International Journal of Analysis and Applications Volume 16, Number 2 (2018), 149-161 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-149 L-DUNFORD–PETTIS AND ALMOST L-DUNFORD–PETTIS SETS IN DUAL BANACH LATTICES HALIMEH ARDAKANI1∗ AND MANIJEH SALIMI2 1Department of Mathematics, Payame Noor University, Iran 2Department of Mathematics, Farhangian University, Iran ∗Corresponding author: halimeh ardakani@yahoo.com Abstract. Following the concept of L–limited sets in dual Banach spaces introduced by Salimi and Mosh- taghioun, we introduce the concepts of L–Dunford–Pettis and almost L–Dunford–Pettis sets in dual Banach lattices and then by a class of operators on Banach lattices, so called disjoint Dunford–Pettis completely continuous operators, we characterize Banach lattices in which almost L–Dunford–Pettis subsets of their dual, coincide with L–Dunford–Pettis sets. 1. Introduction A subset A of a Banach space X is called limited (resp. Dunford–Pettis (DP)), if every weak∗ null (resp. weak null) sequence (x∗n) in X ∗ converges uniformly on A, that is lim n→∞ sup a∈A |〈a,x∗n〉| = 0. Also if A ⊆ X∗ and every weak null sequence (xn) in X converges uniformly on A, we say that A is an L–set. Every relatively compact subset of E is DP. If every DP subset of a Banach space X is relatively compact, then X has the relatively compact DP property (abb. DPrcP). For example, dual Banach spaces with the weak Radon-Nikodym property (see [11], in short WRNP) and Schur spaces (i.e., weak and norm Received 2017-10-15; accepted 2017-12-16; published 2018-03-07. 2010 Mathematics Subject Classification. Primary 46A40; Secondary 46B40, 46B42. Key words and phrases. Dunford–Pettis set; relatively compact Dunford–Pettis property; Dunford–Pettis completely con- tinuous operator. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 149 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-149 Int. J. Anal. Appl. 16 (2) (2018) 150 convergence of sequences in X coincide) have the DPrcP [6]. Also we recall that a Banach space X has the DPrcP if and only if every DP and weakly null sequence (xn) in X is norm null. Recently, the authors in [14], introduced the class of Dunford–Pettis completely continuous (abb. DPcc) operators on Banach spaces. In fact, a bounded linear operator T : X → Y between two Banach spaces is DPcc if it carries DP and weakly null sequences in X to norm null ones in Y . The class of all DPcc operators from X to Y is denoted by DPcc(X,Y ). In this article, by the definition of L–limited sets in [12] in dual Banach spaces, we introduce the concepts of L–DP and almost L–DP sets in Banach lattices and then we obtain Banach lattices in which two classes of sets coincide. Finally by introducing the concept of disjoint DP completely continuous (abb. DPdcc) operators between Banach lattices and positive DPrcP, we obtain some characterizations of them and then the relation between the positive DPrcP of E and DP dcc operators on E is treated. The class of all DPdcc operators from E to Y is denoted by DPdcc(E,Y ). Here, we remember some definitions and terminologies from Banach lattice theory. It is evident that if E is a Banach lattice, then its dual E∗, endowed with the dual norm and pointwise order, is also a Banach lattice. The norm ‖.‖ of a Banach lattice E is order continuous if for each generalized net (xα) such that xα ↓ 0 in E, (xα) converges to 0 for the norm ‖.‖, where the notation xα ↓ 0 means that the net (xα) is decreasing, its infimum exists and inf(xα) = 0. A Banach lattice E is said to be σ–Dedekind complete if every countable subset of E that is bounded above has a supremum. A subset A of E is called solid if |x| ≤ |y| for some y ∈ A implies that x ∈ A and the solid hull of A is the smallest solid set containing A and is exactly the set Sol(A) = {y ∈ E : |y| ≤ |x|, for some x ∈ A}. Throughout this article, X and Y denote the arbitrary Banach spaces and X∗ refers to the dual of the Banach space X. We use Lw∗(X ∗,Y ) for Banach spaces of all bounded weak∗-weak continuous operators from X∗ to Y . Also E and F denote arbitrary Banach lattices and E+ = {x ∈ E : x ≥ 0} refers to the positive cone of the Banach lattice E. BE is the closed unit ball of E. If x is an element of a Banach lattice E, then absolute value of x is denoted by |x|. If a,b belong to E and a ≤ b, the interval [a,b] is the set of all x ∈ E such that a ≤ x ≤ b. A subset of a Banach lattice is called order bounded if it is contained in an order interval. A linear mapping T from E into F is called order bounded if it carries order bounded subsets of E into order bounded sets. We recall from [10] that, an element x belonging to a Riesz space E is discrete, if x > 0 and |y| ≤ x implies y = tx for some real number t. If every order interval [0,y] in E contains a discrete element, then E is said to be a discrete Riesz space. The lattice operations in the Banach lattice E are weakly sequentially continuous if for every weakly null sequence (xn) in E, |xn|→ 0 for σ(E,E∗). We refer the reader for undefined terminologies, to the classical references [1], [2], [10]. Int. J. Anal. Appl. 16 (2) (2018) 151 2. (L)-Dunford–Pettis sets in Banach lattices Following the introducing of the concept L–limited sets in [12], we define L–DP sets and we give some properties of them in Banach spaces and specially in Banach lattices. A norm bounded subset B of a dual Banach space X∗ is said to be an L–limited set if every weakly null and limited sequence (xn) of X converges uniformly to zero on the set B, that is supf∈B |f(xn)|→ 0. Definition 2.1. A norm bounded subset B of a dual Banach space X∗ is said to be an L–DP set if every weakly null and DP sequence (xn) of X converges uniformly to zero on the set B, that is supf∈B |f(xn)|→ 0. It is clear that every L-set set in X∗ is L–DP and every subset of an L–DP set is the same. Also, it is evident that every L–DP set is weak∗ bounded and so is bounded. Similar to [12, Theorem 2.2], we obtain: (a) absolutely closed convex hull of an L–DP set is an L–DP set, (b) relatively weakly compact subsets of dual Banach spaces are L–DP set, (c) every weak∗ null sequence in dual Banach space is an L–DP set. Note that the converse of assertion (b) in general, is false. In fact, the following theorem 2.1, shows that the closed unit ball of `∞ is an L–DP set. Theorem 2.1. A Banach space X has the DPrcP iff every bounded subset of X ∗ is an L–DP set. Proof. Since the Banach space X has the DPrcP iff every DP and weakly null sequence (xn) in X is norm null [14], the proof is clear. � The following Theorem 2.2, gives a necessary and sufficient condition for Banach spaces that L–sets and L–DP sets in its dual coincide. We recall that an operator T : X → Y between two Banach spaces is completely continuous, if T carries weakly null sequences in X to norm null ones, and the class of completely continuous operators is denoted by Cc(X,Y ). Theorem 2.2. A Banach space X has the DP property iff each L–DP set in X∗ is an L–set. Proof. Suppose X has the DP property. Since every weakly null sequence in X is DP so every L–DP set in X∗ is an L-set. Conversely, it is enough to show that for each Banach space Y , Cc(X,Y ) = DPcc(X,Y ) [14, Theorem 1.5]. If T : X → Y is DPcc, it is clear that T∗(BY∗) is an L–DP set. So by hypothesis, it is an L–set and we know that the operator T : X → Y is completely continuous iff T∗(BY∗) is an L–set. � Corollary 2.1. A Banach space with the DPrcP has the DP property if and only if it has the Schur property. Proof. It is clear that the Banach space X has the Schur property if and only if every bounded subset of X∗ is L–set. Now, if X has the DP property and DPrcP, then by Theorem 2.1, unit ball X ∗ is L–DP and so it is an L–set. The converse of the assertion is also clear. � Int. J. Anal. Appl. 16 (2) (2018) 152 Theorem 2.3. Let A be an L–DP subset of a dual Banach lattice E∗ and E has the weakly sequentially continuous lattice operations. Then |A| = {|a| : a ∈ A} is an L– DP set. Proof. We show that every weakly null and DP sequence (xn) in E converges uniformly on |A|, that is, limn→∞ supx∗∈A |〈xn, |x∗|〉| = 0. From [10, Lemma 1.4.4], 〈|xn|, |x∗|〉 = max{〈zn,x∗〉 : |zn| ≤ |xn|} for all n. So, there exists zn ∈ E, such that |zn| ≤ |xn| and 〈|xn|, |x∗|〉 = 〈zn,x∗〉. Since E has the weakly sequentially continuous lattice operations, the sequences (|xn|) and so (zn) are weakly null. Since the set A is L–DP, supx∗∈A |〈zn,x∗〉| → 0. From supx∗∈A|〈xn, |x∗|〉| ≤ supx∗∈A〈|xn|, |x∗|〉, we have supx∗∈A |〈xn, |x∗|〉|→ 0 and then the set |A| is L–DP. � Definition 2.2. A Banach space X has the L–DP property, if every L–DP subset of X∗ is relatively weakly compact. Theorem 2.4. For a Banach space X, the following are equivalent: (a) X has the L–DP property, (b) For each Banach space Y , DPcc(E,Y ) = W(E,Y ), (c) DPcc(X,`∞) = W(X,`∞). Proof. (a) ⇒ (b). Suppose that X has the L-DP property and T : X → Y is DPcc. Thus T∗(BY∗) is an L–DP set. So by hypothesis, it is relatively weakly compact and T is a weakly compact operator. (b) ⇒ (c). It is obvious. (c) ⇒ (a). If X does not have the L–DP property, there exists an L–DP subset A of X∗ that is not relatively weakly compact. So there is a sequence (xn) ⊂ A with no weakly convergent subsequence. Now we show that the operator T : X → `∞ by Tx = (〈x,x∗n〉) , x ∈ X is DPcc, but it is not weakly compact. As (x∗n) ⊂ A is an L–DP set, for every weakly null and DP sequence (xm) in X we have ‖Txm‖ = sup n |〈xm,x∗n〉|→ 0 thus T is a DPcc operator. It is easy to see that T∗(e∗n) = x ∗ n , n ∈ N Thus T∗ is not a weakly compact operator and neithe is T . This finishes the proof. � The classical Banach lattices `p, where 1 ≤ p < ∞ and Schur spaces are discrete KB-space and so they have the DPrcP [3]. The following corollary shows that the classical Banach lattices `p, where 1 < p < ∞ have the L–DP property. Int. J. Anal. Appl. 16 (2) (2018) 153 Corollary 2.2. A Banach space with the DPrcP has the L–DP property if and only if it is reflexive. Proof. If a Banach space X has the DPrcP, then by [14], the identity operator on X is DPcc and so is weakly compact, thanks to the L–DP property of X. Hence X is reflexive. � Recall that a Banach space X is said to have the reciprocal DP property (abb. RDP) if every completely continuous operator on X is weakly compact [7]. Theorem 2.5. If a Banach space X has the L–DP property, then it has the RDP. Proof. For arbitrary Banach space Y , let T : X → Y be a completely continuous operator. Thus it is DPcc and so by Theorem 2.4, T is weakly compact. Hence X has the RDP property. � In the following, we show that the L–DP property is carried by every complemented subspace. Theorem 2.6. If a Banach space X has the L–DP property, then every complemented subspace of X has the L–DP property. Proof. Consider a complemented subspace Y of X and a projection map P : X → Y . Suppose P : Y → `∞ is a DPcc operator, so TP : X → `∞ is also DPcc. Since X has the L–DP property, by Theorem 2.4, TP is weakly compact. Hence T is weakly compact. � The following evident proposition gives a characterization of the L–DP property property by L–DP setes. Proposition 2.1. Let X be a Banach space. Then the following are equivalent: (a) X has the L–DP property, (b) Every L–DP sequence in X∗ is relatively weakly compact. Theorem 2.7. Let E be a Banach lattice with the L–DP property. Then for each f ∈ (E∗)+, [−f,f] is an L–DP set. Proof. It is evident that every L-set in E∗ is an L–DP set. If E is a Banach lattice E with the L–DP property, then every L–set in E∗ is relatively weakly compact and so by [?, Theorem 3.1], E∗ has an order continuous norm. Hence by [2, Theorem 4.9], for each f ∈ (E∗)+, [−f,f] is relatively weakly compact and so it is an L–DP set. � In the rest of this section by using some techniques to those in [4], we investigate additional properties of L–DP sets. Proposition 2.2. Let X be a Banach space and B be a bounded subset of X∗. Then the following are equivalent: Int. J. Anal. Appl. 16 (2) (2018) 154 (a) B is an L–DP set, (b) For each sequence (fn) in B, fn(xn) → 0, for every weakly null and DP sequence (xn) of X. Proof. (a) ⇒ (b). This is from the inequality |fn(xn)| ≤ supf∈B |f(xn)| for each sequence (fn) in B and for every weakly null and DP sequence (xn) of X. (b) ⇒ (a). Assume that B is not an (L) DP set in X∗. Then there exsits an � > 0 and a weakly null and DP sequence (xn) in X such that supf∈B |f(xn)| > � for all n. This implies the existence of a sequence fn in B such that |fn(xn)| > �, for all n. � As in the previous proposition 2.2, we can easily conclude that, for a norm bounded sequence (fn) of X ∗, the subset {fn : n ∈ N} is an L–DP set iff fn(xn) → 0, for every weakly null and DP sequence (xn) of X. Proposition 2.3. Let T be an operator from a Banach space X into a Banach lattice E and f ∈ (E∗)+. Then the following are equivalent: (a) T∗[−f,f] is an L–DP set, (b) For every weakly null and DP sequence (xn) of X, f(|T(xn)|) → 0. Proof. It follows immediately from the equality f(|T(xn)|) = supg∈T∗[−f,f] |g(xn)|. � By taking T = IdE in Proposition 2.3, for each f ∈ (E∗)+, [−f,f] is an L–DP set iff for every weakly null and DP sequence (xn) of E, (|xn|) is weakly null. The next main result, gives an equivalent condition to T∗(B) be an L–DP set, where B is a norm bounded solid subset of E∗ and T is an operator from a Banach space X into a Banach lattice E. Recall that a sequence (xn) in a Banach lattice E is (pairwise) disjoint, if for each i 6= j, |xi|∧ |xj| = 0. Theorem 2.8. Let T be an operator from a Banach space X into a Banach lattice E and B be a norm bounded solid subset of E∗. Then the following are equivalent: (a) T∗(B) is an L–DP set in X∗, (b) T∗[−f,f] and {T∗fn : n ∈ N} are L–DP sets, for each f ∈ B+ and for each norm bounded disjoint sequence (fn) ∈ B+. Proof. The proof is similar to [4, Theorem 2.7]. � By taking T = IdE in Theorem 2.8, we obtain a norm bounded solid subset B of E ∗ is an L–DP set iff [−f,f] and {fn : n ∈ N} are L–DP sets, for each f ∈ B+ and for each disjoint sequence (fn) ∈ B+. The next result characterizes DPcc operators by L–DP sets. Theorem 2.9. For an operator T from a Banach space X into a Banach lattice E, the following are equivalent: Int. J. Anal. Appl. 16 (2) (2018) 155 (a) T is DPcc, (b) T∗(BE∗) is an L– DP set, where BE∗ is the closed unit ball of E ∗, (c) T∗[−f,f] and {T∗fn : n ∈ N} are L–DP sets, for each f ∈ (BE∗)+ and for each norm bounded disjoint sequence (fn) ∈ (BE∗)+, (d) |T(xn)|→ 0 for σ(E,E∗) and fn(Txn) → 0, for every weakly null and DP sequence (xn) in X and for each disjoint sequence (fn) in (BE∗) +. Proof. (a) ⇔ (b). By the equality supf∈T∗(BE∗) |f(xn)| = ‖Txn‖E, T ∗(BE∗) is an L–DP set in X ∗, if and only if, T is a DPcc operator. By Theorem 2.8, the statements (b) and (c) are equivalent and the equivalence (c) ⇔ (d) is a direct consequence of Proposition 2.3. � 3. Almost L-DP sets in Banach lattices In this section we introduce a new class of sets and operators. Definition 3.1. Let E be a Banach lattice and X be a Banach space. Then (a) A norm bounded subset B of a dual Banach lattice E∗ is said to be an almost L–DP set if every disjoint weakly null and DP sequence (xn) of E converges uniformly to zero on the set B, that is supf∈B |f(xn)|→ 0. (b) An operator T from a Banach lattice E into a Banach space X is a disjoint DP completely continuous (abb. DPdcc) operator if the sequence (‖Txn‖) converges to zero for every disjoint weakly null and DP sequence in E. Note that every L–DP set of a dual Banach lattice, is an almost L–DP set, but the converse is false, in general. In fact for many Banach lattices E with the positive DPrcP and without the DPrcP, the closed unit ball of the dual Banach lattice E∗ is an almost L–DP set, but it is not L–DP set. As an example, the closed unit ball B`∞ of `∞ is an almost L–DP set in `∞, but the closed unit ball B(`∞)∗ is not an almost L–DP set in (`∞) ∗. In the following, we give a useful chracterization of almost L-DP sets, that is proved by the method of Proposition 2.2. As we mentioned at the end of the previous section, we use some techniques to those in [4]. Proposition 3.1. Let E be a Banach lattice and B be a norm bounded set in E∗. Then the following are equivalent: (a) B is an almost L–DP set, (b) For each sequence (fn) in B, fn(xn) → 0, for every disjoint weakly null and DP sequence (xn) of E. Int. J. Anal. Appl. 16 (2) (2018) 156 In particular, we obtain: Proposition 3.2. Let E be a Banach lattice and (fn) be a norm bounded sequence in E ∗. Then the following are equivalent: (a) The subset {fn : n ∈ N} is an almost L–DP set, (b) fn(xn) → 0, for every disjoint weakly null and DP sequence (xn) of E. Similar to [4], [−f,f] is an almost L–DP set in E∗, for each f ∈ (E∗)+. Also for an order bounded operator from a Banach lattice E into a Banach lattice F , T∗([−f,f]) is an almost L–DP set, for each f ∈ (F∗)+. Theorem 3.1. Let T be an order bounded operator from a Banach lattice E into a Banach lattice F and B be a norm bounded solid subset of F∗. Then the following are equivalent: (a) T∗(B) is an almost L–DP set in E∗, (b) {T∗fn : n ∈ N} is an almost L–DP set, for each f ∈ B+ and for each disjoint sequence (fn) in B+. (c) fn(Txn) → 0, for every disjoint weakly null and DP sequence (xn) of E+ and for each disjoint sequence (fn) in B +. Proof. The proof is the same as the proof of Theorem 2.9. � By taking T = IdE in Theorem 3.1, we obtain a norm bounded solid subset B of E ∗ is an almost L–DP set iff {fn : n ∈ N} is an almost L–DP set for each disjoint sequence (fn) in B+. The next result characterizes the class of DPdcc operators by almost L– DP sets. Theorem 3.2. For an order bounded operator T from a Banach lattice E into another Banach lattice F , the following are equivalent: (a) T is DPdcc, (b) T∗(BF∗) is an almost L– DP set, where BF∗ is the closed unit ball of F ∗, (c) {T∗(fn) : n ∈ N} is an almost L–DP set for each disjoint sequence (fn) in (BF∗)+, (d) fn(T(xn)) → 0, for every disjoint weakly null and DP sequence (xn) of E+ and for each disjoint sequence (fn) in (BF∗) +. Proof. (a) ⇔ (b). By the equality, supf∈T∗(BF∗) |f(xn)| = ‖Txn‖F , for every sequence (xn) in E, it follows easily that, T∗(BF∗) is an almost L-limited set in E ∗ if and only if T is DPdcc. By Theorem 3.1, the statements (b) and (c) are equivalent and the equivalence (c) ⇔ (d) is a direct consequence of Proposition 3.2. � Int. J. Anal. Appl. 16 (2) (2018) 157 Now the concept of positive DPrcP in Banach lattices is introduced and Banach lattices with the positive DPrcP is characterized. Next we give some properties of DP dcc operators from an arbitrary Banach lattice E to another F , related to the positive DPrcP of the Banach lattice E. Definition 3.2. A Banach lattice E has the positive DPrcP if each weakly null and DP sequence with the positive terms in E is norm null. It is clear that the DPrcP implies the positive DPrcP, but the converse is false, in general. For example, L1[0, 1] has the positive DPrcP without the DPrcP. Theorem 3.3. For a Banach lattice E, the following are equivalent: (a) E has the positive DPrcP, (b) Every weakly null and disjoint DP sequence in E converges to zero in norm. Proof. (a) ⇒ (b). Let (xn) be a weakly null and disjoint DP sequence in E. From [15, Proposition 1.3], the sequence (|xn|) is weakly null and by [8, Lemma 3.7], it is DP in E. From (a), the sequence (|xn|) and so (xn) converges to zero in norm. (b) ⇒ (a). Suppose that infn‖xn‖ = c > 0 for some weakly null and DP sequence (xn) ⊂ E+. Putting yn = c −1xn and using [9, Corollary 5] we find a subsequence (ynk ), a constant d > 0, and a disjoint sequence (zk) of E + such that 0 < zk ≤ ynk and ‖zk‖≥ d. It is clear that disjoint DP sequence (zk) tends weakly to zero, but ‖zk‖≥ d. This fact contradicts the assumption. � Theorem 3.4. A Banach lattice E has the positive DPrcP iff every bounded set in E ∗ is an almost L–DP set. Proof. From Theorem 3.3, a Banach lattice E has the positive DPrcP iff every disjoint weakly null and DP sequence in E is norm null. � Theorem 3.5. Let E be a Banach lattice. Then the following are equivalent: (a) E has the positive DPrcP, (b) For each Banach space Y , DPdcc(E,Y ) = L(E,Y ), (c) DPdcc(E,`∞) = L(E,`∞). Proof. (a) ⇒ (b). If E has the positive DPrcP and (xn) is a weakly null and disjoint DP sequence in E, then by Theorem 3.3, (xn) is norm null and for each bounded operator T on E, ‖Txn‖ → 0; that is, DPdcc(E,F) = L(E,F). (b) ⇒ (c). It is obvious. (c) ⇒ (a). If E does not have the positive DPrcP, then by Theorem 3.3, there exists a weakly null and Int. J. Anal. Appl. 16 (2) (2018) 158 disjoint DP sequence (xn) in E such that ‖xn‖ = 1, for all n. Choose a normalized sequence (x∗n) in E∗ such that |〈xn,x∗n〉| = 1, for all n, and define the operator T : E → `∞ by Tx = (〈x,x∗n〉) , x ∈ E. But T is not DPdcc, since the sequence (xn) is weakly null and disjoint DP and ‖Txn‖≥ 1, for all n. � In the following Theorem 3.6, we show that the positive DPrcP and the DPrcP, coincide in the class of discrete Banach lattices. Let us start with the following lemma. Lemma 3.1. c0 dose not have the positive DPrcP. Proof. It is enough to remember that c0 dose not have the positive Schur property and use the fact that every weakly null sequence in c0 is DP. By [13], a Banach lattice has the positive Schur property, whenever 0 ≤ xn → 0 weakly implies ‖xn‖→ 0 � Now we are able to formulate the following equivalence condition. Theorem 3.6. Let E be a discrete Banach lattice. Then E has the positive DPrcP, if and only if, it has the DPrcP. Proof. Since the positive DPrcP is inherited by closed Riesz subspaces and c0 does not have the positive DPrcP, then E does not contain any order copy of c0. According to [10, Corollary 2.4.12], E is KB space, and so it possesses the DPrcP by [?]. � Corollary 3.1. The dual Banach lattice C(K)∗ has the positive DPrcP, where K is a compact Hausdorff space. Proof. For each positive and weakly null sequence (fn) in C(K) ∗, ‖fn‖ = fn(1K) → 0, where 1K denotes the constant function 1 on K. That is C(K)∗ has the positive DPrcP. On the other hands from [2], the Banach lattice C(K)∗ is discrete and by Theorem 3.6, it has the DPrcP. � Theorem 3.7. Let T : E → X from a Banach lattice E be an operator. Then the following are equivalent: (a) T is DPdcc, (b) the sequence (‖Txn‖) converges to zero for every weakly null and DP sequence in E+, (c) the sequence (‖Txn‖) converges to zero for every disjoint weakly null and DP sequence in E+. Proof. The proof is similar to [5, Theorem 2.2]. � Let M ⊂ L(X,Y ) be a Banach lattice. If M has the positive DPrcP, then by Theorem 3.5 all of the evaluation operators φx and ψy∗ are DP dcc operators, where φx(T) = Tx and ψy∗(T) = T ∗y∗ for x ∈ X, Int. J. Anal. Appl. 16 (2) (2018) 159 y∗ ∈ Y ∗ and T ∈M. Now, we show that the DPaccness of evaluation operators is a sufficient condition for the positive DPrcP of their domain. Theorem 3.8. Let Y has the Schur property and M⊂ L(X,Y ) be a Banach lattice. If for every y∗ ∈ Y ∗, the evaluation operator ψy∗ on M is DPdcc, then M has the positive DPrcP. Proof. If M does not have the positive DPrcP, by Theorem 3.3, there exists a weakly null and disjoint DP sequence (Tn) in M and some � > 0 such that ‖Tn‖ > �, for all n. So there exists a sequence (xn) in BX such that ‖Tn(xn)‖ > �, for all n. On the other hand, the evaluation operator ψy∗ on M is DPdcc for all y∗ ∈ Y ∗ and so ‖T∗n(y∗)‖ = ‖ψy∗(Tn)‖→ 0. Hence |〈Tnxn,y∗〉| ≤ ‖T∗n(y∗)‖→ 0. So the sequence (Tnxn) is weakly null and it is norm null by the Schur property, a fact that is impossible. � Theorem 3.9. Let X has the Schur property and M ⊂ Lw∗(X∗,Y ) be a Banach lattice. If for every x∗ ∈ X∗, the evaluation operator φx∗ on M is DPdcc, then M has the positive DPrcP. Proof. If M does not have the positive DPrcP, by Theorem 3.3, there exists a weakly null and disjoint DP sequence (Tn) in M and some � > 0 such that ‖Tn‖ > �, for all n. On the other hand, the evaluation operator φx∗ on M is DPdcc for all x∗ ∈ X∗ and so ‖Tn(x∗)‖ = ‖φx∗(Tn)‖ → 0. Since ‖T∗n‖ > �, there exists a sequence (y∗n) in BY∗ such that ‖T∗ny∗n‖ > �, for all n. But the Schur property of X shows that the weakly null sequence (T∗ny ∗ n) is norm null, which is a contradiction. � Two final theorems of this section, are a relationship between order weakly compact and M-weakly compact operators with a DPdcc operator. Recall that a continuous operator T : E → X from a Banach lattice E to a Banach space X is order weakly compact if and only if ‖Txn‖→ 0 for every disjoint order bounded sequence (xn) in E [2, Theorem 5.57]. Theorem 3.10. Evere DPdcc operator on a Banach lattice E is order weakly compact. Proof. Let (xn) be an order bounded disjoint sequence of E. It follows from [2] and [?] that (xn) is a weakly null and DP sequence. Since T is DPdcc then, ‖Txn‖→ 0. Hence T is order weakly compact. � An operator T : E → X from a Banach lattice to a Banach space is said to be M-weakly compact if ‖Txn‖→ 0 holds for every norm bounded disjoint sequence (xn) in E [10]. In [14], the authors proved that each DPcc operator from a Banach lattice E to a Banach space X is M-weakly compact when E∗ has an order continuous norm and E has the DP∗ property (that is, every relatively weakly compact set in E is limited). In fact, we have a similar conclusion about DPdcc operators. Theorem 3.11. Let E be a Banach lattice and X be a Banach space. If E∗ has an order continuous norm and E has the DP property, then each DPdcc operator T : E → X is M-weakly compact. Int. J. Anal. Appl. 16 (2) (2018) 160 Proof. Let T : M → X be a DPdcc operator and let (xn) be a bounded disjoint sequence in E. It follows from [10, Corollary 2.9] that (xn) is weakly null and so it is DP by the DP property of E. By our hypothesis on T , we have ‖Txn‖→ 0 and then T is M-weakly compact. � 4. Almost L–DP sets which are L–DP sets As we noted in the beginning of section 3, every L–DP set in the dual Banach lattice E∗, is an almost L–DP set, but the converse is false in general. In this section we characterize Banach lattices in which the class of almost L–DP sets and that of L–DP sets coincide in their dual. Theorem 4.1. For a Banach lattice E, the following are equivalent: (a) Each almost L–DP set in E∗ is an L–DP set, (b) For each Banach space Y , DPdcc(E,Y ) = DPcc(E,Y ), (c) DPdcc(E,`∞) = DPcc(E,`∞). Proof. (a) ⇒ (b). Let T : E → Y be an operator. By the equality sup f∈T∗(BY∗) |f(xn)| = ‖Txn‖Y , for every sequence (xn) in E, it follows easily that, T ∗(BY∗) is an almost L–DP (respectively, L–DP) set in E∗, if and only if, T is a DPdcc (respectively, DPcc) operator. Now, let T be a DPdcc operator. Then T∗(BY∗) is an almost L–DP set in E ∗ and from the hypothesis (a), it is an L–DP set in E∗. Hence T is a DPcc operator. (b) ⇒ (c). It is clear. (c) ⇒ (a). Let B be an almost L–DP set in E∗. To prove that B is an L–DP set, it sufficies to show that fn(xn) → 0 for each sequence (fn) in B and for every weakly null and DP sequence (xn) in E (see Proposition 2.2). Consider the operator S : E → `∞ defined by S(x) = (fn(x))∞n=1, for each x ∈ E. As B is almost L–DP, S is a DPdcc operator. In fact, for every weakly null and disjoint DP sequence (zi) in E, we have ‖Szi‖∞ = ‖fn(zi)∞n=0‖∞ ≤ sup f∈B |f(zi)|→ 0, as i → ∞. It follows that S is a DPdcc operator and so from our hypothesis, S is DPcc. So ‖Sxn‖∞ → 0 and the desired conclusion follows from the inequality |fn(xn)| ≤ ‖Sxn‖∞ for each n. � We recall that, an operator T from a Banach space X into a Banach lattice E is said to be semicompact if for each � > 0 there exists some u ∈ E+ satisfying T(BX) ⊂ [−u,u] + �BE. According to [4, Theorem 4.3], each operator T : E → X is DPdcc, whenever its adjoint T∗ : X∗ → E∗ is semicompact. Int. J. Anal. Appl. 16 (2) (2018) 161 At the end of this section, it should be noted that the adjoint of a DPdcc operator is not necessary DPdcc and vice versa. For example, the identity operator on the Banach lattice `1 is DP dcc (because `1 has the DPrcP, [14]) but its adjoint, Id`∞ : `∞ → `∞, is not DPdcc. In fact, if en = (0, 0, ..., 1, 0, ...) with n’th entry equals to 1 and all others zero, then (en) is an order bounded disjoint sequence of `∞. Hence (en) is weakly null and by [?], it is DP, but ‖Id`∞(en)‖ = ‖en‖∞ = 1 for all n. Also the identity operator on `∞ is not DPdcc but its adjoint is DPdcc, because (`∞) ∗ has DPrcP. Also by Theorem 3.2, B(`∞)∗ is not an almost L–DP set in (`∞) ∗, as noted that in the begining of section 3. References [1] C. D. Aliprantis and O. Burkishaw, Locally Solid Riesz Spaces, Academic Press, New York, London, 1978. [2] C. D. Aliprantis and O. Burkishaw, Positive Operators, Academic Press, New York, London, 1978. [3] B. Aqzzouz and K. 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