Weak compactness and Orlicz spaces

arXiv:0904.2970v1 [math.FA] 20 Apr 2009 Weak ompatness and Orliz spaes Pasal Lefèvre, Daniel Li, Hervé Queéle, Luis Ro dríguez-Piazza June 19, 2013 Abstrat. We give new proofs that some Banah spaes have Peªzy«ski's property (V) . Mathematis Sub jet Classiation. Primary: 46B20; Seondary: 46E30 Key-words. M -ideal; Morse-Transue spae; Orliz spae; Peªzy«ski's property (V) . 1 Intro dution. Reall that a Banah spae X is said to have Peªzy«ski's prop erty (V) if one has a go o d weak-ompatness riterion in the dual spae X∗ of X , namely: every subset A of X∗ is relatively weakly ompat whenever it has the following prop erty (easily seen neessary): lim n→+∞sup x∗∈A|x∗(xn)|= 0 for every weakly unonditionaly Cauhy sequene (xn)n in X ( i.e. suh that Pn≥1|x∗(xn)|<∞ for any x∗∈X∗ ). Equivalently, X has Peªzy«ski's property (V) if and only if for every Banah spae Z and every non-weakly ompat op erator T:X→Z , there exists a subspae X0 , isomorphi to c0 , suh that T is an isomorphism between X0 and T(X0) . Beside the reexive spaes (and in partiular the Lp spaes for 1< p < ∞ ), the spaes C(S) of ontinuous funtions on ompat spaes S have prop erty (V) ; in partiular L∞ has (V) . Another general lass of Banah spaes having prop erty (V) is that of Banah spaes whih are M -ideal in their bidual, i.e. those for whih the anonial deomp osition of their third dual is an ℓ1 deomp osition: X∗∗∗ =X∗⊕1X⊥ (see [8, 9℄). Note that every subspae of a Banah spae M -ideal of its bidual is itself M -ideal of its bidual; hene every suh subspae has prop erty (V) . On the ontrary, a non-reexive Banah spae that do es not ontain c0 annot have prop erty (V) . In partiular, L1 do es not have this prop erty. Thus, the Lp spaes have (V) for 1< p ≤ ∞ , whereas L1 do es not have it. For the 1 Orliz spaes, whih are, in a natural sense, intermediate between L1 and L∞ , D. Leung [12℄ proved, when the dual spae is weakly sequentially omplete, not only that these Orliz spaes have prop erty (V) , but that they atually have the lo al prop erty (V) , i.e. all their ultrap owers have prop erty (V) . D. Leung's pro of uses non trivial prop erties of Banah latties. In this pap er, we shall give an elementary pro of of the (weaker) result that the Orliz spae LΨ has prop erty (V) , when the omplementary funtion of Ψ saties the ∆2 ondition. Aknowledgement. This work was made during the stay in Lens, in May June 2005, of the fourth-named author, as Professeur invité of the Université d'Artois . We are very grateful to the referee for having simplied the pro of of Theorem 2, making it shorter and very more elegant and oneptual, by giving us the statement and the pro of of Prop osition 5. 2 The Morse-Transue spae In this pap er, we shall onsider Orliz spaes dened on a probability spae (Ω,P) , that we shall assume non purely atomi. By an Orliz funtion, we shall understand that Ψ: [0,∞]→[0,∞] is a non-dereasing onvex funtion suh that Ψ(0) = 0 and Ψ(∞) = ∞ . To avoid pathologies, we shall assume that we work with an Orliz funtion Ψ having the following additional prop erties: Ψ is ontinuous at 0 , stritly onvex (hene stritly inreasing), and suh that Ψ(x) x−→ x→∞ ∞. This is essentially to exlude the ase of Ψ(x) = ax . The Orliz spae LΨ(Ω) is the spae of all (equivalene lasses of ) measurable funtions f: Ω →C for whih there is a onstant C > 0 suh that ZΩ Ψ|f(t)| CdP(t)<+∞ and then kfkΨ (the Luxemburg norm ) is the innimum of all p ossible onstants C suh that this integral is ≤1 . To every Orliz funtion is asso iated the omplementary Orliz funtion Φ = Ψ∗: [0,∞]→[0,∞] dened by: Φ(x) = sup y≥0xy −Ψ(y). The extra assumptions on Ψ ensure that Φ is itself stritly onvex. Throughout this pap er, we shall assume that the omplementary Orliz fun- tion satises the ∆2 ondition ( Φ∈∆2 ), i.e. , for some onstant K > 0 , and some x0>0 , we have: Φ(2x)≤KΦ(x),∀x≥x0. 2 This is usually expressed by saying that Ψ satises the ∇2 ondition ( Ψ∈ ∇2 ). This is equivalent to say that for some β > 1 and x0>0 , one has Ψ(x)≤ Ψ(βx)/(2β) for x≥x0 , and that implies that Ψ(x) x−→ x→∞ ∞ . In partiular, this exludes the ase LΨ=L1 . When Φ satises the ∆2 ondition, LΨ is the dual spae of LΦ . We shall denote by MΨ the losure of L∞ in LΨ . Equivalently (see [15℄, page 75), MΨ is the spae of (lasses of ) funtions suh that: ZΩ Ψ|f(t)| CdP(t)<+∞,∀C > 0. This spae is the Morse-Transue spae asso iated to Ψ , and (MΨ)∗=LΦ , isometrially if LΦ is provided with the Orliz norm, and isomorphially if it is equipp ed with the Luxemburg norm (see [15℄, Chapter IV, Theorem 1.7, page 110). We have MΨ=LΨ if and only if Ψ satises the ∆2 ondition, and LΨ is reexive if and only if both Ψ and Φ satisfy the ∆2 ondition. When the omplementary funtion Φ = Ψ∗ of Ψ satises it (but Ψ do es not satisfy this ∆2 ondition, to exlude the reexive ase), we have (see [15℄, Chapter IV, Prop osition 2.8, page 122, and Theorem 2.11, page 123): ( ∗ ) (LΨ)∗= (MΨ)∗⊕1(MΨ)⊥, or, equivalently, (LΨ)∗=LΦ⊕1(MΨ)⊥ , isometrially, with the Orliz norm on LΦ . For all the matter ab out Orliz funtions and Orliz spaes, we refer to [15℄, or to [11℄. It follows from the preeding equation ( ∗ ) that MΨ is an M -ideal in its bidual. Hene MΨ and all its subspaes have Peªzy«ski's prop erty (V) ([8, 9℄; see also [10℄, Chapter II I, Theorem 3.4, and the end of this paper). This result was shown by D. Werner ([19℄; see also [10℄, Chapter I I I, Example 1.4 (d), page 105), by a dierent way, using the ball intersetion prop erty (in these referenes, it is assumed moreover that Ψ do es not satises the ∆2 ondition, but if it satises it, the spae LΨ is reexive, and so the result is obvious). The pro of given in [8, 9℄ of the fat that Banah spaes whih are M -ideal in their bidual have prop erty (V) uses lo al reexivity and the notion of pseudobal l . We are going to give b elow a slightly dierent pro of, whih do es not use this last notion, and seems to us more transparent. Let us note that, however, a stronger property, namely Peªzy«ski's property (u) , was shown sine then to b e satised by the spaes M -ideal of their bidual (see [7℄ and, in a more general setting, [6℄; that follows also from [17℄). Theorem 1 (Godefroy-Saab, [8 , 9℄)℄ Every Banah spae whih is M -ideal in its bidual have property (V) . 3 Pro of. Assume that X∗∗∗ =X∗⊕1X⊥ and let T:X→Y b e a non weakly ompat map. By Gantmaher's Theorem, T∗∗ :X∗∗ →Y∗∗ is not weakly ompat either. This means that T(4)(X(4))6⊆ Y∗∗ . Sine X(4) =X∗∗ ⊕(X∗)⊥ (anonial deomposition of the third dual of X∗ ), there exists some u∈(X∗)⊥ , with kuk= 1 suh that T(4)(u)6= 0 . Now the M -ideal prop erty of X gives X(4) = (X∗)⊥⊕∞X⊥⊥ . It follows that kx+auk= max{kxk,|a|},∀x∈X, ∀a∈C. By lo al reexivity, we an onstrut a sequene (xn)n≥1 in X equivalent to the anonial basis of c0 and suh that kT xnk ≥ δ > 0 for every n≥1 . For that, let 0< δ < kT(4)uk , εn>0 b e suh that (1 −εn)kT(4)uk> δ and Qn≥1(1 + εn)≤2 , Qn≥1(1 −εn)≥1/2 . Assume that x1,...,xn have b een onstruted in suh a way that kT xkk> δ and n Y k=1 (1 −εk) max{|a1|,...,|an|} ≤ ka1x1+···+anxnk ≤ n Y k=1 (1 + εk) max{|a1|,...,|an|} for every salars a1,...,an . Let Vn b e the linear subspae of X(4) generated by {u, x1,...,xn} . By Bellenot's version of the priniple of lo al reexivity ([1℄, Corollary 7), there exists an op erator An:Vn→X suh that kAnk , kA−1 nk are less or equal than (1 + εn+1) , An is the identity on the linear span of {x1,...,xn} and kT(4)uk−kT Anuk≤εn+1kT(4)uk. If xn+1 =Anu , it is now lear that n+1 Y k=1 (1 −εk) max{|a1|,...,|an+1|} ≤ ka1x1+···+an+1xn+1k ≤ n+1 Y k=1 (1 + εk) max{|a1|,...,|an+1|} for every salars a1,...,an+1 and kT xn+1k> δ . Hene 1 2max{|a1|,...,|an|} ≤ ka1x1+···+anxnk ≤ 2 max{|a1|,...,|an|} for every salars a1,...,an . Sine kT xnk> δ , this ends the pro of.  4 3 Peªzy«ski's prop erty (V) for LΨ . As we said, the following result is a partiular ase of that of D. Leung ([12℄), but we shall give an elementary pro of. Theorem 2 ([12 ℄) Suppose that the onjugate funtion Φ of Ψ satises the ∆2 ondition. Then, the spae LΨ has Peªzy«ski's property (V) . As it is well-known (and easy to prove), every dual spae with Peªzy«ski's prop erty (V) is a Grothendiek spae: every weak-star onvergent sequene in its dual is weakly onvergent. Hene, we have: Corollary 3 Suppose that the onjugate funtion Φ of Ψ satises the ∆2 ondition. Then the spae LΨ is a Grothendiek spae. Pro of of Theorem 2. We may assume that LΨ is a real Banah spae. The pro of arises diretly from the two following results, sine E=MΨ is a Banah lattie having prop erty (V) and LΨ= (MΨ)∗∗ . Lemma 4 Suppose that the Orliz funtion Ψ does not satisfy the ∆2 ondition. Then for every sequene (gn)n in the unit bal l of LΨ , there exist a sequene (fn)n in MΨ and a positive funtion g∈LΨ suh that |gn−fn| ≤ g . Prop osition 5 Let E be a Banah lattie that has property (V) . Suppose that for every sequene (x∗∗ n)n in BE∗∗ , there are a sequene (xn)n in E and a positive x∗∗ ∈E∗∗ suh that |x∗∗ n−xn| ≤ x∗∗ . Then E∗∗ has property (V) . Pro of of Lemma 4. Sine, by dominated onvergene, lim t→+∞ZΩ Ψ|gn|1I{|gn|>t}dP= 0, we an hoose, for every n≥1 , a p ositive numb er tn so big that: ZΩ Ψ|gn|1I{|gn|>tn}dP≤1 2n , and, moreover suh that: +∞ X n=1 P(|gn|> tn)<+∞. This last ondition implies, by Borel-Cantelli's lemma, that, almost surely, |gn| ≤ tn for n large enough. Equivalently, by setting: ˜gn=gn1I{|gn|>tn}, 5 we have, almost surely ˜gn= 0 for n large enough. It follows that almost surely supn|˜gn| is attained. Set now: An={ω∈Ω ; |˜g1(ω)|,...,|˜gn−1(ω)|<|˜gn(ω)| and |˜gk(ω)| ≤ |˜gn(ω)|,∀k≥n} ( ω∈An if and only if n is the rst time for whih supk|˜gk(ω)| is attained). The sets An are disjoint and sup n≥1|˜gn|= +∞ X n=1 |˜gn|1IAn. Hene, if we set: g= sup n≥1|˜gn|, we have g∈LΨ , sine, using the disjointness of the An 's: ZΩ Ψ(g)dP= +∞ X n=1 ZAn Ψ(|˜gn|)dP≤ +∞ X n=1 ZΩ Ψ(|˜gn|)dP≤ +∞ X n=1 1 2n= 1. That proves the lemma, by taking fn=gn−˜gn , whih is in L∞⊆MΨ .  Pro of of Prop osition 5. Supp ose that T:E∗∗ →Y is not weakly ompat. Then there exists a sequene (x∗∗ n)n in BE∗∗ suh that (T x∗∗ n)n is not relatively weakly ompat. Cho ose (xn)n and x∗∗ as in the statement of the Prop osition, and set y∗∗ n=x∗∗ n−xn for all n . We have either: (a) (T xn)n is not weakly ompat, or (b) (T y∗∗ n)n is not weakly ompat. If (a) holds, T|E:E→Y is not weakly ompat; hene T|E xes a opy of c0 . If (b) holds, let I b e the losed lattie ideal generated by x∗∗ in E∗∗ , normed so that [−x∗∗, x∗∗] is the unit ball, and let i:I→E∗∗ b e the inlusion map. Sine (y∗∗ n)n lies in [−x∗∗, x∗∗] , T◦i is not weakly ompat. But I is lattie isomorphi to a C(K) spae, and hene has prop erty (V) . Thus T◦i xes a opy of c0 . So T xes a opy of c0 .  Remark. We annot exp et that, for tn big enough, the funtions ˜gn ould have a small norm. For example, let G b e a standard gaussian random variable N(0,1) . For Ψ = Ψ2 ( Ψ2(x) = ex2−1 ), we have, for every t > 0 : ZΩ Ψ2|G|1I{|G|>t} εdP=1 √2πZ|x|>t (ex2/ε2−1) e−x2/2dx = +∞ for every ε < √2 ; that means that kG1I{|G|>t}kΨ2≥√2 for every t > 0 (reall that kGkΨ2=p8/3 : see [13℄, page 31). 6 4 Conluding remarks and questions 1. The full D. Leung's result that LΨ have the loal property (V) , i.e. every ultrap ower of LΨ have the property (V) (see [3℄) annot b e obtained straightforwardly from our pro of. Indeed, sine LΨ= (MΨ)∗∗ is 1 -omplemented in every ultrapower of MΨ , it would sue to prove that every suh ultrapower has prop erty (V) ; but if (MΨ)U∗ ontains (LΦ)U as a w∗ -dense subspae, it is bigger. The ultraprower (LΦ)U is not exatly known in general. In the partiular ase of Ψ = Ψ2 ( Ψ2(x) = ex2−1 ), we have ([4℄, Prop osition 4.1 and Prop osition 4.2): (LΦ2)U∼ =LΦ2(PU)⊕L1(µU). However, sine (LΨ)∗= (LΦ)∗∗ ∼ =LΦ⊕1L1(µ) , all the o dd duals of LΨ an b e written (LΨ)(2n+1) ∼ =(LΨ)∗⊕1L1(µn). Hene we get that all the even duals of LΨ have the prop erty (V) . 2. We an dene the Hardy-Orliz spaes HΨ , in a natural way: it is the subspae of LΨ onsisting of the funtions on the unit irle T=∂D whih have an analyti extension in D ; equivalently, it is the subspae of LΨ whose negative Fourier o eients vanish. In [2℄, J. Bourgain proved that H∞ has prop erty (V) . Do es HΨ have prop erty (V) ? Note that the answer annot follow trivially from our Theorem 2 sine HΨ is omplemented in LΨ if and only if LΨ is reexive: indeed, the Riesz pro jetion from LΨ onto HΨ is b ounded if and only if LΨ is reexive ([18℄; see [16℄, Chapter VI, Theorem 2.8, page 196), and we have: Prop osition 6 Assume that Ψ∈ ∇2 . Then the Hardy-Orliz spae HΨ is omplemented in LΨ if and only if the Riesz projetion is bounded on LΨ . Hene HΨ is omplemented in LΨ if and only if LΨ is reexive. Pro of. Only the neessary ondition needs a pro of. Assume that there is a b ounded pro jetion P from LΨ onto HΨ . For every f∈MΨ , and for every g∈LΦ , the translations t7→ ft and t7→ gt are ontinuous. Hene we an dene ˜ P by setting: h˜ Pf, gi=ZThP(ft), gtidt. One has k˜ PfkΨ≤ kPkkfkΨ , so that ˜ P is bounded from MΨ into LΨ . On the other hand, it is immediate to see that for every trigonometri polynomial f , one has, if en(x) = einx : ˜ P(f) = X n∈Z ˆ f(n)\ P(en)(n)en. Sine P is a pro jetion, we have P(en) = en for n≥0 ; and sine P takes its values in HΨ , we have \ P(en)(k) = 0 for k < 0 ; in partiular \ P(en)(n) = 0 for n < 0 . 7 We get therefore: ˜ P(f) = X n≥0 ˆ f(n)en, that is ˜ P is the restrition to MΨ of the Riesz pro jetion. Hene the Riesz pro jetion is b ounded on MΨ . By taking its bi-adjoint, we get that it is b ounded on LΨ .  In Ryan's pap er ([18℄), it is assumed that Ψ is an N -funtion, that is limx→0Ψ(x) x= 0 . But we may mo dify Ψ on [0,1] to get an N -funtion Ψ1 . Sine we work on a probability spae (Ω,P) , the new spae LΨ1 is equal, as a vetor spae, to LΨ , but with an equivalent norm. Hene Ryan's result remains true without this assumption. Note that, when the probability spae (Ω,P) is separable, sine we have assumed that Ψ∈ ∇2 , the reexivity of LΨ is equivalent to its separability (see [15℄, Chapter I I I, Theorem 5.1, pages 8788). 3. Prop erty (V) allows us to say that LΨ lo oks like Lp , 1< p ≤ ∞ . In some sense, it may b e seen as lose to L∞ when Ψ/∈∆2 , sine it is not reexive. However, from other points of view, it is loser to Lp with p < ∞ ; on the one hand, it is a bidual spae; on the other hand, one has: Prop osition 7 If Ψ∈ ∇2 , then LΨ never has the Dunford-Pettis property. Pro of. We are atually going to show that MΨ do es not have the Dunford- Pettis property. That will prove the prop osition, sine LΨ= (MΨ)∗∗ . Sine Ψ∈ ∇2 , there is some α > 1 and some c > 0 suh that Ψ(x)≥cxα . It follows that LΨ⊆Lα and the natural injetion i:LΨ→Lα is bounded, and hene weakly ompat, sine Lα is reexive. Take now an orthonormal sequene (rn)n≥1 in L2 with onstant modulus equal to 1 (for example, an indep endent sequene of random variables taking the values ±1 eah with probability 1/2 ). One has RΩrnf dP−→ n→+∞0 for every f∈L2 . By density, this remains true for every f∈L1 , and in partiular for every f∈LΦ , sine LΦ⊆L1 . Therefore, (rn)n≥1 weakly onverges to 0 in MΨ . Sine krnkα= 1 , i(rn)n do es not norm-onverge to 0 , and hene the weakly ompat map i:MΨ→Lα is not a Dunford-Pettis op erator. Therefore MΨ do es not have the Dunford-Pettis prop erty.  A slightly dierent way to prove this is to use that for every Banah spae X whih has the Dunford-Pettis prop erty and whih do es not ontain ℓ1 , its dual X∗ has the Shur prop erty ([5, 14℄; see also [13℄, Chapitre 7, Exerie 7.2). But MΨ do es not ontain ℓ1 (b eause all its subspaes have property (V) ; or b eause its dual LΦ is separable). Hene LΦ would have the Shur prop erty. The same argument as ab ove shows that is not the ase. 4. We have required in this pap er that the omplementary funtion Φ satises the ∆2 ondition. Hene, in some sense, the spae LΨ is far from L1 . We may ask what happens when we are in the other side of the sale, namely when LΨ 8 is lose to L1 . But if Ψ satises the ∆2 ondition, then LΨ= (MΦ)∗ and MΦ , b eing M -ideal of its bidual, has property (V) , as said in the Introdution. It follows that LΨ is weakly sequentially omplete (and in fat has prop erty (V∗) ), and if we assume that Φ/∈∆2 (so as LΨ is not reexive), then LΨ do es not have prop erty (V) . Referenes [1℄ S. F. Bellenot, Lo al Reexivity of Normed Spaes, Op erators, and Fréhet Spaes, J. Funt. Anal. 59(1) (1984), 111. [2℄ J. Bourgain, H∞ is a Grothendiek spae, Studia Math. 75 (1983), 193216. [3℄ M. D. Contreras and S. Díaz, Some Banah Spae Prop erties of the Duals of the Disk Algebra and H∞ , Mihigan Math. J. 46 (1999), 123141. [4℄ D. Daunha-Castelle et J.-L. Krivine, Appliations des ultrapro duits à l'étude des espaes et des algèbres de Banah, Studia Math. 41 (1972), 315334. [5℄ H. Fakhoury, Sur les espaes de Banah ne ontenant pas ℓ1(N) , Math. Sand. 41 (1977), 277289. [6℄ G. Go defroy, N. J. Kalton, and P. D. Saphar, Unonditional ideals in Banah spaes, Studia Math. 104, no. 1 (1993), 1359. [7℄ G. Go defroy and D. Li, Banah spaes whih are M -ideals in their bidual have prop erty (u) , Ann. Inst. Fourier (Grenoble) 39, no. 2 (1989), 361371. [8℄ G. Go defroy et P. Saab, Quelques espaes de Banah ayant les propriétés (V) ou (V∗) de A. Peªzy«ski, C.R. Aad. Si. Paris, Série A 303 (1986), 503506. [9℄ G. Go defroy and P. Saab, Weakly unonditionally onvergent series in M - ideals, Math. Sand. 64 (1989), 307318. [10℄ P. Harmand, D. Werner and W. Werner, M -ideals in Banah Spaes and Banah Algebras, Leture Notes in Math. 1547, Springer (1993). [11℄ M. A. Krasnosel'ski and Ya. B. Rutiki, Convex funtions and Orliz spaes (translation), P. Noordho Ltd., Groningen (1961). [12℄ D. Leung, Weak ∗ onvergene in higher duals of Orliz spaes, Pro eedings Amer. Math. So . 103 (3) (1988), 797800. [13℄ D. Li et H. 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