DONG Xiaoli(董小莉),GONG Wanzhong(鞏萬中)

( Department of Mathematics,Anhui Normal University ,Wuhu 241000,China)

Abstract: In this paper some criteria were given about non-l(1)n and locally uniformly non-l(1)n of Orlicz-Bochner sequence spaces endowed with the Luxemburg norm and the Orlicz norm.As corollaries of the main results we get the criteria for Orlicz-Bochner sequence spaces being non-square and being locally uniformly non-square.

Key words: Locally uniformly non-l(1)n ;Non-l(1)n ;Orlicz-Bochner sequence spaces

1.Introduction

Let (X,‖·‖) be a real Banach space.Denote byB(X) andS(X) the unit ball and the unit sphere ofX,respectively.A Banach spaceXis said to be uniformly non-l(1)n(n ≥2,n ∈N) if there existsε ∈(0,1) such that for eachx(1),x(2),··· ,x(n)∈S(X),we have‖x(1)±x(2)±···±x(n)‖≤n(1?ε) for some choice of signs[1?2].The uniformly non-l(1)nproperty plays an important role in probability theory,fixed point theory and many other fields[3–5].A Banach spaceXis called locally uniformly non-l(1)n(n ≥2,n ∈N) if for everyx(1)∈S(X),there existsε ∈(0,1)such that for eachx(2),x(3),··· ,x(n)∈S(X),the inequality‖x(1)±x(2)±···±x(n)‖≤n(1?ε)holds true for some choice of signs[6].A Banach spaceXis called non-l(1)n(n ≥2,n ∈N)if for anyx(1),x(2),··· ,x(n)∈S(X),we have‖x(1)±x(2)±···±x(n)‖

The sufficient and necessary conditions of uniformly non-l(1)n,locally uniformly non-l(1)nand non-l(1)nin Orlicz or Musielak-Orlicz spaces were given by Hudzik and Kami′nska etc[7–11].For Orlicz function spaces with the Orlicz norm or the Luxemburg norm,it is uniformly non-l(1)nif and only if it is reflexive[8].But in Orlicz function spaces with these two different norms,the criteria for the locally uniformly non-l(1)n,as well as the non-l(1)n,are different[8].

CHEN investigated the locally uniformly non-l(1)nproperty of Orlicz-Bochner function spaces endowed with the Luxemburg norm[12].ZHANG and SHANG studied non-squareness and locally uniform non-squareness of Orlicz-Bochner function spaces endowed with both the Luxemburg norm and the Orlicz norm[13–15].GONG,ZHOU and DONG obtained the criteria of Orlicz-Bochner function spaces endowed with the Orlicz norm being locally uniformly nonl(1)nand non-l(1)n[16].Because of the complicated structure of Orlicz-Bochner sequence spaces,the criteria for locally uniform non-l(1)nand non-l(1)nhave not been discussed yet.The aim of this paper is to give sufficient and necessary condition of the locally uniform non-l(1)nand the non-l(1)nproperties of Orlicz-Bochner sequence space equipped with the Luxemburg norm and the Orlicz norm.

Let R be the set of real numbers.A functionM: R→R+is called an Orlicz function ifMis convex,even,M(0) = 0,M(u)>0(u≠ 0) andThe complementary functionNofM,in the sense of Young is defined by

LetMbe an Orlicz function,then its complementary functionNis also an Orlicz function.Mis said to satisfy theδ2-condition,if for someKandr0>0,when|u|≤u0we have

LetXsbe Banach spaces,xs ∈Xs,s= 1,2,···.Forx= (x1,x2,···),we callρM(x) =its modular.The linear space endowed the Luxemburg norm

or the Orlicz norm

forms a Banach space,denoted byl(M)(Xs) orlM(Xs) respectively.In (1.1) the infimum is attained whenk ∈[k?,k??],wherek?= inf{k >0 :ρN(p(kx))≥1},k??= sup{k >0 :ρN(p(kx))≤1}.We called such spaces Orlicz-Bochner sequence spaces.For more references about Orlicz-Bochner spaces and non-l(1)nwe refer to [8,17–20].

2.Some Lemmas

Lemma 2.1[8]The following are equivalent:

1)M ∈δ2;

2) For anyl1>1 andu1>0,there existsε ∈(0,1),such thatM((1+ε)u)≤l1M(u)when|u|≤u1;

3) For anyl2>1 andv0>0,there existsδ >0 such thatN(l2v)≥(l2+δ)N(v) when|v|≤v0.

Lemma 2.2[7,12]LetXbe a Banach space,then

1)Xis non-l(1)nif and only if for allx(1),x(2),··· ,x(n)∈X{0},the inequality

holds for some choice of signs.

2)Xis locally uniformly non-l(1)nif and only if for anyx(1)∈X{0},there isε(x(1)) in(0,1) such that for allx(2),x(3),··· ,x(n)∈X{0},we have

for some choice of signs.

From the proof in [12] we know thatε(x(1)) in Lemma 2.2 2) can be chosen asεin definition.

Lemma 2.3[8]The setQ:=∪{K(x):a ≤‖x‖M ≤b}is bounded for eachb ≥a ≥0 if and only ifN ∈δ2.

Lemma 2.4[8]1)l(M)is non-l(1)nif and only ifM ∈δ2.lMis non-l(1)n.

2)l(M)is locally uniformly non-l(1)nif and only ifM ∈δ2.lMis locally uniformly non-l(1)nif and only ifN ∈δ2.

3.Locally Uniformly Non-l(1)n and Non-l(1)n Properties in l(M)(Xs)

Lemma 3.1LetXsbe locally uniformly non-l(1)n ,x(1)s ∈XsandM ∈δ2,and letd >1,c >1 satisfyThen there existsα(s)∈(0,1),such that for anythe inequality

holds for some choice of signs.

ProofSinceM(αu)<αM(u)wheneveru>0 andα ∈(0,1),and thatMis continuous onthere must existr ∈(0,1) such that for anythere holdsthat is

whenIn virtue ofM ∈δ2there issuch that

when|u|≤d.Combining (3.1) with (3.2) we obtain,for

For anyy(1)s ∈S(Xs),definewhere the infimum is taken over ally(2)s ,y(3)s ,··· ,y(n)sinS(Xs).Denotingthen by the locally uniformly non-l(1)nproperty ofXswe knowb(s)∈(0,1).Therefore it follows from Lemma 2.2 that

for some choice of signs.For the sake of convenience,we shall consider two cases:

Combining the inequality above with (3.4),we can easily verify that

for some choice of signs.

for some choice of signs.

Case IIWithout loss of generality we may assume

Subcase II-1Similarly as in Subcase I-1,we have for some choice of signs.

Subcase II-2For anyx(i)s ∈Xssatisfyand wheni=2,3,··· ,n.Noticing thatwe know

Therefore for any choice of signs,by (3.3),

Finally denote

thenα(s) satisfies the demand.

Theorem 3.1Orlicz-Bochner sequence spacel(M)(Xs) is locally uniformly non-l(1)nif and only ifM ∈δ2andXsis locally uniformly non-l(1)nfor anys ∈N.

Proof(Necessity)l(M)is isometrically isomorphic to a closed subspace ofl(M)(Xs),sol(M)is locally uniformly non-l(1)n,and so by Lemma 2.4M ∈δ2.SimilarlyXsis locally uniformly non-l(1)nfor anys ∈N.

HenceBy Lemma 3.1,fors ∈D,there holds

DenoteThen bythere isk ∈N such thatand writeBin place ofBk.Hence by (3.5) we have

which yields

ThenρMfor some choice of signs.Consequently by the relation between modular and the Luxemburg norm there existsuch that

Hencel(M)(Xs) is locally uniformly non-l(1)n.

Especially forn=2 we have

Corollary 3.1Orlicz-Bochner sequence spacel(M)(Xs) is locally uniformly non-square if and only ifM ∈δ2andXsis locally uniformly non-square for anys ∈N.

Corollary 3.2If 1

Theorem 3.2Orlicz-Bochner sequence spacel(M)(Xs) is non-l(1)nif and only ifM ∈δ2andXsis non-l(1)nfor anys ∈N .

ProofHere we only prove the sufficiency.Supposei= 1,2,··· ,n.DefineBy the convexity ofMwe know the inequality

holds for any choice of signs.

Case Iby Lemma 2.2 and the convexity ofMwe have

for a certain choice of signs.

Combining Case I with Case II we know that there existss ∈N and a certain choice of signs such that

Then

and

that is

Therefore there must exist some choice of signs such that

By the property of Luxemburg norm we havefor some choice of signs,which completes the proof.

Especially forn=2 we have

Corollary 3.3Orlicz-Bochner sequence spacel(M)(Xs) is non-square if and only ifM ∈δ2andXsis non-square for anys ∈N.

Corollary 3.4If 1

4.Locally Uniformly Non-l(1)n and Non-l(1)n Properties in lM(Xs)

Lemma 4.1LetXsbe locally uniformly non-l(1)n,N ∈δ2,10.Then forx(1)s ∈Xs {0}with‖x(1)s ‖≤d,there existsβ(s)∈(0,1),such that for anyx(2)s ,x(3)s ,··· ,x(n)s ∈Xswith‖x(i)s ‖≤d,andk1,k2,··· ,kn ∈(1,l),the following inequality

holds true for some choice of signs,wherek0is defined as

ProofByN ∈δ2and Lemma 2.1 we know

Forη=there existsa ∈(0,1) such that

Fory(1)s ∈S(Xs),define

By the locally uniformly non-l(1)nproperty ofXswe haveFor the clarity we will divide the proof into two cases:

Thus

for some choice of signs.Hence by the convexity ofMand the equality1,we can get

for some choice of signs.

Subcase I-2≤a.Setk0satisfyingFrom the definitions ofk0andwe know

Therefore

Combining (4.1) with the convexity ofMwe can easily get the inequality

holds for any choice of signs.

Case IIWe may assume

Subcase II-1Similarly as in subcase I-1,we can get

for some choice of signs.

Subcase II-2Similarly as in subcase I-2,we obtain

for any choice of signs.

Thereforeβ(s):=maxsatisfies the demand.

Theorem 4.1Orlicz-Bochner sequence spacelM(Xs) is locally uniformly non-l(1)nif and only ifN ∈δ2andXsis locally uniformly non-l(1)nfor anys ∈N.

ProofWe only need to prove the sufficiency.LetandFor anywhere1,2,··· ,n.

Letc >0 andsatisfyρMChoosed >0 such thatand letforj=2,3,··· ,n.By the definition ofAjand

ThusDefineCertainly there isk ∈N such thatIn the sequel we shall denoteand write B in place ofBk.Therefore,by the convexity ofMand Lemma 4.1 we can get,fors ∈B,

Hence combining the inequality with (4.2) we can get

Then we can conclude that there exists a certain choice of signs such that

Thus,by the property of the Orlicz norm,we have

for some choice of signs.ThereforelM(Xs) is locally uniformly non-l(1)n,and the proof is completed.

Corollary 4.1Orlicz-Bochner sequence spacelM(Xs) is locally uniformly non-square if and only ifN ∈δ2andXsis locally uniformly non-square for anys ∈N.

Theorem 4.2Orlicz-Bochner sequence spacelM(Xs) is non-l(1)nif and only ifXsis non-l(1)nfor anys ∈N.

ProofWe only need to prove the sufficiency.Supposefori=1,2,··· ,n,s ∈N.

Case IBy Lemma 2.2 and the convexity ofMwe have,fors ∈

holds for a certain choice of signs.

Case IIFor anythere holdsfor somej ∈{1,2,··· ,n}.Similarly as the proof in Theorem 3.2 we have

Combining Case I with Case II we know that for a certain choice of signs,there holds

for allThen similarly as in the proof of Theorem 3.2 we obtain,for a certain choice of signs,

So there must exist some choice of signs such that

ThereforelM(Xs) is non-l(1)n.

Corollary 4.2Orlicz-Bochner sequence spacelM(Xs) is non-square if and only ifXsis non-square for anys ∈N.