A criterion of weak compactness for operators on subspaces of Orlicz spaces

JOURNAL OF c 200*, Scientific Horizon FUNCTION SPACES AND APPLICATIONS http://www.jfsa.net Volume 6, Number 3 (2008), 277-292 A criterion of weak compactness for operators on subspaces of Orlicz spaces Pascal Lef`evre, Daniel Li, Herv´eQueff´elec and Luis Rodr´ıguez-Piazza (Communicated by Nigel Kalton) 2000 Mathematics Subject Classification. Primary: 46E30; Secondary: 46B20. Keywords and phrases. Morse-Transue space, Orlicz space, weakly compact operators. Abstract. We give a criterion of weak compactness for the operators on the Morse-Transue space MΨ, the subspace of the Orlicz space LΨgenerated by L∞. 1. Introduction and Notation In 1975, C. Niculescu established a characterization of weakly compact operators Tfrom C(S), where Sis a compact space, into a Banach space Z([14, 15], see [3] Theorem 15.2 too): T:C(S)→Zis weakly compact if and only if there exists a Borel probability measure μon Ssuch that, for every >0, there exists a constant C()>0 such that: Tf≤C()fL1(μ)+f∞,∀f∈C(S). The same kind of result was proved by H. Jarchow for C∗-algebras in [7], and by the first author for A(D)andH∞(see [11]). The criterion for H∞ 278 Weak compactness for operators on subspaces of Orlicz spaces played a key role to give an elementary proof of the equivalence between weak compactness and compactness for composition operators on H∞. Beside these spaces, one natural class of Banach spaces is the class of Orlicz spaces LΨ. Unfortunately, we shall see that the above criterion is in general not true for Orlicz spaces. However, it remains true when we restrict ourselves to subspaces of the Morse-Transue space MΨ. This space is the closure of L∞in the Orlicz space LΨ. In this paper, we first give a characterization of the operators from a subspace of MΨwhich fix no copy of c0. When the complementary function of Ψ satisfies Δ2, that gives a criterion of weak compactness. If moreover Ψ satisfies a growth condition, that we call Δ0, the criterion has a more usable formulation, analogous to those described above. As in the case of H∞(but this is far less elementary), this new version obtained for subspaces of Morse-Transue spaces (Theorem 4), combined with a study of generalized Carleson measures, may be used to prove the equivalence between weak compactness and compactness for composition operators on Hardy-Orlicz spaces (see [13]), when Ψ satisfies Δ0. However, we think that this characterization has an intrinsic interest for Orlicz spaces, and will be useful not only for composition operators (see Remark 5 at the end of the paper). In this note, we shall consider Orlicz spaces defined on a probability space (Ω,P), that we shall assume non purely atomic. By an Orlicz function, we shall understand that Ψ: [0,∞]→[0,∞]isa non-decreasing convex function such that Ψ(0) = 0 and Ψ(∞)=∞.To avoid pathologies, we shall assume that we work with an Orlicz function Ψ having the following additional properties: Ψ is continuous at 0, strictly convex (hence strictly increasing), and such that Ψ(x) x−→ x→∞∞. This is essentially to exclude the case of Ψ(x)=ax. The Orlicz space LΨ(Ω) is the space of all (equivalence classes of) measurable functions f:Ω→C for which there is a constant C>0 such that Ω Ψ|f(t)| CdP(t)<+∞ and then fΨ(the Luxemburg norm) is the infimum of all possible constants Csuch that this integral is ≤1. P. Lef`evre et. al. 279 To every Orlicz function is associated the complementary Orlicz function Φ=Ψ ∗:[0,∞]→[0,∞] defined by: Φ(x)=sup y≥0xy −Ψ(y). The extra assumptions on Ψ ensure that Φ is itself strictly convex. Throughout this paper, we shall assume, except explicit mention of the contrary, that the complementary Orlicz function satisfies the Δ2condition (Φ ∈Δ2), i.e.,forsomeconstantK>0, and some x0>0, we have: Φ(2x)≤KΦ(x),∀x≥x0. This is usually expressed by saying that Ψ satisfies the ∇2condition (Ψ ∈∇ 2). This is equivalent to say that for some β>1andx0>0, one has Ψ(x)≤Ψ(βx)/(2β)forx≥x0, and that implies that Ψ(x) x−→ x→∞∞.In particular, this excludes the case LΨ=L1. When Φ satisfies the Δ2condition, LΨis the dual space of LΦ. We shall denote by MΨthe closure of L∞in LΨ. Equivalently (see [16], page 75), MΨis the space of (classes of) functions such that: Ω Ψ|f(t)| CdP(t)<+∞,∀C>0. This space is the Morse-Transue space associated to Ψ, and (MΨ)∗=LΦ, isometrically if LΦis provided with the Orlicz norm, and isomorphically if it is equipped with the Luxemburg norm (see [16], Chapter IV, Theorem 1.7, page 110). We have MΨ=LΨif and only if Ψ satisfies the Δ2condition, and LΨ is reflexive if and only if both Ψ and Φ satisfy the Δ2condition. When the complementary function Φ = Ψ∗of Ψ satisfies it (but Ψ does not satisfy this Δ2condition, to exclude the reflexive case), we have (see [16], Chapter IV, Proposition 2.8, page 122, and Theorem 2.11, page 123): (∗)(LΨ)∗=(MΨ)∗⊕1(MΨ)⊥, or, equivalently, (LΨ)∗=LΦ⊕1(MΨ)⊥, isometrically, with the Orlicz norm on LΦ. For all the matter about Orlicz functions and Orlicz spaces, we refer to [16], or to [9]. 280 Weak compactness for operators on subspaces of Orlicz spaces 2. Main result Our goal in this section is the following criterion of weak compactness for operators. We begin with: Theorem 1. Let Ψbe an arbitrary Orlicz function, and let Xbe a subspace of the Morse-Transue space MΨ. ThenanoperatorT:X→Y from Xinto a Banach space Yfixes no copy of c0if and only if for each ε>0,thereexistsCε>0such that (1) Tf≤CεΩ Ψε|f| fΨdP+εfΨ,∀f∈X. Recall that saying that Tfixes a copy of c0means that there exists a subspace X0of Xisomorphic to c0such that Trealizes an isomorphism between X0and T(X0). Before proving that, we shall give some consequences. First, we have Corollary 2. Assume that the complementary function of Ψhas Δ2 (Ψ ∈∇ 2). Then for every subspace Xof MΨ, and every operator T:X→Y,Tis weakly compact if and only if it satisfies (1). Proof. When the complementary function of Ψ has Δ2, one has the decomposition (∗), which means that MΨis M-ideal in its bidual (see [6, Chapter III]); this result was first shown by D. Werner [17] (see also [6, Chapter III, Example 1.4 (d), page 105]) by a different way, using the ball intersection property; note that in these references, it is moreover assumed that Ψ does not satisfy the Δ2condition, but if it satisfies it, the space LΨis reflexive, and so the result is obvious. But every subspace Xof a Banach space which is M-ideal of its bidual has Pelczy´nki’s property (V) ([4, 5]; see also [6], Chapter III, Theorem 3.4), which means that operators from Xare weakly compact if and only if they fix no copy of c0. With Ψ satisfying the following growth condition, the characterization (1) takes on a more usable form. Definition 3. We say that the Orlicz function Ψsatisfies the Δ0 condition if for some β>1 lim x→+∞ Ψ(βx) Ψ(x)=+∞. This growth condition is a strong negation of the Δ2condition and it implies that the complementary function Φ = Ψ∗of Ψ satisfies the Δ2condition. Note that in the following theorem, we cannot content ourselves with Ψ/∈Δ2(i.e. lim supx→+∞Ψ(βx)/Ψ(x)=+∞), instead of Ψ ∈Δ0(see P. Lef`evre et. al. 281 Remark 3 in Section 3). An interesting question is whether the condition Ψ∈Δ0is actually necessary for this characteriztion. Theorem 4. Assume that Ψsatisfies the Δ0condition, and let Xbe asubspaceofMΨ. Then every linear operator Tmapping Xinto some Banach space Yis weakly compact if and only if for some (and then for all) 1≤p<∞and all ε>0,thereexistsCε>0such that (W) T(f)≤Cεfp+εfΨ,∀f∈X. Remark 5. This theorem extends [12] Theorem II.1. As in the case of C∗-algebras (see [3], Notes and Remarks, Chap. 15), there are miscellaneous applications of such a characterization. Remark 6. Contrary to the Δ2condition where the constant 2 may be replaced by any constant β>1, in this Δ0condition, the constant β cannot be replaced by another, as the following example shows. Example 7. There exists an Orlicz function Ψ such that (2) lim x→+∞ Ψ(5x) Ψ(x)=+∞, but (3) lim inf x→+∞ Ψ(2x) Ψ(x)<+∞. Indeed, let (cn)nbe an increasing sequence of positive numbers such that lim n→∞ cn+1 cn =+∞,takeψ(t)=cnfor t∈(4n,4n+1]andΨ(x)=x 0ψ(t)dt. Then (2) is verified. On the other hand, if xn=2·4n, one has Ψ(xn)≥ cn4n,andΨ(2xn)≤cn4n+1 , so we get (3). Before proving Theorem 4, let us note that it has the following straightforward corollary. Corollary 8. Let Xbe like in Theorem 4, and assume that Fis a family of operators from Xinto a Banach space Ywith the following property: there exists a bounded sequence (gn)nin Xsuch that limn→∞ gn1=0 and such that an operator T∈F is compact whenever lim n→∞Tgn=0. Then every weakly compact operator in T∈F is actually compact. In the forthcoming paper [13], we prove, using a generalization of the notion of Carleson measure, that a composition operator Cφ:HΨ→HΨ 282 Weak compactness for operators on subspaces of Orlicz spaces (HΨis the space of analytic functions on the unit disk Dof the complex plane whose boundary values are in LΨ(∂D), and φ:D→Dis an analytic self-map) is compact whenever lim r→1− sup |ξ|=1 Ψ−11/(1 −r)Cφ(uξ,r)Ψ=0, where uξ,r(z)=1−r 1−¯ ξrz2,|z|<1, and we have lim r→1− sup |ξ|=1 Ψ−11/(1 −r)Cφ(uξ,r)Ψ=0 when Cφis weakly compact and Ψ ∈Δ0. Though the situation does not fit exactly as in Corollary 8 (not because of the space HΨ, which is not a subspace of MΨ: we actually work in HMΨ=HΨ∩MΨsince uξ,r ∈HMΨ, but because of the fact that we ask a uniform limit for |ξ|= 1), the same ideas allow us to get, when Ψ satisfies the condition Δ0,thatCφis compact if and only if it is weakly compact. Proof of Theorem 4. Assume that we have (W). We may assume that p>1, since if (W) is satisfied for some p≥1, it is satisfied for all q≥p. Moreover, we may assume that LΨj →Lpsince Ψ satisfies condition Δ0(since we have: limx→+∞Ψ(x) xr=+∞, for every r>0). Then T(1/Cε)BLp∩(1/ε)BX⊆2BY. Taking the polar of these sets, we get T∗(BY∗)⊆(2Cε)Bj∗[(Lp)∗]+(2ε)BX∗, for every ε>0. By a wellknown lemma of Grothendieck, we get, since Bj∗[(Lp)∗]is weakly compact, that T∗(BY∗) is relatively weakly compact, i.e. T∗, and hence also T,is weakly compact. Conversely, assume in Theorem 4 that Tis weakly compact. We are going to show that (W) is satisfied with p= 1 (hence for all finite p≥1). Let ε>0. Since the Δ0condition implies that the complementary function of Ψ satisfies the Δ2condition, Corollary 2 implies that, when fΨ=1 Tf≤Cε/2Ω Ψ(ε/2)|f|dP+ε/2. As Ψ satisfies the Δ0condition, there is some β>1 such that Ψ(x) Ψ(βx)→0 as x→∞; hence, with κ=ε/2Cε/2,thereexistssomexκ>0such that Ψ(x)≤κΨ(βx)forx≥xκ. By the convexity of Ψ, one has P. Lef`evre et. al. 283 Ψ(x)≤Ψ(xκ) xκx=: Kκxfor 0 ≤x≤xκ. Hence, for every x≥0 Ψ(x)≤κΨ(βx)+Kκx. It follows that, for f∈X,withfΨ=1 Ω Ψ(ε/2)|f|dP≤κΩ Ψβ(ε/2)|f|dP+Kκ ε 2f1≤κ+Kκ ε 2f1 if we have chosen ε≤2/β . Hence Tf≤Cε/2κ+Kκ ε 2f1+ε 2=Cε/2 ε 2Kκf1+Cε/2κ+ε 2=C εf1+ε, which is (W).  Remark 9. The sufficient condition is actually a general fact, which is surely well known (see [11], Theorem 1.1, for a similar result, and [3], Theorem 15.2 for C(K); see also [8], page 81), and has close connection with interpolation (see [2], Proposition 1), but we have found no reference, and so we shall state it separately without proof (the proof follows that given in [3], page 310). Proposition 10. Let T:X→Ybe an operator between two Banach spaces. Assume that there is a Banach space Zand a weakly compact map j:X→Zsuch that: for every ε>0,thereexistsCε>0such that Tx≤CεjxZ+εxX,x∈X. Then Tis weakly compact. Note that, by the Davis-Figiel-Johnson-Pelczy´nski factorization theorem, we may assume that Zis reflexive. We may also assume that jis injective, because ker j⊆ker T,soTinduces a map ˜ T:X/ ker j→Ywith the same property as T. Indeed, if jx =0,thenTx≤εxfor every ε>0, and hence Tx=0. Proof of Theorem 1. Assume first that Tfixes a copy of c0.Thereare hence some δ>0 and a sequence (fn)nin Xequivalent to the canonical basis of c0such that fnΨ=1andTfn≥δ. In particular, there is some M>0 such that, for every choice of εn=±1    N  n=1 εnfn  Ψ≤M, N ≥1. Let (rn)nbe a Rademacher sequence. We have, first by Khintchine’s inequality, then by Jensen’s inequality and Fubini’s Theorem 284 Weak compactness for operators on subspaces of Orlicz spaces Ω Ψ1 M√2N  n=1 |fn|21/2dP≤Ω Ψ1 M1 0 N  n=1 rn(t)fndtdP ≤Ω1 0 Ψ1 M N  n=1 rn(t)fndtdP =1 0Ω Ψ1 M N  n=1 rn(t)fndPdt ≤1. The monotone convergence Theorem gives then Ω Ψ1 M√2∞  n=1 |fn|21/2dP≤1. In particular, ∞ n=1 |fn|2is finite almost everywhere, and hence fn→0 almost everywhere. Since Ψ1 M√2∞ n=1 |fn|21/2∈L1,bytheabove inequalities, Lebesgue’s dominated convergence Theorem gives Ω Ψ|fn| M√2dP−→ n→∞0. But that contradicts (1) with ε≤1/M √2andε<δ,sinceTfn≥δ. The converse follows from the following lemma. Lemma 11. Let Xbe a subspace of MΨ,andlet(hn)nbe a sequence in X,withhnΨ=1 for all n≥1, and such that, for some M>0 Ω Ψ(|hn|/M )dP−→ n→∞0. Then (hn)nhas a subsequence equivalent to the canonical basis of c0. Indeed, if condition (1) is not satisfied, there exist some ε0>0and functions hn∈Xwith hnΨ=1suchthatThn≥2nΩΨ(ε0|hn|)dP+ ε0. That implies that ΩΨ(ε0|hn|)dPtends to 0, so Lemma 11 ensures that (hn)nhas a subsequence, which we shall continue to denote by (hn)n, equivalent to the canonical basis of c0.Then(Thn)nis weakly unconditionally Cauchy. Since Thn≥ε0,(Thn)nhas, by Bessaga- Pelczy´nski’s Theorem, a further subsequence equivalent to the canonical basis of c0. It is then obvious that Trealizes an isomorphism between the spaces generated by these subsequences.  P. Lef`evre et. al. 285 ProofofLemma11. The proof uses the idea of the construction made in the proof of Theorem II.1 in [12], which it generalizes, but with some additional details. By the continuity of Ψ, there exists a>0 such that Ψ(a)=1. Then, since Ψ is increasing, we have, for every g∈L∞,ΩΨa|g| g∞dP≤1, and so gΨ≤(1/a)g∞.Now,choose,foreveryn≥1, positive numbers αn<a/2n+2 such that Ψ(αn/2M)≤1. We are going to construct inductively a subsequence (fn)nof (hn)n,a sequence of functions gn∈L∞and two sequences of positive numbers βn and εn≤min{1/2n+1,M/2n+1}, such that, for every n≥1 (i) if we set M1= 1 and, for n≥2 Mn=max1,Ψg1∞+···+gn−1∞ 2M, then Mnβn≤1/2n+1 ; (ii) fnΨ=1; (iii) fn−gnΨ≤εn,withεnsuch that βnΨ(αn/2εn)≥2; (iv) P({|gn|>α n})≤βn; (v) ˘gnΨ≥1/2, with ˘gn=gn1I{|gn|>αn}. We shall give only the inductive step, since the starting one unfolds identically. Suppose hence that the functions f1,...,f n−1,g1,...,g n−1and the numbers β1,...,β n−1and ε1,...,ε n−1have been constructed. Choose then βn>0 such that Mnβn≤1/2n+1 .NotethatMn≥1 implies that βn≤1/2n+1 .SinceΩΨ(|hk|/M )dP→0asn→∞, we can find fn=hkn such that fnΨ=1,andmoreover P({|fn|>α n/2})≤1 Ψ(αn/2M)Ω Ψ|fn| MdP≤βn 2· Take now εn≤min{1/2n+1,M/2n+1}such that 0 <ε n≤αn/2Ψ−1(2/βn) and gn∈L∞such that fn−gnΨ≤εn. Then, since P({|fn−gn|>α n/2})Ψαn 2εn≤Ω Ψ|fn−gn| εndP≤1, we have P({|gn|>α n})≤P({|fn|>α n/2})+P({|fn−gn|>α n/2}) ≤βn 2+1 Ψ(αn/2εn)≤βn. 292 Weak compactness for operators on subspaces of Orlicz spaces [12] P. Lef`evre, D. Li, H. Queff´elec and L. Rodr´ıguez-Piazza, Some translation-invariant Banach function spaces which contain c0, Studia Math., 163 (2004), 137–155. [13] P. Lef`evre, D. Li, H. Queff´elec and L. Rodr´ıguez-Piazza, Composition operators on Hardy-Orlicz spaces, Memoirs Amer. Math. Soc., to appear. [14] C. P. Niculescu, Absolute continuity and weak compactness, Bull. Amer. Math. Soc., 81 (1975), 1064–1066. [15] C. P. Niculescu, Absolute continuity in Banach space theory,Rev. Roum. Math. Pures Appl., 24 (1979), 413–422. [16] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Pure and Applied Mathematics 146, Marcel Dekker, Inc., 1991. [17] D. Werner, New classes of Banach spaces which are M-ideals in their biduals, Math. Proc. Cambridge Philos. Soc., 111 (1992), 337–354. P. Lef`evre and D. Li Universit´ed’Artois Laboratoire de Math´ematiques de Lens EA 2462 F´ed´eration CNRS Nord-Pas-de-Calais FR 2956 Facult´e des Sciences Jean Perrin, Rue Jean Souvraz, S.P.18 62 307 LENS Cedex FRANCE (E-mail : [email protected]) (E-mail : [email protected]) H. Queff´elec Universit´e des Sciences et Technologies de Lille Laboratoire Paul Painlev´e U.M.R. CNRS 8524 U.F.R. de Math´ematiques, 59 655 VILLENEUVE D’ASCQ Cedex FRANCE (E-mail : queff@math.univ-lille1.fr) Luis Rodr´ıguez-Piazza Universidad de Sevilla Facultad de Matem´aticas, Dpto de An´alisis Matem´atico Apartado de Correos 1160 41 080 SEVILLA SPAIN (E-mail : p[email protected]) (Received : September 2007)