideal if the sequence (pn) is

increasing and bounded from above with pn gt 0 for all n isin N

Corollary 3 If p isin (0infin) then Sappℓp is an operator ideal

Corollary 4 Conditions (b1) and (b2) are satisfied henceSappces((an)(pn)) is an operator ideal

Corollary 5 Assume (pn) is increasing with p0 gt 1 andbounded so S

appces(pn) is an operator ideal

Corollary 6 If p isin (1infin) then Sappcesp is an operator ideal

4 Topological Problem

)e following question arises naturally which sufficientconditions (not necessary) on the sequence space E(weighted Nakano sequence space and generalized Cesarosequence space defined by weighted mean) are the ideal ofthe finite rank operators in the class of Banach spaces densein SE )is gives a negative answer of Rhoades [6] openproblem about the linearity of Eminustype spaces (SE)

Theorem 8 F(X Y) Sℓ(pn )

β(X Y) whenever conditions

(a1) and (a2) are satisfied

Proof First we substantiate that each finite operatorT isin F(X Y) belongs to Sℓ(pn )

β(X Y) Given that

en1113864 1113865nisinN sub ℓ(pn)

β and the space ℓ(pn)

β is linear then for all finiteoperators T isin F(X Y) ie the sequence (sn(T))nisinN con-tains only finitely many numbers different from zeroCurrently we substantiate that Sℓ(pn )

β(X Y) sube F(X Y) On

taking T isin Sℓ(pn )

β(X Y)we obtain (sn(T))infinn0 isin ℓ

(pn)

β hence

ρ((sn(T))infinn0)ltinfin let ε isin (0 1) at that point there exists am isin N minus 0 such that ρ((sn(T))infinnm)lt (ε4C2) for someCge 1 While (sn(T))nisinN is decreasing we get

1113944

2m

nm+1βn s2m(T)( 1113857

pn le 11139442m

nm+1βn sn(T)( 1113857

pn

le 1113944

infin

nm

βn sn(T)( 1113857pn lt

ε4C2

(26)

Hence there exists A isin F2m(X Y) rank Ale 2m and

1113944

3m

n2m+1βn(T minus A)

pn le 11139442m

nm+1βn(T minus A)

pn ltε

4C2 (27)

Since (pn) is a bounded consider

1113944

m

n0βn(T minus A)

pn ltε

4C2 (28)

Let (βn) be monotonic increasing such that there exists aconstant Cge 1 for which β2n+1 leCβn )en we have forngem that

β2m+n le β2m+2n+1 leCβm+n leCβ2n leCβ2n+1 leC2βn (29)

Since (pn) is increasing inequalities (26)ndash(29) give

Journal of Function Spaces 5

d(T A) ρ sn(T minus A)( 1113857infinn0

11139443mminus1

n0βn sn(T minus A)( 1113857

pn + 1113944infin

n3m

βn sn(T minus A)( 1113857pn

le 11139443m

n0βn(T minus A)

pn + 1113944infin

nm

βn+2m sn+2m(T minus A)( 1113857pn+2m

le 3 1113944m

n0βn(T minus A)

pn + C2

1113944

infin

nm

βn sn(T)( 1113857pn lt ε

(30)

Since I3 isin Sℓ(n)

(1)

condition (a1) is not satisfied whichgives a counter example of the converse statement )isfinishes the proof

From )eorem 8 we can say that if (a1) and (a2) aresatisfied then every compact operator would be approxi-mated by finite rank operators and the converse is not alwaystrue

Theorem 9 F(X Y) Sces((an)(pn))(X Y) assume thatstates (b1) and (b2) are fulfilled and the converse is not alwaystrue

Proof Primary since en isin ces((an) (pn)) for each n isin N andthe space ces((an) (pn)) is linear then for every finitemapping T isin F(X Y) ie the sequence (sn(T))nisinN containsonly fnitely many numbers different from zero HenceF(X Y) sube Sces((an)(pn))(X Y) By letting T isinSces((an)(pn))(X Y) we obtain (sn(T))nisinN isin ces((an) (pn))while ρ((αn(T))nisinN)ltinfin Let ε isin (0 1) )en there exists a

number m isin N minus 0 such that ρ((sn(T))infinnm)lt (ε2h+2δC)

for some Cge 1 where δ max 1 1113936infinnm a

pnn1113966 1113967 As sn(T) is

decreasing for every n isin N we obtain

1113944

2m

nm+1an 1113944

n

k0s2m(T)⎛⎝ ⎞⎠

pn

le 11139442m

nm+1an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

le 1113944infin

nm

an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

ltε

2h+2δC

(31)

)en there exists A isin F2m(X Y) and rank Ale 2m and

1113944

3m

n2m+1an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

le 11139442m

nm+1an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

ltε

2h+2δC

(32)

and since (pn) is bounded consider

supnm

infin1113944

m

k0T minus A⎛⎝ ⎞⎠

pn

ltε2hδ

(33)

Hence set

1113944

m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

ltε

2h+3δC (34)

However (pn) is increasing and (an) is decreasing foreach n isin N by using inequalities (31)ndash(34) we have

d(T A) ρ sn(T minus A)( 1113857infinn0

11139443mminus1

n0an 1113944

n

k0sk(T minus A)⎛⎝ ⎞⎠

pn

+ 1113944infin

n3m

an 1113944

n

k0sk(T minus A)

pn⎛⎝ ⎞⎠

le 11139443m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an+2m 1113944

n+2m

k0sk(T minus A)⎛⎝ ⎞⎠

pn+2m

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an 1113944

2mminus 1

k0sk(T minus A) + an 1113944

n+2m

k2m

sk(T minus A)⎛⎝ ⎞⎠

pn

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 11113944

infin

nm

an 1113944

2mminus 1

k0sk(T minus A)⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an 1113944

n+2m

k2m

sk(T minus A)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

le 3 1113944

m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 11113944

infin

nm

an 1113944

m

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944

infin

nm

an 1113944

n

k0sk+2m(T minus A)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 1supinfinnm 1113944

m

k0T minus A)

pn 1113944

infin

nm

an( 1113857pn + 2hminus 1

1113944

infin

nm

an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠lt ε

(35)

Since I3 isin Sces((1)(1)) condition (b2) is not satisfiedwhich gives a counter example of the converse statement)is finishes the proof

From )eorem 9 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 isnot always true

Corollary 7 If (pn) is an increasing with pn gt 0 for all n isin Nand bounded from above then Sl(p) (X Y) F(X Y)

6 Journal of Function Spaces

Corollary 8 If 0ltpltinfin then Slp (X Y) F(X Y)

Corollary 9 Sces(pn)(X Y) F(X Y) if (pn) is increasingwith p0 gt 1 and limn⟶infinsuppn ltinfin

Corollary 10 Scesp(X Y) F(X Y) if 1ltpltinfin

5 Completeness of the Pre-QuasiIdeal Components

For which sequence space E are the components of pre-quasioperator ideal SE complete

Theorem 10 ℓ(pn)

β is a premodular (sss) if conditions (a1)and (a2) are satisfied

Proof We define the functional ρ on ℓ(pn)

β asρ(x) 1113936

infinn0 βn|xn|pn

(i) Evidently ρ(x)ge 0 and ρ(x) 0⟺x θ(ii) )ere is a steady L max 1 supn|λ|pn1113864 1113865ge 1 such that

ρ(λx)le L|λ|ρ(x) for all x isin ℓ(pn)

β and λ isin R(iii) We have the inequality ρ(x + y)leH(ρ(x) + ρ(y))

for all x y isin ℓ(pn)

β

(iv) Clearly follows from inequality (20) of )eorem 6(v) It obtained from (27) )eorem 6 that K0 ge 2(vi) It is clear that F ℓ(pn)

β

(vii) )ere exists a steady 0lt ξ le |λ|p0minus 1 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin R (36)

Theorem 11 ces((an) (pn)) is a premodular (sss) if con-ditions (b1) and (b2) are satisfied

Proof We define the functional ρ on ces((an) (pn)) asρ(x) 1113936

infinn0 (an 1113936

nk0 |xk|)pn

(i) Clearly ρ(x)ge 0 and ρ(x) 0⟺x θ(ii) )ere is a number L max 1 supn|λ|pn1113864 1113865ge 1 with

ρ(λx)le L|λ|ρ(x) for all x isin ces((an) (pn)) andλ isin R

(iii) We have the inequality ρ(x + y)leH(ρ(x) + ρ(y))

for all x y isin ces((an) (pn))(iv) It clearly follows from inequality (24) of )eorem 7(v) It is clear from (27) )eorem 7 that

K0 ge (22hminus 1 + 2hminus 1 + 2h)ge 1(vi) It is clear that F ces((an) (pn))(vii) )ere exists a steady 0 lang ξ le |λ|p0minus 1 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin RWe state the following theorem without proof andthis can be established using standard technique

Theorem 12 Ie function g(P) 9(si(P))infini0 is a pre-quasinorm on SE9

where E9 is a premodular (sss)

Theorem 13 If X and Y are Banach spaces and Eρ is apremodular (sss) then (SEρ

g) where g(T) ρ((sn(T))infinn0)

is a pre-quasi Banach operator ideal

Proof Since Eρ is a premodular (sss) then the functiong(T) ρ((sn(T))infinn0) is a pre-quasi norm on SEρ

Let (Tm)

be a Cauchy sequence in SEρ(X Y) then by utilizing Part

(vii) of Definition 9 and since L(X Y) supe SEρ(X Y) we get

g Ti minus Tj1113872 1113873 ρ sn Ti minus Tj1113872 11138731113872 1113873infinn01113872 1113873ge ρ s0 Ti minus Tj1113872 1113873 0 0 0 1113872 1113873

ρ Ti minus Tj

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868 0 0 0 1113874 1113875ge ξ Tm minus Tj

ρ(1 0 0 0 )

(37)

)en (Tm)misinN is a Cauchy sequence in L(X Y) While thespace L(X Y) is a Banach space so there exists T isin L(X Y)

with limm⟶infinTm minus T 0 and while (sn(Tm))infinn0 isin Eρ foreach m isin N hence using Parts (iii) and (iv) of Definition 9and as ρ is continuous at θ we obtain

g(T) ρ sn(T)( 1113857infinn0( 1113857 ρ sn T minus Tm + Tm( 1113857( 1113857

infinn0( 1113857

leKρ s[n2] T minus Tm( 111385711138721113872 1113873infinn01113873 + Kρ s[n2] Tm( 1113857

infinn01113872 11138731113872 1113873

leKρ Tm minus T

1113872 1113873infinn01113872 1113873 + Kρ sn Tm( 1113857

infinn0( 1113857( 1113857lt ε

(38)

and we have (sn(T))infinn0 isin Eρ then T isin SEρ(X Y)

Corollary 11 If X and Y are Banach spaces and conditions(a1) and (a2) are satisfied then Sℓ(pn)

βis a pre-quasi Banach

operator ideal

Corollary 12 If X and Y are Banach spaces (pn) is in-creasing with pn gt 0 for all n isin N and bounded from abovethen Sℓ(pn ) is a pre-quasi Banach operator ideal

Corollary 13 If X and Y are Banach spaces and 0ltpltinfinthen Sℓp is a pre-quasi Banach operator ideal

Corollary 14 If X and Y are Banach spaces and the con-ditions (b1) and (b2) are satisfied then Sces((an)(pn)) is a pre-quasi Banach operator ideal

Corollary 15 If X and Y are Banach spaces and (pn) isincreasing with p0 gt 1 and limn⟶infinsuppn ltinfin then Sces(pn)

is a pre-quasi Banach operator ideal

Corollary 16 If X and Y are Banach spaces and p isin (1infin)then U

appcesp (X Y) is complete

6 Smallness of the Pre-Quasi BanachOperator Ideal

We give here the sufficient conditions on the weightedNakano sequence space such that the pre-quasi operatorideal formed by the sequence of approximation numbers and

Journal of Function Spaces 7

this sequence space is strictly contained for different weightsand powers

Theorem 14 For any infinite dimensional Banach spaces Xand Y and for any 0ltpn lt qn and 0lt an lt bn for all n isin N it istrue that

Sapp

ℓ pn( )bn( )

(X Y)⫋ Sapp

ℓ qn( )an( )

(X Y)⫋ L(X Y) (39)

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appℓ(pn )

(bn )

(X Y) then (αn(T)) isin ℓ(pn)

(bn) One can see that

1113944

infin

n0an αn(T)( 1113857

qn lt 1113944infin

n0bn αn(T)( 1113857

pn ltinfin (40)

Hence T isin Sappℓ(qn )

(an )

(X Y) Next if we take (an) with

sup an ltinfin (bminus1n ) isin ℓ(qnpn)and μn (1

bn

pn1113968

) So by using)eorem 5 one can find T isin L(X Y) with

116

b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (41)

such that T does not belong to Sappℓ(pn)

(bn)

(X Y) andT isin S

appℓ(qn )

(an )

(X Y)

It is easy to see that Sappℓ(qn)

(an)

(X Y) sub L(X Y) Next if wetake μn (1

anqnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (42)

such that T does not belong to Sappℓ(qn )

(an )

(X Y) )is finishes theproof

Corollary 17 (see [3]) For any infinite dimensional Banachspaces X and Y and for any qgtpgt 0 it is true thatSappℓp (X Y)⫋ S

appℓq (X Y)⫋ L(X Y)

Lemma 2 If (an) is a sequence of positive real with an le 1 forall n isin N (pn) is monotonic increasing bounded sequencewith pn ge 1 for all n isin N or an gt 1 for all n isin N and (pn) ismonotonic decreasing bounded sequence with pn ge 1 for alln isin N one has the following inequality

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k (43)

Proof By using inequality (16) and the sufficient conditionsone has

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

a0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

lt 2suppnminus 1a

pn

0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

2suppnminus 1a

pn

0 + a1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 2suppnminus 1 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 22 suppnminus 1( ) apn

0 + apn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 23 suppnminus 1( ) apn

0 + apn

1 + apn

2 + 1113944n

k3ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

⋮

lt 2(nminus 1s) suppnminus 1( ) apn

0 + apn

1 + apn

2 + middot middot middot + apn

nminus21113872

+ 1113944n

knminus1ak

⎛⎝ ⎞⎠

pn

⎞⎠

lt 2n suppnminus 1( ) 1113944

n

k0a

pn

k lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k

(44)

We give here the sufficient conditions on the generalizedCesaro sequence space defined by weighted mean such thatthe pre-quasi operator ideal formed by the sequence ofapproximation numbers and this sequence space is strictlycontained for different weights and powers

Theorem 15 For any infinite dimensional Banach spaces Xand Y and for any 1ltpn lt qn for all n isin N it is true thatSappces((bn)(pn))(X Y)⫋ S

appces((an)(qn))(X Y)⫋ L(X Y) where (pn)

and (qn) are the monotonic increasing bounded sequences

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appces((bn)(pn))(X Y) then (αn(T)) isin ces((bn) (pn)) One

has

1113944

infin

n0an 1113944

n

k0αk(T)⎛⎝ ⎞⎠

qn

lt 1113944infin

n0bn 1113944

n

k0αk(T)⎛⎝ ⎞⎠

pn

ltinfin (45)

Hence T isin Sappces((an)(qn))(X Y) Next if we take (an) with

sup2(sup qnminus 1)naqnn ltinfin (bminus1

n ) isin ℓ(qnpn) and μn (1bn

pn1113968

) Soby using)eorem 5 and Lemma 2 one can find T isin L(X Y)

with1

16b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (46)

such that T does not belong to Sappces((bn)(pn))(X Y) and

T isin Sappces((an)(qn))(X Y)

8 Journal of Function Spaces

It is easy to see that Sappces((an)(qn))(X Y) sub L(X Y) Next

we take μn (1 an

qnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (47)

such that T does not belong to Sappces((an)(qn))(X Y) )is

finishes the proof

Corollary 18 For any infinite dimensional Banach spaces Xand Y and 1ltplt qltinfin then

Sappcesp

(X Y)⫋ Sappcesq

(X Y)⫋L(X Y) (48)

In this part we give the conditions for which the pre-quasiBanach Operator ideal S

appces((ai)qi)

is small

Theorem 16 If conditions (b1) (b2) and (nan) notin ℓ(pn) aresatisfied then the pre-quasi Banach operator ideal S

appces((ai)pi)

is small

Proof Since (pi) is an increasing sequence with p0 gt 1 and(ai) isin ℓ(pi) take λ (1113936

infini0 a

pi

i )1h )en (Sappces((ai)pi)

g)where g(T) (1λ)(1113936

infini0 (ai 1113936

ij0 αj(T))pi )1h is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appces((ai)pi)

(X Y) L(X Y) then thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y be infinite dimensionalBanach spaces Hence by Dvoretzkyrsquos )eorem [18] form isin N we have quotient spaces XNm and subspaces Mm of

Y which can be mapped onto ℓm2 by isomorphisms Hm and

Am such that HmHminus1m le 2 and AmAminus1

m le 2 Let Im be theidentity map on ℓm

2 Qm be the quotient map from X ontoXNm and Jm be the natural embedding map from Mm intoY Let un be the Bernstein numbers [19] then

1 un Im( 1113857 un AmAminus1m ImHmH

minus1m1113872 1113873

le Am

un A

minus1m ImHm1113872 1113873 H

minus1m

Am

un JmA

minus1m ImHm1113872 1113873 H

minus1m

le Am

dn JmA

minus1m ImHm1113872 1113873 H

minus1m

Am

dn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

le Am

αn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

(49)

for 1le ilem Now

1113944

i

j0ai( 1113857le 1113944

i

j0Am

aiαj JmA

minus1m ImHmQm1113872 1113873 H

minus1m

⟹

(i + 1)ai le Am

ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠ H

minus1m

⟹

(i + 1)ai( 1113857pi le Am

H

minus1m

1113872 1113873

piai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

(50)

)erefore

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1λ

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le 4LC

(51)

Journal of Function Spaces 9

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary and (nan) notin ℓ(pn) )us X and Y both cannot beinfinite dimensional when S

appces((ai)(pi))

(X Y) L(X Y) andhence the result

Theorem 17 If (pi) is an increasing and p0 gt 1 then the pre-quasi Banach operator ideal SKolces(pi)

is small

Corollary 19 If 1ltpltinfin then the quasi Banach operatorideal S

appcesp is small

Corollary 20 If 1ltpltinfin then the quasi Banach operatorideal SKolcesp is small

In this part we give the conditions for which the pre-quasiBanach operator ideal S

appℓ(pn )

β

is small

Theorem 18 If conditions (a1) (a2) and (βn) notin ℓ1 aresatisfied then the pre-quasi Banach operator ideal S

appℓ(pn )

β

issmall

Proof Since conditions (a1) (a2) and (βn) notin ℓ1 are satisfiedthen (S

appℓ(pn )

β

g) where g(T) 1113936infini0 βi(αi(T))pi is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appℓ(pn )

β

(X Y) L(X Y) )en thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y are infinite dimensionalBanach spaces By using inequality (49) and (βn) notin ℓ1 oneobtains

1le Am

H

minus1m

1113872 1113873

pi αi JmAminus1m ImHmQm1113872 11138731113872 1113873

pi⟹

βi le L Am

βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

piH

minus1m

⟹

1113944

m

i0βi le L Am

H

minus1m

1113944

m

i0βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

pi⟹

1113944

m

i0βi le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1113944

m

i0βi le 4LC

(52)

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary )us X and Y both cannot be infinite di-mensional when S

appℓ(pn )

β

(X Y) L(X Y) and hence theresult

Corollary 21 (see [2]) If 0ltpltinfin then the quasi Banachoperator ideal S

appℓp is small

Corollary 22 If 0ltpltinfin then the quasi Banach operatorideal SKolℓp is small

7 Pre-Quasi Simple Banach Operator Ideal

)e following question arises naturally for which weightedNakano sequence space or generalized Cesaro sequencespace defined by weighted mean is the pre-quasi Banachideal simple

Theorem 19 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 Λ Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (53)

Proof Suppose that there exists T isin L(Sℓ(qn )

(an )

Sℓ(pn)

(bn)

) which isnot approximable According to Lemma 1 we can find

X isin L(Sℓ(qn)

(an )

Sℓ(qn )

(an)

) and B isin L(Sℓ(pn )

(bn )

Sℓ(pn )

(bn )

) with BTXIk Ik

)en it follows for all k isin N that

Ik

S

ℓpn( )bn( )

1113944infin

n0bn αn Ik( 1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

1113944kminus 1

n0bn

⎛⎝ ⎞⎠

1suppn

le BTX Ik

S

ℓqn( )an( )

le 1113944infin

n0an αn Ik( 1113857( 1113857

qn⎛⎝ ⎞⎠

1supqn

1113944kminus 1

n0an

⎛⎝ ⎞⎠

1supqn

(54)

But this is impossible

Corollary 23 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 LC Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (55)

Proof Every approximable operator is compact

Theorem 20 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

Λ Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(56)

Proof Suppose that there exists T isin L(Sces((an)(qn))

Sces((bn)(pn))) which is not approximable According toLemma 1 we can find X isin L(Sces((an)(qn)) Sces((an)(qn))) andB isin L(Sces((bn)(pn)) Sces((bn)(pn))) with BTXIk Ik )en itfollows for all k isin N that

10 Journal of Function Spaces

Ik

Sces bn( ) pn( )( )

1113944infin

n0bn 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0bn 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0kbn1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

le BTX Ik

Sces an( ) qn( )( )

le 1113944infin

n0an 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0an 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0kan( 1113857⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

(57)

But this is impossible

Corollary 24 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

LC Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(58)

Theorem 21 For a bounded sequence (pn) with 1lepn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space Sℓ(pn )

(bn )

is simple

Proof Suppose that the closed ideal LC(Sℓ(pn)

(bn)

) contains an

operator Twhich is not approximable According to Lemma

1 we can find X B isin L(Sℓ(pn)

(bn)

) with BTXIk Ik )is means

that ISℓ(pn )

(bn )

isin LC(Sℓ(pn )

(bn )

) Consequently L(Sℓ(pn)

(bn)

) LC(Sℓ(pn)

(bn)

)

)erefore Λ(Sℓ(pn )

(bn )

) is the only nontrivial closed ideal inL(Sℓ(pn )

(bn )

)

Theorem 22 For a bounded sequence (pn) with 1ltpn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space(Sces((bn)(pn))) is simple

Proof Suppose that the closed ideal LC(Sces((bn)(pn))) con-tains an operator Twhich is not approximable According toLemma 1 we can find X B isin L(Sces((bn)(pn))) withBTXIk Ik )is means that ISces((bn)(pn ))

isin LC(Sces((bn)(pn)))Consequently L(Sces((bn)(pn))) LC(Sces((bn)(pn))) )ereforeΛ(Sces((bn)(pn))) is the only nontrivial closed ideal inL(Sces((bn)(pn)))

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to the writing of this paperAll authors read and approved the final manuscript

References

[1] A Pietsch Operator Ideals North-Holland PublishingCompany Amsterdam Netherlands 1980

[2] A Pietsch ldquoSmall ideals of operatorsrdquo Studia Mathematicavol 51 no 3 pp 265ndash267 1974

[3] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbersrdquo in Ieory of Operators inFunctional Spaces pp 206ndash211 Academy of Science SiberianSection Academy of Science Siberian Section NovosibirskRussia 1997 in Russian

[4] N Faried and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 186 2013

[5] N Faried and A A Bakery ldquoSmall operator ideals formed by snumbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1p 357 2018

[6] B E Rhoades ldquoOperators of Andashp typerdquo Atti della AccademiaNazionale dei Lincei Classe di Scienze Fisiche Matematiche eNaturali (VIII) vol 59 no 3-4 pp 238ndash241 1975

[7] A Pietsch Eigenvalues and s-Numbers Cambridge UniversityPress New York NY USA 1986

[8] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 no 4 pp 267ndash278 1974

[9] A Lima and E Oja ldquoIdeals of finite rank operators inter-section properties of balls and the approximation propertyrdquoStudia Mathematica vol 133 pp 175ndash186 1999

[10] S T Chen ldquoGeometry of Orlicz spacesrdquo Dissertations MathInstitute of Mathematics Polish Academy of SciencesWarsaw Poland vol 356 p 356 1996

[11] Y A Cui and H Hudzik ldquoOn the Banach-saks and weakBanach-saks properties of some Banach sequence spacesrdquoActa Scientiarum Mathematicarum vol 65 pp 179ndash1871999

[12] M Sengonul ldquoOn the generalized Nakano sequence spacerdquoIai Journal of Mathematics vol 4 no 2 pp 295ndash304 2006

[13] W Sanhan and S Suantai ldquoOnk-nearly uniform convexproperty in generalized Cesaro sequence spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciencesvol 2003 no 57 pp 3599ndash3607 2003

[14] P Y Lee ldquoCesaro sequence spacesrdquo Mathematical Chroniclevol 13 pp 29ndash45 1984

[15] Y q Lui and P Y LeeMethod of Sequence Spaces Guangdongof science and Technology press Guangdong China 1996 inChinese

[16] J S Shiue ldquoOn the Cesaro sequence spacesrdquo Tamkang Journalof Mathematics vol 1 no 1 pp 19ndash25 1970

[17] B Altay and F Basar ldquoGeneralization of the sequence spaceℓ(p) derived by weighted meansrdquo Journal of MathematicalAnalysis and Applications vol 330 no 1 pp 147ndash185 2007

[18] A Pietsch Operator Ideals VEB Deutscher Verlag derWis-senschaften Berlin Germany 1978

[19] A Pietsch ldquos-numbers of operators in Banach spacesrdquo StudiaMathematica vol 51 no 3 pp 201ndash223 1974

Journal of Function Spaces 11

Definition 5 (see [1]) A bounded linear operatorA E⟶ E (where E is a Banach space) is called compact ifA(B1) has compact closure where B1 denotes the closed unitball of E )e space of all compact operators on E is denotedby Lc(E)

Theorem 2 (see [1]) If E is infinite dimensional Banachspace we have

F(E)⫋Λ(E)⫋Lc(E)⫋ L(E) (8)

Definition 6 (see [1]) Let L be the class of all bounded linearoperators between any arbitrary Banach spaces A subclass Uof L is called an operator ideal if each element U(X Y)

U cap L(X Y) fulfills the following conditions

(i) I5 isin U where 5 represents Banach space of onedimension

(ii) )e space U(X Y) is linear over R(iii) If T isin L(X0 X) V isin U(X Y) and R isin L(Y Y0)

then RVT isin U(X0 Y0) (see [8 9])

)e concept of pre-quasi operator ideal is more generalthan the usual classes of operator ideal

Definition 7 (see [5]) A function g Ω⟶ [0infin) is said tobe a pre-quasi norm on the ideal Ω if the following con-ditions holds

(1) For all T isin Ω(X Y) g(T)ge 0 and g(T) 0 if andonly if T 0

(2) )ere exists a constant Mge 1 such that g(λT) leM|λ|g(T) for all T isin Ω(X Y) and λ isin R

(3) )ere exists a constant Kge 1 such thatg(T1 + T2)leK[g(T1) + g(T2)] for all T1 T2 isin Ω(X Y)

(4) )ere exists a constant Cge 1 such that ifT isin L(X0 X) P isin Ω(X Y) and R isin L(Y Y0) theng(RPT) leCRg(P)T where X0 and Y0 arenormed spaces

Theorem 3 (see [5]) Every quasi norm on the ideal Ω is apre-quasi norm on the ideal Ω

Let p (pn) be a positive real and (βn)nisinN be a sequenceof positive real the weighted Nakano sequence space isdefined by

ℓ pn( )β x xk( 1113857 isin ω ρ(λx)ltinfin for some λgt 01113864 1113865 where ρ(x)

1113944infin

k0βk xk

111386811138681113868111386811138681113868111386811138681113872 1113873

pk

(9)

And (ℓ(pn)

β middot) is a Banach space where

x inf ηgt 0 ρx

η1113888 1113889le 11113896 1113897 (10)

When (pn) is bounded we mark

ℓ pn( )β x xk( 1113857 isin ω 1113944

infin

k0βk xk

11138681113868111386811138681113868111386811138681113868pk ltinfin

⎧⎨

⎩

⎫⎬

⎭ (11)

If βn 1 for all n isin N then ℓ(pn)

β will reduce to ℓ(pn)

studied in [10 11]In [12] Sengonul defined the sequence space as

ces an( 1113857 pn( 1113857( 1113857 x xk( 1113857 isin ω existλgt 0with ρ(λx)ltinfin1113864 1113865

ρ(x) 1113944infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pk

(12)

where (an) and (pn) are the sequences of positive real andpn ge 1 for all n isin N With the norm

x inf ηgt 0 ρx

η1113888 1113889le 11113896 1113897 (13)

Note

(1) Taking an (1(n + 1)) for all n isin N thences((an) (pn)) is reduced to ces(pn) studied bySanhan and Suantai [13]

(2) Taking an (1(n + 1)) and pn p for all n isin Nthen ces((an) (pn)) is reduced to cesp studied bymany authors (see [14ndash16])

Definition 8 (see [5]) Let E be a linear space of sequencesthen E is called a (sss) if

(1) For n isin N en isin E

(2) E is solid ie assuming x (xn) isin w y (yn) isin E

and |xn|le |yn| for all n isin N then x isin E

(3) (x[n2])infinn0 isin E where [n2] indicates the integral

part of [n2] whenever (xn)infinn0 isin E

Definition 9 (see [5]) A subclass of the (sss) is called apremodular (sss) assuming that we have a mapρ E⟶ [0infin[ with the following

(i) For x isin E x θ⟺ ρ(x) 0 with ρ(x)ge 0(ii) For each x isin E and scalar λ we get a real number

Lge 1 for which ρ(λx)le |λ|Lρ(x)

(iii) ρ(x + y)leK(ρ(x) + ρ(y)) for each x y isin E holdsfor a few numbers Kge 1

(iv) For n isin N |xn|le |yn| we obtain ρ(xn)le ρ(yn)

(v) )e inequality ρ(xn)le ρ(x[n2])leK0ρ(xn) holdsfor some numbers K0 ge 1

(vi) F Eρ where F is the space of finite sequences(vii) )ere is a steady ξ gt 0 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin R

Condition (ii) gives the continuity of ρ(x) at θ)e linearspace E enriched with the metric topology formed by thepremodular ρ will be indicated by Eρ Moreover condition(16) in Definition 8 and condition (vi) in Definition 9 explainthat (en)nisinN is a Schauder basis of Eρ

Journal of Function Spaces 3

Notations )e sets SE SE(X Y) SappE and S

appE (X Y) (cf

[5]) as follows

SE ≔ SE(X Y) X andY are Banach spaces1113864 1113865 where

SE(X Y) ≔ T isin L(X Y) si(T)( 1113857(infini0 isin E1113864 1113865Also

SappE ≔ S

appE (X Y) X andY are Banach spaces1113864 1113865 where

SappE (X Y) ≔ T isin L(X Y) αi(T)( 1113857(

infini0 isin E1113864 1113865

(14)

Theorem 4 (see [5]) If E is a (sss) then SE is an operatorideal

Theorem 5 (see [3]) If X and Y are infinite dimensionalBanach spaces and (μi) is a monotonic decreasing sequenceto zero then there exists a bounded linear operator T suchthat

116μ3i le αi(T)le 8μi+1 (15)

Now and after define en 0 0 1 0 0 where 1appears at the nth place for all n isin N and the given inequalitywill be used in the sequel

an + bn

11138681113868111386811138681113868111386811138681113868pn leH an

11138681113868111386811138681113868111386811138681113868pn + bn

11138681113868111386811138681113868111386811138681113868pn1113872 1113873 (16)

whereH max 1 2hminus 11113864 1113865 h supnpn and pn ge 1 for all n isin N(see [17])

3 Main Results

31 Linear Problem We examine here the operator idealscreated by sminusnumbers and also weighted Nakano sequencespace or generalized Cesaro sequence space defined byweighted mean such that those classes of all bounded linearoperators T between arbitrary Banach spaces with (αn(T))

in these sequence spaces type an ideal operator

Theorem 6 ℓ(pn)

β is a (sss) if the following conditions aresatisfied

(a1) Ie sequence (pn) is increasing and bounded fromabove with pn gt 0 for all n isin N

(a2) Either (βn) is monotonic decreasing or monotonicincreasing such that there exists a constant Cge 1 forwhich β2n+1 leCβn

Proof (1) Let x y isin ℓ(pn)

β Since (pn) is bounded we get

1113944

infin

n0βn xn + yn

11138681113868111386811138681113868111386811138681113868pn leH 1113944

infin

n0βn xn

11138681113868111386811138681113868111386811138681113868pn + 1113944infin

n0βn yn

11138681113868111386811138681113868111386811138681113868pn⎛⎝ ⎞⎠ltinfin

(17)

then x + y isin ℓ(pn)

β

(2) Let λ isin R and x isin ℓ(pn)

β Since (pn) is bounded wehave

1113944

infin

n0βn λxn

11138681113868111386811138681113868111386811138681113868pn le sup

n

|λ|pn1113944

infin

n0βn xn

11138681113868111386811138681113868111386811138681113868pn ltinfin (18)

)en λx isin ℓ(pn)

β )erefore by using Parts (1) and (2) wehave that the space ℓ(pn)

β is linear Also en isin ℓ(pn)

β for alln isin N since

1113944

infin

i0βi en(i)

11138681113868111386811138681113868111386811138681113868pi βn (19)

(2) Let |xn|le |yn| for all n isin N and y isin ℓ(pn)

β Since βn gt 0for all n isin N then

1113944

infin

n0βn xn

11138681113868111386811138681113868111386811138681113868pn le 1113944infin

n0βn yn

11138681113868111386811138681113868111386811138681113868pn ltinfin (20)

and we get x isin ℓ(pn)

β

(3) Let (xn) isin ℓ(pn)

β (βn) be an increasing sequence)ere exists Cgt 0 such that β2n+1 leCβn and (pn) beincreasing then we have

1113944

infin

n0βn x[n2]

11138681113868111386811138681113868111386811138681113868pn 1113944infin

n0β2n xn

11138681113868111386811138681113868111386811138681113868p2n + 1113944infin

n0β2n+1 xn

11138681113868111386811138681113868111386811138681113868p2n+1

le 1113944

infin

n0β2n xn

11138681113868111386811138681113868111386811138681113868pn + 1113944

infin

n0β2n+1 xn

11138681113868111386811138681113868111386811138681113868pn le 2C 1113944

infin

n0βn xn

11138681113868111386811138681113868111386811138681113868pn

(21)

and then (x[n2]) isin ℓ(pn)

β

Theorem 7 ces((an) (pn)) is a (sss) if the following con-ditions are satisfied

(b1) Ie sequence (pn) is increasing and bounded withp0 gt 1

(b2) 1113936infinn0 (an)pn ltinfin

Proof (1) Given that x y isin ces((an) (pn)) and λ1 λ2 isin RSince (pn) is bounded we have

1113944

infin

n0an 1113944

n

k0λ1xk + λ2yk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

leH supn

λ11113868111386811138681113868

1113868111386811138681113868pn 1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

⎛⎝

+ supn

λ21113868111386811138681113868

1113868111386811138681113868pn 1113944

infin

n0an 1113944

n

k0yk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

⎞⎠ltinfin

(22)

Hence λ1x + λ2y isin ces((an) (pn)) then the spaceces((an) (pn)) is linear Also prove that em isin ces((an) (pn))

for all m isin N since 1113936infinn0 (an)pn ltinfin So we get

1113944

infin

n0an 1113944

n

k0em(k)

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

1113944infin

nm

an( 1113857pn le 1113944infin

n0an( 1113857

pn ltinfin

(23)

Hence em isin ces((an) (pn))

4 Journal of Function Spaces

(2) Let |xn|le |yn| for all n isin N and y isin ces((an) (pn))Since an gt 0 for all n isin N then

1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 1113944

infin

n0an 1113944

n

k0yk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

ltinfin (24)

and we get x isin ces((an) (pn))

(3) Let (xn) isin ces((an) (pn)) Since (pn) is increasingand the sequence (an) with 1113936

infinn0 (an)pn ltinfin is de-

creasing then we have

1113944

infin

n0an 1113944

n

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

1113944infin

n0a2n 1113944

2n

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

p2n

+ 1113944infin

n0a2n+1 1113944

2n+1

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

p2n+1

le 1113944infin

n0a2n 1113944

2n

k02 xk

11138681113868111386811138681113868111386811138681113868 + xn

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠⎛⎝ ⎞⎠

pn

+ 1113944infin

n0a2n+1 1113944

2n+1

k02 xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 2hminus 11113944

infin

n02an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

+ 1113944infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠ + 1113944infin

n02an 1113944

n

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 2hminus 1 2h+ 11113872 1113873 1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

+ 2h1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 22hminus 1+ 2hminus 1

+ 2h1113872 1113873 1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

ltinfin

(25)

Hence (x[n2]) isin ces((an) (pn))By using)eorem 4 we can get the following corollaries

Corollary 1 Let conditions (a1) and (a2) be satisfied thenSappℓ(pn )

β

be an operator ideal

Corollary 2 Sappℓ(pn ) is an operator ideal if the sequence (pn) is

increasing and bounded from above with pn gt 0 for all n isin N

Corollary 3 If p isin (0infin) then Sappℓp is an operator ideal

Corollary 4 Conditions (b1) and (b2) are satisfied henceSappces((an)(pn)) is an operator ideal

Corollary 5 Assume (pn) is increasing with p0 gt 1 andbounded so S

appces(pn) is an operator ideal

Corollary 6 If p isin (1infin) then Sappcesp is an operator ideal

4 Topological Problem

)e following question arises naturally which sufficientconditions (not necessary) on the sequence space E(weighted Nakano sequence space and generalized Cesarosequence space defined by weighted mean) are the ideal ofthe finite rank operators in the class of Banach spaces densein SE )is gives a negative answer of Rhoades [6] openproblem about the linearity of Eminustype spaces (SE)

Theorem 8 F(X Y) Sℓ(pn )

β(X Y) whenever conditions

(a1) and (a2) are satisfied

Proof First we substantiate that each finite operatorT isin F(X Y) belongs to Sℓ(pn )

β(X Y) Given that

en1113864 1113865nisinN sub ℓ(pn)

β and the space ℓ(pn)

β is linear then for all finiteoperators T isin F(X Y) ie the sequence (sn(T))nisinN con-tains only finitely many numbers different from zeroCurrently we substantiate that Sℓ(pn )

β(X Y) sube F(X Y) On

taking T isin Sℓ(pn )

β(X Y)we obtain (sn(T))infinn0 isin ℓ

(pn)

β hence

ρ((sn(T))infinn0)ltinfin let ε isin (0 1) at that point there exists am isin N minus 0 such that ρ((sn(T))infinnm)lt (ε4C2) for someCge 1 While (sn(T))nisinN is decreasing we get

1113944

2m

nm+1βn s2m(T)( 1113857

pn le 11139442m

nm+1βn sn(T)( 1113857

pn

le 1113944

infin

nm

βn sn(T)( 1113857pn lt

ε4C2

(26)

Hence there exists A isin F2m(X Y) rank Ale 2m and

1113944

3m

n2m+1βn(T minus A)

pn le 11139442m

nm+1βn(T minus A)

pn ltε

4C2 (27)

Since (pn) is a bounded consider

1113944

m

n0βn(T minus A)

pn ltε

4C2 (28)

Let (βn) be monotonic increasing such that there exists aconstant Cge 1 for which β2n+1 leCβn )en we have forngem that

β2m+n le β2m+2n+1 leCβm+n leCβ2n leCβ2n+1 leC2βn (29)

Since (pn) is increasing inequalities (26)ndash(29) give

Journal of Function Spaces 5

d(T A) ρ sn(T minus A)( 1113857infinn0

11139443mminus1

n0βn sn(T minus A)( 1113857

pn + 1113944infin

n3m

βn sn(T minus A)( 1113857pn

le 11139443m

n0βn(T minus A)

pn + 1113944infin

nm

βn+2m sn+2m(T minus A)( 1113857pn+2m

le 3 1113944m

n0βn(T minus A)

pn + C2

1113944

infin

nm

βn sn(T)( 1113857pn lt ε

(30)

Since I3 isin Sℓ(n)

(1)

condition (a1) is not satisfied whichgives a counter example of the converse statement )isfinishes the proof

From )eorem 8 we can say that if (a1) and (a2) aresatisfied then every compact operator would be approxi-mated by finite rank operators and the converse is not alwaystrue

Theorem 9 F(X Y) Sces((an)(pn))(X Y) assume thatstates (b1) and (b2) are fulfilled and the converse is not alwaystrue

Proof Primary since en isin ces((an) (pn)) for each n isin N andthe space ces((an) (pn)) is linear then for every finitemapping T isin F(X Y) ie the sequence (sn(T))nisinN containsonly fnitely many numbers different from zero HenceF(X Y) sube Sces((an)(pn))(X Y) By letting T isinSces((an)(pn))(X Y) we obtain (sn(T))nisinN isin ces((an) (pn))while ρ((αn(T))nisinN)ltinfin Let ε isin (0 1) )en there exists a

number m isin N minus 0 such that ρ((sn(T))infinnm)lt (ε2h+2δC)

for some Cge 1 where δ max 1 1113936infinnm a

pnn1113966 1113967 As sn(T) is

decreasing for every n isin N we obtain

1113944

2m

nm+1an 1113944

n

k0s2m(T)⎛⎝ ⎞⎠

pn

le 11139442m

nm+1an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

le 1113944infin

nm

an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

ltε

2h+2δC

(31)

)en there exists A isin F2m(X Y) and rank Ale 2m and

1113944

3m

n2m+1an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

le 11139442m

nm+1an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

ltε

2h+2δC

(32)

and since (pn) is bounded consider

supnm

infin1113944

m

k0T minus A⎛⎝ ⎞⎠

pn

ltε2hδ

(33)

Hence set

1113944

m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

ltε

2h+3δC (34)

However (pn) is increasing and (an) is decreasing foreach n isin N by using inequalities (31)ndash(34) we have

d(T A) ρ sn(T minus A)( 1113857infinn0

11139443mminus1

n0an 1113944

n

k0sk(T minus A)⎛⎝ ⎞⎠

pn

+ 1113944infin

n3m

an 1113944

n

k0sk(T minus A)

pn⎛⎝ ⎞⎠

le 11139443m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an+2m 1113944

n+2m

k0sk(T minus A)⎛⎝ ⎞⎠

pn+2m

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an 1113944

2mminus 1

k0sk(T minus A) + an 1113944

n+2m

k2m

sk(T minus A)⎛⎝ ⎞⎠

pn

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 11113944

infin

nm

an 1113944

2mminus 1

k0sk(T minus A)⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an 1113944

n+2m

k2m

sk(T minus A)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

le 3 1113944

m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 11113944

infin

nm

an 1113944

m

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944

infin

nm

an 1113944

n

k0sk+2m(T minus A)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 1supinfinnm 1113944

m

k0T minus A)

pn 1113944

infin

nm

an( 1113857pn + 2hminus 1

1113944

infin

nm

an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠lt ε

(35)

Since I3 isin Sces((1)(1)) condition (b2) is not satisfiedwhich gives a counter example of the converse statement)is finishes the proof

From )eorem 9 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 isnot always true

Corollary 7 If (pn) is an increasing with pn gt 0 for all n isin Nand bounded from above then Sl(p) (X Y) F(X Y)

6 Journal of Function Spaces

Corollary 8 If 0ltpltinfin then Slp (X Y) F(X Y)

Corollary 9 Sces(pn)(X Y) F(X Y) if (pn) is increasingwith p0 gt 1 and limn⟶infinsuppn ltinfin

Corollary 10 Scesp(X Y) F(X Y) if 1ltpltinfin

5 Completeness of the Pre-QuasiIdeal Components

For which sequence space E are the components of pre-quasioperator ideal SE complete

Theorem 10 ℓ(pn)

β is a premodular (sss) if conditions (a1)and (a2) are satisfied

Proof We define the functional ρ on ℓ(pn)

β asρ(x) 1113936

infinn0 βn|xn|pn

(i) Evidently ρ(x)ge 0 and ρ(x) 0⟺x θ(ii) )ere is a steady L max 1 supn|λ|pn1113864 1113865ge 1 such that

ρ(λx)le L|λ|ρ(x) for all x isin ℓ(pn)

β and λ isin R(iii) We have the inequality ρ(x + y)leH(ρ(x) + ρ(y))

for all x y isin ℓ(pn)

β

(iv) Clearly follows from inequality (20) of )eorem 6(v) It obtained from (27) )eorem 6 that K0 ge 2(vi) It is clear that F ℓ(pn)

β

(vii) )ere exists a steady 0lt ξ le |λ|p0minus 1 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin R (36)

Theorem 11 ces((an) (pn)) is a premodular (sss) if con-ditions (b1) and (b2) are satisfied

Proof We define the functional ρ on ces((an) (pn)) asρ(x) 1113936

infinn0 (an 1113936

nk0 |xk|)pn

(i) Clearly ρ(x)ge 0 and ρ(x) 0⟺x θ(ii) )ere is a number L max 1 supn|λ|pn1113864 1113865ge 1 with

ρ(λx)le L|λ|ρ(x) for all x isin ces((an) (pn)) andλ isin R

(iii) We have the inequality ρ(x + y)leH(ρ(x) + ρ(y))

for all x y isin ces((an) (pn))(iv) It clearly follows from inequality (24) of )eorem 7(v) It is clear from (27) )eorem 7 that

K0 ge (22hminus 1 + 2hminus 1 + 2h)ge 1(vi) It is clear that F ces((an) (pn))(vii) )ere exists a steady 0 lang ξ le |λ|p0minus 1 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin RWe state the following theorem without proof andthis can be established using standard technique

Theorem 12 Ie function g(P) 9(si(P))infini0 is a pre-quasinorm on SE9

where E9 is a premodular (sss)

Theorem 13 If X and Y are Banach spaces and Eρ is apremodular (sss) then (SEρ

g) where g(T) ρ((sn(T))infinn0)

is a pre-quasi Banach operator ideal

Proof Since Eρ is a premodular (sss) then the functiong(T) ρ((sn(T))infinn0) is a pre-quasi norm on SEρ

Let (Tm)

be a Cauchy sequence in SEρ(X Y) then by utilizing Part

(vii) of Definition 9 and since L(X Y) supe SEρ(X Y) we get

g Ti minus Tj1113872 1113873 ρ sn Ti minus Tj1113872 11138731113872 1113873infinn01113872 1113873ge ρ s0 Ti minus Tj1113872 1113873 0 0 0 1113872 1113873

ρ Ti minus Tj

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868 0 0 0 1113874 1113875ge ξ Tm minus Tj

ρ(1 0 0 0 )

(37)

)en (Tm)misinN is a Cauchy sequence in L(X Y) While thespace L(X Y) is a Banach space so there exists T isin L(X Y)

with limm⟶infinTm minus T 0 and while (sn(Tm))infinn0 isin Eρ foreach m isin N hence using Parts (iii) and (iv) of Definition 9and as ρ is continuous at θ we obtain

g(T) ρ sn(T)( 1113857infinn0( 1113857 ρ sn T minus Tm + Tm( 1113857( 1113857

infinn0( 1113857

leKρ s[n2] T minus Tm( 111385711138721113872 1113873infinn01113873 + Kρ s[n2] Tm( 1113857

infinn01113872 11138731113872 1113873

leKρ Tm minus T

1113872 1113873infinn01113872 1113873 + Kρ sn Tm( 1113857

infinn0( 1113857( 1113857lt ε

(38)

and we have (sn(T))infinn0 isin Eρ then T isin SEρ(X Y)

Corollary 11 If X and Y are Banach spaces and conditions(a1) and (a2) are satisfied then Sℓ(pn)

βis a pre-quasi Banach

operator ideal

Corollary 12 If X and Y are Banach spaces (pn) is in-creasing with pn gt 0 for all n isin N and bounded from abovethen Sℓ(pn ) is a pre-quasi Banach operator ideal

Corollary 13 If X and Y are Banach spaces and 0ltpltinfinthen Sℓp is a pre-quasi Banach operator ideal

Corollary 14 If X and Y are Banach spaces and the con-ditions (b1) and (b2) are satisfied then Sces((an)(pn)) is a pre-quasi Banach operator ideal

Corollary 15 If X and Y are Banach spaces and (pn) isincreasing with p0 gt 1 and limn⟶infinsuppn ltinfin then Sces(pn)

is a pre-quasi Banach operator ideal

Corollary 16 If X and Y are Banach spaces and p isin (1infin)then U

appcesp (X Y) is complete

6 Smallness of the Pre-Quasi BanachOperator Ideal

We give here the sufficient conditions on the weightedNakano sequence space such that the pre-quasi operatorideal formed by the sequence of approximation numbers and

Journal of Function Spaces 7

this sequence space is strictly contained for different weightsand powers

Theorem 14 For any infinite dimensional Banach spaces Xand Y and for any 0ltpn lt qn and 0lt an lt bn for all n isin N it istrue that

Sapp

ℓ pn( )bn( )

(X Y)⫋ Sapp

ℓ qn( )an( )

(X Y)⫋ L(X Y) (39)

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appℓ(pn )

(bn )

(X Y) then (αn(T)) isin ℓ(pn)

(bn) One can see that

1113944

infin

n0an αn(T)( 1113857

qn lt 1113944infin

n0bn αn(T)( 1113857

pn ltinfin (40)

Hence T isin Sappℓ(qn )

(an )

(X Y) Next if we take (an) with

sup an ltinfin (bminus1n ) isin ℓ(qnpn)and μn (1

bn

pn1113968

) So by using)eorem 5 one can find T isin L(X Y) with

116

b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (41)

such that T does not belong to Sappℓ(pn)

(bn)

(X Y) andT isin S

appℓ(qn )

(an )

(X Y)

It is easy to see that Sappℓ(qn)

(an)

(X Y) sub L(X Y) Next if wetake μn (1

anqnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (42)

such that T does not belong to Sappℓ(qn )

(an )

(X Y) )is finishes theproof

Corollary 17 (see [3]) For any infinite dimensional Banachspaces X and Y and for any qgtpgt 0 it is true thatSappℓp (X Y)⫋ S

appℓq (X Y)⫋ L(X Y)

Lemma 2 If (an) is a sequence of positive real with an le 1 forall n isin N (pn) is monotonic increasing bounded sequencewith pn ge 1 for all n isin N or an gt 1 for all n isin N and (pn) ismonotonic decreasing bounded sequence with pn ge 1 for alln isin N one has the following inequality

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k (43)

Proof By using inequality (16) and the sufficient conditionsone has

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

a0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

lt 2suppnminus 1a

pn

0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

2suppnminus 1a

pn

0 + a1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 2suppnminus 1 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 22 suppnminus 1( ) apn

0 + apn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 23 suppnminus 1( ) apn

0 + apn

1 + apn

2 + 1113944n

k3ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

⋮

lt 2(nminus 1s) suppnminus 1( ) apn

0 + apn

1 + apn

2 + middot middot middot + apn

nminus21113872

+ 1113944n

knminus1ak

⎛⎝ ⎞⎠

pn

⎞⎠

lt 2n suppnminus 1( ) 1113944

n

k0a

pn

k lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k

(44)

We give here the sufficient conditions on the generalizedCesaro sequence space defined by weighted mean such thatthe pre-quasi operator ideal formed by the sequence ofapproximation numbers and this sequence space is strictlycontained for different weights and powers

Theorem 15 For any infinite dimensional Banach spaces Xand Y and for any 1ltpn lt qn for all n isin N it is true thatSappces((bn)(pn))(X Y)⫋ S

appces((an)(qn))(X Y)⫋ L(X Y) where (pn)

and (qn) are the monotonic increasing bounded sequences

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appces((bn)(pn))(X Y) then (αn(T)) isin ces((bn) (pn)) One

has

1113944

infin

n0an 1113944

n

k0αk(T)⎛⎝ ⎞⎠

qn

lt 1113944infin

n0bn 1113944

n

k0αk(T)⎛⎝ ⎞⎠

pn

ltinfin (45)

Hence T isin Sappces((an)(qn))(X Y) Next if we take (an) with

sup2(sup qnminus 1)naqnn ltinfin (bminus1

n ) isin ℓ(qnpn) and μn (1bn

pn1113968

) Soby using)eorem 5 and Lemma 2 one can find T isin L(X Y)

with1

16b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (46)

such that T does not belong to Sappces((bn)(pn))(X Y) and

T isin Sappces((an)(qn))(X Y)

8 Journal of Function Spaces

It is easy to see that Sappces((an)(qn))(X Y) sub L(X Y) Next

we take μn (1 an

qnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (47)

such that T does not belong to Sappces((an)(qn))(X Y) )is

finishes the proof

Corollary 18 For any infinite dimensional Banach spaces Xand Y and 1ltplt qltinfin then

Sappcesp

(X Y)⫋ Sappcesq

(X Y)⫋L(X Y) (48)

In this part we give the conditions for which the pre-quasiBanach Operator ideal S

appces((ai)qi)

is small

Theorem 16 If conditions (b1) (b2) and (nan) notin ℓ(pn) aresatisfied then the pre-quasi Banach operator ideal S

appces((ai)pi)

is small

Proof Since (pi) is an increasing sequence with p0 gt 1 and(ai) isin ℓ(pi) take λ (1113936

infini0 a

pi

i )1h )en (Sappces((ai)pi)

g)where g(T) (1λ)(1113936

infini0 (ai 1113936

ij0 αj(T))pi )1h is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appces((ai)pi)

(X Y) L(X Y) then thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y be infinite dimensionalBanach spaces Hence by Dvoretzkyrsquos )eorem [18] form isin N we have quotient spaces XNm and subspaces Mm of

Y which can be mapped onto ℓm2 by isomorphisms Hm and

Am such that HmHminus1m le 2 and AmAminus1

m le 2 Let Im be theidentity map on ℓm

2 Qm be the quotient map from X ontoXNm and Jm be the natural embedding map from Mm intoY Let un be the Bernstein numbers [19] then

1 un Im( 1113857 un AmAminus1m ImHmH

minus1m1113872 1113873

le Am

un A

minus1m ImHm1113872 1113873 H

minus1m

Am

un JmA

minus1m ImHm1113872 1113873 H

minus1m

le Am

dn JmA

minus1m ImHm1113872 1113873 H

minus1m

Am

dn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

le Am

αn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

(49)

for 1le ilem Now

1113944

i

j0ai( 1113857le 1113944

i

j0Am

aiαj JmA

minus1m ImHmQm1113872 1113873 H

minus1m

⟹

(i + 1)ai le Am

ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠ H

minus1m

⟹

(i + 1)ai( 1113857pi le Am

H

minus1m

1113872 1113873

piai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

(50)

)erefore

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1λ

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le 4LC

(51)

Journal of Function Spaces 9

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary and (nan) notin ℓ(pn) )us X and Y both cannot beinfinite dimensional when S

appces((ai)(pi))

(X Y) L(X Y) andhence the result

Theorem 17 If (pi) is an increasing and p0 gt 1 then the pre-quasi Banach operator ideal SKolces(pi)

is small

Corollary 19 If 1ltpltinfin then the quasi Banach operatorideal S

appcesp is small

Corollary 20 If 1ltpltinfin then the quasi Banach operatorideal SKolcesp is small

In this part we give the conditions for which the pre-quasiBanach operator ideal S

appℓ(pn )

β

is small

Theorem 18 If conditions (a1) (a2) and (βn) notin ℓ1 aresatisfied then the pre-quasi Banach operator ideal S

appℓ(pn )

β

issmall

Proof Since conditions (a1) (a2) and (βn) notin ℓ1 are satisfiedthen (S

appℓ(pn )

β

g) where g(T) 1113936infini0 βi(αi(T))pi is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appℓ(pn )

β

(X Y) L(X Y) )en thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y are infinite dimensionalBanach spaces By using inequality (49) and (βn) notin ℓ1 oneobtains

1le Am

H

minus1m

1113872 1113873

pi αi JmAminus1m ImHmQm1113872 11138731113872 1113873

pi⟹

βi le L Am

βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

piH

minus1m

⟹

1113944

m

i0βi le L Am

H

minus1m

1113944

m

i0βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

pi⟹

1113944

m

i0βi le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1113944

m

i0βi le 4LC

(52)

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary )us X and Y both cannot be infinite di-mensional when S

appℓ(pn )

β

(X Y) L(X Y) and hence theresult

Corollary 21 (see [2]) If 0ltpltinfin then the quasi Banachoperator ideal S

appℓp is small

Corollary 22 If 0ltpltinfin then the quasi Banach operatorideal SKolℓp is small

7 Pre-Quasi Simple Banach Operator Ideal

)e following question arises naturally for which weightedNakano sequence space or generalized Cesaro sequencespace defined by weighted mean is the pre-quasi Banachideal simple

Theorem 19 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 Λ Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (53)

Proof Suppose that there exists T isin L(Sℓ(qn )

(an )

Sℓ(pn)

(bn)

) which isnot approximable According to Lemma 1 we can find

X isin L(Sℓ(qn)

(an )

Sℓ(qn )

(an)

) and B isin L(Sℓ(pn )

(bn )

Sℓ(pn )

(bn )

) with BTXIk Ik

)en it follows for all k isin N that

Ik

S

ℓpn( )bn( )

1113944infin

n0bn αn Ik( 1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

1113944kminus 1

n0bn

⎛⎝ ⎞⎠

1suppn

le BTX Ik

S

ℓqn( )an( )

le 1113944infin

n0an αn Ik( 1113857( 1113857

qn⎛⎝ ⎞⎠

1supqn

1113944kminus 1

n0an

⎛⎝ ⎞⎠

1supqn

(54)

But this is impossible

Corollary 23 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 LC Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (55)

Proof Every approximable operator is compact

Theorem 20 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

Λ Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(56)

Proof Suppose that there exists T isin L(Sces((an)(qn))

Sces((bn)(pn))) which is not approximable According toLemma 1 we can find X isin L(Sces((an)(qn)) Sces((an)(qn))) andB isin L(Sces((bn)(pn)) Sces((bn)(pn))) with BTXIk Ik )en itfollows for all k isin N that

10 Journal of Function Spaces

Ik

Sces bn( ) pn( )( )

1113944infin

n0bn 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0bn 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0kbn1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

le BTX Ik

Sces an( ) qn( )( )

le 1113944infin

n0an 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0an 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0kan( 1113857⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

(57)

But this is impossible

Corollary 24 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

LC Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(58)

Theorem 21 For a bounded sequence (pn) with 1lepn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space Sℓ(pn )

(bn )

is simple

Proof Suppose that the closed ideal LC(Sℓ(pn)

(bn)

) contains an

operator Twhich is not approximable According to Lemma

1 we can find X B isin L(Sℓ(pn)

(bn)

) with BTXIk Ik )is means

that ISℓ(pn )

(bn )

isin LC(Sℓ(pn )

(bn )

) Consequently L(Sℓ(pn)

(bn)

) LC(Sℓ(pn)

(bn)

)

)erefore Λ(Sℓ(pn )

(bn )

) is the only nontrivial closed ideal inL(Sℓ(pn )

(bn )

)

Theorem 22 For a bounded sequence (pn) with 1ltpn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space(Sces((bn)(pn))) is simple

Proof Suppose that the closed ideal LC(Sces((bn)(pn))) con-tains an operator Twhich is not approximable According toLemma 1 we can find X B isin L(Sces((bn)(pn))) withBTXIk Ik )is means that ISces((bn)(pn ))

isin LC(Sces((bn)(pn)))Consequently L(Sces((bn)(pn))) LC(Sces((bn)(pn))) )ereforeΛ(Sces((bn)(pn))) is the only nontrivial closed ideal inL(Sces((bn)(pn)))

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to the writing of this paperAll authors read and approved the final manuscript

References

[1] A Pietsch Operator Ideals North-Holland PublishingCompany Amsterdam Netherlands 1980

[2] A Pietsch ldquoSmall ideals of operatorsrdquo Studia Mathematicavol 51 no 3 pp 265ndash267 1974

[3] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbersrdquo in Ieory of Operators inFunctional Spaces pp 206ndash211 Academy of Science SiberianSection Academy of Science Siberian Section NovosibirskRussia 1997 in Russian

[4] N Faried and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 186 2013

[5] N Faried and A A Bakery ldquoSmall operator ideals formed by snumbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1p 357 2018

[6] B E Rhoades ldquoOperators of Andashp typerdquo Atti della AccademiaNazionale dei Lincei Classe di Scienze Fisiche Matematiche eNaturali (VIII) vol 59 no 3-4 pp 238ndash241 1975

[7] A Pietsch Eigenvalues and s-Numbers Cambridge UniversityPress New York NY USA 1986

[8] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 no 4 pp 267ndash278 1974

[9] A Lima and E Oja ldquoIdeals of finite rank operators inter-section properties of balls and the approximation propertyrdquoStudia Mathematica vol 133 pp 175ndash186 1999

[10] S T Chen ldquoGeometry of Orlicz spacesrdquo Dissertations MathInstitute of Mathematics Polish Academy of SciencesWarsaw Poland vol 356 p 356 1996

[11] Y A Cui and H Hudzik ldquoOn the Banach-saks and weakBanach-saks properties of some Banach sequence spacesrdquoActa Scientiarum Mathematicarum vol 65 pp 179ndash1871999

[12] M Sengonul ldquoOn the generalized Nakano sequence spacerdquoIai Journal of Mathematics vol 4 no 2 pp 295ndash304 2006

[13] W Sanhan and S Suantai ldquoOnk-nearly uniform convexproperty in generalized Cesaro sequence spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciencesvol 2003 no 57 pp 3599ndash3607 2003

[14] P Y Lee ldquoCesaro sequence spacesrdquo Mathematical Chroniclevol 13 pp 29ndash45 1984

[15] Y q Lui and P Y LeeMethod of Sequence Spaces Guangdongof science and Technology press Guangdong China 1996 inChinese

[16] J S Shiue ldquoOn the Cesaro sequence spacesrdquo Tamkang Journalof Mathematics vol 1 no 1 pp 19ndash25 1970

[17] B Altay and F Basar ldquoGeneralization of the sequence spaceℓ(p) derived by weighted meansrdquo Journal of MathematicalAnalysis and Applications vol 330 no 1 pp 147ndash185 2007

[18] A Pietsch Operator Ideals VEB Deutscher Verlag derWis-senschaften Berlin Germany 1978

[19] A Pietsch ldquos-numbers of operators in Banach spacesrdquo StudiaMathematica vol 51 no 3 pp 201ndash223 1974

Journal of Function Spaces 11

Notations )e sets SE SE(X Y) SappE and S

appE (X Y) (cf

[5]) as follows

SE ≔ SE(X Y) X andY are Banach spaces1113864 1113865 where

SE(X Y) ≔ T isin L(X Y) si(T)( 1113857(infini0 isin E1113864 1113865Also

SappE ≔ S

appE (X Y) X andY are Banach spaces1113864 1113865 where

SappE (X Y) ≔ T isin L(X Y) αi(T)( 1113857(

infini0 isin E1113864 1113865

(14)

Theorem 4 (see [5]) If E is a (sss) then SE is an operatorideal

Theorem 5 (see [3]) If X and Y are infinite dimensionalBanach spaces and (μi) is a monotonic decreasing sequenceto zero then there exists a bounded linear operator T suchthat

116μ3i le αi(T)le 8μi+1 (15)

Now and after define en 0 0 1 0 0 where 1appears at the nth place for all n isin N and the given inequalitywill be used in the sequel

an + bn

11138681113868111386811138681113868111386811138681113868pn leH an

11138681113868111386811138681113868111386811138681113868pn + bn

11138681113868111386811138681113868111386811138681113868pn1113872 1113873 (16)

whereH max 1 2hminus 11113864 1113865 h supnpn and pn ge 1 for all n isin N(see [17])

3 Main Results

31 Linear Problem We examine here the operator idealscreated by sminusnumbers and also weighted Nakano sequencespace or generalized Cesaro sequence space defined byweighted mean such that those classes of all bounded linearoperators T between arbitrary Banach spaces with (αn(T))

in these sequence spaces type an ideal operator

Theorem 6 ℓ(pn)

β is a (sss) if the following conditions aresatisfied

(a1) Ie sequence (pn) is increasing and bounded fromabove with pn gt 0 for all n isin N

(a2) Either (βn) is monotonic decreasing or monotonicincreasing such that there exists a constant Cge 1 forwhich β2n+1 leCβn

Proof (1) Let x y isin ℓ(pn)

β Since (pn) is bounded we get

1113944

infin

n0βn xn + yn

11138681113868111386811138681113868111386811138681113868pn leH 1113944

infin

n0βn xn

11138681113868111386811138681113868111386811138681113868pn + 1113944infin

n0βn yn

11138681113868111386811138681113868111386811138681113868pn⎛⎝ ⎞⎠ltinfin

(17)

then x + y isin ℓ(pn)

β

(2) Let λ isin R and x isin ℓ(pn)

β Since (pn) is bounded wehave

1113944

infin

n0βn λxn

11138681113868111386811138681113868111386811138681113868pn le sup

n

|λ|pn1113944

infin

n0βn xn

11138681113868111386811138681113868111386811138681113868pn ltinfin (18)

)en λx isin ℓ(pn)

β )erefore by using Parts (1) and (2) wehave that the space ℓ(pn)

β is linear Also en isin ℓ(pn)

β for alln isin N since

1113944

infin

i0βi en(i)

11138681113868111386811138681113868111386811138681113868pi βn (19)

(2) Let |xn|le |yn| for all n isin N and y isin ℓ(pn)

β Since βn gt 0for all n isin N then

1113944

infin

n0βn xn

11138681113868111386811138681113868111386811138681113868pn le 1113944infin

n0βn yn

11138681113868111386811138681113868111386811138681113868pn ltinfin (20)

and we get x isin ℓ(pn)

β

(3) Let (xn) isin ℓ(pn)

β (βn) be an increasing sequence)ere exists Cgt 0 such that β2n+1 leCβn and (pn) beincreasing then we have

1113944

infin

n0βn x[n2]

11138681113868111386811138681113868111386811138681113868pn 1113944infin

n0β2n xn

11138681113868111386811138681113868111386811138681113868p2n + 1113944infin

n0β2n+1 xn

11138681113868111386811138681113868111386811138681113868p2n+1

le 1113944

infin

n0β2n xn

11138681113868111386811138681113868111386811138681113868pn + 1113944

infin

n0β2n+1 xn

11138681113868111386811138681113868111386811138681113868pn le 2C 1113944

infin

n0βn xn

11138681113868111386811138681113868111386811138681113868pn

(21)

and then (x[n2]) isin ℓ(pn)

β

Theorem 7 ces((an) (pn)) is a (sss) if the following con-ditions are satisfied

(b1) Ie sequence (pn) is increasing and bounded withp0 gt 1

(b2) 1113936infinn0 (an)pn ltinfin

Proof (1) Given that x y isin ces((an) (pn)) and λ1 λ2 isin RSince (pn) is bounded we have

1113944

infin

n0an 1113944

n

k0λ1xk + λ2yk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

leH supn

λ11113868111386811138681113868

1113868111386811138681113868pn 1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

⎛⎝

+ supn

λ21113868111386811138681113868

1113868111386811138681113868pn 1113944

infin

n0an 1113944

n

k0yk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

⎞⎠ltinfin

(22)

Hence λ1x + λ2y isin ces((an) (pn)) then the spaceces((an) (pn)) is linear Also prove that em isin ces((an) (pn))

for all m isin N since 1113936infinn0 (an)pn ltinfin So we get

1113944

infin

n0an 1113944

n

k0em(k)

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

1113944infin

nm

an( 1113857pn le 1113944infin

n0an( 1113857

pn ltinfin

(23)

Hence em isin ces((an) (pn))

4 Journal of Function Spaces

(2) Let |xn|le |yn| for all n isin N and y isin ces((an) (pn))Since an gt 0 for all n isin N then

1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 1113944

infin

n0an 1113944

n

k0yk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

ltinfin (24)

and we get x isin ces((an) (pn))

(3) Let (xn) isin ces((an) (pn)) Since (pn) is increasingand the sequence (an) with 1113936

infinn0 (an)pn ltinfin is de-

creasing then we have

1113944

infin

n0an 1113944

n

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

1113944infin

n0a2n 1113944

2n

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

p2n

+ 1113944infin

n0a2n+1 1113944

2n+1

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

p2n+1

le 1113944infin

n0a2n 1113944

2n

k02 xk

11138681113868111386811138681113868111386811138681113868 + xn

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠⎛⎝ ⎞⎠

pn

+ 1113944infin

n0a2n+1 1113944

2n+1

k02 xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 2hminus 11113944

infin

n02an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

+ 1113944infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠ + 1113944infin

n02an 1113944

n

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 2hminus 1 2h+ 11113872 1113873 1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

+ 2h1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 22hminus 1+ 2hminus 1

+ 2h1113872 1113873 1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

ltinfin

(25)

Hence (x[n2]) isin ces((an) (pn))By using)eorem 4 we can get the following corollaries

Corollary 1 Let conditions (a1) and (a2) be satisfied thenSappℓ(pn )

β

be an operator ideal

Corollary 2 Sappℓ(pn ) is an operator ideal if the sequence (pn) is

increasing and bounded from above with pn gt 0 for all n isin N

Corollary 3 If p isin (0infin) then Sappℓp is an operator ideal

Corollary 4 Conditions (b1) and (b2) are satisfied henceSappces((an)(pn)) is an operator ideal

Corollary 5 Assume (pn) is increasing with p0 gt 1 andbounded so S

appces(pn) is an operator ideal

Corollary 6 If p isin (1infin) then Sappcesp is an operator ideal

4 Topological Problem

)e following question arises naturally which sufficientconditions (not necessary) on the sequence space E(weighted Nakano sequence space and generalized Cesarosequence space defined by weighted mean) are the ideal ofthe finite rank operators in the class of Banach spaces densein SE )is gives a negative answer of Rhoades [6] openproblem about the linearity of Eminustype spaces (SE)

Theorem 8 F(X Y) Sℓ(pn )

β(X Y) whenever conditions

(a1) and (a2) are satisfied

Proof First we substantiate that each finite operatorT isin F(X Y) belongs to Sℓ(pn )

β(X Y) Given that

en1113864 1113865nisinN sub ℓ(pn)

β and the space ℓ(pn)

β is linear then for all finiteoperators T isin F(X Y) ie the sequence (sn(T))nisinN con-tains only finitely many numbers different from zeroCurrently we substantiate that Sℓ(pn )

β(X Y) sube F(X Y) On

taking T isin Sℓ(pn )

β(X Y)we obtain (sn(T))infinn0 isin ℓ

(pn)

β hence

ρ((sn(T))infinn0)ltinfin let ε isin (0 1) at that point there exists am isin N minus 0 such that ρ((sn(T))infinnm)lt (ε4C2) for someCge 1 While (sn(T))nisinN is decreasing we get

1113944

2m

nm+1βn s2m(T)( 1113857

pn le 11139442m

nm+1βn sn(T)( 1113857

pn

le 1113944

infin

nm

βn sn(T)( 1113857pn lt

ε4C2

(26)

Hence there exists A isin F2m(X Y) rank Ale 2m and

1113944

3m

n2m+1βn(T minus A)

pn le 11139442m

nm+1βn(T minus A)

pn ltε

4C2 (27)

Since (pn) is a bounded consider

1113944

m

n0βn(T minus A)

pn ltε

4C2 (28)

Let (βn) be monotonic increasing such that there exists aconstant Cge 1 for which β2n+1 leCβn )en we have forngem that

β2m+n le β2m+2n+1 leCβm+n leCβ2n leCβ2n+1 leC2βn (29)

Since (pn) is increasing inequalities (26)ndash(29) give

Journal of Function Spaces 5

d(T A) ρ sn(T minus A)( 1113857infinn0

11139443mminus1

n0βn sn(T minus A)( 1113857

pn + 1113944infin

n3m

βn sn(T minus A)( 1113857pn

le 11139443m

n0βn(T minus A)

pn + 1113944infin

nm

βn+2m sn+2m(T minus A)( 1113857pn+2m

le 3 1113944m

n0βn(T minus A)

pn + C2

1113944

infin

nm

βn sn(T)( 1113857pn lt ε

(30)

Since I3 isin Sℓ(n)

(1)

condition (a1) is not satisfied whichgives a counter example of the converse statement )isfinishes the proof

From )eorem 8 we can say that if (a1) and (a2) aresatisfied then every compact operator would be approxi-mated by finite rank operators and the converse is not alwaystrue

Theorem 9 F(X Y) Sces((an)(pn))(X Y) assume thatstates (b1) and (b2) are fulfilled and the converse is not alwaystrue

Proof Primary since en isin ces((an) (pn)) for each n isin N andthe space ces((an) (pn)) is linear then for every finitemapping T isin F(X Y) ie the sequence (sn(T))nisinN containsonly fnitely many numbers different from zero HenceF(X Y) sube Sces((an)(pn))(X Y) By letting T isinSces((an)(pn))(X Y) we obtain (sn(T))nisinN isin ces((an) (pn))while ρ((αn(T))nisinN)ltinfin Let ε isin (0 1) )en there exists a

number m isin N minus 0 such that ρ((sn(T))infinnm)lt (ε2h+2δC)

for some Cge 1 where δ max 1 1113936infinnm a

pnn1113966 1113967 As sn(T) is

decreasing for every n isin N we obtain

1113944

2m

nm+1an 1113944

n

k0s2m(T)⎛⎝ ⎞⎠

pn

le 11139442m

nm+1an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

le 1113944infin

nm

an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

ltε

2h+2δC

(31)

)en there exists A isin F2m(X Y) and rank Ale 2m and

1113944

3m

n2m+1an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

le 11139442m

nm+1an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

ltε

2h+2δC

(32)

and since (pn) is bounded consider

supnm

infin1113944

m

k0T minus A⎛⎝ ⎞⎠

pn

ltε2hδ

(33)

Hence set

1113944

m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

ltε

2h+3δC (34)

However (pn) is increasing and (an) is decreasing foreach n isin N by using inequalities (31)ndash(34) we have

d(T A) ρ sn(T minus A)( 1113857infinn0

11139443mminus1

n0an 1113944

n

k0sk(T minus A)⎛⎝ ⎞⎠

pn

+ 1113944infin

n3m

an 1113944

n

k0sk(T minus A)

pn⎛⎝ ⎞⎠

le 11139443m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an+2m 1113944

n+2m

k0sk(T minus A)⎛⎝ ⎞⎠

pn+2m

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an 1113944

2mminus 1

k0sk(T minus A) + an 1113944

n+2m

k2m

sk(T minus A)⎛⎝ ⎞⎠

pn

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 11113944

infin

nm

an 1113944

2mminus 1

k0sk(T minus A)⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an 1113944

n+2m

k2m

sk(T minus A)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

le 3 1113944

m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 11113944

infin

nm

an 1113944

m

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944

infin

nm

an 1113944

n

k0sk+2m(T minus A)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 1supinfinnm 1113944

m

k0T minus A)

pn 1113944

infin

nm

an( 1113857pn + 2hminus 1

1113944

infin

nm

an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠lt ε

(35)

Since I3 isin Sces((1)(1)) condition (b2) is not satisfiedwhich gives a counter example of the converse statement)is finishes the proof

From )eorem 9 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 isnot always true

Corollary 7 If (pn) is an increasing with pn gt 0 for all n isin Nand bounded from above then Sl(p) (X Y) F(X Y)

6 Journal of Function Spaces

Corollary 8 If 0ltpltinfin then Slp (X Y) F(X Y)

Corollary 9 Sces(pn)(X Y) F(X Y) if (pn) is increasingwith p0 gt 1 and limn⟶infinsuppn ltinfin

Corollary 10 Scesp(X Y) F(X Y) if 1ltpltinfin

5 Completeness of the Pre-QuasiIdeal Components

For which sequence space E are the components of pre-quasioperator ideal SE complete

Theorem 10 ℓ(pn)

β is a premodular (sss) if conditions (a1)and (a2) are satisfied

Proof We define the functional ρ on ℓ(pn)

β asρ(x) 1113936

infinn0 βn|xn|pn

(i) Evidently ρ(x)ge 0 and ρ(x) 0⟺x θ(ii) )ere is a steady L max 1 supn|λ|pn1113864 1113865ge 1 such that

ρ(λx)le L|λ|ρ(x) for all x isin ℓ(pn)

β and λ isin R(iii) We have the inequality ρ(x + y)leH(ρ(x) + ρ(y))

for all x y isin ℓ(pn)

β

(iv) Clearly follows from inequality (20) of )eorem 6(v) It obtained from (27) )eorem 6 that K0 ge 2(vi) It is clear that F ℓ(pn)

β

(vii) )ere exists a steady 0lt ξ le |λ|p0minus 1 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin R (36)

Theorem 11 ces((an) (pn)) is a premodular (sss) if con-ditions (b1) and (b2) are satisfied

Proof We define the functional ρ on ces((an) (pn)) asρ(x) 1113936

infinn0 (an 1113936

nk0 |xk|)pn

(i) Clearly ρ(x)ge 0 and ρ(x) 0⟺x θ(ii) )ere is a number L max 1 supn|λ|pn1113864 1113865ge 1 with

ρ(λx)le L|λ|ρ(x) for all x isin ces((an) (pn)) andλ isin R

(iii) We have the inequality ρ(x + y)leH(ρ(x) + ρ(y))

for all x y isin ces((an) (pn))(iv) It clearly follows from inequality (24) of )eorem 7(v) It is clear from (27) )eorem 7 that

K0 ge (22hminus 1 + 2hminus 1 + 2h)ge 1(vi) It is clear that F ces((an) (pn))(vii) )ere exists a steady 0 lang ξ le |λ|p0minus 1 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin RWe state the following theorem without proof andthis can be established using standard technique

Theorem 12 Ie function g(P) 9(si(P))infini0 is a pre-quasinorm on SE9

where E9 is a premodular (sss)

Theorem 13 If X and Y are Banach spaces and Eρ is apremodular (sss) then (SEρ

g) where g(T) ρ((sn(T))infinn0)

is a pre-quasi Banach operator ideal

Proof Since Eρ is a premodular (sss) then the functiong(T) ρ((sn(T))infinn0) is a pre-quasi norm on SEρ

Let (Tm)

be a Cauchy sequence in SEρ(X Y) then by utilizing Part

(vii) of Definition 9 and since L(X Y) supe SEρ(X Y) we get

g Ti minus Tj1113872 1113873 ρ sn Ti minus Tj1113872 11138731113872 1113873infinn01113872 1113873ge ρ s0 Ti minus Tj1113872 1113873 0 0 0 1113872 1113873

ρ Ti minus Tj

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868 0 0 0 1113874 1113875ge ξ Tm minus Tj

ρ(1 0 0 0 )

(37)

)en (Tm)misinN is a Cauchy sequence in L(X Y) While thespace L(X Y) is a Banach space so there exists T isin L(X Y)

with limm⟶infinTm minus T 0 and while (sn(Tm))infinn0 isin Eρ foreach m isin N hence using Parts (iii) and (iv) of Definition 9and as ρ is continuous at θ we obtain

g(T) ρ sn(T)( 1113857infinn0( 1113857 ρ sn T minus Tm + Tm( 1113857( 1113857

infinn0( 1113857

leKρ s[n2] T minus Tm( 111385711138721113872 1113873infinn01113873 + Kρ s[n2] Tm( 1113857

infinn01113872 11138731113872 1113873

leKρ Tm minus T

1113872 1113873infinn01113872 1113873 + Kρ sn Tm( 1113857

infinn0( 1113857( 1113857lt ε

(38)

and we have (sn(T))infinn0 isin Eρ then T isin SEρ(X Y)

Corollary 11 If X and Y are Banach spaces and conditions(a1) and (a2) are satisfied then Sℓ(pn)

βis a pre-quasi Banach

operator ideal

Corollary 12 If X and Y are Banach spaces (pn) is in-creasing with pn gt 0 for all n isin N and bounded from abovethen Sℓ(pn ) is a pre-quasi Banach operator ideal

Corollary 13 If X and Y are Banach spaces and 0ltpltinfinthen Sℓp is a pre-quasi Banach operator ideal

Corollary 14 If X and Y are Banach spaces and the con-ditions (b1) and (b2) are satisfied then Sces((an)(pn)) is a pre-quasi Banach operator ideal

Corollary 15 If X and Y are Banach spaces and (pn) isincreasing with p0 gt 1 and limn⟶infinsuppn ltinfin then Sces(pn)

is a pre-quasi Banach operator ideal

Corollary 16 If X and Y are Banach spaces and p isin (1infin)then U

appcesp (X Y) is complete

6 Smallness of the Pre-Quasi BanachOperator Ideal

We give here the sufficient conditions on the weightedNakano sequence space such that the pre-quasi operatorideal formed by the sequence of approximation numbers and

Journal of Function Spaces 7

this sequence space is strictly contained for different weightsand powers

Theorem 14 For any infinite dimensional Banach spaces Xand Y and for any 0ltpn lt qn and 0lt an lt bn for all n isin N it istrue that

Sapp

ℓ pn( )bn( )

(X Y)⫋ Sapp

ℓ qn( )an( )

(X Y)⫋ L(X Y) (39)

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appℓ(pn )

(bn )

(X Y) then (αn(T)) isin ℓ(pn)

(bn) One can see that

1113944

infin

n0an αn(T)( 1113857

qn lt 1113944infin

n0bn αn(T)( 1113857

pn ltinfin (40)

Hence T isin Sappℓ(qn )

(an )

(X Y) Next if we take (an) with

sup an ltinfin (bminus1n ) isin ℓ(qnpn)and μn (1

bn

pn1113968

) So by using)eorem 5 one can find T isin L(X Y) with

116

b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (41)

such that T does not belong to Sappℓ(pn)

(bn)

(X Y) andT isin S

appℓ(qn )

(an )

(X Y)

It is easy to see that Sappℓ(qn)

(an)

(X Y) sub L(X Y) Next if wetake μn (1

anqnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (42)

such that T does not belong to Sappℓ(qn )

(an )

(X Y) )is finishes theproof

Corollary 17 (see [3]) For any infinite dimensional Banachspaces X and Y and for any qgtpgt 0 it is true thatSappℓp (X Y)⫋ S

appℓq (X Y)⫋ L(X Y)

Lemma 2 If (an) is a sequence of positive real with an le 1 forall n isin N (pn) is monotonic increasing bounded sequencewith pn ge 1 for all n isin N or an gt 1 for all n isin N and (pn) ismonotonic decreasing bounded sequence with pn ge 1 for alln isin N one has the following inequality

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k (43)

Proof By using inequality (16) and the sufficient conditionsone has

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

a0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

lt 2suppnminus 1a

pn

0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

2suppnminus 1a

pn

0 + a1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 2suppnminus 1 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 22 suppnminus 1( ) apn

0 + apn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 23 suppnminus 1( ) apn

0 + apn

1 + apn

2 + 1113944n

k3ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

⋮

lt 2(nminus 1s) suppnminus 1( ) apn

0 + apn

1 + apn

2 + middot middot middot + apn

nminus21113872

+ 1113944n

knminus1ak

⎛⎝ ⎞⎠

pn

⎞⎠

lt 2n suppnminus 1( ) 1113944

n

k0a

pn

k lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k

(44)

We give here the sufficient conditions on the generalizedCesaro sequence space defined by weighted mean such thatthe pre-quasi operator ideal formed by the sequence ofapproximation numbers and this sequence space is strictlycontained for different weights and powers

Theorem 15 For any infinite dimensional Banach spaces Xand Y and for any 1ltpn lt qn for all n isin N it is true thatSappces((bn)(pn))(X Y)⫋ S

appces((an)(qn))(X Y)⫋ L(X Y) where (pn)

and (qn) are the monotonic increasing bounded sequences

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appces((bn)(pn))(X Y) then (αn(T)) isin ces((bn) (pn)) One

has

1113944

infin

n0an 1113944

n

k0αk(T)⎛⎝ ⎞⎠

qn

lt 1113944infin

n0bn 1113944

n

k0αk(T)⎛⎝ ⎞⎠

pn

ltinfin (45)

Hence T isin Sappces((an)(qn))(X Y) Next if we take (an) with

sup2(sup qnminus 1)naqnn ltinfin (bminus1

n ) isin ℓ(qnpn) and μn (1bn

pn1113968

) Soby using)eorem 5 and Lemma 2 one can find T isin L(X Y)

with1

16b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (46)

such that T does not belong to Sappces((bn)(pn))(X Y) and

T isin Sappces((an)(qn))(X Y)

8 Journal of Function Spaces

It is easy to see that Sappces((an)(qn))(X Y) sub L(X Y) Next

we take μn (1 an

qnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (47)

such that T does not belong to Sappces((an)(qn))(X Y) )is

finishes the proof

Corollary 18 For any infinite dimensional Banach spaces Xand Y and 1ltplt qltinfin then

Sappcesp

(X Y)⫋ Sappcesq

(X Y)⫋L(X Y) (48)

In this part we give the conditions for which the pre-quasiBanach Operator ideal S

appces((ai)qi)

is small

Theorem 16 If conditions (b1) (b2) and (nan) notin ℓ(pn) aresatisfied then the pre-quasi Banach operator ideal S

appces((ai)pi)

is small

Proof Since (pi) is an increasing sequence with p0 gt 1 and(ai) isin ℓ(pi) take λ (1113936

infini0 a

pi

i )1h )en (Sappces((ai)pi)

g)where g(T) (1λ)(1113936

infini0 (ai 1113936

ij0 αj(T))pi )1h is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appces((ai)pi)

(X Y) L(X Y) then thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y be infinite dimensionalBanach spaces Hence by Dvoretzkyrsquos )eorem [18] form isin N we have quotient spaces XNm and subspaces Mm of

Y which can be mapped onto ℓm2 by isomorphisms Hm and

Am such that HmHminus1m le 2 and AmAminus1

m le 2 Let Im be theidentity map on ℓm

2 Qm be the quotient map from X ontoXNm and Jm be the natural embedding map from Mm intoY Let un be the Bernstein numbers [19] then

1 un Im( 1113857 un AmAminus1m ImHmH

minus1m1113872 1113873

le Am

un A

minus1m ImHm1113872 1113873 H

minus1m

Am

un JmA

minus1m ImHm1113872 1113873 H

minus1m

le Am

dn JmA

minus1m ImHm1113872 1113873 H

minus1m

Am

dn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

le Am

αn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

(49)

for 1le ilem Now

1113944

i

j0ai( 1113857le 1113944

i

j0Am

aiαj JmA

minus1m ImHmQm1113872 1113873 H

minus1m

⟹

(i + 1)ai le Am

ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠ H

minus1m

⟹

(i + 1)ai( 1113857pi le Am

H

minus1m

1113872 1113873

piai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

(50)

)erefore

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1λ

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le 4LC

(51)

Journal of Function Spaces 9

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary and (nan) notin ℓ(pn) )us X and Y both cannot beinfinite dimensional when S

appces((ai)(pi))

(X Y) L(X Y) andhence the result

Theorem 17 If (pi) is an increasing and p0 gt 1 then the pre-quasi Banach operator ideal SKolces(pi)

is small

Corollary 19 If 1ltpltinfin then the quasi Banach operatorideal S

appcesp is small

Corollary 20 If 1ltpltinfin then the quasi Banach operatorideal SKolcesp is small

In this part we give the conditions for which the pre-quasiBanach operator ideal S

appℓ(pn )

β

is small

Theorem 18 If conditions (a1) (a2) and (βn) notin ℓ1 aresatisfied then the pre-quasi Banach operator ideal S

appℓ(pn )

β

issmall

Proof Since conditions (a1) (a2) and (βn) notin ℓ1 are satisfiedthen (S

appℓ(pn )

β

g) where g(T) 1113936infini0 βi(αi(T))pi is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appℓ(pn )

β

(X Y) L(X Y) )en thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y are infinite dimensionalBanach spaces By using inequality (49) and (βn) notin ℓ1 oneobtains

1le Am

H

minus1m

1113872 1113873

pi αi JmAminus1m ImHmQm1113872 11138731113872 1113873

pi⟹

βi le L Am

βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

piH

minus1m

⟹

1113944

m

i0βi le L Am

H

minus1m

1113944

m

i0βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

pi⟹

1113944

m

i0βi le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1113944

m

i0βi le 4LC

(52)

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary )us X and Y both cannot be infinite di-mensional when S

appℓ(pn )

β

(X Y) L(X Y) and hence theresult

Corollary 21 (see [2]) If 0ltpltinfin then the quasi Banachoperator ideal S

appℓp is small

Corollary 22 If 0ltpltinfin then the quasi Banach operatorideal SKolℓp is small

7 Pre-Quasi Simple Banach Operator Ideal

)e following question arises naturally for which weightedNakano sequence space or generalized Cesaro sequencespace defined by weighted mean is the pre-quasi Banachideal simple

Theorem 19 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 Λ Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (53)

Proof Suppose that there exists T isin L(Sℓ(qn )

(an )

Sℓ(pn)

(bn)

) which isnot approximable According to Lemma 1 we can find

X isin L(Sℓ(qn)

(an )

Sℓ(qn )

(an)

) and B isin L(Sℓ(pn )

(bn )

Sℓ(pn )

(bn )

) with BTXIk Ik

)en it follows for all k isin N that

Ik

S

ℓpn( )bn( )

1113944infin

n0bn αn Ik( 1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

1113944kminus 1

n0bn

⎛⎝ ⎞⎠

1suppn

le BTX Ik

S

ℓqn( )an( )

le 1113944infin

n0an αn Ik( 1113857( 1113857

qn⎛⎝ ⎞⎠

1supqn

1113944kminus 1

n0an

⎛⎝ ⎞⎠

1supqn

(54)

But this is impossible

Corollary 23 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 LC Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (55)

Proof Every approximable operator is compact

Theorem 20 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

Λ Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(56)

Proof Suppose that there exists T isin L(Sces((an)(qn))

Sces((bn)(pn))) which is not approximable According toLemma 1 we can find X isin L(Sces((an)(qn)) Sces((an)(qn))) andB isin L(Sces((bn)(pn)) Sces((bn)(pn))) with BTXIk Ik )en itfollows for all k isin N that

10 Journal of Function Spaces

Ik

Sces bn( ) pn( )( )

1113944infin

n0bn 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0bn 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0kbn1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

le BTX Ik

Sces an( ) qn( )( )

le 1113944infin

n0an 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0an 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0kan( 1113857⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

(57)

But this is impossible

Corollary 24 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

LC Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(58)

Theorem 21 For a bounded sequence (pn) with 1lepn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space Sℓ(pn )

(bn )

is simple

Proof Suppose that the closed ideal LC(Sℓ(pn)

(bn)

) contains an

operator Twhich is not approximable According to Lemma

1 we can find X B isin L(Sℓ(pn)

(bn)

) with BTXIk Ik )is means

that ISℓ(pn )

(bn )

isin LC(Sℓ(pn )

(bn )

) Consequently L(Sℓ(pn)

(bn)

) LC(Sℓ(pn)

(bn)

)

)erefore Λ(Sℓ(pn )

(bn )

) is the only nontrivial closed ideal inL(Sℓ(pn )

(bn )

)

Theorem 22 For a bounded sequence (pn) with 1ltpn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space(Sces((bn)(pn))) is simple

Proof Suppose that the closed ideal LC(Sces((bn)(pn))) con-tains an operator Twhich is not approximable According toLemma 1 we can find X B isin L(Sces((bn)(pn))) withBTXIk Ik )is means that ISces((bn)(pn ))

isin LC(Sces((bn)(pn)))Consequently L(Sces((bn)(pn))) LC(Sces((bn)(pn))) )ereforeΛ(Sces((bn)(pn))) is the only nontrivial closed ideal inL(Sces((bn)(pn)))

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to the writing of this paperAll authors read and approved the final manuscript

References

[1] A Pietsch Operator Ideals North-Holland PublishingCompany Amsterdam Netherlands 1980

[2] A Pietsch ldquoSmall ideals of operatorsrdquo Studia Mathematicavol 51 no 3 pp 265ndash267 1974

[3] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbersrdquo in Ieory of Operators inFunctional Spaces pp 206ndash211 Academy of Science SiberianSection Academy of Science Siberian Section NovosibirskRussia 1997 in Russian

[4] N Faried and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 186 2013

[5] N Faried and A A Bakery ldquoSmall operator ideals formed by snumbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1p 357 2018

[6] B E Rhoades ldquoOperators of Andashp typerdquo Atti della AccademiaNazionale dei Lincei Classe di Scienze Fisiche Matematiche eNaturali (VIII) vol 59 no 3-4 pp 238ndash241 1975

[7] A Pietsch Eigenvalues and s-Numbers Cambridge UniversityPress New York NY USA 1986

[8] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 no 4 pp 267ndash278 1974

[9] A Lima and E Oja ldquoIdeals of finite rank operators inter-section properties of balls and the approximation propertyrdquoStudia Mathematica vol 133 pp 175ndash186 1999

[10] S T Chen ldquoGeometry of Orlicz spacesrdquo Dissertations MathInstitute of Mathematics Polish Academy of SciencesWarsaw Poland vol 356 p 356 1996

[11] Y A Cui and H Hudzik ldquoOn the Banach-saks and weakBanach-saks properties of some Banach sequence spacesrdquoActa Scientiarum Mathematicarum vol 65 pp 179ndash1871999

[12] M Sengonul ldquoOn the generalized Nakano sequence spacerdquoIai Journal of Mathematics vol 4 no 2 pp 295ndash304 2006

[13] W Sanhan and S Suantai ldquoOnk-nearly uniform convexproperty in generalized Cesaro sequence spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciencesvol 2003 no 57 pp 3599ndash3607 2003

[14] P Y Lee ldquoCesaro sequence spacesrdquo Mathematical Chroniclevol 13 pp 29ndash45 1984

[15] Y q Lui and P Y LeeMethod of Sequence Spaces Guangdongof science and Technology press Guangdong China 1996 inChinese

[16] J S Shiue ldquoOn the Cesaro sequence spacesrdquo Tamkang Journalof Mathematics vol 1 no 1 pp 19ndash25 1970

[17] B Altay and F Basar ldquoGeneralization of the sequence spaceℓ(p) derived by weighted meansrdquo Journal of MathematicalAnalysis and Applications vol 330 no 1 pp 147ndash185 2007

[18] A Pietsch Operator Ideals VEB Deutscher Verlag derWis-senschaften Berlin Germany 1978

[19] A Pietsch ldquos-numbers of operators in Banach spacesrdquo StudiaMathematica vol 51 no 3 pp 201ndash223 1974

Journal of Function Spaces 11

(2) Let |xn|le |yn| for all n isin N and y isin ces((an) (pn))Since an gt 0 for all n isin N then

1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 1113944

infin

n0an 1113944

n

k0yk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

ltinfin (24)

and we get x isin ces((an) (pn))

(3) Let (xn) isin ces((an) (pn)) Since (pn) is increasingand the sequence (an) with 1113936

infinn0 (an)pn ltinfin is de-

creasing then we have

1113944

infin

n0an 1113944

n

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

1113944infin

n0a2n 1113944

2n

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

p2n

+ 1113944infin

n0a2n+1 1113944

2n+1

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

p2n+1

le 1113944infin

n0a2n 1113944

2n

k02 xk

11138681113868111386811138681113868111386811138681113868 + xn

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠⎛⎝ ⎞⎠

pn

+ 1113944infin

n0a2n+1 1113944

2n+1

k02 xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 2hminus 11113944

infin

n02an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

+ 1113944infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠ + 1113944infin

n02an 1113944

n

k0x[k2]

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 2hminus 1 2h+ 11113872 1113873 1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

+ 2h1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

le 22hminus 1+ 2hminus 1

+ 2h1113872 1113873 1113944

infin

n0an 1113944

n

k0xk

11138681113868111386811138681113868111386811138681113868⎛⎝ ⎞⎠

pn

ltinfin

(25)

Hence (x[n2]) isin ces((an) (pn))By using)eorem 4 we can get the following corollaries

Corollary 1 Let conditions (a1) and (a2) be satisfied thenSappℓ(pn )

β

be an operator ideal

Corollary 2 Sappℓ(pn ) is an operator ideal if the sequence (pn) is

increasing and bounded from above with pn gt 0 for all n isin N

Corollary 3 If p isin (0infin) then Sappℓp is an operator ideal

Corollary 4 Conditions (b1) and (b2) are satisfied henceSappces((an)(pn)) is an operator ideal

Corollary 5 Assume (pn) is increasing with p0 gt 1 andbounded so S

appces(pn) is an operator ideal

Corollary 6 If p isin (1infin) then Sappcesp is an operator ideal

4 Topological Problem

)e following question arises naturally which sufficientconditions (not necessary) on the sequence space E(weighted Nakano sequence space and generalized Cesarosequence space defined by weighted mean) are the ideal ofthe finite rank operators in the class of Banach spaces densein SE )is gives a negative answer of Rhoades [6] openproblem about the linearity of Eminustype spaces (SE)

Theorem 8 F(X Y) Sℓ(pn )

β(X Y) whenever conditions

(a1) and (a2) are satisfied

Proof First we substantiate that each finite operatorT isin F(X Y) belongs to Sℓ(pn )

β(X Y) Given that

en1113864 1113865nisinN sub ℓ(pn)

β and the space ℓ(pn)

β is linear then for all finiteoperators T isin F(X Y) ie the sequence (sn(T))nisinN con-tains only finitely many numbers different from zeroCurrently we substantiate that Sℓ(pn )

β(X Y) sube F(X Y) On

taking T isin Sℓ(pn )

β(X Y)we obtain (sn(T))infinn0 isin ℓ

(pn)

β hence

ρ((sn(T))infinn0)ltinfin let ε isin (0 1) at that point there exists am isin N minus 0 such that ρ((sn(T))infinnm)lt (ε4C2) for someCge 1 While (sn(T))nisinN is decreasing we get

1113944

2m

nm+1βn s2m(T)( 1113857

pn le 11139442m

nm+1βn sn(T)( 1113857

pn

le 1113944

infin

nm

βn sn(T)( 1113857pn lt

ε4C2

(26)

Hence there exists A isin F2m(X Y) rank Ale 2m and

1113944

3m

n2m+1βn(T minus A)

pn le 11139442m

nm+1βn(T minus A)

pn ltε

4C2 (27)

Since (pn) is a bounded consider

1113944

m

n0βn(T minus A)

pn ltε

4C2 (28)

Let (βn) be monotonic increasing such that there exists aconstant Cge 1 for which β2n+1 leCβn )en we have forngem that

β2m+n le β2m+2n+1 leCβm+n leCβ2n leCβ2n+1 leC2βn (29)

Since (pn) is increasing inequalities (26)ndash(29) give

Journal of Function Spaces 5

d(T A) ρ sn(T minus A)( 1113857infinn0

11139443mminus1

n0βn sn(T minus A)( 1113857

pn + 1113944infin

n3m

βn sn(T minus A)( 1113857pn

le 11139443m

n0βn(T minus A)

pn + 1113944infin

nm

βn+2m sn+2m(T minus A)( 1113857pn+2m

le 3 1113944m

n0βn(T minus A)

pn + C2

1113944

infin

nm

βn sn(T)( 1113857pn lt ε

(30)

Since I3 isin Sℓ(n)

(1)

condition (a1) is not satisfied whichgives a counter example of the converse statement )isfinishes the proof

From )eorem 8 we can say that if (a1) and (a2) aresatisfied then every compact operator would be approxi-mated by finite rank operators and the converse is not alwaystrue

Theorem 9 F(X Y) Sces((an)(pn))(X Y) assume thatstates (b1) and (b2) are fulfilled and the converse is not alwaystrue

Proof Primary since en isin ces((an) (pn)) for each n isin N andthe space ces((an) (pn)) is linear then for every finitemapping T isin F(X Y) ie the sequence (sn(T))nisinN containsonly fnitely many numbers different from zero HenceF(X Y) sube Sces((an)(pn))(X Y) By letting T isinSces((an)(pn))(X Y) we obtain (sn(T))nisinN isin ces((an) (pn))while ρ((αn(T))nisinN)ltinfin Let ε isin (0 1) )en there exists a

number m isin N minus 0 such that ρ((sn(T))infinnm)lt (ε2h+2δC)

for some Cge 1 where δ max 1 1113936infinnm a

pnn1113966 1113967 As sn(T) is

decreasing for every n isin N we obtain

1113944

2m

nm+1an 1113944

n

k0s2m(T)⎛⎝ ⎞⎠

pn

le 11139442m

nm+1an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

le 1113944infin

nm

an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

ltε

2h+2δC

(31)

)en there exists A isin F2m(X Y) and rank Ale 2m and

1113944

3m

n2m+1an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

le 11139442m

nm+1an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

ltε

2h+2δC

(32)

and since (pn) is bounded consider

supnm

infin1113944

m

k0T minus A⎛⎝ ⎞⎠

pn

ltε2hδ

(33)

Hence set

1113944

m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

ltε

2h+3δC (34)

However (pn) is increasing and (an) is decreasing foreach n isin N by using inequalities (31)ndash(34) we have

d(T A) ρ sn(T minus A)( 1113857infinn0

11139443mminus1

n0an 1113944

n

k0sk(T minus A)⎛⎝ ⎞⎠

pn

+ 1113944infin

n3m

an 1113944

n

k0sk(T minus A)

pn⎛⎝ ⎞⎠

le 11139443m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an+2m 1113944

n+2m

k0sk(T minus A)⎛⎝ ⎞⎠

pn+2m

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an 1113944

2mminus 1

k0sk(T minus A) + an 1113944

n+2m

k2m

sk(T minus A)⎛⎝ ⎞⎠

pn

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 11113944

infin

nm

an 1113944

2mminus 1

k0sk(T minus A)⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an 1113944

n+2m

k2m

sk(T minus A)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

le 3 1113944

m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 11113944

infin

nm

an 1113944

m

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944

infin

nm

an 1113944

n

k0sk+2m(T minus A)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 1supinfinnm 1113944

m

k0T minus A)

pn 1113944

infin

nm

an( 1113857pn + 2hminus 1

1113944

infin

nm

an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠lt ε

(35)

Since I3 isin Sces((1)(1)) condition (b2) is not satisfiedwhich gives a counter example of the converse statement)is finishes the proof

From )eorem 9 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 isnot always true

Corollary 7 If (pn) is an increasing with pn gt 0 for all n isin Nand bounded from above then Sl(p) (X Y) F(X Y)

6 Journal of Function Spaces

Corollary 8 If 0ltpltinfin then Slp (X Y) F(X Y)

Corollary 9 Sces(pn)(X Y) F(X Y) if (pn) is increasingwith p0 gt 1 and limn⟶infinsuppn ltinfin

Corollary 10 Scesp(X Y) F(X Y) if 1ltpltinfin

5 Completeness of the Pre-QuasiIdeal Components

For which sequence space E are the components of pre-quasioperator ideal SE complete

Theorem 10 ℓ(pn)

β is a premodular (sss) if conditions (a1)and (a2) are satisfied

Proof We define the functional ρ on ℓ(pn)

β asρ(x) 1113936

infinn0 βn|xn|pn

(i) Evidently ρ(x)ge 0 and ρ(x) 0⟺x θ(ii) )ere is a steady L max 1 supn|λ|pn1113864 1113865ge 1 such that

ρ(λx)le L|λ|ρ(x) for all x isin ℓ(pn)

β and λ isin R(iii) We have the inequality ρ(x + y)leH(ρ(x) + ρ(y))

for all x y isin ℓ(pn)

β

(iv) Clearly follows from inequality (20) of )eorem 6(v) It obtained from (27) )eorem 6 that K0 ge 2(vi) It is clear that F ℓ(pn)

β

(vii) )ere exists a steady 0lt ξ le |λ|p0minus 1 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin R (36)

Theorem 11 ces((an) (pn)) is a premodular (sss) if con-ditions (b1) and (b2) are satisfied

Proof We define the functional ρ on ces((an) (pn)) asρ(x) 1113936

infinn0 (an 1113936

nk0 |xk|)pn

(i) Clearly ρ(x)ge 0 and ρ(x) 0⟺x θ(ii) )ere is a number L max 1 supn|λ|pn1113864 1113865ge 1 with

ρ(λx)le L|λ|ρ(x) for all x isin ces((an) (pn)) andλ isin R

(iii) We have the inequality ρ(x + y)leH(ρ(x) + ρ(y))

for all x y isin ces((an) (pn))(iv) It clearly follows from inequality (24) of )eorem 7(v) It is clear from (27) )eorem 7 that

K0 ge (22hminus 1 + 2hminus 1 + 2h)ge 1(vi) It is clear that F ces((an) (pn))(vii) )ere exists a steady 0 lang ξ le |λ|p0minus 1 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin RWe state the following theorem without proof andthis can be established using standard technique

Theorem 12 Ie function g(P) 9(si(P))infini0 is a pre-quasinorm on SE9

where E9 is a premodular (sss)

Theorem 13 If X and Y are Banach spaces and Eρ is apremodular (sss) then (SEρ

g) where g(T) ρ((sn(T))infinn0)

is a pre-quasi Banach operator ideal

Proof Since Eρ is a premodular (sss) then the functiong(T) ρ((sn(T))infinn0) is a pre-quasi norm on SEρ

Let (Tm)

be a Cauchy sequence in SEρ(X Y) then by utilizing Part

(vii) of Definition 9 and since L(X Y) supe SEρ(X Y) we get

g Ti minus Tj1113872 1113873 ρ sn Ti minus Tj1113872 11138731113872 1113873infinn01113872 1113873ge ρ s0 Ti minus Tj1113872 1113873 0 0 0 1113872 1113873

ρ Ti minus Tj

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868 0 0 0 1113874 1113875ge ξ Tm minus Tj

ρ(1 0 0 0 )

(37)

)en (Tm)misinN is a Cauchy sequence in L(X Y) While thespace L(X Y) is a Banach space so there exists T isin L(X Y)

with limm⟶infinTm minus T 0 and while (sn(Tm))infinn0 isin Eρ foreach m isin N hence using Parts (iii) and (iv) of Definition 9and as ρ is continuous at θ we obtain

g(T) ρ sn(T)( 1113857infinn0( 1113857 ρ sn T minus Tm + Tm( 1113857( 1113857

infinn0( 1113857

leKρ s[n2] T minus Tm( 111385711138721113872 1113873infinn01113873 + Kρ s[n2] Tm( 1113857

infinn01113872 11138731113872 1113873

leKρ Tm minus T

1113872 1113873infinn01113872 1113873 + Kρ sn Tm( 1113857

infinn0( 1113857( 1113857lt ε

(38)

and we have (sn(T))infinn0 isin Eρ then T isin SEρ(X Y)

Corollary 11 If X and Y are Banach spaces and conditions(a1) and (a2) are satisfied then Sℓ(pn)

βis a pre-quasi Banach

operator ideal

Corollary 12 If X and Y are Banach spaces (pn) is in-creasing with pn gt 0 for all n isin N and bounded from abovethen Sℓ(pn ) is a pre-quasi Banach operator ideal

Corollary 13 If X and Y are Banach spaces and 0ltpltinfinthen Sℓp is a pre-quasi Banach operator ideal

Corollary 14 If X and Y are Banach spaces and the con-ditions (b1) and (b2) are satisfied then Sces((an)(pn)) is a pre-quasi Banach operator ideal

Corollary 15 If X and Y are Banach spaces and (pn) isincreasing with p0 gt 1 and limn⟶infinsuppn ltinfin then Sces(pn)

is a pre-quasi Banach operator ideal

Corollary 16 If X and Y are Banach spaces and p isin (1infin)then U

appcesp (X Y) is complete

6 Smallness of the Pre-Quasi BanachOperator Ideal

We give here the sufficient conditions on the weightedNakano sequence space such that the pre-quasi operatorideal formed by the sequence of approximation numbers and

Journal of Function Spaces 7

this sequence space is strictly contained for different weightsand powers

Theorem 14 For any infinite dimensional Banach spaces Xand Y and for any 0ltpn lt qn and 0lt an lt bn for all n isin N it istrue that

Sapp

ℓ pn( )bn( )

(X Y)⫋ Sapp

ℓ qn( )an( )

(X Y)⫋ L(X Y) (39)

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appℓ(pn )

(bn )

(X Y) then (αn(T)) isin ℓ(pn)

(bn) One can see that

1113944

infin

n0an αn(T)( 1113857

qn lt 1113944infin

n0bn αn(T)( 1113857

pn ltinfin (40)

Hence T isin Sappℓ(qn )

(an )

(X Y) Next if we take (an) with

sup an ltinfin (bminus1n ) isin ℓ(qnpn)and μn (1

bn

pn1113968

) So by using)eorem 5 one can find T isin L(X Y) with

116

b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (41)

such that T does not belong to Sappℓ(pn)

(bn)

(X Y) andT isin S

appℓ(qn )

(an )

(X Y)

It is easy to see that Sappℓ(qn)

(an)

(X Y) sub L(X Y) Next if wetake μn (1

anqnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (42)

such that T does not belong to Sappℓ(qn )

(an )

(X Y) )is finishes theproof

Corollary 17 (see [3]) For any infinite dimensional Banachspaces X and Y and for any qgtpgt 0 it is true thatSappℓp (X Y)⫋ S

appℓq (X Y)⫋ L(X Y)

Lemma 2 If (an) is a sequence of positive real with an le 1 forall n isin N (pn) is monotonic increasing bounded sequencewith pn ge 1 for all n isin N or an gt 1 for all n isin N and (pn) ismonotonic decreasing bounded sequence with pn ge 1 for alln isin N one has the following inequality

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k (43)

Proof By using inequality (16) and the sufficient conditionsone has

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

a0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

lt 2suppnminus 1a

pn

0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

2suppnminus 1a

pn

0 + a1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 2suppnminus 1 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 22 suppnminus 1( ) apn

0 + apn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 23 suppnminus 1( ) apn

0 + apn

1 + apn

2 + 1113944n

k3ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

⋮

lt 2(nminus 1s) suppnminus 1( ) apn

0 + apn

1 + apn

2 + middot middot middot + apn

nminus21113872

+ 1113944n

knminus1ak

⎛⎝ ⎞⎠

pn

⎞⎠

lt 2n suppnminus 1( ) 1113944

n

k0a

pn

k lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k

(44)

We give here the sufficient conditions on the generalizedCesaro sequence space defined by weighted mean such thatthe pre-quasi operator ideal formed by the sequence ofapproximation numbers and this sequence space is strictlycontained for different weights and powers

Theorem 15 For any infinite dimensional Banach spaces Xand Y and for any 1ltpn lt qn for all n isin N it is true thatSappces((bn)(pn))(X Y)⫋ S

appces((an)(qn))(X Y)⫋ L(X Y) where (pn)

and (qn) are the monotonic increasing bounded sequences

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appces((bn)(pn))(X Y) then (αn(T)) isin ces((bn) (pn)) One

has

1113944

infin

n0an 1113944

n

k0αk(T)⎛⎝ ⎞⎠

qn

lt 1113944infin

n0bn 1113944

n

k0αk(T)⎛⎝ ⎞⎠

pn

ltinfin (45)

Hence T isin Sappces((an)(qn))(X Y) Next if we take (an) with

sup2(sup qnminus 1)naqnn ltinfin (bminus1

n ) isin ℓ(qnpn) and μn (1bn

pn1113968

) Soby using)eorem 5 and Lemma 2 one can find T isin L(X Y)

with1

16b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (46)

such that T does not belong to Sappces((bn)(pn))(X Y) and

T isin Sappces((an)(qn))(X Y)

8 Journal of Function Spaces

It is easy to see that Sappces((an)(qn))(X Y) sub L(X Y) Next

we take μn (1 an

qnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (47)

such that T does not belong to Sappces((an)(qn))(X Y) )is

finishes the proof

Corollary 18 For any infinite dimensional Banach spaces Xand Y and 1ltplt qltinfin then

Sappcesp

(X Y)⫋ Sappcesq

(X Y)⫋L(X Y) (48)

In this part we give the conditions for which the pre-quasiBanach Operator ideal S

appces((ai)qi)

is small

Theorem 16 If conditions (b1) (b2) and (nan) notin ℓ(pn) aresatisfied then the pre-quasi Banach operator ideal S

appces((ai)pi)

is small

Proof Since (pi) is an increasing sequence with p0 gt 1 and(ai) isin ℓ(pi) take λ (1113936

infini0 a

pi

i )1h )en (Sappces((ai)pi)

g)where g(T) (1λ)(1113936

infini0 (ai 1113936

ij0 αj(T))pi )1h is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appces((ai)pi)

(X Y) L(X Y) then thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y be infinite dimensionalBanach spaces Hence by Dvoretzkyrsquos )eorem [18] form isin N we have quotient spaces XNm and subspaces Mm of

Y which can be mapped onto ℓm2 by isomorphisms Hm and

Am such that HmHminus1m le 2 and AmAminus1

m le 2 Let Im be theidentity map on ℓm

2 Qm be the quotient map from X ontoXNm and Jm be the natural embedding map from Mm intoY Let un be the Bernstein numbers [19] then

1 un Im( 1113857 un AmAminus1m ImHmH

minus1m1113872 1113873

le Am

un A

minus1m ImHm1113872 1113873 H

minus1m

Am

un JmA

minus1m ImHm1113872 1113873 H

minus1m

le Am

dn JmA

minus1m ImHm1113872 1113873 H

minus1m

Am

dn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

le Am

αn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

(49)

for 1le ilem Now

1113944

i

j0ai( 1113857le 1113944

i

j0Am

aiαj JmA

minus1m ImHmQm1113872 1113873 H

minus1m

⟹

(i + 1)ai le Am

ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠ H

minus1m

⟹

(i + 1)ai( 1113857pi le Am

H

minus1m

1113872 1113873

piai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

(50)

)erefore

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1λ

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le 4LC

(51)

Journal of Function Spaces 9

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary and (nan) notin ℓ(pn) )us X and Y both cannot beinfinite dimensional when S

appces((ai)(pi))

(X Y) L(X Y) andhence the result

Theorem 17 If (pi) is an increasing and p0 gt 1 then the pre-quasi Banach operator ideal SKolces(pi)

is small

Corollary 19 If 1ltpltinfin then the quasi Banach operatorideal S

appcesp is small

Corollary 20 If 1ltpltinfin then the quasi Banach operatorideal SKolcesp is small

In this part we give the conditions for which the pre-quasiBanach operator ideal S

appℓ(pn )

β

is small

Theorem 18 If conditions (a1) (a2) and (βn) notin ℓ1 aresatisfied then the pre-quasi Banach operator ideal S

appℓ(pn )

β

issmall

Proof Since conditions (a1) (a2) and (βn) notin ℓ1 are satisfiedthen (S

appℓ(pn )

β

g) where g(T) 1113936infini0 βi(αi(T))pi is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appℓ(pn )

β

(X Y) L(X Y) )en thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y are infinite dimensionalBanach spaces By using inequality (49) and (βn) notin ℓ1 oneobtains

1le Am

H

minus1m

1113872 1113873

pi αi JmAminus1m ImHmQm1113872 11138731113872 1113873

pi⟹

βi le L Am

βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

piH

minus1m

⟹

1113944

m

i0βi le L Am

H

minus1m

1113944

m

i0βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

pi⟹

1113944

m

i0βi le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1113944

m

i0βi le 4LC

(52)

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary )us X and Y both cannot be infinite di-mensional when S

appℓ(pn )

β

(X Y) L(X Y) and hence theresult

Corollary 21 (see [2]) If 0ltpltinfin then the quasi Banachoperator ideal S

appℓp is small

Corollary 22 If 0ltpltinfin then the quasi Banach operatorideal SKolℓp is small

7 Pre-Quasi Simple Banach Operator Ideal

)e following question arises naturally for which weightedNakano sequence space or generalized Cesaro sequencespace defined by weighted mean is the pre-quasi Banachideal simple

Theorem 19 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 Λ Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (53)

Proof Suppose that there exists T isin L(Sℓ(qn )

(an )

Sℓ(pn)

(bn)

) which isnot approximable According to Lemma 1 we can find

X isin L(Sℓ(qn)

(an )

Sℓ(qn )

(an)

) and B isin L(Sℓ(pn )

(bn )

Sℓ(pn )

(bn )

) with BTXIk Ik

)en it follows for all k isin N that

Ik

S

ℓpn( )bn( )

1113944infin

n0bn αn Ik( 1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

1113944kminus 1

n0bn

⎛⎝ ⎞⎠

1suppn

le BTX Ik

S

ℓqn( )an( )

le 1113944infin

n0an αn Ik( 1113857( 1113857

qn⎛⎝ ⎞⎠

1supqn

1113944kminus 1

n0an

⎛⎝ ⎞⎠

1supqn

(54)

But this is impossible

Corollary 23 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 LC Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (55)

Proof Every approximable operator is compact

Theorem 20 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

Λ Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(56)

Proof Suppose that there exists T isin L(Sces((an)(qn))

Sces((bn)(pn))) which is not approximable According toLemma 1 we can find X isin L(Sces((an)(qn)) Sces((an)(qn))) andB isin L(Sces((bn)(pn)) Sces((bn)(pn))) with BTXIk Ik )en itfollows for all k isin N that

10 Journal of Function Spaces

Ik

Sces bn( ) pn( )( )

1113944infin

n0bn 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0bn 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0kbn1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

le BTX Ik

Sces an( ) qn( )( )

le 1113944infin

n0an 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0an 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0kan( 1113857⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

(57)

But this is impossible

Corollary 24 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

LC Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(58)

Theorem 21 For a bounded sequence (pn) with 1lepn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space Sℓ(pn )

(bn )

is simple

Proof Suppose that the closed ideal LC(Sℓ(pn)

(bn)

) contains an

operator Twhich is not approximable According to Lemma

1 we can find X B isin L(Sℓ(pn)

(bn)

) with BTXIk Ik )is means

that ISℓ(pn )

(bn )

isin LC(Sℓ(pn )

(bn )

) Consequently L(Sℓ(pn)

(bn)

) LC(Sℓ(pn)

(bn)

)

)erefore Λ(Sℓ(pn )

(bn )

) is the only nontrivial closed ideal inL(Sℓ(pn )

(bn )

)

Theorem 22 For a bounded sequence (pn) with 1ltpn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space(Sces((bn)(pn))) is simple

Proof Suppose that the closed ideal LC(Sces((bn)(pn))) con-tains an operator Twhich is not approximable According toLemma 1 we can find X B isin L(Sces((bn)(pn))) withBTXIk Ik )is means that ISces((bn)(pn ))

isin LC(Sces((bn)(pn)))Consequently L(Sces((bn)(pn))) LC(Sces((bn)(pn))) )ereforeΛ(Sces((bn)(pn))) is the only nontrivial closed ideal inL(Sces((bn)(pn)))

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to the writing of this paperAll authors read and approved the final manuscript

References

[1] A Pietsch Operator Ideals North-Holland PublishingCompany Amsterdam Netherlands 1980

[2] A Pietsch ldquoSmall ideals of operatorsrdquo Studia Mathematicavol 51 no 3 pp 265ndash267 1974

[3] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbersrdquo in Ieory of Operators inFunctional Spaces pp 206ndash211 Academy of Science SiberianSection Academy of Science Siberian Section NovosibirskRussia 1997 in Russian

[4] N Faried and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 186 2013

[5] N Faried and A A Bakery ldquoSmall operator ideals formed by snumbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1p 357 2018

[6] B E Rhoades ldquoOperators of Andashp typerdquo Atti della AccademiaNazionale dei Lincei Classe di Scienze Fisiche Matematiche eNaturali (VIII) vol 59 no 3-4 pp 238ndash241 1975

[7] A Pietsch Eigenvalues and s-Numbers Cambridge UniversityPress New York NY USA 1986

[8] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 no 4 pp 267ndash278 1974

[9] A Lima and E Oja ldquoIdeals of finite rank operators inter-section properties of balls and the approximation propertyrdquoStudia Mathematica vol 133 pp 175ndash186 1999

[10] S T Chen ldquoGeometry of Orlicz spacesrdquo Dissertations MathInstitute of Mathematics Polish Academy of SciencesWarsaw Poland vol 356 p 356 1996

[11] Y A Cui and H Hudzik ldquoOn the Banach-saks and weakBanach-saks properties of some Banach sequence spacesrdquoActa Scientiarum Mathematicarum vol 65 pp 179ndash1871999

[12] M Sengonul ldquoOn the generalized Nakano sequence spacerdquoIai Journal of Mathematics vol 4 no 2 pp 295ndash304 2006

[13] W Sanhan and S Suantai ldquoOnk-nearly uniform convexproperty in generalized Cesaro sequence spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciencesvol 2003 no 57 pp 3599ndash3607 2003

[14] P Y Lee ldquoCesaro sequence spacesrdquo Mathematical Chroniclevol 13 pp 29ndash45 1984

[15] Y q Lui and P Y LeeMethod of Sequence Spaces Guangdongof science and Technology press Guangdong China 1996 inChinese

[16] J S Shiue ldquoOn the Cesaro sequence spacesrdquo Tamkang Journalof Mathematics vol 1 no 1 pp 19ndash25 1970

[17] B Altay and F Basar ldquoGeneralization of the sequence spaceℓ(p) derived by weighted meansrdquo Journal of MathematicalAnalysis and Applications vol 330 no 1 pp 147ndash185 2007

[18] A Pietsch Operator Ideals VEB Deutscher Verlag derWis-senschaften Berlin Germany 1978

[19] A Pietsch ldquos-numbers of operators in Banach spacesrdquo StudiaMathematica vol 51 no 3 pp 201ndash223 1974

Journal of Function Spaces 11

d(T A) ρ sn(T minus A)( 1113857infinn0

11139443mminus1

n0βn sn(T minus A)( 1113857

pn + 1113944infin

n3m

βn sn(T minus A)( 1113857pn

le 11139443m

n0βn(T minus A)

pn + 1113944infin

nm

βn+2m sn+2m(T minus A)( 1113857pn+2m

le 3 1113944m

n0βn(T minus A)

pn + C2

1113944

infin

nm

βn sn(T)( 1113857pn lt ε

(30)

Since I3 isin Sℓ(n)

(1)

condition (a1) is not satisfied whichgives a counter example of the converse statement )isfinishes the proof

From )eorem 8 we can say that if (a1) and (a2) aresatisfied then every compact operator would be approxi-mated by finite rank operators and the converse is not alwaystrue

Theorem 9 F(X Y) Sces((an)(pn))(X Y) assume thatstates (b1) and (b2) are fulfilled and the converse is not alwaystrue

Proof Primary since en isin ces((an) (pn)) for each n isin N andthe space ces((an) (pn)) is linear then for every finitemapping T isin F(X Y) ie the sequence (sn(T))nisinN containsonly fnitely many numbers different from zero HenceF(X Y) sube Sces((an)(pn))(X Y) By letting T isinSces((an)(pn))(X Y) we obtain (sn(T))nisinN isin ces((an) (pn))while ρ((αn(T))nisinN)ltinfin Let ε isin (0 1) )en there exists a

number m isin N minus 0 such that ρ((sn(T))infinnm)lt (ε2h+2δC)

for some Cge 1 where δ max 1 1113936infinnm a

pnn1113966 1113967 As sn(T) is

decreasing for every n isin N we obtain

1113944

2m

nm+1an 1113944

n

k0s2m(T)⎛⎝ ⎞⎠

pn

le 11139442m

nm+1an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

le 1113944infin

nm

an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

ltε

2h+2δC

(31)

)en there exists A isin F2m(X Y) and rank Ale 2m and

1113944

3m

n2m+1an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

le 11139442m

nm+1an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

ltε

2h+2δC

(32)

and since (pn) is bounded consider

supnm

infin1113944

m

k0T minus A⎛⎝ ⎞⎠

pn

ltε2hδ

(33)

Hence set

1113944

m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

ltε

2h+3δC (34)

However (pn) is increasing and (an) is decreasing foreach n isin N by using inequalities (31)ndash(34) we have

d(T A) ρ sn(T minus A)( 1113857infinn0

11139443mminus1

n0an 1113944

n

k0sk(T minus A)⎛⎝ ⎞⎠

pn

+ 1113944infin

n3m

an 1113944

n

k0sk(T minus A)

pn⎛⎝ ⎞⎠

le 11139443m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an+2m 1113944

n+2m

k0sk(T minus A)⎛⎝ ⎞⎠

pn+2m

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an 1113944

2mminus 1

k0sk(T minus A) + an 1113944

n+2m

k2m

sk(T minus A)⎛⎝ ⎞⎠

pn

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 11113944

infin

nm

an 1113944

2mminus 1

k0sk(T minus A)⎛⎝ ⎞⎠

pn

+ 1113944infin

nm

an 1113944

n+2m

k2m

sk(T minus A)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

le 3 1113944

m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 11113944

infin

nm

an 1113944

m

k0T minus A⎛⎝ ⎞⎠

pn

+ 1113944

infin

nm

an 1113944

n

k0sk+2m(T minus A)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

le 3 1113944m

n0an 1113944

n

k0T minus A⎛⎝ ⎞⎠

pn

+ 2hminus 1supinfinnm 1113944

m

k0T minus A)

pn 1113944

infin

nm

an( 1113857pn + 2hminus 1

1113944

infin

nm

an 1113944

n

k0sk(T)⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠lt ε

(35)

Since I3 isin Sces((1)(1)) condition (b2) is not satisfiedwhich gives a counter example of the converse statement)is finishes the proof

From )eorem 9 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 isnot always true

Corollary 7 If (pn) is an increasing with pn gt 0 for all n isin Nand bounded from above then Sl(p) (X Y) F(X Y)

6 Journal of Function Spaces

Corollary 8 If 0ltpltinfin then Slp (X Y) F(X Y)

Corollary 9 Sces(pn)(X Y) F(X Y) if (pn) is increasingwith p0 gt 1 and limn⟶infinsuppn ltinfin

Corollary 10 Scesp(X Y) F(X Y) if 1ltpltinfin

5 Completeness of the Pre-QuasiIdeal Components

For which sequence space E are the components of pre-quasioperator ideal SE complete

Theorem 10 ℓ(pn)

β is a premodular (sss) if conditions (a1)and (a2) are satisfied

Proof We define the functional ρ on ℓ(pn)

β asρ(x) 1113936

infinn0 βn|xn|pn

(i) Evidently ρ(x)ge 0 and ρ(x) 0⟺x θ(ii) )ere is a steady L max 1 supn|λ|pn1113864 1113865ge 1 such that

ρ(λx)le L|λ|ρ(x) for all x isin ℓ(pn)

β and λ isin R(iii) We have the inequality ρ(x + y)leH(ρ(x) + ρ(y))

for all x y isin ℓ(pn)

β

(iv) Clearly follows from inequality (20) of )eorem 6(v) It obtained from (27) )eorem 6 that K0 ge 2(vi) It is clear that F ℓ(pn)

β

(vii) )ere exists a steady 0lt ξ le |λ|p0minus 1 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin R (36)

Theorem 11 ces((an) (pn)) is a premodular (sss) if con-ditions (b1) and (b2) are satisfied

Proof We define the functional ρ on ces((an) (pn)) asρ(x) 1113936

infinn0 (an 1113936

nk0 |xk|)pn

(i) Clearly ρ(x)ge 0 and ρ(x) 0⟺x θ(ii) )ere is a number L max 1 supn|λ|pn1113864 1113865ge 1 with

ρ(λx)le L|λ|ρ(x) for all x isin ces((an) (pn)) andλ isin R

(iii) We have the inequality ρ(x + y)leH(ρ(x) + ρ(y))

for all x y isin ces((an) (pn))(iv) It clearly follows from inequality (24) of )eorem 7(v) It is clear from (27) )eorem 7 that

K0 ge (22hminus 1 + 2hminus 1 + 2h)ge 1(vi) It is clear that F ces((an) (pn))(vii) )ere exists a steady 0 lang ξ le |λ|p0minus 1 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin RWe state the following theorem without proof andthis can be established using standard technique

Theorem 12 Ie function g(P) 9(si(P))infini0 is a pre-quasinorm on SE9

where E9 is a premodular (sss)

Theorem 13 If X and Y are Banach spaces and Eρ is apremodular (sss) then (SEρ

g) where g(T) ρ((sn(T))infinn0)

is a pre-quasi Banach operator ideal

Proof Since Eρ is a premodular (sss) then the functiong(T) ρ((sn(T))infinn0) is a pre-quasi norm on SEρ

Let (Tm)

be a Cauchy sequence in SEρ(X Y) then by utilizing Part

(vii) of Definition 9 and since L(X Y) supe SEρ(X Y) we get

g Ti minus Tj1113872 1113873 ρ sn Ti minus Tj1113872 11138731113872 1113873infinn01113872 1113873ge ρ s0 Ti minus Tj1113872 1113873 0 0 0 1113872 1113873

ρ Ti minus Tj

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868 0 0 0 1113874 1113875ge ξ Tm minus Tj

ρ(1 0 0 0 )

(37)

)en (Tm)misinN is a Cauchy sequence in L(X Y) While thespace L(X Y) is a Banach space so there exists T isin L(X Y)

with limm⟶infinTm minus T 0 and while (sn(Tm))infinn0 isin Eρ foreach m isin N hence using Parts (iii) and (iv) of Definition 9and as ρ is continuous at θ we obtain

g(T) ρ sn(T)( 1113857infinn0( 1113857 ρ sn T minus Tm + Tm( 1113857( 1113857

infinn0( 1113857

leKρ s[n2] T minus Tm( 111385711138721113872 1113873infinn01113873 + Kρ s[n2] Tm( 1113857

infinn01113872 11138731113872 1113873

leKρ Tm minus T

1113872 1113873infinn01113872 1113873 + Kρ sn Tm( 1113857

infinn0( 1113857( 1113857lt ε

(38)

and we have (sn(T))infinn0 isin Eρ then T isin SEρ(X Y)

Corollary 11 If X and Y are Banach spaces and conditions(a1) and (a2) are satisfied then Sℓ(pn)

βis a pre-quasi Banach

operator ideal

Corollary 12 If X and Y are Banach spaces (pn) is in-creasing with pn gt 0 for all n isin N and bounded from abovethen Sℓ(pn ) is a pre-quasi Banach operator ideal

Corollary 13 If X and Y are Banach spaces and 0ltpltinfinthen Sℓp is a pre-quasi Banach operator ideal

Corollary 14 If X and Y are Banach spaces and the con-ditions (b1) and (b2) are satisfied then Sces((an)(pn)) is a pre-quasi Banach operator ideal

Corollary 15 If X and Y are Banach spaces and (pn) isincreasing with p0 gt 1 and limn⟶infinsuppn ltinfin then Sces(pn)

is a pre-quasi Banach operator ideal

Corollary 16 If X and Y are Banach spaces and p isin (1infin)then U

appcesp (X Y) is complete

6 Smallness of the Pre-Quasi BanachOperator Ideal

We give here the sufficient conditions on the weightedNakano sequence space such that the pre-quasi operatorideal formed by the sequence of approximation numbers and

Journal of Function Spaces 7

this sequence space is strictly contained for different weightsand powers

Theorem 14 For any infinite dimensional Banach spaces Xand Y and for any 0ltpn lt qn and 0lt an lt bn for all n isin N it istrue that

Sapp

ℓ pn( )bn( )

(X Y)⫋ Sapp

ℓ qn( )an( )

(X Y)⫋ L(X Y) (39)

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appℓ(pn )

(bn )

(X Y) then (αn(T)) isin ℓ(pn)

(bn) One can see that

1113944

infin

n0an αn(T)( 1113857

qn lt 1113944infin

n0bn αn(T)( 1113857

pn ltinfin (40)

Hence T isin Sappℓ(qn )

(an )

(X Y) Next if we take (an) with

sup an ltinfin (bminus1n ) isin ℓ(qnpn)and μn (1

bn

pn1113968

) So by using)eorem 5 one can find T isin L(X Y) with

116

b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (41)

such that T does not belong to Sappℓ(pn)

(bn)

(X Y) andT isin S

appℓ(qn )

(an )

(X Y)

It is easy to see that Sappℓ(qn)

(an)

(X Y) sub L(X Y) Next if wetake μn (1

anqnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (42)

such that T does not belong to Sappℓ(qn )

(an )

(X Y) )is finishes theproof

Corollary 17 (see [3]) For any infinite dimensional Banachspaces X and Y and for any qgtpgt 0 it is true thatSappℓp (X Y)⫋ S

appℓq (X Y)⫋ L(X Y)

Lemma 2 If (an) is a sequence of positive real with an le 1 forall n isin N (pn) is monotonic increasing bounded sequencewith pn ge 1 for all n isin N or an gt 1 for all n isin N and (pn) ismonotonic decreasing bounded sequence with pn ge 1 for alln isin N one has the following inequality

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k (43)

Proof By using inequality (16) and the sufficient conditionsone has

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

a0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

lt 2suppnminus 1a

pn

0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

2suppnminus 1a

pn

0 + a1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 2suppnminus 1 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 22 suppnminus 1( ) apn

0 + apn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 23 suppnminus 1( ) apn

0 + apn

1 + apn

2 + 1113944n

k3ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

⋮

lt 2(nminus 1s) suppnminus 1( ) apn

0 + apn

1 + apn

2 + middot middot middot + apn

nminus21113872

+ 1113944n

knminus1ak

⎛⎝ ⎞⎠

pn

⎞⎠

lt 2n suppnminus 1( ) 1113944

n

k0a

pn

k lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k

(44)

We give here the sufficient conditions on the generalizedCesaro sequence space defined by weighted mean such thatthe pre-quasi operator ideal formed by the sequence ofapproximation numbers and this sequence space is strictlycontained for different weights and powers

Theorem 15 For any infinite dimensional Banach spaces Xand Y and for any 1ltpn lt qn for all n isin N it is true thatSappces((bn)(pn))(X Y)⫋ S

appces((an)(qn))(X Y)⫋ L(X Y) where (pn)

and (qn) are the monotonic increasing bounded sequences

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appces((bn)(pn))(X Y) then (αn(T)) isin ces((bn) (pn)) One

has

1113944

infin

n0an 1113944

n

k0αk(T)⎛⎝ ⎞⎠

qn

lt 1113944infin

n0bn 1113944

n

k0αk(T)⎛⎝ ⎞⎠

pn

ltinfin (45)

Hence T isin Sappces((an)(qn))(X Y) Next if we take (an) with

sup2(sup qnminus 1)naqnn ltinfin (bminus1

n ) isin ℓ(qnpn) and μn (1bn

pn1113968

) Soby using)eorem 5 and Lemma 2 one can find T isin L(X Y)

with1

16b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (46)

such that T does not belong to Sappces((bn)(pn))(X Y) and

T isin Sappces((an)(qn))(X Y)

8 Journal of Function Spaces

It is easy to see that Sappces((an)(qn))(X Y) sub L(X Y) Next

we take μn (1 an

qnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (47)

such that T does not belong to Sappces((an)(qn))(X Y) )is

finishes the proof

Corollary 18 For any infinite dimensional Banach spaces Xand Y and 1ltplt qltinfin then

Sappcesp

(X Y)⫋ Sappcesq

(X Y)⫋L(X Y) (48)

In this part we give the conditions for which the pre-quasiBanach Operator ideal S

appces((ai)qi)

is small

Theorem 16 If conditions (b1) (b2) and (nan) notin ℓ(pn) aresatisfied then the pre-quasi Banach operator ideal S

appces((ai)pi)

is small

Proof Since (pi) is an increasing sequence with p0 gt 1 and(ai) isin ℓ(pi) take λ (1113936

infini0 a

pi

i )1h )en (Sappces((ai)pi)

g)where g(T) (1λ)(1113936

infini0 (ai 1113936

ij0 αj(T))pi )1h is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appces((ai)pi)

(X Y) L(X Y) then thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y be infinite dimensionalBanach spaces Hence by Dvoretzkyrsquos )eorem [18] form isin N we have quotient spaces XNm and subspaces Mm of

Y which can be mapped onto ℓm2 by isomorphisms Hm and

Am such that HmHminus1m le 2 and AmAminus1

m le 2 Let Im be theidentity map on ℓm

2 Qm be the quotient map from X ontoXNm and Jm be the natural embedding map from Mm intoY Let un be the Bernstein numbers [19] then

1 un Im( 1113857 un AmAminus1m ImHmH

minus1m1113872 1113873

le Am

un A

minus1m ImHm1113872 1113873 H

minus1m

Am

un JmA

minus1m ImHm1113872 1113873 H

minus1m

le Am

dn JmA

minus1m ImHm1113872 1113873 H

minus1m

Am

dn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

le Am

αn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

(49)

for 1le ilem Now

1113944

i

j0ai( 1113857le 1113944

i

j0Am

aiαj JmA

minus1m ImHmQm1113872 1113873 H

minus1m

⟹

(i + 1)ai le Am

ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠ H

minus1m

⟹

(i + 1)ai( 1113857pi le Am

H

minus1m

1113872 1113873

piai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

(50)

)erefore

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1λ

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le 4LC

(51)

Journal of Function Spaces 9

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary and (nan) notin ℓ(pn) )us X and Y both cannot beinfinite dimensional when S

appces((ai)(pi))

(X Y) L(X Y) andhence the result

Theorem 17 If (pi) is an increasing and p0 gt 1 then the pre-quasi Banach operator ideal SKolces(pi)

is small

Corollary 19 If 1ltpltinfin then the quasi Banach operatorideal S

appcesp is small

Corollary 20 If 1ltpltinfin then the quasi Banach operatorideal SKolcesp is small

In this part we give the conditions for which the pre-quasiBanach operator ideal S

appℓ(pn )

β

is small

Theorem 18 If conditions (a1) (a2) and (βn) notin ℓ1 aresatisfied then the pre-quasi Banach operator ideal S

appℓ(pn )

β

issmall

Proof Since conditions (a1) (a2) and (βn) notin ℓ1 are satisfiedthen (S

appℓ(pn )

β

g) where g(T) 1113936infini0 βi(αi(T))pi is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appℓ(pn )

β

(X Y) L(X Y) )en thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y are infinite dimensionalBanach spaces By using inequality (49) and (βn) notin ℓ1 oneobtains

1le Am

H

minus1m

1113872 1113873

pi αi JmAminus1m ImHmQm1113872 11138731113872 1113873

pi⟹

βi le L Am

βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

piH

minus1m

⟹

1113944

m

i0βi le L Am

H

minus1m

1113944

m

i0βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

pi⟹

1113944

m

i0βi le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1113944

m

i0βi le 4LC

(52)

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary )us X and Y both cannot be infinite di-mensional when S

appℓ(pn )

β

(X Y) L(X Y) and hence theresult

Corollary 21 (see [2]) If 0ltpltinfin then the quasi Banachoperator ideal S

appℓp is small

Corollary 22 If 0ltpltinfin then the quasi Banach operatorideal SKolℓp is small

7 Pre-Quasi Simple Banach Operator Ideal

)e following question arises naturally for which weightedNakano sequence space or generalized Cesaro sequencespace defined by weighted mean is the pre-quasi Banachideal simple

Theorem 19 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 Λ Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (53)

Proof Suppose that there exists T isin L(Sℓ(qn )

(an )

Sℓ(pn)

(bn)

) which isnot approximable According to Lemma 1 we can find

X isin L(Sℓ(qn)

(an )

Sℓ(qn )

(an)

) and B isin L(Sℓ(pn )

(bn )

Sℓ(pn )

(bn )

) with BTXIk Ik

)en it follows for all k isin N that

Ik

S

ℓpn( )bn( )

1113944infin

n0bn αn Ik( 1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

1113944kminus 1

n0bn

⎛⎝ ⎞⎠

1suppn

le BTX Ik

S

ℓqn( )an( )

le 1113944infin

n0an αn Ik( 1113857( 1113857

qn⎛⎝ ⎞⎠

1supqn

1113944kminus 1

n0an

⎛⎝ ⎞⎠

1supqn

(54)

But this is impossible

Corollary 23 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 LC Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (55)

Proof Every approximable operator is compact

Theorem 20 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

Λ Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(56)

Proof Suppose that there exists T isin L(Sces((an)(qn))

Sces((bn)(pn))) which is not approximable According toLemma 1 we can find X isin L(Sces((an)(qn)) Sces((an)(qn))) andB isin L(Sces((bn)(pn)) Sces((bn)(pn))) with BTXIk Ik )en itfollows for all k isin N that

10 Journal of Function Spaces

Ik

Sces bn( ) pn( )( )

1113944infin

n0bn 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0bn 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0kbn1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

le BTX Ik

Sces an( ) qn( )( )

le 1113944infin

n0an 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0an 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0kan( 1113857⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

(57)

But this is impossible

Corollary 24 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

LC Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(58)

Theorem 21 For a bounded sequence (pn) with 1lepn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space Sℓ(pn )

(bn )

is simple

Proof Suppose that the closed ideal LC(Sℓ(pn)

(bn)

) contains an

operator Twhich is not approximable According to Lemma

1 we can find X B isin L(Sℓ(pn)

(bn)

) with BTXIk Ik )is means

that ISℓ(pn )

(bn )

isin LC(Sℓ(pn )

(bn )

) Consequently L(Sℓ(pn)

(bn)

) LC(Sℓ(pn)

(bn)

)

)erefore Λ(Sℓ(pn )

(bn )

) is the only nontrivial closed ideal inL(Sℓ(pn )

(bn )

)

Theorem 22 For a bounded sequence (pn) with 1ltpn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space(Sces((bn)(pn))) is simple

Proof Suppose that the closed ideal LC(Sces((bn)(pn))) con-tains an operator Twhich is not approximable According toLemma 1 we can find X B isin L(Sces((bn)(pn))) withBTXIk Ik )is means that ISces((bn)(pn ))

isin LC(Sces((bn)(pn)))Consequently L(Sces((bn)(pn))) LC(Sces((bn)(pn))) )ereforeΛ(Sces((bn)(pn))) is the only nontrivial closed ideal inL(Sces((bn)(pn)))

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to the writing of this paperAll authors read and approved the final manuscript

References

[1] A Pietsch Operator Ideals North-Holland PublishingCompany Amsterdam Netherlands 1980

[2] A Pietsch ldquoSmall ideals of operatorsrdquo Studia Mathematicavol 51 no 3 pp 265ndash267 1974

[3] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbersrdquo in Ieory of Operators inFunctional Spaces pp 206ndash211 Academy of Science SiberianSection Academy of Science Siberian Section NovosibirskRussia 1997 in Russian

[4] N Faried and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 186 2013

[5] N Faried and A A Bakery ldquoSmall operator ideals formed by snumbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1p 357 2018

[6] B E Rhoades ldquoOperators of Andashp typerdquo Atti della AccademiaNazionale dei Lincei Classe di Scienze Fisiche Matematiche eNaturali (VIII) vol 59 no 3-4 pp 238ndash241 1975

[7] A Pietsch Eigenvalues and s-Numbers Cambridge UniversityPress New York NY USA 1986

[8] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 no 4 pp 267ndash278 1974

[9] A Lima and E Oja ldquoIdeals of finite rank operators inter-section properties of balls and the approximation propertyrdquoStudia Mathematica vol 133 pp 175ndash186 1999

[10] S T Chen ldquoGeometry of Orlicz spacesrdquo Dissertations MathInstitute of Mathematics Polish Academy of SciencesWarsaw Poland vol 356 p 356 1996

[11] Y A Cui and H Hudzik ldquoOn the Banach-saks and weakBanach-saks properties of some Banach sequence spacesrdquoActa Scientiarum Mathematicarum vol 65 pp 179ndash1871999

[12] M Sengonul ldquoOn the generalized Nakano sequence spacerdquoIai Journal of Mathematics vol 4 no 2 pp 295ndash304 2006

[13] W Sanhan and S Suantai ldquoOnk-nearly uniform convexproperty in generalized Cesaro sequence spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciencesvol 2003 no 57 pp 3599ndash3607 2003

[14] P Y Lee ldquoCesaro sequence spacesrdquo Mathematical Chroniclevol 13 pp 29ndash45 1984

[15] Y q Lui and P Y LeeMethod of Sequence Spaces Guangdongof science and Technology press Guangdong China 1996 inChinese

[16] J S Shiue ldquoOn the Cesaro sequence spacesrdquo Tamkang Journalof Mathematics vol 1 no 1 pp 19ndash25 1970

[17] B Altay and F Basar ldquoGeneralization of the sequence spaceℓ(p) derived by weighted meansrdquo Journal of MathematicalAnalysis and Applications vol 330 no 1 pp 147ndash185 2007

[18] A Pietsch Operator Ideals VEB Deutscher Verlag derWis-senschaften Berlin Germany 1978

[19] A Pietsch ldquos-numbers of operators in Banach spacesrdquo StudiaMathematica vol 51 no 3 pp 201ndash223 1974

Journal of Function Spaces 11

Corollary 8 If 0ltpltinfin then Slp (X Y) F(X Y)

Corollary 9 Sces(pn)(X Y) F(X Y) if (pn) is increasingwith p0 gt 1 and limn⟶infinsuppn ltinfin

Corollary 10 Scesp(X Y) F(X Y) if 1ltpltinfin

5 Completeness of the Pre-QuasiIdeal Components

For which sequence space E are the components of pre-quasioperator ideal SE complete

Theorem 10 ℓ(pn)

β is a premodular (sss) if conditions (a1)and (a2) are satisfied

Proof We define the functional ρ on ℓ(pn)

β asρ(x) 1113936

infinn0 βn|xn|pn

(i) Evidently ρ(x)ge 0 and ρ(x) 0⟺x θ(ii) )ere is a steady L max 1 supn|λ|pn1113864 1113865ge 1 such that

ρ(λx)le L|λ|ρ(x) for all x isin ℓ(pn)

β and λ isin R(iii) We have the inequality ρ(x + y)leH(ρ(x) + ρ(y))

for all x y isin ℓ(pn)

β

(iv) Clearly follows from inequality (20) of )eorem 6(v) It obtained from (27) )eorem 6 that K0 ge 2(vi) It is clear that F ℓ(pn)

β

(vii) )ere exists a steady 0lt ξ le |λ|p0minus 1 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin R (36)

Theorem 11 ces((an) (pn)) is a premodular (sss) if con-ditions (b1) and (b2) are satisfied

Proof We define the functional ρ on ces((an) (pn)) asρ(x) 1113936

infinn0 (an 1113936

nk0 |xk|)pn

(i) Clearly ρ(x)ge 0 and ρ(x) 0⟺x θ(ii) )ere is a number L max 1 supn|λ|pn1113864 1113865ge 1 with

ρ(λx)le L|λ|ρ(x) for all x isin ces((an) (pn)) andλ isin R

(iii) We have the inequality ρ(x + y)leH(ρ(x) + ρ(y))

for all x y isin ces((an) (pn))(iv) It clearly follows from inequality (24) of )eorem 7(v) It is clear from (27) )eorem 7 that

K0 ge (22hminus 1 + 2hminus 1 + 2h)ge 1(vi) It is clear that F ces((an) (pn))(vii) )ere exists a steady 0 lang ξ le |λ|p0minus 1 such that

ρ(λ 0 0 0 )ge ξ|λ|ρ(1 0 0 0 ) for any λ isin RWe state the following theorem without proof andthis can be established using standard technique

Theorem 12 Ie function g(P) 9(si(P))infini0 is a pre-quasinorm on SE9

where E9 is a premodular (sss)

Theorem 13 If X and Y are Banach spaces and Eρ is apremodular (sss) then (SEρ

g) where g(T) ρ((sn(T))infinn0)

is a pre-quasi Banach operator ideal

Proof Since Eρ is a premodular (sss) then the functiong(T) ρ((sn(T))infinn0) is a pre-quasi norm on SEρ

Let (Tm)

be a Cauchy sequence in SEρ(X Y) then by utilizing Part

(vii) of Definition 9 and since L(X Y) supe SEρ(X Y) we get

g Ti minus Tj1113872 1113873 ρ sn Ti minus Tj1113872 11138731113872 1113873infinn01113872 1113873ge ρ s0 Ti minus Tj1113872 1113873 0 0 0 1113872 1113873

ρ Ti minus Tj

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868 0 0 0 1113874 1113875ge ξ Tm minus Tj

ρ(1 0 0 0 )

(37)

)en (Tm)misinN is a Cauchy sequence in L(X Y) While thespace L(X Y) is a Banach space so there exists T isin L(X Y)

with limm⟶infinTm minus T 0 and while (sn(Tm))infinn0 isin Eρ foreach m isin N hence using Parts (iii) and (iv) of Definition 9and as ρ is continuous at θ we obtain

g(T) ρ sn(T)( 1113857infinn0( 1113857 ρ sn T minus Tm + Tm( 1113857( 1113857

infinn0( 1113857

leKρ s[n2] T minus Tm( 111385711138721113872 1113873infinn01113873 + Kρ s[n2] Tm( 1113857

infinn01113872 11138731113872 1113873

leKρ Tm minus T

1113872 1113873infinn01113872 1113873 + Kρ sn Tm( 1113857

infinn0( 1113857( 1113857lt ε

(38)

and we have (sn(T))infinn0 isin Eρ then T isin SEρ(X Y)

Corollary 11 If X and Y are Banach spaces and conditions(a1) and (a2) are satisfied then Sℓ(pn)

βis a pre-quasi Banach

operator ideal

Corollary 12 If X and Y are Banach spaces (pn) is in-creasing with pn gt 0 for all n isin N and bounded from abovethen Sℓ(pn ) is a pre-quasi Banach operator ideal

Corollary 13 If X and Y are Banach spaces and 0ltpltinfinthen Sℓp is a pre-quasi Banach operator ideal

Corollary 14 If X and Y are Banach spaces and the con-ditions (b1) and (b2) are satisfied then Sces((an)(pn)) is a pre-quasi Banach operator ideal

Corollary 15 If X and Y are Banach spaces and (pn) isincreasing with p0 gt 1 and limn⟶infinsuppn ltinfin then Sces(pn)

is a pre-quasi Banach operator ideal

Corollary 16 If X and Y are Banach spaces and p isin (1infin)then U

appcesp (X Y) is complete

6 Smallness of the Pre-Quasi BanachOperator Ideal

We give here the sufficient conditions on the weightedNakano sequence space such that the pre-quasi operatorideal formed by the sequence of approximation numbers and

Journal of Function Spaces 7

this sequence space is strictly contained for different weightsand powers

Theorem 14 For any infinite dimensional Banach spaces Xand Y and for any 0ltpn lt qn and 0lt an lt bn for all n isin N it istrue that

Sapp

ℓ pn( )bn( )

(X Y)⫋ Sapp

ℓ qn( )an( )

(X Y)⫋ L(X Y) (39)

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appℓ(pn )

(bn )

(X Y) then (αn(T)) isin ℓ(pn)

(bn) One can see that

1113944

infin

n0an αn(T)( 1113857

qn lt 1113944infin

n0bn αn(T)( 1113857

pn ltinfin (40)

Hence T isin Sappℓ(qn )

(an )

(X Y) Next if we take (an) with

sup an ltinfin (bminus1n ) isin ℓ(qnpn)and μn (1

bn

pn1113968

) So by using)eorem 5 one can find T isin L(X Y) with

116

b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (41)

such that T does not belong to Sappℓ(pn)

(bn)

(X Y) andT isin S

appℓ(qn )

(an )

(X Y)

It is easy to see that Sappℓ(qn)

(an)

(X Y) sub L(X Y) Next if wetake μn (1

anqnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (42)

such that T does not belong to Sappℓ(qn )

(an )

(X Y) )is finishes theproof

Corollary 17 (see [3]) For any infinite dimensional Banachspaces X and Y and for any qgtpgt 0 it is true thatSappℓp (X Y)⫋ S

appℓq (X Y)⫋ L(X Y)

Lemma 2 If (an) is a sequence of positive real with an le 1 forall n isin N (pn) is monotonic increasing bounded sequencewith pn ge 1 for all n isin N or an gt 1 for all n isin N and (pn) ismonotonic decreasing bounded sequence with pn ge 1 for alln isin N one has the following inequality

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k (43)

Proof By using inequality (16) and the sufficient conditionsone has

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

a0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

lt 2suppnminus 1a

pn

0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

2suppnminus 1a

pn

0 + a1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 2suppnminus 1 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 22 suppnminus 1( ) apn

0 + apn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 23 suppnminus 1( ) apn

0 + apn

1 + apn

2 + 1113944n

k3ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

⋮

lt 2(nminus 1s) suppnminus 1( ) apn

0 + apn

1 + apn

2 + middot middot middot + apn

nminus21113872

+ 1113944n

knminus1ak

⎛⎝ ⎞⎠

pn

⎞⎠

lt 2n suppnminus 1( ) 1113944

n

k0a

pn

k lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k

(44)

We give here the sufficient conditions on the generalizedCesaro sequence space defined by weighted mean such thatthe pre-quasi operator ideal formed by the sequence ofapproximation numbers and this sequence space is strictlycontained for different weights and powers

Theorem 15 For any infinite dimensional Banach spaces Xand Y and for any 1ltpn lt qn for all n isin N it is true thatSappces((bn)(pn))(X Y)⫋ S

appces((an)(qn))(X Y)⫋ L(X Y) where (pn)

and (qn) are the monotonic increasing bounded sequences

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appces((bn)(pn))(X Y) then (αn(T)) isin ces((bn) (pn)) One

has

1113944

infin

n0an 1113944

n

k0αk(T)⎛⎝ ⎞⎠

qn

lt 1113944infin

n0bn 1113944

n

k0αk(T)⎛⎝ ⎞⎠

pn

ltinfin (45)

Hence T isin Sappces((an)(qn))(X Y) Next if we take (an) with

sup2(sup qnminus 1)naqnn ltinfin (bminus1

n ) isin ℓ(qnpn) and μn (1bn

pn1113968

) Soby using)eorem 5 and Lemma 2 one can find T isin L(X Y)

with1

16b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (46)

such that T does not belong to Sappces((bn)(pn))(X Y) and

T isin Sappces((an)(qn))(X Y)

8 Journal of Function Spaces

It is easy to see that Sappces((an)(qn))(X Y) sub L(X Y) Next

we take μn (1 an

qnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (47)

such that T does not belong to Sappces((an)(qn))(X Y) )is

finishes the proof

Corollary 18 For any infinite dimensional Banach spaces Xand Y and 1ltplt qltinfin then

Sappcesp

(X Y)⫋ Sappcesq

(X Y)⫋L(X Y) (48)

In this part we give the conditions for which the pre-quasiBanach Operator ideal S

appces((ai)qi)

is small

Theorem 16 If conditions (b1) (b2) and (nan) notin ℓ(pn) aresatisfied then the pre-quasi Banach operator ideal S

appces((ai)pi)

is small

Proof Since (pi) is an increasing sequence with p0 gt 1 and(ai) isin ℓ(pi) take λ (1113936

infini0 a

pi

i )1h )en (Sappces((ai)pi)

g)where g(T) (1λ)(1113936

infini0 (ai 1113936

ij0 αj(T))pi )1h is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appces((ai)pi)

(X Y) L(X Y) then thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y be infinite dimensionalBanach spaces Hence by Dvoretzkyrsquos )eorem [18] form isin N we have quotient spaces XNm and subspaces Mm of

Y which can be mapped onto ℓm2 by isomorphisms Hm and

Am such that HmHminus1m le 2 and AmAminus1

m le 2 Let Im be theidentity map on ℓm

2 Qm be the quotient map from X ontoXNm and Jm be the natural embedding map from Mm intoY Let un be the Bernstein numbers [19] then

1 un Im( 1113857 un AmAminus1m ImHmH

minus1m1113872 1113873

le Am

un A

minus1m ImHm1113872 1113873 H

minus1m

Am

un JmA

minus1m ImHm1113872 1113873 H

minus1m

le Am

dn JmA

minus1m ImHm1113872 1113873 H

minus1m

Am

dn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

le Am

αn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

(49)

for 1le ilem Now

1113944

i

j0ai( 1113857le 1113944

i

j0Am

aiαj JmA

minus1m ImHmQm1113872 1113873 H

minus1m

⟹

(i + 1)ai le Am

ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠ H

minus1m

⟹

(i + 1)ai( 1113857pi le Am

H

minus1m

1113872 1113873

piai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

(50)

)erefore

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1λ

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le 4LC

(51)

Journal of Function Spaces 9

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary and (nan) notin ℓ(pn) )us X and Y both cannot beinfinite dimensional when S

appces((ai)(pi))

(X Y) L(X Y) andhence the result

Theorem 17 If (pi) is an increasing and p0 gt 1 then the pre-quasi Banach operator ideal SKolces(pi)

is small

Corollary 19 If 1ltpltinfin then the quasi Banach operatorideal S

appcesp is small

Corollary 20 If 1ltpltinfin then the quasi Banach operatorideal SKolcesp is small

In this part we give the conditions for which the pre-quasiBanach operator ideal S

appℓ(pn )

β

is small

Theorem 18 If conditions (a1) (a2) and (βn) notin ℓ1 aresatisfied then the pre-quasi Banach operator ideal S

appℓ(pn )

β

issmall

Proof Since conditions (a1) (a2) and (βn) notin ℓ1 are satisfiedthen (S

appℓ(pn )

β

g) where g(T) 1113936infini0 βi(αi(T))pi is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appℓ(pn )

β

(X Y) L(X Y) )en thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y are infinite dimensionalBanach spaces By using inequality (49) and (βn) notin ℓ1 oneobtains

1le Am

H

minus1m

1113872 1113873

pi αi JmAminus1m ImHmQm1113872 11138731113872 1113873

pi⟹

βi le L Am

βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

piH

minus1m

⟹

1113944

m

i0βi le L Am

H

minus1m

1113944

m

i0βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

pi⟹

1113944

m

i0βi le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1113944

m

i0βi le 4LC

(52)

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary )us X and Y both cannot be infinite di-mensional when S

appℓ(pn )

β

(X Y) L(X Y) and hence theresult

Corollary 21 (see [2]) If 0ltpltinfin then the quasi Banachoperator ideal S

appℓp is small

Corollary 22 If 0ltpltinfin then the quasi Banach operatorideal SKolℓp is small

7 Pre-Quasi Simple Banach Operator Ideal

)e following question arises naturally for which weightedNakano sequence space or generalized Cesaro sequencespace defined by weighted mean is the pre-quasi Banachideal simple

Theorem 19 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 Λ Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (53)

Proof Suppose that there exists T isin L(Sℓ(qn )

(an )

Sℓ(pn)

(bn)

) which isnot approximable According to Lemma 1 we can find

X isin L(Sℓ(qn)

(an )

Sℓ(qn )

(an)

) and B isin L(Sℓ(pn )

(bn )

Sℓ(pn )

(bn )

) with BTXIk Ik

)en it follows for all k isin N that

Ik

S

ℓpn( )bn( )

1113944infin

n0bn αn Ik( 1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

1113944kminus 1

n0bn

⎛⎝ ⎞⎠

1suppn

le BTX Ik

S

ℓqn( )an( )

le 1113944infin

n0an αn Ik( 1113857( 1113857

qn⎛⎝ ⎞⎠

1supqn

1113944kminus 1

n0an

⎛⎝ ⎞⎠

1supqn

(54)

But this is impossible

Corollary 23 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 LC Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (55)

Proof Every approximable operator is compact

Theorem 20 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

Λ Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(56)

Proof Suppose that there exists T isin L(Sces((an)(qn))

Sces((bn)(pn))) which is not approximable According toLemma 1 we can find X isin L(Sces((an)(qn)) Sces((an)(qn))) andB isin L(Sces((bn)(pn)) Sces((bn)(pn))) with BTXIk Ik )en itfollows for all k isin N that

10 Journal of Function Spaces

Ik

Sces bn( ) pn( )( )

1113944infin

n0bn 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0bn 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0kbn1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

le BTX Ik

Sces an( ) qn( )( )

le 1113944infin

n0an 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0an 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0kan( 1113857⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

(57)

But this is impossible

Corollary 24 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

LC Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(58)

Theorem 21 For a bounded sequence (pn) with 1lepn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space Sℓ(pn )

(bn )

is simple

Proof Suppose that the closed ideal LC(Sℓ(pn)

(bn)

) contains an

operator Twhich is not approximable According to Lemma

1 we can find X B isin L(Sℓ(pn)

(bn)

) with BTXIk Ik )is means

that ISℓ(pn )

(bn )

isin LC(Sℓ(pn )

(bn )

) Consequently L(Sℓ(pn)

(bn)

) LC(Sℓ(pn)

(bn)

)

)erefore Λ(Sℓ(pn )

(bn )

) is the only nontrivial closed ideal inL(Sℓ(pn )

(bn )

)

Theorem 22 For a bounded sequence (pn) with 1ltpn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space(Sces((bn)(pn))) is simple

Proof Suppose that the closed ideal LC(Sces((bn)(pn))) con-tains an operator Twhich is not approximable According toLemma 1 we can find X B isin L(Sces((bn)(pn))) withBTXIk Ik )is means that ISces((bn)(pn ))

isin LC(Sces((bn)(pn)))Consequently L(Sces((bn)(pn))) LC(Sces((bn)(pn))) )ereforeΛ(Sces((bn)(pn))) is the only nontrivial closed ideal inL(Sces((bn)(pn)))

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to the writing of this paperAll authors read and approved the final manuscript

References

[1] A Pietsch Operator Ideals North-Holland PublishingCompany Amsterdam Netherlands 1980

[2] A Pietsch ldquoSmall ideals of operatorsrdquo Studia Mathematicavol 51 no 3 pp 265ndash267 1974

[3] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbersrdquo in Ieory of Operators inFunctional Spaces pp 206ndash211 Academy of Science SiberianSection Academy of Science Siberian Section NovosibirskRussia 1997 in Russian

[4] N Faried and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 186 2013

[5] N Faried and A A Bakery ldquoSmall operator ideals formed by snumbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1p 357 2018

[6] B E Rhoades ldquoOperators of Andashp typerdquo Atti della AccademiaNazionale dei Lincei Classe di Scienze Fisiche Matematiche eNaturali (VIII) vol 59 no 3-4 pp 238ndash241 1975

[7] A Pietsch Eigenvalues and s-Numbers Cambridge UniversityPress New York NY USA 1986

[8] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 no 4 pp 267ndash278 1974

[9] A Lima and E Oja ldquoIdeals of finite rank operators inter-section properties of balls and the approximation propertyrdquoStudia Mathematica vol 133 pp 175ndash186 1999

[10] S T Chen ldquoGeometry of Orlicz spacesrdquo Dissertations MathInstitute of Mathematics Polish Academy of SciencesWarsaw Poland vol 356 p 356 1996

[11] Y A Cui and H Hudzik ldquoOn the Banach-saks and weakBanach-saks properties of some Banach sequence spacesrdquoActa Scientiarum Mathematicarum vol 65 pp 179ndash1871999

[12] M Sengonul ldquoOn the generalized Nakano sequence spacerdquoIai Journal of Mathematics vol 4 no 2 pp 295ndash304 2006

[13] W Sanhan and S Suantai ldquoOnk-nearly uniform convexproperty in generalized Cesaro sequence spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciencesvol 2003 no 57 pp 3599ndash3607 2003

[14] P Y Lee ldquoCesaro sequence spacesrdquo Mathematical Chroniclevol 13 pp 29ndash45 1984

[15] Y q Lui and P Y LeeMethod of Sequence Spaces Guangdongof science and Technology press Guangdong China 1996 inChinese

[16] J S Shiue ldquoOn the Cesaro sequence spacesrdquo Tamkang Journalof Mathematics vol 1 no 1 pp 19ndash25 1970

[17] B Altay and F Basar ldquoGeneralization of the sequence spaceℓ(p) derived by weighted meansrdquo Journal of MathematicalAnalysis and Applications vol 330 no 1 pp 147ndash185 2007

[18] A Pietsch Operator Ideals VEB Deutscher Verlag derWis-senschaften Berlin Germany 1978

[19] A Pietsch ldquos-numbers of operators in Banach spacesrdquo StudiaMathematica vol 51 no 3 pp 201ndash223 1974

Journal of Function Spaces 11

this sequence space is strictly contained for different weightsand powers

Theorem 14 For any infinite dimensional Banach spaces Xand Y and for any 0ltpn lt qn and 0lt an lt bn for all n isin N it istrue that

Sapp

ℓ pn( )bn( )

(X Y)⫋ Sapp

ℓ qn( )an( )

(X Y)⫋ L(X Y) (39)

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appℓ(pn )

(bn )

(X Y) then (αn(T)) isin ℓ(pn)

(bn) One can see that

1113944

infin

n0an αn(T)( 1113857

qn lt 1113944infin

n0bn αn(T)( 1113857

pn ltinfin (40)

Hence T isin Sappℓ(qn )

(an )

(X Y) Next if we take (an) with

sup an ltinfin (bminus1n ) isin ℓ(qnpn)and μn (1

bn

pn1113968

) So by using)eorem 5 one can find T isin L(X Y) with

116

b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (41)

such that T does not belong to Sappℓ(pn)

(bn)

(X Y) andT isin S

appℓ(qn )

(an )

(X Y)

It is easy to see that Sappℓ(qn)

(an)

(X Y) sub L(X Y) Next if wetake μn (1

anqnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (42)

such that T does not belong to Sappℓ(qn )

(an )

(X Y) )is finishes theproof

Corollary 17 (see [3]) For any infinite dimensional Banachspaces X and Y and for any qgtpgt 0 it is true thatSappℓp (X Y)⫋ S

appℓq (X Y)⫋ L(X Y)

Lemma 2 If (an) is a sequence of positive real with an le 1 forall n isin N (pn) is monotonic increasing bounded sequencewith pn ge 1 for all n isin N or an gt 1 for all n isin N and (pn) ismonotonic decreasing bounded sequence with pn ge 1 for alln isin N one has the following inequality

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k (43)

Proof By using inequality (16) and the sufficient conditionsone has

1113944

n

k0ak

⎛⎝ ⎞⎠

pn

a0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

lt 2suppnminus 1a

pn

0 + 1113944

n

k1ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

2suppnminus 1a

pn

0 + a1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 2suppnminus 1 2suppnminus 1a

pn

0 + 2suppnminus 1a

pn

1 + 1113944

n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

lt 22 suppnminus 1( ) apn

0 + apn

1 + 1113944n

k2ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

lt 23 suppnminus 1( ) apn

0 + apn

1 + apn

2 + 1113944n

k3ak

⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

⋮

lt 2(nminus 1s) suppnminus 1( ) apn

0 + apn

1 + apn

2 + middot middot middot + apn

nminus21113872

+ 1113944n

knminus1ak

⎛⎝ ⎞⎠

pn

⎞⎠

lt 2n suppnminus 1( ) 1113944

n

k0a

pn

k lt 2n suppnminus 1( ) 1113944

n

k0a

pk

k

(44)

We give here the sufficient conditions on the generalizedCesaro sequence space defined by weighted mean such thatthe pre-quasi operator ideal formed by the sequence ofapproximation numbers and this sequence space is strictlycontained for different weights and powers

Theorem 15 For any infinite dimensional Banach spaces Xand Y and for any 1ltpn lt qn for all n isin N it is true thatSappces((bn)(pn))(X Y)⫋ S

appces((an)(qn))(X Y)⫋ L(X Y) where (pn)

and (qn) are the monotonic increasing bounded sequences

Proof Let X and Y be infinite dimensional Banach spacesand for any 0ltpn lt qn and 0lt an lt bn for all n isin N ifT isin S

appces((bn)(pn))(X Y) then (αn(T)) isin ces((bn) (pn)) One

has

1113944

infin

n0an 1113944

n

k0αk(T)⎛⎝ ⎞⎠

qn

lt 1113944infin

n0bn 1113944

n

k0αk(T)⎛⎝ ⎞⎠

pn

ltinfin (45)

Hence T isin Sappces((an)(qn))(X Y) Next if we take (an) with

sup2(sup qnminus 1)naqnn ltinfin (bminus1

n ) isin ℓ(qnpn) and μn (1bn

pn1113968

) Soby using)eorem 5 and Lemma 2 one can find T isin L(X Y)

with1

16b3n

p3n1113968 le αn(T)le

8bn+1

pn+11113968 (46)

such that T does not belong to Sappces((bn)(pn))(X Y) and

T isin Sappces((an)(qn))(X Y)

8 Journal of Function Spaces

It is easy to see that Sappces((an)(qn))(X Y) sub L(X Y) Next

we take μn (1 an

qnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (47)

such that T does not belong to Sappces((an)(qn))(X Y) )is

finishes the proof

Corollary 18 For any infinite dimensional Banach spaces Xand Y and 1ltplt qltinfin then

Sappcesp

(X Y)⫋ Sappcesq

(X Y)⫋L(X Y) (48)

In this part we give the conditions for which the pre-quasiBanach Operator ideal S

appces((ai)qi)

is small

Theorem 16 If conditions (b1) (b2) and (nan) notin ℓ(pn) aresatisfied then the pre-quasi Banach operator ideal S

appces((ai)pi)

is small

Proof Since (pi) is an increasing sequence with p0 gt 1 and(ai) isin ℓ(pi) take λ (1113936

infini0 a

pi

i )1h )en (Sappces((ai)pi)

g)where g(T) (1λ)(1113936

infini0 (ai 1113936

ij0 αj(T))pi )1h is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appces((ai)pi)

(X Y) L(X Y) then thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y be infinite dimensionalBanach spaces Hence by Dvoretzkyrsquos )eorem [18] form isin N we have quotient spaces XNm and subspaces Mm of

Y which can be mapped onto ℓm2 by isomorphisms Hm and

Am such that HmHminus1m le 2 and AmAminus1

m le 2 Let Im be theidentity map on ℓm

2 Qm be the quotient map from X ontoXNm and Jm be the natural embedding map from Mm intoY Let un be the Bernstein numbers [19] then

1 un Im( 1113857 un AmAminus1m ImHmH

minus1m1113872 1113873

le Am

un A

minus1m ImHm1113872 1113873 H

minus1m

Am

un JmA

minus1m ImHm1113872 1113873 H

minus1m

le Am

dn JmA

minus1m ImHm1113872 1113873 H

minus1m

Am

dn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

le Am

αn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

(49)

for 1le ilem Now

1113944

i

j0ai( 1113857le 1113944

i

j0Am

aiαj JmA

minus1m ImHmQm1113872 1113873 H

minus1m

⟹

(i + 1)ai le Am

ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠ H

minus1m

⟹

(i + 1)ai( 1113857pi le Am

H

minus1m

1113872 1113873

piai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

(50)

)erefore

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1λ

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le 4LC

(51)

Journal of Function Spaces 9

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary and (nan) notin ℓ(pn) )us X and Y both cannot beinfinite dimensional when S

appces((ai)(pi))

(X Y) L(X Y) andhence the result

Theorem 17 If (pi) is an increasing and p0 gt 1 then the pre-quasi Banach operator ideal SKolces(pi)

is small

Corollary 19 If 1ltpltinfin then the quasi Banach operatorideal S

appcesp is small

Corollary 20 If 1ltpltinfin then the quasi Banach operatorideal SKolcesp is small

In this part we give the conditions for which the pre-quasiBanach operator ideal S

appℓ(pn )

β

is small

Theorem 18 If conditions (a1) (a2) and (βn) notin ℓ1 aresatisfied then the pre-quasi Banach operator ideal S

appℓ(pn )

β

issmall

Proof Since conditions (a1) (a2) and (βn) notin ℓ1 are satisfiedthen (S

appℓ(pn )

β

g) where g(T) 1113936infini0 βi(αi(T))pi is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appℓ(pn )

β

(X Y) L(X Y) )en thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y are infinite dimensionalBanach spaces By using inequality (49) and (βn) notin ℓ1 oneobtains

1le Am

H

minus1m

1113872 1113873

pi αi JmAminus1m ImHmQm1113872 11138731113872 1113873

pi⟹

βi le L Am

βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

piH

minus1m

⟹

1113944

m

i0βi le L Am

H

minus1m

1113944

m

i0βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

pi⟹

1113944

m

i0βi le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1113944

m

i0βi le 4LC

(52)

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary )us X and Y both cannot be infinite di-mensional when S

appℓ(pn )

β

(X Y) L(X Y) and hence theresult

Corollary 21 (see [2]) If 0ltpltinfin then the quasi Banachoperator ideal S

appℓp is small

Corollary 22 If 0ltpltinfin then the quasi Banach operatorideal SKolℓp is small

7 Pre-Quasi Simple Banach Operator Ideal

)e following question arises naturally for which weightedNakano sequence space or generalized Cesaro sequencespace defined by weighted mean is the pre-quasi Banachideal simple

Theorem 19 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 Λ Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (53)

Proof Suppose that there exists T isin L(Sℓ(qn )

(an )

Sℓ(pn)

(bn)

) which isnot approximable According to Lemma 1 we can find

X isin L(Sℓ(qn)

(an )

Sℓ(qn )

(an)

) and B isin L(Sℓ(pn )

(bn )

Sℓ(pn )

(bn )

) with BTXIk Ik

)en it follows for all k isin N that

Ik

S

ℓpn( )bn( )

1113944infin

n0bn αn Ik( 1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

1113944kminus 1

n0bn

⎛⎝ ⎞⎠

1suppn

le BTX Ik

S

ℓqn( )an( )

le 1113944infin

n0an αn Ik( 1113857( 1113857

qn⎛⎝ ⎞⎠

1supqn

1113944kminus 1

n0an

⎛⎝ ⎞⎠

1supqn

(54)

But this is impossible

Corollary 23 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 LC Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (55)

Proof Every approximable operator is compact

Theorem 20 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

Λ Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(56)

Proof Suppose that there exists T isin L(Sces((an)(qn))

Sces((bn)(pn))) which is not approximable According toLemma 1 we can find X isin L(Sces((an)(qn)) Sces((an)(qn))) andB isin L(Sces((bn)(pn)) Sces((bn)(pn))) with BTXIk Ik )en itfollows for all k isin N that

10 Journal of Function Spaces

Ik

Sces bn( ) pn( )( )

1113944infin

n0bn 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0bn 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0kbn1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

le BTX Ik

Sces an( ) qn( )( )

le 1113944infin

n0an 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0an 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0kan( 1113857⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

(57)

But this is impossible

Corollary 24 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

LC Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(58)

Theorem 21 For a bounded sequence (pn) with 1lepn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space Sℓ(pn )

(bn )

is simple

Proof Suppose that the closed ideal LC(Sℓ(pn)

(bn)

) contains an

operator Twhich is not approximable According to Lemma

1 we can find X B isin L(Sℓ(pn)

(bn)

) with BTXIk Ik )is means

that ISℓ(pn )

(bn )

isin LC(Sℓ(pn )

(bn )

) Consequently L(Sℓ(pn)

(bn)

) LC(Sℓ(pn)

(bn)

)

)erefore Λ(Sℓ(pn )

(bn )

) is the only nontrivial closed ideal inL(Sℓ(pn )

(bn )

)

Theorem 22 For a bounded sequence (pn) with 1ltpn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space(Sces((bn)(pn))) is simple

Proof Suppose that the closed ideal LC(Sces((bn)(pn))) con-tains an operator Twhich is not approximable According toLemma 1 we can find X B isin L(Sces((bn)(pn))) withBTXIk Ik )is means that ISces((bn)(pn ))

isin LC(Sces((bn)(pn)))Consequently L(Sces((bn)(pn))) LC(Sces((bn)(pn))) )ereforeΛ(Sces((bn)(pn))) is the only nontrivial closed ideal inL(Sces((bn)(pn)))

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to the writing of this paperAll authors read and approved the final manuscript

References

[1] A Pietsch Operator Ideals North-Holland PublishingCompany Amsterdam Netherlands 1980

[2] A Pietsch ldquoSmall ideals of operatorsrdquo Studia Mathematicavol 51 no 3 pp 265ndash267 1974

[3] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbersrdquo in Ieory of Operators inFunctional Spaces pp 206ndash211 Academy of Science SiberianSection Academy of Science Siberian Section NovosibirskRussia 1997 in Russian

[4] N Faried and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 186 2013

[5] N Faried and A A Bakery ldquoSmall operator ideals formed by snumbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1p 357 2018

[6] B E Rhoades ldquoOperators of Andashp typerdquo Atti della AccademiaNazionale dei Lincei Classe di Scienze Fisiche Matematiche eNaturali (VIII) vol 59 no 3-4 pp 238ndash241 1975

[7] A Pietsch Eigenvalues and s-Numbers Cambridge UniversityPress New York NY USA 1986

[8] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 no 4 pp 267ndash278 1974

[9] A Lima and E Oja ldquoIdeals of finite rank operators inter-section properties of balls and the approximation propertyrdquoStudia Mathematica vol 133 pp 175ndash186 1999

[10] S T Chen ldquoGeometry of Orlicz spacesrdquo Dissertations MathInstitute of Mathematics Polish Academy of SciencesWarsaw Poland vol 356 p 356 1996

[11] Y A Cui and H Hudzik ldquoOn the Banach-saks and weakBanach-saks properties of some Banach sequence spacesrdquoActa Scientiarum Mathematicarum vol 65 pp 179ndash1871999

[12] M Sengonul ldquoOn the generalized Nakano sequence spacerdquoIai Journal of Mathematics vol 4 no 2 pp 295ndash304 2006

[13] W Sanhan and S Suantai ldquoOnk-nearly uniform convexproperty in generalized Cesaro sequence spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciencesvol 2003 no 57 pp 3599ndash3607 2003

[14] P Y Lee ldquoCesaro sequence spacesrdquo Mathematical Chroniclevol 13 pp 29ndash45 1984

[15] Y q Lui and P Y LeeMethod of Sequence Spaces Guangdongof science and Technology press Guangdong China 1996 inChinese

[16] J S Shiue ldquoOn the Cesaro sequence spacesrdquo Tamkang Journalof Mathematics vol 1 no 1 pp 19ndash25 1970

[17] B Altay and F Basar ldquoGeneralization of the sequence spaceℓ(p) derived by weighted meansrdquo Journal of MathematicalAnalysis and Applications vol 330 no 1 pp 147ndash185 2007

[18] A Pietsch Operator Ideals VEB Deutscher Verlag derWis-senschaften Berlin Germany 1978

[19] A Pietsch ldquos-numbers of operators in Banach spacesrdquo StudiaMathematica vol 51 no 3 pp 201ndash223 1974

Journal of Function Spaces 11

It is easy to see that Sappces((an)(qn))(X Y) sub L(X Y) Next

we take μn (1 an

qnradic

) So by using )eorem 5 one can findT isin L(X Y) with

116

a3nq3nradic le αn(T)le

8an+1

qn+1radic (47)

such that T does not belong to Sappces((an)(qn))(X Y) )is

finishes the proof

Corollary 18 For any infinite dimensional Banach spaces Xand Y and 1ltplt qltinfin then

Sappcesp

(X Y)⫋ Sappcesq

(X Y)⫋L(X Y) (48)

In this part we give the conditions for which the pre-quasiBanach Operator ideal S

appces((ai)qi)

is small

Theorem 16 If conditions (b1) (b2) and (nan) notin ℓ(pn) aresatisfied then the pre-quasi Banach operator ideal S

appces((ai)pi)

is small

Proof Since (pi) is an increasing sequence with p0 gt 1 and(ai) isin ℓ(pi) take λ (1113936

infini0 a

pi

i )1h )en (Sappces((ai)pi)

g)where g(T) (1λ)(1113936

infini0 (ai 1113936

ij0 αj(T))pi )1h is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appces((ai)pi)

(X Y) L(X Y) then thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y be infinite dimensionalBanach spaces Hence by Dvoretzkyrsquos )eorem [18] form isin N we have quotient spaces XNm and subspaces Mm of

Y which can be mapped onto ℓm2 by isomorphisms Hm and

Am such that HmHminus1m le 2 and AmAminus1

m le 2 Let Im be theidentity map on ℓm

2 Qm be the quotient map from X ontoXNm and Jm be the natural embedding map from Mm intoY Let un be the Bernstein numbers [19] then

1 un Im( 1113857 un AmAminus1m ImHmH

minus1m1113872 1113873

le Am

un A

minus1m ImHm1113872 1113873 H

minus1m

Am

un JmA

minus1m ImHm1113872 1113873 H

minus1m

le Am

dn JmA

minus1m ImHm1113872 1113873 H

minus1m

Am

dn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

le Am

αn JmA

minus1m ImHmQm1113872 1113873 H

minus1m

(49)

for 1le ilem Now

1113944

i

j0ai( 1113857le 1113944

i

j0Am

aiαj JmA

minus1m ImHmQm1113872 1113873 H

minus1m

⟹

(i + 1)ai le Am

ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠ H

minus1m

⟹

(i + 1)ai( 1113857pi le Am

H

minus1m

1113872 1113873

piai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

(50)

)erefore

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

1λ

1113944

m

i0ai 1113944

i

j0αj JmA

minus1m ImHmQm1113872 1113873⎛⎝ ⎞⎠

pi

⎡⎢⎢⎣ ⎤⎥⎥⎦

1h

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1λ

1113944

m

i0(i + 1)ai( 1113857

pi⎛⎝ ⎞⎠

1h

le 4LC

(51)

Journal of Function Spaces 9

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary and (nan) notin ℓ(pn) )us X and Y both cannot beinfinite dimensional when S

appces((ai)(pi))

(X Y) L(X Y) andhence the result

Theorem 17 If (pi) is an increasing and p0 gt 1 then the pre-quasi Banach operator ideal SKolces(pi)

is small

Corollary 19 If 1ltpltinfin then the quasi Banach operatorideal S

appcesp is small

Corollary 20 If 1ltpltinfin then the quasi Banach operatorideal SKolcesp is small

In this part we give the conditions for which the pre-quasiBanach operator ideal S

appℓ(pn )

β

is small

Theorem 18 If conditions (a1) (a2) and (βn) notin ℓ1 aresatisfied then the pre-quasi Banach operator ideal S

appℓ(pn )

β

issmall

Proof Since conditions (a1) (a2) and (βn) notin ℓ1 are satisfiedthen (S

appℓ(pn )

β

g) where g(T) 1113936infini0 βi(αi(T))pi is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appℓ(pn )

β

(X Y) L(X Y) )en thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y are infinite dimensionalBanach spaces By using inequality (49) and (βn) notin ℓ1 oneobtains

1le Am

H

minus1m

1113872 1113873

pi αi JmAminus1m ImHmQm1113872 11138731113872 1113873

pi⟹

βi le L Am

βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

piH

minus1m

⟹

1113944

m

i0βi le L Am

H

minus1m

1113944

m

i0βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

pi⟹

1113944

m

i0βi le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1113944

m

i0βi le 4LC

(52)

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary )us X and Y both cannot be infinite di-mensional when S

appℓ(pn )

β

(X Y) L(X Y) and hence theresult

Corollary 21 (see [2]) If 0ltpltinfin then the quasi Banachoperator ideal S

appℓp is small

Corollary 22 If 0ltpltinfin then the quasi Banach operatorideal SKolℓp is small

7 Pre-Quasi Simple Banach Operator Ideal

)e following question arises naturally for which weightedNakano sequence space or generalized Cesaro sequencespace defined by weighted mean is the pre-quasi Banachideal simple

Theorem 19 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 Λ Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (53)

Proof Suppose that there exists T isin L(Sℓ(qn )

(an )

Sℓ(pn)

(bn)

) which isnot approximable According to Lemma 1 we can find

X isin L(Sℓ(qn)

(an )

Sℓ(qn )

(an)

) and B isin L(Sℓ(pn )

(bn )

Sℓ(pn )

(bn )

) with BTXIk Ik

)en it follows for all k isin N that

Ik

S

ℓpn( )bn( )

1113944infin

n0bn αn Ik( 1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

1113944kminus 1

n0bn

⎛⎝ ⎞⎠

1suppn

le BTX Ik

S

ℓqn( )an( )

le 1113944infin

n0an αn Ik( 1113857( 1113857

qn⎛⎝ ⎞⎠

1supqn

1113944kminus 1

n0an

⎛⎝ ⎞⎠

1supqn

(54)

But this is impossible

Corollary 23 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 LC Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (55)

Proof Every approximable operator is compact

Theorem 20 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

Λ Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(56)

Proof Suppose that there exists T isin L(Sces((an)(qn))

Sces((bn)(pn))) which is not approximable According toLemma 1 we can find X isin L(Sces((an)(qn)) Sces((an)(qn))) andB isin L(Sces((bn)(pn)) Sces((bn)(pn))) with BTXIk Ik )en itfollows for all k isin N that

10 Journal of Function Spaces

Ik

Sces bn( ) pn( )( )

1113944infin

n0bn 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0bn 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0kbn1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

le BTX Ik

Sces an( ) qn( )( )

le 1113944infin

n0an 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0an 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0kan( 1113857⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

(57)

But this is impossible

Corollary 24 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

LC Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(58)

Theorem 21 For a bounded sequence (pn) with 1lepn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space Sℓ(pn )

(bn )

is simple

Proof Suppose that the closed ideal LC(Sℓ(pn)

(bn)

) contains an

operator Twhich is not approximable According to Lemma

1 we can find X B isin L(Sℓ(pn)

(bn)

) with BTXIk Ik )is means

that ISℓ(pn )

(bn )

isin LC(Sℓ(pn )

(bn )

) Consequently L(Sℓ(pn)

(bn)

) LC(Sℓ(pn)

(bn)

)

)erefore Λ(Sℓ(pn )

(bn )

) is the only nontrivial closed ideal inL(Sℓ(pn )

(bn )

)

Theorem 22 For a bounded sequence (pn) with 1ltpn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space(Sces((bn)(pn))) is simple

Proof Suppose that the closed ideal LC(Sces((bn)(pn))) con-tains an operator Twhich is not approximable According toLemma 1 we can find X B isin L(Sces((bn)(pn))) withBTXIk Ik )is means that ISces((bn)(pn ))

isin LC(Sces((bn)(pn)))Consequently L(Sces((bn)(pn))) LC(Sces((bn)(pn))) )ereforeΛ(Sces((bn)(pn))) is the only nontrivial closed ideal inL(Sces((bn)(pn)))

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to the writing of this paperAll authors read and approved the final manuscript

References

[1] A Pietsch Operator Ideals North-Holland PublishingCompany Amsterdam Netherlands 1980

[2] A Pietsch ldquoSmall ideals of operatorsrdquo Studia Mathematicavol 51 no 3 pp 265ndash267 1974

[3] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbersrdquo in Ieory of Operators inFunctional Spaces pp 206ndash211 Academy of Science SiberianSection Academy of Science Siberian Section NovosibirskRussia 1997 in Russian

[4] N Faried and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 186 2013

[5] N Faried and A A Bakery ldquoSmall operator ideals formed by snumbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1p 357 2018

[6] B E Rhoades ldquoOperators of Andashp typerdquo Atti della AccademiaNazionale dei Lincei Classe di Scienze Fisiche Matematiche eNaturali (VIII) vol 59 no 3-4 pp 238ndash241 1975

[7] A Pietsch Eigenvalues and s-Numbers Cambridge UniversityPress New York NY USA 1986

[8] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 no 4 pp 267ndash278 1974

[9] A Lima and E Oja ldquoIdeals of finite rank operators inter-section properties of balls and the approximation propertyrdquoStudia Mathematica vol 133 pp 175ndash186 1999

[10] S T Chen ldquoGeometry of Orlicz spacesrdquo Dissertations MathInstitute of Mathematics Polish Academy of SciencesWarsaw Poland vol 356 p 356 1996

[11] Y A Cui and H Hudzik ldquoOn the Banach-saks and weakBanach-saks properties of some Banach sequence spacesrdquoActa Scientiarum Mathematicarum vol 65 pp 179ndash1871999

[12] M Sengonul ldquoOn the generalized Nakano sequence spacerdquoIai Journal of Mathematics vol 4 no 2 pp 295ndash304 2006

[13] W Sanhan and S Suantai ldquoOnk-nearly uniform convexproperty in generalized Cesaro sequence spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciencesvol 2003 no 57 pp 3599ndash3607 2003

[14] P Y Lee ldquoCesaro sequence spacesrdquo Mathematical Chroniclevol 13 pp 29ndash45 1984

[15] Y q Lui and P Y LeeMethod of Sequence Spaces Guangdongof science and Technology press Guangdong China 1996 inChinese

[16] J S Shiue ldquoOn the Cesaro sequence spacesrdquo Tamkang Journalof Mathematics vol 1 no 1 pp 19ndash25 1970

[17] B Altay and F Basar ldquoGeneralization of the sequence spaceℓ(p) derived by weighted meansrdquo Journal of MathematicalAnalysis and Applications vol 330 no 1 pp 147ndash185 2007

[18] A Pietsch Operator Ideals VEB Deutscher Verlag derWis-senschaften Berlin Germany 1978

[19] A Pietsch ldquos-numbers of operators in Banach spacesrdquo StudiaMathematica vol 51 no 3 pp 201ndash223 1974

Journal of Function Spaces 11

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary and (nan) notin ℓ(pn) )us X and Y both cannot beinfinite dimensional when S

appces((ai)(pi))

(X Y) L(X Y) andhence the result

Theorem 17 If (pi) is an increasing and p0 gt 1 then the pre-quasi Banach operator ideal SKolces(pi)

is small

Corollary 19 If 1ltpltinfin then the quasi Banach operatorideal S

appcesp is small

Corollary 20 If 1ltpltinfin then the quasi Banach operatorideal SKolcesp is small

In this part we give the conditions for which the pre-quasiBanach operator ideal S

appℓ(pn )

β

is small

Theorem 18 If conditions (a1) (a2) and (βn) notin ℓ1 aresatisfied then the pre-quasi Banach operator ideal S

appℓ(pn )

β

issmall

Proof Since conditions (a1) (a2) and (βn) notin ℓ1 are satisfiedthen (S

appℓ(pn )

β

g) where g(T) 1113936infini0 βi(αi(T))pi is a pre-quasi

Banach operator ideal Let X and Y be any two Banachspaces Suppose that S

appℓ(pn )

β

(X Y) L(X Y) )en thereexists a constant Cgt 0 such that g(T)leCT for allT isin L(X Y) Assume that X and Y are infinite dimensionalBanach spaces By using inequality (49) and (βn) notin ℓ1 oneobtains

1le Am

H

minus1m

1113872 1113873

pi αi JmAminus1m ImHmQm1113872 11138731113872 1113873

pi⟹

βi le L Am

βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

piH

minus1m

⟹

1113944

m

i0βi le L Am

H

minus1m

1113944

m

i0βi αi JmA

minus1m ImHmQm1113872 11138731113872 1113873

pi⟹

1113944

m

i0βi le L Am

H

minus1m

g JmA

minus1m ImHmQm1113872 1113873⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m ImHmQm

⟹

1113944

m

i0βi le LC Am

H

minus1m

JmA

minus1m

Im

HmQm

LC Am

H

minus1m

A

minus1m

Im

Hm

⟹

1113944

m

i0βi le 4LC

(52)

for some Lge 1 )us we arrive at a contradiction since m isan arbitrary )us X and Y both cannot be infinite di-mensional when S

appℓ(pn )

β

(X Y) L(X Y) and hence theresult

Corollary 21 (see [2]) If 0ltpltinfin then the quasi Banachoperator ideal S

appℓp is small

Corollary 22 If 0ltpltinfin then the quasi Banach operatorideal SKolℓp is small

7 Pre-Quasi Simple Banach Operator Ideal

)e following question arises naturally for which weightedNakano sequence space or generalized Cesaro sequencespace defined by weighted mean is the pre-quasi Banachideal simple

Theorem 19 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 Λ Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (53)

Proof Suppose that there exists T isin L(Sℓ(qn )

(an )

Sℓ(pn)

(bn)

) which isnot approximable According to Lemma 1 we can find

X isin L(Sℓ(qn)

(an )

Sℓ(qn )

(an)

) and B isin L(Sℓ(pn )

(bn )

Sℓ(pn )

(bn )

) with BTXIk Ik

)en it follows for all k isin N that

Ik

S

ℓpn( )bn( )

1113944infin

n0bn αn Ik( 1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

1113944kminus 1

n0bn

⎛⎝ ⎞⎠

1suppn

le BTX Ik

S

ℓqn( )an( )

le 1113944infin

n0an αn Ik( 1113857( 1113857

qn⎛⎝ ⎞⎠

1supqn

1113944kminus 1

n0an

⎛⎝ ⎞⎠

1supqn

(54)

But this is impossible

Corollary 23 If (pn) and (qn) are bounded sequences with1lepn lt qn and 0lt an lt bn for all n isin N then

L Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 LC Sℓ qn( )

an( )

Sℓ pn( )

bn( )

1113888 1113889 (55)

Proof Every approximable operator is compact

Theorem 20 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

Λ Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(56)

Proof Suppose that there exists T isin L(Sces((an)(qn))

Sces((bn)(pn))) which is not approximable According toLemma 1 we can find X isin L(Sces((an)(qn)) Sces((an)(qn))) andB isin L(Sces((bn)(pn)) Sces((bn)(pn))) with BTXIk Ik )en itfollows for all k isin N that

10 Journal of Function Spaces

Ik

Sces bn( ) pn( )( )

1113944infin

n0bn 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0bn 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0kbn1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

le BTX Ik

Sces an( ) qn( )( )

le 1113944infin

n0an 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0an 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0kan( 1113857⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

(57)

But this is impossible

Corollary 24 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

LC Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(58)

Theorem 21 For a bounded sequence (pn) with 1lepn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space Sℓ(pn )

(bn )

is simple

Proof Suppose that the closed ideal LC(Sℓ(pn)

(bn)

) contains an

operator Twhich is not approximable According to Lemma

1 we can find X B isin L(Sℓ(pn)

(bn)

) with BTXIk Ik )is means

that ISℓ(pn )

(bn )

isin LC(Sℓ(pn )

(bn )

) Consequently L(Sℓ(pn)

(bn)

) LC(Sℓ(pn)

(bn)

)

)erefore Λ(Sℓ(pn )

(bn )

) is the only nontrivial closed ideal inL(Sℓ(pn )

(bn )

)

Theorem 22 For a bounded sequence (pn) with 1ltpn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space(Sces((bn)(pn))) is simple

Proof Suppose that the closed ideal LC(Sces((bn)(pn))) con-tains an operator Twhich is not approximable According toLemma 1 we can find X B isin L(Sces((bn)(pn))) withBTXIk Ik )is means that ISces((bn)(pn ))

isin LC(Sces((bn)(pn)))Consequently L(Sces((bn)(pn))) LC(Sces((bn)(pn))) )ereforeΛ(Sces((bn)(pn))) is the only nontrivial closed ideal inL(Sces((bn)(pn)))

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to the writing of this paperAll authors read and approved the final manuscript

References

[1] A Pietsch Operator Ideals North-Holland PublishingCompany Amsterdam Netherlands 1980

[2] A Pietsch ldquoSmall ideals of operatorsrdquo Studia Mathematicavol 51 no 3 pp 265ndash267 1974

[3] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbersrdquo in Ieory of Operators inFunctional Spaces pp 206ndash211 Academy of Science SiberianSection Academy of Science Siberian Section NovosibirskRussia 1997 in Russian

[4] N Faried and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 186 2013

[5] N Faried and A A Bakery ldquoSmall operator ideals formed by snumbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1p 357 2018

[6] B E Rhoades ldquoOperators of Andashp typerdquo Atti della AccademiaNazionale dei Lincei Classe di Scienze Fisiche Matematiche eNaturali (VIII) vol 59 no 3-4 pp 238ndash241 1975

[7] A Pietsch Eigenvalues and s-Numbers Cambridge UniversityPress New York NY USA 1986

[8] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 no 4 pp 267ndash278 1974

[9] A Lima and E Oja ldquoIdeals of finite rank operators inter-section properties of balls and the approximation propertyrdquoStudia Mathematica vol 133 pp 175ndash186 1999

[10] S T Chen ldquoGeometry of Orlicz spacesrdquo Dissertations MathInstitute of Mathematics Polish Academy of SciencesWarsaw Poland vol 356 p 356 1996

[11] Y A Cui and H Hudzik ldquoOn the Banach-saks and weakBanach-saks properties of some Banach sequence spacesrdquoActa Scientiarum Mathematicarum vol 65 pp 179ndash1871999

[12] M Sengonul ldquoOn the generalized Nakano sequence spacerdquoIai Journal of Mathematics vol 4 no 2 pp 295ndash304 2006

[13] W Sanhan and S Suantai ldquoOnk-nearly uniform convexproperty in generalized Cesaro sequence spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciencesvol 2003 no 57 pp 3599ndash3607 2003

[14] P Y Lee ldquoCesaro sequence spacesrdquo Mathematical Chroniclevol 13 pp 29ndash45 1984

[15] Y q Lui and P Y LeeMethod of Sequence Spaces Guangdongof science and Technology press Guangdong China 1996 inChinese

[16] J S Shiue ldquoOn the Cesaro sequence spacesrdquo Tamkang Journalof Mathematics vol 1 no 1 pp 19ndash25 1970

[17] B Altay and F Basar ldquoGeneralization of the sequence spaceℓ(p) derived by weighted meansrdquo Journal of MathematicalAnalysis and Applications vol 330 no 1 pp 147ndash185 2007

[18] A Pietsch Operator Ideals VEB Deutscher Verlag derWis-senschaften Berlin Germany 1978

[19] A Pietsch ldquos-numbers of operators in Banach spacesrdquo StudiaMathematica vol 51 no 3 pp 201ndash223 1974

Journal of Function Spaces 11

Ik

Sces bn( ) pn( )( )

1113944infin

n0bn 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0bn 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

pn

⎛⎝ ⎞⎠

1sup pn

1113944infin

n0kbn1113857( 1113857

pn⎛⎝ ⎞⎠

1sup pn

le BTX Ik

Sces an( ) qn( )( )

le 1113944infin

n0an 1113944

n

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0an 1113944

kminus 1

i0αi Ik( 1113857⎛⎝ ⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

1113944infin

n0kan( 1113857⎞⎠

qn

⎛⎝ ⎞⎠

1sup qn

(57)

But this is impossible

Corollary 24 If (pn) and (qn) are bounded sequences with1ltpn lt qn and 0lt an lt bn for all n isin N then

L Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

LC Sces an( ) qn( )( ) Sces bn( ) pn( )( )1113874 1113875

(58)

Theorem 21 For a bounded sequence (pn) with 1lepn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space Sℓ(pn )

(bn )

is simple

Proof Suppose that the closed ideal LC(Sℓ(pn)

(bn)

) contains an

operator Twhich is not approximable According to Lemma

1 we can find X B isin L(Sℓ(pn)

(bn)

) with BTXIk Ik )is means

that ISℓ(pn )

(bn )

isin LC(Sℓ(pn )

(bn )

) Consequently L(Sℓ(pn)

(bn)

) LC(Sℓ(pn)

(bn)

)

)erefore Λ(Sℓ(pn )

(bn )

) is the only nontrivial closed ideal inL(Sℓ(pn )

(bn )

)

Theorem 22 For a bounded sequence (pn) with 1ltpn ltinfinand bn gt 0 for all n isin N the pre-quasi Banach space(Sces((bn)(pn))) is simple

Proof Suppose that the closed ideal LC(Sces((bn)(pn))) con-tains an operator Twhich is not approximable According toLemma 1 we can find X B isin L(Sces((bn)(pn))) withBTXIk Ik )is means that ISces((bn)(pn ))

isin LC(Sces((bn)(pn)))Consequently L(Sces((bn)(pn))) LC(Sces((bn)(pn))) )ereforeΛ(Sces((bn)(pn))) is the only nontrivial closed ideal inL(Sces((bn)(pn)))

Data Availability

No data were used to support this study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to the writing of this paperAll authors read and approved the final manuscript

References

[1] A Pietsch Operator Ideals North-Holland PublishingCompany Amsterdam Netherlands 1980

[2] A Pietsch ldquoSmall ideals of operatorsrdquo Studia Mathematicavol 51 no 3 pp 265ndash267 1974

[3] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbersrdquo in Ieory of Operators inFunctional Spaces pp 206ndash211 Academy of Science SiberianSection Academy of Science Siberian Section NovosibirskRussia 1997 in Russian

[4] N Faried and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalitiesand Applications vol 2013 no 1 p 186 2013

[5] N Faried and A A Bakery ldquoSmall operator ideals formed by snumbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1p 357 2018

[6] B E Rhoades ldquoOperators of Andashp typerdquo Atti della AccademiaNazionale dei Lincei Classe di Scienze Fisiche Matematiche eNaturali (VIII) vol 59 no 3-4 pp 238ndash241 1975

[7] A Pietsch Eigenvalues and s-Numbers Cambridge UniversityPress New York NY USA 1986

[8] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 no 4 pp 267ndash278 1974

[9] A Lima and E Oja ldquoIdeals of finite rank operators inter-section properties of balls and the approximation propertyrdquoStudia Mathematica vol 133 pp 175ndash186 1999

[10] S T Chen ldquoGeometry of Orlicz spacesrdquo Dissertations MathInstitute of Mathematics Polish Academy of SciencesWarsaw Poland vol 356 p 356 1996

[11] Y A Cui and H Hudzik ldquoOn the Banach-saks and weakBanach-saks properties of some Banach sequence spacesrdquoActa Scientiarum Mathematicarum vol 65 pp 179ndash1871999

[12] M Sengonul ldquoOn the generalized Nakano sequence spacerdquoIai Journal of Mathematics vol 4 no 2 pp 295ndash304 2006

[13] W Sanhan and S Suantai ldquoOnk-nearly uniform convexproperty in generalized Cesaro sequence spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciencesvol 2003 no 57 pp 3599ndash3607 2003

[14] P Y Lee ldquoCesaro sequence spacesrdquo Mathematical Chroniclevol 13 pp 29ndash45 1984

[15] Y q Lui and P Y LeeMethod of Sequence Spaces Guangdongof science and Technology press Guangdong China 1996 inChinese

[16] J S Shiue ldquoOn the Cesaro sequence spacesrdquo Tamkang Journalof Mathematics vol 1 no 1 pp 19ndash25 1970

[17] B Altay and F Basar ldquoGeneralization of the sequence spaceℓ(p) derived by weighted meansrdquo Journal of MathematicalAnalysis and Applications vol 330 no 1 pp 147ndash185 2007

[18] A Pietsch Operator Ideals VEB Deutscher Verlag derWis-senschaften Berlin Germany 1978

[19] A Pietsch ldquos-numbers of operators in Banach spacesrdquo StudiaMathematica vol 51 no 3 pp 201ndash223 1974

Journal of Function Spaces 11