Weak compactness and fixed point property for affine mappings

Weak Compactness and Fixed Point Property for Affine Mappings T. Dom´ınguez Benavides and M. A. Jap´on Pineda1 Address: Departamento de An´alisis Matem´atico, University of Seville, 41080 Seville, Spain. E-mail: [email protected], jap[email protected] and S. Prus2 Address: Department of Mathematics, M. Curie-SkÃlodowska University, 20-031 Lublin, Poland. E-mail: [email protected] Version: December 22, 2001 It is shown that a closed convex bounded subset of a Banach space is weakly compact if and only if it has the generic fixed point property for continuous affine mappings. The class of continuous affine mappings can be replaced by the class of affine mappings which are uniformly Lipschitzian with some constant M > 1 in the case of c0, the class of affine mappings which are uniformly Lipschitzian with some constant M > √6 in the case of quasi-reflexive James’ space Jand the class of nonexpansive affine mappings in the case of L-embedded spaces. 1. INTRODUCTION P. K. Lin and Y. Sternfeld [10] gave the complete characterization of norm compactness for convex subsets of a Banach space in terms of a fixed point property. They proved that if a convex set Kis not compact, then there exists a Lipschitzian mapping f:K→Kwith inf{kx−f(x)k:x∈ K}>0. It follows that a convex set in a Banach space has the fixed point property for Lipschitzian mappings if and only if it is compact. In this paper we study similar problems for weak compactness. Let Xbe a Banach space. We say that a closed convex bounded subset Cof Xhas the generic fixed point property for a class of mappings, if every mapping from a convex closed subset of Cinto itself belonging to this class has a fixed point. We will show that weak compactness of convex sets can be characterized in terms of the generic fixed point property for some classes 1The first and the second author were partially supported by project BFM 2000-0344 and FQM-127. 2The third author was partially supported by KBN grant NO 2P03A02915. 1 of affine mappings. A continuous affine self-mapping of a closed convex set is weakly continuous. The well-known Schauder-Tychonoff theorem (see [4, p. 74]) shows therefore that a continuous affine self-mapping of a convex weakly compact subset Cof a Banach space Xhas a fixed point (see also [14]). To complete the characterization, in a closed convex bounded but not weakly compact set Cwe construct a closed convex subset K which admits a continuous affine self-mapping without a fixed point. Thus, in general a closed convex bounded subset of a Banach space is weakly compact if and only if it has the generic fixed point property for continuous affine mappings. Moreover, in some spaces it is possible to replace the class of all continuous affine mappings by a smaller one. For instance, in [3] convex weakly compact subsets of the space L1[0,1] are characterized as the only ones which have the generic fixed point property for nonexpansive (i.e. Lipschitzian with constant 1) affine mappings. A similar result was proved for the preduals of semi-finite von Neumann algebra equipped with a faithful normal semi-finite trace. In this paper we prove that a closed convex bounded subset Cof c0is weakly compact if and only if Chas the generic fixed point property for affine mappings which are uniformly Lipschitzian. In fact, we prove that these mappings can be chosen with Lipschitz constant arbitrarily close to 1. As far as we know, it is an open problem if the constant can be chosen equal to 1. Since B. Maurey [13] proved that convex weakly compact subsets of c0 have the generic fixed point property for nonexpansive mappings, a positive answer to the above problem would give the inverse of Maurey’s result (see [12] for related results). The main tools for proving our results are basic sequences equivalent to the summing basis of c0. We will show that such a sequence can be extracted from any sequence (xn) in c0which converges weak?in `∞to an element x∈`∞\c0. The summing basis can be also considered in generalized James’ spaces Jp, 1 <p<∞. Their definition extends that of quasi-reflexive James’ space J, which is J2in this notation. We will show that a convex closed bounded subset Cof Jpis weakly compact if and only if there is M > 31/p21/q, where 1/p + 1/q = 1, such that Chas the generic fixed point property for uniformly Lipschitzian affine mappings with constant M. In the last section we will extend the results given in [3] to a larger class of spaces, the so-called L-embedded Banach spaces. 2. PRELIMINARIES The notation and terminology used in this paper are standard. They can be found for instance in [11] and [2]. For convenience of the reader we recall the basic definitions. Let Cbe a nonempty subset of a Banach space. The convex hull of Cwill be denoted by co C. Let us recall that a 2 self-mapping Tof a convex set Cis said to be affine if T(λx + (1 −λ)y) = λTx + (1 −λ)Ty whenever x, y ∈Cand λ∈[0,1]. A mapping T:C→Cis nonexpansive if kTx −Tyk ≤ kx−yk for all x, y ∈C. We say that Tis uniformly Lipschitzian with a constant Mif kTnx−Tnyk ≤ Mkx−yk for every n∈Nand all x, y ∈C. Basic sequences will be our main tool in this paper. Let (xn) be a sequence in a Banach space X. Its closed linear span will be denoted by [xn]. Let us recall that (xn) is a basic sequence if each x∈Xhas a unique expansion of the form x=P∞ n=1 tnxnfor some scalars t1, t2, . . . . Then the projections Pndefined on [xn] by the formula PnÃ∞ X i=1 tixi!= n X i=1 tixi are uniformly bounded and S= sup{kPnk:n∈N}is called the basis constant of (xn) (see [11]). It is clear that inf ©kx−yk:x∈[xi]n i=1,kxk ≥ a, y ∈[xi]∞ i=n+1, n ∈Nª≥a S.(1) for every a > 0. Additionally, we put Rn=Id[xn]−Pnand S+({xn}) = (x= ∞ X n=1 tnxn:tn≥0 for every n∈N, ∞ X n=1 tn= 1). In the sequel we will use the following fact. Fact 2.1. Let (xn) be a bounded sequence in a Banach space Xwithout weak convergent subsequences. Then (i) there exist a subsequence (xnk) and a functional f∈X?such that (xnk) is a basic sequence and a= inf{f(xnk) : k∈N}>0. Consequently, setting g= (1/a)fand yk= (a/f(xnk))xnk, we have g(yk) = 1 for every k∈N. (ii) co(yk) = S+({yk}). Proof. (i) By [7], (xn) has a basic subsequence (xnk). Our assumption guarantees that (xnk) does not weakly converge to zero. Passing to a subsequence, we can therefore find f∈X?so that inf{f(xnk) : k∈N}>0. (ii) is trivial. 3 Remark 2.2. The reasoning in the proof of (i) works only for real spaces. In the case of a complex space it is necessary to replace the functional f by its real or imaginary part. A sequence (yk) of nonzero vectors of Xis said to be a block basic sequence of a basic sequence (xn) if there exist a sequence (αn) of scalars and an increasing sequence of integers 0 ≤p1< p2< . . . such that yk= pk+1 X i=pk+1 αixi for every k. Clearly, (yk) is also a basic sequence and the basis constant of (yk) does not exceed that of (xn). Let (xn) and (yn) be basic sequences. We say that (xn) is equivalent to (yn) provided that a series P∞ n=1 tnxnconverges if and only if P∞ n=1 tnyn converges. This is the case if and only if there exist constants M1, M2∈ (0,∞) such that M1° ° ° ° ° ∞ X n=1 tnxn° ° ° ° ° ≤° ° ° ° ° ∞ X n=1 tnyn° ° ° ° ° ≤M2° ° ° ° ° ∞ X n=1 tnxn° ° ° ° ° (2) for every sequence (tn) of scalars such that the above series converge. We say that (xn) is λ-equivalent to (yn) if M2/M1≤λ. Clearly, the relation of λ-equivalence is symmetric. We will apply the following result (see [11, Proposition 1.a.9]). Theorem 2.3. Let (xn)be a basic sequence with the basis constant K in a Banach space Xand let M= inf{kxnk:n∈N}>0. If (yn)is a sequence in Xsuch that s= ∞ X n=1 kxn−ynk<M 2K, then (2) holds with M1= 1−2Ks/M and M2= 1+2Ks/M. Consequently, (yn)is a basic sequence (1 + 2Ks/M)(1 −2Ks/M)−1-equivalent to (xn). 3. CHARACTERIZATION OF WEAKLY COMPACT CONVEX SETS Let (en) be the standard basis of the space c0. The sequence of vectors σn=Pn k=1 ek= (1, . . . , 1,0,0...) is called the summing basis. It is easy to see that ° ° ° ° ° ∞ X n=1 tnσn° ° ° ° ° = sup n∈N¯¯¯¯¯ ∞ X k=n tk¯¯¯¯¯ for every sequence (tn) of scalars such that the series P∞ n=1 tnconverges. 4 Let 1 < p < ∞. By Jpwe denote the space of all sequences x= (x(n)) of real numbers such that limn→∞ x(n) = 0 and kxk= sup Ãm−1 X k=1 |x(qk)−x(qk+1)|p!1/p <∞ where the supremum is taken over all finite sequences q1<· · · < qmof positive integers. In case p= 2 this gives us the well-known definition of James’ space (see [6]). Let Pnbe the projection associated to the standard basis (en) of Jp. Then Pnx= (x(1), . . . , x(n),0,0, . . . ) for every x∈Jp. Using this formula, we extend Pnto the linear space of all sequences. For each p, the space Jpis not reflexive and J?? pis the space of all convergent sequences xsuch that kxkJ?? p= sup n∈N kPnxkJp is finite (see [11, Proposition 1.b.2]). Moreover, Jpdoes not contain c0and `1isomorphically. As in the case of c0, the vectors σn= (1, . . . , 1,0,0, . . . ) form a basis of Jp. If (tk) is a sequence of scalars such that the series P∞ k=1 tkσkconverges in Jp, then ° ° ° ° ° ∞ X k=1 tkσk° ° ° ° ° = sup   m−1 X k=1 ¯¯¯¯¯¯ qk+1−1 X i=qk ti¯¯¯¯¯¯ p  1/p where the supremum is taken over all finite sequences q1<· · · < qmof positive integers. Proposition 3.1. (a) Let Cbe a closed convex bounded set in a Banach space Xand (en)be the natural basis of `1. If Cis not weakly compact, then Ccontains a basic sequence (yn)such that there is an affine homeomorphism φ:S+({yn})→S+({en})with φ(yn) = enfor every n∈N. (b) Let Cbe a closed convex bounded set in Jpand (σn)be the summing basis of Jp. If Cis not weakly compact, then Ccontains a basic sequence (yn)equivalent to (σn). In particular, Ccontains a closed convex subset K=S+({yn})which is bi-Lipschitz homeomorphic to S+({σn}). (c) Let Cbe a closed convex bounded set in c0and (σn)be the summing basis of c0. If Cis not weakly compact, then Ccontains a basic sequence (yn)equivalent to (σn). In particular, Ccontains a closed convex subset K=S+({yn})which is bi-Lipschitz homeomorphic to S+({σn}). Proof. (a) Translating the set C, we can assume that 0 ∈C. Then Fact 2.1 gives a basic sequence (yn) in Cand a functional g∈X?with 5 g(yn) = 1 for every n∈N. Let K=S+({yn}) and φ:K→S+({en}) be the affine mapping such that φ(yn) = enfor every n∈N. We will check that φis a homeomorphism. Take x=P∞ n=1 tnyn∈Kand ² > 0. Fix an index msuch that P∞ n=m+1 tn< ²/4 and put δ=²/(8mkgk(S+ 1)) where Sis the basis constant of (yn). If u=P∞ n=1 bnyn∈Kis such that kx−uk< δ, then |tn−bn|<2Skgkδfor every n∈Nand hence, Pm n=1 |tn−bn|< ²/4. Next, ¯¯¯¯¯ ∞ X n=m+1 (bn−tn)¯¯¯¯¯ =|g(Rm(u−x))| ≤ kgkkRmkku−xk<² 4, and therefore ∞ X n=m+1 bn≤¯¯¯¯¯ ∞ X n=m+1 (bn−tn)¯¯¯¯¯ + ∞ X n=m+1 tn<² 2. Finally, we have kφ(u)−φ(x)k`1= ∞ X n=1 |bn−tn| ≤ m X n=1 |bn−tn|+ ∞ X n=m+1 bn+ ∞ X n=m+1 tn< ², which shows that φis continuous. Clearly, kφ−1(u)−φ−1(v)k ≤ maxn∈Nkynkku−vk`1for all u, v ∈ S+({en}). Thus φ−1is continuous. (b) Since Cis not weakly compact, there exists a sequence (xn) in C such that (xn) converges weak?in J?? pto some x∈J?? p\Jp. Passing to a subsequence, we can assume that (xn) is a basic sequence (see [7]). Let S be its basis constant. We put M1= inf{kxnk:n∈N},M2= sup{kxnk:n∈N}. Since xis a convergent sequence and x /∈Jp,L= limk→∞ |x(k)|>0. Given ²∈(0,1), we set M= (1 −²/16)Land γk=M1²(S(²+ 16))−12−k−2for k∈N. It is easy to see that there exists m0∈Nsuch that if m0≤q1<· · · < qm, then m−1 X k=1 |x(qk)−x(qk+1)|p≤µL² 16 ¶p .(3) Next, we choose two increasing sequences (mk) and (nk) so that m1≥m0, kPmk(xnk−x)k< γk,kRmk+1 (xnk)k< γk+1, and |x(j)|> M for every j≥m1. We put uk=Pmkx,vk= (Pmk+1 −Pmk)(xnk) and wk=uk+vk. Then kwk−xnkk ≤kPmk(uk−xnk)k+kRmk(vk−xnk)k =kPmk(x−xnk)k+kRmk+1 (xnk)k<2γk 6 for every k. Applying Theorem 2.3, we see that (wk) is a basic sequence (1+²/8)-equivalent to (xnk). Let K1denote the basis constant of (wk). We choose a sequence (pn) of nonnegative integers such that ∆k>2(M2K12k+1 (²+ 16)/(M²))qfor every kwhere ∆k=pk+1 −pkand 1/p + 1/q = 1. Let zk=1 ∆k pk+1 X i=pk+1 ui, z0 k=1 ∆k pk+1 X i=pk+1 wi. Then kz0 kk ≥ |z0 k(mpk+1)|> M and it is easy to see that kz0 k−zkk=1 ∆k° ° ° ° ° ° pk+1 X i=pk+1 vi° ° ° ° ° ° ≤21 q ∆k  pk+1 X i=pk+1 kvikp  1/p <M² K1(²+ 16)2k+1 for every k. Theorem 2.3 shows that (zk) is a basic sequence (1 + ²/8)- equivalent to (z0 k). Consequently, (zk) is (1 + ²/8)2-equivalent to a block basic sequence (yn) of (xnk) whose terms belong to co{xnk}. We will show that (zk) is equivalent to (σk). To this end let us fix a sequence (tk) such that the series P∞ k=1 tkσkconverges and put N= kP∞ k=1 tkσkk,y=P∞ k=1 tkzk. We take a finite sequence q1<· · · < qmof positive integers. By A1we denote the set of all 1 ≤j < m such that there exists k≥2 with qj≤mpk< qj+1 and let A2be the set of the remaining indices. Given j∈A1, we find k≥1, 0 ≤i1≤∆k−1, l≥2 and 0 ≤i2≤ ∆l−1 such that mpk+i1< qj≤mpk+i1+1 < mpl+i2< qj+1 ≤mpl+i2+1. Then |y(qj)−y(qj+1)|=¯¯¯¯¯ x(qj)Ãλtk+ l−1 X i=k+1 ti+ (1 −µ)tl! + (x(qj)−x(qj+1)) õtl+ ∞ X i=l+1 ti!¯¯¯¯¯ ≤|x(qj)|max (¯¯¯¯¯ l−υ X i=k+ν ti¯¯¯¯¯ :ν, υ = 0,1) +|x(qj)−x(qj+1)|sup n∈N¯¯¯¯¯ ∞ X i=n ti¯¯¯¯¯ . where λ= 1 −i1/∆k,µ= 1 −i2/∆l. This gives us the estimate |y(qj)−y(qj+1)| ≤ M2¯¯¯¯¯¯ lj X i=kj ti¯¯¯¯¯¯ +|x(qj)−x(qj+1)|N(4) for some k≤kj≤lj≤l. Observe that X j∈A1 ¯¯¯¯¯¯ lj X i=kj ti¯¯¯¯¯¯ p ≤2Np.(5) 7 Let us now consider the set A2. We decompose it into disjoint intervals Ak 2where A1 2={j∈A2: 1 ≤qj< qj+1 ≤mp2}and Ak 2={j∈A2:mpk< qj< qj+1 ≤mpk+1 }for k≥2. Assume that Ak 2is not empty. It is easy to see that if j∈Ak 2, then |y(qj)−y(qj+1)| ≤ λk j|x(qj)tk|+|x(qj)−x(qj+1)|N(6) for some nonnegative λk jsuch that Pj∈Ak 2λk j≤1. Clearly, ∞ X k=1 X j∈Ak 2 (λk j|tk|)p≤ ∞ X k=1 |tk|p≤Np.(7) Here we regard sums over the empty set as zero. Using (4), (5), (6), and (7), we obtain   m−1 X j=1 |y(qj)−y(qj+1)|p  1/p ≤M2Ã2Np+ ∞ X k=1 |tk|p!1/p +Nkxk ≤M2(31/p + 1)N. (8) We now set rk=mpk+1 for k∈N. If i < j, then |y(ri)−y(rj)|=¯¯¯¯¯¯ x(ri)Ãj−1 X k=i tk!+ (x(ri)−x(rj))   ∞ X k=j tk ¯¯¯¯¯¯ ≥ |x(ri)|¯¯¯¯¯ j−1 X k=i tk¯¯¯¯¯ − |x(ri)−x(rj)|¯¯¯¯¯¯ ∞ X k=j tk¯¯¯¯¯¯ ≥M¯¯¯¯¯ j−1 X k=i tk¯¯¯¯¯ − |x(ri)−x(rj)|N. This and (3) show that kyk ≥ Ãm−1 X k=1 |y(rqk)−y(rqk+1 )|p!1/p ≥M  m−1 X k=1 ¯¯¯¯¯¯ qk+1−1 X i=qk ti¯¯¯¯¯¯ p  1/p −L² 16 N for every sequence q1<· · · < qmof positive integers. Hence kyk ≥ NµM−L² 16 ¶=NL ³1−² 8´.(9) This completes the proof of (b). (c) To prove (c) we can apply an argument similar to that in the proof of (b) (now L= lim supk→∞ |x(k)|>0). In this way it is not difficult to 8 construct a basic sequence (zk) which is M2/M equivalent to the summing basis (σn) of c0and such that (zk) is (1 + ²/8)2equivalent to a block basic (yn) of (xnk) whose terms belong to co{xnk}. In the special case when Cis the unit ball of a nonreflexive space part (a) of Proposition 3.1 was obtained in [14] (see also [15]). We will generalize another result from [14]. Let C6=∅be a convex subset of a Banach space and T:C→Cbe an affine mapping. We put θ(T) = inf nlim inf n→∞ kx−Tnyk:x, y ∈Co. From the first part of the proof of [14, Theorem 3] we see that inf{kx−Txk: x∈C}= 0. In spite of this fact, if Cis not weakly compact, it is possible to construct a set K⊂Cand a continuous affine mapping T:K→K such that θ(T)>0. In particular, Tfails to have fixed points. Let (xn) be a bounded basic sequence. The right shift T0with respect to (xn) is the mapping defined by the formula T0Ã∞ X n=1 tnxn!= ∞ X n=1 tnxn+1. By the bilateral shift T1with respect to (xn) we in turn mean the mapping T1Ã∞ X n=1 tnxn!=t2x1+ ∞ X k=1 t2k−1x2k+1 + ∞ X k=2 t2kx2k−2. Clearly, T0and T1are affine self-mappings of S+({xn}) and T1is onto. Theorem 3.2. (a) Let Cbe a closed convex bounded set in a Banach space X. If Cis not weakly compact, then there are a closed convex subset K⊂Cand an affine continuous mapping T:K→Ksuch that T(K) = K and θ(T)>0. (b) Let C⊂Jpbe a closed convex bounded set. If Cis not weakly compact, then there are a closed convex subset K⊂Cand an affine uniformly Lipschitzian mapping T:K→Ksuch that θ(T)>0. (c) Let C⊂c0be a closed convex bounded set. If Cis not weakly compact, then there are a closed convex subset K⊂Cand an affine uniformly Lipschitzian mapping T:K→Ksuch that θ(T)>0. Proof. (a) Let (en) be the standard basis of `1and T1:S+({en})→ S+({en}) be the bilateral shift. Fact 2.1 gives us a sequence (yn) in Cand a functional g∈X?. Let Sbe the basis constant of (yn) and K=S+({yn}). The proof of Proposition 3.1 (a) shows that the affine mapping φ:K→ S+({en}) such that φ(yn) = enis a homeomorphism. Define T:K→K by the formula T=φ−1T1φ. Let x=P∞ n=1 tnynand y=P∞ n=1 bnyn 9 [15] V. D. Milman, Geometric theory of Banach spaces. Part I. The theory of basis and minimal systems, Russ. Math. Surveys 25 (1970), 111–170 (translated from Usp. Mat. Nauk 25 (1970), 113–173). [16] H. Pfitzner, L-embedded Banach spaces and measure topology, Israel J. Math, to appear. 16