Reflexivity and the Fixed-Point Property for Nonexpansive Maps

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

200, 653]662 Ž1996.

0229

Reflexivity and the Fixed-Point Property for Nonexpansive Maps P. N. Dowling,U C. J. Lennard,† and B. Turett ‡, 1 *Miami Uni¨ ersity, Oxford, Ohio 45056; † Uni¨ ersity of Pittsburgh, Pittsburgh, Pennsyl¨ ania 15260; ‡ Oakland Uni¨ ersity, Rochester, Michigan 48309 and Uni¨ ersity College, Galway, Ireland Submitted by Richard M. Aron Received April 19, 1995

Connections between reflexivity and the fixed-point property for nonexpansive self-mappings of nonempty, closed, bounded, convex subsets of a Banach space are investigated. In particular, it is shown that l 1 Ž G . for uncountable sets G and l` cannot even be renormed to have the fixed-point property. As a consequence, if an Orlicz space on a finite measure space that is not purely atomic is endowed with the Orlicz norm, the Orlicz space has the fixed-point property exactly when it is reflexive. Q 1996 Academic Press, Inc.

The problem of classifying the family of Banach spaces for which every nonexpansive self-mapping of a nonempty, closed, bounded, convex subset of the Banach space has a fixed point has been intensively studied for almost 40 years. Such Banach spaces are said to have the fixed-point property for nonexpansi¨ e mappings, or the fixed-point property for short. Despite many impressive results stating that certain classes of Banach spaces have the fixed-point property, the problem is, in some ways, as open today as it was 30 years ago. It is easy to give examples of nonexpansive mappings, in fact, isometries on subsets of c 0 and l 1 without fixed points. ŽSee, for example, the recent books w2, 9x.. Since every classical nonreflexive Banach space contains an isometric copy of one of these two spaces, it follows that all of the classical nonreflexive spaces fail to have the fixed-point property. It has also been known for some time that uniformly rotund Banach spaces have the fixed point property w3, 10, 12x; thus, all of the classical reflexive spaces have the 1

Supported in part by an Oakland University Research Fellowship. 653 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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DOWLING, LENNARD, AND TURETT

fixed-point property. It is, therefore, natural to question if a relationship exists between Banach spaces with the fixed-point property and reflexivity, and it is questions of this nature that we address in this paper. To be exact, does the fixed-point property imply reflexivity and do reflexive Banach spaces have the fixed-point property? With regard to the second question, it should be noted that it is not even known if superreflexive spaces have the fixed-point property. In considering the first question, if the fixed-point property implies reflexivity, it is tempting to try to construct nonreflexive spaces with the fixed-point property. Where, however, should one look for such spaces? Perhaps the most natural place to look is among the renormings of c 0 or l 1 since these are, in a sense, the simplest of the nonreflexive Banach spaces. There is another advantage of considering these two classical spaces. If it can be shown that neither of these spaces can be renormed to have the fixed-point property, it would follow that the fixed point property in a Banach lattice or in a Banach space with an unconditional basis would imply reflexivity w4x. Thus, success in considering the isomorphism classes of these two spaces would have extremely pleasant consequences for several large classes of Banach spaces. For the time being, we restrict our attention to renormings of l 1. It is known w11x that any renorming of l 1 contains almost isometric copies of l 1. It is reasonable therefore to try to perturb the classical example of a nonexpansive map without a fixed point in l 1 to get a similar example in any renorming of l 1. Unfortunately, we have not had success in doing this in general. However, if the copy of l 1 is ‘‘good enough,’’ then such a perturbation can be made to work. DEFINITION 1. A Banach space X contains an asymptotically isometric copy of l 1 if, for every sequence Ž « n . decreasing to 0, there exists a sequence Ž x n . of norm-one elements in X such that Ý nŽ1 y « n .< a n < F 5Ý n a n x n 5 F Ý n < a n < for all sequences Ž a n . of real numbers. It has been shown in w6x that, if a Banach space contains an asymptotically isometric copy of l 1, then the Banach space fails the fixed-point property. Since there exist renormings of l 1 such that the renorming contains no asymptotically isometric copy of l 1 w7x, the above fact cannot be used to show that all renormings of l 1 fail the fixed-point property. It does, however, suggest another question: Which Banach spaces contain asymptotically isometric copies of l 1 ? These spaces would fail to have the fixed-point property. In fact, in considering the answer to this question, we are led to Banach spaces which cannot even be renormed to have the fixed-point property.

REFLEXIVITY AND THE FIXED-POINT PROPERTY

655

THEOREM 2. If G is uncountable, then any renorming of l 1 Ž G . contains an asymptotically isometric copy of l 1. Consequently, if G is uncountable, l 1 Ž G . cannot be renormed to ha¨ e the fixed-point property. Note that the result in w7x mentioned prior to the statement of the theorem shows that the uncountability of the index set is crucial. Proof. Let eg be the element in l 1 Ž G . with eg Žg . s 1 and eg Ž a . s 0 if a / g . Let A ? A be a norm equivalent to the l 1 norm on l 1 Ž G .. Specifically, there exists m, M ) 0 such that mÝg g F < cg < F AÝg g F cg eg A F M Ýg g F < cg < for all finite sets F in G. Note that if there exists a sequence Ž eg j . such that Nj ' A eg jA decreases to m, then `

Ý js1

m Nj

< cg < F j

`

Ý cg js1

eg j j

Nj

F

`

Ý < cg < . j

js1

Since mrNj increases to 1, a subsequence of the sequence Ž eg j . will provide an asymptotically isometric copy of l 1 in the A ? A-renorming of l 1 Ž G .. Thus, to complete the proof, we need only show the existence of a sequence with the described property. Actually the sequence constructed will be a certain block basic sequence of Ž eg .. Let A be an uncountable subset of G and define m A ' infAÝ b g B c b e b A : Ý b g B < c b < s 1, B a finite subset of A4 . Note that m F m A F M for all uncountable subsets A of G and that m A increases as A decreases. Let Ž Aa .a - v 1 be a decreasing chain of uncountable subsets of G, where v 1 is the first uncountable ordinal. Then Ž m A a .a - v 1 is a nondecreasing transfinite sequence of real numbers and, hence, eventually constant. Thus there exists a 0 such that if a G a 0 , then m A a s m A a ' m 0 . 0 Consider Aa 0 . There exist a natural number n1 and real numbers c j0 and 1 < c j0 < s 1 and m 0 F elements g j0 in G for j s 1, . . . , n1 such that Ý njs1 n 0 1 AÝ js1 c j eg 0 A F m 0 q 1. Since F a Aa s B, there exists a 1 G a 0 such j that g j0 f Aa 1 for j s 1, . . . , n1. Since m A a s m 0 , there exist a natural 1 number n 2 and real numbers c 1j and elements g j1 in G for j s 1, . . . , n 2 2 2 c 1 e 1 A F m q 1 . Continue in < c 1j < s 1 and m 0 F AÝ njs1 such that Ý njs1 j gj 0 2 this matter to obtain a block basic sequence Ž x k . of Ž eg ., where x k s k c k e k . Then Ý njs1 j gj m0 Ý < ck < F k

Ý ck x k k

F M Ý < ck < k

and A x k Ax m 0

as k ­`.

656

DOWLING, LENNARD, AND TURETT

Thus, by the reasoning supplied earlier, the sequence Ž x krA x k A. generates an asymptotically isometric copy of l 1 in l 1 Ž G ., and this completes the proof. Since Pełczynski ´ w15x has shown that, for separable Banach spaces X, X contains an isomorphic copy of l 1 if and only if X U contains isomorphic copies of l 1 Ž G . for some uncountable G, the following corollary is immediate. COROLLARY 3. If X is a separable Banach space containing an isomorphic copy of l 1, then X U cannot be renormed to ha¨ e the fixed-point property. In particular, l` cannot be renormed to ha¨ e the fixed-point property. Note that, with the above hypotheses, it may be possible to renorm X U to have the weak fixed-point property, i.e., that nonexpansive self-mappings of weakly compact convex subsets have fixed points. For example, since l` can be renormed to have normal structure w19x, l` can be renormed to have the weak fixed-point property. In certain circumstances, the separability of X in Corollary 3 is unnecessary. COROLLARY 4. If a Banach space X contains a complemented copy of l 1, then X U cannot be renormed to ha¨ e the fixed-point property. Corollary 4 follows immediately from the second statement in Theorem 3 since X U will contain a copy of l` whenever X contains a complemented copy of l 1, regardless of the separability of X w5x. Of course, there do exist spaces Že.g., C w0, 1x. which contain copies of l 1 , but which contain no complemented copies of l 1. The usefulness of the above results in relating the fixed-point property with reflexivity will now be illustrated in the context of Orlicz spaces. The basic definitions and facts about Orlicz spaces may be found in w13 or 16x. We recall a few specific results that we shall need. First, if Ž V, S, m . is a finite nonatomic measure space, the Orlicz space LF Ž m . is reflexive if and only if the Young function F and its conjugate function C both satisfy a D 2-condition for large values. ŽThis is a result due to Luxemburg w14, 16x.. Recall also that an Orlicz space LF Ž m . can be endowed with either of two equivalent norms: the Luxemburg norm 5 f 5 F s inf r ) 0 : HF Ž frr . d m F 14 or the Orlicz norm NF Ž?. s sup Hfg d m : H C Ž g . d m F 14 . ŽNote, our notation is the opposite of the notation in w16x.. We shall also use that, if the measure space is finite and not purely atomic, the Young function F satisfies a D 2-condition for large values of its argument if and only if the Orlicz space LF Ž m ., endowed with the Luxemburg norm contains no isometric copy of l` w17x. Since the exact nature of the isometry will prove useful in the following, let us briefly describe how l` embeds in Ž LF Ž m .,

REFLEXIVITY AND THE FIXED-POINT PROPERTY

657

5 ? 5 F . when F fails to satisfy a D 2-condition for large values. In this setting, one can construct a sequence of norm-one functions Ž g n . such that the g n’s have disjoint support, HF Ž g n . d m decreases to 0, and Ý`ns1 g n is also norm-one. The map which sends a bounded sequence Ž a n . to Ý a n g n is then an isometry of l` into LF Ž m .. Finally, it is useful to note that 5 f 5 F F NF Ž f . for every f g LF Ž m . and that if f / 0, the inequality is strict w13, p. 78; 16, p. 73x. A connection between reflexivity and the fixed-point property in Orlicz spaces can now be stated. THEOREM 5. Let Ž V, S, m . be a finite measure space that is not purely atomic. The Orlicz space LF Ž m ., endowed with the Orlicz norm, has the fixed-point property if and only if it is reflexi¨ e. Proof of Necessity. Suppose that Ž LF Ž m ., NF Ž?.. has the fixed-point property but that LF Ž m . is not reflexive. Then, by Luxemburg’s result mentioned above, either the Young function F or its conjugate function C fails to satisfy a D 2-condition for large values. Suppose F fails to satisfy the D 2-condition for large values. Then l` embeds isometrically into LF Ž m . with the Luxemburg norm and, since the Luxemburg and the Orlicz norms are equivalent, l` embeds isomorphically into LF Ž m . with the Orlicz norm. But this would give a renorming of l` with the fixed-point property, a contradiction to Corollary 3. The only other possibility is that C fails to satisfy the D 2-condition for large values while F does. Let Ž g n . denote the functions in Ž LC , 5 ? 5 C . s Ž LF Ž m ., NF Ž?..U giving rise to the isometry of l` in Ž LC , 5 ? 5 C .. Choose functions Ž f n . in LF Ž m . such that NF Ž f n . s 1, the support of f n is contained in the support of g n , and Hg n f n d m ) 1 y e n , where Ž e n . is a sequence decreasing to 0. Then, since 5Ý nŽsgn a n . g n 5 C s 1, NF

ž Ý a f / G H ž Ý Ž sgn a . g / ž Ý a f / dm n n

n

n

n

n

s

n n

n

Ý < a n
G

Ý Ž 1 y en . < a n < . n

Thus, since NF Ž f n . s 1,

Ý Ž 1 y e n . < a n < F NF Ý a n f n n

ž

n

/ F Ý < a <. n

n

658

DOWLING, LENNARD, AND TURETT

Thus, if C fails the D 2-condition for large values, Ž LF Ž m ., NF Ž?.. contains an asymptotically isometric copy of l 1 and, thus, fails to have the fixed-point property. This contradicts the hypothesis and the proof that the fixed-point property implies the reflexivity is complete. Before proving the converse, it is useful to note an alternative formula for computing the Orlicz norm and a sufficient condition for a Banach lattice to have the weak fixed-point property. It is known that NF Ž f . s inf k ) 0 Ž1rk .Ž1 q HV F Ž kf . d m . for each f g LF Ž m . w13, 16x. This formula, sometimes called Amemiya’s formula, allows the computation of the Orlicz norm without reference to the conjugate Young function. Note that the infimum is actually attained w16x. In the setting of Banach lattices, a sufficient condition that ensures the weak fixed-point property is known w2x. A Banach lattice X is uniformly monotone if, for every e ) 0, there exists d ) 0 such that 5 x 5 G 1 q d whenever x G y G 0, 5 x y y 5 G e , and 5 y 5 s 1. It is known that, if X is the Banach lattice with a uniformly monotone norm and l 1 is not finitely representable in X, then X has the weak fixed-point property. We now return to the proof of Theorem 5. Proof of Sufficiency. Suppose LF Ž m . is reflexive. Let e ) 0 and f, g g L m . such that g G f G 0, NF Ž g y f . G e , and NF Ž f . s 1. In order to show that LF Ž m . is uniformly monotone, we need to show that NF Ž g . ) 1 q d for some d depending only on e . Since F satisfies a D 2-condition for large values, convergence in norm is equivalent to convergence in mean w13, p. 76; 16, p. 83x. Thus there exists h Ž e . ) 0, depending on e , but not depending on f or g, such that HV F Ž g y f . d m ) h Ž e .. By Amemiya’s formula for the Orlicz norm of g, choose k ) 0 such that NF Ž g . s Ž1rk .Ž1 q HV F Ž k g . d m .. Assume first that k G 1. In this case, FŽ

NF Ž g . s G

1

k 1

k q

ž ž

1q

HVF Ž k g . d m

1q

HVF Ž k f . d m

1

k

/ /

HVF Ž k Ž g y f . . d m ,

G NF Ž f . q

HVF Ž g y f . d m ,

G 1 q hŽ e . .

since f , g y f G 0, since k G 1,

REFLEXIVITY AND THE FIXED-POINT PROPERTY

659

In the case that k F 1, consider two possibilities: either NF Ž g . G 3r2 Žin which case there is nothing to prove. or 1 F NF Ž g . F 3r2. In the latter case, it is clear that k G 2r3. Again, the equivalence of convergence in mean and norm convergence in the presence of a D 2-condition for large values yields that there exists j ) 0 such that HV F Ž2 gr3. d m ) j for all g with norm at least 1. Then, NF Ž g . s

1

k

ž

1q

HVF Ž k g . d m

G1q

HVF Ž k g . d m

G1q

HVF Ž 2 gr3. d m

/

G1qj. Setting d Ž e . s minh Ž e ., j , 12 4 yields that a reflexive Orlicz space with the Orlicz norm is uniformly monotone. ŽIn fact, the proof shows a bit more: Only the fact that F satisfies a D 2-condition for large values is used to get the conclusion.. Since a reflexive Orlicz space is also superreflexive w1; 4, p. 174; 16, p. 297x, l 1 is not finitely representable in LF Ž m .. Thus, by the previously mentioned theorem, the reflexive Orlicz space LF Ž m ., endowed with the Orlicz norm, has the Žweak. fixed-point property. This completes the proof of Theorem 5. The sufficiency in Theorem 5 may also be derived from a result of Wang and Shi w18x which states that an Orlicz space with the Orlicz norm has uniform normal structure exactly when it is reflexive. Whether a theorem analogous to Theorem 5 is true if the Orlicz norm is replaced by the Luxemburg norm is still open. In the proof of the necessity, it was noted, in the case that the conjugate function C fails to satisfy a D 2-condition for large values while F satisfies one, that the Orlicz space endowed with the Orlicz norm contains an asymptotically isometric copy of l 1 and, thus, fails to have the fixed point property. In the same setting, the analogous proof provides copies of l 1 in Ž LF Ž m ., 5 ? 5 F ., but it does not show that these copies are asymptotically isometric. So, although the analogous theorem may be true, a different proof will be needed in the second case in the necessity. The other direction of the theorem remains valid even if the Orlicz norm is replaced by the Luxemburg norm. The proof of this fact is modelled on the proof of Proposition 1 in w8x. THEOREM 6. Let Ž V, S, m . be a finite measure space that is not purely atomic. If F satisfies a D 2-condition for large ¨ alues then the Orlicz space

660

DOWLING, LENNARD, AND TURETT

LF Ž m ., endowed with the Luxemburg norm, is uniformly monotone. Consequently, if Ž V, S, m . is a finite measure space that is not purely atomic, a reflexi¨ e Orlicz space LF Ž m . with the Luxemburg norm has the fixed-point property. Proof. Suppose F satisfies a D 2-condition for large values and let K ) 0. As noted in w8x, a classical argument shows that there exists M ) 0 such that F Ž l t . F l M F Ž t . if t G K and l G 1. Let 0 - e - 1 and set a s Ž er2. M . Choose K ) 0 such that F Ž K . s arŽŽ a q 2. m Ž V .. ' h. Now, using the first stated fact, choose M ) 0 such that F Ž l t . F l M F Ž t . if t G K er2 and l G 1. With l s 2re and u s l t, it follows that

a M F Ž u . F F Ž e ur2.

if u G K .

Ž ).

It is then straightforward to see that there exist 0 - d - 1 such that, for u G K, F Ž urŽ1 q d .. G Ž1 y ar2. F Ž u.. Now let g, f g LF Ž m ., g G f G 0, 5 f 5 F s 1, and 5 g 5 F G e . Then, since F satisfies a D 2-condition for large values, HV F Ž f . d m s 1 and HV F ŽŽ g y f .re . d m G 1. Since g y f G 0, g

HVF ž 1 q d / d m G HVF

ž

f

/

1qd

dm q

HVF

ž

gyf 1qd

/

dm .

Setting S s  w g V : g y f G e K 4 and computing each of the integrals on the right-hand side separately yields

HVF

ž

gyf 1qd

/

dm G

HVF

G

HSF

G

e

ž ž

gyf 2

/

e gyf e

2

M

dm

/

dm

gyf

ž /H ž e / m Ž. e ž / ž H ž e / m/ e ž / Ž hm Ž . . 2

S

F

M

G

1y

2

VRS

F

M

G

2

1y

s a Ž 1 y hm Ž V . .

by )

d

V

gyf

d

REFLEXIVITY AND THE FIXED-POINT PROPERTY

661

and, with T s  v g V : f Ž v . - K 4 ,

HVF

ž

f 1qd

/

dm s

HTF

G

HTF

ž ž

f 1qd f 1qd

G0q 1y

ž

G 1y

ž

a 2

/Ž

a 2

/ /

dm q

HVRTF

dm q 1 y

/ž

ž

1y

ž

a 2

f 1qd

/H

VRT

HTF Ž f . d m

/

dm

F Ž f . dm

/

1 y hm Ž V . . .

So, combining the above computations yields that

a

g

HVF ž 1 q d / d m G ž 1 y 2 / Ž 1 y hm Ž V . . q a Ž 1 y hm Ž V . . s 1q

ž

a 2

/Ž

1 y hm Ž V . .

s 1. Thus, 5 g 5 F G 1 q d which completes the proof of Theorem 6. Finally, let us note that similar arguments hold if a Banach space contains ‘‘good’’ copies of c 0 . A Banach space is said to have asymptotically isometric copies of c 0 if, for every sequence Ž e n . decreasing to 0, there exists a sequence Ž x n . in the unit sphere of X such that max ng F Ž1 y e n .< a n < F 5Ý ng F a n x n 5 F max ng F Ž1 q e n .< a n < for all finite subsets F of natural numbers. PROPOSITION 7. If a Banach space X contains an asymptotically isometric copy of c 0 , then X fails to ha¨ e the fixed-point property. Proof. Assume that X contains an asymptotically isometric copy of c 0 and let Ž x n . be the unit vector basis of c 0 as above. Define C s  Ý`ns 1 a n x n : Ž a n . g c 0 , 0 F a n F 14 . Let Ž e n . be a sequence which decreases to 0 such that e n - 14 for each n and Ý`ns1 e n - `. Then Ł nŽ1 y 2 e n . / 0. Define T ŽÝ`ns1 a n x n . s 14 Ž1 y a 1 . x 1 q Ý`ns1Ž1 y 2 e n . a n x nq1. It is straightforward to check that T is a nonexpansive self-mapping of C without a fixed point. Although a result analogous to Theorem 2 can be shown concerning lattice renormings of c 0 Ž G . for uncountable G, this is not pursued here since the authors know of no significant consequences of this fact.

662

DOWLING, LENNARD, AND TURETT

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