We establish necessary and sufficient conditions under which positive almost limited operators between Banach lattices have almost limited adjoints and positive operators with almost limited adjoints are themselves almost limited. Also, we give some consequences.peerReviewe

El Fahri, K.

H'michane, J.

Moussa, M.

Extracta Mathematicae

English

Dehesa. Repositorio Institucional de la Universidad de Extremadura

E extracta mathematicae Vol. 29, Núm. 1 - 2, 65 – 71 (2014)Duality of Almost Limited OperatorsJ. H’michane1,, K. El Fahri2,, M. Moussa2,1Université Moulay Ismail, Faculté des Sciences, Département de Mathématiques,B.P. 11201 Zitoune, Meknèse, Moroccohm1982jad@gmail.com2Université Ibn Tofail, Faculté des Sciences, Département de Mathématiques,B.P. 133, Kénitra, Moroccokamalelfahri@gmail.com, mohammed.moussa09@gmail.comPresented by Manuel González Received July 7, 2014Abstract : We establish necessary and sufficient conditions under which positive almost lim-ited operators between Banach lattices have almost limited adjoints and positive operatorswith almost limited adjoints are themselves almost limited. Also, we give some consequences.Key words: Almost limited operator, almost limited set, order continuous norm, dual posi-tive Schur property.AMS Subject Class. (2010): 46B42, 47B60, 47B65.1. IntroductionRecall from [5] that an operator T from a Banach space X into Banachlattice E is called almost limited if T (BX) is an almost limited set in E,equivalently, ∥T ′(fn)∥ −→ 0 for every disjoint weak* null sequence (fn) in E′.We denote by La−lim(X,E) the classe of almost limited operators.In this paper, we investigate the direct and reciprocal duality property ofalmost limited operators. We will prove that if E and F are Banach latticessuch that F is Dedekind σ−complete then, each positive almost limited op-erator T : E −→ F admits an almost limited adjoint T ′ if, and only if, E′has the dual positive Schur property or the norm of F is order continuous(Theorem 3.5). Next, we will establish that if E and F are Banach latticessuch that F has the property (d ) then, each positive operator T : E −→ Fis almost limited whenever its adjoint T ′ is almost limited if, and only if, thenorm of E′ is order continuous or F has the dual positive Schur property(Theorem 3.7).The article is organized as follows, after the introduction section, we give inpreliminaries section all notations and definitions of Banach lattice theory that6566 j. h’michane, k. el fahri, m. moussawe will need in this paper. In the main results section, we study in the firstsubsection the direct duality property for the class of almost limited operatorsand in the second subsection we study the reciprocal duality property of thatclass of operators.2. PreliminaryLet us recall from [2] that a norm bounded subset A of a Banach latticeE is said to be almost limited, if every disjoint weak* null sequence (fn) inE′ converges uniformly to zero on A. Clearly that all relatively compact setsand all limited sets in a Banach lattice are almost limited. The converse doesnot hold in general. For example, the closed unit ball of the Banach latticeBℓ∞ is almost limited, but it is neither relatively compact nor limited.To state our results, we need to fix some notations and recall some defi-nitions. A Banach lattice is a Banach space (E, ∥.∥) such that E is a vectorlattice and its norm satisfies the following property: for each x, y ∈ E suchthat |x| ≤ |y|, we have ∥x∥ ≤ ∥y∥. A norm ∥ · ∥ of a Banach lattice E is ordercontinuous if for each generalized sequence (xα) such that xα ↓ 0 in E, (xα)converges to 0 for the norm ∥ · ∥ where the notation xα ↓ 0 means that (xα)is decreasing, its infimum exists and inf(xα) = 0.Note that if E is a Banach lattice, its topological dual E′, endowed with thedual norm and the dual order, is also a Banach lattice. Also, a vector latticeE is Dedekind σ-complete if every majorized countable nonempty subset ofE has a supremum. We will use the term operator T : E −→ F to mean abounded linear mapping. It is positive if T (x) ≥ 0 in F whenever x ≥ 0 in E.Note that each positive linear mapping on a Banach lattice is continuous. Ifan operator T : E −→ F is positive then, its adjoint T ′ : F ′ −→ E′ is likewisepositive, where T ′ is defined by T ′(f)(x) = f(T (x)) for each f ∈ F ′ and foreach x ∈ E.A Banach lattice E has:- the property (d ) if, |fn| ∧ |fm| = 0 in E′ and fn −→ 0 in the σ(E′, E)-topology of E′ implies |fn| −→ 0 in the σ(E′, E)-topology of E′. It should benoted, by [7, Proposition 1.4], that if E is Dedekind σ-complete then, E hasproperty (d ). But the converse is false. In fact, the Banach lattice ℓ∞/c0 hasthe property (d ) but is not Dedekind σ-complete [7, Remark 1.5].- the dual positive Schur property if, ∥fn∥ → 0 for every weak* null se-quence (fn) ⊂ (E′)+, equivalently, ∥fn∥ → 0 for every weak* null sequence(fn) ⊂ (E′)+ consisting of pairwise disjoint terms [7, Proposition 2.3].duality of almost limited operators 67- the dual Schur property if, ∥fn∥ → 0 for every weak* null sequence(fn) ⊂ E′ consisting of pairwise disjoint terms [5, Definition 3.2].For terminologies concerning Banach lattice theory and positive operatorswe refer the reader to the excellent book of Aliprantis-Burkinshaw [1].3. Mains resultsFirstly, we give the following definitions,Definition 3.1. Let E and F be two Banach spaces and U(E,F ) a classof operators from E to F .• We shall say that the class U(E,F ) satisfies the direct duality property,if we have T ∈ U(E,F ) =⇒ T ′ ∈ U(F ′, E′).• By duality, we say that the class U(E,F ) satisfies the reciprocal dualityproperty, if we have T ′ ∈ U(F ′, E′) =⇒ T ∈ U(E,F ).3.1. Direct duality property for almost limited operators.Note that there exists an almost limited operator whose adjoint is not al-most limited. Indeed, the identity operator of the Banach lattice ℓ∞ is almostlimited (because ℓ∞ has the dual Schur property), but its adjoint, which is theidentity operator of the Banach lattice (ℓ∞)′, is not almost limited (because(ℓ∞)′ does not have the dual Schur property [7, Proposition 2.1]).To give our first main result we will need the following Lemmas,Lemma 3.2. [4, Proposition 2.1] An operator T : X −→ E from a Banachspace X into a Banach lattice E with the property (d ) is almost limited if,and only if, ∥T ′(fn)∥ −→ 0 for every weak* null sequence (fn) in E′ consistingof positive pairwise disjoint elements.Lemma 3.3. [4, Remark 2.4] Let E and F be two Banach lattice and letX be a Banach space. If E has the dual positive Schur property (for example,E = ℓ∞), F has the property (d ) and S is positive then, T = S ◦ R : X −→E −→ F is an almost limited operator.Lemma 3.4. A Banach lattice E does not have the dual positive Schurproperty if, and only if, there exists a positive disjoint weak* null sequence(fn) of E′, (yn) ⊂ B+E and some ε > 0 such that fn(yn) ≥ ε for all n.68 j. h’michane, k. el fahri, m. moussaProof. If E does not have the dual positive Schur property, we know thatthere exists a positive disjoint weak* null sequence (fn) of E′ which is notnorm convergent to 0. By passing to a subsequence if necessarily, we mayassume that there exists some ε > 0 with ∥fn∥ > ε for all n. As ∥fn∥ =sup{fn(y) : y ∈ B+F }, for each n there exist yn ∈ B+F such that fn(yn) ≥ ε.The converse is easy.In the following theorem we give necessary and sufficient conditions of Ba-nach lattices under which the adjoint of each positive almost limited operatorT : E −→ F is also almost limited.Theorem 3.5. Let E and F be two Banach lattices such that F isDedekind σ−complete. The following conditions are equivalent:1. For each positive operator T ∈La−lim(E,F ) we have T ′∈La−lim(F ′, E′);2. At least one of the following conditions is valid:(a) E′ has the dual positive Schur property;(b) The norm of F is order continuous.Proof. (1) =⇒ (2) Assume by way of contradiction that E′ does not havethe dual positive Schur property and the norm of F is not order continuous.We have to construct a positive operator T ∈ La−lim(E,F ) with an adjointT ′ /∈ La−lim(F ′, E′). Indeed, since the norm of F is not order continuous then,by [6, Corollary 2.4.3] we may assume that ℓ∞ is a closed sublattice of F . AsE′ does not have the dual positive Schur property, Lemma 3.4 implies thatthere exists a positive disjoint weak* null sequence (fn) of E′′, (gn) ⊂ B+E′and some ε > 0 such that fn(gn) ≥ ε for all n. Consider the positive operatorT : E −→ ℓ∞ ⊂ F defined byT (x) = (gn(x))∞n=1for all x ∈ E. By Lemma 3.3, T is an almost limited operator. But T ′ is notalmost limited. In fact, note thatT ′ : (ℓ∞)′ −→ E′,(λn)∞n=1 7→∞∑n=1λngnduality of almost limited operators 69and for every n, we have∥T ′′(fn)∥ = supx∈B(ℓ∞)′|T ′′(fn)(x)| = supx∈B(ℓ∞)′|fn(T ′(x))|≥ fn(T ′(en)) = fn(gn) ≥ ε,(where (en)∞n=1 is the canonical basis of ℓ1 ⊂ (ℓ∞)′). As (fn) is a disjointweak* null sequence of E′′, we conclude that T ′ is not almost limited.(2.a)=⇒(1) Obvious.(2.b)=⇒(1) Let T : E −→ F be a positive almost limited operator. Since E′has the property (d ) (because E′ is Dedekind σ-complete) then, by Lemma 3.2it suffices to show that ∥T ′′(fn)∥ −→ 0 for each positive disjoint weak* nullsequence (fn) ⊂ E′′. It is clear that 0 ≤ T ′′(fn) −→ 0 for σ(F ′′, F ′). By[3, Corollary 2.7], it suffices to show that T ′′(fn)(gn) −→ 0 for each disjointnorm bounded sequence (gn) ⊂ (F ′)+. As the norm of F is order continuous,it follows from [6, Corollary 2.4.3] that gn −→ 0 for σ(F ′, F ). Now, since Tis almost limited, we have ∥T ′(gn)∥ −→ 0. As fn −→ 0 for σ(E′′, E′), (fn) isnorm bounded. Hence T ′′(fn)(gn) = fn(T′(gn)) ≤ ∥fn∥∥T ′(gn)∥ −→ 0. Thiscomplete the proof.Corollary 3.6. Let E be a Dedekind σ−complete Banach lattice. Then,the following statements are equivalent:1. For each positive operator T ∈La−lim(E,E) we have T ′∈La−lim(E′, E′);2. The norm of E is order continuous.3.2. Reciprocal duality property for almost limited opera-tors. In this section, we characterize Banach lattice under which each pos-itive operator is almost limited whenever it’s adjoint is almost limited. Notethat there exist an operator which is not almost limited while its adjoint isalmost limited. Indeed, the identity operator of the Banach lattice ℓ1, is notalmost limited (because ℓ1 does not have the dual Schur property), howeverits adjoint, which is the identity operator of the Banach lattice ℓ∞, is almostlimited.Theorem 3.7. Let E and F be two Banach lattices such that F has theproperty (d ). Then, the following assertions are equivalent:1. For each positive operator T : E −→ F such that T ′ ∈ La−lim(F ′, E′)we have T ∈ La−lim(E,F );70 j. h’michane, k. el fahri, m. moussa2. One of the following is valid:(a) The norm of E′ is order continuous;(b) F has the dual positive Schur property.Proof. Assume by way of contradiction that the norm of E′ is not or-der continuous and F does not have the dual positive Schur property. Wehave to construct a positive operator T /∈ La−lim(E,F ) but its adjoint T ′ ∈La−lim(F′, E′).Indeed, since the the norm of E′ is not order continuous, it follows fromTheorem 2.4.14 and Proposition 2.3.11 in [6] that E contains a sub-latticeisomorphic to ℓ1 and there exists a positive projection P : E −→ ℓ1. On theother hand, since F does not have the dual positive Schur property, Lemma 3.4implies that there exists a positive disjoint weak* null sequence (fn) of F′,(yn) ⊂ B+F and some ε > 0 such that fn(yn) ≥ ε for all n. Consider the positiveoperator S : ℓ1 −→ F defined by S((αi)) =∑∞i=1 αiyi for each (αi) ∈ ℓ1.Now, we consider the positive operator T = S ◦ P : E −→ ℓ1 −→ F . ByLemma 3.4, T ′ : F ′ −→ ℓ∞ −→ E′ is almost limited. But T is not almostlimited. Indeed, the sequence (fn) is a positive disjoint weak* null sequencein F ′. As the operator P : E −→ ℓ1 is surjective, there exists δ > 0 such thatδBℓ1 ⊂ P (BE). Hence∥T ′(fn)∥ = supx∈BE|T ′(fn)(x)| = supx∈BE|fn(T (x))|= supx∈BE|fn ◦ S(P (x))| ≥ δ |fn ◦ S(en)| ≥ δ |fn(yn)| ≥ δ ε(where (en)∞n=1 is the canonical basis of ℓ1). Then ∥T ′(fn)∥ > δ ε for all n.On the other hand, since (fn) is a positive disjoint weak* null sequence in F′,we conclude that T is not almost limited.(2.b)=⇒(1) Obvious.(2.a)=⇒(1) Let T : E −→ F be a positive operator such that its adjoint T ′ isalmost limited. Since F has the property (d ) then, by Lemma 3.2 it sufficesto show that ∥T ′(fn)∥ −→ 0 for each positive disjoint weak* null sequence(fn) ⊂ F ′.It is clear that 0 ≤ T ′(fn) −→ 0 for σ(E′, E). By [3, Corollary 2.7], itsuffices to show that T ′(fn)(xn) −→ 0 for each disjoint norm bounded se-quence (xn) ⊂ E+. As the norm of E′ is order continuous, it follows from[6, Theorem 2.4.14] that xn −→ 0 for σ(E,E′). As the canonical embed-ding τ : E −→ E′′ is a lattice homomorphism then, (τ(xn)) is a positiveduality of almost limited operators 71disjoint weak* null sequence in E′′. Now, since T ′ is almost limited, we ob-tain ∥T ′′(τ(xn))∥ −→ 0. As fn −→ 0 for σ(F ′, F ), (fn) is norm bounded andT ′(fn)(xn) = τ(xn)(T′(fn)) = T′′(τ(xn))(fn) ≤ ∥T ′′(τ(xn))∥∥fn∥ −→ 0 andthis proves that T is almost limited.Corollary 3.8. Let E be a Banach lattice with the property (d ). Thenthe following statements are equivalent:1. For each positive operator T : E −→ E such that T ′ ∈ La−lim(E′, E′)we have T ∈ La−lim(E,E);2. The norm of E′ is order continuous.References[1] C.D. Aliprantis, O. Burkinshaw, “Positive Operators”, Reprint of the1985 original, Springer, Dordrecht, 2006.[2] J.X. Chen, Z.L. Chen, G.X. Ji, Almost limited sets in Banach lattices, J.Math. Anal. Appl. 412 (1) (2014), 547 – 553.[3] P.G. Dodds, D.H. Fremlin, Compact operators in Banach lattices, IsraelJ. Math. 34 (4) (1979), 287 – 320.[4] A. Elbour, N. Machrafi, M. Moussa, Weak compactness of almostlimited operators, http://arxiv.org/abs/1403.3348.[5] N. Machrafi, A. Elbour, M. Moussa, Some characterizations of almostlimited sets and applications, http://arxiv.org/abs/1312.2770.[6] P. Meyer-Nieberg, “Banach Lattices”, Universitext, Springer-Verlag, Berlin,1991.[7] W. Wnuk, On the dual positive Schur property in Banach lattices, Positivity17 (3) (2013), 759 -û 773.

Duality of almost limited operators

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Duality of almost limited operators

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