International Journal of Analysis and Applications Volume 16, Number 1 (2018), 50-61 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-50 ON THE LIMITED p-SCHUR PROPERTY OF SOME OPERATOR SPACES M.B. DEHGHANI1, S.M. MOSHTAGHIOUN1,∗ AND M. DEHGHANI2 1Department of Mathematics, yazd University, P. O. Box 89195-741, Yazd, Iran 2Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, P. O. Box 87317-53153, Kashan, Iran ∗Corresponding author: moshtagh@yazd.ac.ir Abstract. We introduce and study the notion of limited p-Schur property (1 ≤ p ≤∞) of Banach spaces. Also, we establish some necessary and sufficient conditions under which some operator spaces have the limited p-Schur property. In particular, we prove that if X and Y are two Banach spaces such that X contains no copy of `1 and Y has the limited p-Schur property, then K(X, Y ) (the space of all compact operators from X into Y ) has the limited p-Schur property. 1. Introduction A non-empty subset K of a Banach space X is said to be limited (resp Dunford-Pettis (DP)), if for every weak∗-null (resp. weakly null) sequence (x∗n) in the dual space X ∗ of X converges uniformly on K, that is, lim n→∞ sup x∈K |〈x,x∗n〉| = 0 where 〈x,x∗〉 denotes the duality between x ∈ X and x∗ ∈ X∗. In particular, a sequence (xn) ⊂ X is limited if and only if 〈xn,x∗n〉→ 0, for all weak∗-null sequences (x∗n) in X∗. Received 11th September, 2017; accepted 27th November, 2017; published 3rd January, 2018. 2010 Mathematics Subject Classification. 47L05; 46B25. Key words and phrases. Schur property; p-Schur property; limited p-Schur property; limited p-converging; weakly p-compact. 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. 50 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-50 Int. J. Anal. Appl. 16 (1) (2018) 51 A subset K of a Banach space X is a limited set if and only if for any Banach space Y , every pointwise convergent sequence (Tn) ⊂ L(X,Y ) converges uniformly on K, where L(X,Y ) denoted the space of all bounded operators from X into Y [17, Corollary 1.1.2]. It is easily seen that every relatively compact subset of a Banach space is limited. But the converse is not true, in general. If every limited subset of Banach space X is relatively compact, then X has the Gelfand-Phillips (GP) property. For example, the classical Banach space c0 and `1 have the GP property and every reflexive space and dual space containing no copy of `1 have the same property. A sequence (xn) in Banach space X is called weakly p-summable with 1 ≤ p < ∞, if for each x∗ ∈ X∗, the sequence (〈xn,x∗〉) ∈ `p and a sequence (xn) in X is said to be weakly p-convergent to x ∈ X if the sequence (xn − x) ∈ `weakp (X), where `weakp (X) denoted the space of all weakly p-summable sequence in X. Also a bounded set K in a Banach space is said to be relatively weakly p-compact, 1 ≤ p ≤ ∞ if every sequence in K has a weakly p-convergent subsequence. If the limit point of each weakly p-convergent subsequence is in K, then we call K weakly p-compact set. Also, a Banach space X is weakly p-compact if the closed unit ball BX of X is a weakly p-compact set. An operator T ∈ L(X,Y ) is said to be p-converging if it transfers weakly p-summable sequence into norm null sequences. The class of all p-converging operators from X into Y is denoted by Cp(X,Y ). An operator T ∈ L(X,Y ) is limited p-converging if it transfers limited and weakly p-summable sequences into norm null sequences. we denote the space of all limited p-converging operators from X into Y by Clp(X,Y ) [7]. A Banach space X has the Schur property if every weakly null sequence in X converges in norm. The simplest Banach space with the Schur property is `1. Also a banach space X has the p-Schur property (1 ≤ p ≤∞) if every weakly p-summable subset of X is compact. In other words, if 1 ≤ p < ∞, X has the p-Schur property if and only if every sequence (xn) ∈ `weakp (X) is a norm null sequence, for example, `p has the 1-Schur property. Moreover, X has the ∞-Schur property if and only if every sequence in cweak0 (X) in norm null where cweak0 (X) containing all weakly null sequences in X. So ∞-Schur property coincides with the Schur property. Also one note that every Schur space has the p-Schur property [6]. The reader is referred to [2, 11, 14–16] for more information about these concepts. In this note, we study the limited p-Schur property of some operator spaces, specially, the space of compact operators. We prove that if X and Y are two Banach spaces such that X contains no copy of `1 and Y has the limited p-Schur property, then K(X,Y ) has the limited p-Schur property. Finally, we conclude that if (Xα)α∈I are Banach spaces and X = (⊕α∈IXα)1 their `1-sum, then the space X has the p-Schur property if and only if each factor Xα has the same property. Int. J. Anal. Appl. 16 (1) (2018) 52 2. Main results Recall that the Banach space X has the limited p-Schur property if every limited weakly p-compact subset of X is relatively compact. More precisely, the Banach space X has the limited p-Schur property if and only if every limited sequence (xn) ∈ `weakp (X) is norm null. It is easy to see that every Banach space with the p-Schur property and every Banach space with GP property is limited p-Schur [7]. Moreover, a Banach space X has the GP property if and only if every limited weakly null sequence in X is norm null [2, Proposition 6.8]. Therefore the limited Schur (i.e., limited ∞-Schur ) property is equivalent to the GP property. Also, if a Banach space X have the limited p-Schur and DP∗p properties, then it has the p-Schur property. Indeed, a Banach space X is said to have the DP∗-property of order p (DP∗p ) if all weakly p-compact sets in X are limited [10]. Recall that if M is a closed subspace of L(X,Y ), then for arbitrary elements x ∈ X and y∗ ∈ Y ∗, the evaluation operators φx : M → Y and ψy∗ : M → X∗ on M are defined by φx(T) = Tx, ψy∗ (T) = T ∗y∗, (T ∈ M). Theorem 2.1. Let X and Y be two Banach spaces such that X is weakly p-compact and Y has the p-Schur property. Then L(X,Y ) has the limited p-Schur property. Proof. Suppose that (Tn) is a limited weakly p-summable sequence in L(X,Y ). We have to prove that (Tn) is norm null. We first observe that for every x ∈ X the evaluation operator φx from L(X,Y ) to Y maps the sequence (Tn) to the sequence (Tnx). So the latter is also a limited weakly p-summable sequence in Y . Therefore ‖Tnx‖→ 0, since Y has the limited p-Schur property. Now, suppose that (Tn) is not norm null. Then there is a sequence (xn) in X and ε > 0 such that ‖Tnxn‖ > 2ε, for all n ∈ N. Since X is weakly p-compact we may assume that there exists x ∈ X such that (xn − x) ∈ `weakp (X). As ‖Tnx‖ → 0, we may finally suppose that fn = ‖Tnxn −Tnx‖ > ε for all n ∈ N. Now, choose functional y∗n in BY ∗ so that 〈Tnxn −Tnx,y∗n〉 = fn, and define Λn ∈ L(X,Y )∗ by 〈T, Λn〉 = 〈Txn −Tx,y∗n〉, for all T ∈ L(X,Y ). Since |〈T, Λn〉| ≤ ‖Txn −Tx‖ → 0, because (xn −x) ∈ `weakp (X), we see that (Λn) is a weak∗-null sequence. But 〈Tn, Λn〉 = fn > ε > 0 for all n ∈ N. Contradicting the assumption that (Tn) is limited. � Corollary 2.1. Let X and Y be two Banach spaces. If X is reflexive and Y has the Schur property, then L(X,Y ) has the GP property. Int. J. Anal. Appl. 16 (1) (2018) 53 Proof. Let p = ∞ in Theorem 2.1. � Corollary 2.2. Let X and Y be two Banach spaces. If X is a weakly p-compact and Y ∗ has the p-Schur property, then (X⊗̂πY )∗ has the limited p-Schur property. Proof. It follows easily from the fact that L(X,Y ∗) = (X⊗̂πY )∗. � Corollary 2.3. Let X and Y be two Banach spaces. If X∗ has the p-Schur property and Y ∗ is weakly p-compact, then L(X,Y ) has the limited p-Schur property. Proof. The mapping T 7→ T∗ maps L(X,Y ) onto a closed subspace of L(Y ∗,X∗), which has the limited p-Schur property by virtue of Theorem 2.1. � In the following theorem we give a necessary and sufficient condition for which a Banach space has the limited p-Schur property. Theorem 2.2. The Banach space X has the limited p-Schur property if and only if L(X,Y ) = Clp(X,Y ), for every Banach space Y . Proof. Suppose that X has the limited p-Schur property. If T ∈ L(X,Y ) and (xn) ∈ `weakp (X) is a limited sequence, then ‖xn‖→ 0. Hence ‖Txn‖→ 0. Conversely, If Y = X, then the identity operator on X is belongs to Clp. Therefore X has the limited p-Scuhr property. � Similarly, we can prove that the Banach space X has the limited p-Schur property if and only if L(Y,X) = Clp(Y,X) for every Banach space Y . Theorem 2.3. If X∗ has the limited p-Schur property and Y has the Schur property, then L(X,Y ) has the limited p-Schur property. Proof. Since X∗ has the limited p-Schur property, Theorem 2.2 implies that each ψy∗ : L(X,Y ) → X∗ is limited p-converging. It follows that L(X,Y ) has the limited p-Schur property. In fact, if L(X,Y ) does not have the limited p-Schur property, then there exists a limited weakly p-summable sequence (Tn) ⊆ L(X,Y ) such that ‖Tn‖ > ε for all n ∈ N and some ε > 0. Choose a sequence xn ∈ BX such that ‖Tnxn‖ > ε. On the other hand, ψy∗ is limited p-converging, for all y ∗ ∈ Y ∗. Therefore ‖T∗ny∗‖ = ‖ψy∗Tn‖ → 0. It follows that |〈Tnxn,y∗〉| ≤ ‖T∗ny ∗‖‖xn‖→ 0. Hence (Tnxn) is weakly null and so is norm null. This contradiction shows that L(X,Y ) has the limited p-Schur property. � Int. J. Anal. Appl. 16 (1) (2018) 54 Example 2.1. If X∗ has the limited p-Schur property, then `weak1 (X ∗) has the same property. Indeed, if one denote `weak ∗ 1 (X ∗) as the space of all sequences (x∗n) ⊂ X∗ such that (〈x,x∗n〉) ∈ `1, for all x ∈ X, then by [5, P. 427], `weak1 (X ∗) = `weak ∗ 1 (X ∗). Also, `weak ∗ 1 (X ∗) is isometrically isomorphism to L(X,`1); see e.g., [8, Proposition 19.4.3]. Since `1 has the Schur property, it follows that L(X,`1) = ` weak 1 (X ∗) has the limited p-Schur property. If we take p = ∞ in Theorem 2.3 we obtain the following result. Corollary 2.4. If X∗ has the GP property and Y has the Schur property, then L(X,Y ) has the GP property. Theorem 2.4. Let X and Y be Banach spaces. If X has the limited p-Schur property and Y has the GP property, then the space Kw∗ (X ∗,Y ) of all compact weak∗-weak continuous operators from X∗ into Y has the limited p-Schur property. Proof. Let (Tn) be a limited weakly p-summable sequence in Kw∗ (X ∗,Y ). We have to show that ‖Tn‖→ 0. We can choose a sequence (x∗n) in X ∗ such that ‖x∗n‖ = 1 and ‖Tnx∗n‖ ≥ 1 2 ‖Tn‖ for all n ∈ N. Now, we prove that (Tnx ∗ n) is weakly null limited sequence in Y . Fix any y ∗ ∈ Y ∗. Then for all T ∈ Kw∗ (X∗,Y ), the operator y∗◦T is a weak∗ continuous linear functional on X∗ so that y∗◦T ∈ X ⊂ X∗∗. Thus the operator T 7→ y∗ ◦T from Kw∗ (X∗,Y ) into X shows that the sequence (y∗ ◦Tn) is limited weakly p-summable in X. So ‖y∗ ◦Tn‖→ 0 and for each y∗ ∈ Y ∗ we have 〈y∗,Tnx∗n〉 = 〈y ∗ ◦Tn,x∗n〉→ 0 and so (Tnx ∗ n) is weakly null. Now, assume that (y∗n) is a weak ∗-null sequence in Y ∗ and define a sequence (Λn) in Kw∗ (X ∗,Y )∗ by 〈T, Λn〉 = 〈Tx∗n,y∗n〉. If T ∈ Kw∗ (X∗,Y ), then T(BX∗ ) is relatively compact and so it is a limited set in Y . It follows that lim n→∞ sup x∗∈BX∗ 〈Tx∗,y∗n〉 = 0. Therefore ‖y∗n ◦ T‖ → 0. Thus 〈y∗n ◦ T,x∗n〉 → 0 and so (Λn) is weak∗-null in Kw∗ (X∗,Y )∗. Since (Tn) is limited, we have 〈Tnx∗n,y ∗ n〉 = 〈Tn, Λn〉→ 0 and so (Tnx ∗ n) is limited. Finally, the GP property of Y yields that ‖Tnx∗n‖→ 0 which implies ‖Tn‖→ 0. � Note that the map T 7→ T∗∗ is an isometric isomorphism from K(X,Y ) into Kw∗ (X∗,Y ). Therefore we have the following result. Corollary 2.5. Let X and Y be two Banach spaces. If X∗ has the limited p-Schur property and Y has the GP property, then K(X,Y ) has the limited p-Schur property. Int. J. Anal. Appl. 16 (1) (2018) 55 Since X⊗̂εY may be identified with a closed subspace of Kw∗ (X∗,Y ) via the isometric embedding X⊗̂εY ↪→ Kw∗ (X∗,Y ) which is defined by x ⊗ y 7→ θx⊗y, where θx⊗y(x∗) = 〈x,x∗〉y, we have the fol- lowing corollary. Corollary 2.6. If X has the limited p-Schur property and Y has the GP property, then injective tensor product X⊗̂εY has the limited p-Schur property. Theorem 2.5. [9, 13] Let X and Y be two Banach spaces and M ⊆ K(X,Y ) such that for all x ∈ X, M(x) := {Tx : T ∈ M} is relatively compact in Y . Then under each of the following conditions, M is a relatively compact subset of K(X,Y ). (a) X∗∗ has the GP property and for every weak∗-null sequence (x∗∗n ) ⊆ X∗∗, (T∗∗x∗∗n ) is norm null uniformly with respect T ∈ M. (b) X contains no copy of `1 and for every weakly null sequence (xn) ⊆ X, (Txn) is norm null uniformly with respect T ∈ M. Recall that the operator T ∈ L(X,Y ) is said to be limited operator if T(BX) is a limited set in Y . The class of all limited operator from X into Y is denoted by L(X,Y ). On the other hand, T ∈ L(X,Y ) if and only if T∗ : Y ∗ → X∗ is weak∗-norm sequential continuous cf. [2]. Theorem 2.6. Let X be a Banach space such that X∗ has the GP property. If F is a closed subspace of K(X,Y ) and for every x∗∗ ∈ X∗∗, the evaluation operator φx∗∗ on F is limited p-converging, then F has the limited p-Schur property. Proof. First, observe that the evaluation operator φx∗∗ , as a generalization of φx is denoted by φx∗∗ (T) = T∗∗x∗∗, for all T ∈ M and x∗∗ ∈ X∗∗. Let M ⊂ F be a limited weakly p-compact set. Since for every x ∈ X, the evaluation map φx is limited p- converging, we conclude that M(x) = {Tx : T ∈ M} is relatively compact. Since the adjoint of every limited operator is weak∗-norm sequentially continuous, it follows that for every compact operator T ∈ K(X,Y ), the operator T∗ is also compact and so is limited. This shows that T∗∗ is weak∗-norm sequentially continuous and therefore for each weak∗-null sequence (x∗∗n ) in X ∗∗, the sequence (T∗∗x∗∗n ) is norm null, that is φx∗∗ is a pointwise norm null sequence of bounded linear operators. Hence (φx∗∗n ) converges uniformly on the limited set M [17, Corollary 1.1.2]. It follows that lim n→∞ sup T∈M ‖φx∗∗n (T)‖ = 0. Then by Theorem 2.5 (a) M is relatively compact and so F has the p-Schur property. � If one use Theorem 2.5 (b) instead of Theorem 2.5 (a), we can prove the following theorem. Int. J. Anal. Appl. 16 (1) (2018) 56 Theorem 2.7. Let X be a Banach space containing no copy of `1. If F is a closed subspace of K(X,Y ) such that for each x ∈ X, the evaluation operator φx is limited p-converging, then F has the limited p-Schur property. Recall that a subset H of L(X,Y ) is uniformly completely continuous, if for every weakly null sequence (xn) in X, lim n→∞ sup T∈H ‖Txn‖ = 0. We remember the following theorem, which has a main role in the proof of the Theorem 2.9. Theorem 2.8. [13] If X contains no copy of `1, then a subset H ⊆ K(X,Y ) is relatively compact if and only if H is uniformly completely continuous and for each x ∈ X, the set φx(H) is relatively compact in Y . Theorem 2.9. If X contains no copy of `1 and Y has the limited p-Schur property, then K(X,Y ) has the limited p-Schur property. Proof. If Y has the limited p-Schur property, then Theorem 2.2 shows that each φx : K(X,Y ) → Y is limited p-converging. Now, suppose that H ⊂ K(X,Y ) is a limited weakly p-compact set. Therefore φx(H) is relatively compact for all x ∈ X. On the other hand, if (xn) is weakly null in X, then complete continuity of each operator T ∈ H implies that ‖φxn (T)‖ = ‖Txn‖ → 0. Therefore (φxn ) is a norm null sequence at each element T ∈ H and then it is uniformly convergent on the limited set H [17, Corolarry 1.1.2]. Hence lim n→∞ sup T∈H ‖Txn‖ = lim n→∞ sup T∈H ‖φxn (T)‖ = 0. This shows that H is uniformly completely continuous. Hence Theorem 2.5 (a) shows that H is relatively compact in K(X,Y ) and so K(X,Y ) has the limited p-Schur property. � Recall that if 1 ≤ p ≤ ∞, the Banach space X has the Dunford-Pettis property of order p (DPp) if for each Banach space Y, every weakly compact operator T : X → Y is p-converging. For more information about DPp property of Banach spaces the reader is referred to [3]. Corollary 2.7. If 2 < q < ∞ and 1 q + 1 q∗ = 1, then (`q⊗̂ε`q)∗ and (`q⊗̂π`q)∗ have the limited p-Schur property, for all 1 < p < q. Proof. Since 1 < q∗ < 2 and q∗ < q < ∞, by Pitt’s Theorem; (see [1, Theorem 2.1.4]), every bounded operator T : `q → `q∗ is compact. Therefore (`q⊗̂π`q)∗ = L(`q,`q∗ ) = K(`q,`q∗ ) and (`q⊗̂ε`q)∗ = I(`q,`q∗ ) ⊂ K(`q,`q∗ ), where I(`q,`q∗ ) is the space of all integral operators from `q into `q∗ [5, P. 119]. Hence it is enough to show that K(`q,`q∗ ) has the limited p-Schur property, for all 1 < p < q. In fact, by [3, Example 3.3] `q∗ has the DPp property, for all 1 < p < q. It follows from [6, Theorem 2.31] that `q∗ has the (limited) Int. J. Anal. Appl. 16 (1) (2018) 57 p-Schur property, for all 1 < p < q. On the other hand, `q contains no copy of `1. Therefore Theorem 2.9 (or Corollary 2.5) shows that K(`q,`q∗ ) has the limited p-Schur property, for all 1 < p < q. � We also notice that by Theorem 2.2, if the closed subspace M of L(X,Y ) has the limited p-Schur property, then all operators on M, such as evaluation operators, are limited p-converging. Therefore the converse of Theorem 2.6 is also true. Moreover, in the following two theorems 2.11 and 2.12, we will give another sufficient conditions for the limited p-Schur property of closed subspace M of some operator spaces with respect to the limited p-converging of evaluation operators. To obtain our next result we need the following well known theorem. Theorem 2.10. [9] Let X and Y be two Banach spaces and H be a subset of L(X,Y ) such that (1) H(BX) = {Tx : T ∈ H,x ∈ BX} is relatively compact. (2) ψy∗ (H) is relatively compact for all y ∗ ∈ Y ∗. Then H is relatively compact. Theorem 2.11. Let M be a closed linear subspace of L(X,Y ) such that the closed linear span of the set M(X) = {Tx : T ∈ M,x ∈ X} has the GP property. If all evaluation operator ψy∗ are limited p-converging, then M has the limited p-Schur property. Proof. Suppose that H is a limited weakly p-compact subset of M. By Theorem 2.10, it is enough to show that H(BX) and all ψy∗ (H) are relatively compact in Y and X ∗, respectively. For every y∗ ∈ Y ∗, the evaluation operator ψy∗ is limited p-converging. Therefore ψy∗ (H) is relatively compact. On the other hand, if (y∗n) is a weak ∗-null sequence in Y ∗, then the weak∗-norm sequential continuity of the adjoint of eah T ∈ H implies that ‖ψy∗n (T)‖ = ‖T ∗y∗n‖→ 0 as n →∞. Therefore (ψy∗n ) converges pointwise on H an so it is converges uniformly on the subset H of M. Hence sup{|〈Tx,y∗n〉| : T ∈ H,x ∈ BX} = sup{|〈x,T ∗y∗n〉| : T ∈ H,x ∈ BX} = sup T∈H ‖T∗y∗n‖→ 0. Thus H(BX) is limited and so is relatively compact. � Now, we give a sufficient condition for the limited p-Schur property of subspaces of Lw∗ (X ∗,Y ) of all bounded weak∗-weak continuous operator from X∗ to Y . Clearly, if T ∈ Lw∗ (X∗,Y ), then T∗ transfers Y ∗ into X. The proof of this theorem is similar to the proof of Theorem 3.6 of [6]. So we omit its proof. Theorem 2.12. Let X and Y be Banach spaces such that X has the Schur property. If M is a closed subspace of Lw∗ (X ∗,Y ) such that every evaluation operator φx∗ is limited p-converging on M, then M has the limited p–Schur property. Int. J. Anal. Appl. 16 (1) (2018) 58 Recall that according to [6], a bounded subset K of a Banach space X is p-Limited if lim n sup x∈K |〈x,x∗n〉| = 0, for every (x∗n) ∈ `weakp (X∗) . A subset K of a dual space X∗ of X is Lp-set if lim n sup x∗∈K |〈xn,x∗〉| = 0 for every sequence (xn) ∈ `weakp (X). Also, a sequence (x∗n) in X ∗ is an Lp-set if and only if limn→∞〈xn,x∗n〉 = 0 for all (xn) ∈ `weakp (X) [7]. It is clear that for every limited subset and every p-limited subset of a dual space is an Lp-set. Moreover, the following result has been proved in [7]. Theorem 2.13. A Banach space X is weakly p-compact if and only if every Lp-set in X ∗ is relatively compact. Theorem 2.14. Let X and Y be Banach spaces. If X contains no copy of `1, Y ∗ is weakly p-compact and for every h ∈ L(X,Y ∗∗), for every weakly null sequence (xn) ⊂ X, the sequence (hxn) is an Lp-set, then K(X,Y ) has the GP property and so has the limited p-Schur property. Proof. Let M ⊂ K(X,Y ) be a limited set. We have to prove that M is relatively compact. Since M(x) = {Tx : T ∈ M} is a limited set in Y and so is an Lp-set, therefore M(x) is a relatively compact set, by Theorem 2.13. Assume that condition (b) of Theorem 2.5 in not verified. So there are a positive number ε, a weakly null sequence (xn) ⊂ X and a sequence (Tn) ⊂ M such that for all n ∈ N, ||Tnxn|| > ε. Now we prove that (Tnxn) is weakly null. For every y ∗ ∈ Y ∗, the set {T∗ny∗ : n ∈ N} is a Dunford-Pettis subset of X∗. Since (xn) is weakly null, it follows that 〈Tnxn,y∗〉 = 〈T∗ny ∗,xn〉→ 0 for every y∗ ∈ Y ∗. So the sequence (Tnxn) is weakly null. Now, we prove that (Tnxn) is a p-limited set. Suppose that (y ∗ n) ∈ `weakp (Y ∗) and h ∈ (X⊗̂πY ∗)∗ = L(X,Y ∗∗). As (hxn) is an Lp-set in Y ∗∗ we have h(xn ⊗ y∗n) = 〈hxn,y∗n〉 → 0 and so (xn ⊗ y∗n) is weakly null in X ⊗π Y ∗. Since X⊗̂πY ∗ embeds into K(X,Y )∗, it follows that (xn ⊗y∗n) is also weakly null in space K(X,Y )∗. Then it must be that lim n→∞ 〈Tnxn,y∗n〉 = lim n→∞ 〈Tn,xn ⊗π y∗n〉 = 0, because (Tn) is a limited set and so is a DP set. So we have actually proved that (Tnxn) is a p-limited set and so Lp-set. It follows from Theorem 2.13 that it must be a relatively compact set. Since it is a weakly null sequence, there is a norm null subsequence and it is a contradiction. � In [18] the authors have been proved that for Banach spaces (Xα)α∈I, if X = (⊕α∈IXα)1 is their `1-direct sum, then X has the Schur property if and only if each factor Xα has the same property. Here, by a similar idea, we prove that the same condition holds for (limited) p-Schur property. Int. J. Anal. Appl. 16 (1) (2018) 59 Theorem 2.15. Let (Xα)α∈I be Banach spaces and X = (⊕α∈IXα)1. Then the space X has the p-Schur property if and only if each Xα has the p-Schur property. Proof. If X = (⊕α∈IXα)1 has the p-Schur property, then clearly, every closed subspace of X has the p-Schur property. Hence each Xα has the p-Schur property. On the other hand, a straightforward computations shows that a Banach space has the p-Schur property if and only if all of its closed separable subspaces have the p-Schur property. Therefore we can assume that each Xα is separable and take I = N. Hence X = (⊕Xk)1 is separable and so has the GP property. If (xn) ∈ `weakp (X), where xn = (bn,k)k∈N, then (bn,k) ∈ `weakp (Xk) for all k ∈ N. Since Xk has the p-Schur property, therefore ||bn,k|| → 0 as n → ∞, for all k ∈ N. We have to prove that ||xn|| → 0 or the weakly null sequence (xn) is relatively compact. Let {fn}n∈N be a w∗-null sequence in BX∗ . If we show that lim n→∞ 〈xn,fn〉 = 0, then the proof is completed, thanks to the GP property of X. Each fn is of the form fn = (an,k)k∈N and for all k ∈ N, an,k w∗−→ 0 in X∗k as n → ∞. To prove that lim n→∞ 〈xn,fn〉 = 0, it is enough to show that sup n ∑ k>M |〈an,k,bn,k〉|→ 0 as M →∞. Therefore we have to show that for each ε > 0 there exists M ∈ N such that ∑ k>M |〈an,k,bn,k〉| < ε, (2.1) for all sufficiently large enough n ∈ N. Let (2.1) is false. Then there is an ε > 0 such that ∑ k>M |〈an,k,bn,k〉| ≥ ε, (2.2) for all M ∈ N and some sufficiently large enough n ∈ N. Consider a sequence of positive number, (δk) such that ∑∞ k=1 δk < ε 4 . By the technique given in the proof of main theorem of [18] one can construct two strictly increasing sequences, (nk)k≥1 and (Mk)k≥0 such that (1) ∑ j>Mk ||bnk,j|| ≤ δk for each k ≥ 1 (2) Mk−1∑ j=1 |〈an,j,bnk−1,j〉| ≤ δk for each n ≥ nk (3) ∑ j>Mk−1 |〈ank,j,bnk,j〉| ≥ ε. Now, let us choose a sequence (λj) such that |λj| = 1, for all j and λj〈ank,j,bnk,j〉 = |〈ank,j,bnk,j〉|, where k ≥ 1 and Mk−1 + 1 ≤ j ≤ Mk. Let h = (hj)j≥1 = (λ1an1,1,λ2an1,2, ...,λM1an1,M1,λM1+1an2,M1+1, ...). Int. J. Anal. Appl. 16 (1) (2018) 60 Then ||h|| = sup j≥1 ||hj|| ≤ 1 and 〈h,xnk〉 = ∞∑ j=1 〈hj,bnk,j〉 = k−1∑ i=1 Mi∑ j=Mi−1+1 λj〈ani,j,bni,j〉 + Mk∑ j=Mk−1+1 |〈ank,j,bnk,j〉| + ∞∑ j=Mk λj〈ank,j,bnk,j〉. with due attention to ||ank,j|| ≤ 1 and inequalities (1), (2) and (3): |〈h,xnk〉| ≥ − k−1∑ i=1 δi + Mk∑ j=Mk−1+1 |〈ank,j,bnk,j〉|− δk ≥ − k−1∑ i=1 δi + ∑ j>Mk−1 |〈ank,j,bnk,j〉|− ∑ j≤Mk−1 |〈ank,j,bnk,j〉|− δk ≥ − k−1∑ i=1 δi + ∑ j>Mk−1 |〈ank,j,bnk,j〉|− 2δk ≥ ε− k−1∑ i=1 δi − 2δk ≥ ε− 2 ∞∑ i=1 δi > ε 2 . This contradiction shows that (2.1) is true. So lim M→∞ sup n∈N M∑ k=1 |〈an,k,bn,k〉| = 0 Therefore lim n→∞ ∑∞ k=1 |〈an,k,bn,k〉| = ∑∞ k=1 limn→∞ |〈an,k,bn,k〉| = 0 Since |〈fn,xn〉| ≤ ∑∞ k=1 |〈an,k,bn,k〉| we conclude that lim n→∞ |〈fn,xn〉| = 0 and so ||xn||→ 0. � By a similar technique we have the following theorem. Theorem 2.16. Suppose that (Xα)α∈I are Banach spaces and X = (⊕α∈IXα)1. 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