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
Pn̰
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
T0̰
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
T1̰
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