`International Journal of Mathematics and Mathematical SciencesVolume 2011, Article ID 287145, 12 pageshttp://dx.doi.org/10.1155/2011/287145`
Research Article

Comparison between Certain Equivalent Norms Regarding Some Familiar Properties Implying WFPP

Centro de Investigación en Matemáticas (CIMAT), Apartado Postal 402, 36000 Guanajuato, GTO, Mexico

Received 13 December 2010; Accepted 1 March 2011

Copyright © 2011 Helga Fetter and Berta 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.

Abstract

In a Banach space with a basis we define a similar norm to the norm shown by Lin to make 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. in [1] defined a norm in which was used by Lin [2] to exhibit an equivalent norm which makes into a space with the fixed point property (FPP). A similar norm can be defined in every Banach space with a basis. Since with 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 property (WFPP). 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 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. Let be a Banach space with a basis . Let and be the projection . The basis is called premonotone if and monotone if where for every and for every .

Definition 2.2. Let be a Banach space with a basis and with and . Let . Then if is a norm in which we will call -norm.
Clearly
Observe that, if is a basis in , then it is always premonotone in and, if is monotone in , then it is also monotone in . Also observe that since for every we have that , there exists such that .
Next we define the properties related to wfpp we are going to analyze. The definition of GGLD is not the original one found in [3], but an equivalent one found in [4].

Definition 2.3. Let be a Banach space.(1) has the Opial property if for every weakly null sequence and for every , ,(2)If has a basis, it has the generalized Gossez-Lami Dozo property (GGLD) [4] if, for every weakly null normalized block basic sequence , we have that (3)A bounded sequence is called diametral if has weak normal structure (WNS) if there is no weakly null diametral nonzero sequence in .(4)The coefficient , related to uniform nonsquareness, since if and only if is uniformly nonsquare, is given by (5)The coefficient [5] is defined by (6)Coefficients and [6] are defined as follows: for each It is known that and the Opial property implies wfpp. Also
First we will show that the Opial property is inherited from to and that has GGLD if and only if has GGLD. In order to achieve this, we need the following result shown in [7].

Lemma 2.4. Let be a Banach space with a premonotone basis . Then(1) if converges weakly to , if and only if ,(2) if converges weakly to 0 and , there exists a subsequence of such that .

Lemma 2.5. Let be a Banach space with a premonotone basis . If has the Opial property, then also has the Opial property, but the converse is false.

Proof. Let be weakly null in and . Then, by Lemma 2.4 and by (2.2),
It is known that, for , has the Opial property. Consider any -norm in with the canonical basis , and let be such that. Then, for , Thus, does not have the Opial property.

Lemma 2.6. Let be a Banach space with a premonotone basis . Then, has GGLD if and only if has GGLD.

Proof. Let be a weakly null normalized block basic sequence. By Lemma 2.4, exists if and only if exists and in this case . Also The above equality follows immediately from the following inequality, for , This proves the lemma.

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

Lemma 2.7. Let be the space with the norm , where , and Then has WNS.

Proof. Let be a weakly null nonzero sequence. We may assume that and that there exists a block basic sequence with . Suppose that with for and that . Let be such that . Let , with and . Then, Hence, and cannot be a diametral sequence, since for a diametral sequence it is true that for every .

Lemma 2.8. Let be an increasing sequence with . Then there is a space with WNS, such that with the -norm does not have WNS.

Proof. Let be a subsequence of such that, if , then . Observe that . Let be the space with the norm , where . By Lemma 2.7, since , we know that has WNS. Now let be the -norm in with respect to . Let ; then , and thus and, if , Therefore, since and , we have that . Also, if , , Hence, is diametral in .

The above example is another proof of the fact that for every there are Banach spaces and with so that has WNS but does not.

With regard to the coefficient , we will see that, if has a premonotone basis and , then and we will show a sufficient condition for the reverse implication. For this we need the following lemma.

Lemma 2.9. Let be a Banach space with a basis. If then .

Proof. It is clear that .
Now let , , , and , with such that . By passing to a subsequence, we may assume that there exist a block basic sequence with and such that and . Then, Let ; then , and we conclude that , thus proving the assertion.

Similarly one can prove that, if is a space with a basis,

It is known (see [6, 8]) that, if , are Banach spaces, then So, if is a Banach space with a basis, with and , and is with the -norm, then , and if , . Similarly, if , then and, if , then . But the next proposition shows that in fact always implies . For the coefficient , in general neither implies nor the other way round, as we will see in Examples 2.16 and 2.17.

Proposition 2.10. Suppose that is a Banach space with a premonotone basis . Then implies that .

Proof. Let and . Suppose that . Then, for every . Let and with finite support, , and let be a weakly null block basic sequence with such that and . Then we may suppose that for every and . Hence, We may also assume that the supports of and are disjoint. Let and .
Suppose that for some . It is not possible that , because this would mean that So, by passing to a subsequence if necessary, we may assume that for every we have that . Thus, Since , by passing to the limit, we obtain that Similarly, there exists with Suppose that , and let and . Then, since the basis is premonotone, and ; thus, Therefore, and . Further, since , and .
Hence, Similarly, We deduce that , and letting tend to zero we obtain and .

Examples 2.14 and 2.17 exhibit spaces in which and . There is however a special case in which .

Recall that a basis of a Banach space is 1-spreading if, whenever and is a subsequence of and .

Proposition 2.11. If is a Banach space with a premonotone 1-spreading basis, then .

Proof. Let be the translation given by , and let ,, , , where has finite support and is a weakly null block basic sequence such that for every . Let ; then there exists such that for the supports of and are disjoint, and thus, since the basis is 1-spreading, and . Then, , and for Hence, , and, by passing to the limit as tends to infinity, we get and thus the desired result.

Similarly to Propositions 2.10 and 2.11 we can prove the following.

Proposition 2.12. Suppose that is a Banach space with a premonotone basis . Then implies that and, if the basis is premonotone and 1-spreading, then .

Corollary 2.13. If is a Banach space with a premonotone 1-spreading basis, then if and only if ; also if and only if .

Next we will show an example of a space without a 1-spreading basis, such that , but and thus .

Example 2.14. Let be with the following norm: Let denote the canonical basis of . Then for every , , and thus and . On the other hand let with finite support and and suppose that is a block basic sequence with for , with and the support of does not intersect the support of . Then, for every and , Thus, . Hence and, by (2.9), .

With regard to the coefficient we have the following which is proved similarly to Proposition 2.11.

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

In general neither implies nor the other way round, as the following examples show.

Example 2.16. Let and , where and . Then, if , but for every , .
Since , it is easy to see that for , Thus, , and by (2.22), since , we obtain that.
Now let , , , and . Then but .

This last example is another proof of the known fact that for every there are Banach spaces and with , but . In the following example we exhibit a space with such that .

Example 2.17. Let , where, for , . Then , and, if is such that and , and thus .
Obviously if is the canonical basis of , and , then for and thus and .
Suppose now that Then there exist sequences , such that and so that
By passing to a subsequence, we may assume that, for every , , and , and that the subsequence satisfies one of the next three cases:(1) and for every ,(2) for every ,(3) and for every .
Observe that
Case 1. Let , where is the -norm. Then Since , then, by (2.2), and, by (2.22), since and this is a contradiction.Case 2. Let . Suppose that , , , and Squaring both inequalities and adding them, since , we get By passing to the limit as , since , we obtain that , and this contradicts .Case 3. Let . Suppose that and and Then, by (2.38),
Since in if , one has that implies , then Also Let ; then By (2.38),, and since, for we have, using (2.42), that where and But therefore, and, taking the limit as , we get which is a contradiction.
Hence, , and, by (2.10), we have that .

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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