MethodsofFunctional AnalysisandTopology Vol. 22(2016), no. 4,pp. 387–392 L-DUNFORD-PETTIS PROPERTY IN BANACH SPACES A.RETBIANDB.ELWAHBI 7 1 Abstract. In this paper, we introduce and study the concept of L-Dunford-Pettis 0 setsandL-Dunford-PettispropertyinBanachspaces. Next,wegiveacharacteriza- 2 tion of the L-Dunford-Pettis property with respect to some well-known geometric propertiesofBanachspaces. Finally,somecomplementabilityofoperatorsonBanach n spaceswiththeL-Dunford-Pettispropertyarealsoinvestigated. a J 2 ] 1. Introduction and notation A F A norm bounded subset A of a Banach space X is called Dunford-Pettis (DP for h. short) if every weakly null sequence (fn) in X′ convergeuniformly to zero on A, that is, t limn→∞supx∈A|fn(x)|=0. a An operator T between two Banach spaces X and Y is completely continuous if T m maps weakly null sequences into norm null ones. [ Recallfrom[11],thatanoperatorT :X →Y betweentwoBanachspacesisDunford- 1 Pettiscompletely continuous(abb. DPcc)if itcarriesaweaklynullsequence,whichis a v DPsetinX tonormnullonesinY. Itisclearthateverycompletelycontinuousoperator 2 is DPcc. Also every weakly compact operator is DPcc (see Corollary 1.1 of [11]). 5 A Banach space X has: 5 0 – arelativelycompactDunford-Pettisproperty(DPrcPforshort)ifeveryDunford- 0 Pettis set in X is relatively compact [5]. For example, every Schur spaces have . the DPrcP. 1 0 – aGrothendieckproperty(oraBanachspaceX isaGrothendieckspace)ifweak⋆ 7 and weak convergence of sequences in X′ coincide. For example, each reflexive 1 space is a Grothendieck space. : v – aDunford-Pettisproperty(DPpropertyforshort)ifeveryweaklycompactopera- i torT fromX intoanotherBanachspaceY iscompletelycontinuous,equivalently, X if every relatively weakly compact subset of X is DP. r a – a reciprocal Dunford-Pettis property (RDP property for short) if every com- pletely continuous operator on X is weakly compact. A subspace X of a Banach space X is complemented if there exists a projection P 1 from X to X (see page 9 of [2]). 1 Recall from [1], that a Banach lattice is a Banach space (E,k·k) such that E is a vector lattice and its norm satisfies the following property: for each x,y ∈ E such that |x|≤|y|, we have kxk≤kyk. We denote by c , ℓ1, and ℓ∞ the Banach spaces of all sequences converging to zero, 0 all absolutely summable sequences, and all bounded sequences, respectively. Let us recall that a norm bounded subset A of a Banach space X′ is called L-set if every weakly null sequence (x ) in X converge uniformly to zero on A, that is, n 2010 Mathematics Subject Classification. 46A40,46B40,46H42. Key words and phrases. Dunford-Pettis set, Dunford-Pettis relatively compact property, Dunford- Pettiscompletelycontinuous operator. 387 388 A.RETBIANDB.ELWAHBI limn→∞supf∈A|f(xn)| =0. Note also that a Banach space X has the RDP property if and only if every L-set in X′ is relatively weakly compact. In his paper, G. Emmanuelle in [4] used the concept of L-set to characterize Banach spaces not containing ℓ1, and gave several consequences concerning Dunford-Pettis sets. Later, the idea of L-set is also used to establish a dual characterizationof the Dunford- Pettis property [6]. The aim of this paper is to introduce and study the notion of L-Dunford-Pettis set in a Banach space, which is related to the Dunford-Pettis set (Definition 2.1), and note thateveryL-setinatopologicaldualofaBanachspaceisL-Dunford-Pettisset(Proposi- tion2.3). Afterthat,weintroducetheL-Dunford-PettispropertyinBanachspacewhich is sharedby those Banachspaces whose L-Dunford-Pettissubsets of his topologicaldual are relatively weakly compact (Definition 2.6). Next, we obtain some important con- sequences. More precisely, a characterizations of L-Dunford-Pettis property in Banach spaces in terms of DPcc and weakly compact operators (Theorem 2.7), the relation be- tweenL-Dunford-PettispropertywithDPandGrothendieckproperties(Theorem2.8),a new characterizationsof Banachspace with DPrcP (resp, reflexive Banach space) (The- orem 2.5) (resp, Corollary 2.10). Finally, we investigate the complementability of the class of weakly compact operators from X into ℓ∞ in the class of DPcc from X into ℓ∞ (Theorem 2.13 and Corollary 2.14). Thenotationsandterminologiesarestandard. WeusethesymbolsX,Y forarbitrary Banach spaces. We denoted the closed unit ball of X by B , the topological dual of X X by X′ and T′ : Y′ → X′ refers to the adjoint of a bounded linear operator T : X → Y. We refer the reader for undefined terminologies to the references [1, 8, 9]. 2. Main results Definition 2.1. LetX be aBanachspace. Anormbounded subsetAofthe dualspace X′ is called an L-Dunford-Pettis set, if every weakly null sequence (x ), which is a DP n set in X converges uniformly to zero on A, that is, limn→∞supf∈A|f(xn)|=0. For a proof of the next Proposition, we need the following Lemma which is just Lemma 1.3 of [11]. Lemma 2.2. A sequence (x ) in X is DP if and only if f (x )→0 as n→∞for every n n n weakly null sequence (f ) in X′. n The following Proposition gives some additional properties of L-Dunford-Pettis sets in a topological dual Banach space. Proposition 2.3. Let X be a Banach space. Then (1) every subset of an L-Dunford-Pettis set in X′ is L-Dunford-Pettis, (2) every L-set in X′ is L-Dunford-Pettis, (3) relatively weakly compact subset of X′ is L-Dunford-Pettis, (4) absolutely closed convex hull of an L-Dunford-Pettis set in X′ is L-Dunford- Pettis. Proof. (1) and (2) are obvious. (3) Suppose A ⊂ X′ is relatively weakly compact but it is not an L-Dunford-Pettis set. Then, there exists a weakly null sequence (x ), which is a DP set in X, a sequence n (f ) in A andan ǫ>0 such that |f (x )|>ǫ for all integer n. As A is relatively weakly n n n compact, there exists a subsequence (g ) of (f ) that converges weakly to an element g n n in X′. But from |g (x )|≤|(g −g)(x )|+|g(x )| n n n n n L-DUNFORD-PETTIS PROPERTY IN BANACH SPACES 389 and Lemma 2.2, we obtain that |g (x )|→0 as n→∞. This is a contradiction. n n (4)Let A be a L-Dunford-Pettisset inX′, and(x ) be a weaklynull sequence,which n is a DP set in X. Since sup |f(x )|=sup |f(x )| f∈aco(A) n f∈A n n n for each n, where aco(A) = {P λ x :x ∈A,∀i,P |λ |≤1} is the absolutely i=1 i i i i=1 i closed convex hull of A (see [1, pp. 148, 151]), then it is clear that aco(A) is L-Dunford- Pettis set in X′. (cid:3) We need the following Lemma which is just Lemma 1.2 of [11]. Lemma 2.4. A Banach space X has the DPrcP if and only if any weakly null sequence, which is a DP set in X is norm null. FromLemma2.4,weobtainthefollowingcharacterizationofDPrcPinaBanachspace in terms of an L-Dunford-Pettis set of his topological dual. Theorem 2.5. A Banach space X has the DPrcP if and only if every bounded subset of X′ is an L-Dunford-Pettis set. Proof. (⇐) Let (x ) be a weakly null sequence, which is a DP set in X. As n kx k=sup |f(x )| n f∈BX′ n for each n, and by our hypothesis, we see that kx k →0 as n →∞. By Lemma 2.4 we n deduce that X has the DPrcP. (⇒) Assume by way of contradiction that there exist a bounded subset A, which is notanL-Dunford-PettissetofX′. Then,thereexistsaweaklynullsequence(x ),which n is a Dunford-Pettis set of X such that sup |f(x )| > ǫ > 0 for some ǫ > 0 and each f∈A n n. Hence, for every n there exists some f in A such that |f (x )|>ǫ. n n n On the other hand, since (f ) ⊂ A, there exist some M > 0 such that kf k ≤ M n n X′ for all n. Thus, |f (x )|≤Mkx k n n n for each n, then by our hypothesis and Lemma 2.4, we have |f (x )| → 0 as n → ∞, n n which is impossible. This completes the proof. (cid:3) Remark 1. Note by Proposition2.3 assertion(3) that every relatively weakly compact subsetofatopologicaldualBanachspaceisL-Dunford-Pettis. Theconverseisnottruein general. Infact,theclosedunitballBℓ∞ ofℓ∞isL-Dunford-Pettisset(seeTheorem2.5), but it is not relatively weakly compact. We make the following definition. Definition 2.6. A Banach space X has the L-Dunford-Pettis property, if every L- Dunford-Pettis set in X′ is relatively weakly compact. As is known a DPcc operator is not weakly compact in general. For example, the identity operator Idℓ1 :ℓ1 →ℓ1 is DPcc, but it is not weakly compact. Inthe followingTheorem,wegiveacharacterizationsofL-Dunford-Pettispropertyof Banach space in terms of DPcc and weakly compact operators. Theorem 2.7. Let X be a Banach space, then the following assertions are equivalent: (1) X has the L-Dunford-Pettis property, (2) for each Banach space Y, every DPcc operator from X into Y is weakly compact, (3) every DPcc operator from X into ℓ∞ is weakly compact. 390 A.RETBIANDB.ELWAHBI Proof. (1) ⇒(2) Suppose that X has the L-Dunford-Pettis property and T : X → Y is DPcc operator. Thus T′(BY′) is an L-Dunford-Pettis set in X′. So by hypothesis, it is relatively weakly compact and T is a weakly compact operator. (2)⇒(3) Obvious. (3)⇒(1)IfX doesnothavetheL-Dunford-Pettisproperty,thereexistsanL-Dunford- Pettis subset A of X′ that is not relatively weakly compact. So there is a sequence (f ) ⊆ A with no weakly convergent subsequence. Now, we show that the operator n T : X → ℓ∞ defined by T(x) = (f (x)) for all x ∈ X is DPcc but it is not weakly n compact. As (f ) ⊆ A is L-Dunford-Pettis set, for every weakly null sequence (x ), n m which is a DP set in X we have kT(x )k=sup |f (x )|→0, as m→∞, m n n m so T is a Dunford-Pettis completely continuous operator. We have T′((λ )∞ ) = n n=1 P∞ λ f for every (λ )∞ ∈ ℓ1 ⊂ (ℓ∞)′. If e′ is the usual basis element in ℓ1 n=1 n n n n=1 n then T′(e′ )=f , for all n∈N. Thus, T′ is not a weakly compact operator and neither n n is T. This finishes the proof. (cid:3) Theorem 2.8. Let E be a Banach lattice. If E has both properties of DP and Grothendieck, then it has the L-Dunford-Pettis pro- perty. Proof. Suppose that T :E →Y is DPcc operator. As E has the DP property, it follows from Theorem 1.5 [11] that T is completely continuous. On the other hand, ℓ1 is not a Grothendieck space and Grothendieck property is carried by complemented subspaces. Hence the Grothendieck space E does not have any complemented copy of ℓ1. By [10], E has the RDP property and so the completely continuous operator T is weakly compact. From Theorem 2.7 we deduce that E has the L-Dunford-Pettis property. (cid:3) Remark 2. Since ℓ∞ has the Grothendieck and DP properties, it has the L-Dunford- Pettis property. LetusrecallthatK isaninfinitecompactHausdorffspaceifitisacompactHausdorff space, which contains infinitely many points. ForaninfinitecompactHausdorffspaceK,wehavethefollowingresultfortheBanach space C(K) of all continuous functions on K with supremum norm. Corollary 2.9. If C(K) contains no complemented copy of c , then it has L-Dunford- 0 Pettis property. Proof. Since C(K) contains no complemented copy of c , it is a Grothendieck space [3]. 0 On the other hand, C(K) be a Banach lattice with the DP property, and by Theorem 2.8 we deduce that C(K) has L-Dunford-Pettis property. (cid:3) Corollary 2.10. A DPrc space has the L-Dunford-Pettis property if and only if it is reflexive. Proof. (⇒)IfaBanachspaceX hastheDPrcP,thenbyTheorem1.3of[11],theidentity operator Id on X is DPcc. As X has the L-Dunford-Pettis property, it follows from X Theorem 2.7 that Id is weakly compact, and hence X is reflexive. X (⇐) Obvious. (cid:3) Remark 3. Note that the Banach space ℓ1 is not reflexive and has the DPrcP, then from Corollary 2.10, we conclude that ℓ1 does not have the L-Dunford-Pettis property. Theorem 2.11. If a Banach space X has the L-Dunford-Pettis property, then every complemented subspace of X has the L-Dunford-Pettis property. L-DUNFORD-PETTIS PROPERTY IN BANACH SPACES 391 Proof. Consider a complemented subspace X of X and a projection map P : X →X . 1 1 Suppose T : X → ℓ∞ is DPcc operator, then TP : X →ℓ∞ is also DPcc. Since X has 1 L-Dunford-Pettis, by Theorem 2.7, TP is weakly compact. Hence T is weakly compact, also from Theorem 2.7 we conclude that X has L-Dunford-Pettis, and this completes 1 the proof. (cid:3) Let X be a Banach space. We denote by L(X,ℓ∞) the class of all bounded linear operators from X into ℓ∞, by W(X,ℓ∞) the class of all weakly compact operators from X into ℓ∞, and by DPcc(X,ℓ∞) the class of all Dunford-Pettis completely continuous operators from X into ℓ∞. RecallthatBahreiniin[2]investigatedthecomplementabilityofW(X,ℓ∞)inL(X,ℓ∞), and she proved that if X is not a reflexive Banach space, then W(X,ℓ∞) is not com- plemented in L(X,ℓ∞). In the next Theorem, we establish the complementability of W(X,ℓ∞) in DPcc(X,ℓ∞). We need the following lemma of [7] . Lemma 2.12. Let X be a separable Banach space, and φ:ℓ∞ →L(X,ℓ∞) is a bounded linear operator with φ(e )=0 for all n, where e is the usual basis element in c . Then n n 0 there is an infinite subset M of N such that for each α ∈ ℓ∞(M), φ(α) = 0, where ℓ∞(M) is the set of all α=(α )∈ℓ∞ with α =0 for each n∈/ M. n n Theorem 2.13. If X does not have the L-Dunford-Pettis property, then W(X,ℓ∞) is not complemented in DPcc(X,ℓ∞). Proof. ConsiderasubsetAofX′ thatis L-Dunford-Pettisbutitis notrelativelyweakly compact. So there is a sequence (f ) in A such that has no weakly convergent subse- n quence. Hence S : X → ℓ∞ defined by S(x) = (f (x)) is an DPcc operator but it is n not weakly compact. Choose a bounded sequence (x ) in B such that (S(x )) has no n X n weakly convergent subsequence. Let X = hx i, the closed linear span of the sequence 1 n (x ) in X. It follows that X is a separable subspace of X such that S/X is not a n 1 1 weakly compact operator. If g = f /X , we have (g ) ⊆ X′ is bounded and has no n n 1 n 1 weakly convergent subsequence. Now define the operator T : ℓ∞ → DPcc(X,ℓ∞) by T(α)(x) = (α f (x)), where n n x∈X and α=(α )∈ℓ∞. Then n kT(α)(x)k=sup |α f (x)|≤kαk.kf k.kxk<∞. n n n n We claim that T(α)∈DPcc(X,ℓ∞) for each α=(α )∈ℓ∞. n Let α = (α ) ∈ ℓ∞ and let (x ) be a weakly null sequence, which is a DP set in X. n m As (f ) is L-Dunford-Pettis set sup |f (x )|→0 as m→∞. So we have n n n m kT(α)(x )k=sup |α f (x )|≤kαk.sup |f (x )|→0, m n n n m n n m as m → ∞. Then this finishes the proof that T is a well-defined operator from ℓ∞ into DPcc(X,ℓ∞). Let R:DPcc(X,ℓ∞)→DPcc(X ,ℓ∞) be the restriction map and define 1 φ:ℓ∞ →DPcc(X ,ℓ∞) by φ=RT. 1 Now suppose that W(X,ℓ∞) is complemented in DPcc(X,ℓ∞) and P :DPcc(X,ℓ∞)→W(X,ℓ∞) is a projection. Define ψ : ℓ∞ →W(X ,ℓ∞) by ψ = RPT. Note that as T(e ) is a one 1 n rank operator, we have T(e )∈W(X,ℓ∞). Hence n ψ(e )=RPT(e )=RT(e )=φ(e ) n n n n 392 A.RETBIANDB.ELWAHBI for all n∈N. From Lemma 2.12, there is an infinite set M ⊆N such that ψ(α)=φ(α) for all α∈ℓ∞(M). Thus φ(χ ) is a weakly compactoperator. Onthe other hand, if e′ M n is the usual basis element of ℓ1, for each x∈X and each n∈M, we have 1 (φ(χ ))′(e′ )(x)=f (x). M n n Therefore (φ(χ ))′(e′ ) = f /X = g for all n ∈ M. Thus (φ(χ ))′ is not a weakly M n n 1 n M compact operator and neither is φ(χ ). This contradiction ends the proof. (cid:3) M As a consequence of Theorem 2.7 and Theorem 2.13, we obtain the following result. Corollary 2.14. LetX beaBanach space. Then thefollowing assertions areequivalent: (1) X has the L-Dunford-Pettis property, (2) W(X,ℓ∞)=DPcc(X,ℓ∞), (3) W(X,ℓ∞) is complemented in DPcc(X,ℓ∞). References 1. C.D.AliprantisandO.Burkinshaw,Positive operators, Springer,Dordrecht,2006. 2. M.BahreiniEsfahani,Complemented subspacesof bounded linearoperators,Ph.D.thesis,Uni- versityofNorthTexas,2003. 3. P.Cembranos,C(K,E)containsacomplementedcopyofc0,Proc.Amer.Math.Soc.91(1984), no.4,556–558. 4. G.Emmanuele,AdualcharacterizationofBanachspacesnotcontainingl1,Bull.PolishAcad. Sci.Math.34(1986), no.3-4,155–160. 5. G.Emmanuele,BanachspacesinwhichDunford-Pettissetsarerelativelycompact,Arch.Math. 58(1992), no.5,477–485. 6. I. Ghenciu and P. Lewis, The Dunford-Pettis property, the Gelfand-Phillips property, and L- sets,Colloq.Math.106(2006), no.2,311–324. 7. 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