International Journal of Mathematical Archive-5(12), 2014, Available online through ISSN

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1 International Journal of Mathematical Archive-5(12), 2014, Available online through ISSN ABOUT WEAK ALMOST LIMITED OPERATORS A. Retbi* and B. El Wahbi Université Ibn Tofail, Faculté des Sciences, Département de Mathématiques, B.P. 133, Kénitra, Morocco. (Received On: ; Revised & Accepted On: ) ABSTRACT We give a characterization of pairs of Banach lattices E, F for which every positive weak almost limited operator T: E F is limited. Also, we characterize Banach space X and Banach lattice F for which each weak almost limited operator T: X F is Dunford-Pettis, and we derive some consequences Mathematics Subject Classication: 46A40, 46B40, 46E42. Key words and phrases: weak almost limited operator, limited operator, Dunford-Pettis operator, Schur property, dual positive dual Schur property, order continuous norm. 1. INTRODUCTION Throughout this paper X, Y will denote real Banach spaces, and E, F will denote real Banach lattices. B X is the closed unit ball of X. We will use the term operator T: X Y between two Banach spaces to mean a bounded linear mapping. A norm bounded set A in a Banach space X is called limited, if every weak* null sequence (f n ) in X converges uniformly to zero on A, that is, lim nn ssssss xx AA f n (x) = 0. An operator T: X Y is said to be limited whenever TT(B X ) is a limited set in Y, equivalently, whenever TT (f n ) 0 for every weak* null sequence (f n ) in Y. From [3] a norm bounded subset A of a Banach lattice E is said to be almost limited, if every disjoint weak* null sequence (f n ) in E converges uniformly to zero on A. By [9], an operator T: X F is called almost limited if TT(B X ) is a almost limited set in F, equivalently, whenever TT (f n ) 0 for every disjoint weak* null sequence (f n ) in F. It is a class which contains that of limited. Recall from [6] an operator T : X F from a Banach space X into a Banach lattice F is called weak almost limited if T carries each relatively weakly compact set in X to an almost limited set in F, equivalently, whenever (ff nn (xx nn )) 0 for every weakly null sequence (xx nn ) in X and every disjoint weak* null sequence (f n ) in F. An operator T from a Banach space X into another Y is called Dunford-Pettis if TT(x n ) 0 for every weakly null sequence (x n ) in X [1]. Note that every limited (resp. almost limited, Dunord-Pettis) operator T: X F is weak almost limited. Recall that a nonzero element x of a vector lattice G is discrete if the order ideal generated by x equals the subspace generated by x. The vector lattice G is discrete, if it admits a complete disjoint system of discrete elements. A Banach space X has - the Schur property, if xx nn 0 for every weak null sequence (xx nn ) XX. - the Dunford-Pettis* property (DP* property for short), if xx nn ww 0 in X and ffnn 0 in X imply (ff nn (xx nn )) 0. A Banach lattice E has - the weak Dunford-Pettis property (wdp* property for short), if every relatively weakly compact set in E is almost limited, equivalently, whenever (ff nn (xx nn )) 0 for every weakly null sequence (xx nn ) in E and for every disjoint weak* null sequence (ff nn ) in E [3]. Corresponding Author: A. Retbi* Université Ibn Tofail, Faculté des Sciences, Département de Mathématiques, B.P. 133, Kénitra, Morocco. abderrahmanretbi@hotmail.com International Journal of Mathematical Archive- 5(12), Dec

2 - the dual positive Schur property, if ff nn 0 for every weak* null sequence (xx nn ) (EE ) +, equivalently, whenever ff nn 0 for every weak* null sequence (xx nn ) (EE ) + consisting of pairwise disjoint terms see Proposition 2.3 in [12]. - the property (d) whenever ff nn ff mm = 0 and ff nn 0 in E imply ff nn 0. Note by Proposition 1.4 of [12], that every σσ-dedekind complete Banach lattice has the property (d) but the converse is not true in general. In fact, the Banach lattice l /cc 0 has the property (d) but it is not σσ-dedekind complete see Remark 1.5 in [12]. Recall that the lattice operations of a Banach lattice E are weak* sequentially continuous, whenever ff nn 0 in E imply ff nn 0. For terminology concerning Banach lattice theory and positive operators we refer the reader to [1, 10]. In this paper, we characterize pairs of Banach lattices E and F such that each positive weak almost limited operator T: E F is limited (Theorem 2.2). Next, we establish a characterization of Banach space X and Banach lattice F for which every weak almost limited operator T: X F is Dunford-Pettis (Theorem 2.5). As consequences, we will give some interesting results. 2. MAIN RESULTS Note that a Banach lattice E has wdp* property if and only if the identity operator I: E E is weak almost limited. Also, every limited operator T: X E is weak almost limited, but the converse is not true in general. In fact, the identity operator II: l l is weak almost limited (because l has the wdp* property) but it fail to be limited (because the closed unit ball BB l is not limited set in l ). If E is the topological dual of a Banach lattice E and if ff EE, the null ideal of F is defined by NN ff = {xx EE: ff ( xx ) = 0}, the carrier CC ff of ff EE is defined by CC ff = (NN ff ) dd = {xx EE: uu vv = 0, ffffff aaaaaa vv NN ff }. To give a first major result, we need the following lemma. Lemma 2.1: ([2], Lemma 2.4). Let E be a Banach lattice. If E does not have an order continuous norm, then there exists a disjoint sequence (uu nn ) of a positive elements in E with uu nn 1 or all n and that for some 0 f E and some ε > 0, we have ff(uu nn ) > εε for all n. Moreover, the components ff nn of ff in the cariers CC uunn form an order bounded disjoint sequence in (EE ) + such that ff nn (uu nn ) = ff(uu nn ) for all n and ff nn (uu mm ) = 0 if nn mm. Our first major result is given by the following theorem. Theorem 2.2: Let E and F be two Banach lattices such that F is discrete with order continuous. Then the following assertions are equivalent: (1) Each order bounded weak almost limited operator from E into F is limited. (2) Each positive weak almost limited operator from E into F is limited. (3) One of the following assertions is valid: (a) the norm of E is order continuous, (b) F has the dual positive Schur property. Proof: (1) (2): Obvious. (2) (3): Assume by way of contradiction that F does not have the positive dual Schur property and the norm of E is not order continuous. We have to construct a positive weak almost limited which is not limited. since the norm of E is not order continuous, it follows from Lemma 2.1 that there exists a disjoint sequence (uu nn ) of a positive elements in E with uu nn 1 or all n and that for some 0 f E and some ε > 0, we have ff(uu nn ) > εε for all n. Moreover, the components ff nn of ff in the cariers CC uunn form an order bounded disjoint sequence in (EE ) + such that ff nn (uu nn ) = ff(uu nn ) for all n and ff nn (uu mm ) = 0 if nn mm. Note that 0 ff nn ff and ff nn ff nn (uu nn ) = ff(uu nn ) > 0 holds for all n. Now, we define the operator PP 1 : E l 1 by PP 1 (xx) = (ff nn (xx)) nn=1 for all xx EE. 2014, IJMA. All Rights Reserved 76

3 Since nn=1 ff nn (xx) nn=1 ff nn ( xx ) ff( xx ) holds for each xx EE, the Operator PP 1 is well defined and is positive. On the other hand, since F does not have the dual positive Schur property, then there is a weak* null sequence (gg nn ) (FF ) + such that (gg nn ) is not norm null. By choosing a subsequence we may suppose that there is εε > 0 with gg nn εε > 0 for all n. From the equality gg nn = sup {gg nn (yy) : y BB + + FF }, there exists a sequence (yy nn ) BB FF such that gg nn (yy nn ) εε holds for all n. Now, consider the operator PP 2 : l 1 F defined by PP 2 ((λλ nn ) nn=1 ) = nn=1 λλ nn yy nn for each (λλ nn ) nn=1 l 1. Now, consider the positive operator TT = PP 2 οοοο 1 : E F, so that T(x) = nn=1 ff nn (xx) yy nn for each xx EE. Since l 1 has the Schur property then, T is Dunford-Pettis and hence T is weak almost limited but is not limited. Indeed, note that for the weak* null sequence (gg nn ) (FF ) +, we have for every n, TT (gg nn )= kk=1 gg nn (yy kk ) ff kk gg nn (yy nn ) ff nn 0 Thus, TT (gg nn ) gg nn (yy nn ) ff nn = gg nn (yy nn ) ff nn ε. εε for every n, this prove that T is not limited. (a) (1): Let (xx nn ) be a norm bounded disjoint sequence in EE + and (ff nn ) be a disjoint weak* null sequence in (FF ) +. Since the norm of EE is order continuous then by Theorem of [10] the sequence (xx nn ) is weakly nul. We have T an order bounded weak almost limited operator then, by Theorem 2.4 of [6] ff nn (TT(xx nn )) 0 for every norm bounded disjoint sequence (xx nn ) in EE + and every disjoint weak* null sequence (ff nn ) in (FF ) + and hence by proposition 4.4 of [9] we conclude that T is almost limited, i.e., T(BB EE ) is almost limited. Since the norm of F is order continuous, by Remark 2.7(2) of [3] the solid hull S of T(BB EE ) is almost limited. As FF is discrete with an order continuous norm, then by Theorem 3.1 of [4] the lattice operations of FF are weak* sequentially continuous. So by Theorem 2.6 of [9] S (and hence T(BB EE ) itself) is limited. Therefore, T is limited. (b) (1): In this case, every operator TT : E F is limited. In fact, let (ff nn ) in FF be a weak* null sequence. Since FF is discrete with an order continuous norm, then by Theorem 3.1 of [4] the lattice operations of FF are weak* sequentially continuous, this implies that the positive sequence ( ff nn ) FF is weak* null. So by the dual positive Schur property of FF, ff nn 0, and hence TT (gg nn ) 0, as desired. In particular, if E = F, we have Corollary 2.3: Let E be a Banach lattice discrete with order continuous. Then the following assertions are equivalent: (1) Each positive weak almost limited operator from E into E is limited. (2) The norm of E is order continuous Proof: Note that if E has the dual positive Schur property then, the norm of E is order continuous, the result follows from Theorem 2.2. Corollary 2.4: Let F be a Banach lattices discrete with order continuous. F has the dual positive Schur property if and only if every positive operator TT : l 1 F is limited. Proof: Since l does not an order continuous norm, then the result follows from Theorem 2.2. Note that there exists a weak almost limited operator which is not Dunford-Pettis. Indeed, the identity operator of the Banach space l is weak almost limited (because l has the wdp* property) but it fail to be Dunford-Pettis (because l does not have the Schur property). Our second major result is given by the following theorem. Theorem 2.5: Let X be a Banach space and F be Banach lattice such that F is Dedekind σσ-complete. Then the following assertions are equivalent: (1) Each weak almost limited operator T from X into F is Dunford-Pettis. (2) One of the following assertions is valid: (a) X has the Schur property, (b) the norm of F is order continuous. Proof: (1) (2): Assume by way of contradiction that X does not have the Schur property and the norm of F is not order continuous. We have to construct a positive weak almost limited operator which is not Dunford-Pettis. As X does not have the Schur property, then there exists a weakly null sequence (xx nn ) in XX which is not norm null. By choosing a subsequence we may suppose that there is εε > 0 with xx nn εε > 0 for all n. From the equality xx nn = sup { ff(xxnn) : f X, f = 1 }, there exists a sequence (ffnn) XX such that ffnn = 1 and ffnn(xxnn) εε holds for all n. Now, consider the operator RR: XX l defined by RR(xx) = (ff nn (xx)) nn=1 for all xx XX. 2014, IJMA. All Rights Reserved 77

4 On the other hand, since the norm of FF is not order continuous, it follows from Theorem 4.51 of [1] that l is lattice embeddable in F, i.e., there exists a lattice homomorphism S: l F and there exists two positive constants M and mm satisfying mm (λλ kk ) kk SS(λλ kk ) kk FF M (λλ kk ) kk for all (λλ kk ) kk l. Put TT = SSSSSS, and note that T is weak almost limitedoperator see Remark 2.3 of [6]. However, for a weakly null sequence (xx nn ) X, we have TT(xx nn ) SS((ff kk (xx nn )) kk ((ff kk (xx nn )) kk mm ff nn (xx nn ) mmε for every n. This show that T is not Dunford-Pettis, and we are done. (a) (1): In this case, each operator TT: XX FF is Dunford-Pettis. (b) (1): Let (xx nn ) XX be a weakly null sequence. We shall show that TT(xx nn ) 0. By corollary 2.6 of [5], it suffices to proof that TT(xx nn ) ww 0 and ff nn TT(xx nn ) 0 for every disjoint and norm bounded sequence (ff nn ) (FF ) +. Let ff (FF ) + and by Theorem 1.23 of [1] there exists some gg [ ff, ff ] with ff( TT(xx nn ) ) = gg(tt(xx nn )). Since xx ww nn 0 then ff( TT(xx nn ) ) = gg TT(xx nn ) =T (g) (xx nn ) 0, Thus (ff nn ) (FF ) + be a disjoint and norm bounded sequence. TT(xx nn ) ww 0. On the other hand, let As the norm of F is order continuous, then by corollary of [10] ff nn 0. Now, since T is weak almost limited then, (ff nn (xx nn )) 0. This completes the proof. Corollary 2.6: Let E be a Banach lattices Dedekind σσ-complete. Then the following assertions are equivalent. (1) Each weak almost limited operator T: E E is Dunford-Pettis. (2) The norm of E is order continuous. Proof: Note that if E has the Schur property then, the norm of E is order continuous, the result follows from theorem 2.5. Corollary 2.7: A Banach space X has the Schur property if and only if every operator T: X l is Dunford-Pettis. Proof: Note that l is Dedekind σσ-complete and dos not have an order continuous norm, then from theorem 2.5, X has the Schur property if and only if every weak almost limited operator T: X l is DunfordPettis, and by Remark 2.3 of [6] every operator T: X l is weak almost limited. This complete the proof. Corollary 2.8: Let F be a Banach lattice Dedekind σσ-complete. The norm of F is order continuous if and only if every operator T: l F is Dunford-Pettis. Proof: Note that l dos not have the Schur property, then from theorem 2.5, the norm of F is order continuous if and only if every weak almost limited operator T : l F is Dunford-Pettis, and by Remark 2.3 of [6] every operator T : l F is weak almost limited. This complete the proof. REFERENCES 1. Aliprantis C.D. and Burkinshaw O., Positive operators. Reprint of the 1985 original. Springer, Dordrecht, Belmesnaoui Aqzzouz, Khalid Bouras and Aziz Elbour, Some Generalizations on positive Dunford-Pettis operators, Result. Math. 54 (2009), Chen J.X., Chen Z.L., Ji G.X.: Almost limited sets in Banach lattices, J. Math. Anal. Appl. 412 (2014) Z. L. Chen and A. W. Wickstead., Equalities involving the modulus of an operator, Mathematical Proceedings of the Royal Irish Academy, 99A (1), (1999). 5. P.G. Dodds, and D.H. Fremlin, Compact operators on Banach lattices, Israel J. Math. 34 (1979), Elbour A., Machra N., Moussa M., On the class of weak almost limited operators. arxiv: v1 [math.fa] 1 Mar A. El Kaddouri, J. Hmichane, K. Bouras, M. Moussa., On the class of weak* Dunford-Pettis operators, Rend. Circ. Mat. Palermo (2) 62 (2013), Khalid Bouras. Mohamed Moussa, On the class of positive almost Dunford-Pettis operators, Positivity, DOI /s , (2012), 12 pages. 9. Machrafi N., Elbour A., Moussa M.: Some characterizations of almost limited sets and applications , IJMA. All Rights Reserved 78

5 10. Meyer-Nieberg P., Banach lattices. Universitext. Springer-Verlag, Berlin, H. H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, Berlin and New York, Wnuk W.: On the dual positive Schur property in Banach lattices. Positivity (2013) 17: Wnuk W., Banach Lattices with the weak Dunford-Pettis Property. Atti Sem. Mat.Fis. Univ. Modena 42(1), (1994). 14. W. Wnuk, Banach Lattices with Order Continuous Norms. Polish Scientic Publishers, Warsaw, Source of support: Nil, Conflict of interest: None Declared [Copy right This is an Open Access article distributed under the terms of the International Journal of Mathematical Archive (IJMA), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.] 2014, IJMA. All Rights Reserved 79