It is proved that the family of all equivalent norms in c0 without asymptotically isometric copies of c0 and failing to have the fixed point property is dense in the set of all renormings of c0.Dirección General de Enseñanza SuperiorJunta de Andaluci

Barrera Cuevas, Alfredo

Japón Pineda, María de los Ángeles

English

idUS. Depósito de Investigación Universidad de Sevilla

Fixed Point Theory, 14(2013), No. 1, 53-58http://www.math.ubbcluj.ro/∼nodeacj/sfptcj.htmlDENSITY OF THE SET OF RENORMINGS IN c0 WITHOUTASYMPTOTICALLY ISOMETRIC COPIES OF c0 ANDFAILING TO HAVE THE FIXED POINT PROPERTYA. BARRERA-CUEVAS AND M. A. JAPÓNDedicated to K. Goebel on the occasion of his retirement, and to L. Ciric, W.A. Kirkand I. A. Rus on the occasion of their 75th birthdayDepartamento de Análisis MatemáticoFacultad de Matemáticas, Universidad de SevillaTarfia s/n, 41012, Sevilla, EspañaE-mail: japon@us.es, gauss alf@hotmail.comAbstract. It is proved that the family of all equivalent norms in c0 without asymptotically isometriccopies of c0 and failing to have the fixed point property is dense in the set of all renormings of c0.Key Words and Phrases: Asymptotically isometric copies, fixed point theory, renorming theory,nonexpansive mappings.2010 Mathematics Subject Classification: 46B03, 46B20, 47H09, 47H10.1. IntroductionA Banach space is said to have the fixed point property (FPP) if every nonexpansivemapping defined from a closed convex bounded subset into itself has a fixed point.Recall that a mapping defined on a subset of a Banach space is nonexpansive if‖Tx − Ty‖ ≤ ‖x − y‖ for all x, y in the domain of T . It is well-known that thegeometry of the Banach space plays a fundamental role to assure the FPP (see themonographs [8], [10] and the references therein). It is also classical that the sequenceBanach spaces `1 and c0 fail to have the FPP. In 1997, P. Dowling, C. Lennard andB. Turett [2] proved that a Banach space fails to have the FPP whenever this spacecontains a sequence which generates either an asymptotically isometric copy of c0or an asymptotically isometric copy of `1 (see also Chapter 9 in [10]). In the lastyears, asymptotically isometric copies of c0 and `1 have become very helpful in thedevelopment of the fixed point theory for nonexpansive mappings. Actually, manyclasses of nonreflexive Banach spaces are known to fail the FPP because it is provedthat they contain one of such copies (see for instance [3], [5], [6] or [13]).It is noteworthy that the non-expansiveness condition strongly depends on thegiven norm in the Banach space X. If T is nonexpansive for a norm ‖ · ‖, T may failThe authors are partially supported by DGES, Grant MTM-2012-34847-C02-01 and Junta deAndalućıa, Grant FQM-127.5354 A. BARRERA-CUEVAS AND M. A. JAPÓNto be nonexpansive for another equivalent norm |·| in X. In fact, it was proved by P.K.Lin [12] that `1 can be renormed to have the fixed point property for nonexpansivemappings and it is still an open problem whether there exists an equivalent norm in c0satisfying such condition. On the other hand, not all equivalent norms in c0 containan asymptotically isometric copy of c0 [2]. In fact, in [9] a dense family of equivalentnorms in c0 failing to have an asymptotically isometric copy of c0 is given. In [1] T.Domı́nguez-Benavides proved that the set of all equivalent norms in c0 which fail tohave the FPP is also dense in the set of all renormings of c0. The main object of thispaper is to unify these two results. That is, we will prove that for every equivalentnorm in c0, we can obtain a renorming of c0 as close as the first one and we like andwhich fails both, to have the FPP and to have an asymptotically isometric copy ofc0, which improves the density results given for the Banach space c0 in [1] and [9].2. PreliminariesIn this section we introduce the definitions, notation and the known results thatwe will use in the proof of the main theorem.Definition 2.1. A Banach space (X, ‖·‖) is said to have an asymptotically isometriccopy of c0 if there are a null sequence (εn) in (0, 1) and a sequence (xn) in X so thatsupn∈N(1− εn)|tn| ≤∥∥∥∥∥∞∑n=1tnxn∥∥∥∥∥ ≤ supn∈N |tn|,for all (tn) ∈ c0. In this case we say that X contains a c0-a.i. copy.Given (X, ‖ · ‖) a Banach space we define by P(X) the set of all equivalent normson X endowed with the metric [7]:ρ(p, q) = sup{|p(x)− q(x)| : ‖x‖ ≤ 1}, if p, q ∈ P(X).We consider the Banach space c0 endowed with its usual norm given by ‖x‖∞ =supn |tn| whenever x = (tn)n ∈ c0.Similarly to [9], we set:PFPP (c0) := {p ∈ P(c0) : (c0, p) has the FPP },andP0(c0) := {p ∈ P(c0) : (c0, p) fails to have a c0-a.i. copy }.We know that PFPP (c0) ⊂ P0(c0) [2] although it is an open problem whetherPFPP (c0) could be nonempty. On the other hand, P0(c0) \ PFPP (c0) 6= ∅. Indeed,let us consider the following equivalent norm on c0:|x|∞ := supn|tn|+∞∑n=1|tn|2nif x = (tn)n ∈ c0. It can be proved that (c0, |·|∞) fails to have both, an asymptoticallyisometric copy of c0 [2], and the FPP [5].We recall James’ Lemma:DENSITY OF THE SET OF RENORMINGS IN c0 55Lemma 2.2. [11] Let (X, ‖ · ‖) be a Banach space which contains a subspace isomor-phic to c0. Then for any positive number δ > 0 there is a sequence (yn) in X suchthat(1− δ) supn|tn| <∥∥∥∥∥∑ntnyn∥∥∥∥∥ ≤ supn |tn|for all sequences (tn) ∈ c0.In [1] T. Domı́nguez-Benavides proved that P(c0) \ PFPP (c0) is dense in P(c0).The main object of this manuscript is to prove that P0(c0) \ PFPP (c0) is also densein P(c0), that is, that for every equivalent norm p in c0 we are able to find anotherequivalent norm q as close as p as we like in the ρ metric such that q fails to haveboth, the FPP and to have an asymptotically isometric copy of c0. Notice thatP0(c0) \ PFPP (c0) is a set of equivalent norms of c0 which is strictly contained inP(c0) \ PFPP (c0).3. Main resultTheorem 3.1. The set P0(c0) \ PFPP (c0) is dense in P(c0).Proof. Let p be any norm in P(c0) and ε > 0. Take a constant b such that p(x) ≤b‖x‖∞ for all x ∈ c0 and δ a positive number such thatbδ1− δ<ε2Using Lemma 2.2 we obtain a sequence (yn) in c0 such that(1− δ) supn|tn| ≤ p(∑ntnyn)≤ supn|tn|for all (tn) ∈ c0.Moreover, following James’ proof (see [11], Lemma 2.2) se can achieve that thesequence (yn)n is finitely supported and that supp(yn) ∩ supp(ym) = ∅ si n 6= m,where by supp(y) we denote the support of the vector y. Moreover, from the proof ofLemma 2.2 in [11] we can also deduce that ‖yn‖∞ = ‖ym‖∞ if n, m ∈ N.Let us define Y := span〈yn〉 which is a subspace of c0. We define in the subspace Ythe equivalent norm |y| := supn |tn| whenever y =∑ntnyn. According to the aboveinequalities we have(1− δ)|y| ≤ p(y) ≤ |y|for all y ∈ Y .Since Y is a closed subspace of c0, applying Lemma 2.2 in [1], we can extend thenorm | · | to the whole Banach space c0 keeping the same equivalent constants, thatis, there exists a norm q in c0 such that q(y) = |y| whenever y ∈ Y and(1− δ)q(x) ≤ p(x) ≤ q(x),for all x ∈ c0.56 A. BARRERA-CUEVAS AND M. A. JAPÓNUsing now Theorem 4.1 in [9], for every λ > 0, the norm q + λ| · |∞ fails to havean asymptotically isometric copy of c0, where | · |∞ was defined in Section 2. Let usprove that (c0, q + λ| · |∞) also fails to have the FPP for every λ > 0:For x =∑ntnen ∈ c0, we setη(x) :=∑n|tn|2n.With this notation |x|∞ = ‖x‖∞ + η(x) for all x ∈ c0. Notice that η(ax + by) =|a|η(x)+ |b|η(y) if x, y are two vectors in c0 with disjoint supports and a, b ∈ R. Sincethe sequence (yn) is bounded and disjointly supported, we can check that limn η(yn) =0. Hence, without loss of generality, we can extract a subsequence(ynk)k ⊂ (yn)n suchthat η(yn(k+1)) ≤ η(ynk) for all k ∈ N.Define the setC := {∞∑k=1tkynk : 0 ≤ tk ≤ 1},which is a closed convex bounded subset of c0 and it is contained in Y . Let us considerthe mapping T : C → C given byT (∞∑k=1tkynk) = y1 +∞∑k=1tnyn(k+1)It is easy to check that T is fixed point free since (ynk)k is a basic sequence. Let usprove that T is nonexpansive for the norm q + λ| · |∞.Firstly, T is nonexpansive for the q norm: set x =∞∑k=1tkynk , y =∞∑k=1skynk twovectors in C.q(Tx− Ty) = |Tx− Ty| = supk|tk − sk| = |x− y| = q(x− y).On the other hand, T is also nonexpansive for the | · |∞ norm. Indeed:|Tx− Ty|∞ =∣∣∣∣∣∞∑k=1(tk − sk)yn(k+1)∣∣∣∣∣∞=∥∥∥∥∥∞∑k=1(tk − sk)yn(k+1)∥∥∥∥∥∞+ η( ∞∑k=1(tk − sk)yn(k+1))= supk|tk − sk|‖yn(k+1)‖∞ +∞∑k=1|tk − sk|η(yn(k+1))≤ supk|tk − sk|‖ynk‖∞ +∞∑k=1|tk − sk|η(ynk)=∥∥∥∥∥∞∑k=1(tk − sk)ynk∥∥∥∥∥∞+ η( ∞∑k=1(tk − sk)ynk)DENSITY OF THE SET OF RENORMINGS IN c0 57=∣∣∣∣∣∞∑k=1(tk − sk)ynk∣∣∣∣∣∞= |x− y|∞.The above shows that T is nonexpansive for the norm q + λ| · |∞ for every λ > 0.Consider 0 < λ < ε/4 and take x ∈ c0 with ‖x‖∞ ≤ 1:|p(x)− q(x)− λ|x|∞| ≤ |p(x)− q(x)|+ λ|x|∞≤ 11−δ p(x)− p(x) + λ|x|∞≤ δ1−δ b‖x‖∞ + 2λ‖x‖∞< εwhich implies that ρ(p, q + λ| · |∞) < ε and this finishes the proof.Remark 3.2. A classical way to measure the distance between two isomorphic Ba-nach spaces is to consider the Banach-Mazur distance defined byd(X, Y ) = inf{‖T‖‖T−1‖ : T : X → Y isomorphism }.From the proof of Theorem 3.1 and by using the identity operator between c0 endowedwith two equivalent norms, we can derive that for every renorming of c0 we can find anisomorphic Banach space without the FPP, failing to have an asymptotically isometriccopy of c0 and whose Banach-Mazur distance is as close to one as we like.References[1] T. Domı́nguez Benavides, Distortion and stability of the fixed point property for non-expansivemappings, Nonlinear Anal., 75(2012), 3229-3234.[2] P.N. Dowling, C.J. Lennard, B. Turett, Asymptotically perturbed norms of classical sequencespaces with applications to fixed point theory, Proceedings of Workshop on Fixed Point Theory(Kazimierz Dolny, 1997), Ann. Univ. Mariae Curie-Sklodowska Sect. A, 51(1997), no. 2, 67–98.[3] P.N. Dowling, C.J. Lennard, Every nonreflexive subspace of L1[0, 1] fails the fixed point property,Proc. Amer. Math. Soc., 125(1997), 443-446.[4] P.N. Dowling, C.J. Lennard, B. Turett, Asymptotically Isometric Copies of c0 in Banach Spaces,J. Math. Anal. Appl., 219(1998), 377-391.[5] P.N. Dowling, P.K. Lin, B. Turett, Failure of the FPP inside an asymptotically isometric-freecopy of c0, Nonlinear Anal., 73(2010), 1175-1179.[6] P.N. Dowling, N. Randrianantoanina, Spaces of compact operators on a Hilbert space with thefixed point property, J. Funct. Anal., 168(1)(1999), 111-120.[7] M. Fabian, L. Zajicek, V. Zizler, On residuality of the set of rotund norms on a Banach space,Math. Ann., 258(1982), 349-351.[8] K. Goebel, W. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990.[9] C. Hernández-Linares, M. Japón, C. Lennard, Renormings failing to have asymptotically isomet-ric copies of `1 or c0, Nonlinear Anal. (to appear) http://dx.doi.org/10.1016/j.na.2012.09.006.[10] W.A. Kirk, B. Sims (Ed.), Handbook of Metric Fixed Point Theory, Kluwer Academic Publish-ers, 2001.[11] R.C. James, Uniformly non-square Banach spaces, Ann. of Math., 80(1964), no. 2, 542-550.[12] P.K. Lin, There is an equivalent norm on `1 that has the fixed point property, Nonlinear Anal.,68(2008), no. 8, 2303-2308.[13] H. Pfitzner, A note on asymptotically isometric copies of `1 and c0, Proc. Amer. Math. Soc.,129(5)(2001), 1367-1373.Received: August 30, 2012; Accepted: October 31, 2012.58 A. BARRERA-CUEVAS AND M. A. JAPÓN

Density of the set of renormings in c0 without asymptotically isometric copies of c0 and failing to have the fixed point property

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Density of the set of renormings in c0 without asymptotically isometric copies of c0 and failing to have the fixed point property

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