Comparison between Certain Equivalent Norms Regarding Some Familiar Properties Implying WFPP

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Volume 2011, Article ID 287145,12pages doi:10.1155/2011/287145

Research Article

Comparison between Certain Equivalent Norms

Regarding Some Familiar Properties Implying

WFPP

Helga Fetter and Berta Gamboa de Buen

Centro de Investigaci´on en Matem´aticas (CIMAT), Apartado Postal 402, 36000 Guanajuato, GTO, Mexico

Correspondence should be addressed to Helga Fetter,[email protected]

Received 13 December 2010; Accepted 1 March 2011

Academic Editor: Enrique Llorens-Fuster

Copyrightq2011 H. Fetter and B. Gamboa de Buen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In a Banach space with a basis we define a similar norm to the norm shown by Lin to makel1

into a space with FPP and make a comparative study of certain geometric properties such as the Opial property, WNS, and uniform nonsquareness of the original space and the space with the new norm.

1. Introduction

Dowling et al. in1defined a norm inl1which was used by Lin2to exhibit an equivalent

norm which makesl1into a space with the fixed point propertyFPP. A similar norm can be

defined in every Banach spaceXwith a basis. Sincel1with its usual norm does not have FPP,

we asked ourselves if this norm in these spaces would also improve properties that imply the weak fixed point propertyWFPP. We found out that in some instances it does, in some cases the original norm has better properties, and in some cases you cannot compare them. We give several examples to illustrate our assertions.

2. The

Γ

Norm in a Banach Space

We start by giving the definition of the generalization of the norm used by Lin in a Banach space with a basis.

Definition 2.1. LetX, · be a Banach space with a basis{en}. Letx ∞_i1xiei ∈ X and

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ifQnx ≥ Qn1xand monotone if Pnx ≤ Pn1xwherePnx I−Qn1x for every

x∈Xand for everyn∈Æ.

Definition 2.2. LetX, · be a Banach space with a basis{en}andΓ {γn} ⊂Êwith 0< γn<

γn1and limnγn1. Letx∞_i1xiei∈X. Then if

|x|sup

n γnQnx, 2.1

| · |is a norm inXwhich we will callΓ-norm.

Clearly

γ1x ≤ |x| ≤

sup

n Qn

x. 2.2

Observe that, if{en}is a basis inX, · , then it is always premonotone inX,| · |

and, if {en} is monotone in X, · , then it is also monotone inX,| · |. Also observe that since for everyx ∈ X we have that limnγnQnx 0, there existsn0 such that|x|

γn0Qn0x.

Next we define the properties related to wfpp we are going to analyze. The definition of GGLD is not the original one found in3, but an equivalent one found in4.

Definition 2.3. LetYbe a Banach space.

1Y has the Opial property if for every weakly null sequence{xn} ⊂Yand for every

x∈Y, _{x /}0,

lim sup

n

xn<lim sup

n

xn−x. _2.3

2IfY has a basis, it has the generalized Gossez-Lami Dozo propertyGGLD 4if, for every weakly null normalized block basic sequence{yn}, we have that

lim

n sup_i,j≥nyi−yj>1. 2.4

3A bounded sequence{yn} ⊂Yis called diametral if

lim

n d

yn1,conv

yi

n

i1 diam

yn

∞

n1. 2.5

Y has weak normal structureWNSif there is no weakly null diametral nonzero sequence inY.

4The coeﬃcientJY, related to uniform nonsquareness, sinceJY <2 if and only ifY is uniformly nonsquare, is given by

JY supminxy,x−y :x, y∈BY

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5The coeﬃcientRY 5is defined by

RY sup

lim inf

n xnx:{x},{xn} ⊂BY, xn−→w 0

. 2.7

6Coeﬃcients RWa, Yand MWY 6are defined as follows: for eacha >0

RWa, Y sup

min

lim inf

n xnx,lim infn xn−x

:x ≤a,{xn} ⊂BY, xn−→

w 0

,

MWY sup

1a

RWa, Y :a >0

.

2.8

It is known that GGLD⇒WNS⇒wfpp and the Opial property implies wfpp. Also

RY<2⇒MWY>1⇒wfpp, 2.9

JY<2⇒MWY>1⇒wfpp. 2.10

First we will show that the Opial property is inherited fromX,| · |toX, · and thatX, · has GGLD if and only ifX,| · |has GGLD. In order to achieve this, we need the following result shown in7.

Lemma 2.4. LetX, · be a Banach space with a premonotone basis{en}. Then

1if{xn}converges weakly tox, limnxn−xaif and only if limn|xn−x|a,

2if{xn}converges weakly to 0 and limnlimrxn−xra, there exists a subsequence{yn}

of{xn}such that limnlimr|yn−yr|a.

Lemma 2.5. LetX,·be a Banach space with a premonotone basis{en}. IfX,|·|has the Opial

property, thenX, · also has the Opial property, but the converse is false.

Proof. Let{xn}be weakly null inXandx∈X, x /0. Then, byLemma 2.4and by2.2,

lim sup

n xn

lim sup

n |xn|<

lim sup

n |xn−x| ≤

lim sup

n xn−x. 2.11

It is known that, for 1< p <∞,lp, · has the Opial property. Consider anyΓ-norm

| · |inlpwith the canonical basis{en}, and letδ > 0 be such that δ < γ₂p−γ₁p/γ₁p

1/p

. Then, forn≥2,

|δe1en|max

γ1δp11/p, γn

γn,

lim

n |en|limn γnlimn |δe1en|1.

2.12

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Lemma 2.6. LetX,·be a Banach space with a premonotone basis{en}. Then,X,·has GGLD

if and only ifX,| · |has GGLD.

Proof. Let{yn}be a weakly null normalized block basic sequence. ByLemma 2.4, limnyn

exists if and only if limn|yn|exists and in this case limnynlimn|yn|. Also

lim

n _i,j≥nsupyi−yjlimn sup_i,j≥nyi−yj. 2.13

The above equality follows immediately from the following inequality, fori,j≥n:

_y_i₋_y_j_≤_y_i₋_y_j_≤ 1

γn

_y_i₋_y_j_. _2.14

This proves the lemma.

Now we will show that there exists a space with WNS such that with theΓ-norm it does not have WNS.

Lemma 2.7. LetXbe the spacec0 with the normxsup|bi|

_∞

i1εi|bi|, wherex

_∞

i1biei,

εi>0 and∞_i1εi<∞. ThenXhas WNS.

Proof. Let{xn} ⊂Xbe a weakly null nonzero sequence. We may assume thatx1/0 and that

there exists a block basic sequence{un} ⊂Xwithun−xn →

n→ ∞0. Suppose thatun qn

ipnaiei

withpn ≤qn< pn1forn∈Æand thatx1

_∞

i1biei. Letkbe such thatk_i1εi|bi|δ /0. Let

ε < δ/2,s > kwithQsx1< εandn > s. Then,

x1−unε≥ Psx1−unmaxPsx1_∞,un_∞

s

i1

εi|bi|

qn

ipn

εi|ai| ≥ unδ. 2.15

Hence, lim sup_nx1−xn ≥ lim supnxnδ/2 and {xn} cannot be a diametral sequence,

since for a diametral sequence {xn}it is true that limnx−xn diam{xn}for every x ∈

conv{xn}.

Lemma 2.8. LetΓ {γn} ⊂0,1be an increasing sequence with limnγn1. Then there is a space

X, · with WNS, such thatXwith theΓ-norm| · |does not have WNS.

Proof. Let{γnj}_jbe a subsequence of{γn}such that, ifεnj 1/31/γnj−1, then

_∞

i1εni <∞.

Observe thatεnj < εnj1. LetX be the spacec0with the normx sup|ai|

_∞

i1εni|ani|

_∞

i1,i /nj|ai|/2i, wherex

_∞

i1aiei. ByLemma 2.7, since

_∞

i1εni 1/2i < ∞, we know

thatX has WNS. Now let| · |be theΓ-norm inX with respect to{γi}. Letuj enj; then

γnj 1/13εnj, and thus

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and, ifj < m,

_u_j₋_u_m_max_γ_nj₁_ε_nj_ε_nm_{, γ}_nm₁_ε_nm

≤maxγnj

12εnj

, γnm1εnm

<1.

2.17

Therefore, since 0 ∈ conv{un} and limn|un| 1, we have that diam|·|{un} 1. Also, if 0≤λi,

_n−₁

i1 λi1,

|un| ≤

n−1

i1

λiui−un

≤1. 2.18

Hence,{un}is diametral inX,| · |.

The above example is another proof of the fact that for everyε >0 there are Banach spacesXandYwithdX, Y<1εso thatXhas WNS butYdoes not.

With regard to the coeﬃcient MWX, we will see that, ifXhas a premonotone basis and MWX, · >1, then MWX,| · |>1 and we will show a suﬃcient condition for the reverse implication. For this we need the following lemma.

Lemma 2.9. LetXbe a Banach space with a basis. If

RW1a, X sup

minlim infuny,lim infun−y :un−→

w 0,

{un} ⊂BX is a block basic sequence,

_y_≤_a_{and support of}_y _{is finite}_,

2.19

then RWa, X RW1a, X.

Proof. It is clear that RW1a, X≤RWa, X.

Now letε >0,x∈X,x ≤a, and{xn} ⊂BX, withxn →

w 0 such that minlimnxn

x,limnxn−x > RWa, X−ε. By passing to a subsequence, we may assume that there

exist a block basic sequence{un} ⊂BXwithxn−un< εandm∈Æsuch thatx−Pmx< ε

andPmx<1εx ≤a1ε. Then,

Pmx

1εun

≥ xnx − xn−un − x−Pmx − Pmx ε

1ε,

Pmx

1ε−un

≥ xn−x − xn−un − x−Pmx − Pmx₁ε

ε.

2.20

Lety Pmx/1ε; theny ≤a, and we conclude that RW1a, X ≥RWa, X−3aε,

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Similarly one can prove that, ifXis a space with a basis,

RY sup

lim inf

n unx:{x},{un} ⊂BY, un−→w 0,

{un}is a block basic sequence and support ofxis finite

.

2.21

It is knownsee6,8that, ifX,Y are Banach spaces, then

MWX≤MWYdX, Y, JX≤JYdX, Y. 2.22

So, ifXis a Banach space with a basis,Γ {γn}with 0 ≤γn ≤ γn1 ≤1 andγ1 >1/MWX,

andY isX with theΓ-norm, then MWY ≥ MWX/dX, Y ≥ γ1MWX > 1, and ifγ1 >

1/MWY, MWX > 1. Similarly, ifγ1 > JX/2, thenJY < 2 and, ifγ1 > JY/2, then

JX<2. But the next proposition shows that in fact MWX>1 always implies MWY>1. For the coeﬃcientJ, in general neitherJX<2 impliesJY<2 nor the other way round, as we will see in Examples2.16and2.17.

Proposition 2.10. Suppose thatXis a Banach space with a premonotone basis {en}. Then MW

MWX, · >1 implies that MW1MWX,| · |>1.

Proof. Let RWa,X,· Raand RWa,X,|·| R1a. Suppose that MW11. Then,

R1a 1afor everya >0. Leta >0 and 0< ε < a, y∈Xwith finite support,|y| ≤a, and let{un}be a weakly null block basic sequence with|un| ≤1 such that limn|uny|>1a−ε

and limn|un−y|>1a−ε. Then we may suppose that for everyn, |uny|>1a−ε

and|un−y|>1a−ε. Hence,

|un| ≥1−ε, y≥a−ε. 2.23

We may also assume that the supports ofyandunare disjoint. Letun rn_ilnaieiandy

r

i1biei.

Suppose that|uny|γmnrn_imnaibieifor somemn≤rn. It is not possible that

mn> r, because this would mean that

1a−ε≤unyγmn

rn

mn

aiei

≤ |un| ≤1. 2.24

So, by passing to a subsequence if necessary, we may assume that for everynwe have that

mni0≤r. Thus,

1a−ε≤unyγi0

rn

ii0

aibiei

≤γi0

rn

iln

aiei

r

ii0

biei

≤ γi0

γln a.

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Since limnγln 1, by passing to the limit, we obtain that

γi0≥1−ε. 2.26

Similarly, there existsγi1≥1−εwith

1a−ε≤un−y≤ γi1

γln a. 2.27

Suppose that i1 ≥ i0, and let y0

_r

ii0biei and y1

_r

ii1biei. Then, since the basis is

premonotone, 1a−ε ≤ |uny| |uny0| ≤ uny0and 1a−ε ≤ |un−y| |un−y1| ≤ un−y1 ≤ un−y0; thus,

a−ε≤y0≤y≤a. 2.28

Therefore,

a−ε≤y0≤y0≤ 1

γi0

_y₀_≤ a

γi0 ≤

a

1−ε 2.29

and|y0 −a| ≤max{ε, aε/1−ε}. Further, sinceγi0≤γln,

1−ε≤ |un| ≤ un ≤ 1

γln|un| ≤

1

γln ≤

1

1−ε 2.30

and|un −1| ≤ε/1−ε. Hence,

uunn

y0

_y₀a

≥uny0− |un −1| −y0−a

≥1a−ε− ε

1−ε −max

ε, aε

1−ε

.

2.31

Similarly,

uunn−

y0

_y₀a≥1a−ε− ε

1−ε−max

ε, aε

1−ε

. 2.32

We deduce that minlim infnun/un y0/y0a,lim infnun/un−y0/y0a ≥

1a−ε−ε/1−ε−max{ε, aε/1−ε}, and lettingεtend to zero we obtainRa 1aand

MW1.

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Recall that a basis{en}of a Banach spaceXis 1-spreading if, wheneverx∞_i1aiei∈

Xand{eni}iis a subsequence of{en},

_∞

i1aieni ∈Xand∞_i1aiei∞_i1aieni.

Proposition 2.11. IfXis a Banach space with a premonotone 1-spreading basis, thenMWX,·≥

MWX,| · |.

Proof. LetTm : X → X be the translation given byTm∞i1aiei ∞i1aieim, and leta >

0,y∈X,y ≤a,{un} ⊂BX, whereyhas finite support and{un}is a weakly null block basic

sequence such that minyun,y−un > RWa,X, · −εfor everyn. Letm ∈ Æ;

then there existsN ∈ Æ such that forn > N the supports ofTmy andun are disjoint, and

thus, since the basis is 1-spreading,y−unTmy−unandyunTmyun. Then,

|Tmy| ≤ Tmy ≤a,|un| ≤ un ≤1 and forn > N

_T_m_y₋_u_n_≥_γ_m_T_m_y₋_u_n_γ_m_y₋_u_n_,

_T_m_y_u_n_≥_γ_m_T_m_y_u_n_γ_m_y_u_n_. 2.33

Hence, RWa,X,| · | ≥ γmRWa,X, · , and, by passing to the limit asmtends to

infinity, we get RWa,X,| · |≥RWa,X, · and thus the desired result.

Similarly to Propositions2.10and2.11we can prove the following.

Proposition 2.12. Suppose that X is a Banach space with a premonotone basis {en}. Then R RX, · <2 implies thatR1 RX,| · | <2 and, if the basis is premonotone and 1-spreading,

thenRRX, · ≤RX,| · | R1.

Corollary 2.13. IfXis a Banach space with a premonotone 1-spreading basis, thenMW1>1 if and

only ifMW >1; alsoR1<2 if and only ifR <2.

Next we will show an example of a space without a 1-spreading basis, such thatR2, MW1 butR1<2 and thus MW1>1.

Example 2.14. LetXbec0with the following norm:

{an}|a1|max

i≥2 |ai|. 2.34

Let{en}denote the canonical basis ofc0. Then for everya >0,ae1enae1−en1a,

and thusR 2 and MW 1. On the other hand letx ∈Xwith finite support and|x| ≤1 and suppose that{un}is a block basic sequence with|un| ≤1 forn ∈ N,un rn_imnaiei

withmn≤rn< mn1and the support ofxdoes not intersect the support ofun. Then, for every

m≥2 andn≥m, γmQmxunmaxγmQmxc0, γmunc0≤1,

γ1xun ≤1γ1unc0 ≤1

γ1

γmn. 2.35

Thus, lim inf|xun| ≤1γ1. HenceR1≤1γ1<2 and, by2.9, MW1>1.

With regard to the coeﬃcientJ we have the following which is proved similarly to

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Proposition 2.15. Suppose that X is a Banach space with a premonotone 1-spreading basis

{en}. ThenJX, · ≤JX,| · |.

In general neitherJX, · <2 impliesJX,| · |<2 nor the other way round, as the following examples show.

Example 2.16. Let 1> μ≥1/√2 andX Ê

2

μ

2l2, whereÊ

2

μ Ê

2_,_·

μandx1, x2μ

max|x1|,|x2|, μ|x1||x2|. Then, ifμ≤γ2/γ1γ2,JX<2 but for everyΓ,JX,| · | 2.

Sinceμ≥1/√2, it is easy to see that forx x1, x2∈Ê

2_,

μ

1−2μ2μ2x2≤ xμ≤μ

√

2x2. _2.36

Thus,dl2, X≤dÊ

2_,

Ê

2

μ √

2

1−2μ2μ2_<√_{2, and by}_2.22_{, since}_J_l2 √_{2, we obtain}

thatJX<2.

Now let Γ {γn}, μ ≤ γ2/γ1 γ2, x 1/γ1,1/γ2,0,0, . . ., and y −1/γ1,1/γ2,0,0, . . .. Then|x||y|1 but|xy||x−y|2.

This last example is another proof of the known fact that for every ε > 0 there are Banach spacesXandYwithdX, Y<1ε,JX<2 butJY 2. In the following example we exhibit a spaceXwithJX 2 such thatJX,| · |<2.

Example 2.17. LetX l2, · , where, forx an ∈ l2,x |a1| ∞_i2a2i

1/2

. Then

JX 2,MWX 1, and, ifΓis such thatγ2>1/

√

2 andγ1/γ2<1/

√

2,JX,| · |<2 and

thus MWX, | · |>1.

Obviously if{en}is the canonical basis ofl2, anda >0, thenae1enae1−en

1aforn >1 and thusJX 2 and MWX 1.

Suppose now thatJY JX,| · | 2. Then there exist sequences{xn},{yn} ⊂BY

such that limn|xnyn|limn|xn−yn|2 and{mn},{ln} ⊂Æso that

_x_n_y_n_γ_mn_Q_mn

xnyn ,

_x_n₋_y_n_γ_ln_Q_ln

xn−yn .

2.37

By passing to a subsequence, we may assume that, for everyn∈Æ,e ∗

1xn≥0,e∗1yn≥

0 ande∗₁xn≥e∗₁yn, and that the subsequence satisfies one of the next three cases:

1mn>1 andln>1 for everyn,

2mnln1 for everyn,

3mn1 andln>1 for everyn.

Observe that

γmQm

xy ≥2−ε impliesγmQmx ≥1−εand γmQm

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Case 1. LetZ l2,| · |_Γ1, where| · |_Γ1is theΓ1{γi}∞_i₂-norm. Then

|Q2xn|_Γ1≤1, Q2yn_Γ1≤1,

_Q₂

xnyn _Γ1 ≥2−ε, Q2

xn−yn _Γ1≥2−ε.

2.39

Since 1/γ2 <

√

2, then, by2.2,dZ, l2 <√2 and, by2.22, sinceJl2 2, JZ< 2 and this is a contradiction.

Case 2. Letε >0. Suppose thatx∞_i1aiei,y∞_i1biei∈BY,a1>0, b1>0,a1> b1and

γ1

_∞

i2

aibi2

1/2

≥2−ε−γ1a1b1,

γ1

_∞

i2

ai−bi2

1/2

≥2−ε−γ1a1−b1.

2.40

Squaring both inequalities and adding them, sinceγ2∞_i2a2i

1/2_{≤ |}

x|, we get

2γ2 1

γ2 2

≥γ₁2

∞

i2

a2_i b2_i≥2−ε−γ1a1 2γ12b21≥

2−ε−γ1a1 2. 2.41

By passing to the limit asε → 0, sinceγ1a1≤1, we obtain that

√

2γ1/γ2≥2−γ1a1 ≥1, and

this contradictsγ1/γ2<1/

√

2.

Case 3. Let 2> ε >0. Suppose thatx∞i1aiei, y

_∞

i1biei∈BY, a1 >0, b1 ≥0, a1 > b1and

|xy| γ1|a1b1| ∞_i2aibi2

1/2

≥ 2−εand|x−y| γm∞_imai−bi2

1/2

≥

2−ε. Then, by2.38,

γm

_∞

im

a2_i

1/2

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Since inl2ifu, v∈Bl2, one has thatu−v ≥ δimpliesuv ≤2

1−δ/22, then

γm

_∞

imaibi2

1/2

≤√4ε−ε2_. _Also

2−ε≤γ1

⎛

⎝_|_a₁_b₁_|

_m−₁

i2

aibi2

1/2

_∞

im

aibi2

1/2⎞

⎠

≤γ1|a1b1|γ1

_m−₁

i2

aibi2

1/2

γ1

γm

4ε−ε2

≤γ1|a1b1|γ1

_m−₁

i2

aibi2

1/2

4ε−ε2_.

2.43

Letφε√4ε−ε2_{; then}

2−φ≤γ1|a1b1|γ1

_m−₁

i2

aibi2

1/2

. 2.44

By2.38, γ1|a1|γ1

_m−₁

i2 a2i

1/2

≥1−φ, and since, forA, B >0,

AB1/2−A1/2 B

AB1/2A1/2, 2.45

we have, using2.42, that

1≥γ1|a1|γ1

_∞

i2

a2i

1/2

≥γ1|a1|γ1

_m−₁

i2

a2i

1−ε2 γm2

1/2

γ1|a1|γ1

_m−₁

i2

a2i

1/2

γ11−ε2

γm2

1

AB1/2A1/2

≥1−φγ11−ε

2

γm2

1

AB1/2A1/2,

2.46

whereA m−_i₂1a2

i

1/2

andB 1−ε2/γ2

m. But

AB1/2A1/2≤

1

γ2 2

1−ε2

γ2

m

1/2

1

γ2 ≤

3

γ2

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therefore,

γ1< γ1

γ2

m

≤ φ

1−ε2

3

γ2, 2.48

and, taking the limit asε → 0, we getγ10 which is a contradiction.

Hence,JX,| · |<2, and, by2.10, we have that MWX,| · |>1.

Acknowledgments

The authors thank the referees for their detailed review and valuable observations. This work was partially funded by Grant SEP CONACYT 102380.

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