Banach Lattices and the Weak Fixed Point Property

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1 Baach Lattices ad the Weak Fixed Poit Property Tim Dalby ad Brailey Sims School of Mathematics, Statistics ad Computer Sciece, The Uiversity of New Eglad, NSW 2351, Australia, School of Mathematical ad Physical Scieces, The Uiversity of Newcastle, NSW 2308, Australia, ABSTRACT Baach space properties that imply the weak fixed poit property are ivestigated i a Baach lattice settig Mathematics Subject Classificatio: 46B20, 47H10 1. INTRODUCTION A Baach space is said to have the weak fixed poit property (w-fpp) if every oexpasive mappig o every oempty weak compact covex set has a fixed poit. The weak fixed poit property ad Baach lattices has ot bee the subject of may papers i the last twety or so years; see Sie [23], Soardi [24], Maurey [15], Elto et al. [7], Borwei ad Sims [3], Li [12], Sims [19] ad [20], ad Khamsi ad Turpi [11]. This is despite the fact that may examples have a order theoretic ature, see for example Borwei ad Sims [3]. 1

2 Hopig to geerate reewed iterest i the w-fpp ad Baach lattices, this paper revisits the property of weak orthogoality from Borwei ad Sims [3] ad Sims [19]. We the cosider Baach lattices with uiformly mootoe orm, a property that was exploited i Elto et al. [7]. Alog the way, other properties kow to be associated with the w-fpp i Baach spaces are studied i the cotext of Baach lattices. 2. PRELIMINARIES The usual approach to provig that a particular Baach space has the w-fpp is to assume that it does ot have this property ad obtai a cotradictio. Thus there is a oempty weak compact covex set C with a fixed poit free oexpasive mappig T where T : C C. Usig the weak compactess of C ad the oexpasiveess of T it ca be show that there exists, i C, a weak ull sequece with certai properties ivolvig the orm. So most approaches to the w-fpp problem have ivolved weak ull sequeces ad their relatioship to the orm. I Baach lattices, the lattice structure ca be added to this mix. The followig defiitios reflect this situatio. Opial s coditio, from Opial [17], states if x 0 ad x 0 the lim sup x < lim sup x x. Nostrict Opial coditio has the strict iequality replaced by. Uiform Opial s coditio i Prus [18] is a stregtheig of Opial s coditio: for every ɛ > 0 there is a r > 0 such that 1 + r lim if x + x for each x X with x 1 ad each sequece (x ), x lim if x 1. 0, with 2

3 There is a Opial s modulus, itroduced i Li et al. [13], defied as r(c) := if{lim if for c > 0. x + x 1 : x c, x 0 ad lim if x 1} X has the uiform Opial s coditio if ad oly if r(c) > 0 for all c > 0 ad the ostrict Opial coditio if ad oly if r(c) 0 for all c > 0, see Dalby [5]. A slightly differet but related approach produces the followig property, due to Sims [22]. A Baach space has property(k) if there exists K [0, 1) such that wheever x 0, x 1 ad lim if x x 1 we have x K. If K is ot the same across X but depeds o the sequece (x ), the the coditio is called property (k). Property(K) with K = 0 is equivalet to Opial s coditio ad Dalby [5] showed that a Baach space has property(k) if ad oly if r(1) > 0. ormal structure. Sims [22] proved that property(k) implies weak Next some defiitios for Baach lattices. A Baach lattice is said to be weakly orthogoal if wheever x 0 the lim x x = 0 for all x X. Sims [20] showed that weakly orthogoal Baach lattices have the w-fpp ad a Baach space X has the w-fpp if there exists a weakly orthogoal Baach lattice Y with d(x, Y ) < 5 1 where d(x, Y ) is the Baach-Mazur distace betwee X ad Y. I [4], Dalby exteded the distace to Note that Borwei ad Sims [3] used a slightly weaker defiitio of weak orthogoality, amely whe x 0 the lim m lim x m x = 0. 3

4 It has become the practice to use the stroger defiitio whe referrig to weak orthogoality, see for example Sims [19] ad Garcia-Falset [8]. The orm of a Baach lattice is said to be uiformly mootoe if give ɛ > 0 there is a δ > 0 such that if x, y 0 with y = 1 ad x + y 1 + δ the x ɛ. A equivalet defiitio is: There exists a strictly icreasig cotiuous fuctio δ o [0,1] with δ(0) = 0 so that if x, y 0 with 1 = y x the x + y 1 + δ( x ). Birkhoff [2] was resposible for the first versio ad the secod versio appeared i Katzelso ad Tzafriri [10]. Akcoglu ad Suchesto [1] cosidered both these ad several other formulatios. They showed that the two defiitios are equivalet ad i Orlicz fuctio spaces they are equivalet to the 2 coditio. Note that a Baach lattice, X, that has a uiformly mootoe orm is weak sequetially complete so it caot cotai a subspace isomorphic to c 0. I particular, X has order cotiuous orm. Recall that the orm is said to be order cotiuous if if{ x : x A} = 0 for every dowward directed set A X such that if(a) = 0. The orm of a Baach lattice is said to be strictly mootoe if x > y 0 implies x > y. X havig a uiformly mootoe orm is equivalet to X := l (X)/c 0 (X) havig a strictly mootoe orm. See for example Elto et al. [7]. Fially, a Baach lattice has a p-superadditive orm if ( x p + y p ) 1/p x + y for all disjoit x, y. A p-superadditive orm is a uiformly mootoe orm. 4

5 The w-fpp is separably determied, see for example Goebel ad Kirk [9]. So throughout this paper X will be assumed to be a ifiite dimesioal separable Baach lattice. So if X is σ-dedekid complete the the orm is order cotiuous, see Lidestrauss ad Tzafriri [14] or Meyer-Nieberg [16] for details. 3. RESULTS First a result cocerig the ostrict Opial coditio. Propositio 1. If X is a Baach lattice with order cotiuous orm the X satisfies the ostrict Opial coditio for positive weak ull sequeces. Proof: Let x 0 where x 0 for all, the by propositio of Meyer- Nieberg [16], there exists a disjoit sequece, (x ), of positive elemets i B X such that lim sup x (x ) = lim sup x. Order cotiuity of the orm implies that x lim sup 0. So for ay x X x + x lim sup x (x + x) x (x ) = lim sup = lim sup x. The situatio ivolvig Opial s coditio ad uiform Opial s coditio is left for the momet util weak orthogoality has bee dealt with. Propositio 2. If X is a Baach lattice with order cotiuous orm the the lattice operatios are weak sequetially cotiuous if ad oly if X is weakly orthogoal, ad hece has the w-fpp. 5

6 Proof: ( ) Let x 0 the x 0. For x X let y := x x. The y 0 ad 0 y x. If lim y 0 the by takig subsequeces we have if y α for some α > 0. Let z := y /α the z 0, 0 z x /α ad z 1 for all. By corollary of Meyer-Nieberg [16], for 0 < β < 1 there exists a subsequece (z k ) ad disjoit (w k ) such that 0 w k z k ad w k β > 0 for all k. So 0 w k x /α for all k. The the order cotiuous orm ad theorem of Meyer-Nieberg [16] meas w k 0, a cotradictio. ( ) This follows the ideas cotaied i the proof of propositio i Meyer-Nieberg [16]. That is, let x 0 ad by theorem of [16], it suffices to show that x k 0 for every subsequece such that ( x k ) is weak Cauchy. From lemma of [16], there exists a icreasig, positive sequece (y k ) such that x k y k 0. Weak orthogoality implies that lim k x k x = 0 for all x X ad Therefore lim k x k y k x = 0 for all x X. x k x + x k y k x 0. 6

7 But x k x + x k y k x ( x k + x k y k ) x y k x = y k x 0. Therefore lim k y k x = 0. But sice (y k ) is icreasig so is (y k x ) which meas that y k = 0 for all k ad so x k 0. So Baach lattices with lattice operatios weak sequetially cotiuous ad order cotiuous orm have the w-fpp, as do Baach spaces whose Baach- Mazur distace from such lattices is less tha The lattice operatios of ay abstract M space are weak sequetially cotiuous. See for example Meyer-Nieberg [16], propositio But lemma 1.b.10 of Lidestrauss ad Tzafriri [14] states that a abstract M space has order cotiuous orm if ad oly if it is order isometric to c 0 (Γ), for some idex set Γ. So propositio 2 icludes c 0 but excludes ay M space with a order uit, for example C(K) where K is a ifiite compact Hausdorff space. Also see Borwei ad Sims [3] for further cosequeces of propositio 2. If the Baach lattice is atomic the by propositio of Meyer-Nieberg [16] the lattice operatios are weak sequetially cotiuous ad so we have the followig corollary. Corollary 3. Let X be a atomic Baach lattice with order cotiuous orm the X is weakly orthogoal, ad hece has the w-fpp. 7

8 It is well kow that if X is a Baach lattice ad c 0 X (by Lidestrauss ad Tzafriri [14] this is equivalet to X ot cotaiig a sublattice order isomorphic to c 0 ) the X has order cotiuous orm which leads to the followig. Corollary 4.. Let X be a atomic Baach lattice where c 0 X the X is weakly orthogoal, ad hece has the w-fpp. Khamsi ad Turpi [11] cosidered Baach spaces with a vector lattice structure satisfyig: (α) (x + y + ad x y ) x y, x, y X; (β) for some real costat k < 2, x y x k y, x, y X. Istead of the weak topology, the topology, τ, studied was the coarsest topology o X for which the map x x u is cotiuous at 0 for every u X, u 0. Khamsi ad Turpi showed that every oexpasive map o every oempty τ-compact covex subset has a fixed poit. For weakly orthogoal Baach lattices this is the w-fpp result of Sims [19]. Garcia-Falset [8] exteded this set up to have k 2 but required the additioal coditio of the alterate-sigs Baach-Saks property. I this paper Garcia-Falset called a Baach space, X, weakly orthogoal if X satisfies (α) ad (β) ad if for each weakly ull sequece (x ) i X, lim x x = 0 for every x X. To obtai the w-fpp the additioal coditio was the weak Baach-Saks property. Related to the foregoig is the followig. Questio: If X is a weakly orthogoal Baach lattice with orm, does X with the ew orm x 1 := x + x satisfy the w-fpp? It is straight forward to show that 1 is a equivalet Baach space orm that satisfies (α) ad (β). So (X, 1 ) satisfies the w-fpp if X has the weak Baach-Saks 8

9 property. To obtai Opial s coditio, the coditio that X must have a order cotiuous orm has to be stregtheed to X havig a uiformly mootoe orm. Note that a Baach lattice that is weakly orthogoal has the Baach space property, WORTH: if x 0 the lim sup x x = lim sup x + x for all x X. This i tur implies the ostrict Opial coditio. Propositio 5. If X is a Baach lattice with uiformly mootoe orm ad whose lattice operatios are weak sequetially cotiuous the X satisfies Opial s coditio. Proof: Recall that a uiformly mootoe orm implies that c 0 X ad so by propositio 2, X is weakly orthogoal. Thus X has WORTH. Assume that X does ot satisfy Opial s coditio the there exists x 0 ad a ozero x X such that lim sup x lim sup x + x. Sice X satisfies the ostrict Opial coditio we have lim sup x = lim sup x + x. Without loss of geerality we may assume lim x = lim x + x = 1 ad if x > 0. So x / x 0 ad x / x 0. Nostrict Opial coditio implies 9

10 1 lim sup x / x + x x / x + x lim sup = lim sup lim sup x / x x x / x + x. usig WORTH Therefore Also 1 lim sup x / x + x = lim sup x / x + x. lim sup x / x + x lim sup x / x x + lim x + x = lim 1/ x 1 x + lim x + x = 1. Thus usig the weak lower semi-cotiuity of the orm 1 = lim sup x / x + x x. This meas that x x / x for all. The uiformly mootoe orm meas there exists a strictly icreasig cotiuous fuctio δ o [0, 1] where Lettig we have x / x + x 1 + δ( x ) for all. 1 = lim sup x / x + x 1 + δ( x ) > 1. 10

11 A cotradictio. It ca be show that a Baach lattice satisfyig the coditios of propositio 5 has property(k) ad so has weak ormal structure. Uiform Opial s coditio ca be foud by usig the spaces l p, 1 < p <, as guides. Propositio 6. If X is a Baach lattice with p-superadditive orm, 1 < p <, ad whose lattices operatios are weak sequetially cotiuous the X satisfies the uiform Opial s coditio with r(c) (1 + c p ) 1/p 1. Proof: Recall that a orm is p-superadditive if x p + y p x + y p for all disjoit x, y. It ca be show that this coditio is equivalet to the same iequality where x ad y are merely 0. See for example propositio of Meyer-Nieberg [16]. Let x 0, lim if x 1 ad x c > 0. The So x p + x p x + x p for all. ( ) 1/p lim if x p + x p lim if x + x. Usig weak orthogoality ad a similar argumet to that i propositio 5 we have ad thus lim if lim if x + x = lim if x + x x + x (1 + c p ) 1/p = 1 + [(1 + c p ) 1/p 1]. 11

12 Which meas that X satisfies the uiform Opial s coditio with r(c) (1 + c p ) 1/p 1. This propositio covers the cases of l p, 1 < p <. Note that if the orm is p-additive ad X is atomic ad separable the X is isometrically isomorphic to l p. It is a log stadig cojecture i metric fixed poit theory that reflexivity ad the fixed poit property are liked. This meas that the presece or absece of c 0 ad l 1 is of iterest, which leads to the followig propositio. Propositio 7. Let X be a Baach space. If c 0 X the X does ot have property(k). Proof: c 0 X if ad oly if there exists a sequece (ɛ ) i (0, 1) where ɛ 0 ad a sequece (x ) i X such that (1 ɛ ) sup k t k k= t k x k (1+ɛ ) sup t k for all (t k ) c 0, for all N. k Without loss of geerality ɛ 0. Note that x 0 ad lim x = 1. Fix N the 1 ɛ x x k 1 + ɛ for all k >. So 1 ɛ lim k x x k 1 + ɛ. Therefore r( x ) lim k x x k 1 for all ad r( x ) ɛ for all. Takig we have r(1) 0 which implies r(1) = 0. property(k) if ad oly if r(1) > 0. But X has 12

13 Aother way of viewig this is: if X has a equivalet orm which satisfies property(k) the X does ot cotai a isomorphic copy of c 0. Note: Thus if X is a Baach lattice where l 1 X ad X has property(k) the X is reflexive. Dalby [6] showed that if X statisfies the coditio that R(X ) < 2 ad has the ostrict *Opial property the X satisfied property(k). So if X is a Baach lattice with X order cotiuous, R(X ) < 2 ad havig the ostrict *Opial property the X is reflexive. 13

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