Research ArticleSmall Pre-Quasi Banach Operator Ideals of Type Orlicz-CesaacuteroMean Sequence Spaces

Awad A Bakery 12 andMustafa M Mohammed 13

1Department of Mathematics Faculty of Science and Arts University of Jeddah(UJ) PO Box 355 Code 21921 Khulais Saudi Arabia2Department of Mathematics Faculty of Science Ain Shams University PO Box 1156 Abbassia Cairo 11566 Egypt3Department of Statistics Faculty of Science Sudan University of Science amp Technology Khartoum Sudan

Correspondence should be addressed to Mustafa M Mohammed mustastagmailcom

Received 11 February 2019 Accepted 15 April 2019 Published 2 May 2019

Guest Editor Tuncer Acar

Copyright copy 2019 Awad A Bakery and Mustafa M Mohammed This 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

In this paper we give the sufficient conditions on Orlicz-Cesaro mean sequence spaces 119888119890119904120593 where 120593 is an Orlicz function such thatthe class 119878119888119890119904120593 of all bounded linear operators between arbitrary Banach spaces with its sequence of 119904minusnumbers which belong to119888119890119904120593 forms an operator ideal The completeness and denseness of its ideal components are specified and 119878119888119890119904120593 constructs a pre-quasiBanach operator ideal Some inclusion relations between the pre-quasi operator ideals and the inclusion relations for their duals areexplained Moreover we have presented the sufficient conditions on 119888119890119904120593 such that the pre-quasi Banach operator ideal generatedby approximation number is small The above results coincide with that known for 119888119890119904119901 (1 lt 119901 lt infin)

1 Introduction

Throughout the paper by 119908 we mean the space of all realsequences R the real numbers and N = 0 1 2 andL(119883 119884) the space of all bounded linear operators from anormed space 119883 into a normed space 119884 The operator idealstheory takes an importance in functional analysis since ithas numerous applications in fixed point theorem geometryof Banach spaces spectral theory eigenvalue distributionstheorem etc Some of the operator ideals in the class ofnormed spaces or Banach spaces in functional analysis arecharacterized by various scalar sequence spaces For examplethe ideal of compact operators is defined by kolmogorovnumbers and the space 1198880 of convergent to zero sequencesPietsch [1] inspected the operator ideals framed by theapproximation numbers and the classical sequence spaceℓ119901(0 lt 119901 lt infin) He proved that the ideals of Hilbert Schmidtoperators and nuclear operators between Hilbert spaces aredefined by ℓ2 and ℓ1 respectively and the sequence ofapproximation numbers In [2] Faried and Bakery examinedthe operator ideals developed by generalized Cesaro Orliczsequence spaces ℓ119872 and the approximation numbers In [3]

Faried and Bakery studied the operator ideals constructedby 119904minus numbers generalized Cesaro and Orlicz sequencespaces ℓ119872 and show that the operator ideal formed by theprevious sequence spaces and approximation numbers issmall under certain conditions Also summation process andsequences spaces applications are closely related to Korovkintype approximation theorems and linear positive operatorsstudied by Costarelli and Vinti [4] and Altomare [5]The ideaof this paper is to examine a generalized class 119878119888119890119904120593 by usingOrlicz-Cesaro mean sequence spaces 119888119890119904120593 and the sequenceof 119904-numbers for which 119878119888119890119904120593 constructs an operator idealThe components of 119878119888119890119904120593 as a pre-quasi Banach operator idealcontaining finite dimensional operators as a dense subset andits completeness are proved The inclusion relations betweenthe pre-quasi operator ideals and the inclusion relations fortheir duals are determined Finally we show that the pre-quasi Banach operator ideal formed by the approximationnumbers and 119888119890119904120593 is small under certain conditions Theseresults coincide with that known for 119888119890119904119901 (1 lt 119901 lt infin) in[3] Furthermore we give some examples which support ourmain results

HindawiJournal of Function SpacesVolume 2019 Article ID 7265010 9 pageshttpsdoiorg10115520197265010

2 Journal of Function Spaces

2 Definitions and Preliminaries

Definition 1 (see [6]) The sequence (119904119899(119879))infin119899=0 for all 119879 isinL(119883 119884) is named an 119904-function and the number 119904119899(119879) iscalled the 119899119905ℎ 119904- number of 119879 if the following are satisfied

(a) monotonicity 119879 = 1199040(119879) ge 1199041(119879) ge 1199042(119879) ge sdot sdot sdot ge0 for all 119879 isin L(119883 119884)(b) additivity 119904119898+119899minus1(1198791 + 1198792) le 119904119898(1198791) + 119904119899(1198792) for all1198791 1198792 isin L(119883 119884)119898 119899 isin N(c) property of ideal 119904119899(119877119879119875) le 119877119904119899(119879) 119875 for all119875 isin L(1198830 119883) 119879 isin L(119883 119884) and 119877 isin L(119884 1198840) where1198830 and 1198840 are normed spaces(d) 119904119899(120573119879) = |120573|119904119899(119879) for every 119879 isin L(119883 119884) 120573 isin R(e) rank property if rank(119879) le 119899 then 119904119899(119879) = 0 for every119879 isin L(119883 119884)(f) property of norming

119904119894 (119868119895) = 1 if 119894 lt 1198950 if 119894 ge 119895 (1)

where 119868119895 is the identity operator on R119895

There are a few instances of 119904-numbers we notice theaccompanying conditions

(1) The n-th approximation number denoted by 120572119899(119879)is defined by 120572119899(119879) = inf 119879 minus 119861 119861 isinL(119883 119884) and rank(119861) le 119899

(2) Then-thHilbert number denoted by ℎ119899(119879) is definedby

ℎ119899 (119879) = sup 120572119899 (119860119879119861) 1003817100381710038171003817119860 119884 997888rarr ℓ21003817100381710038171003817le 1 and 1003817100381710038171003817119861 ℓ2 997888rarr 1198831003817100381710038171003817 le 1 (2)

(3) The n-th Weyl number denoted by 119909119899(119879) is definedby

119909119899 (119879) = inf 120572119899 (119879119861) 1003817100381710038171003817119861 ℓ2 997888rarr 1198831003817100381710038171003817 le 1 (3)

(4) The n-th Kolmogorov number denoted by 119889119899(119879) isdefined by

119889119899 (119879) = infdim119884le119899

sup119909le1

inf119910isin119884

1003817100381710038171003817119879119909 minus 1199101003817100381710038171003817 (4)

(5) The n-th Gelrsquofand number denoted by 119888119899(119879) isdefined by 119888119899(119879) = 120572119899(119869119884119879) where 119869119884 is a metricinjection from the space119884 to a higher space 119897infin(Ψ) foran adequate index setΨ This number is independentof the choice of the higher space 119897infin(Ψ)

(6) The n-th Chang number denoted by 119910119899(119879) is definedby

119910119899 (119879) = inf 120572119899 (119860119879) 1003817100381710038171003817119860 119884 997888rarr ℓ21003817100381710038171003817 le 1 (5)

Remark 2 (see [6]) Among all the 119904-number sequencescharacterized above it is easy to check that the approximationnumber120572119899(119879) is the largest and theHilbert number ℎ119899(119879) isthe smallest 119904-number sequence ie ℎ119899(119879) le 119904119899(119879) le 120572119899(119879)for any bounded linear operator119879 If119879 is defined on aHilbertspace and compact then all the 119904-numbers correspond withthe eigenvalues of |119879| where |119879| = (119879lowast119879)12Theorem 3 ([6] p115) Let 119879 isin L(119883 119884) Then

ℎ119899 (119879) le 119909119899 (119879) le 119888119899 (119879) le 120572119899 (119879) ℎ119899 (119879) le 119910119899 (119879) le 119889119899 (119879) le 120572119899 (119879) (6)

Theorem 4 ([6] p90) An 119904-number sequence is called injec-tive if for any metric injection 119870 isin L(119884 1198840) 119904119899(119879) = 119904119899(119870119879)for all 119879 isin L(119883 119884)Theorem 5 ([6] p95) An 119904-number sequence is called surjec-tive if for any metric surjection 119875 isin L(1198830 119883) 119904119899(119879) = 119904119899(119879119875)for all 119879 isin L(119883 119884)Theorem 6 ([6] pp90-94) The Weyl and Gelrsquofand numbersare injective

Theorem 7 ([6] pp95) TheChang and Kolmogorov numbersare surjective

Definition 8 A finite rank operator is a bounded linearoperator whose dimension of the range space is finite

Definition 9 ((dual 119904-numbers) [7]) For each 119904-numbersequence 119904 = (119904119899) a dual 119904-number function 119904119889 = (119904119889119899) isdefined by

119904119889119899 (119879) = 119904119899 (1198791015840) for all 119879 isin L (119883 119884) (7)

where 1198791015840 is the dual of 119879Definition 10 ([8] p152)) An 119904-number sequence is calledsymmetric if 119904119899(119879) ge 119904119899(1198791015840) for all 119879 isin L(119883 119884) If 119904119899(119879) =119904119899(1198791015840) then the 119904-number sequence is said to be completelysymmetric

Presently we express some known results of dual of an 119904-number sequence

Theorem 11 ([8] p152) The approximation numbers aresymmetric ie 120572119899(1198791015840) le 120572119899(119879) for 119879 isin L(119883 119884)Remark 12 (see [9]) 120572119899(119879) = 120572119899(1198791015840) for every compactoperator 119879Theorem 13 ([8] p153) Let 119879 isin L(119883 119884) Then

119888119899 (1198791015840) le 119889119899 (119879) 119888119899 (119879) = 119889119899 (1198791015840) (8)

In addition if 119879 is a compact operator then 119889119899(119879) = 119888119899(1198791015840)

Journal of Function Spaces 3

Theorem 14 ([6] p96) Let 119879 isin L(119883 119884) Then

119910119899 (1198791015840) le 119909119899 (119879) 119909119899 (119879) = 119910119899 (1198791015840) (9)

ie Chang numbers and Weyl numbers are dual to each other

Theorem 15 ([8] p153) The Hilbert numbers are completelysymmetric ie ℎ119899(1198791015840) = ℎ119899(119879) for all 119879 isin L(119883 119884)Definition 16 (see [10 11]) The operator ideal U flU(119883 119884) 119883 119886119899119889 119884 119886119903119890 119861119886119899119886119888ℎ 119878119901119886119888119890119904 is a subclass of lin-ear bounded operators such that its components U(119883 119884)which are subsets of L(119883 119884) fulfill the accompanying con-ditions

(i) 119868119860 isin U where 119860 indicates one dimensional Banachspace where U sub L

(ii) For 1198791 1198792 isin U(119883 119884) then 12057311198791 + 12057321198792 isin U(119883 119884) forany scalars 1205731 1205732

(iii) If 119879 isin L(1198830 119883) 119877 isin U(119883 119884) and 119875 isin L(119884 1198840) then119875119877119879 isin U(1198830 1198840)Definition 17 (see [12 13]) An Orlicz function is a function120593 [0infin) 997888rarr [0infin) which is nondecreasing convex andcontinuous with 120593(0) = 0 and 120593(119909) gt 0 for 119909 gt 0 andlim119909997888rarrinfin120593(119909) = infin

Definition 18 An Orlicz function 120593 is said to satisfy Δ 2-condition for every values of 119909 ge 0 if there is 119886 gt 0 suchthat 120593(2119909) le 119886120593(119909) The Δ 2-condition is corresponding to120593(119898119909) le 119886119898120593(119909) for every values of119898 gt 1 and 119909

Lindenstrauss and Tzafriri [14] utilized the idea of anOlicz function to define Orlicz sequence space

ℓ120593 = 119909 isin 120596 120588 (120582119909) lt infin for some 120582 gt 0where 120588 (119909) = infinsum

119896=0

120593 (10038161003816100381610038161199091198961003816100381610038161003816) (10)

(ℓ120593 ) is a Banach space with the Luxemburg norm

119909ℓ120593 = inf 120582 gt 0 120588 (120582minus1119909) le 1 (11)

Every Orlicz sequence space contains a subspace that isisomorphic to ℓ119901 for some 1 le 119901 lt infin or 1198880 ([15] Theorem4a9)

In the recent past lot of work has been done on sequencespaces defined by Orlicz functions by Altin et al [16] Et etal ([17 18]) Tripathy et al ([19ndash21]) and Mohiuddine et al([22ndash25])

Given an Orlicz function 120593 the Orlicz-Cesaro meansequence spaces is defined by

119888119890119904120593 = 119906 = (119906119894) isin 120596 120588 (120573119906) lt infin for some 120573 gt 0 120588 (119906) = infinsum

119894=0

120601(sum119894119895=0

1003816100381610038161003816100381611990611989510038161003816100381610038161003816119894 + 1 ) (12)

(119888119890119904120593 ) is a Banach space with the Luxemburg norm givenby

119906119888119890119904120593 = inf 120573 gt 0 120588 (120573minus1119906) le 1 (13)

It seems that Orlicz-Cesaro mean sequence spaces 119888119890119904120593appeared for the first time in 1988 when Lim and Yee foundtheir dual spaces [26] Recently Cui Hudzik Petrot Suantaiand Szymaszkiewicz obtained important properties of spaces119888119890119904120593 [27] In 2007 Maligranda Petrot and Suantai showedthat 119888119890119904120593 is not B-convex if 120593 isin Δ 2 and 119888119890119904120593 = 0 [28]The extreme points and strong 119883-points of 119888119890119904120593 have beencharacterized by Foralewski Hudzik and Szymaszkiewicz in[29] In the case when 120593(119906) = 119906119901 1 le 119901 lt infin the space 119888119890119904120593is just a Cesaro sequence space 119888119890119904119901 with the norm given by

119906119888119890119904119901 = [[

infinsum119894=0

(sum119894119895=0

1003816100381610038161003816100381611990611989510038161003816100381610038161003816119894 + 1 )119901]]

1119901

(14)

It is well known that 1198881198901199041 = 0 [30]Definition 19 (see [31]) TheMatuszewska Orlicz lower index120572120593 of an Orlicz function 120593 is defined as follows

120572120593 = sup 119901 gt 0 exist119870gt0 forall0lt120582119905le1120593 (120582119905) le 119870119905119901120593 (120582) (15)

Theorem 20 (see [31]) For any Orlicz function 120593 we have120572120593 gt 1 if and only if ℓ120593 sub 119888119890119904120593 In particular if 120572120593 gt 1 then119888119890119904120593 = 0Theorem 21 (see [31]) Let 1205931 and 1205932 be Orlicz functions Ifthere exist 119887 1199050 gt 0 such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905)for all 119905 isin [0 1199050] then 1198881198901199041205931 sub 1198881198901199041205932 Theorem 22 (see [31]) Let 1205931 and 1205932 be Orlicz functions and1205721205931 gt 1 then 1198881198901199041205931 sub 1198881198901199041205932 if and only if there exist 119887 1199050 gt 0such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905) for all 119905 isin [0 1199050]Definition 23 (see [2]) A class of linear sequence spacesE is called a special space of sequences (sss) having threeproperties

(1) 119890119894 isin E for all 119894 isin N(2) if 119909 = (119909119894) isin 119908 119910 = (119910119894) isin E and |119909119894| le |119910119894| for every119894 isin N then 119909 isin E ldquoie E is solidrdquo(3) if (119909119894)infin119894=0 isin E then (119909[1198942])infin119894=0 isin E wherever [1198942]

means the integral part of 1198942Definition 24 (see [2]) A subclass of the special space ofsequences is called a premodular (sss) if there is a function984858 E 997888rarr [0infin[ fulfilling the accompanying conditions

(i) 984858(119909) ge 0 for each 119909 isin E and 984858(119909) = 0 lArrrArr 119909 = 120579where 120579 is the zero element of E

(ii) there exists 119871 ge 1 such that 984858(120582119909) le 119871|120582|984858(119909) for all119909 isin E and for any scalar 120582(iii) for some 119870 ge 1 we have 984858(119909 + 119910) le 119870(984858(119909) + 984858(119910))

for every 119909 119910 isin E

4 Journal of Function Spaces

(iv) if |119909119894| le |119910119894| for all 119894 isin N then 984858((119909119894)) le 984858((119910119894))(v) for some 1198700 ge 1 we have

984858 ((119909119894)) le 984858 ((119909[1198942])) le 1198700984858 ((119909119894)) (16)

(vi) the set of all finite sequences is 984858-dense in E Thismeans for each 119909 = (119909119894)infin119894=119900 isin E and for each 120576 gt 0there exists119898 isin N such that 984858((119909119894)infin119894=119898) lt 120576

(vii) there exists a constant 120585 gt 0 such that984858(120582 0 0 0 ) ge 120585|120582|984858(1 0 0 0 ) for any 120582 isin R

We denote (E984858 984858) for the linear spaceE equippedwith themetrizable topology generated by 984858Theorem 25 (see [32]) If 119883 119884 are infinite dimensionalBanach spaces and 120582119894 is a monotonic decreasing sequence tozero then there exists a bounded linear operator 119879 such that

1161205823119894 le 120572119894 (119879) le 8120582119894+1 (17)

Notations 26 (see [3])

119878E fl 119878E(119883 119884) 119883 and 119884 are Banach Spaceswhere119878E(119883 119884) fl 119879 isin L(119883 119884) ((119904119894(119879))infin119894=0 isin E Also119878119886119901119901E fl 119878119886119901119901E (119883 119884) 119883 and 119884 are Banach Spaceswhere119878119886119901119901E (119883 119884) fl 119879 isin L(119883 119884) ((120572119894(119879))infin119894=0 isin E

Theorem 27 (see [3]) If E is a (sss) then 119878E is an operatorideal

The concept of pre-quasi operator ideal which is moregeneral than the usual classes of operator ideal

Definition 28 (see [3]) A function 119892 Ω 997888rarr [0infin) issaid to be a pre-quasi norm on the ideal Ω fulfilling theaccompanying conditions

(1) for all119879 isin Ω(119883 119884) 119892(119879) ge 0 and 119892(119879) = 0 if and onlyif 119879 = 0

(2) there exists a constant 119871 ge 1 such that 119892(120573119879) le119871|120573|119892(119879) for all 119879 isin Ω(119883 119884) and 120573 isin R(3) there exists a constant 119870 ge 1 such that 119892(1198791 + 1198792) le119870[119892(1198791) + 119892(1198792)] for all 1198791 1198792 isin Ω(119883 119884)(4) there exists a constant 119862 ge 1 such that if 119875 isin

L(1198830 119883) 119877 isin Ω(119883 119884) and 119879 isin L(119884 1198840) then119892(119879119877119875) le 119862 119879119892(119877) 119875 where 1198830 and 1198840 arenormed spaces

Theorem 29 (see [3]) Every quasi norm on the ideal Ω is apre-quasi norm on the ideal Ω

Here and after we define 119890119894 = 0 0 1 0 0 where1 appears at the 119894119905ℎ place for all 119894 isin N

3 Main Results

We give here the conditions onOrlicz-Cesaromean sequencespaces 119888119890119904120593 such that the class 119878119888119890119904120593 of all bounded linearoperators between arbitrary Banach spaces with its sequenceof 119904minusnumbers which belong to 119888119890119904120593 forms an operator ideal

Theorem30 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then 119878119888119890119904120593 is an operator ideal

Proof (1-i) Let 119909 119910 isin 119888119890119904120593 Since 120593 is nondecreasing convexand satisfying Δ 2-condition we get for some 119896 gt 0 that

infinsum119899=0

120593(sum119899119894=0

1003816100381610038161003816119909119894 + 1199101198941003816100381610038161003816119899 + 1 )le 119896 [infinsum

119899=0

120593(sum119899119894=0

10038161003816100381610038161199091198941003816100381610038161003816119899 + 1 ) + infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199101198941003816100381610038161003816119899 + 1 )] lt infin(18)

then 119909 + 119910 isin 119888119890119904120593(1-ii) Let 120582 isin R and 119909 isin 119888119890119904120593 and since 120593 is convex and

satisfying Δ 2-condition we get for some 119896 gt 0 thatinfinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161205821199091198941003816100381610038161003816119899 + 1 ) le |120582| 119896infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199091198941003816100381610038161003816119899 + 1 ) lt infin (19)

then120582119909 isin 119888119890119904120593 from (1-i) and (1-ii) 119888119890119904120593 is a linear space Since119890119899 isin ℓ120593 for all 119899 isin N and 120572120593 gt 1 then fromTheorem 20 weget 119890119899 isin 119888119890119904120593 for all 119899 isin N(2) Let |119909119899| le |119910119899| for all 119899 isin N and 119910 isin 119888119890119904120593 since 120593 isnondecreasing then we have

infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199091198941003816100381610038161003816119899 + 1 ) le infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199101198941003816100381610038161003816119899 + 1 ) lt infin (20)

and we get 119909 isin 119888119890119904120593(3) Let (119909119899) isin 119888119890119904120593 Since 120593 is satisfying Δ 2-condition weget for some 119896 gt 0 that

infinsum119899=0

120593(sum119899119894=0

1003816100381610038161003816119909[1198942]1003816100381610038161003816119899 + 1 ) le (119896 + 1) infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199091198941003816100381610038161003816119899 + 1 ) lt infin (21)

then (119909[1198992]) isin 119888119890119904120593Then 119888119890119904120593 is a (sss) hence byTheorem 27119878119888119890119904120593 is an operator ideal

Corollary 31 119878119888119890119904119902 is an operator ideal if 1 lt 119902 lt infin

We give the conditions on Orlicz-Cesaro mean sequencespaces 119888119890119904120593 such that the ideal of the finite rank operators isdense in 119878119888119890119904120593(119883 119884)Theorem 32 119878119888119890119904120593(119883 119884) = 119865(X 119884) if 120593 is an Orlicz functionsatisfying Δ 2-condition and 120572120593 gt 1Proof Let us define 984858(119906) = suminfin

119894=0 120593(sum119894119895=0 |119906119895|(119894 + 1)) on 119888119890119904120593

First we have to show that 119865(119883 119884) sube 119878119888119890119904120593(119883 119884) Since120572120593 gt 1 we have 119890119894 isin 119888119890119904120593 for each 119894 isin N and 120593 is an

Journal of Function Spaces 5

Orlicz function satisfying Δ 2-condition so for each finiteoperator119875 isin 119865(119883 119884) ie we obtain (119904119894(119875))infin119894=0 which containsonly finitely many terms different from zero hence 119875 isin119878119888119890119904120593(119883 119884) Currently we prove that 119878119888119890119904120593(119883 119884) sube 119865(119883 119884)let 119875 isin 119878119888119890119904120593(119883 119884) we have (119904119894(119875))infin119894=0 isin 119888119890119904120593 and hence984858(119904119894(119875))infin119894=0 lt infin By taking 120576 isin (0 1) hence there exists a1198940 isin N minus 0 such that 984858((119904119894(119875))infin119894=1198940) lt 12057691205751198622 for some 119888 ge 1where 120575 = max1 suminfin

119894=1198940120593(1(119894 + 1)) As 119904119894(119875) is decreasing

for every 119894 isin N and 120593 is nondecreasing we have

1198940120593 (11990421198940 (119875)) le21198940sum

119894=1198940+1

120593(sum119894119895=0 119904119895 (119875)119894 + 1 )

le infinsum119894=1198940

120593(sum119894119895=0 119904119895 (119875)119894 + 1 ) lt 12057691205751198622

(22)

Hence there exists 119861 isin 11986521198940(119883 119884) such that rank 119861 le 21198940 and1198940120593 (119875 minus 119861) le 21198940sum

119894=1198940+1

120593(sum119894119895=0 119875 minus 119861119894 + 1 ) lt 12057691205751198622

(23)

Since 120593 is right continuous at 0 and nondecreasing then onconsidering this

119875 minus 119861 lt 120576611986221198940120575 (24)

Let 1198961 gt 0 1198962 gt 0 and 119862 = max1 1198961 1198962 since 120593 is Orliczfunction and by using (22) (23) and (24) we have

119889 (119875 119861) = 984858 (119904119894 (119875 minus 119861))infin119894=0 = 31198940minus1sum119894=0

120593

sdot (sum119894119895=0 119904119895 (119875 minus 119861)119894 + 1 ) + infinsum

119894=31198940

120593(sum119894119895=0 119904119895 (119875 minus 119861)119894 + 1 )

le 31198940minus1sum119894=0

120593(sum119894119895=0 119875 minus 119861119894 + 1 ) + infinsum

119894=1198940

120593

sdot (sum119894+21198940119895=0 119904119895 (119875 minus 119861)119894 + 1 ) le 31198940120593 (119875 minus 119861) + infinsum

119894=1198940

120593

sdot (sum119894+21198940119895=0 119904119895 (119875 minus 119861)119894 + 1 ) le 31198940120593 (119875 minus 119861)

+ infinsum119894=1198940

120593(sum21198940minus1119895=0 119904119895 (119875 minus 119861) + sum119894+21198940

119895=21198940119904119895 (119875 minus 119861)119894 + 1 )

le 31198940120593 (119875 minus 119861) + 1198961 [[infinsum119894=1198940

120593(sum21198940minus1119895=0 119904119895 (119875 minus 119861)119894 + 1 )

+ infinsum119894=1198940

120593(sum119894+21198940119895=21198940

119904119895 (119875 minus 119861)119894 + 1 )]] le 31198940120593 (119875 minus 119861)

+ 1198961 [[infinsum119894=1198940

120593(sum21198940minus1119895=0 119875 minus 119861

119894 + 1 )

+ infinsum119894=1198940

120593(sum119894119895=0 119904119895+21198940 (119875 minus 119861)119894 + 1 )]] le 31198940120593 (119875 minus 119861)

+ 2119894011989611198962 119875 minus 119861 infinsum119894=1198940

120593( 1119894 + 1) + 1198961infinsum119894=1198940

120593

sdot (sum119894119895=0 119904119895 (119875)119894 + 1 ) le 31198940120593 (119875 minus 119861) + 211989401198622 119875

minus 119861 infinsum119894=1198940

120593( 1119894 + 1) + 119862infinsum119894=1198940

120593(sum119894119895=0 119904119895 (119875)119894 + 1 ) lt 120576

(25)

Corollary 33 119878119888119890119904119901(119883 119884) = 119865(119883 119884) if 1 lt 119901 lt infin

We express the accompanying theorem without verifica-tion these can be set up utilizing standard procedure

Theorem34 The function 119892(119875) = suminfin119894=0 120593(sum119894

119895=0 |119904119895(119875)|(119894+1))is a pre-quasi norm on 119878119888119890119904120593 if 120593 is an Orlicz function satisfyingΔ 2-condition and 120572120593 gt 1

We give the sufficient conditions on Orlicz-Cesaro meansequence spaces 119888119890119904120593 such that the components of the pre-quasi operator ideal 119878119888119890119904120593 are complete

Theorem 35 If 119883 and 119884 are Banach spaces 120593 is anOrlicz function satisfying Δ 2-condition and 120572120593 gt 1 then(119878119888119890119904120593(119883 119884) 119892) is a pre-quasi Banach operator idealProof Since 120593 is an Orlicz function satisfying Δ 2-condition then the function 119892(119875) = 984858((119904119899(119875))infin119899=0) =suminfin

119899=0 120593(sum119899119898=0 |119904119898(119875)|(119899 + 1)) is a pre-quasi norm on119878119888119890119904120593 Let (119875119898) be a Cauchy sequence in 119878119888119890119904120593(119883 119884) Since

L(119883 119884) supe 119878119888119890119904120593(119883 119884) and 120572120593 gt 1 we can find a constant120585 gt 0 such that

119892 (119875119894 minus 119875119895) = 984858 ((119904119899 (119875119894 minus 119875119895))infin119899=0)ge 984858 (1199040 (119875119894 minus 119875119895) 0 0 0 )= 984858 (10038171003817100381710038171003817119875119894 minus 11987511989510038171003817100381710038171003817 0 0 0 )ge 120585 10038171003817100381710038171003817119875119894 minus 11987511989510038171003817100381710038171003817 984858 (1 0 0 0 )

(26)

then (119875119898)119898isinN is also a Cauchy sequence in L(119883 119884) Whilethe space L(119883 119884) is a Banach space there exists 119875 isin L(119883 119884)

6 Journal of Function Spaces

such that lim119898997888rarrinfin 119875119898minus119875 = 0 while (119904119899(119875119898))infin119899=0 isin 119888119890119904120593 forevery 119898 isin N Since 984858 is continuous at 120579 and for some 119870 ge 1we obtain

119892 (119875) = 984858 ((119904119899 (119875))infin119899=0) = 984858 ((119904119899 (119875 minus 119875119898 + 119875119898))infin119899=0)le 119870984858 ((119904[1198992] (119875 minus 119875119898))infin119899=0)+ 119870984858 ((120572[1198992] (119875119898)infin119899=0))

le 119870984858 ((1003817100381710038171003817119875119898 minus 1198751003817100381710038171003817)infin119899=0) + 119870984858 ((119904119899 (119875119898)infin119899=0))lt infin

(27)

we have (119904119899(119875))infin119899=0 isin 119888119890119904120593 and then 119875 isin 119878119888119890119904120593(119883 119884)Corollary 36 If 119883 and 119884 are Banach spaces and 1 lt 119902 ltinfin then (119878119888119890119904119902(119883 119884) 119892) is quasi Banach operator ideal where119892(119875) = 984858((119904119899(119875))infin119899=0) = [suminfin

119899=0(sum119899119898=0 |119904119898(119875)|(119899 + 1))119902]1119902

Theorem 37 Let 1205931 1205932 be Orlicz functions and 1205721205931 gt 1 Forany infinite dimensional Banach spaces 119883 119884 and if there exist119887 1199050 gt 0 such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905) for all 119905 isin[0 1199050] it is true that

1198781198861199011199011198881198901199041205931(119883 119884) ⫋ 1198781198861199011199011198881198901199041205932

(119883 119884) ⫋ L (119883 119884) (28)

Proof Let119883 and119884 be infinite dimensional Banach spaces andthere exist 119887 1199050 gt 0 such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905)for all 119905 isin [0 1199050] if 119875 isin 1198781198861199011199011198881198901199041205931

(119883 119884) then (120572119899(119875)) isin 1198881198901199041205931 From Theorems 21 22 and 25 we have 1198881198901199041205931 sub 1198881198901199041205932 hence119875 isin 1198781198861199011199011198881198901199041205932

(119883 119884) It is easy to see that 1198781198861199011199011198881198901199041205932(119883 119884) sub L(119883 119884)

Corollary 38 For any infinite dimensional Banach spaces 119883119884 and 1 lt 119901 lt 119902 lt infin then 119878119886119901119901119888119890119904119901(119883 119884) ⫋ 119878119886119901119901119888119890119904119902

(119883 119884) ⫋L(119883 119884)

We now study some properties of the pre-quasi Banachoperator ideal 119878119888119890119904120593 Theorem 39 The pre-quasi Banach operator ideal (119878119888119890119904120593 119892) isinjective if the 119904-number sequence is injective

Proof Let 119879 isin L(119883 119884) and 119875 isin L(119884 1198840) be any metric injec-tion Assume that 119875119879 isin 119878119888119890119904120593(119883 1198840) then 984858(119904119899(119875119879)) lt infinSince the 119904-number sequence is injective we have 119904119899(119875119879) =119904119899(119879) for all 119879 isin L(119883 119884) 119899 isin N So 984858(119904119899(119879)) = 984858(119904119899(119875119879)) ltinfin Hence 119879 isin 119878119888119890119904120593(119883 119884) and clearly 119892(119879) = 119892(119875119879) isverified

Remark 40 The pre-quasi Banach operator ideal (119878119882119890119910119897119888119890119904120593

119892)and the pre-quasi Banach operator ideal (119878119866119890119897119888119890119904120593

119892) are injectivepre-quasi Banach operator ideal

Theorem 41 The pre-quasi Banach operator ideal (119878119888119890119904120593 119892) issurjective if the 119904-number sequence is surjective

Proof Let 119879 isin L(119883 119884) and 119875 isin L(1198830 119883) be any metricsurjection Suppose that 119879119875 isin 119878119888119890119904120593(1198830 119884) then 984858(119904119899(119879119875)) ltinfin Since the 119904-number sequence is surjective we have119904119899(119879119875) = 119904119899(119879) for all 119879 isin L(119883 119884) 119899 isin N So 984858(119904119899(119879)) =984858(119904119899(119879119875)) lt infin Hence 119879 isin 119878119888119890119904120593(119883 119884) and clearly 119892(119879) =119892(119879119875) is verifiedRemark 42 The pre-quasi Banach operator ideal (119878119862ℎ119886119899119892119888119890119904120593

119892) and the pre-quasi Banach operator ideal (119878119870119900119897

119888119890119904120593 119892) are

surjective pre-quasi Banach operator ideal

Likewise we have the accompanying inclusion relationsbetween the pre-quasi Banach operator ideals

Theorem 43 (1) 119878119886119901119901119888119890119904120593sube 119878119870119900119897

119888119890119904120593sube 119878119862ℎ119886119899119892119888119890119904120593

sube 119878119867119894119897119887119888119890119904120593

(2) 119878119886119901119901119888119890119904120593

sube 119878119866119890119897119888119890119904120593sube 119878119882119890119910119897

119888119890119904120593sube 119878119867119894119897119887

119888119890119904120593

Proof Since ℎ119899(119879) le 119910119899(119879) le 119889119899(119879) le 120572119899(119879) and ℎ119899(119879) le119909119899(119879) le 119888119899(119879) le 120572119899(119879) for every 119899 isin N and 984858 isnondecreasing we obtain

984858 (ℎ119899 (119879)) le 984858 (119910119899 (119879)) le 984858 (119889119899 (119879)) le 984858 (120572119899 (119879)) 984858 (ℎ119899 (119879)) le 984858 (119909119899 (119879)) le 984858 (119888119899 (119879)) le 984858 (120572119899 (119879)) (29)

Hence the result is as follows

We presently express the dual of the pre-quasi operatorideal formed by different 119904minus number sequences

Theorem 44 The pre-quasi operator ideal 119878119867119894119897119887119888119890119904120593

is completelysymmetric and the pre-quasi operator ideal 119878119886119901119901119888119890119904120593

is symmetric

Proof Since ℎ119899(1198791015840) = ℎ119899(119879) and 120572119899(1198791015840) le 120572119899(119879) for all 119879 isinL(119883 119884) we have 119878119867119894119897119887

119888119890119904120593= (119878119867119894119897119887

119888119890119904120593)1015840 and 119878119886119901119901119888119890119904120593

sube (119878119886119901119901119888119890119904120593)1015840

In perspective on Theorem 13 we express the followingresult without proof

Theorem 45 The pre-quasi operator ideal 119878119870119900119897119888119890119904120593

sube (119878119866119890119897119888119890119904120593)1015840 and

119878119866119890119897119888119890119904120593= (119878119870119900119897

119888119890119904120593)1015840 In addition if 119879 is a compact operator from 119883

to 119884 then 119878119870119900119897119888119890119904120593

= (119878119866119890119897119888119890119904120593)1015840

In perspective on Theorem 14 we express the followingresult without proof

Theorem 46 The pre-quasi operator ideal 119878119862ℎ119886119899119892119888119890119904120593= (119878119882119890119910119897

119888119890119904120593)1015840

and 119878119882119890119910119897119888119890119904120593

= (119878119862ℎ119886119899119892119888119890119904120593)1015840

Theorem47 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then the pre-quasi Banach operator ideal 119878119886119901119901119888119890119904120593

issmall

Proof Since 120593 is an Orlicz function and 120572120593 gt 1 take120573 = suminfin119894=0 120593(1(119894 + 1)) Then (119878119886119901119901119888119890119904120593

119892) where 119892(119879) =

Journal of Function Spaces 7

984858((120572119899(119879))infin119899=0) = (1120573)suminfin119899=0 120593(sum119899

119898=0 120572119898(119879)(119899 + 1)) is a pre-quasi Banach operator ideal Let119883 and 119884 be any two Banachspaces Assume that 119878119886119901119901119888119890119904120593

(119883 119884) = L(119883 119884) then there existsa constant 119862 gt 0 such that 119892(119879) le 119862119879 for all 119879 isin L(119883 119884)Suppose that119883 and119884 are infinite dimensional Banach spacesThen by Dvoretzkyrsquos theorem [8] for119898 isin N we have quotientspaces 119883119872119898 and subspaces 119873119898 of 119884 which can be mappedonto ℓ1198982 by isomorphisms 119881119898 and 119861119898 such that 119881119898119881minus1

119898 le2 and 119861119898119861minus1119898 le 2 Consider 119868119898 be the identity map onℓ1198982 119875119898 be the quotient map from 119883 onto 119883119872119898 and 119876119898 be

the natural embedding map from 119873119898 into 119884 Let V119899 be theBernstein numbers [7] then

1 = V119899 (119868119898) = V119899 (119861119898119861minus1119898 119868119898119881119898119881minus1

119898 )le 1003817100381710038171003817119861119898

1003817100381710038171003817 V119899 (119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817= 1003817100381710038171003817119861119898

1003817100381710038171003817 V119899 (119876119898119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817le 1003817100381710038171003817119861119898

1003817100381710038171003817 119889119899 (119876119898119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817= 1003817100381710038171003817119861119898

1003817100381710038171003817 119889119899 (119876119898119861minus1119898 119868119898119881119898119876119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817le 1003817100381710038171003817119861119898

1003817100381710038171003817 120572119899 (119876119898119861minus1119898 119868119898119881119898119876119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817

(30)

for 1 le 119894 le 119898 Now since 120593 is nondecreasing and having Δ 2-condition we have

119894sum119895=0

(1) le 119894sum119895=0

10038171003817100381710038171198611198981003817100381710038171003817 120572119895 (119876119898119861minus1

119898 119868119898119881119898119875119898) 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 997904rArr1119894 + 1 (119894 + 1) le 1003817100381710038171003817119861119898

1003817100381710038171003817( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898))

sdot 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 997904rArr120593 (1) le 119871 (1003817100381710038171003817119861119898

1003817100381710038171003817 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817)sdot 120593( 1119894 + 1

119894sum119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898))

(31)

Therefore

119898sum119894=0

120593 (1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817119898sum119894=0

120593

sdot ( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898)) 997904rArr

120593 (1)120573 (119898 + 1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 1120573119898sum119894=0

120593

sdot ( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898)) 997904rArr

120593 (1)120573 (119898 + 1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 119892 (119876119898119861minus1119898 119868119898119881119898119875119898) 997904rArr

120593 (1)120573 (119898 + 1) le 119871119862 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 10038171003817100381710038171003817119876119898119861minus1119898 11986811989811988111989811987511989810038171003817100381710038171003817 997904rArr

120593 (1)120573 (119898 + 1) le 119871119862 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 10038171003817100381710038171003817119876119898119861minus1119898

10038171003817100381710038171003817 10038171003817100381710038171198681198981003817100381710038171003817 10038171003817100381710038171198811198981198751198981003817100381710038171003817= 119871119862 1003817100381710038171003817119861119898

1003817100381710038171003817 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 10038171003817100381710038171003817119861minus1119898

10038171003817100381710038171003817 10038171003817100381710038171198681198981003817100381710038171003817 10038171003817100381710038171198811198981003817100381710038171003817 997904rArr120593 (1)120573 (119898 + 1) le 4119871119862

(32)

for some 119871 ge 1 Thus we arrive at a contradiction since 119898 isarbitrary Hence119883 and119884 both cannot be infinite dimensionalwhen 119878119886119901119901119888119890119904120593

(119883 119884) = L(119883 119884)Theorem48 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then the pre-quasi Banach operator ideal 119878119870119900119897

119888119890119904120593is

small

Corollary 49 If 119901 isin (1infin) then the quasi Banach operatorideal 119878119886119901119901119888119890119904119901

is small

Corollary 50 If 119901 isin (1infin) then the quasi Banach operatorideal 119878119870119900119897

119888119890119904119901is small

4 Examples

We give some examples which support our main results

Example 1 Let 120593 be an Orlicz function the subspace 119888119890119904ℎ120593 ofall order continuous elements of 119888119890119904120593 is defined as [27]

119888119890119904ℎ120593= 119909 isin 119888119890119904120593 forall119896gt0 exist119899119896isinN

infinsum119899=119899119896

120593(119896119899119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816) lt infin (33)

If 120593 is an Orlicz function satisfying Δ 2-condition and 120572120593 gt 1then the following conditions are satisfied

(1) 119878119888119890119904ℎ120593 is an operator ideal

(2) 119878119888119890119904ℎ120593(119883 119884) = 119865(119883 119884)(3) If 119883 and 119884 are Banach spaces then (119878119888119890119904ℎ120593(119883 119884) 119892) is

pre-quasi Banach operator ideal

(4) The pre-quasi Banach operator ideal 119878119886119901119901119888119890119904ℎ120593

is small

Proof Since 120593 is an Orlicz function satisfying Δ 2-conditionand 120572120593 gt 1 then from Theorem (5) in [31] we have 119888119890119904ℎ120593 =119888119890119904120593 which completes the proof

8 Journal of Function Spaces

Example 2 Let 120593 be defined as

120593 (119905) = 119886119897119905119897 + 119886119897minus1119905119897minus1 + + 1198861119905where 119886119894 gt 0 for all 1 le 119894 le 119897 119897 isin N 119897 gt 1 and 119905 ge 0 (34)

It is clear that 120593 is an Orlicz function and 120572120593 = 119897 gt 1 Also 120593is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) le 2119897 lt infin (35)

Then the following conditions are satisfied

(1) 119878119888119890119904120593 is an operator ideal

(2) 119878119888119890119904120593(119883 119884) = 119865(119883 119884)(3) If 119883 and 119884 are Banach spaces then (119878119888119890119904120593(119883 119884) 119892) is

pre-quasi Banach operator ideal(4) The pre-quasi Banach operator ideal 119878119886119901119901119888119890119904120593

is small

In the following two examples we will explain the impor-tance of the sufficient conditions

Example 3 Let 120593 be defined as

120593 (119905) =

0 if 119905 = 0minus119905ln 119905 if 119905 isin (0 1119890 ] 321198901199052 minus 119905 + 12119890 if 119905 isin (1119890 infin)

(36)

It is clear that 120593 is an Orlicz function Since suminfin119899=1 120593(1119899) =suminfin

119899=1(1119899 ln 119899) = infin hence 119888119890119904120593 = 0 The space 119878119888119890119904120593 is notoperator ideal since 119868119870 notin 119878119888119890119904120593 Also since120593 is convex functionand for 119901 gt 1 we have

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

1199051minus119901 ln 120582ln 120582119905

= lim119905997888rarr0+

(1 minus 119901) 1199051minus119901 ln 120582 = infin (37)

for all 120582 isin (0 1] then 120572120593 = 1 Although 120593 is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) = lim sup119905997888rarr0+

2 ln 119905ln 2119905 le 2 lt infin (38)

Example 4 Let 120593(119906) = int119906

0119891(119905)119889119905 where 119891(119905) is defined as

119891 (119905)

=

0 if 119905 = 01119899 if 119905 isin [ 1(119899 + 1) 1119899) for 119899 = 1 2 3 119905 if 119905 isin [1infin)

(39)

It is clear that 120593 is an Orlicz function Let 119879 isin 119878119888119890119904120593 with119904119899(119879) = 1119899 for all 119899 isin N We have for 119899 gt 2 that120593 (119904119899 (2119879)) = int2119899

0119891 (119905) 119889119905 gt int2119899

1119899119891 (119905) 119889119905

gt int1(119899minus1)

1119899119891 (119905) 119889119905 gt 1119899 (119899 minus 1)

119899120593 (119904119899 (119879)) = 119899int1119899

0119891 (119905) 119889119905

lt 119899 sup0le119905le1119899

119891 (119905) int1119899

01 119889119905 lt 1119899 (119899 minus 1)

(40)

Hence 2119879 notin 119878119888119890119904120593 so the space 119878119888119890119904120593 is not operator ideal and120593 notin Δ 2 Also since 120593 is convex function and for 119901 gt 1 wehave

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

119905minus119901 = infin (41)

for all 120582 isin (0 1] then 120572120593 = 1Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

The authors received no financial support for the researchauthorship and or publication of this article

Conflicts of Interest

The authors declare that 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 vol 20 North-Holland PublishingCompany Amsterdam The Netherlands 1980

[2] N F Mohamed and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalities andApplications vol 2013 article 186 2013

[3] N Faried and A A Bakery ldquoSmall operator ideals formed bys numbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1 article357 2018

[4] D Costarelli and G Vinti ldquoA quantitative estimate for the sam-pling kantorovich series in terms of the modulus of continuityin orlicz spacesrdquo Constructive Mathematical Analysis vol 2 no1 pp 8ndash14 2019

Journal of Function Spaces 9

[5] F Altomare ldquoIterates of markov operators and construc-tive approximation of semigroupsrdquo Constructive MathematicalAnalysis vol 2 no 1 pp 22ndash39 2019

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

[7] A Pietsch ldquos-numbers of operators in banach spacesrdquo StudiaMathematica vol 51 pp 201ndash223 1974

[8] A Pietsch Operator Ideals VEB Deutscher Verlag DerWis-senschaften Berlin Germany 1978

[9] C V Hutton ldquoOn the approximation numbers of an operatorand its adjointrdquo Mathematische Annalen vol 210 pp 277ndash2801974

[10] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 pp 267ndash278 1974

[11] A Lima and E Oja ldquoIdeals of finite rank operators intersectionproperties of balls and the approximation propertyrdquo StudiaMathematica vol 133 no 2 pp 175ndash186 1999

[12] M A Krasnoselskii and Y B Rutickii Convex Functions andOrlicz Spaces Gorningen Netherlands 1961

[13] W Orlicz and U Raume ldquoLMrdquo Bulletin International delrsquoAcademie Polonaise des Sciences et des Lettres Serie A pp 93ndash107 1936

[14] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971

[15] J Lindenstrauss and L Tzafriri Classical Banach Spaces vol92 of I Sequence spaces Ergebnisse der Mathematik und ihrerGrenzgebiete Springer-Verlag Berlin Germany 1977

[16] Y Altin M Et and B C Tripathy ldquoThe sequence space|119873119901|(119872 119903 119902 119904) on seminormed spacesrdquo Applied Mathematicsand Computation vol 154 no 2 pp 423ndash430 2004

[17] M Et L P Lee and B C Tripathy ldquoStrongly almost (119881 120582)(119903)-summable sequences defined by Orlicz functionsrdquo HokkaidoMathematical Journal vol 35 no 1 pp 197ndash213 2006

[18] M Et Y Altin B Choudhary and B C Tripathy ldquoOn someclasses of sequences defined by sequences of Orlicz functionsrdquoMathematical Inequalities amp Applications vol 9 no 2 pp 335ndash342 2006

[19] B C Tripathy and S Mahanta ldquoOn a class of differencesequences related to the l119901 space defined by Orlicz functionsrdquoMathematica Slovaca vol 57 no 2 pp 171ndash178 2007

[20] B C Tripathy and H Dutta ldquoOn some new paranormed differ-ence sequence spaces defined by Orlicz functionsrdquo KyungpookMathematical Journal vol 50 no 1 pp 59ndash69 2010

[21] B C Tripathy and B Hazarika ldquoI-convergent sequences spacesdefined by Orlicz functionrdquo Acta Mathematicae ApplicataeSinica vol 27 no 1 pp 149ndash154 2011

[22] S A Mohiuddine and B Hazarika ldquoSome classes of ideal con-vergent sequences and generalized difference matrix operatorrdquoFilomat vol 31 no 6 pp 1827ndash1834 2017

[23] S A Mohiuddine and K Raj ldquoVector valued Orlicz-Lorentzsequence spaces and their operator idealsrdquo Journal of NonlinearSciences and Applications JNSA vol 10 no 2 pp 338ndash353 2017

[24] S Abdul Mohiuddine K Raj and A Alotaibi ldquoGeneralizedspaces of double sequences for Orlicz functions and bounded-regular matrices over n-normed spacesrdquo Journal of Inequalitiesand Applications vol 2014 article 332 2014

[25] S A Mohiuddine S K Sharma and D A Abuzaid ldquoSomeseminormed difference sequence spaces over n-normed spacesdefined by a musielak-orlicz function of order (120572120573)rdquo Journal ofFunction Spaces vol 2018 Article ID 4312817 11 pages 2018

[26] S-K Lim and P Y Lee ldquoAnOrlicz extension of Cesaro sequencespacesrdquo Roczniki Polskiego Towarzystwa Matematycznego SeriaI Commentationes Mathematicae Prace Matematyczne vol 28no 1 pp 117ndash128 1988

[27] Y Cui H Hudzik N Petrot S Suantai and A SzymaszkiewiczldquoBasic topological and geometric properties of Cesaro-Orliczspacesrdquo The Proceedings of the Indian Academy of Sciences ndashMathematical Sciences vol 115 no 4 pp 461ndash476 2005

[28] LMaligranda N Petrot and S Suantai ldquoOn the James constantand B-convexity of Cesaro and Cesaro-Orlicz sequence spacesrdquoJournal of Mathematical Analysis and Applications vol 326 pp312ndash326 2007

[29] P Foralewski H Hudzik and A Szymaszkiewicz ldquoLocalrotundity structure of Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 345 no 1 pp410ndash419 2008

[30] G M Leibowitz ldquoA note on the Cesaro sequence spacesrdquoTamkang Journal of Mathematics vol 2 no 2 pp 151ndash157 1971

[31] D Kubiak ldquoA note on Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 349 no 1 pp291ndash296 2009

[32] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbers (in Russian)rdquo in Theoryof Operators in Functional Spaces pp 206ndash211 Academy ofScience Siberian section Novosibirsk Russia 1977

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Journal of Function Spaces 3

Theorem 14 ([6] p96) Let 119879 isin L(119883 119884) Then

119910119899 (1198791015840) le 119909119899 (119879) 119909119899 (119879) = 119910119899 (1198791015840) (9)

ie Chang numbers and Weyl numbers are dual to each other

Theorem 15 ([8] p153) The Hilbert numbers are completelysymmetric ie ℎ119899(1198791015840) = ℎ119899(119879) for all 119879 isin L(119883 119884)Definition 16 (see [10 11]) The operator ideal U flU(119883 119884) 119883 119886119899119889 119884 119886119903119890 119861119886119899119886119888ℎ 119878119901119886119888119890119904 is a subclass of lin-ear bounded operators such that its components U(119883 119884)which are subsets of L(119883 119884) fulfill the accompanying con-ditions

(i) 119868119860 isin U where 119860 indicates one dimensional Banachspace where U sub L

(ii) For 1198791 1198792 isin U(119883 119884) then 12057311198791 + 12057321198792 isin U(119883 119884) forany scalars 1205731 1205732

(iii) If 119879 isin L(1198830 119883) 119877 isin U(119883 119884) and 119875 isin L(119884 1198840) then119875119877119879 isin U(1198830 1198840)Definition 17 (see [12 13]) An Orlicz function is a function120593 [0infin) 997888rarr [0infin) which is nondecreasing convex andcontinuous with 120593(0) = 0 and 120593(119909) gt 0 for 119909 gt 0 andlim119909997888rarrinfin120593(119909) = infin

Definition 18 An Orlicz function 120593 is said to satisfy Δ 2-condition for every values of 119909 ge 0 if there is 119886 gt 0 suchthat 120593(2119909) le 119886120593(119909) The Δ 2-condition is corresponding to120593(119898119909) le 119886119898120593(119909) for every values of119898 gt 1 and 119909

Lindenstrauss and Tzafriri [14] utilized the idea of anOlicz function to define Orlicz sequence space

ℓ120593 = 119909 isin 120596 120588 (120582119909) lt infin for some 120582 gt 0where 120588 (119909) = infinsum

119896=0

120593 (10038161003816100381610038161199091198961003816100381610038161003816) (10)

(ℓ120593 ) is a Banach space with the Luxemburg norm

119909ℓ120593 = inf 120582 gt 0 120588 (120582minus1119909) le 1 (11)

Every Orlicz sequence space contains a subspace that isisomorphic to ℓ119901 for some 1 le 119901 lt infin or 1198880 ([15] Theorem4a9)

In the recent past lot of work has been done on sequencespaces defined by Orlicz functions by Altin et al [16] Et etal ([17 18]) Tripathy et al ([19ndash21]) and Mohiuddine et al([22ndash25])

Given an Orlicz function 120593 the Orlicz-Cesaro meansequence spaces is defined by

119888119890119904120593 = 119906 = (119906119894) isin 120596 120588 (120573119906) lt infin for some 120573 gt 0 120588 (119906) = infinsum

119894=0

120601(sum119894119895=0

1003816100381610038161003816100381611990611989510038161003816100381610038161003816119894 + 1 ) (12)

(119888119890119904120593 ) is a Banach space with the Luxemburg norm givenby

119906119888119890119904120593 = inf 120573 gt 0 120588 (120573minus1119906) le 1 (13)

It seems that Orlicz-Cesaro mean sequence spaces 119888119890119904120593appeared for the first time in 1988 when Lim and Yee foundtheir dual spaces [26] Recently Cui Hudzik Petrot Suantaiand Szymaszkiewicz obtained important properties of spaces119888119890119904120593 [27] In 2007 Maligranda Petrot and Suantai showedthat 119888119890119904120593 is not B-convex if 120593 isin Δ 2 and 119888119890119904120593 = 0 [28]The extreme points and strong 119883-points of 119888119890119904120593 have beencharacterized by Foralewski Hudzik and Szymaszkiewicz in[29] In the case when 120593(119906) = 119906119901 1 le 119901 lt infin the space 119888119890119904120593is just a Cesaro sequence space 119888119890119904119901 with the norm given by

119906119888119890119904119901 = [[

infinsum119894=0

(sum119894119895=0

1003816100381610038161003816100381611990611989510038161003816100381610038161003816119894 + 1 )119901]]

1119901

(14)

It is well known that 1198881198901199041 = 0 [30]Definition 19 (see [31]) TheMatuszewska Orlicz lower index120572120593 of an Orlicz function 120593 is defined as follows

120572120593 = sup 119901 gt 0 exist119870gt0 forall0lt120582119905le1120593 (120582119905) le 119870119905119901120593 (120582) (15)

Theorem 20 (see [31]) For any Orlicz function 120593 we have120572120593 gt 1 if and only if ℓ120593 sub 119888119890119904120593 In particular if 120572120593 gt 1 then119888119890119904120593 = 0Theorem 21 (see [31]) Let 1205931 and 1205932 be Orlicz functions Ifthere exist 119887 1199050 gt 0 such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905)for all 119905 isin [0 1199050] then 1198881198901199041205931 sub 1198881198901199041205932 Theorem 22 (see [31]) Let 1205931 and 1205932 be Orlicz functions and1205721205931 gt 1 then 1198881198901199041205931 sub 1198881198901199041205932 if and only if there exist 119887 1199050 gt 0such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905) for all 119905 isin [0 1199050]Definition 23 (see [2]) A class of linear sequence spacesE is called a special space of sequences (sss) having threeproperties

(1) 119890119894 isin E for all 119894 isin N(2) if 119909 = (119909119894) isin 119908 119910 = (119910119894) isin E and |119909119894| le |119910119894| for every119894 isin N then 119909 isin E ldquoie E is solidrdquo(3) if (119909119894)infin119894=0 isin E then (119909[1198942])infin119894=0 isin E wherever [1198942]

means the integral part of 1198942Definition 24 (see [2]) A subclass of the special space ofsequences is called a premodular (sss) if there is a function984858 E 997888rarr [0infin[ fulfilling the accompanying conditions

(i) 984858(119909) ge 0 for each 119909 isin E and 984858(119909) = 0 lArrrArr 119909 = 120579where 120579 is the zero element of E

(ii) there exists 119871 ge 1 such that 984858(120582119909) le 119871|120582|984858(119909) for all119909 isin E and for any scalar 120582(iii) for some 119870 ge 1 we have 984858(119909 + 119910) le 119870(984858(119909) + 984858(119910))

for every 119909 119910 isin E

4 Journal of Function Spaces

(iv) if |119909119894| le |119910119894| for all 119894 isin N then 984858((119909119894)) le 984858((119910119894))(v) for some 1198700 ge 1 we have

984858 ((119909119894)) le 984858 ((119909[1198942])) le 1198700984858 ((119909119894)) (16)

(vi) the set of all finite sequences is 984858-dense in E Thismeans for each 119909 = (119909119894)infin119894=119900 isin E and for each 120576 gt 0there exists119898 isin N such that 984858((119909119894)infin119894=119898) lt 120576

(vii) there exists a constant 120585 gt 0 such that984858(120582 0 0 0 ) ge 120585|120582|984858(1 0 0 0 ) for any 120582 isin R

We denote (E984858 984858) for the linear spaceE equippedwith themetrizable topology generated by 984858Theorem 25 (see [32]) If 119883 119884 are infinite dimensionalBanach spaces and 120582119894 is a monotonic decreasing sequence tozero then there exists a bounded linear operator 119879 such that

1161205823119894 le 120572119894 (119879) le 8120582119894+1 (17)

Notations 26 (see [3])

119878E fl 119878E(119883 119884) 119883 and 119884 are Banach Spaceswhere119878E(119883 119884) fl 119879 isin L(119883 119884) ((119904119894(119879))infin119894=0 isin E Also119878119886119901119901E fl 119878119886119901119901E (119883 119884) 119883 and 119884 are Banach Spaceswhere119878119886119901119901E (119883 119884) fl 119879 isin L(119883 119884) ((120572119894(119879))infin119894=0 isin E

Theorem 27 (see [3]) If E is a (sss) then 119878E is an operatorideal

The concept of pre-quasi operator ideal which is moregeneral than the usual classes of operator ideal

Definition 28 (see [3]) A function 119892 Ω 997888rarr [0infin) issaid to be a pre-quasi norm on the ideal Ω fulfilling theaccompanying conditions

(1) for all119879 isin Ω(119883 119884) 119892(119879) ge 0 and 119892(119879) = 0 if and onlyif 119879 = 0

(2) there exists a constant 119871 ge 1 such that 119892(120573119879) le119871|120573|119892(119879) for all 119879 isin Ω(119883 119884) and 120573 isin R(3) there exists a constant 119870 ge 1 such that 119892(1198791 + 1198792) le119870[119892(1198791) + 119892(1198792)] for all 1198791 1198792 isin Ω(119883 119884)(4) there exists a constant 119862 ge 1 such that if 119875 isin

L(1198830 119883) 119877 isin Ω(119883 119884) and 119879 isin L(119884 1198840) then119892(119879119877119875) le 119862 119879119892(119877) 119875 where 1198830 and 1198840 arenormed spaces

Theorem 29 (see [3]) Every quasi norm on the ideal Ω is apre-quasi norm on the ideal Ω

Here and after we define 119890119894 = 0 0 1 0 0 where1 appears at the 119894119905ℎ place for all 119894 isin N

3 Main Results

We give here the conditions onOrlicz-Cesaromean sequencespaces 119888119890119904120593 such that the class 119878119888119890119904120593 of all bounded linearoperators between arbitrary Banach spaces with its sequenceof 119904minusnumbers which belong to 119888119890119904120593 forms an operator ideal

Theorem30 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then 119878119888119890119904120593 is an operator ideal

Proof (1-i) Let 119909 119910 isin 119888119890119904120593 Since 120593 is nondecreasing convexand satisfying Δ 2-condition we get for some 119896 gt 0 that

infinsum119899=0

120593(sum119899119894=0

1003816100381610038161003816119909119894 + 1199101198941003816100381610038161003816119899 + 1 )le 119896 [infinsum

119899=0

120593(sum119899119894=0

10038161003816100381610038161199091198941003816100381610038161003816119899 + 1 ) + infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199101198941003816100381610038161003816119899 + 1 )] lt infin(18)

then 119909 + 119910 isin 119888119890119904120593(1-ii) Let 120582 isin R and 119909 isin 119888119890119904120593 and since 120593 is convex and

satisfying Δ 2-condition we get for some 119896 gt 0 thatinfinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161205821199091198941003816100381610038161003816119899 + 1 ) le |120582| 119896infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199091198941003816100381610038161003816119899 + 1 ) lt infin (19)

then120582119909 isin 119888119890119904120593 from (1-i) and (1-ii) 119888119890119904120593 is a linear space Since119890119899 isin ℓ120593 for all 119899 isin N and 120572120593 gt 1 then fromTheorem 20 weget 119890119899 isin 119888119890119904120593 for all 119899 isin N(2) Let |119909119899| le |119910119899| for all 119899 isin N and 119910 isin 119888119890119904120593 since 120593 isnondecreasing then we have

infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199091198941003816100381610038161003816119899 + 1 ) le infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199101198941003816100381610038161003816119899 + 1 ) lt infin (20)

and we get 119909 isin 119888119890119904120593(3) Let (119909119899) isin 119888119890119904120593 Since 120593 is satisfying Δ 2-condition weget for some 119896 gt 0 that

infinsum119899=0

120593(sum119899119894=0

1003816100381610038161003816119909[1198942]1003816100381610038161003816119899 + 1 ) le (119896 + 1) infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199091198941003816100381610038161003816119899 + 1 ) lt infin (21)

then (119909[1198992]) isin 119888119890119904120593Then 119888119890119904120593 is a (sss) hence byTheorem 27119878119888119890119904120593 is an operator ideal

Corollary 31 119878119888119890119904119902 is an operator ideal if 1 lt 119902 lt infin

We give the conditions on Orlicz-Cesaro mean sequencespaces 119888119890119904120593 such that the ideal of the finite rank operators isdense in 119878119888119890119904120593(119883 119884)Theorem 32 119878119888119890119904120593(119883 119884) = 119865(X 119884) if 120593 is an Orlicz functionsatisfying Δ 2-condition and 120572120593 gt 1Proof Let us define 984858(119906) = suminfin

119894=0 120593(sum119894119895=0 |119906119895|(119894 + 1)) on 119888119890119904120593

First we have to show that 119865(119883 119884) sube 119878119888119890119904120593(119883 119884) Since120572120593 gt 1 we have 119890119894 isin 119888119890119904120593 for each 119894 isin N and 120593 is an

Journal of Function Spaces 5

Orlicz function satisfying Δ 2-condition so for each finiteoperator119875 isin 119865(119883 119884) ie we obtain (119904119894(119875))infin119894=0 which containsonly finitely many terms different from zero hence 119875 isin119878119888119890119904120593(119883 119884) Currently we prove that 119878119888119890119904120593(119883 119884) sube 119865(119883 119884)let 119875 isin 119878119888119890119904120593(119883 119884) we have (119904119894(119875))infin119894=0 isin 119888119890119904120593 and hence984858(119904119894(119875))infin119894=0 lt infin By taking 120576 isin (0 1) hence there exists a1198940 isin N minus 0 such that 984858((119904119894(119875))infin119894=1198940) lt 12057691205751198622 for some 119888 ge 1where 120575 = max1 suminfin

119894=1198940120593(1(119894 + 1)) As 119904119894(119875) is decreasing

for every 119894 isin N and 120593 is nondecreasing we have

1198940120593 (11990421198940 (119875)) le21198940sum

119894=1198940+1

120593(sum119894119895=0 119904119895 (119875)119894 + 1 )

le infinsum119894=1198940

120593(sum119894119895=0 119904119895 (119875)119894 + 1 ) lt 12057691205751198622

(22)

Hence there exists 119861 isin 11986521198940(119883 119884) such that rank 119861 le 21198940 and1198940120593 (119875 minus 119861) le 21198940sum

119894=1198940+1

120593(sum119894119895=0 119875 minus 119861119894 + 1 ) lt 12057691205751198622

(23)

Since 120593 is right continuous at 0 and nondecreasing then onconsidering this

119875 minus 119861 lt 120576611986221198940120575 (24)

Let 1198961 gt 0 1198962 gt 0 and 119862 = max1 1198961 1198962 since 120593 is Orliczfunction and by using (22) (23) and (24) we have

119889 (119875 119861) = 984858 (119904119894 (119875 minus 119861))infin119894=0 = 31198940minus1sum119894=0

120593

sdot (sum119894119895=0 119904119895 (119875 minus 119861)119894 + 1 ) + infinsum

119894=31198940

120593(sum119894119895=0 119904119895 (119875 minus 119861)119894 + 1 )

le 31198940minus1sum119894=0

120593(sum119894119895=0 119875 minus 119861119894 + 1 ) + infinsum

119894=1198940

120593

sdot (sum119894+21198940119895=0 119904119895 (119875 minus 119861)119894 + 1 ) le 31198940120593 (119875 minus 119861) + infinsum

119894=1198940

120593

sdot (sum119894+21198940119895=0 119904119895 (119875 minus 119861)119894 + 1 ) le 31198940120593 (119875 minus 119861)

+ infinsum119894=1198940

120593(sum21198940minus1119895=0 119904119895 (119875 minus 119861) + sum119894+21198940

119895=21198940119904119895 (119875 minus 119861)119894 + 1 )

le 31198940120593 (119875 minus 119861) + 1198961 [[infinsum119894=1198940

120593(sum21198940minus1119895=0 119904119895 (119875 minus 119861)119894 + 1 )

+ infinsum119894=1198940

120593(sum119894+21198940119895=21198940

119904119895 (119875 minus 119861)119894 + 1 )]] le 31198940120593 (119875 minus 119861)

+ 1198961 [[infinsum119894=1198940

120593(sum21198940minus1119895=0 119875 minus 119861

119894 + 1 )

+ infinsum119894=1198940

120593(sum119894119895=0 119904119895+21198940 (119875 minus 119861)119894 + 1 )]] le 31198940120593 (119875 minus 119861)

+ 2119894011989611198962 119875 minus 119861 infinsum119894=1198940

120593( 1119894 + 1) + 1198961infinsum119894=1198940

120593

sdot (sum119894119895=0 119904119895 (119875)119894 + 1 ) le 31198940120593 (119875 minus 119861) + 211989401198622 119875

minus 119861 infinsum119894=1198940

120593( 1119894 + 1) + 119862infinsum119894=1198940

120593(sum119894119895=0 119904119895 (119875)119894 + 1 ) lt 120576

(25)

Corollary 33 119878119888119890119904119901(119883 119884) = 119865(119883 119884) if 1 lt 119901 lt infin

We express the accompanying theorem without verifica-tion these can be set up utilizing standard procedure

Theorem34 The function 119892(119875) = suminfin119894=0 120593(sum119894

119895=0 |119904119895(119875)|(119894+1))is a pre-quasi norm on 119878119888119890119904120593 if 120593 is an Orlicz function satisfyingΔ 2-condition and 120572120593 gt 1

We give the sufficient conditions on Orlicz-Cesaro meansequence spaces 119888119890119904120593 such that the components of the pre-quasi operator ideal 119878119888119890119904120593 are complete

Theorem 35 If 119883 and 119884 are Banach spaces 120593 is anOrlicz function satisfying Δ 2-condition and 120572120593 gt 1 then(119878119888119890119904120593(119883 119884) 119892) is a pre-quasi Banach operator idealProof Since 120593 is an Orlicz function satisfying Δ 2-condition then the function 119892(119875) = 984858((119904119899(119875))infin119899=0) =suminfin

119899=0 120593(sum119899119898=0 |119904119898(119875)|(119899 + 1)) is a pre-quasi norm on119878119888119890119904120593 Let (119875119898) be a Cauchy sequence in 119878119888119890119904120593(119883 119884) Since

L(119883 119884) supe 119878119888119890119904120593(119883 119884) and 120572120593 gt 1 we can find a constant120585 gt 0 such that

119892 (119875119894 minus 119875119895) = 984858 ((119904119899 (119875119894 minus 119875119895))infin119899=0)ge 984858 (1199040 (119875119894 minus 119875119895) 0 0 0 )= 984858 (10038171003817100381710038171003817119875119894 minus 11987511989510038171003817100381710038171003817 0 0 0 )ge 120585 10038171003817100381710038171003817119875119894 minus 11987511989510038171003817100381710038171003817 984858 (1 0 0 0 )

(26)

then (119875119898)119898isinN is also a Cauchy sequence in L(119883 119884) Whilethe space L(119883 119884) is a Banach space there exists 119875 isin L(119883 119884)

6 Journal of Function Spaces

such that lim119898997888rarrinfin 119875119898minus119875 = 0 while (119904119899(119875119898))infin119899=0 isin 119888119890119904120593 forevery 119898 isin N Since 984858 is continuous at 120579 and for some 119870 ge 1we obtain

119892 (119875) = 984858 ((119904119899 (119875))infin119899=0) = 984858 ((119904119899 (119875 minus 119875119898 + 119875119898))infin119899=0)le 119870984858 ((119904[1198992] (119875 minus 119875119898))infin119899=0)+ 119870984858 ((120572[1198992] (119875119898)infin119899=0))

le 119870984858 ((1003817100381710038171003817119875119898 minus 1198751003817100381710038171003817)infin119899=0) + 119870984858 ((119904119899 (119875119898)infin119899=0))lt infin

(27)

we have (119904119899(119875))infin119899=0 isin 119888119890119904120593 and then 119875 isin 119878119888119890119904120593(119883 119884)Corollary 36 If 119883 and 119884 are Banach spaces and 1 lt 119902 ltinfin then (119878119888119890119904119902(119883 119884) 119892) is quasi Banach operator ideal where119892(119875) = 984858((119904119899(119875))infin119899=0) = [suminfin

119899=0(sum119899119898=0 |119904119898(119875)|(119899 + 1))119902]1119902

Theorem 37 Let 1205931 1205932 be Orlicz functions and 1205721205931 gt 1 Forany infinite dimensional Banach spaces 119883 119884 and if there exist119887 1199050 gt 0 such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905) for all 119905 isin[0 1199050] it is true that

1198781198861199011199011198881198901199041205931(119883 119884) ⫋ 1198781198861199011199011198881198901199041205932

(119883 119884) ⫋ L (119883 119884) (28)

Proof Let119883 and119884 be infinite dimensional Banach spaces andthere exist 119887 1199050 gt 0 such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905)for all 119905 isin [0 1199050] if 119875 isin 1198781198861199011199011198881198901199041205931

(119883 119884) then (120572119899(119875)) isin 1198881198901199041205931 From Theorems 21 22 and 25 we have 1198881198901199041205931 sub 1198881198901199041205932 hence119875 isin 1198781198861199011199011198881198901199041205932

(119883 119884) It is easy to see that 1198781198861199011199011198881198901199041205932(119883 119884) sub L(119883 119884)

Corollary 38 For any infinite dimensional Banach spaces 119883119884 and 1 lt 119901 lt 119902 lt infin then 119878119886119901119901119888119890119904119901(119883 119884) ⫋ 119878119886119901119901119888119890119904119902

(119883 119884) ⫋L(119883 119884)

We now study some properties of the pre-quasi Banachoperator ideal 119878119888119890119904120593 Theorem 39 The pre-quasi Banach operator ideal (119878119888119890119904120593 119892) isinjective if the 119904-number sequence is injective

Proof Let 119879 isin L(119883 119884) and 119875 isin L(119884 1198840) be any metric injec-tion Assume that 119875119879 isin 119878119888119890119904120593(119883 1198840) then 984858(119904119899(119875119879)) lt infinSince the 119904-number sequence is injective we have 119904119899(119875119879) =119904119899(119879) for all 119879 isin L(119883 119884) 119899 isin N So 984858(119904119899(119879)) = 984858(119904119899(119875119879)) ltinfin Hence 119879 isin 119878119888119890119904120593(119883 119884) and clearly 119892(119879) = 119892(119875119879) isverified

Remark 40 The pre-quasi Banach operator ideal (119878119882119890119910119897119888119890119904120593

119892)and the pre-quasi Banach operator ideal (119878119866119890119897119888119890119904120593

119892) are injectivepre-quasi Banach operator ideal

Theorem 41 The pre-quasi Banach operator ideal (119878119888119890119904120593 119892) issurjective if the 119904-number sequence is surjective

Proof Let 119879 isin L(119883 119884) and 119875 isin L(1198830 119883) be any metricsurjection Suppose that 119879119875 isin 119878119888119890119904120593(1198830 119884) then 984858(119904119899(119879119875)) ltinfin Since the 119904-number sequence is surjective we have119904119899(119879119875) = 119904119899(119879) for all 119879 isin L(119883 119884) 119899 isin N So 984858(119904119899(119879)) =984858(119904119899(119879119875)) lt infin Hence 119879 isin 119878119888119890119904120593(119883 119884) and clearly 119892(119879) =119892(119879119875) is verifiedRemark 42 The pre-quasi Banach operator ideal (119878119862ℎ119886119899119892119888119890119904120593

119892) and the pre-quasi Banach operator ideal (119878119870119900119897

119888119890119904120593 119892) are

surjective pre-quasi Banach operator ideal

Likewise we have the accompanying inclusion relationsbetween the pre-quasi Banach operator ideals

Theorem 43 (1) 119878119886119901119901119888119890119904120593sube 119878119870119900119897

119888119890119904120593sube 119878119862ℎ119886119899119892119888119890119904120593

sube 119878119867119894119897119887119888119890119904120593

(2) 119878119886119901119901119888119890119904120593

sube 119878119866119890119897119888119890119904120593sube 119878119882119890119910119897

119888119890119904120593sube 119878119867119894119897119887

119888119890119904120593

Proof Since ℎ119899(119879) le 119910119899(119879) le 119889119899(119879) le 120572119899(119879) and ℎ119899(119879) le119909119899(119879) le 119888119899(119879) le 120572119899(119879) for every 119899 isin N and 984858 isnondecreasing we obtain

984858 (ℎ119899 (119879)) le 984858 (119910119899 (119879)) le 984858 (119889119899 (119879)) le 984858 (120572119899 (119879)) 984858 (ℎ119899 (119879)) le 984858 (119909119899 (119879)) le 984858 (119888119899 (119879)) le 984858 (120572119899 (119879)) (29)

Hence the result is as follows

We presently express the dual of the pre-quasi operatorideal formed by different 119904minus number sequences

Theorem 44 The pre-quasi operator ideal 119878119867119894119897119887119888119890119904120593

is completelysymmetric and the pre-quasi operator ideal 119878119886119901119901119888119890119904120593

is symmetric

Proof Since ℎ119899(1198791015840) = ℎ119899(119879) and 120572119899(1198791015840) le 120572119899(119879) for all 119879 isinL(119883 119884) we have 119878119867119894119897119887

119888119890119904120593= (119878119867119894119897119887

119888119890119904120593)1015840 and 119878119886119901119901119888119890119904120593

sube (119878119886119901119901119888119890119904120593)1015840

In perspective on Theorem 13 we express the followingresult without proof

Theorem 45 The pre-quasi operator ideal 119878119870119900119897119888119890119904120593

sube (119878119866119890119897119888119890119904120593)1015840 and

119878119866119890119897119888119890119904120593= (119878119870119900119897

119888119890119904120593)1015840 In addition if 119879 is a compact operator from 119883

to 119884 then 119878119870119900119897119888119890119904120593

= (119878119866119890119897119888119890119904120593)1015840

In perspective on Theorem 14 we express the followingresult without proof

Theorem 46 The pre-quasi operator ideal 119878119862ℎ119886119899119892119888119890119904120593= (119878119882119890119910119897

119888119890119904120593)1015840

and 119878119882119890119910119897119888119890119904120593

= (119878119862ℎ119886119899119892119888119890119904120593)1015840

Theorem47 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then the pre-quasi Banach operator ideal 119878119886119901119901119888119890119904120593

issmall

Proof Since 120593 is an Orlicz function and 120572120593 gt 1 take120573 = suminfin119894=0 120593(1(119894 + 1)) Then (119878119886119901119901119888119890119904120593

119892) where 119892(119879) =

Journal of Function Spaces 7

984858((120572119899(119879))infin119899=0) = (1120573)suminfin119899=0 120593(sum119899

119898=0 120572119898(119879)(119899 + 1)) is a pre-quasi Banach operator ideal Let119883 and 119884 be any two Banachspaces Assume that 119878119886119901119901119888119890119904120593

(119883 119884) = L(119883 119884) then there existsa constant 119862 gt 0 such that 119892(119879) le 119862119879 for all 119879 isin L(119883 119884)Suppose that119883 and119884 are infinite dimensional Banach spacesThen by Dvoretzkyrsquos theorem [8] for119898 isin N we have quotientspaces 119883119872119898 and subspaces 119873119898 of 119884 which can be mappedonto ℓ1198982 by isomorphisms 119881119898 and 119861119898 such that 119881119898119881minus1

119898 le2 and 119861119898119861minus1119898 le 2 Consider 119868119898 be the identity map onℓ1198982 119875119898 be the quotient map from 119883 onto 119883119872119898 and 119876119898 be

the natural embedding map from 119873119898 into 119884 Let V119899 be theBernstein numbers [7] then

1 = V119899 (119868119898) = V119899 (119861119898119861minus1119898 119868119898119881119898119881minus1

119898 )le 1003817100381710038171003817119861119898

1003817100381710038171003817 V119899 (119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817= 1003817100381710038171003817119861119898

1003817100381710038171003817 V119899 (119876119898119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817le 1003817100381710038171003817119861119898

1003817100381710038171003817 119889119899 (119876119898119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817= 1003817100381710038171003817119861119898

1003817100381710038171003817 119889119899 (119876119898119861minus1119898 119868119898119881119898119876119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817le 1003817100381710038171003817119861119898

1003817100381710038171003817 120572119899 (119876119898119861minus1119898 119868119898119881119898119876119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817

(30)

for 1 le 119894 le 119898 Now since 120593 is nondecreasing and having Δ 2-condition we have

119894sum119895=0

(1) le 119894sum119895=0

10038171003817100381710038171198611198981003817100381710038171003817 120572119895 (119876119898119861minus1

119898 119868119898119881119898119875119898) 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 997904rArr1119894 + 1 (119894 + 1) le 1003817100381710038171003817119861119898

1003817100381710038171003817( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898))

sdot 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 997904rArr120593 (1) le 119871 (1003817100381710038171003817119861119898

1003817100381710038171003817 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817)sdot 120593( 1119894 + 1

119894sum119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898))

(31)

Therefore

119898sum119894=0

120593 (1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817119898sum119894=0

120593

sdot ( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898)) 997904rArr

120593 (1)120573 (119898 + 1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 1120573119898sum119894=0

120593

sdot ( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898)) 997904rArr

120593 (1)120573 (119898 + 1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 119892 (119876119898119861minus1119898 119868119898119881119898119875119898) 997904rArr

120593 (1)120573 (119898 + 1) le 119871119862 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 10038171003817100381710038171003817119876119898119861minus1119898 11986811989811988111989811987511989810038171003817100381710038171003817 997904rArr

120593 (1)120573 (119898 + 1) le 119871119862 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 10038171003817100381710038171003817119876119898119861minus1119898

10038171003817100381710038171003817 10038171003817100381710038171198681198981003817100381710038171003817 10038171003817100381710038171198811198981198751198981003817100381710038171003817= 119871119862 1003817100381710038171003817119861119898

1003817100381710038171003817 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 10038171003817100381710038171003817119861minus1119898

10038171003817100381710038171003817 10038171003817100381710038171198681198981003817100381710038171003817 10038171003817100381710038171198811198981003817100381710038171003817 997904rArr120593 (1)120573 (119898 + 1) le 4119871119862

(32)

for some 119871 ge 1 Thus we arrive at a contradiction since 119898 isarbitrary Hence119883 and119884 both cannot be infinite dimensionalwhen 119878119886119901119901119888119890119904120593

(119883 119884) = L(119883 119884)Theorem48 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then the pre-quasi Banach operator ideal 119878119870119900119897

119888119890119904120593is

small

Corollary 49 If 119901 isin (1infin) then the quasi Banach operatorideal 119878119886119901119901119888119890119904119901

is small

Corollary 50 If 119901 isin (1infin) then the quasi Banach operatorideal 119878119870119900119897

119888119890119904119901is small

4 Examples

We give some examples which support our main results

Example 1 Let 120593 be an Orlicz function the subspace 119888119890119904ℎ120593 ofall order continuous elements of 119888119890119904120593 is defined as [27]

119888119890119904ℎ120593= 119909 isin 119888119890119904120593 forall119896gt0 exist119899119896isinN

infinsum119899=119899119896

120593(119896119899119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816) lt infin (33)

If 120593 is an Orlicz function satisfying Δ 2-condition and 120572120593 gt 1then the following conditions are satisfied

(1) 119878119888119890119904ℎ120593 is an operator ideal

(2) 119878119888119890119904ℎ120593(119883 119884) = 119865(119883 119884)(3) If 119883 and 119884 are Banach spaces then (119878119888119890119904ℎ120593(119883 119884) 119892) is

pre-quasi Banach operator ideal

(4) The pre-quasi Banach operator ideal 119878119886119901119901119888119890119904ℎ120593

is small

Proof Since 120593 is an Orlicz function satisfying Δ 2-conditionand 120572120593 gt 1 then from Theorem (5) in [31] we have 119888119890119904ℎ120593 =119888119890119904120593 which completes the proof

8 Journal of Function Spaces

Example 2 Let 120593 be defined as

120593 (119905) = 119886119897119905119897 + 119886119897minus1119905119897minus1 + + 1198861119905where 119886119894 gt 0 for all 1 le 119894 le 119897 119897 isin N 119897 gt 1 and 119905 ge 0 (34)

It is clear that 120593 is an Orlicz function and 120572120593 = 119897 gt 1 Also 120593is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) le 2119897 lt infin (35)

Then the following conditions are satisfied

(1) 119878119888119890119904120593 is an operator ideal

(2) 119878119888119890119904120593(119883 119884) = 119865(119883 119884)(3) If 119883 and 119884 are Banach spaces then (119878119888119890119904120593(119883 119884) 119892) is

pre-quasi Banach operator ideal(4) The pre-quasi Banach operator ideal 119878119886119901119901119888119890119904120593

is small

In the following two examples we will explain the impor-tance of the sufficient conditions

Example 3 Let 120593 be defined as

120593 (119905) =

0 if 119905 = 0minus119905ln 119905 if 119905 isin (0 1119890 ] 321198901199052 minus 119905 + 12119890 if 119905 isin (1119890 infin)

(36)

It is clear that 120593 is an Orlicz function Since suminfin119899=1 120593(1119899) =suminfin

119899=1(1119899 ln 119899) = infin hence 119888119890119904120593 = 0 The space 119878119888119890119904120593 is notoperator ideal since 119868119870 notin 119878119888119890119904120593 Also since120593 is convex functionand for 119901 gt 1 we have

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

1199051minus119901 ln 120582ln 120582119905

= lim119905997888rarr0+

(1 minus 119901) 1199051minus119901 ln 120582 = infin (37)

for all 120582 isin (0 1] then 120572120593 = 1 Although 120593 is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) = lim sup119905997888rarr0+

2 ln 119905ln 2119905 le 2 lt infin (38)

Example 4 Let 120593(119906) = int119906

0119891(119905)119889119905 where 119891(119905) is defined as

119891 (119905)

=

0 if 119905 = 01119899 if 119905 isin [ 1(119899 + 1) 1119899) for 119899 = 1 2 3 119905 if 119905 isin [1infin)

(39)

It is clear that 120593 is an Orlicz function Let 119879 isin 119878119888119890119904120593 with119904119899(119879) = 1119899 for all 119899 isin N We have for 119899 gt 2 that120593 (119904119899 (2119879)) = int2119899

0119891 (119905) 119889119905 gt int2119899

1119899119891 (119905) 119889119905

gt int1(119899minus1)

1119899119891 (119905) 119889119905 gt 1119899 (119899 minus 1)

119899120593 (119904119899 (119879)) = 119899int1119899

0119891 (119905) 119889119905

lt 119899 sup0le119905le1119899

119891 (119905) int1119899

01 119889119905 lt 1119899 (119899 minus 1)

(40)

Hence 2119879 notin 119878119888119890119904120593 so the space 119878119888119890119904120593 is not operator ideal and120593 notin Δ 2 Also since 120593 is convex function and for 119901 gt 1 wehave

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

119905minus119901 = infin (41)

for all 120582 isin (0 1] then 120572120593 = 1Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

The authors received no financial support for the researchauthorship and or publication of this article

Conflicts of Interest

The authors declare that 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 vol 20 North-Holland PublishingCompany Amsterdam The Netherlands 1980

[2] N F Mohamed and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalities andApplications vol 2013 article 186 2013

[3] N Faried and A A Bakery ldquoSmall operator ideals formed bys numbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1 article357 2018

[4] D Costarelli and G Vinti ldquoA quantitative estimate for the sam-pling kantorovich series in terms of the modulus of continuityin orlicz spacesrdquo Constructive Mathematical Analysis vol 2 no1 pp 8ndash14 2019

Journal of Function Spaces 9

[5] F Altomare ldquoIterates of markov operators and construc-tive approximation of semigroupsrdquo Constructive MathematicalAnalysis vol 2 no 1 pp 22ndash39 2019

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

[7] A Pietsch ldquos-numbers of operators in banach spacesrdquo StudiaMathematica vol 51 pp 201ndash223 1974

[8] A Pietsch Operator Ideals VEB Deutscher Verlag DerWis-senschaften Berlin Germany 1978

[9] C V Hutton ldquoOn the approximation numbers of an operatorand its adjointrdquo Mathematische Annalen vol 210 pp 277ndash2801974

[10] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 pp 267ndash278 1974

[11] A Lima and E Oja ldquoIdeals of finite rank operators intersectionproperties of balls and the approximation propertyrdquo StudiaMathematica vol 133 no 2 pp 175ndash186 1999

[12] M A Krasnoselskii and Y B Rutickii Convex Functions andOrlicz Spaces Gorningen Netherlands 1961

[13] W Orlicz and U Raume ldquoLMrdquo Bulletin International delrsquoAcademie Polonaise des Sciences et des Lettres Serie A pp 93ndash107 1936

[14] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971

[15] J Lindenstrauss and L Tzafriri Classical Banach Spaces vol92 of I Sequence spaces Ergebnisse der Mathematik und ihrerGrenzgebiete Springer-Verlag Berlin Germany 1977

[16] Y Altin M Et and B C Tripathy ldquoThe sequence space|119873119901|(119872 119903 119902 119904) on seminormed spacesrdquo Applied Mathematicsand Computation vol 154 no 2 pp 423ndash430 2004

[17] M Et L P Lee and B C Tripathy ldquoStrongly almost (119881 120582)(119903)-summable sequences defined by Orlicz functionsrdquo HokkaidoMathematical Journal vol 35 no 1 pp 197ndash213 2006

[18] M Et Y Altin B Choudhary and B C Tripathy ldquoOn someclasses of sequences defined by sequences of Orlicz functionsrdquoMathematical Inequalities amp Applications vol 9 no 2 pp 335ndash342 2006

[19] B C Tripathy and S Mahanta ldquoOn a class of differencesequences related to the l119901 space defined by Orlicz functionsrdquoMathematica Slovaca vol 57 no 2 pp 171ndash178 2007

[20] B C Tripathy and H Dutta ldquoOn some new paranormed differ-ence sequence spaces defined by Orlicz functionsrdquo KyungpookMathematical Journal vol 50 no 1 pp 59ndash69 2010

[21] B C Tripathy and B Hazarika ldquoI-convergent sequences spacesdefined by Orlicz functionrdquo Acta Mathematicae ApplicataeSinica vol 27 no 1 pp 149ndash154 2011

[22] S A Mohiuddine and B Hazarika ldquoSome classes of ideal con-vergent sequences and generalized difference matrix operatorrdquoFilomat vol 31 no 6 pp 1827ndash1834 2017

[23] S A Mohiuddine and K Raj ldquoVector valued Orlicz-Lorentzsequence spaces and their operator idealsrdquo Journal of NonlinearSciences and Applications JNSA vol 10 no 2 pp 338ndash353 2017

[24] S Abdul Mohiuddine K Raj and A Alotaibi ldquoGeneralizedspaces of double sequences for Orlicz functions and bounded-regular matrices over n-normed spacesrdquo Journal of Inequalitiesand Applications vol 2014 article 332 2014

[25] S A Mohiuddine S K Sharma and D A Abuzaid ldquoSomeseminormed difference sequence spaces over n-normed spacesdefined by a musielak-orlicz function of order (120572120573)rdquo Journal ofFunction Spaces vol 2018 Article ID 4312817 11 pages 2018

[26] S-K Lim and P Y Lee ldquoAnOrlicz extension of Cesaro sequencespacesrdquo Roczniki Polskiego Towarzystwa Matematycznego SeriaI Commentationes Mathematicae Prace Matematyczne vol 28no 1 pp 117ndash128 1988

[27] Y Cui H Hudzik N Petrot S Suantai and A SzymaszkiewiczldquoBasic topological and geometric properties of Cesaro-Orliczspacesrdquo The Proceedings of the Indian Academy of Sciences ndashMathematical Sciences vol 115 no 4 pp 461ndash476 2005

[28] LMaligranda N Petrot and S Suantai ldquoOn the James constantand B-convexity of Cesaro and Cesaro-Orlicz sequence spacesrdquoJournal of Mathematical Analysis and Applications vol 326 pp312ndash326 2007

[29] P Foralewski H Hudzik and A Szymaszkiewicz ldquoLocalrotundity structure of Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 345 no 1 pp410ndash419 2008

[30] G M Leibowitz ldquoA note on the Cesaro sequence spacesrdquoTamkang Journal of Mathematics vol 2 no 2 pp 151ndash157 1971

[31] D Kubiak ldquoA note on Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 349 no 1 pp291ndash296 2009

[32] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbers (in Russian)rdquo in Theoryof Operators in Functional Spaces pp 206ndash211 Academy ofScience Siberian section Novosibirsk Russia 1977

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4 Journal of Function Spaces

(iv) if |119909119894| le |119910119894| for all 119894 isin N then 984858((119909119894)) le 984858((119910119894))(v) for some 1198700 ge 1 we have

984858 ((119909119894)) le 984858 ((119909[1198942])) le 1198700984858 ((119909119894)) (16)

(vi) the set of all finite sequences is 984858-dense in E Thismeans for each 119909 = (119909119894)infin119894=119900 isin E and for each 120576 gt 0there exists119898 isin N such that 984858((119909119894)infin119894=119898) lt 120576

(vii) there exists a constant 120585 gt 0 such that984858(120582 0 0 0 ) ge 120585|120582|984858(1 0 0 0 ) for any 120582 isin R

We denote (E984858 984858) for the linear spaceE equippedwith themetrizable topology generated by 984858Theorem 25 (see [32]) If 119883 119884 are infinite dimensionalBanach spaces and 120582119894 is a monotonic decreasing sequence tozero then there exists a bounded linear operator 119879 such that

1161205823119894 le 120572119894 (119879) le 8120582119894+1 (17)

Notations 26 (see [3])

119878E fl 119878E(119883 119884) 119883 and 119884 are Banach Spaceswhere119878E(119883 119884) fl 119879 isin L(119883 119884) ((119904119894(119879))infin119894=0 isin E Also119878119886119901119901E fl 119878119886119901119901E (119883 119884) 119883 and 119884 are Banach Spaceswhere119878119886119901119901E (119883 119884) fl 119879 isin L(119883 119884) ((120572119894(119879))infin119894=0 isin E

Theorem 27 (see [3]) If E is a (sss) then 119878E is an operatorideal

The concept of pre-quasi operator ideal which is moregeneral than the usual classes of operator ideal

Definition 28 (see [3]) A function 119892 Ω 997888rarr [0infin) issaid to be a pre-quasi norm on the ideal Ω fulfilling theaccompanying conditions

(1) for all119879 isin Ω(119883 119884) 119892(119879) ge 0 and 119892(119879) = 0 if and onlyif 119879 = 0

(2) there exists a constant 119871 ge 1 such that 119892(120573119879) le119871|120573|119892(119879) for all 119879 isin Ω(119883 119884) and 120573 isin R(3) there exists a constant 119870 ge 1 such that 119892(1198791 + 1198792) le119870[119892(1198791) + 119892(1198792)] for all 1198791 1198792 isin Ω(119883 119884)(4) there exists a constant 119862 ge 1 such that if 119875 isin

L(1198830 119883) 119877 isin Ω(119883 119884) and 119879 isin L(119884 1198840) then119892(119879119877119875) le 119862 119879119892(119877) 119875 where 1198830 and 1198840 arenormed spaces

Theorem 29 (see [3]) Every quasi norm on the ideal Ω is apre-quasi norm on the ideal Ω

Here and after we define 119890119894 = 0 0 1 0 0 where1 appears at the 119894119905ℎ place for all 119894 isin N

3 Main Results

We give here the conditions onOrlicz-Cesaromean sequencespaces 119888119890119904120593 such that the class 119878119888119890119904120593 of all bounded linearoperators between arbitrary Banach spaces with its sequenceof 119904minusnumbers which belong to 119888119890119904120593 forms an operator ideal

Theorem30 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then 119878119888119890119904120593 is an operator ideal

Proof (1-i) Let 119909 119910 isin 119888119890119904120593 Since 120593 is nondecreasing convexand satisfying Δ 2-condition we get for some 119896 gt 0 that

infinsum119899=0

120593(sum119899119894=0

1003816100381610038161003816119909119894 + 1199101198941003816100381610038161003816119899 + 1 )le 119896 [infinsum

119899=0

120593(sum119899119894=0

10038161003816100381610038161199091198941003816100381610038161003816119899 + 1 ) + infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199101198941003816100381610038161003816119899 + 1 )] lt infin(18)

then 119909 + 119910 isin 119888119890119904120593(1-ii) Let 120582 isin R and 119909 isin 119888119890119904120593 and since 120593 is convex and

satisfying Δ 2-condition we get for some 119896 gt 0 thatinfinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161205821199091198941003816100381610038161003816119899 + 1 ) le |120582| 119896infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199091198941003816100381610038161003816119899 + 1 ) lt infin (19)

then120582119909 isin 119888119890119904120593 from (1-i) and (1-ii) 119888119890119904120593 is a linear space Since119890119899 isin ℓ120593 for all 119899 isin N and 120572120593 gt 1 then fromTheorem 20 weget 119890119899 isin 119888119890119904120593 for all 119899 isin N(2) Let |119909119899| le |119910119899| for all 119899 isin N and 119910 isin 119888119890119904120593 since 120593 isnondecreasing then we have

infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199091198941003816100381610038161003816119899 + 1 ) le infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199101198941003816100381610038161003816119899 + 1 ) lt infin (20)

and we get 119909 isin 119888119890119904120593(3) Let (119909119899) isin 119888119890119904120593 Since 120593 is satisfying Δ 2-condition weget for some 119896 gt 0 that

infinsum119899=0

120593(sum119899119894=0

1003816100381610038161003816119909[1198942]1003816100381610038161003816119899 + 1 ) le (119896 + 1) infinsum119899=0

120593(sum119899119894=0

10038161003816100381610038161199091198941003816100381610038161003816119899 + 1 ) lt infin (21)

then (119909[1198992]) isin 119888119890119904120593Then 119888119890119904120593 is a (sss) hence byTheorem 27119878119888119890119904120593 is an operator ideal

Corollary 31 119878119888119890119904119902 is an operator ideal if 1 lt 119902 lt infin

We give the conditions on Orlicz-Cesaro mean sequencespaces 119888119890119904120593 such that the ideal of the finite rank operators isdense in 119878119888119890119904120593(119883 119884)Theorem 32 119878119888119890119904120593(119883 119884) = 119865(X 119884) if 120593 is an Orlicz functionsatisfying Δ 2-condition and 120572120593 gt 1Proof Let us define 984858(119906) = suminfin

119894=0 120593(sum119894119895=0 |119906119895|(119894 + 1)) on 119888119890119904120593

First we have to show that 119865(119883 119884) sube 119878119888119890119904120593(119883 119884) Since120572120593 gt 1 we have 119890119894 isin 119888119890119904120593 for each 119894 isin N and 120593 is an

Journal of Function Spaces 5

Orlicz function satisfying Δ 2-condition so for each finiteoperator119875 isin 119865(119883 119884) ie we obtain (119904119894(119875))infin119894=0 which containsonly finitely many terms different from zero hence 119875 isin119878119888119890119904120593(119883 119884) Currently we prove that 119878119888119890119904120593(119883 119884) sube 119865(119883 119884)let 119875 isin 119878119888119890119904120593(119883 119884) we have (119904119894(119875))infin119894=0 isin 119888119890119904120593 and hence984858(119904119894(119875))infin119894=0 lt infin By taking 120576 isin (0 1) hence there exists a1198940 isin N minus 0 such that 984858((119904119894(119875))infin119894=1198940) lt 12057691205751198622 for some 119888 ge 1where 120575 = max1 suminfin

119894=1198940120593(1(119894 + 1)) As 119904119894(119875) is decreasing

for every 119894 isin N and 120593 is nondecreasing we have

1198940120593 (11990421198940 (119875)) le21198940sum

119894=1198940+1

120593(sum119894119895=0 119904119895 (119875)119894 + 1 )

le infinsum119894=1198940

120593(sum119894119895=0 119904119895 (119875)119894 + 1 ) lt 12057691205751198622

(22)

Hence there exists 119861 isin 11986521198940(119883 119884) such that rank 119861 le 21198940 and1198940120593 (119875 minus 119861) le 21198940sum

119894=1198940+1

120593(sum119894119895=0 119875 minus 119861119894 + 1 ) lt 12057691205751198622

(23)

Since 120593 is right continuous at 0 and nondecreasing then onconsidering this

119875 minus 119861 lt 120576611986221198940120575 (24)

Let 1198961 gt 0 1198962 gt 0 and 119862 = max1 1198961 1198962 since 120593 is Orliczfunction and by using (22) (23) and (24) we have

119889 (119875 119861) = 984858 (119904119894 (119875 minus 119861))infin119894=0 = 31198940minus1sum119894=0

120593

sdot (sum119894119895=0 119904119895 (119875 minus 119861)119894 + 1 ) + infinsum

119894=31198940

120593(sum119894119895=0 119904119895 (119875 minus 119861)119894 + 1 )

le 31198940minus1sum119894=0

120593(sum119894119895=0 119875 minus 119861119894 + 1 ) + infinsum

119894=1198940

120593

sdot (sum119894+21198940119895=0 119904119895 (119875 minus 119861)119894 + 1 ) le 31198940120593 (119875 minus 119861) + infinsum

119894=1198940

120593

sdot (sum119894+21198940119895=0 119904119895 (119875 minus 119861)119894 + 1 ) le 31198940120593 (119875 minus 119861)

+ infinsum119894=1198940

120593(sum21198940minus1119895=0 119904119895 (119875 minus 119861) + sum119894+21198940

119895=21198940119904119895 (119875 minus 119861)119894 + 1 )

le 31198940120593 (119875 minus 119861) + 1198961 [[infinsum119894=1198940

120593(sum21198940minus1119895=0 119904119895 (119875 minus 119861)119894 + 1 )

+ infinsum119894=1198940

120593(sum119894+21198940119895=21198940

119904119895 (119875 minus 119861)119894 + 1 )]] le 31198940120593 (119875 minus 119861)

+ 1198961 [[infinsum119894=1198940

120593(sum21198940minus1119895=0 119875 minus 119861

119894 + 1 )

+ infinsum119894=1198940

120593(sum119894119895=0 119904119895+21198940 (119875 minus 119861)119894 + 1 )]] le 31198940120593 (119875 minus 119861)

+ 2119894011989611198962 119875 minus 119861 infinsum119894=1198940

120593( 1119894 + 1) + 1198961infinsum119894=1198940

120593

sdot (sum119894119895=0 119904119895 (119875)119894 + 1 ) le 31198940120593 (119875 minus 119861) + 211989401198622 119875

minus 119861 infinsum119894=1198940

120593( 1119894 + 1) + 119862infinsum119894=1198940

120593(sum119894119895=0 119904119895 (119875)119894 + 1 ) lt 120576

(25)

Corollary 33 119878119888119890119904119901(119883 119884) = 119865(119883 119884) if 1 lt 119901 lt infin

We express the accompanying theorem without verifica-tion these can be set up utilizing standard procedure

Theorem34 The function 119892(119875) = suminfin119894=0 120593(sum119894

119895=0 |119904119895(119875)|(119894+1))is a pre-quasi norm on 119878119888119890119904120593 if 120593 is an Orlicz function satisfyingΔ 2-condition and 120572120593 gt 1

We give the sufficient conditions on Orlicz-Cesaro meansequence spaces 119888119890119904120593 such that the components of the pre-quasi operator ideal 119878119888119890119904120593 are complete

Theorem 35 If 119883 and 119884 are Banach spaces 120593 is anOrlicz function satisfying Δ 2-condition and 120572120593 gt 1 then(119878119888119890119904120593(119883 119884) 119892) is a pre-quasi Banach operator idealProof Since 120593 is an Orlicz function satisfying Δ 2-condition then the function 119892(119875) = 984858((119904119899(119875))infin119899=0) =suminfin

119899=0 120593(sum119899119898=0 |119904119898(119875)|(119899 + 1)) is a pre-quasi norm on119878119888119890119904120593 Let (119875119898) be a Cauchy sequence in 119878119888119890119904120593(119883 119884) Since

L(119883 119884) supe 119878119888119890119904120593(119883 119884) and 120572120593 gt 1 we can find a constant120585 gt 0 such that

119892 (119875119894 minus 119875119895) = 984858 ((119904119899 (119875119894 minus 119875119895))infin119899=0)ge 984858 (1199040 (119875119894 minus 119875119895) 0 0 0 )= 984858 (10038171003817100381710038171003817119875119894 minus 11987511989510038171003817100381710038171003817 0 0 0 )ge 120585 10038171003817100381710038171003817119875119894 minus 11987511989510038171003817100381710038171003817 984858 (1 0 0 0 )

(26)

then (119875119898)119898isinN is also a Cauchy sequence in L(119883 119884) Whilethe space L(119883 119884) is a Banach space there exists 119875 isin L(119883 119884)

6 Journal of Function Spaces

such that lim119898997888rarrinfin 119875119898minus119875 = 0 while (119904119899(119875119898))infin119899=0 isin 119888119890119904120593 forevery 119898 isin N Since 984858 is continuous at 120579 and for some 119870 ge 1we obtain

119892 (119875) = 984858 ((119904119899 (119875))infin119899=0) = 984858 ((119904119899 (119875 minus 119875119898 + 119875119898))infin119899=0)le 119870984858 ((119904[1198992] (119875 minus 119875119898))infin119899=0)+ 119870984858 ((120572[1198992] (119875119898)infin119899=0))

le 119870984858 ((1003817100381710038171003817119875119898 minus 1198751003817100381710038171003817)infin119899=0) + 119870984858 ((119904119899 (119875119898)infin119899=0))lt infin

(27)

we have (119904119899(119875))infin119899=0 isin 119888119890119904120593 and then 119875 isin 119878119888119890119904120593(119883 119884)Corollary 36 If 119883 and 119884 are Banach spaces and 1 lt 119902 ltinfin then (119878119888119890119904119902(119883 119884) 119892) is quasi Banach operator ideal where119892(119875) = 984858((119904119899(119875))infin119899=0) = [suminfin

119899=0(sum119899119898=0 |119904119898(119875)|(119899 + 1))119902]1119902

Theorem 37 Let 1205931 1205932 be Orlicz functions and 1205721205931 gt 1 Forany infinite dimensional Banach spaces 119883 119884 and if there exist119887 1199050 gt 0 such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905) for all 119905 isin[0 1199050] it is true that

1198781198861199011199011198881198901199041205931(119883 119884) ⫋ 1198781198861199011199011198881198901199041205932

(119883 119884) ⫋ L (119883 119884) (28)

Proof Let119883 and119884 be infinite dimensional Banach spaces andthere exist 119887 1199050 gt 0 such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905)for all 119905 isin [0 1199050] if 119875 isin 1198781198861199011199011198881198901199041205931

(119883 119884) then (120572119899(119875)) isin 1198881198901199041205931 From Theorems 21 22 and 25 we have 1198881198901199041205931 sub 1198881198901199041205932 hence119875 isin 1198781198861199011199011198881198901199041205932

(119883 119884) It is easy to see that 1198781198861199011199011198881198901199041205932(119883 119884) sub L(119883 119884)

Corollary 38 For any infinite dimensional Banach spaces 119883119884 and 1 lt 119901 lt 119902 lt infin then 119878119886119901119901119888119890119904119901(119883 119884) ⫋ 119878119886119901119901119888119890119904119902

(119883 119884) ⫋L(119883 119884)

We now study some properties of the pre-quasi Banachoperator ideal 119878119888119890119904120593 Theorem 39 The pre-quasi Banach operator ideal (119878119888119890119904120593 119892) isinjective if the 119904-number sequence is injective

Proof Let 119879 isin L(119883 119884) and 119875 isin L(119884 1198840) be any metric injec-tion Assume that 119875119879 isin 119878119888119890119904120593(119883 1198840) then 984858(119904119899(119875119879)) lt infinSince the 119904-number sequence is injective we have 119904119899(119875119879) =119904119899(119879) for all 119879 isin L(119883 119884) 119899 isin N So 984858(119904119899(119879)) = 984858(119904119899(119875119879)) ltinfin Hence 119879 isin 119878119888119890119904120593(119883 119884) and clearly 119892(119879) = 119892(119875119879) isverified

Remark 40 The pre-quasi Banach operator ideal (119878119882119890119910119897119888119890119904120593

119892)and the pre-quasi Banach operator ideal (119878119866119890119897119888119890119904120593

119892) are injectivepre-quasi Banach operator ideal

Theorem 41 The pre-quasi Banach operator ideal (119878119888119890119904120593 119892) issurjective if the 119904-number sequence is surjective

Proof Let 119879 isin L(119883 119884) and 119875 isin L(1198830 119883) be any metricsurjection Suppose that 119879119875 isin 119878119888119890119904120593(1198830 119884) then 984858(119904119899(119879119875)) ltinfin Since the 119904-number sequence is surjective we have119904119899(119879119875) = 119904119899(119879) for all 119879 isin L(119883 119884) 119899 isin N So 984858(119904119899(119879)) =984858(119904119899(119879119875)) lt infin Hence 119879 isin 119878119888119890119904120593(119883 119884) and clearly 119892(119879) =119892(119879119875) is verifiedRemark 42 The pre-quasi Banach operator ideal (119878119862ℎ119886119899119892119888119890119904120593

119892) and the pre-quasi Banach operator ideal (119878119870119900119897

119888119890119904120593 119892) are

surjective pre-quasi Banach operator ideal

Likewise we have the accompanying inclusion relationsbetween the pre-quasi Banach operator ideals

Theorem 43 (1) 119878119886119901119901119888119890119904120593sube 119878119870119900119897

119888119890119904120593sube 119878119862ℎ119886119899119892119888119890119904120593

sube 119878119867119894119897119887119888119890119904120593

(2) 119878119886119901119901119888119890119904120593

sube 119878119866119890119897119888119890119904120593sube 119878119882119890119910119897

119888119890119904120593sube 119878119867119894119897119887

119888119890119904120593

Proof Since ℎ119899(119879) le 119910119899(119879) le 119889119899(119879) le 120572119899(119879) and ℎ119899(119879) le119909119899(119879) le 119888119899(119879) le 120572119899(119879) for every 119899 isin N and 984858 isnondecreasing we obtain

984858 (ℎ119899 (119879)) le 984858 (119910119899 (119879)) le 984858 (119889119899 (119879)) le 984858 (120572119899 (119879)) 984858 (ℎ119899 (119879)) le 984858 (119909119899 (119879)) le 984858 (119888119899 (119879)) le 984858 (120572119899 (119879)) (29)

Hence the result is as follows

We presently express the dual of the pre-quasi operatorideal formed by different 119904minus number sequences

Theorem 44 The pre-quasi operator ideal 119878119867119894119897119887119888119890119904120593

is completelysymmetric and the pre-quasi operator ideal 119878119886119901119901119888119890119904120593

is symmetric

Proof Since ℎ119899(1198791015840) = ℎ119899(119879) and 120572119899(1198791015840) le 120572119899(119879) for all 119879 isinL(119883 119884) we have 119878119867119894119897119887

119888119890119904120593= (119878119867119894119897119887

119888119890119904120593)1015840 and 119878119886119901119901119888119890119904120593

sube (119878119886119901119901119888119890119904120593)1015840

In perspective on Theorem 13 we express the followingresult without proof

Theorem 45 The pre-quasi operator ideal 119878119870119900119897119888119890119904120593

sube (119878119866119890119897119888119890119904120593)1015840 and

119878119866119890119897119888119890119904120593= (119878119870119900119897

119888119890119904120593)1015840 In addition if 119879 is a compact operator from 119883

to 119884 then 119878119870119900119897119888119890119904120593

= (119878119866119890119897119888119890119904120593)1015840

In perspective on Theorem 14 we express the followingresult without proof

Theorem 46 The pre-quasi operator ideal 119878119862ℎ119886119899119892119888119890119904120593= (119878119882119890119910119897

119888119890119904120593)1015840

and 119878119882119890119910119897119888119890119904120593

= (119878119862ℎ119886119899119892119888119890119904120593)1015840

Theorem47 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then the pre-quasi Banach operator ideal 119878119886119901119901119888119890119904120593

issmall

Proof Since 120593 is an Orlicz function and 120572120593 gt 1 take120573 = suminfin119894=0 120593(1(119894 + 1)) Then (119878119886119901119901119888119890119904120593

119892) where 119892(119879) =

Journal of Function Spaces 7

984858((120572119899(119879))infin119899=0) = (1120573)suminfin119899=0 120593(sum119899

119898=0 120572119898(119879)(119899 + 1)) is a pre-quasi Banach operator ideal Let119883 and 119884 be any two Banachspaces Assume that 119878119886119901119901119888119890119904120593

(119883 119884) = L(119883 119884) then there existsa constant 119862 gt 0 such that 119892(119879) le 119862119879 for all 119879 isin L(119883 119884)Suppose that119883 and119884 are infinite dimensional Banach spacesThen by Dvoretzkyrsquos theorem [8] for119898 isin N we have quotientspaces 119883119872119898 and subspaces 119873119898 of 119884 which can be mappedonto ℓ1198982 by isomorphisms 119881119898 and 119861119898 such that 119881119898119881minus1

119898 le2 and 119861119898119861minus1119898 le 2 Consider 119868119898 be the identity map onℓ1198982 119875119898 be the quotient map from 119883 onto 119883119872119898 and 119876119898 be

the natural embedding map from 119873119898 into 119884 Let V119899 be theBernstein numbers [7] then

1 = V119899 (119868119898) = V119899 (119861119898119861minus1119898 119868119898119881119898119881minus1

119898 )le 1003817100381710038171003817119861119898

1003817100381710038171003817 V119899 (119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817= 1003817100381710038171003817119861119898

1003817100381710038171003817 V119899 (119876119898119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817le 1003817100381710038171003817119861119898

1003817100381710038171003817 119889119899 (119876119898119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817= 1003817100381710038171003817119861119898

1003817100381710038171003817 119889119899 (119876119898119861minus1119898 119868119898119881119898119876119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817le 1003817100381710038171003817119861119898

1003817100381710038171003817 120572119899 (119876119898119861minus1119898 119868119898119881119898119876119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817

(30)

for 1 le 119894 le 119898 Now since 120593 is nondecreasing and having Δ 2-condition we have

119894sum119895=0

(1) le 119894sum119895=0

10038171003817100381710038171198611198981003817100381710038171003817 120572119895 (119876119898119861minus1

119898 119868119898119881119898119875119898) 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 997904rArr1119894 + 1 (119894 + 1) le 1003817100381710038171003817119861119898

1003817100381710038171003817( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898))

sdot 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 997904rArr120593 (1) le 119871 (1003817100381710038171003817119861119898

1003817100381710038171003817 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817)sdot 120593( 1119894 + 1

119894sum119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898))

(31)

Therefore

119898sum119894=0

120593 (1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817119898sum119894=0

120593

sdot ( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898)) 997904rArr

120593 (1)120573 (119898 + 1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 1120573119898sum119894=0

120593

sdot ( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898)) 997904rArr

120593 (1)120573 (119898 + 1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 119892 (119876119898119861minus1119898 119868119898119881119898119875119898) 997904rArr

120593 (1)120573 (119898 + 1) le 119871119862 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 10038171003817100381710038171003817119876119898119861minus1119898 11986811989811988111989811987511989810038171003817100381710038171003817 997904rArr

120593 (1)120573 (119898 + 1) le 119871119862 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 10038171003817100381710038171003817119876119898119861minus1119898

10038171003817100381710038171003817 10038171003817100381710038171198681198981003817100381710038171003817 10038171003817100381710038171198811198981198751198981003817100381710038171003817= 119871119862 1003817100381710038171003817119861119898

1003817100381710038171003817 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 10038171003817100381710038171003817119861minus1119898

10038171003817100381710038171003817 10038171003817100381710038171198681198981003817100381710038171003817 10038171003817100381710038171198811198981003817100381710038171003817 997904rArr120593 (1)120573 (119898 + 1) le 4119871119862

(32)

for some 119871 ge 1 Thus we arrive at a contradiction since 119898 isarbitrary Hence119883 and119884 both cannot be infinite dimensionalwhen 119878119886119901119901119888119890119904120593

(119883 119884) = L(119883 119884)Theorem48 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then the pre-quasi Banach operator ideal 119878119870119900119897

119888119890119904120593is

small

Corollary 49 If 119901 isin (1infin) then the quasi Banach operatorideal 119878119886119901119901119888119890119904119901

is small

Corollary 50 If 119901 isin (1infin) then the quasi Banach operatorideal 119878119870119900119897

119888119890119904119901is small

4 Examples

We give some examples which support our main results

Example 1 Let 120593 be an Orlicz function the subspace 119888119890119904ℎ120593 ofall order continuous elements of 119888119890119904120593 is defined as [27]

119888119890119904ℎ120593= 119909 isin 119888119890119904120593 forall119896gt0 exist119899119896isinN

infinsum119899=119899119896

120593(119896119899119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816) lt infin (33)

If 120593 is an Orlicz function satisfying Δ 2-condition and 120572120593 gt 1then the following conditions are satisfied

(1) 119878119888119890119904ℎ120593 is an operator ideal

(2) 119878119888119890119904ℎ120593(119883 119884) = 119865(119883 119884)(3) If 119883 and 119884 are Banach spaces then (119878119888119890119904ℎ120593(119883 119884) 119892) is

pre-quasi Banach operator ideal

(4) The pre-quasi Banach operator ideal 119878119886119901119901119888119890119904ℎ120593

is small

Proof Since 120593 is an Orlicz function satisfying Δ 2-conditionand 120572120593 gt 1 then from Theorem (5) in [31] we have 119888119890119904ℎ120593 =119888119890119904120593 which completes the proof

8 Journal of Function Spaces

Example 2 Let 120593 be defined as

120593 (119905) = 119886119897119905119897 + 119886119897minus1119905119897minus1 + + 1198861119905where 119886119894 gt 0 for all 1 le 119894 le 119897 119897 isin N 119897 gt 1 and 119905 ge 0 (34)

It is clear that 120593 is an Orlicz function and 120572120593 = 119897 gt 1 Also 120593is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) le 2119897 lt infin (35)

Then the following conditions are satisfied

(1) 119878119888119890119904120593 is an operator ideal

(2) 119878119888119890119904120593(119883 119884) = 119865(119883 119884)(3) If 119883 and 119884 are Banach spaces then (119878119888119890119904120593(119883 119884) 119892) is

pre-quasi Banach operator ideal(4) The pre-quasi Banach operator ideal 119878119886119901119901119888119890119904120593

is small

In the following two examples we will explain the impor-tance of the sufficient conditions

Example 3 Let 120593 be defined as

120593 (119905) =

0 if 119905 = 0minus119905ln 119905 if 119905 isin (0 1119890 ] 321198901199052 minus 119905 + 12119890 if 119905 isin (1119890 infin)

(36)

It is clear that 120593 is an Orlicz function Since suminfin119899=1 120593(1119899) =suminfin

119899=1(1119899 ln 119899) = infin hence 119888119890119904120593 = 0 The space 119878119888119890119904120593 is notoperator ideal since 119868119870 notin 119878119888119890119904120593 Also since120593 is convex functionand for 119901 gt 1 we have

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

1199051minus119901 ln 120582ln 120582119905

= lim119905997888rarr0+

(1 minus 119901) 1199051minus119901 ln 120582 = infin (37)

for all 120582 isin (0 1] then 120572120593 = 1 Although 120593 is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) = lim sup119905997888rarr0+

2 ln 119905ln 2119905 le 2 lt infin (38)

Example 4 Let 120593(119906) = int119906

0119891(119905)119889119905 where 119891(119905) is defined as

119891 (119905)

=

0 if 119905 = 01119899 if 119905 isin [ 1(119899 + 1) 1119899) for 119899 = 1 2 3 119905 if 119905 isin [1infin)

(39)

It is clear that 120593 is an Orlicz function Let 119879 isin 119878119888119890119904120593 with119904119899(119879) = 1119899 for all 119899 isin N We have for 119899 gt 2 that120593 (119904119899 (2119879)) = int2119899

0119891 (119905) 119889119905 gt int2119899

1119899119891 (119905) 119889119905

gt int1(119899minus1)

1119899119891 (119905) 119889119905 gt 1119899 (119899 minus 1)

119899120593 (119904119899 (119879)) = 119899int1119899

0119891 (119905) 119889119905

lt 119899 sup0le119905le1119899

119891 (119905) int1119899

01 119889119905 lt 1119899 (119899 minus 1)

(40)

Hence 2119879 notin 119878119888119890119904120593 so the space 119878119888119890119904120593 is not operator ideal and120593 notin Δ 2 Also since 120593 is convex function and for 119901 gt 1 wehave

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

119905minus119901 = infin (41)

for all 120582 isin (0 1] then 120572120593 = 1Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

The authors received no financial support for the researchauthorship and or publication of this article

Conflicts of Interest

The authors declare that 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 vol 20 North-Holland PublishingCompany Amsterdam The Netherlands 1980

[2] N F Mohamed and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalities andApplications vol 2013 article 186 2013

[3] N Faried and A A Bakery ldquoSmall operator ideals formed bys numbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1 article357 2018

[4] D Costarelli and G Vinti ldquoA quantitative estimate for the sam-pling kantorovich series in terms of the modulus of continuityin orlicz spacesrdquo Constructive Mathematical Analysis vol 2 no1 pp 8ndash14 2019

Journal of Function Spaces 9

[5] F Altomare ldquoIterates of markov operators and construc-tive approximation of semigroupsrdquo Constructive MathematicalAnalysis vol 2 no 1 pp 22ndash39 2019

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

[7] A Pietsch ldquos-numbers of operators in banach spacesrdquo StudiaMathematica vol 51 pp 201ndash223 1974

[8] A Pietsch Operator Ideals VEB Deutscher Verlag DerWis-senschaften Berlin Germany 1978

[9] C V Hutton ldquoOn the approximation numbers of an operatorand its adjointrdquo Mathematische Annalen vol 210 pp 277ndash2801974

[10] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 pp 267ndash278 1974

[11] A Lima and E Oja ldquoIdeals of finite rank operators intersectionproperties of balls and the approximation propertyrdquo StudiaMathematica vol 133 no 2 pp 175ndash186 1999

[12] M A Krasnoselskii and Y B Rutickii Convex Functions andOrlicz Spaces Gorningen Netherlands 1961

[13] W Orlicz and U Raume ldquoLMrdquo Bulletin International delrsquoAcademie Polonaise des Sciences et des Lettres Serie A pp 93ndash107 1936

[14] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971

[15] J Lindenstrauss and L Tzafriri Classical Banach Spaces vol92 of I Sequence spaces Ergebnisse der Mathematik und ihrerGrenzgebiete Springer-Verlag Berlin Germany 1977

[16] Y Altin M Et and B C Tripathy ldquoThe sequence space|119873119901|(119872 119903 119902 119904) on seminormed spacesrdquo Applied Mathematicsand Computation vol 154 no 2 pp 423ndash430 2004

[17] M Et L P Lee and B C Tripathy ldquoStrongly almost (119881 120582)(119903)-summable sequences defined by Orlicz functionsrdquo HokkaidoMathematical Journal vol 35 no 1 pp 197ndash213 2006

[18] M Et Y Altin B Choudhary and B C Tripathy ldquoOn someclasses of sequences defined by sequences of Orlicz functionsrdquoMathematical Inequalities amp Applications vol 9 no 2 pp 335ndash342 2006

[19] B C Tripathy and S Mahanta ldquoOn a class of differencesequences related to the l119901 space defined by Orlicz functionsrdquoMathematica Slovaca vol 57 no 2 pp 171ndash178 2007

[20] B C Tripathy and H Dutta ldquoOn some new paranormed differ-ence sequence spaces defined by Orlicz functionsrdquo KyungpookMathematical Journal vol 50 no 1 pp 59ndash69 2010

[21] B C Tripathy and B Hazarika ldquoI-convergent sequences spacesdefined by Orlicz functionrdquo Acta Mathematicae ApplicataeSinica vol 27 no 1 pp 149ndash154 2011

[22] S A Mohiuddine and B Hazarika ldquoSome classes of ideal con-vergent sequences and generalized difference matrix operatorrdquoFilomat vol 31 no 6 pp 1827ndash1834 2017

[23] S A Mohiuddine and K Raj ldquoVector valued Orlicz-Lorentzsequence spaces and their operator idealsrdquo Journal of NonlinearSciences and Applications JNSA vol 10 no 2 pp 338ndash353 2017

[24] S Abdul Mohiuddine K Raj and A Alotaibi ldquoGeneralizedspaces of double sequences for Orlicz functions and bounded-regular matrices over n-normed spacesrdquo Journal of Inequalitiesand Applications vol 2014 article 332 2014

[25] S A Mohiuddine S K Sharma and D A Abuzaid ldquoSomeseminormed difference sequence spaces over n-normed spacesdefined by a musielak-orlicz function of order (120572120573)rdquo Journal ofFunction Spaces vol 2018 Article ID 4312817 11 pages 2018

[26] S-K Lim and P Y Lee ldquoAnOrlicz extension of Cesaro sequencespacesrdquo Roczniki Polskiego Towarzystwa Matematycznego SeriaI Commentationes Mathematicae Prace Matematyczne vol 28no 1 pp 117ndash128 1988

[27] Y Cui H Hudzik N Petrot S Suantai and A SzymaszkiewiczldquoBasic topological and geometric properties of Cesaro-Orliczspacesrdquo The Proceedings of the Indian Academy of Sciences ndashMathematical Sciences vol 115 no 4 pp 461ndash476 2005

[28] LMaligranda N Petrot and S Suantai ldquoOn the James constantand B-convexity of Cesaro and Cesaro-Orlicz sequence spacesrdquoJournal of Mathematical Analysis and Applications vol 326 pp312ndash326 2007

[29] P Foralewski H Hudzik and A Szymaszkiewicz ldquoLocalrotundity structure of Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 345 no 1 pp410ndash419 2008

[30] G M Leibowitz ldquoA note on the Cesaro sequence spacesrdquoTamkang Journal of Mathematics vol 2 no 2 pp 151ndash157 1971

[31] D Kubiak ldquoA note on Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 349 no 1 pp291ndash296 2009

[32] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbers (in Russian)rdquo in Theoryof Operators in Functional Spaces pp 206ndash211 Academy ofScience Siberian section Novosibirsk Russia 1977

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Journal of Function Spaces 5

Orlicz function satisfying Δ 2-condition so for each finiteoperator119875 isin 119865(119883 119884) ie we obtain (119904119894(119875))infin119894=0 which containsonly finitely many terms different from zero hence 119875 isin119878119888119890119904120593(119883 119884) Currently we prove that 119878119888119890119904120593(119883 119884) sube 119865(119883 119884)let 119875 isin 119878119888119890119904120593(119883 119884) we have (119904119894(119875))infin119894=0 isin 119888119890119904120593 and hence984858(119904119894(119875))infin119894=0 lt infin By taking 120576 isin (0 1) hence there exists a1198940 isin N minus 0 such that 984858((119904119894(119875))infin119894=1198940) lt 12057691205751198622 for some 119888 ge 1where 120575 = max1 suminfin

119894=1198940120593(1(119894 + 1)) As 119904119894(119875) is decreasing

for every 119894 isin N and 120593 is nondecreasing we have

1198940120593 (11990421198940 (119875)) le21198940sum

119894=1198940+1

120593(sum119894119895=0 119904119895 (119875)119894 + 1 )

le infinsum119894=1198940

120593(sum119894119895=0 119904119895 (119875)119894 + 1 ) lt 12057691205751198622

(22)

Hence there exists 119861 isin 11986521198940(119883 119884) such that rank 119861 le 21198940 and1198940120593 (119875 minus 119861) le 21198940sum

119894=1198940+1

120593(sum119894119895=0 119875 minus 119861119894 + 1 ) lt 12057691205751198622

(23)

Since 120593 is right continuous at 0 and nondecreasing then onconsidering this

119875 minus 119861 lt 120576611986221198940120575 (24)

Let 1198961 gt 0 1198962 gt 0 and 119862 = max1 1198961 1198962 since 120593 is Orliczfunction and by using (22) (23) and (24) we have

119889 (119875 119861) = 984858 (119904119894 (119875 minus 119861))infin119894=0 = 31198940minus1sum119894=0

120593

sdot (sum119894119895=0 119904119895 (119875 minus 119861)119894 + 1 ) + infinsum

119894=31198940

120593(sum119894119895=0 119904119895 (119875 minus 119861)119894 + 1 )

le 31198940minus1sum119894=0

120593(sum119894119895=0 119875 minus 119861119894 + 1 ) + infinsum

119894=1198940

120593

sdot (sum119894+21198940119895=0 119904119895 (119875 minus 119861)119894 + 1 ) le 31198940120593 (119875 minus 119861) + infinsum

119894=1198940

120593

sdot (sum119894+21198940119895=0 119904119895 (119875 minus 119861)119894 + 1 ) le 31198940120593 (119875 minus 119861)

+ infinsum119894=1198940

120593(sum21198940minus1119895=0 119904119895 (119875 minus 119861) + sum119894+21198940

119895=21198940119904119895 (119875 minus 119861)119894 + 1 )

le 31198940120593 (119875 minus 119861) + 1198961 [[infinsum119894=1198940

120593(sum21198940minus1119895=0 119904119895 (119875 minus 119861)119894 + 1 )

+ infinsum119894=1198940

120593(sum119894+21198940119895=21198940

119904119895 (119875 minus 119861)119894 + 1 )]] le 31198940120593 (119875 minus 119861)

+ 1198961 [[infinsum119894=1198940

120593(sum21198940minus1119895=0 119875 minus 119861

119894 + 1 )

+ infinsum119894=1198940

120593(sum119894119895=0 119904119895+21198940 (119875 minus 119861)119894 + 1 )]] le 31198940120593 (119875 minus 119861)

+ 2119894011989611198962 119875 minus 119861 infinsum119894=1198940

120593( 1119894 + 1) + 1198961infinsum119894=1198940

120593

sdot (sum119894119895=0 119904119895 (119875)119894 + 1 ) le 31198940120593 (119875 minus 119861) + 211989401198622 119875

minus 119861 infinsum119894=1198940

120593( 1119894 + 1) + 119862infinsum119894=1198940

120593(sum119894119895=0 119904119895 (119875)119894 + 1 ) lt 120576

(25)

Corollary 33 119878119888119890119904119901(119883 119884) = 119865(119883 119884) if 1 lt 119901 lt infin

We express the accompanying theorem without verifica-tion these can be set up utilizing standard procedure

Theorem34 The function 119892(119875) = suminfin119894=0 120593(sum119894

119895=0 |119904119895(119875)|(119894+1))is a pre-quasi norm on 119878119888119890119904120593 if 120593 is an Orlicz function satisfyingΔ 2-condition and 120572120593 gt 1

We give the sufficient conditions on Orlicz-Cesaro meansequence spaces 119888119890119904120593 such that the components of the pre-quasi operator ideal 119878119888119890119904120593 are complete

Theorem 35 If 119883 and 119884 are Banach spaces 120593 is anOrlicz function satisfying Δ 2-condition and 120572120593 gt 1 then(119878119888119890119904120593(119883 119884) 119892) is a pre-quasi Banach operator idealProof Since 120593 is an Orlicz function satisfying Δ 2-condition then the function 119892(119875) = 984858((119904119899(119875))infin119899=0) =suminfin

119899=0 120593(sum119899119898=0 |119904119898(119875)|(119899 + 1)) is a pre-quasi norm on119878119888119890119904120593 Let (119875119898) be a Cauchy sequence in 119878119888119890119904120593(119883 119884) Since

L(119883 119884) supe 119878119888119890119904120593(119883 119884) and 120572120593 gt 1 we can find a constant120585 gt 0 such that

119892 (119875119894 minus 119875119895) = 984858 ((119904119899 (119875119894 minus 119875119895))infin119899=0)ge 984858 (1199040 (119875119894 minus 119875119895) 0 0 0 )= 984858 (10038171003817100381710038171003817119875119894 minus 11987511989510038171003817100381710038171003817 0 0 0 )ge 120585 10038171003817100381710038171003817119875119894 minus 11987511989510038171003817100381710038171003817 984858 (1 0 0 0 )

(26)

then (119875119898)119898isinN is also a Cauchy sequence in L(119883 119884) Whilethe space L(119883 119884) is a Banach space there exists 119875 isin L(119883 119884)

6 Journal of Function Spaces

such that lim119898997888rarrinfin 119875119898minus119875 = 0 while (119904119899(119875119898))infin119899=0 isin 119888119890119904120593 forevery 119898 isin N Since 984858 is continuous at 120579 and for some 119870 ge 1we obtain

119892 (119875) = 984858 ((119904119899 (119875))infin119899=0) = 984858 ((119904119899 (119875 minus 119875119898 + 119875119898))infin119899=0)le 119870984858 ((119904[1198992] (119875 minus 119875119898))infin119899=0)+ 119870984858 ((120572[1198992] (119875119898)infin119899=0))

le 119870984858 ((1003817100381710038171003817119875119898 minus 1198751003817100381710038171003817)infin119899=0) + 119870984858 ((119904119899 (119875119898)infin119899=0))lt infin

(27)

we have (119904119899(119875))infin119899=0 isin 119888119890119904120593 and then 119875 isin 119878119888119890119904120593(119883 119884)Corollary 36 If 119883 and 119884 are Banach spaces and 1 lt 119902 ltinfin then (119878119888119890119904119902(119883 119884) 119892) is quasi Banach operator ideal where119892(119875) = 984858((119904119899(119875))infin119899=0) = [suminfin

119899=0(sum119899119898=0 |119904119898(119875)|(119899 + 1))119902]1119902

Theorem 37 Let 1205931 1205932 be Orlicz functions and 1205721205931 gt 1 Forany infinite dimensional Banach spaces 119883 119884 and if there exist119887 1199050 gt 0 such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905) for all 119905 isin[0 1199050] it is true that

1198781198861199011199011198881198901199041205931(119883 119884) ⫋ 1198781198861199011199011198881198901199041205932

(119883 119884) ⫋ L (119883 119884) (28)

Proof Let119883 and119884 be infinite dimensional Banach spaces andthere exist 119887 1199050 gt 0 such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905)for all 119905 isin [0 1199050] if 119875 isin 1198781198861199011199011198881198901199041205931

(119883 119884) then (120572119899(119875)) isin 1198881198901199041205931 From Theorems 21 22 and 25 we have 1198881198901199041205931 sub 1198881198901199041205932 hence119875 isin 1198781198861199011199011198881198901199041205932

(119883 119884) It is easy to see that 1198781198861199011199011198881198901199041205932(119883 119884) sub L(119883 119884)

Corollary 38 For any infinite dimensional Banach spaces 119883119884 and 1 lt 119901 lt 119902 lt infin then 119878119886119901119901119888119890119904119901(119883 119884) ⫋ 119878119886119901119901119888119890119904119902

(119883 119884) ⫋L(119883 119884)

We now study some properties of the pre-quasi Banachoperator ideal 119878119888119890119904120593 Theorem 39 The pre-quasi Banach operator ideal (119878119888119890119904120593 119892) isinjective if the 119904-number sequence is injective

Proof Let 119879 isin L(119883 119884) and 119875 isin L(119884 1198840) be any metric injec-tion Assume that 119875119879 isin 119878119888119890119904120593(119883 1198840) then 984858(119904119899(119875119879)) lt infinSince the 119904-number sequence is injective we have 119904119899(119875119879) =119904119899(119879) for all 119879 isin L(119883 119884) 119899 isin N So 984858(119904119899(119879)) = 984858(119904119899(119875119879)) ltinfin Hence 119879 isin 119878119888119890119904120593(119883 119884) and clearly 119892(119879) = 119892(119875119879) isverified

Remark 40 The pre-quasi Banach operator ideal (119878119882119890119910119897119888119890119904120593

119892)and the pre-quasi Banach operator ideal (119878119866119890119897119888119890119904120593

119892) are injectivepre-quasi Banach operator ideal

Theorem 41 The pre-quasi Banach operator ideal (119878119888119890119904120593 119892) issurjective if the 119904-number sequence is surjective

Proof Let 119879 isin L(119883 119884) and 119875 isin L(1198830 119883) be any metricsurjection Suppose that 119879119875 isin 119878119888119890119904120593(1198830 119884) then 984858(119904119899(119879119875)) ltinfin Since the 119904-number sequence is surjective we have119904119899(119879119875) = 119904119899(119879) for all 119879 isin L(119883 119884) 119899 isin N So 984858(119904119899(119879)) =984858(119904119899(119879119875)) lt infin Hence 119879 isin 119878119888119890119904120593(119883 119884) and clearly 119892(119879) =119892(119879119875) is verifiedRemark 42 The pre-quasi Banach operator ideal (119878119862ℎ119886119899119892119888119890119904120593

119892) and the pre-quasi Banach operator ideal (119878119870119900119897

119888119890119904120593 119892) are

surjective pre-quasi Banach operator ideal

Likewise we have the accompanying inclusion relationsbetween the pre-quasi Banach operator ideals

Theorem 43 (1) 119878119886119901119901119888119890119904120593sube 119878119870119900119897

119888119890119904120593sube 119878119862ℎ119886119899119892119888119890119904120593

sube 119878119867119894119897119887119888119890119904120593

(2) 119878119886119901119901119888119890119904120593

sube 119878119866119890119897119888119890119904120593sube 119878119882119890119910119897

119888119890119904120593sube 119878119867119894119897119887

119888119890119904120593

Proof Since ℎ119899(119879) le 119910119899(119879) le 119889119899(119879) le 120572119899(119879) and ℎ119899(119879) le119909119899(119879) le 119888119899(119879) le 120572119899(119879) for every 119899 isin N and 984858 isnondecreasing we obtain

984858 (ℎ119899 (119879)) le 984858 (119910119899 (119879)) le 984858 (119889119899 (119879)) le 984858 (120572119899 (119879)) 984858 (ℎ119899 (119879)) le 984858 (119909119899 (119879)) le 984858 (119888119899 (119879)) le 984858 (120572119899 (119879)) (29)

Hence the result is as follows

We presently express the dual of the pre-quasi operatorideal formed by different 119904minus number sequences

Theorem 44 The pre-quasi operator ideal 119878119867119894119897119887119888119890119904120593

is completelysymmetric and the pre-quasi operator ideal 119878119886119901119901119888119890119904120593

is symmetric

Proof Since ℎ119899(1198791015840) = ℎ119899(119879) and 120572119899(1198791015840) le 120572119899(119879) for all 119879 isinL(119883 119884) we have 119878119867119894119897119887

119888119890119904120593= (119878119867119894119897119887

119888119890119904120593)1015840 and 119878119886119901119901119888119890119904120593

sube (119878119886119901119901119888119890119904120593)1015840

In perspective on Theorem 13 we express the followingresult without proof

Theorem 45 The pre-quasi operator ideal 119878119870119900119897119888119890119904120593

sube (119878119866119890119897119888119890119904120593)1015840 and

119878119866119890119897119888119890119904120593= (119878119870119900119897

119888119890119904120593)1015840 In addition if 119879 is a compact operator from 119883

to 119884 then 119878119870119900119897119888119890119904120593

= (119878119866119890119897119888119890119904120593)1015840

In perspective on Theorem 14 we express the followingresult without proof

Theorem 46 The pre-quasi operator ideal 119878119862ℎ119886119899119892119888119890119904120593= (119878119882119890119910119897

119888119890119904120593)1015840

and 119878119882119890119910119897119888119890119904120593

= (119878119862ℎ119886119899119892119888119890119904120593)1015840

Theorem47 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then the pre-quasi Banach operator ideal 119878119886119901119901119888119890119904120593

issmall

Proof Since 120593 is an Orlicz function and 120572120593 gt 1 take120573 = suminfin119894=0 120593(1(119894 + 1)) Then (119878119886119901119901119888119890119904120593

119892) where 119892(119879) =

Journal of Function Spaces 7

984858((120572119899(119879))infin119899=0) = (1120573)suminfin119899=0 120593(sum119899

119898=0 120572119898(119879)(119899 + 1)) is a pre-quasi Banach operator ideal Let119883 and 119884 be any two Banachspaces Assume that 119878119886119901119901119888119890119904120593

(119883 119884) = L(119883 119884) then there existsa constant 119862 gt 0 such that 119892(119879) le 119862119879 for all 119879 isin L(119883 119884)Suppose that119883 and119884 are infinite dimensional Banach spacesThen by Dvoretzkyrsquos theorem [8] for119898 isin N we have quotientspaces 119883119872119898 and subspaces 119873119898 of 119884 which can be mappedonto ℓ1198982 by isomorphisms 119881119898 and 119861119898 such that 119881119898119881minus1

119898 le2 and 119861119898119861minus1119898 le 2 Consider 119868119898 be the identity map onℓ1198982 119875119898 be the quotient map from 119883 onto 119883119872119898 and 119876119898 be

the natural embedding map from 119873119898 into 119884 Let V119899 be theBernstein numbers [7] then

1 = V119899 (119868119898) = V119899 (119861119898119861minus1119898 119868119898119881119898119881minus1

119898 )le 1003817100381710038171003817119861119898

1003817100381710038171003817 V119899 (119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817= 1003817100381710038171003817119861119898

1003817100381710038171003817 V119899 (119876119898119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817le 1003817100381710038171003817119861119898

1003817100381710038171003817 119889119899 (119876119898119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817= 1003817100381710038171003817119861119898

1003817100381710038171003817 119889119899 (119876119898119861minus1119898 119868119898119881119898119876119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817le 1003817100381710038171003817119861119898

1003817100381710038171003817 120572119899 (119876119898119861minus1119898 119868119898119881119898119876119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817

(30)

for 1 le 119894 le 119898 Now since 120593 is nondecreasing and having Δ 2-condition we have

119894sum119895=0

(1) le 119894sum119895=0

10038171003817100381710038171198611198981003817100381710038171003817 120572119895 (119876119898119861minus1

119898 119868119898119881119898119875119898) 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 997904rArr1119894 + 1 (119894 + 1) le 1003817100381710038171003817119861119898

1003817100381710038171003817( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898))

sdot 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 997904rArr120593 (1) le 119871 (1003817100381710038171003817119861119898

1003817100381710038171003817 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817)sdot 120593( 1119894 + 1

119894sum119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898))

(31)

Therefore

119898sum119894=0

120593 (1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817119898sum119894=0

120593

sdot ( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898)) 997904rArr

120593 (1)120573 (119898 + 1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 1120573119898sum119894=0

120593

sdot ( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898)) 997904rArr

120593 (1)120573 (119898 + 1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 119892 (119876119898119861minus1119898 119868119898119881119898119875119898) 997904rArr

120593 (1)120573 (119898 + 1) le 119871119862 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 10038171003817100381710038171003817119876119898119861minus1119898 11986811989811988111989811987511989810038171003817100381710038171003817 997904rArr

120593 (1)120573 (119898 + 1) le 119871119862 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 10038171003817100381710038171003817119876119898119861minus1119898

10038171003817100381710038171003817 10038171003817100381710038171198681198981003817100381710038171003817 10038171003817100381710038171198811198981198751198981003817100381710038171003817= 119871119862 1003817100381710038171003817119861119898

1003817100381710038171003817 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 10038171003817100381710038171003817119861minus1119898

10038171003817100381710038171003817 10038171003817100381710038171198681198981003817100381710038171003817 10038171003817100381710038171198811198981003817100381710038171003817 997904rArr120593 (1)120573 (119898 + 1) le 4119871119862

(32)

for some 119871 ge 1 Thus we arrive at a contradiction since 119898 isarbitrary Hence119883 and119884 both cannot be infinite dimensionalwhen 119878119886119901119901119888119890119904120593

(119883 119884) = L(119883 119884)Theorem48 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then the pre-quasi Banach operator ideal 119878119870119900119897

119888119890119904120593is

small

Corollary 49 If 119901 isin (1infin) then the quasi Banach operatorideal 119878119886119901119901119888119890119904119901

is small

Corollary 50 If 119901 isin (1infin) then the quasi Banach operatorideal 119878119870119900119897

119888119890119904119901is small

4 Examples

We give some examples which support our main results

Example 1 Let 120593 be an Orlicz function the subspace 119888119890119904ℎ120593 ofall order continuous elements of 119888119890119904120593 is defined as [27]

119888119890119904ℎ120593= 119909 isin 119888119890119904120593 forall119896gt0 exist119899119896isinN

infinsum119899=119899119896

120593(119896119899119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816) lt infin (33)

If 120593 is an Orlicz function satisfying Δ 2-condition and 120572120593 gt 1then the following conditions are satisfied

(1) 119878119888119890119904ℎ120593 is an operator ideal

(2) 119878119888119890119904ℎ120593(119883 119884) = 119865(119883 119884)(3) If 119883 and 119884 are Banach spaces then (119878119888119890119904ℎ120593(119883 119884) 119892) is

pre-quasi Banach operator ideal

(4) The pre-quasi Banach operator ideal 119878119886119901119901119888119890119904ℎ120593

is small

Proof Since 120593 is an Orlicz function satisfying Δ 2-conditionand 120572120593 gt 1 then from Theorem (5) in [31] we have 119888119890119904ℎ120593 =119888119890119904120593 which completes the proof

8 Journal of Function Spaces

Example 2 Let 120593 be defined as

120593 (119905) = 119886119897119905119897 + 119886119897minus1119905119897minus1 + + 1198861119905where 119886119894 gt 0 for all 1 le 119894 le 119897 119897 isin N 119897 gt 1 and 119905 ge 0 (34)

It is clear that 120593 is an Orlicz function and 120572120593 = 119897 gt 1 Also 120593is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) le 2119897 lt infin (35)

Then the following conditions are satisfied

(1) 119878119888119890119904120593 is an operator ideal

(2) 119878119888119890119904120593(119883 119884) = 119865(119883 119884)(3) If 119883 and 119884 are Banach spaces then (119878119888119890119904120593(119883 119884) 119892) is

pre-quasi Banach operator ideal(4) The pre-quasi Banach operator ideal 119878119886119901119901119888119890119904120593

is small

In the following two examples we will explain the impor-tance of the sufficient conditions

Example 3 Let 120593 be defined as

120593 (119905) =

0 if 119905 = 0minus119905ln 119905 if 119905 isin (0 1119890 ] 321198901199052 minus 119905 + 12119890 if 119905 isin (1119890 infin)

(36)

It is clear that 120593 is an Orlicz function Since suminfin119899=1 120593(1119899) =suminfin

119899=1(1119899 ln 119899) = infin hence 119888119890119904120593 = 0 The space 119878119888119890119904120593 is notoperator ideal since 119868119870 notin 119878119888119890119904120593 Also since120593 is convex functionand for 119901 gt 1 we have

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

1199051minus119901 ln 120582ln 120582119905

= lim119905997888rarr0+

(1 minus 119901) 1199051minus119901 ln 120582 = infin (37)

for all 120582 isin (0 1] then 120572120593 = 1 Although 120593 is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) = lim sup119905997888rarr0+

2 ln 119905ln 2119905 le 2 lt infin (38)

Example 4 Let 120593(119906) = int119906

0119891(119905)119889119905 where 119891(119905) is defined as

119891 (119905)

=

0 if 119905 = 01119899 if 119905 isin [ 1(119899 + 1) 1119899) for 119899 = 1 2 3 119905 if 119905 isin [1infin)

(39)

It is clear that 120593 is an Orlicz function Let 119879 isin 119878119888119890119904120593 with119904119899(119879) = 1119899 for all 119899 isin N We have for 119899 gt 2 that120593 (119904119899 (2119879)) = int2119899

0119891 (119905) 119889119905 gt int2119899

1119899119891 (119905) 119889119905

gt int1(119899minus1)

1119899119891 (119905) 119889119905 gt 1119899 (119899 minus 1)

119899120593 (119904119899 (119879)) = 119899int1119899

0119891 (119905) 119889119905

lt 119899 sup0le119905le1119899

119891 (119905) int1119899

01 119889119905 lt 1119899 (119899 minus 1)

(40)

Hence 2119879 notin 119878119888119890119904120593 so the space 119878119888119890119904120593 is not operator ideal and120593 notin Δ 2 Also since 120593 is convex function and for 119901 gt 1 wehave

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

119905minus119901 = infin (41)

for all 120582 isin (0 1] then 120572120593 = 1Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

The authors received no financial support for the researchauthorship and or publication of this article

Conflicts of Interest

The authors declare that 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 vol 20 North-Holland PublishingCompany Amsterdam The Netherlands 1980

[2] N F Mohamed and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalities andApplications vol 2013 article 186 2013

[3] N Faried and A A Bakery ldquoSmall operator ideals formed bys numbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1 article357 2018

[4] D Costarelli and G Vinti ldquoA quantitative estimate for the sam-pling kantorovich series in terms of the modulus of continuityin orlicz spacesrdquo Constructive Mathematical Analysis vol 2 no1 pp 8ndash14 2019

Journal of Function Spaces 9

[5] F Altomare ldquoIterates of markov operators and construc-tive approximation of semigroupsrdquo Constructive MathematicalAnalysis vol 2 no 1 pp 22ndash39 2019

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

[7] A Pietsch ldquos-numbers of operators in banach spacesrdquo StudiaMathematica vol 51 pp 201ndash223 1974

[8] A Pietsch Operator Ideals VEB Deutscher Verlag DerWis-senschaften Berlin Germany 1978

[9] C V Hutton ldquoOn the approximation numbers of an operatorand its adjointrdquo Mathematische Annalen vol 210 pp 277ndash2801974

[10] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 pp 267ndash278 1974

[11] A Lima and E Oja ldquoIdeals of finite rank operators intersectionproperties of balls and the approximation propertyrdquo StudiaMathematica vol 133 no 2 pp 175ndash186 1999

[12] M A Krasnoselskii and Y B Rutickii Convex Functions andOrlicz Spaces Gorningen Netherlands 1961

[13] W Orlicz and U Raume ldquoLMrdquo Bulletin International delrsquoAcademie Polonaise des Sciences et des Lettres Serie A pp 93ndash107 1936

[14] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971

[15] J Lindenstrauss and L Tzafriri Classical Banach Spaces vol92 of I Sequence spaces Ergebnisse der Mathematik und ihrerGrenzgebiete Springer-Verlag Berlin Germany 1977

[16] Y Altin M Et and B C Tripathy ldquoThe sequence space|119873119901|(119872 119903 119902 119904) on seminormed spacesrdquo Applied Mathematicsand Computation vol 154 no 2 pp 423ndash430 2004

[17] M Et L P Lee and B C Tripathy ldquoStrongly almost (119881 120582)(119903)-summable sequences defined by Orlicz functionsrdquo HokkaidoMathematical Journal vol 35 no 1 pp 197ndash213 2006

[18] M Et Y Altin B Choudhary and B C Tripathy ldquoOn someclasses of sequences defined by sequences of Orlicz functionsrdquoMathematical Inequalities amp Applications vol 9 no 2 pp 335ndash342 2006

[19] B C Tripathy and S Mahanta ldquoOn a class of differencesequences related to the l119901 space defined by Orlicz functionsrdquoMathematica Slovaca vol 57 no 2 pp 171ndash178 2007

[20] B C Tripathy and H Dutta ldquoOn some new paranormed differ-ence sequence spaces defined by Orlicz functionsrdquo KyungpookMathematical Journal vol 50 no 1 pp 59ndash69 2010

[21] B C Tripathy and B Hazarika ldquoI-convergent sequences spacesdefined by Orlicz functionrdquo Acta Mathematicae ApplicataeSinica vol 27 no 1 pp 149ndash154 2011

[22] S A Mohiuddine and B Hazarika ldquoSome classes of ideal con-vergent sequences and generalized difference matrix operatorrdquoFilomat vol 31 no 6 pp 1827ndash1834 2017

[23] S A Mohiuddine and K Raj ldquoVector valued Orlicz-Lorentzsequence spaces and their operator idealsrdquo Journal of NonlinearSciences and Applications JNSA vol 10 no 2 pp 338ndash353 2017

[24] S Abdul Mohiuddine K Raj and A Alotaibi ldquoGeneralizedspaces of double sequences for Orlicz functions and bounded-regular matrices over n-normed spacesrdquo Journal of Inequalitiesand Applications vol 2014 article 332 2014

[25] S A Mohiuddine S K Sharma and D A Abuzaid ldquoSomeseminormed difference sequence spaces over n-normed spacesdefined by a musielak-orlicz function of order (120572120573)rdquo Journal ofFunction Spaces vol 2018 Article ID 4312817 11 pages 2018

[26] S-K Lim and P Y Lee ldquoAnOrlicz extension of Cesaro sequencespacesrdquo Roczniki Polskiego Towarzystwa Matematycznego SeriaI Commentationes Mathematicae Prace Matematyczne vol 28no 1 pp 117ndash128 1988

[27] Y Cui H Hudzik N Petrot S Suantai and A SzymaszkiewiczldquoBasic topological and geometric properties of Cesaro-Orliczspacesrdquo The Proceedings of the Indian Academy of Sciences ndashMathematical Sciences vol 115 no 4 pp 461ndash476 2005

[28] LMaligranda N Petrot and S Suantai ldquoOn the James constantand B-convexity of Cesaro and Cesaro-Orlicz sequence spacesrdquoJournal of Mathematical Analysis and Applications vol 326 pp312ndash326 2007

[29] P Foralewski H Hudzik and A Szymaszkiewicz ldquoLocalrotundity structure of Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 345 no 1 pp410ndash419 2008

[30] G M Leibowitz ldquoA note on the Cesaro sequence spacesrdquoTamkang Journal of Mathematics vol 2 no 2 pp 151ndash157 1971

[31] D Kubiak ldquoA note on Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 349 no 1 pp291ndash296 2009

[32] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbers (in Russian)rdquo in Theoryof Operators in Functional Spaces pp 206ndash211 Academy ofScience Siberian section Novosibirsk Russia 1977

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6 Journal of Function Spaces

such that lim119898997888rarrinfin 119875119898minus119875 = 0 while (119904119899(119875119898))infin119899=0 isin 119888119890119904120593 forevery 119898 isin N Since 984858 is continuous at 120579 and for some 119870 ge 1we obtain

119892 (119875) = 984858 ((119904119899 (119875))infin119899=0) = 984858 ((119904119899 (119875 minus 119875119898 + 119875119898))infin119899=0)le 119870984858 ((119904[1198992] (119875 minus 119875119898))infin119899=0)+ 119870984858 ((120572[1198992] (119875119898)infin119899=0))

le 119870984858 ((1003817100381710038171003817119875119898 minus 1198751003817100381710038171003817)infin119899=0) + 119870984858 ((119904119899 (119875119898)infin119899=0))lt infin

(27)

we have (119904119899(119875))infin119899=0 isin 119888119890119904120593 and then 119875 isin 119878119888119890119904120593(119883 119884)Corollary 36 If 119883 and 119884 are Banach spaces and 1 lt 119902 ltinfin then (119878119888119890119904119902(119883 119884) 119892) is quasi Banach operator ideal where119892(119875) = 984858((119904119899(119875))infin119899=0) = [suminfin

119899=0(sum119899119898=0 |119904119898(119875)|(119899 + 1))119902]1119902

Theorem 37 Let 1205931 1205932 be Orlicz functions and 1205721205931 gt 1 Forany infinite dimensional Banach spaces 119883 119884 and if there exist119887 1199050 gt 0 such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905) for all 119905 isin[0 1199050] it is true that

1198781198861199011199011198881198901199041205931(119883 119884) ⫋ 1198781198861199011199011198881198901199041205932

(119883 119884) ⫋ L (119883 119884) (28)

Proof Let119883 and119884 be infinite dimensional Banach spaces andthere exist 119887 1199050 gt 0 such that 1205932(1199050) gt 0 and 1205932(119905) le 1205931(119887119905)for all 119905 isin [0 1199050] if 119875 isin 1198781198861199011199011198881198901199041205931

(119883 119884) then (120572119899(119875)) isin 1198881198901199041205931 From Theorems 21 22 and 25 we have 1198881198901199041205931 sub 1198881198901199041205932 hence119875 isin 1198781198861199011199011198881198901199041205932

(119883 119884) It is easy to see that 1198781198861199011199011198881198901199041205932(119883 119884) sub L(119883 119884)

Corollary 38 For any infinite dimensional Banach spaces 119883119884 and 1 lt 119901 lt 119902 lt infin then 119878119886119901119901119888119890119904119901(119883 119884) ⫋ 119878119886119901119901119888119890119904119902

(119883 119884) ⫋L(119883 119884)

We now study some properties of the pre-quasi Banachoperator ideal 119878119888119890119904120593 Theorem 39 The pre-quasi Banach operator ideal (119878119888119890119904120593 119892) isinjective if the 119904-number sequence is injective

Proof Let 119879 isin L(119883 119884) and 119875 isin L(119884 1198840) be any metric injec-tion Assume that 119875119879 isin 119878119888119890119904120593(119883 1198840) then 984858(119904119899(119875119879)) lt infinSince the 119904-number sequence is injective we have 119904119899(119875119879) =119904119899(119879) for all 119879 isin L(119883 119884) 119899 isin N So 984858(119904119899(119879)) = 984858(119904119899(119875119879)) ltinfin Hence 119879 isin 119878119888119890119904120593(119883 119884) and clearly 119892(119879) = 119892(119875119879) isverified

Remark 40 The pre-quasi Banach operator ideal (119878119882119890119910119897119888119890119904120593

119892)and the pre-quasi Banach operator ideal (119878119866119890119897119888119890119904120593

119892) are injectivepre-quasi Banach operator ideal

Theorem 41 The pre-quasi Banach operator ideal (119878119888119890119904120593 119892) issurjective if the 119904-number sequence is surjective

Proof Let 119879 isin L(119883 119884) and 119875 isin L(1198830 119883) be any metricsurjection Suppose that 119879119875 isin 119878119888119890119904120593(1198830 119884) then 984858(119904119899(119879119875)) ltinfin Since the 119904-number sequence is surjective we have119904119899(119879119875) = 119904119899(119879) for all 119879 isin L(119883 119884) 119899 isin N So 984858(119904119899(119879)) =984858(119904119899(119879119875)) lt infin Hence 119879 isin 119878119888119890119904120593(119883 119884) and clearly 119892(119879) =119892(119879119875) is verifiedRemark 42 The pre-quasi Banach operator ideal (119878119862ℎ119886119899119892119888119890119904120593

119892) and the pre-quasi Banach operator ideal (119878119870119900119897

119888119890119904120593 119892) are

surjective pre-quasi Banach operator ideal

Likewise we have the accompanying inclusion relationsbetween the pre-quasi Banach operator ideals

Theorem 43 (1) 119878119886119901119901119888119890119904120593sube 119878119870119900119897

119888119890119904120593sube 119878119862ℎ119886119899119892119888119890119904120593

sube 119878119867119894119897119887119888119890119904120593

(2) 119878119886119901119901119888119890119904120593

sube 119878119866119890119897119888119890119904120593sube 119878119882119890119910119897

119888119890119904120593sube 119878119867119894119897119887

119888119890119904120593

Proof Since ℎ119899(119879) le 119910119899(119879) le 119889119899(119879) le 120572119899(119879) and ℎ119899(119879) le119909119899(119879) le 119888119899(119879) le 120572119899(119879) for every 119899 isin N and 984858 isnondecreasing we obtain

984858 (ℎ119899 (119879)) le 984858 (119910119899 (119879)) le 984858 (119889119899 (119879)) le 984858 (120572119899 (119879)) 984858 (ℎ119899 (119879)) le 984858 (119909119899 (119879)) le 984858 (119888119899 (119879)) le 984858 (120572119899 (119879)) (29)

Hence the result is as follows

We presently express the dual of the pre-quasi operatorideal formed by different 119904minus number sequences

Theorem 44 The pre-quasi operator ideal 119878119867119894119897119887119888119890119904120593

is completelysymmetric and the pre-quasi operator ideal 119878119886119901119901119888119890119904120593

is symmetric

Proof Since ℎ119899(1198791015840) = ℎ119899(119879) and 120572119899(1198791015840) le 120572119899(119879) for all 119879 isinL(119883 119884) we have 119878119867119894119897119887

119888119890119904120593= (119878119867119894119897119887

119888119890119904120593)1015840 and 119878119886119901119901119888119890119904120593

sube (119878119886119901119901119888119890119904120593)1015840

In perspective on Theorem 13 we express the followingresult without proof

Theorem 45 The pre-quasi operator ideal 119878119870119900119897119888119890119904120593

sube (119878119866119890119897119888119890119904120593)1015840 and

119878119866119890119897119888119890119904120593= (119878119870119900119897

119888119890119904120593)1015840 In addition if 119879 is a compact operator from 119883

to 119884 then 119878119870119900119897119888119890119904120593

= (119878119866119890119897119888119890119904120593)1015840

In perspective on Theorem 14 we express the followingresult without proof

Theorem 46 The pre-quasi operator ideal 119878119862ℎ119886119899119892119888119890119904120593= (119878119882119890119910119897

119888119890119904120593)1015840

and 119878119882119890119910119897119888119890119904120593

= (119878119862ℎ119886119899119892119888119890119904120593)1015840

Theorem47 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then the pre-quasi Banach operator ideal 119878119886119901119901119888119890119904120593

issmall

Proof Since 120593 is an Orlicz function and 120572120593 gt 1 take120573 = suminfin119894=0 120593(1(119894 + 1)) Then (119878119886119901119901119888119890119904120593

119892) where 119892(119879) =

Journal of Function Spaces 7

984858((120572119899(119879))infin119899=0) = (1120573)suminfin119899=0 120593(sum119899

119898=0 120572119898(119879)(119899 + 1)) is a pre-quasi Banach operator ideal Let119883 and 119884 be any two Banachspaces Assume that 119878119886119901119901119888119890119904120593

(119883 119884) = L(119883 119884) then there existsa constant 119862 gt 0 such that 119892(119879) le 119862119879 for all 119879 isin L(119883 119884)Suppose that119883 and119884 are infinite dimensional Banach spacesThen by Dvoretzkyrsquos theorem [8] for119898 isin N we have quotientspaces 119883119872119898 and subspaces 119873119898 of 119884 which can be mappedonto ℓ1198982 by isomorphisms 119881119898 and 119861119898 such that 119881119898119881minus1

119898 le2 and 119861119898119861minus1119898 le 2 Consider 119868119898 be the identity map onℓ1198982 119875119898 be the quotient map from 119883 onto 119883119872119898 and 119876119898 be

the natural embedding map from 119873119898 into 119884 Let V119899 be theBernstein numbers [7] then

1 = V119899 (119868119898) = V119899 (119861119898119861minus1119898 119868119898119881119898119881minus1

119898 )le 1003817100381710038171003817119861119898

1003817100381710038171003817 V119899 (119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817= 1003817100381710038171003817119861119898

1003817100381710038171003817 V119899 (119876119898119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817le 1003817100381710038171003817119861119898

1003817100381710038171003817 119889119899 (119876119898119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817= 1003817100381710038171003817119861119898

1003817100381710038171003817 119889119899 (119876119898119861minus1119898 119868119898119881119898119876119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817le 1003817100381710038171003817119861119898

1003817100381710038171003817 120572119899 (119876119898119861minus1119898 119868119898119881119898119876119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817

(30)

for 1 le 119894 le 119898 Now since 120593 is nondecreasing and having Δ 2-condition we have

119894sum119895=0

(1) le 119894sum119895=0

10038171003817100381710038171198611198981003817100381710038171003817 120572119895 (119876119898119861minus1

119898 119868119898119881119898119875119898) 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 997904rArr1119894 + 1 (119894 + 1) le 1003817100381710038171003817119861119898

1003817100381710038171003817( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898))

sdot 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 997904rArr120593 (1) le 119871 (1003817100381710038171003817119861119898

1003817100381710038171003817 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817)sdot 120593( 1119894 + 1

119894sum119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898))

(31)

Therefore

119898sum119894=0

120593 (1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817119898sum119894=0

120593

sdot ( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898)) 997904rArr

120593 (1)120573 (119898 + 1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 1120573119898sum119894=0

120593

sdot ( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898)) 997904rArr

120593 (1)120573 (119898 + 1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 119892 (119876119898119861minus1119898 119868119898119881119898119875119898) 997904rArr

120593 (1)120573 (119898 + 1) le 119871119862 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 10038171003817100381710038171003817119876119898119861minus1119898 11986811989811988111989811987511989810038171003817100381710038171003817 997904rArr

120593 (1)120573 (119898 + 1) le 119871119862 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 10038171003817100381710038171003817119876119898119861minus1119898

10038171003817100381710038171003817 10038171003817100381710038171198681198981003817100381710038171003817 10038171003817100381710038171198811198981198751198981003817100381710038171003817= 119871119862 1003817100381710038171003817119861119898

1003817100381710038171003817 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 10038171003817100381710038171003817119861minus1119898

10038171003817100381710038171003817 10038171003817100381710038171198681198981003817100381710038171003817 10038171003817100381710038171198811198981003817100381710038171003817 997904rArr120593 (1)120573 (119898 + 1) le 4119871119862

(32)

for some 119871 ge 1 Thus we arrive at a contradiction since 119898 isarbitrary Hence119883 and119884 both cannot be infinite dimensionalwhen 119878119886119901119901119888119890119904120593

(119883 119884) = L(119883 119884)Theorem48 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then the pre-quasi Banach operator ideal 119878119870119900119897

119888119890119904120593is

small

Corollary 49 If 119901 isin (1infin) then the quasi Banach operatorideal 119878119886119901119901119888119890119904119901

is small

Corollary 50 If 119901 isin (1infin) then the quasi Banach operatorideal 119878119870119900119897

119888119890119904119901is small

4 Examples

We give some examples which support our main results

Example 1 Let 120593 be an Orlicz function the subspace 119888119890119904ℎ120593 ofall order continuous elements of 119888119890119904120593 is defined as [27]

119888119890119904ℎ120593= 119909 isin 119888119890119904120593 forall119896gt0 exist119899119896isinN

infinsum119899=119899119896

120593(119896119899119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816) lt infin (33)

If 120593 is an Orlicz function satisfying Δ 2-condition and 120572120593 gt 1then the following conditions are satisfied

(1) 119878119888119890119904ℎ120593 is an operator ideal

(2) 119878119888119890119904ℎ120593(119883 119884) = 119865(119883 119884)(3) If 119883 and 119884 are Banach spaces then (119878119888119890119904ℎ120593(119883 119884) 119892) is

pre-quasi Banach operator ideal

(4) The pre-quasi Banach operator ideal 119878119886119901119901119888119890119904ℎ120593

is small

Proof Since 120593 is an Orlicz function satisfying Δ 2-conditionand 120572120593 gt 1 then from Theorem (5) in [31] we have 119888119890119904ℎ120593 =119888119890119904120593 which completes the proof

8 Journal of Function Spaces

Example 2 Let 120593 be defined as

120593 (119905) = 119886119897119905119897 + 119886119897minus1119905119897minus1 + + 1198861119905where 119886119894 gt 0 for all 1 le 119894 le 119897 119897 isin N 119897 gt 1 and 119905 ge 0 (34)

It is clear that 120593 is an Orlicz function and 120572120593 = 119897 gt 1 Also 120593is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) le 2119897 lt infin (35)

Then the following conditions are satisfied

(1) 119878119888119890119904120593 is an operator ideal

(2) 119878119888119890119904120593(119883 119884) = 119865(119883 119884)(3) If 119883 and 119884 are Banach spaces then (119878119888119890119904120593(119883 119884) 119892) is

pre-quasi Banach operator ideal(4) The pre-quasi Banach operator ideal 119878119886119901119901119888119890119904120593

is small

In the following two examples we will explain the impor-tance of the sufficient conditions

Example 3 Let 120593 be defined as

120593 (119905) =

0 if 119905 = 0minus119905ln 119905 if 119905 isin (0 1119890 ] 321198901199052 minus 119905 + 12119890 if 119905 isin (1119890 infin)

(36)

It is clear that 120593 is an Orlicz function Since suminfin119899=1 120593(1119899) =suminfin

119899=1(1119899 ln 119899) = infin hence 119888119890119904120593 = 0 The space 119878119888119890119904120593 is notoperator ideal since 119868119870 notin 119878119888119890119904120593 Also since120593 is convex functionand for 119901 gt 1 we have

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

1199051minus119901 ln 120582ln 120582119905

= lim119905997888rarr0+

(1 minus 119901) 1199051minus119901 ln 120582 = infin (37)

for all 120582 isin (0 1] then 120572120593 = 1 Although 120593 is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) = lim sup119905997888rarr0+

2 ln 119905ln 2119905 le 2 lt infin (38)

Example 4 Let 120593(119906) = int119906

0119891(119905)119889119905 where 119891(119905) is defined as

119891 (119905)

=

0 if 119905 = 01119899 if 119905 isin [ 1(119899 + 1) 1119899) for 119899 = 1 2 3 119905 if 119905 isin [1infin)

(39)

It is clear that 120593 is an Orlicz function Let 119879 isin 119878119888119890119904120593 with119904119899(119879) = 1119899 for all 119899 isin N We have for 119899 gt 2 that120593 (119904119899 (2119879)) = int2119899

0119891 (119905) 119889119905 gt int2119899

1119899119891 (119905) 119889119905

gt int1(119899minus1)

1119899119891 (119905) 119889119905 gt 1119899 (119899 minus 1)

119899120593 (119904119899 (119879)) = 119899int1119899

0119891 (119905) 119889119905

lt 119899 sup0le119905le1119899

119891 (119905) int1119899

01 119889119905 lt 1119899 (119899 minus 1)

(40)

Hence 2119879 notin 119878119888119890119904120593 so the space 119878119888119890119904120593 is not operator ideal and120593 notin Δ 2 Also since 120593 is convex function and for 119901 gt 1 wehave

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

119905minus119901 = infin (41)

for all 120582 isin (0 1] then 120572120593 = 1Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

The authors received no financial support for the researchauthorship and or publication of this article

Conflicts of Interest

The authors declare that 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 vol 20 North-Holland PublishingCompany Amsterdam The Netherlands 1980

[2] N F Mohamed and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalities andApplications vol 2013 article 186 2013

[3] N Faried and A A Bakery ldquoSmall operator ideals formed bys numbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1 article357 2018

[4] D Costarelli and G Vinti ldquoA quantitative estimate for the sam-pling kantorovich series in terms of the modulus of continuityin orlicz spacesrdquo Constructive Mathematical Analysis vol 2 no1 pp 8ndash14 2019

Journal of Function Spaces 9

[5] F Altomare ldquoIterates of markov operators and construc-tive approximation of semigroupsrdquo Constructive MathematicalAnalysis vol 2 no 1 pp 22ndash39 2019

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

[7] A Pietsch ldquos-numbers of operators in banach spacesrdquo StudiaMathematica vol 51 pp 201ndash223 1974

[8] A Pietsch Operator Ideals VEB Deutscher Verlag DerWis-senschaften Berlin Germany 1978

[9] C V Hutton ldquoOn the approximation numbers of an operatorand its adjointrdquo Mathematische Annalen vol 210 pp 277ndash2801974

[10] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 pp 267ndash278 1974

[11] A Lima and E Oja ldquoIdeals of finite rank operators intersectionproperties of balls and the approximation propertyrdquo StudiaMathematica vol 133 no 2 pp 175ndash186 1999

[12] M A Krasnoselskii and Y B Rutickii Convex Functions andOrlicz Spaces Gorningen Netherlands 1961

[13] W Orlicz and U Raume ldquoLMrdquo Bulletin International delrsquoAcademie Polonaise des Sciences et des Lettres Serie A pp 93ndash107 1936

[14] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971

[15] J Lindenstrauss and L Tzafriri Classical Banach Spaces vol92 of I Sequence spaces Ergebnisse der Mathematik und ihrerGrenzgebiete Springer-Verlag Berlin Germany 1977

[16] Y Altin M Et and B C Tripathy ldquoThe sequence space|119873119901|(119872 119903 119902 119904) on seminormed spacesrdquo Applied Mathematicsand Computation vol 154 no 2 pp 423ndash430 2004

[17] M Et L P Lee and B C Tripathy ldquoStrongly almost (119881 120582)(119903)-summable sequences defined by Orlicz functionsrdquo HokkaidoMathematical Journal vol 35 no 1 pp 197ndash213 2006

[18] M Et Y Altin B Choudhary and B C Tripathy ldquoOn someclasses of sequences defined by sequences of Orlicz functionsrdquoMathematical Inequalities amp Applications vol 9 no 2 pp 335ndash342 2006

[19] B C Tripathy and S Mahanta ldquoOn a class of differencesequences related to the l119901 space defined by Orlicz functionsrdquoMathematica Slovaca vol 57 no 2 pp 171ndash178 2007

[20] B C Tripathy and H Dutta ldquoOn some new paranormed differ-ence sequence spaces defined by Orlicz functionsrdquo KyungpookMathematical Journal vol 50 no 1 pp 59ndash69 2010

[21] B C Tripathy and B Hazarika ldquoI-convergent sequences spacesdefined by Orlicz functionrdquo Acta Mathematicae ApplicataeSinica vol 27 no 1 pp 149ndash154 2011

[22] S A Mohiuddine and B Hazarika ldquoSome classes of ideal con-vergent sequences and generalized difference matrix operatorrdquoFilomat vol 31 no 6 pp 1827ndash1834 2017

[23] S A Mohiuddine and K Raj ldquoVector valued Orlicz-Lorentzsequence spaces and their operator idealsrdquo Journal of NonlinearSciences and Applications JNSA vol 10 no 2 pp 338ndash353 2017

[24] S Abdul Mohiuddine K Raj and A Alotaibi ldquoGeneralizedspaces of double sequences for Orlicz functions and bounded-regular matrices over n-normed spacesrdquo Journal of Inequalitiesand Applications vol 2014 article 332 2014

[25] S A Mohiuddine S K Sharma and D A Abuzaid ldquoSomeseminormed difference sequence spaces over n-normed spacesdefined by a musielak-orlicz function of order (120572120573)rdquo Journal ofFunction Spaces vol 2018 Article ID 4312817 11 pages 2018

[26] S-K Lim and P Y Lee ldquoAnOrlicz extension of Cesaro sequencespacesrdquo Roczniki Polskiego Towarzystwa Matematycznego SeriaI Commentationes Mathematicae Prace Matematyczne vol 28no 1 pp 117ndash128 1988

[27] Y Cui H Hudzik N Petrot S Suantai and A SzymaszkiewiczldquoBasic topological and geometric properties of Cesaro-Orliczspacesrdquo The Proceedings of the Indian Academy of Sciences ndashMathematical Sciences vol 115 no 4 pp 461ndash476 2005

[28] LMaligranda N Petrot and S Suantai ldquoOn the James constantand B-convexity of Cesaro and Cesaro-Orlicz sequence spacesrdquoJournal of Mathematical Analysis and Applications vol 326 pp312ndash326 2007

[29] P Foralewski H Hudzik and A Szymaszkiewicz ldquoLocalrotundity structure of Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 345 no 1 pp410ndash419 2008

[30] G M Leibowitz ldquoA note on the Cesaro sequence spacesrdquoTamkang Journal of Mathematics vol 2 no 2 pp 151ndash157 1971

[31] D Kubiak ldquoA note on Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 349 no 1 pp291ndash296 2009

[32] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbers (in Russian)rdquo in Theoryof Operators in Functional Spaces pp 206ndash211 Academy ofScience Siberian section Novosibirsk Russia 1977

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Submit your manuscripts atwwwhindawicom

Journal of Function Spaces 7

984858((120572119899(119879))infin119899=0) = (1120573)suminfin119899=0 120593(sum119899

119898=0 120572119898(119879)(119899 + 1)) is a pre-quasi Banach operator ideal Let119883 and 119884 be any two Banachspaces Assume that 119878119886119901119901119888119890119904120593

(119883 119884) = L(119883 119884) then there existsa constant 119862 gt 0 such that 119892(119879) le 119862119879 for all 119879 isin L(119883 119884)Suppose that119883 and119884 are infinite dimensional Banach spacesThen by Dvoretzkyrsquos theorem [8] for119898 isin N we have quotientspaces 119883119872119898 and subspaces 119873119898 of 119884 which can be mappedonto ℓ1198982 by isomorphisms 119881119898 and 119861119898 such that 119881119898119881minus1

119898 le2 and 119861119898119861minus1119898 le 2 Consider 119868119898 be the identity map onℓ1198982 119875119898 be the quotient map from 119883 onto 119883119872119898 and 119876119898 be

the natural embedding map from 119873119898 into 119884 Let V119899 be theBernstein numbers [7] then

1 = V119899 (119868119898) = V119899 (119861119898119861minus1119898 119868119898119881119898119881minus1

119898 )le 1003817100381710038171003817119861119898

1003817100381710038171003817 V119899 (119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817= 1003817100381710038171003817119861119898

1003817100381710038171003817 V119899 (119876119898119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817le 1003817100381710038171003817119861119898

1003817100381710038171003817 119889119899 (119876119898119861minus1119898 119868119898119881119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817= 1003817100381710038171003817119861119898

1003817100381710038171003817 119889119899 (119876119898119861minus1119898 119868119898119881119898119876119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817le 1003817100381710038171003817119861119898

1003817100381710038171003817 120572119899 (119876119898119861minus1119898 119868119898119881119898119876119898) 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817

(30)

for 1 le 119894 le 119898 Now since 120593 is nondecreasing and having Δ 2-condition we have

119894sum119895=0

(1) le 119894sum119895=0

10038171003817100381710038171198611198981003817100381710038171003817 120572119895 (119876119898119861minus1

119898 119868119898119881119898119875119898) 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 997904rArr1119894 + 1 (119894 + 1) le 1003817100381710038171003817119861119898

1003817100381710038171003817( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898))

sdot 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 997904rArr120593 (1) le 119871 (1003817100381710038171003817119861119898

1003817100381710038171003817 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817)sdot 120593( 1119894 + 1

119894sum119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898))

(31)

Therefore

119898sum119894=0

120593 (1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817119898sum119894=0

120593

sdot ( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898)) 997904rArr

120593 (1)120573 (119898 + 1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 1120573119898sum119894=0

120593

sdot ( 1119894 + 1119894sum

119895=0

120572119895 (119876119898119861minus1119898 119868119898119881119898119875119898)) 997904rArr

120593 (1)120573 (119898 + 1) le 119871 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 119892 (119876119898119861minus1119898 119868119898119881119898119875119898) 997904rArr

120593 (1)120573 (119898 + 1) le 119871119862 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 10038171003817100381710038171003817119876119898119861minus1119898 11986811989811988111989811987511989810038171003817100381710038171003817 997904rArr

120593 (1)120573 (119898 + 1) le 119871119862 10038171003817100381710038171198611198981003817100381710038171003817 10038171003817100381710038171003817119881minus1

119898

10038171003817100381710038171003817 10038171003817100381710038171003817119876119898119861minus1119898

10038171003817100381710038171003817 10038171003817100381710038171198681198981003817100381710038171003817 10038171003817100381710038171198811198981198751198981003817100381710038171003817= 119871119862 1003817100381710038171003817119861119898

1003817100381710038171003817 10038171003817100381710038171003817119881minus1119898

10038171003817100381710038171003817 10038171003817100381710038171003817119861minus1119898

10038171003817100381710038171003817 10038171003817100381710038171198681198981003817100381710038171003817 10038171003817100381710038171198811198981003817100381710038171003817 997904rArr120593 (1)120573 (119898 + 1) le 4119871119862

(32)

for some 119871 ge 1 Thus we arrive at a contradiction since 119898 isarbitrary Hence119883 and119884 both cannot be infinite dimensionalwhen 119878119886119901119901119888119890119904120593

(119883 119884) = L(119883 119884)Theorem48 If120593 is anOrlicz function satisfyingΔ 2-conditionand 120572120593 gt 1 then the pre-quasi Banach operator ideal 119878119870119900119897

119888119890119904120593is

small

Corollary 49 If 119901 isin (1infin) then the quasi Banach operatorideal 119878119886119901119901119888119890119904119901

is small

Corollary 50 If 119901 isin (1infin) then the quasi Banach operatorideal 119878119870119900119897

119888119890119904119901is small

4 Examples

We give some examples which support our main results

Example 1 Let 120593 be an Orlicz function the subspace 119888119890119904ℎ120593 ofall order continuous elements of 119888119890119904120593 is defined as [27]

119888119890119904ℎ120593= 119909 isin 119888119890119904120593 forall119896gt0 exist119899119896isinN

infinsum119899=119899119896

120593(119896119899119899sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816) lt infin (33)

If 120593 is an Orlicz function satisfying Δ 2-condition and 120572120593 gt 1then the following conditions are satisfied

(1) 119878119888119890119904ℎ120593 is an operator ideal

(2) 119878119888119890119904ℎ120593(119883 119884) = 119865(119883 119884)(3) If 119883 and 119884 are Banach spaces then (119878119888119890119904ℎ120593(119883 119884) 119892) is

pre-quasi Banach operator ideal

(4) The pre-quasi Banach operator ideal 119878119886119901119901119888119890119904ℎ120593

is small

Proof Since 120593 is an Orlicz function satisfying Δ 2-conditionand 120572120593 gt 1 then from Theorem (5) in [31] we have 119888119890119904ℎ120593 =119888119890119904120593 which completes the proof

8 Journal of Function Spaces

Example 2 Let 120593 be defined as

120593 (119905) = 119886119897119905119897 + 119886119897minus1119905119897minus1 + + 1198861119905where 119886119894 gt 0 for all 1 le 119894 le 119897 119897 isin N 119897 gt 1 and 119905 ge 0 (34)

It is clear that 120593 is an Orlicz function and 120572120593 = 119897 gt 1 Also 120593is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) le 2119897 lt infin (35)

Then the following conditions are satisfied

(1) 119878119888119890119904120593 is an operator ideal

(2) 119878119888119890119904120593(119883 119884) = 119865(119883 119884)(3) If 119883 and 119884 are Banach spaces then (119878119888119890119904120593(119883 119884) 119892) is

pre-quasi Banach operator ideal(4) The pre-quasi Banach operator ideal 119878119886119901119901119888119890119904120593

is small

In the following two examples we will explain the impor-tance of the sufficient conditions

Example 3 Let 120593 be defined as

120593 (119905) =

0 if 119905 = 0minus119905ln 119905 if 119905 isin (0 1119890 ] 321198901199052 minus 119905 + 12119890 if 119905 isin (1119890 infin)

(36)

It is clear that 120593 is an Orlicz function Since suminfin119899=1 120593(1119899) =suminfin

119899=1(1119899 ln 119899) = infin hence 119888119890119904120593 = 0 The space 119878119888119890119904120593 is notoperator ideal since 119868119870 notin 119878119888119890119904120593 Also since120593 is convex functionand for 119901 gt 1 we have

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

1199051minus119901 ln 120582ln 120582119905

= lim119905997888rarr0+

(1 minus 119901) 1199051minus119901 ln 120582 = infin (37)

for all 120582 isin (0 1] then 120572120593 = 1 Although 120593 is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) = lim sup119905997888rarr0+

2 ln 119905ln 2119905 le 2 lt infin (38)

Example 4 Let 120593(119906) = int119906

0119891(119905)119889119905 where 119891(119905) is defined as

119891 (119905)

=

0 if 119905 = 01119899 if 119905 isin [ 1(119899 + 1) 1119899) for 119899 = 1 2 3 119905 if 119905 isin [1infin)

(39)

It is clear that 120593 is an Orlicz function Let 119879 isin 119878119888119890119904120593 with119904119899(119879) = 1119899 for all 119899 isin N We have for 119899 gt 2 that120593 (119904119899 (2119879)) = int2119899

0119891 (119905) 119889119905 gt int2119899

1119899119891 (119905) 119889119905

gt int1(119899minus1)

1119899119891 (119905) 119889119905 gt 1119899 (119899 minus 1)

119899120593 (119904119899 (119879)) = 119899int1119899

0119891 (119905) 119889119905

lt 119899 sup0le119905le1119899

119891 (119905) int1119899

01 119889119905 lt 1119899 (119899 minus 1)

(40)

Hence 2119879 notin 119878119888119890119904120593 so the space 119878119888119890119904120593 is not operator ideal and120593 notin Δ 2 Also since 120593 is convex function and for 119901 gt 1 wehave

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

119905minus119901 = infin (41)

for all 120582 isin (0 1] then 120572120593 = 1Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

The authors received no financial support for the researchauthorship and or publication of this article

Conflicts of Interest

The authors declare that 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 vol 20 North-Holland PublishingCompany Amsterdam The Netherlands 1980

[2] N F Mohamed and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalities andApplications vol 2013 article 186 2013

[3] N Faried and A A Bakery ldquoSmall operator ideals formed bys numbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1 article357 2018

[4] D Costarelli and G Vinti ldquoA quantitative estimate for the sam-pling kantorovich series in terms of the modulus of continuityin orlicz spacesrdquo Constructive Mathematical Analysis vol 2 no1 pp 8ndash14 2019

Journal of Function Spaces 9

[5] F Altomare ldquoIterates of markov operators and construc-tive approximation of semigroupsrdquo Constructive MathematicalAnalysis vol 2 no 1 pp 22ndash39 2019

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

[7] A Pietsch ldquos-numbers of operators in banach spacesrdquo StudiaMathematica vol 51 pp 201ndash223 1974

[8] A Pietsch Operator Ideals VEB Deutscher Verlag DerWis-senschaften Berlin Germany 1978

[9] C V Hutton ldquoOn the approximation numbers of an operatorand its adjointrdquo Mathematische Annalen vol 210 pp 277ndash2801974

[10] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 pp 267ndash278 1974

[11] A Lima and E Oja ldquoIdeals of finite rank operators intersectionproperties of balls and the approximation propertyrdquo StudiaMathematica vol 133 no 2 pp 175ndash186 1999

[12] M A Krasnoselskii and Y B Rutickii Convex Functions andOrlicz Spaces Gorningen Netherlands 1961

[13] W Orlicz and U Raume ldquoLMrdquo Bulletin International delrsquoAcademie Polonaise des Sciences et des Lettres Serie A pp 93ndash107 1936

[14] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971

[15] J Lindenstrauss and L Tzafriri Classical Banach Spaces vol92 of I Sequence spaces Ergebnisse der Mathematik und ihrerGrenzgebiete Springer-Verlag Berlin Germany 1977

[16] Y Altin M Et and B C Tripathy ldquoThe sequence space|119873119901|(119872 119903 119902 119904) on seminormed spacesrdquo Applied Mathematicsand Computation vol 154 no 2 pp 423ndash430 2004

[17] M Et L P Lee and B C Tripathy ldquoStrongly almost (119881 120582)(119903)-summable sequences defined by Orlicz functionsrdquo HokkaidoMathematical Journal vol 35 no 1 pp 197ndash213 2006

[18] M Et Y Altin B Choudhary and B C Tripathy ldquoOn someclasses of sequences defined by sequences of Orlicz functionsrdquoMathematical Inequalities amp Applications vol 9 no 2 pp 335ndash342 2006

[19] B C Tripathy and S Mahanta ldquoOn a class of differencesequences related to the l119901 space defined by Orlicz functionsrdquoMathematica Slovaca vol 57 no 2 pp 171ndash178 2007

[20] B C Tripathy and H Dutta ldquoOn some new paranormed differ-ence sequence spaces defined by Orlicz functionsrdquo KyungpookMathematical Journal vol 50 no 1 pp 59ndash69 2010

[21] B C Tripathy and B Hazarika ldquoI-convergent sequences spacesdefined by Orlicz functionrdquo Acta Mathematicae ApplicataeSinica vol 27 no 1 pp 149ndash154 2011

[22] S A Mohiuddine and B Hazarika ldquoSome classes of ideal con-vergent sequences and generalized difference matrix operatorrdquoFilomat vol 31 no 6 pp 1827ndash1834 2017

[23] S A Mohiuddine and K Raj ldquoVector valued Orlicz-Lorentzsequence spaces and their operator idealsrdquo Journal of NonlinearSciences and Applications JNSA vol 10 no 2 pp 338ndash353 2017

[24] S Abdul Mohiuddine K Raj and A Alotaibi ldquoGeneralizedspaces of double sequences for Orlicz functions and bounded-regular matrices over n-normed spacesrdquo Journal of Inequalitiesand Applications vol 2014 article 332 2014

[25] S A Mohiuddine S K Sharma and D A Abuzaid ldquoSomeseminormed difference sequence spaces over n-normed spacesdefined by a musielak-orlicz function of order (120572120573)rdquo Journal ofFunction Spaces vol 2018 Article ID 4312817 11 pages 2018

[26] S-K Lim and P Y Lee ldquoAnOrlicz extension of Cesaro sequencespacesrdquo Roczniki Polskiego Towarzystwa Matematycznego SeriaI Commentationes Mathematicae Prace Matematyczne vol 28no 1 pp 117ndash128 1988

[27] Y Cui H Hudzik N Petrot S Suantai and A SzymaszkiewiczldquoBasic topological and geometric properties of Cesaro-Orliczspacesrdquo The Proceedings of the Indian Academy of Sciences ndashMathematical Sciences vol 115 no 4 pp 461ndash476 2005

[28] LMaligranda N Petrot and S Suantai ldquoOn the James constantand B-convexity of Cesaro and Cesaro-Orlicz sequence spacesrdquoJournal of Mathematical Analysis and Applications vol 326 pp312ndash326 2007

[29] P Foralewski H Hudzik and A Szymaszkiewicz ldquoLocalrotundity structure of Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 345 no 1 pp410ndash419 2008

[30] G M Leibowitz ldquoA note on the Cesaro sequence spacesrdquoTamkang Journal of Mathematics vol 2 no 2 pp 151ndash157 1971

[31] D Kubiak ldquoA note on Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 349 no 1 pp291ndash296 2009

[32] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbers (in Russian)rdquo in Theoryof Operators in Functional Spaces pp 206ndash211 Academy ofScience Siberian section Novosibirsk Russia 1977

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Journal of Function Spaces

Example 2 Let 120593 be defined as

120593 (119905) = 119886119897119905119897 + 119886119897minus1119905119897minus1 + + 1198861119905where 119886119894 gt 0 for all 1 le 119894 le 119897 119897 isin N 119897 gt 1 and 119905 ge 0 (34)

It is clear that 120593 is an Orlicz function and 120572120593 = 119897 gt 1 Also 120593is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) le 2119897 lt infin (35)

Then the following conditions are satisfied

(1) 119878119888119890119904120593 is an operator ideal

(2) 119878119888119890119904120593(119883 119884) = 119865(119883 119884)(3) If 119883 and 119884 are Banach spaces then (119878119888119890119904120593(119883 119884) 119892) is

pre-quasi Banach operator ideal(4) The pre-quasi Banach operator ideal 119878119886119901119901119888119890119904120593

is small

In the following two examples we will explain the impor-tance of the sufficient conditions

Example 3 Let 120593 be defined as

120593 (119905) =

0 if 119905 = 0minus119905ln 119905 if 119905 isin (0 1119890 ] 321198901199052 minus 119905 + 12119890 if 119905 isin (1119890 infin)

(36)

It is clear that 120593 is an Orlicz function Since suminfin119899=1 120593(1119899) =suminfin

119899=1(1119899 ln 119899) = infin hence 119888119890119904120593 = 0 The space 119878119888119890119904120593 is notoperator ideal since 119868119870 notin 119878119888119890119904120593 Also since120593 is convex functionand for 119901 gt 1 we have

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

1199051minus119901 ln 120582ln 120582119905

= lim119905997888rarr0+

(1 minus 119901) 1199051minus119901 ln 120582 = infin (37)

for all 120582 isin (0 1] then 120572120593 = 1 Although 120593 is satisfying Δ 2-condition since

lim sup119905997888rarr0+

120593 (2119905)120593 (119905) = lim sup119905997888rarr0+

2 ln 119905ln 2119905 le 2 lt infin (38)

Example 4 Let 120593(119906) = int119906

0119891(119905)119889119905 where 119891(119905) is defined as

119891 (119905)

=

0 if 119905 = 01119899 if 119905 isin [ 1(119899 + 1) 1119899) for 119899 = 1 2 3 119905 if 119905 isin [1infin)

(39)

It is clear that 120593 is an Orlicz function Let 119879 isin 119878119888119890119904120593 with119904119899(119879) = 1119899 for all 119899 isin N We have for 119899 gt 2 that120593 (119904119899 (2119879)) = int2119899

0119891 (119905) 119889119905 gt int2119899

1119899119891 (119905) 119889119905

gt int1(119899minus1)

1119899119891 (119905) 119889119905 gt 1119899 (119899 minus 1)

119899120593 (119904119899 (119879)) = 119899int1119899

0119891 (119905) 119889119905

lt 119899 sup0le119905le1119899

119891 (119905) int1119899

01 119889119905 lt 1119899 (119899 minus 1)

(40)

Hence 2119879 notin 119878119888119890119904120593 so the space 119878119888119890119904120593 is not operator ideal and120593 notin Δ 2 Also since 120593 is convex function and for 119901 gt 1 wehave

lim119905997888rarr0+

120593 (120582119905)120593 (120582) 119905119901 = lim119905997888rarr0+

119905minus119901 = infin (41)

for all 120582 isin (0 1] then 120572120593 = 1Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

The authors received no financial support for the researchauthorship and or publication of this article

Conflicts of Interest

The authors declare that 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 vol 20 North-Holland PublishingCompany Amsterdam The Netherlands 1980

[2] N F Mohamed and A A Bakery ldquoMappings of type Orlicz andgeneralized Cesaro sequence spacerdquo Journal of Inequalities andApplications vol 2013 article 186 2013

[3] N Faried and A A Bakery ldquoSmall operator ideals formed bys numbers on generalized Cesaro and Orlicz sequence spacesrdquoJournal of Inequalities and Applications vol 2018 no 1 article357 2018

[4] D Costarelli and G Vinti ldquoA quantitative estimate for the sam-pling kantorovich series in terms of the modulus of continuityin orlicz spacesrdquo Constructive Mathematical Analysis vol 2 no1 pp 8ndash14 2019

Journal of Function Spaces 9

[5] F Altomare ldquoIterates of markov operators and construc-tive approximation of semigroupsrdquo Constructive MathematicalAnalysis vol 2 no 1 pp 22ndash39 2019

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

[7] A Pietsch ldquos-numbers of operators in banach spacesrdquo StudiaMathematica vol 51 pp 201ndash223 1974

[8] A Pietsch Operator Ideals VEB Deutscher Verlag DerWis-senschaften Berlin Germany 1978

[9] C V Hutton ldquoOn the approximation numbers of an operatorand its adjointrdquo Mathematische Annalen vol 210 pp 277ndash2801974

[10] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 pp 267ndash278 1974

[11] A Lima and E Oja ldquoIdeals of finite rank operators intersectionproperties of balls and the approximation propertyrdquo StudiaMathematica vol 133 no 2 pp 175ndash186 1999

[12] M A Krasnoselskii and Y B Rutickii Convex Functions andOrlicz Spaces Gorningen Netherlands 1961

[13] W Orlicz and U Raume ldquoLMrdquo Bulletin International delrsquoAcademie Polonaise des Sciences et des Lettres Serie A pp 93ndash107 1936

[14] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971

[15] J Lindenstrauss and L Tzafriri Classical Banach Spaces vol92 of I Sequence spaces Ergebnisse der Mathematik und ihrerGrenzgebiete Springer-Verlag Berlin Germany 1977

[16] Y Altin M Et and B C Tripathy ldquoThe sequence space|119873119901|(119872 119903 119902 119904) on seminormed spacesrdquo Applied Mathematicsand Computation vol 154 no 2 pp 423ndash430 2004

[17] M Et L P Lee and B C Tripathy ldquoStrongly almost (119881 120582)(119903)-summable sequences defined by Orlicz functionsrdquo HokkaidoMathematical Journal vol 35 no 1 pp 197ndash213 2006

[18] M Et Y Altin B Choudhary and B C Tripathy ldquoOn someclasses of sequences defined by sequences of Orlicz functionsrdquoMathematical Inequalities amp Applications vol 9 no 2 pp 335ndash342 2006

[19] B C Tripathy and S Mahanta ldquoOn a class of differencesequences related to the l119901 space defined by Orlicz functionsrdquoMathematica Slovaca vol 57 no 2 pp 171ndash178 2007

[20] B C Tripathy and H Dutta ldquoOn some new paranormed differ-ence sequence spaces defined by Orlicz functionsrdquo KyungpookMathematical Journal vol 50 no 1 pp 59ndash69 2010

[21] B C Tripathy and B Hazarika ldquoI-convergent sequences spacesdefined by Orlicz functionrdquo Acta Mathematicae ApplicataeSinica vol 27 no 1 pp 149ndash154 2011

[22] S A Mohiuddine and B Hazarika ldquoSome classes of ideal con-vergent sequences and generalized difference matrix operatorrdquoFilomat vol 31 no 6 pp 1827ndash1834 2017

[23] S A Mohiuddine and K Raj ldquoVector valued Orlicz-Lorentzsequence spaces and their operator idealsrdquo Journal of NonlinearSciences and Applications JNSA vol 10 no 2 pp 338ndash353 2017

[24] S Abdul Mohiuddine K Raj and A Alotaibi ldquoGeneralizedspaces of double sequences for Orlicz functions and bounded-regular matrices over n-normed spacesrdquo Journal of Inequalitiesand Applications vol 2014 article 332 2014

[25] S A Mohiuddine S K Sharma and D A Abuzaid ldquoSomeseminormed difference sequence spaces over n-normed spacesdefined by a musielak-orlicz function of order (120572120573)rdquo Journal ofFunction Spaces vol 2018 Article ID 4312817 11 pages 2018

[26] S-K Lim and P Y Lee ldquoAnOrlicz extension of Cesaro sequencespacesrdquo Roczniki Polskiego Towarzystwa Matematycznego SeriaI Commentationes Mathematicae Prace Matematyczne vol 28no 1 pp 117ndash128 1988

[27] Y Cui H Hudzik N Petrot S Suantai and A SzymaszkiewiczldquoBasic topological and geometric properties of Cesaro-Orliczspacesrdquo The Proceedings of the Indian Academy of Sciences ndashMathematical Sciences vol 115 no 4 pp 461ndash476 2005

[28] LMaligranda N Petrot and S Suantai ldquoOn the James constantand B-convexity of Cesaro and Cesaro-Orlicz sequence spacesrdquoJournal of Mathematical Analysis and Applications vol 326 pp312ndash326 2007

[29] P Foralewski H Hudzik and A Szymaszkiewicz ldquoLocalrotundity structure of Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 345 no 1 pp410ndash419 2008

[30] G M Leibowitz ldquoA note on the Cesaro sequence spacesrdquoTamkang Journal of Mathematics vol 2 no 2 pp 151ndash157 1971

[31] D Kubiak ldquoA note on Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 349 no 1 pp291ndash296 2009

[32] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbers (in Russian)rdquo in Theoryof Operators in Functional Spaces pp 206ndash211 Academy ofScience Siberian section Novosibirsk Russia 1977

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Journal of Function Spaces 9

[5] F Altomare ldquoIterates of markov operators and construc-tive approximation of semigroupsrdquo Constructive MathematicalAnalysis vol 2 no 1 pp 22ndash39 2019

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

[7] A Pietsch ldquos-numbers of operators in banach spacesrdquo StudiaMathematica vol 51 pp 201ndash223 1974

[8] A Pietsch Operator Ideals VEB Deutscher Verlag DerWis-senschaften Berlin Germany 1978

[9] C V Hutton ldquoOn the approximation numbers of an operatorand its adjointrdquo Mathematische Annalen vol 210 pp 277ndash2801974

[10] N J Kalton ldquoSpaces of compact operatorsrdquo MathematischeAnnalen vol 208 pp 267ndash278 1974

[11] A Lima and E Oja ldquoIdeals of finite rank operators intersectionproperties of balls and the approximation propertyrdquo StudiaMathematica vol 133 no 2 pp 175ndash186 1999

[12] M A Krasnoselskii and Y B Rutickii Convex Functions andOrlicz Spaces Gorningen Netherlands 1961

[13] W Orlicz and U Raume ldquoLMrdquo Bulletin International delrsquoAcademie Polonaise des Sciences et des Lettres Serie A pp 93ndash107 1936

[14] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971

[15] J Lindenstrauss and L Tzafriri Classical Banach Spaces vol92 of I Sequence spaces Ergebnisse der Mathematik und ihrerGrenzgebiete Springer-Verlag Berlin Germany 1977

[16] Y Altin M Et and B C Tripathy ldquoThe sequence space|119873119901|(119872 119903 119902 119904) on seminormed spacesrdquo Applied Mathematicsand Computation vol 154 no 2 pp 423ndash430 2004

[17] M Et L P Lee and B C Tripathy ldquoStrongly almost (119881 120582)(119903)-summable sequences defined by Orlicz functionsrdquo HokkaidoMathematical Journal vol 35 no 1 pp 197ndash213 2006

[18] M Et Y Altin B Choudhary and B C Tripathy ldquoOn someclasses of sequences defined by sequences of Orlicz functionsrdquoMathematical Inequalities amp Applications vol 9 no 2 pp 335ndash342 2006

[19] B C Tripathy and S Mahanta ldquoOn a class of differencesequences related to the l119901 space defined by Orlicz functionsrdquoMathematica Slovaca vol 57 no 2 pp 171ndash178 2007

[20] B C Tripathy and H Dutta ldquoOn some new paranormed differ-ence sequence spaces defined by Orlicz functionsrdquo KyungpookMathematical Journal vol 50 no 1 pp 59ndash69 2010

[21] B C Tripathy and B Hazarika ldquoI-convergent sequences spacesdefined by Orlicz functionrdquo Acta Mathematicae ApplicataeSinica vol 27 no 1 pp 149ndash154 2011

[22] S A Mohiuddine and B Hazarika ldquoSome classes of ideal con-vergent sequences and generalized difference matrix operatorrdquoFilomat vol 31 no 6 pp 1827ndash1834 2017

[23] S A Mohiuddine and K Raj ldquoVector valued Orlicz-Lorentzsequence spaces and their operator idealsrdquo Journal of NonlinearSciences and Applications JNSA vol 10 no 2 pp 338ndash353 2017

[24] S Abdul Mohiuddine K Raj and A Alotaibi ldquoGeneralizedspaces of double sequences for Orlicz functions and bounded-regular matrices over n-normed spacesrdquo Journal of Inequalitiesand Applications vol 2014 article 332 2014

[25] S A Mohiuddine S K Sharma and D A Abuzaid ldquoSomeseminormed difference sequence spaces over n-normed spacesdefined by a musielak-orlicz function of order (120572120573)rdquo Journal ofFunction Spaces vol 2018 Article ID 4312817 11 pages 2018

[26] S-K Lim and P Y Lee ldquoAnOrlicz extension of Cesaro sequencespacesrdquo Roczniki Polskiego Towarzystwa Matematycznego SeriaI Commentationes Mathematicae Prace Matematyczne vol 28no 1 pp 117ndash128 1988

[27] Y Cui H Hudzik N Petrot S Suantai and A SzymaszkiewiczldquoBasic topological and geometric properties of Cesaro-Orliczspacesrdquo The Proceedings of the Indian Academy of Sciences ndashMathematical Sciences vol 115 no 4 pp 461ndash476 2005

[28] LMaligranda N Petrot and S Suantai ldquoOn the James constantand B-convexity of Cesaro and Cesaro-Orlicz sequence spacesrdquoJournal of Mathematical Analysis and Applications vol 326 pp312ndash326 2007

[29] P Foralewski H Hudzik and A Szymaszkiewicz ldquoLocalrotundity structure of Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 345 no 1 pp410ndash419 2008

[30] G M Leibowitz ldquoA note on the Cesaro sequence spacesrdquoTamkang Journal of Mathematics vol 2 no 2 pp 151ndash157 1971

[31] D Kubiak ldquoA note on Cesaro-Orlicz sequence spacesrdquo Journalof Mathematical Analysis and Applications vol 349 no 1 pp291ndash296 2009

[32] B M Makarov and N Faried ldquoSome properties of operatorideals constructed by s numbers (in Russian)rdquo in Theoryof Operators in Functional Spaces pp 206ndash211 Academy ofScience Siberian section Novosibirsk Russia 1977

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom