The Szlenk index and the fixed point property under renorming

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 268270, 9pages doi:10.1155/2010/268270 Research Article The Szlenk Index and the Fixed Point Property under Renorming T. Dom´ ınguez Benavides Facultad de Matem´ aticas, Universidad de Sevilla, P.O. Box 1160, Sevilla 41080, Spain Correspondence should be addressed to T. Dom´ ınguez Benavides, [email protected] Received 25 November 2009; Accepted 19 January 2010 Academic Editor: Tomonari Suzuki Copyright q2010 T. Dom´ ınguez Benavides. 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. Assume that Xis a Banach space such that its Szlenk index SzXis less than or equal to the first infinite ordinal ω. We prove that Xcan be renormed in such a way that Xwith the resultant norm satisfies RX<2, where R·is the Garc´ ıa-Falset coefficient. This leads us to prove that if Xis a Banach space which can be continuously embedded in a Banach space Ywith SzY≤ω, then, Xcan be renormed to satisfy the w-FPP. This result can be applied to Banach spaces which can be embedded in CK,whereKis a scattered compact topological space such that Kω∅. Furthermore, for a Banach space X, ·, we consider a distance in the space Pof all norms in X which are equivalent to ·for which Pbecomes a Baire space.IfSzX≤ω, we show that for almost all norms in the sense of porosityin P,Xsatisfies the w-FPP. For general reflexive spaces independently of the Szlenk index, we prove another strong generic result in the sense of Baire category. 1. Introduction Assume that X, ·is a Banach space. The most common aim of the Renorming Theory is to find an equivalent norm which satisfies or which does not satisfycertain specific properties. A detailed account of this topic can be found in the monographs 1–3. This paper focuses on the Renorming Theory in connection with the Fixed Point Theory. It is usually said that a Banach space Xsatisfies the weak Fixed Point Property w-FPPif for every convex weakly compact subset Cof X, each nonexpansive mapping T:C→Chas a fixed point. Many geometrical properties of Xuniform convexity, uniform smoothness, uniform convexity in every direction, uniform non-squareness, normal structure, etc.are known to imply the w-FPP see, e.g., 4–6and references therein. However, no characterization of the w-FPP in terms of these properties is known. Therefore, we can regard the w-FPP as an intrinsic property of a Banach space. Since the w-FPP is not preserved under isomorphisms, a very 2 Fixed Point Theory and Applications natural question in Renorming Theory and Fixed Point Theory would be the following: let Xbe a Banach space. Is it possible to renorm Xso that the resultant space has the w-FPP? This is not generally the case. Indeed, Partington 7,8has proved that every renorming of ∞Γ for an uncountable set Γand any renorming of ∞/c0contains an isometric copy of ∞and, consequently, it fails the w-FPP due to Alspach example 9. Thus, it would be interesting to identify some classes of Banach spaces which can be renormed to satisfy the w-FPP. For instance, Day et al. 10have proved that every separable Banach space has a UCED renorming. Since uniform convexity in every direction implies normal structure and this property implies the w-FPP see, e.g., 4, we obtain that any separable Banach space can be renormed to satisfy the w-FPP. These arguments do not work for nonseparable spaces because, as mentioned above, there are some Banach spaces which cannot be renormed to satisfy the w-FPP. In fact, in 10, it is shown that c0Γ has no UCED renorming if Γis uncountable. Since in 11an example is given of a reflexive Banach spaces which does not admit any UCED renorming, the following question, which appears in 12, Open Question VIand 1, Problem VII.3and which remained unanswered for a long time, seems to be very natural: can any reflexive Banach space be renormed to satisfy the w-FPP? In 13it is shown that this is indeed the case. Actually, the following result is proved in 13: assume that Xis a Banach space such that there exists a bounded one-one linear operator from Xinto c0Γ. Then, Xhas an equivalent norm which satisfies the w-FPP. This embedding property is satisfied by a very general class of Banach spaces, for instance subspaces of a space with Markushevich basis, as WCG spaces and so separable and reflexive spaces, dual of separable spaces as ∞, and so forth. The proof of the result in 13is strongly based upon some specific properties of the space c0Γ, specially the equality Rc0Γ  1, where R·is Garc´ ıa-Falset’s coefficient 14. It must be noted that any Banach space Ysuch that RY<2 satisfies the w-FPP see 15. Thus, it would be natural to extend the above result to any Banach space which can be embedded in more general Banach spaces than c0Γ, but still satisfying RY<2. In 16 we prove this extension in the following sense: assume that Yis a Banach space such that RY<2, where R·is Garc´ ıa-Falset’s coefficient, and Xis a Banach space which can be continuously embedded in Y. Then, Xcan be renormed to satisfy the w-FPP. In this paper we will use the Szlenk index to show a wide class of Banach spaces X which can be renormed to satisfy RX<2. The Szlenk index SzX17is an ordinal number which was introduced to prove that there is no separable reflexive Banach space universal for the class of all separable reflexive Banach spaces. Later, this index has been used in various areas of the geometry of Banach spaces see 18for a survey about it. Recently, Raja 19 has proved that if Xis an Asplund space and SzX≤ω, then there is an equivalent norm on Xsuch that the dual norm on X∗is UKK∗. We will show in this paper that this fact leads us to prove RX<2 when Xis endowed with this norm. On the other hand, if we endow Γwith the discrete topology and denote by Kthe one-point compactification of Γ, then c0Γ is isometrically contained in CK, where K is a topological compact space which satisfies K2∅. Thus, if a Banach space can be continuously embedded in c0Γ then, it can also be embedded in CK, where Kis a scattered compact topological space such that Kω∅. Since CKsatisfies the w-FPP 20when Kis a scattered compact topological space Ksuch that Kω∅, another natural question would be the following: assume that Xis a Banach space which can be continuously embedded in CK for some Kas above. Can Xbe renormed to satisfy the w-FPP? Using the results about the Szlenk index and the main result in 16, we can prove that this is indeed the case. Nominally, since SzCK ≤ωif and only ifKis as above, we obtain the following: let CKbe the Fixed Point Theory and Applications 3 space of real continuous functions defined on a scattered compact topological space Ksuch that Kω∅. Then, it can be renormed in such a way that RCK,·<2where ·is the new normand the dual norm is UKK∗. In order to better understand the relevance of this result, note that in the metrizable case, if Kω∅, then CKis isomorphic to c0and, consequently, there exists an equivalent norm ·such that RCK,·1. From this result and the main result in 16, we can easily deduce that if a Banach space can be continuously embedded in CK,Kas above, then it can be renormed to satisfy the w-FPP. In 16the same result for CKwas obtained by a direct and very technical method. This is a strict improvement of the result in 13, because, as proved in 21, when Kis a Ciesielski-Pol’s compact, then K3∅,butCKcannot be continuously embedded in c0Γ for any set Γ. In the last section, for a Banach space X, ·, we consider a metric in the space Pof all norms in Xwhich are equivalent to ·, and note that Pbecomes a Baire space for the corresponding metric topology. If SzX≤ω, we show that for almost all norms in the sense of porosityin P,Xsatisfies the w-FPP. We finish with another strong generic result in the sense of Baire category for general reflexive spaces without any assumption on the Szlenk index. 2. Szlenk Index and Fixed Points We start reminding some definition and stating the previous results which we will use. Definition 2.1. Let Mbe a topological space and AasubsetofM.ThesetAis said to be perfect if it is closed and has no isolated point, that is, Ais equal to the set of its own accumulation points. The space Mis said to be scattered if it contains no perfect nonvoid subset. If Ais a subset of a topological space M, the derived set of Ais the set A1of all accumulation points of A.Ifαis an ordinal number, we define the αth-derived set by transfinite induction: A0A, Aα1Aα1,A λ α<λ Aα,2.1 where λis a limit ordinal. Let us recall the definition of Garc´ ıa-Falset’s coefficient. Definition 2.2 see 14.LetXbe a Banach space. The coefficient RXis defined by RXsuplim inf xnx:xnis weakly null with xn≤1,x1.2.2 Theorem 2.3 see 15.Let Xbe a Banach space such that RX<2. Then, Xsatisfies the w-FPP. Theorem 2.4 see 16.Let Ybe a Banach space such that RY<2. Assume that Xis another Banach space, such that there exists a continuous one-to-one mapping J:X→Y. Then, Xcan be renormed to satisfy the w-FPP. Definition 2.5. Let Xbe a Banach space with dual X∗. We say that the dual norm is UKK∗if for every ε>0 there is θε>0 such that every u∈BX∗with u>1−θεhas a weak∗open neighborhood Uwith diam BX∗∩U<ε. 4 Fixed Point Theory and Applications We remind the definition of the Szlenk index. Following the survey 18, we consider a more general definition than that in 17. However, both definitions are identical for separable spaces which do not contain 1. Definition 2.6. Let Xbe a Banach space and X∗its dual. For any bounded subset A⊂X∗,we define a Szlenk derivation by A ε{u∈A: for every w∗-neighborhood Uof u,diamA∩ U≥ε}. By iteration, the sets Aγ εare defined for any ordinal number γ, taking intersection in the case of limit ordinals. The indices SzXεare ordinal numbers defined as SzXεinfγ:BX∗γ ε∅2.3 if such an ordinal exists. Otherwise, we write SzXε∞. Finally the Szlenk index is defined by SzXsupε>0SzXε. Remark 2.7. It is known see 18, Theorem 2or 1, Theorem 5.2 that SzX/ ∞if and only if Xis an Asplund space. Since our results apply for Banach spaces satisfying SzX≤ω,from now on, we will only consider Asplund spaces. Theorem 2.8 see 19.Let Xbe an Asplund space with SzX≤ω. Then, there is an equivalent norm on Xsuch that the dual norm on X∗is UKK∗. Let Kbe a compact topological space. It is known see, e.g., 1, Lemma 8.3 that CK is an Asplund space if and only if Kis scattered. For special scattered sets, we have a more precise result. Theorem 2.9 see 18, Theorem 24.Let Kbe a scattered compact space. The following assertions are equivalent: iSzCK ≤ω, iiKω∅. We will use the equivalent definition of the UKK∗property given by the following lemma. Lemma 2.10. Assume that Xis a Banach space. Then the dual norm is UKK∗if and only if for every ε>0,thereexistsδ>0such that if {uα}is a net in the unit ball of X∗convergent to uin the weak∗ topology such that limαuα−u>ε,thenu<1−δ. Proof. Assume that the above condition is satisfied and let ε>0. Suppose that diam U∩B∗ X>ε for every open neighborhood of uin the weak∗-topology. We can choose uU∈BX∗∩Usuch that uU−u>ε/3. Then, {uU}is a net in BX∗convergent to uin the weak∗-topology. Taking a subnet {uα}of {uU}such that limαuα−uexists, we obtain u≤1−δε/3. Conversely, assume that the dual norm is UKK∗.Let{uα}be a net in BX∗convergent to uin the weak∗- topology such that limαuα−u>ε.LetUbe an open neighborhood of uin the weak∗- topology. There exist α0such that for every α≥α0we have uα−u>εand uα∈U.Thus diam U∩BX∗>ε, which implies u≤1−θε. Remark 2.11. Note that the above notion implies the sequential-UKK∗condition, that is, the dual norm is sequentially-UKK∗if for every ε>0, there exists δ>0 such that if {un}is a Fixed Point Theory and Applications 5 sequence in the unit ball of X∗convergent to uin the weak∗topology such that un−u>ε, then u<1−δ. Both conditions are equivalent if either Xis separable and, consequently, the weak∗-topology restricted to bounded subsets of X∗is metrizableor Xis reflexive due to the angelicity of weak compact sets. Theorem 2.12. Let Xbe an Asplund space with SzX≤ω. Then, there is an equivalent norm |·| on Xsuch that RX, |·| <2and, hence, X, |·|satisfies the w-FPP. Proof. By Theorem 2.8, there exists an equivalent norm on X, such that the dual norm satisfies the UKK∗property. We follow an argument inspired on that in the proof of Proposition III.11 in 15. Assume that {xn}is a weakly null sequence in BXand x∈BX. For every n∈N, choose un∈SX∗such that unxxnxxn. Taking a subsequence, if necessary, we can assume that limnxxndoes exist. Let {unα}be a subnet of {un}which is weak∗-convergent to uand such that limαunα−uexists. Assume ≤1/2 and choose an arbitrary η>0. Since {xnα}is a weakly null net, there exists α0such that |uxnα|<η/2, unα−u<1/2η and |unαx−ux|<η/2 for every α≥α0. Thus, we have xnαxunαxnαunαx uxunα−uxnαunα−uxuxnα ≤uunα−u2η ≤11 22η, 2.4 which implies that limnxnx≤3/2. If >1/2, from Lemma 2.10 we have that u< 1−δ1/2. Since xnαxunαxnαx≤|unαx||unαxnα|≤1|unαx|,2.5 we have lim inf nxnxlim αxnαx≤1|ux|≤1u≤11−δ1 2<2−δ1 2.2.6 Thus, RX<max{3/2,2−δ1/2}. Remarks 2.13. 1Following an argument as in the proof of Proposition III.11 in 15, we can also obtain the condition RX<2 under the following more general assumption which is usually denoted as w-UKK∗property: there exist ∈0,1and δ>0 such that if {uα}is a net in the unit ball of X∗convergent to uin the weak∗-topology and such that limαuα−u>, then u<1−δ. However, this condition does not yield to an improvement of the above theorem, because if X∗satisfies the w-UKK∗property, there is a renorming of Xsuch that the dual norm satisfies the UKK∗property. Indeed, it is easy to check that the w-UKK∗property implies that the Szlenk index SzXis finite for some ∈0,1. Since the function SzXis submultiplicative 18,Proposition4, we have that SzXn≤SzXnand thus SzXis finite for every positive . Thus, the existence of an equivalent norm in Xsuch that the dual norm satisfies the UKK∗property is a consequence of Theorem 2.8. 6 Fixed Point Theory and Applications 2We can also deduce some fixed point properties for the dual norm. First of all, we should mention that if Xis an Asplund space, then X∗can be continuously embedded in c0Γ for some set Γ22. Thus, by the main result in 13,X∗has an equivalent in general nondualnorm which satisfies the w-FPP. On the other hand, we know see 23, Corollary 5.10 that property UKK∗implies that the coefficient w∗CSX∗is greater than 1, where w∗CSX∗inflimn/ mun−um limnun,2.7 and the infimum is taken over all weak∗-null sequences {un}in X∗such that both limits exist and limnun/ 0. This condition implies that every separable weak∗-compact subset of X∗ has normal structure see 24, Theorem 2or 23, Proposition 5.3.Thus,X∗admits a dual equivalent norm such that if Tis a nonexpansive mapping defined from a separable weak∗- compact convex subset Cof X∗into C, then Thas a fixed point see 24, Theorem 1.IfXis reflexive, the separability assumption can be removed, because the condition WCSX∗> 1 implies normal structure for weakly compact subsets of X∗and we recover the first mentioned renorming result now, for a dual norm because any equivalent norm is a dual norm in a reflexive space 25. However, in this case we obtain a stronger result because we have an equivalent norm in Xsuch that Xendowed with the new norm satisfies the wFPP and X∗endowed with the dual norm satisfies the w-FPP either Theorem 3.4 in the last section will show a different way to prove a stronger result. Also in the reflexive case, since X∗is nearly uniform convex, we can also assure that X∗satisfies the w-FPP for nonexpansive multivalued mappings with compact convex valuessee, e.g., 26. Theorem 2.12 jointly with 16, Theorem 2.5yields to the main result in this paper. Theorem 2.14. Let Ybe a Banach space with SzY≤ω. Assume that Xis another Banach space, such that there exists a continuous one-to-one mapping J:X→Y. Then, Xcan be renormed to satisfy the w-FPP. Assume that Γis an uncountable set. We can consider that Γis endowed with the discrete topology. Let Kbe the one-point compactification of Γ. Then, c0Γ,· ∞is isomorphic to CK,· ∞by defining S:CK→c0Γ by Sxγ  xγ−x∞. Thus any space which can be continuously embedded in c0Γ,·∞, can be also embedded in CK,· ∞, where K2∅. From Theorems 2.9 and 2.14, we obtain the following result which strictly improves the main result in 13, because as mentioned in the introduction and proved in 21, there exists a compact set Ciesielski-Pol’s compact, such that K3∅,but CKcannot be continuously embedded in c0Γ for any set Γ. The same result is proved in 16using a direct but very technical argument. Corollary 2.15. Let Xbe a Banach space which can be continuously embedded in CK,· ∞for some compact set Ksuch that Kω∅. Then, Xcan be renormed to satisfy the w-FPP. 3. Genericity of the w-FPP and Szlenk Index Following the approach in 27, for a Banach space X, ·, with closed unit ball B, we denote by Pthe Baire space of all equivalent norms with the metric ρp, qsup{|px−qx|:x∈ B}. Fixed Point Theory and Applications 7 In a Baire space, we can regard first category sets as negligible sets. However, we can also consider a deeper notion of negligible set. We should remember that a set Ain a topological space Xis nowhere dense if its closure has empty interior. If Xis a metric space, this fact means that for every x∈Aand r>0, there exists y∈Xand r>0 such that By,r⊂Bx,r\A. A more strict condition is the following. Definition 3.1. Let Mbe a metric space. A subset Aof Mis said to be porous if there exist 0<β≤1andr0>0 such that for every x∈Aand 0 <r≤r0, there exists y∈Xsuch that By,βr⊂Bx, r∩M\A.AsubsetAof Mis called σ-porous if Ais the union of a countable family of porous sets. Porous and σ-porous set can be considered “small” in M. In particular a σ-porous set is obviously of Baire first category and, for MRn,aσ-porous set is a null set with respect to the Lebesgue measure. In 28, Theorem 14,itisprovedthatifXis a Banach space such that RX<2, then there exists a σ-porous set A⊂Psuch that if q∈P\Athe space X, qsatisfies the w-FPP. From this and Theorem 2.12, we easily obtain the following generic result. Corollary 3.2. Assume that Xis a Banach space with SzX≤ωand Pis the set of all norms in Xwhich are equivalent to the original norm with the metric ρp, qsup{|px−qx|:x∈B}. Then, there exists a σ-porous set A⊂Psuch that if q∈P\Athe space X, qsatisfies the w-FPP. In particular, we obtain the following generic result, which can be regarded as an improvement of the result in 20about the w-FPP in CK. Corollary 3.3. Assume that Kω∅and Pis the set of all norms in CKwhich are equivalent to the supremum norm with the metric ρp,qsup{|px−qx|:x∈B}. Then, there exists a σ-porous set A⊂Psuch that if q∈P\A, the space CK,qsatisfies the w-FPP. For general reflexive spaces independently of the Szlenk index, we can use the main result in 29to prove a strong generic result in the sense of the Remarks 2.13.Ifpis a norm in a Banach space X, we will denote by p∗the dual norm on the dual space X∗and by Qthe Baire space of all equivalent norms to · ∗with the metric ρr, ssup{|ru−su|:u∗≤ 1}. Theorem 3.4. Let X, ·be a reflexive space. There exists a residual subset R0of P(i.e., P\R0is of Baire fist category) such that for every p∈R 0, the spaces X, pand (X∗,p∗satisfy both the w-FPP. Proof. By 29, Corollary 2.5, there exist a residual subset Rof Pand another residual subset Sin Qsuch that if p∈Rand s∈S, the spaces X, pand X∗,ssatisfy the w-FPP. We claim that the mapping h:P→Qdefined by hpp∗is an homeomorphism from Ponto Q. Indeed, this mapping is clearly one-one. Moreover, his onto because any equivalent norm in a reflexive space is a dual norm 25. It is enough to prove that his continuous because h−1 is similar to h. Fixed p∈Pand >0. Denote by athe positive number inf{px:x1}. Assume that ρp, q<δ:min{a2/4,a/2}.Notethatqx≥px−δx≥ax/2 for every x∈X. Furthermore, px≤1 implies qx 1δx≤1,3.1 8 Fixed Point Theory and Applications and, analogously, qx≤1 implies px 1δx≤1.3.2 Assume that u∗≤1andqx≤1. We have |ux|≤   ux 1δx       ux−x 1δx    ≤sup uy :py≤1   x−x 1δx    ≤p∗uδx2≤p∗uε. 3.3 Thus q∗u<p ∗uε. Analogously, p∗u<q ∗uεwhich implies |p∗u−q∗u|<ε for every uin the unit ball of X∗,·∗,thatis,ρp∗,q∗<ε. Finally, defining R0R∩h−1S, we conclude the proof. Remark 3.5. We do not know if a porous version of the above theorem does hold. In fact, we do not know either if Corollary 2.5 in 29holds in the sense of porosity. 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