Weak compactness and fixed point property for affine mappings

Weak Compac ness and Fixed Poin P ope y o A ine Mappings T. Dom´ınguez Bena ides and M. A. Jap´on Pineda1 Add ess: Depa amen o de An´alisis Ma em´a ico, Uni e si y o Se ille, 41080 Se ille, Spain. E-mail: [email p o ec ed], jap[email p o ec ed] and S. P us2 Add ess: Depa men o Ma hema ics, M. Cu ie-SkÃlodowska Uni e si y, 20-031 Lublin, Poland. E-mail: [email p o ec ed] Ve sion: Decembe 22, 2001 I is shown ha a closed con ex bounded subse o a Banach space is weakly compac i and only i i has he gene ic ixed poin p ope y o con inuous a ine mappings. The class o con inuous a ine mappings can be eplaced by he class o a ine mappings which a e uni o mly Lipschi zian wi h some cons an M > 1 in he case o c0, he class o a ine mappings which a e uni o mly Lipschi zian wi h some cons an M > √6 in he case o quasi- e lexi e James’ space Jand he class o nonexpansi e a ine mappings in he case o L-embedded spaces. 1. INTRODUCTION P. K. Lin and Y. S e n eld [10] ga e he comple e cha ac e iza ion o no m compac ness o con ex subse s o a Banach space in e ms o a ixed poin p ope y. They p o ed ha i a con ex se Kis no compac , hen he e exis s a Lipschi zian mapping :K→Kwi h in {kx− (x)k:x∈ K}>0. I ollows ha a con ex se in a Banach space has he ixed poin p ope y o Lipschi zian mappings i and only i i is compac . In his pape we s udy simila p oblems o weak compac ness. Le Xbe a Banach space. We say ha a closed con ex bounded subse Co Xhas he gene ic ixed poin p ope y o a class o mappings, i e e y mapping om a con ex closed subse o Cin o i sel belonging o his class has a ixed poin . We will show ha weak compac ness o con ex se s can be cha ac e ized in e ms o he gene ic ixed poin p ope y o some classes 1The i s and he second au ho we e pa ially suppo ed by p ojec BFM 2000-0344 and FQM-127. 2The hi d au ho was pa ially suppo ed by KBN g an NO 2P03A02915. 1 o a ine mappings. A con inuous a ine sel -mapping o a closed con ex se is weakly con inuous. The well-known Schaude -Tychono heo em (see [4, p. 74]) shows he e o e ha a con inuous a ine sel -mapping o a con ex weakly compac subse Co a Banach space Xhas a ixed poin (see also [14]). To comple e he cha ac e iza ion, in a closed con ex bounded bu no weakly compac se Cwe cons uc a closed con ex subse K which admi s a con inuous a ine sel -mapping wi hou a ixed poin . Thus, in gene al a closed con ex bounded subse o a Banach space is weakly compac i and only i i has he gene ic ixed poin p ope y o con inuous a ine mappings. Mo eo e , in some spaces i is possible o eplace he class o all con in- uous a ine mappings by a smalle one. Fo ins ance, in [3] con ex weakly compac subse s o he space L1[0,1] a e cha ac e ized as he only ones which ha e he gene ic ixed poin p ope y o nonexpansi e (i.e. Lips- chi zian wi h cons an 1) a ine mappings. A simila esul was p o ed o he p eduals o semi- ini e on Neumann algeb a equipped wi h a ai h ul no mal semi- ini e ace. In his pape we p o e ha a closed con ex bounded subse Co c0is weakly compac i and only i Chas he gene ic ixed poin p ope y o a ine mappings which a e uni o mly Lipschi zian. In ac , we p o e ha hese mappings can be chosen wi h Lipschi z cons an a bi a ily close o 1. As a as we know, i is an open p oblem i he cons an can be chosen equal o 1. Since B. Mau ey [13] p o ed ha con ex weakly compac subse s o c0 ha e he gene ic ixed poin p ope y o nonexpansi e mappings, a posi i e answe o he abo e p oblem would gi e he in e se o Mau ey’s esul (see [12] o ela ed esul s). The main ools o p o ing ou esul s a e basic sequences equi alen o he summing basis o c0. We will show ha such a sequence can be ex ac ed om any sequence (xn) in c0which con e ges weak?in `∞ o an elemen x∈`∞ c0. The summing basis can be also conside ed in gene alized James’ spaces Jp, 1 <p<∞. Thei de ini ion ex ends ha o quasi- e lexi e James’ space J, which is J2in his no a ion. We will show ha a con ex closed bounded subse Co Jpis weakly compac i and only i he e is M > 31/p21/q, whe e 1/p + 1/q = 1, such ha Chas he gene ic ixed poin p ope y o uni o mly Lipschi zian a ine mappings wi h cons an M. In he las sec ion we will ex end he esul s gi en in [3] o a la ge class o spaces, he so-called L-embedded Banach spaces. 2. PRELIMINARIES The no a ion and e minology used in his pape a e s anda d. They can be ound o ins ance in [11] and [2]. Fo con enience o he eade we ecall he basic de ini ions. Le Cbe a nonemp y subse o a Banach space. The con ex hull o Cwill be deno ed by co C. Le us ecall ha a 2 sel -mapping To a con ex se Cis said o be a ine i T(λx + (1 −λ)y) = λTx + (1 −λ)Ty whene e x, y ∈Cand λ∈[0,1]. A mapping T:C→Cis nonexpansi e i kTx −Tyk ≤ kx−yk o all x, y ∈C. We say ha Tis uni o mly Lipschi zian wi h a cons an Mi kTnx−Tnyk ≤ Mkx−yk o e e y n∈Nand all x, y ∈C. Basic sequences will be ou main ool in his pape . Le (xn) be a sequence in a Banach space X. I s closed linea span will be deno ed by [xn]. Le us ecall ha (xn) is a basic sequence i each x∈Xhas a unique expansion o he o m x=P∞ n=1 nxn o some scala s 1, 2, . . . . Then he p ojec ions Pnde ined on [xn] by he o mula PnÃ∞ X i=1 ixi!= n X i=1 ixi a e uni o mly bounded and S= sup{kPnk:n∈N}is called he basis cons an o (xn) (see [11]). I is clea ha in ©kx−yk:x∈[xi]n i=1,kxk ≥ a, y ∈[xi]∞ i=n+1, n ∈Nª≥a S.(1) o e e y a > 0. Addi ionally, we pu Rn=Id[xn]−Pnand S+({xn}) = (x= ∞ X n=1 nxn: n≥0 o e e y n∈N, ∞ X n=1 n= 1). In he sequel we will use he ollowing ac . Fac 2.1. Le (xn) be a bounded sequence in a Banach space Xwi hou weak con e gen subsequences. Then (i) he e exis a subsequence (xnk) and a unc ional ∈X?such ha (xnk) is a basic sequence and a= in { (xnk) : k∈N}>0. Consequen ly, se ing g= (1/a) and yk= (a/ (xnk))xnk, we ha e g(yk) = 1 o e e y k∈N. (ii) co(yk) = S+({yk}). P oo . (i) By [7], (xn) has a basic subsequence (xnk). Ou assump ion gua an ees ha (xnk) does no weakly con e ge o ze o. Passing o a subsequence, we can he e o e ind ∈X?so ha in { (xnk) : k∈N}>0. (ii) is i ial. 3 Rema k 2.2. The easoning in he p oo o (i) wo ks only o eal spaces. In he case o a complex space i is necessa y o eplace he unc ional by i s eal o imagina y pa . A sequence (yk) o nonze o ec o s o Xis said o be a block basic sequence o a basic sequence (xn) i he e exis a sequence (αn) o scala s and an inc easing sequence o in ege s 0 ≤p1< p2< . . . such ha yk= pk+1 X i=pk+1 αixi o e e y k. Clea ly, (yk) is also a basic sequence and he basis cons an o (yk) does no exceed ha o (xn). Le (xn) and (yn) be basic sequences. We say ha (xn) is equi alen o (yn) p o ided ha a se ies P∞ n=1 nxncon e ges i and only i P∞ n=1 nyn con e ges. This is he case i and only i he e exis cons an s M1, M2∈ (0,∞) such ha M1° ° ° ° ° ∞ X n=1 nxn° ° ° ° ° ≤° ° ° ° ° ∞ X n=1 nyn° ° ° ° ° ≤M2° ° ° ° ° ∞ X n=1 nxn° ° ° ° ° (2) o e e y sequence ( n) o scala s such ha he abo e se ies con e ge. We say ha (xn) is λ-equi alen o (yn) i M2/M1≤λ. Clea ly, he ela ion o λ-equi alence is symme ic. We will apply he ollowing esul (see [11, P oposi ion 1.a.9]). Theo em 2.3. Le (xn)be a basic sequence wi h he basis cons an K in a Banach space Xand le M= in {kxnk:n∈N}>0. I (yn)is a sequence in Xsuch ha s= ∞ X n=1 kxn−ynk<M 2K, hen (2) holds wi h M1= 1−2Ks/M and M2= 1+2Ks/M. Consequen ly, (yn)is a basic sequence (1 + 2Ks/M)(1 −2Ks/M)−1-equi alen o (xn). 3. CHARACTERIZATION OF WEAKLY COMPACT CONVEX SETS Le (en) be he s anda d basis o he space c0. The sequence o ec o s σn=Pn k=1 ek= (1, . . . , 1,0,0...) is called he summing basis. I is easy o see ha ° ° ° ° ° ∞ X n=1 nσn° ° ° ° ° = sup n∈N¯¯¯¯¯ ∞ X k=n k¯¯¯¯¯ o e e y sequence ( n) o scala s such ha he se ies P∞ n=1 ncon e ges. 4 Le 1 < p < ∞. By Jpwe deno e he space o all sequences x= (x(n)) o eal numbe s such ha limn→∞ x(n) = 0 and kxk= sup Ãm−1 X k=1 |x(qk)−x(qk+1)|p!1/p <∞ whe e he sup emum is aken o e all ini e sequences q1<· · · < qmo posi i e in ege s. In case p= 2 his gi es us he well-known de ini ion o James’ space (see [6]). Le Pnbe he p ojec ion associa ed o he s anda d basis (en) o Jp. Then Pnx= (x(1), . . . , x(n),0,0, . . . ) o e e y x∈Jp. Using his o mula, we ex end Pn o he linea space o all sequences. Fo each p, he space Jpis no e lexi e and J?? pis he space o all con e gen sequences xsuch ha kxkJ?? p= sup n∈N kPnxkJp is ini e (see [11, P oposi ion 1.b.2]). Mo eo e , Jpdoes no con ain c0and `1isomo phically. As in he case o c0, he ec o s σn= (1, . . . , 1,0,0, . . . ) o m a basis o Jp. I ( k) is a sequence o scala s such ha he se ies P∞ k=1 kσkcon e ges in Jp, hen ° ° ° ° ° ∞ X k=1 kσk° ° ° ° ° = sup   m−1 X k=1 ¯¯¯¯¯¯ qk+1−1 X i=qk i¯¯¯¯¯¯ p  1/p whe e he sup emum is aken o e all ini e sequences q1<· · · < qmo posi i e in ege s. P oposi ion 3.1. (a) Le Cbe a closed con ex bounded se in a Ba- nach space Xand (en)be he na u al basis o `1. I Cis no weakly compac , hen Ccon ains a basic sequence (yn)such ha he e is an a ine homeo- mo phism φ:S+({yn})→S+({en})wi h φ(yn) = en o e e y n∈N. (b) Le Cbe a closed con ex bounded se in Jpand (σn)be he summing basis o Jp. I Cis no weakly compac , hen Ccon ains a basic sequence (yn)equi alen o (σn). In pa icula , Ccon ains a closed con ex subse K=S+({yn})which is bi-Lipschi z homeomo phic o S+({σn}). (c) Le Cbe a closed con ex bounded se in c0and (σn)be he summing basis o c0. I Cis no weakly compac , hen Ccon ains a basic sequence (yn)equi alen o (σn). In pa icula , Ccon ains a closed con ex subse K=S+({yn})which is bi-Lipschi z homeomo phic o S+({σn}). P oo . (a) T ansla ing he se C, we can assume ha 0 ∈C. Then Fac 2.1 gi es a basic sequence (yn) in Cand a unc ional g∈X?wi h 5 g(yn) = 1 o e e y n∈N. Le K=S+({yn}) and φ:K→S+({en}) be he a ine mapping such ha φ(yn) = en o e e y n∈N. We will check ha φis a homeomo phism. Take x=P∞ n=1 nyn∈Kand ² > 0. Fix an index msuch ha P∞ n=m+1 n< ²/4 and pu δ=²/(8mkgk(S+ 1)) whe e Sis he basis cons an o (yn). I u=P∞ n=1 bnyn∈Kis such ha kx−uk< δ, hen | n−bn|<2Skgkδ o e e y n∈Nand hence, Pm n=1 | n−bn|< ²/4. Nex , ¯¯¯¯¯ ∞ X n=m+1 (bn− n)¯¯¯¯¯ =|g(Rm(u−x))| ≤ kgkkRmkku−xk<² 4, and he e o e ∞ X n=m+1 bn≤¯¯¯¯¯ ∞ X n=m+1 (bn− n)¯¯¯¯¯ + ∞ X n=m+1 n<² 2. Finally, we ha e kφ(u)−φ(x)k`1= ∞ X n=1 |bn− n| ≤ m X n=1 |bn− n|+ ∞ X n=m+1 bn+ ∞ X n=m+1 n< ², which shows ha φis con inuous. Clea ly, kφ−1(u)−φ−1( )k ≤ maxn∈Nkynkku− k`1 o all u, ∈ S+({en}). Thus φ−1is con inuous. (b) Since Cis no weakly compac , he e exis s a sequence (xn) in C such ha (xn) con e ges weak?in J?? p o some x∈J?? p Jp. Passing o a subsequence, we can assume ha (xn) is a basic sequence (see [7]). Le S be i s basis cons an . We pu M1= in {kxnk:n∈N},M2= sup{kxnk:n∈N}. Since xis a con e gen sequence and x /∈Jp,L= limk→∞ |x(k)|>0. Gi en ²∈(0,1), we se M= (1 −²/16)Land γk=M1²(S(²+ 16))−12−k−2 o k∈N. I is easy o see ha he e exis s m0∈Nsuch ha i m0≤q1<· · · < qm, hen m−1 X k=1 |x(qk)−x(qk+1)|p≤µL² 16 ¶p .(3) Nex , we choose wo inc easing sequences (mk) and (nk) so ha m1≥m0, kPmk(xnk−x)k< γk,kRmk+1 (xnk)k< γk+1, and |x(j)|> M o e e y j≥m1. We pu uk=Pmkx, k= (Pmk+1 −Pmk)(xnk) and wk=uk+ k. Then kwk−xnkk ≤kPmk(uk−xnk)k+kRmk( k−xnk)k =kPmk(x−xnk)k+kRmk+1 (xnk)k<2γk 6 o e e y k. Applying Theo em 2.3, we see ha (wk) is a basic sequence (1+²/8)-equi alen o (xnk). Le K1deno e he basis cons an o (wk). We choose a sequence (pn) o nonnega i e in ege s such ha ∆k>2(M2K12k+1 (²+ 16)/(M²))q o e e y kwhe e ∆k=pk+1 −pkand 1/p + 1/q = 1. Le 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 i is easy o see ha kz0 k−zkk=1 ∆k° ° ° ° ° ° pk+1 X i=pk+1 i° ° ° ° ° ° ≤21 q ∆k  pk+1 X i=pk+1 k ikp  1/p <M² K1(²+ 16)2k+1 o e e y k. Theo em 2.3 shows ha (zk) is a basic sequence (1 + ²/8)- equi alen o (z0 k). Consequen ly, (zk) is (1 + ²/8)2-equi alen o a block basic sequence (yn) o (xnk) whose e ms belong o co{xnk}. We will show ha (zk) is equi alen o (σk). To his end le us ix a sequence ( k) such ha he se ies P∞ k=1 kσkcon e ges and pu N= kP∞ k=1 kσkk,y=P∞ k=1 kzk. We ake a ini e sequence q1<· · · < qmo posi i e in ege s. By A1we deno e he se o all 1 ≤j < m such ha he e exis s k≥2 wi h qj≤mpk< qj+1 and le A2be he se o he emaining indices. Gi en j∈A1, we ind k≥1, 0 ≤i1≤∆k−1, l≥2 and 0 ≤i2≤ ∆l−1 such ha mpk+i1< qj≤mpk+i1+1 < mpl+i2< qj+1 ≤mpl+i2+1. Then |y(qj)−y(qj+1)|=¯¯¯¯¯ x(qj)Ãλ k+ l−1 X i=k+1 i+ (1 −µ) l! + (x(qj)−x(qj+1)) õ l+ ∞ X i=l+1 i!¯¯¯¯¯ ≤|x(qj)|max (¯¯¯¯¯ l−υ X i=k+ν i¯¯¯¯¯ :ν, υ = 0,1) +|x(qj)−x(qj+1)|sup n∈N¯¯¯¯¯ ∞ X i=n i¯¯¯¯¯ . whe e λ= 1 −i1/∆k,µ= 1 −i2/∆l. This gi es us he es ima e |y(qj)−y(qj+1)| ≤ M2¯¯¯¯¯¯ lj X i=kj i¯¯¯¯¯¯ +|x(qj)−x(qj+1)|N(4) o some k≤kj≤lj≤l. Obse e ha X j∈A1 ¯¯¯¯¯¯ lj X i=kj i¯¯¯¯¯¯ p ≤2Np.(5) 7 Le us now conside he se A2. We decompose i in o disjoin in e als Ak 2whe e A1 2={j∈A2: 1 ≤qj< qj+1 ≤mp2}and Ak 2={j∈A2:mpk< qj< qj+1 ≤mpk+1 } o k≥2. Assume ha Ak 2is no emp y. I is easy o see ha i j∈Ak 2, hen |y(qj)−y(qj+1)| ≤ λk j|x(qj) k|+|x(qj)−x(qj+1)|N(6) o some nonnega i e λk jsuch ha Pj∈Ak 2λk j≤1. Clea ly, ∞ X k=1 X j∈Ak 2 (λk j| k|)p≤ ∞ X k=1 | k|p≤Np.(7) He e we ega d sums o e he emp y se as ze o. Using (4), (5), (6), and (7), we ob ain   m−1 X j=1 |y(qj)−y(qj+1)|p  1/p ≤M2Ã2Np+ ∞ X k=1 | k|p!1/p +Nkxk ≤M2(31/p + 1)N. (8) We now se k=mpk+1 o k∈N. I i < j, hen |y( i)−y( j)|=¯¯¯¯¯¯ x( i)Ãj−1 X k=i k!+ (x( i)−x( j))   ∞ X k=j k ¯¯¯¯¯¯ ≥ |x( i)|¯¯¯¯¯ j−1 X k=i k¯¯¯¯¯ − |x( i)−x( j)|¯¯¯¯¯¯ ∞ X k=j k¯¯¯¯¯¯ ≥M¯¯¯¯¯ j−1 X k=i k¯¯¯¯¯ − |x( i)−x( j)|N. This and (3) show ha kyk ≥ Ãm−1 X k=1 |y( qk)−y( qk+1 )|p!1/p ≥M  m−1 X k=1 ¯¯¯¯¯¯ qk+1−1 X i=qk i¯¯¯¯¯¯ p  1/p −L² 16 N o e e y sequence q1<· · · < qmo posi i e in ege s. Hence kyk ≥ NµM−L² 16 ¶=NL ³1−² 8´.(9) This comple es he p oo o (b). (c) To p o e (c) we can apply an a gumen simila o ha in he p oo o (b) (now L= lim supk→∞ |x(k)|>0). In his way i is no di icul o 8 cons uc a basic sequence (zk) which is M2/M equi alen o he summing basis (σn) o c0and such ha (zk) is (1 + ²/8)2equi alen o a block basic (yn) o (xnk) whose e ms belong o co{xnk}. In he special case when Cis he uni ball o a non e lexi e space pa (a) o P oposi ion 3.1 was ob ained in [14] (see also [15]). We will gene alize ano he esul om [14]. Le C6=∅be a con ex subse o a Banach space and T:C→Cbe an a ine mapping. We pu θ(T) = in nlim in n→∞ kx−Tnyk:x, y ∈Co. F om he i s pa o he p oo o [14, Theo em 3] we see ha in {kx−Txk: x∈C}= 0. In spi e o his ac , i Cis no weakly compac , i is possible o cons uc a se K⊂Cand a con inuous a ine mapping T:K→K such ha θ(T)>0. In pa icula , T ails o ha e ixed poin s. Le (xn) be a bounded basic sequence. The igh shi T0wi h espec o (xn) is he mapping de ined by he o mula T0Ã∞ X n=1 nxn!= ∞ X n=1 nxn+1. By he bila e al shi T1wi h espec o (xn) we in u n mean he mapping T1Ã∞ X n=1 nxn!= 2x1+ ∞ X k=1 2k−1x2k+1 + ∞ X k=2 2kx2k−2. Clea ly, T0and T1a e a ine sel -mappings o S+({xn}) and T1is on o. Theo em 3.2. (a) Le Cbe a closed con ex bounded se in a Banach space X. I Cis no weakly compac , hen he e a e a closed con ex subse K⊂Cand an a ine con inuous mapping T:K→Ksuch ha T(K) = K and θ(T)>0. (b) Le C⊂Jpbe a closed con ex bounded se . I Cis no weakly com- pac , hen he e a e a closed con ex subse K⊂Cand an a ine uni o mly Lipschi zian mapping T:K→Ksuch ha θ(T)>0. (c) Le C⊂c0be a closed con ex bounded se . I Cis no weakly com- pac , hen he e a e a closed con ex subse K⊂Cand an a ine uni o mly Lipschi zian mapping T:K→Ksuch ha θ(T)>0. P oo . (a) Le (en) be he s anda d basis o `1and T1:S+({en})→ S+({en}) be he bila e al shi . Fac 2.1 gi es us a sequence (yn) in Cand a unc ional g∈X?. Le Sbe he basis cons an o (yn) and K=S+({yn}). The p oo o P oposi ion 3.1 (a) shows ha he a ine mapping φ:K→ S+({en}) such ha φ(yn) = enis a homeomo phism. De ine T:K→K by he o mula T=φ−1T1φ. Le x=P∞ n=1 nynand y=P∞ n=1 bnyn 9 [15] V. D. Milman, Geome ic heo y o Banach spaces. Pa I. The heo y o basis and minimal sys ems, Russ. Ma h. Su eys 25 (1970), 111–170 ( ansla ed om Usp. Ma . Nauk 25 (1970), 113–173). [16] H. P i zne , L-embedded Banach spaces and measu e opology, Is ael J. Ma h, o appea . 16