# Weak compactness of AM-compact operators.

1 Introduction and notation

Recall that an operator T from a Banach lattice E into a Banach space X is called AM-compact if the image of each order bounded subset of E is a relatively compact subset of X. Note that an AM-compact operator is not necessary weakly compact. In fact, the identity operator of the Banach lattice [l.sup.1], is AM-compact but it is not weakly compact. Conversely, a weakly compact operator is not necessary AM-compact. For an example, the identity operator of the Banach lattice [L.sup.2]([0,1]) is weakly compact but it is not AM-compact. If not, for each x [member of] [L.sup.2]([0,1]), the order interval [0,x] would be norm compact, and hence [L.sup.2]([0,1]) would be discrete, and this is false.

Note that none of the two classes satisfies the problem of domination [2,7], but while the class of weakly compact operators satisfies the problem of duality that of AM-compact operators does not satisfy it [8,17].

Our objective in this paper is to investigate Banach lattices on which each AM-compact operator is weakly compact and in another paper, we will look at the reciprocal problem. In fact, in this paper, we will establish that if E is a Banach lattice and X is a Banach space such that each AM-compact operator T: E [right arrow] X is weakly compact, then the norm of E' is order continuous or X is reflexive. And conversely, if E is a KB-space, then each AM-compact operator T: E [right arrow] X is weakly compact if E is order continuous or X is reflexive. Next, we will give a necessary and sufficient condition for which the second power of an AM-compact operator (resp. operator of strong type B) is weakly compact.

To state our results, we need to fix some notations and recall some definitions. A Banach lattice is a Banach space (E,[parallel]*[parallel]) such that E is a vector lattice and its norm satisfies the following property: for each x, y [member of] E such that [absolute value of x] [less than or equal to] [absolute value of y], we have [parallel]x[parallel] [less than or equal to] [parallel]y[parallel]. If E is a Banach lattice, its topological dual E', endowed with the dual norm and the dual order, is also a Banach lattice.

We refer to [1] for unexplained terminology on Banach lattice theory.

2 Main results

We will use the term operator T: E [right arrow] F between two Banach lattices to mean a bounded linear mapping. It is positive if T(x) [greater than or equal to] 0 in F whenever x [greater than or equal to] 0 in E. The operator T is regular if T = [T.sub.1] - [T.sub.2] where [T.sub.1] and [T.sub.2] are positive operators from E into F. Note that each positive linear mapping on a Banach lattice is continuous.

Let us recall that a norm [parallel]*[parallel] of a Banach lattice E is order continuous if for each generalized sequence ([x.sub.[alpha]]) such that [x.sub.[alpha]] [down arrow] 0 in E, the sequence ([x.sub.[alpha]]) converges to 0 for the norm [parallel]*[parallel] where the notation [x.sub.[alpha]] [down arrow] 0 means that the sequence ([x.sub.[alpha]]) is decreasing, its infimum exists and inf ([x.sub.[alpha]]) = 0. A Banach lattice E is said to be a KB-space whenever every increasing norm bounded sequence of [E.sup.+] is norm convergent. As an example, each reflexive Banach lattice is a KB-space. Our following result gives necessary conditions under which each AM-compact operator is weakly compact.

Theorem 2.1. Let E be a Banach lattice and X a Banach space. If each AM-compact operator T: E [right arrow] X is weakly compact, then one of the following assertions is valid:

1. the norm of E' is order continuous,

2. X is reflexive.

Proof. Assume that the norm of E' is not order continuous. It follows from Theorem 2.4.14 and Proposition 2.3.11 of [14] that E contains a sublattice isomorphic to [l.sup.1] and there exists a positive projection P: E [right arrow] [l.sup.1].

To finish the proof we have to show that X is reflexive. By the Eberlein-Smulian's Theorem it suffices to show that every sequence ([x.sub.n]) in the closed unit ball of X has a subsequence, that we note also by ([x.sub.n]), which converges weakly to an element of X. Consider the operator T: [l.sup.1] [right arrow] X defined by T (([[lambda].sub.i])) = [[infinity].summation over (i=1)] [[lambda].sub.i][x.sub.i] for each ([[lambda].sub.i]) [member of] [l.sup.1]. The composed operator T * P: E [right arrow] [l.sup.1] [right arrow] X is AM-compact (because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is AM-compact) and hence by our hypothesis T * P is weakly compact. If we note by ([e.sub.n]) the sequence with all terms zero and the nth equals 1, then the sequence ([x.sub.n]) = ((T * P) ([e.sub.n])) has a subsequence which converges weakly to an element of X. This ends the proof.

Remark 2.2. The necessary condition (2) in Theorem 2.1 is sufficient, but the condition (1) is not. In fact, the identity operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the Banach lattice [c.sub.0] is AM-compact and the norm of ([c.sub.0])' = [l.sup.1] is order continuous, but [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not weakly compact.

Recall from [3] that a subset A of a Banach lattice E is called b-order bounded if it is order bounded in the topological bidual E". It is clear that every order bounded subset of E is b-order bounded. However, the converse is not true in general.

A Banach lattice E is said to have the (b)-property if A [subset] E is order bounded in E whenever it is order bounded in its topological bidual E".

An operator T from a Banach lattice E into a Banach space X is said to be b-weakly compact whenever T carries each b-order bounded subset of E into a relatively weakly compact subset of X. Note that each weakly compact operator is b-weakly compact but the converse may be false in general. For an example, the identity operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is b-weakly compact but it is not weakly compact. For more information on b-weakly compact operators see [9], [10], [3], [6], [4].

Conversely, we have the following result.

Theorem 2.3. Let E be a Banach lattice and X a Banach space. Then each AM-compact operator T: E [right arrow] X is weakly compact if one of the following assertions is valid:

1. E is reflexive,

2. X is reflexive.

Whenever E is a KB-space, then by using Theorem 2.1 and Theorem 2.3, we establish that the two necessary conditions of Theorem 2.1 become sufficient.

Theorem 2.4. Let E be a KB-space and let X be a Banach space. Then the following assertions are equivalent:

1. each AM-compact operator T: E [right arrow] X is weakly compact,

2. one of the following assertions holds:

(a) the norm of E' is order continuous,

(b) X is reflexive.

Proof. (1) [??] (2) Follows from Theorem 2.1.

(2) [??] (1) Follows from Theorem 2.3.

Remark 2.5. The two properties "the norm of E' is order continuous" and "E is a KB-space" are independent. In fact, there exists a KB-space E such that the norm of its topological dual E' is not order continuous. For example, the Banach lattice [l.sup.1] is a KB-space but ([l.sup.1])' = [l.sup.[infinity]] does not have an order continuous norm. And conversely, there exists a Banach lattice E which is not a KB-space but the norm of its topological dual E' is order continuous. For example, the Banach lattice [c.sub.0] is not a KB-space but the norm of ([c.sub.0])' = [l.sup.1] is order continuous norm.

Remark 2.6. The assumption "E is a KB-space" is essential in Theorem 2.4. For instance, for p > 1 the operator [T.sub.p]: [X.sub.P] [right arrow] [c.sub.0] constructed in [13] is AM-compact which is not weakly compact where the Banach lattice [X.sub.P] as defined in [13]. However, the norm of ([X.sub.P])' is order continuous. Note that the Banach lattice [X.sub.P] is not a KB-space. Otherwise, the operator [T.sub.p]: [X.sub.P] [right arrow] [c.sub.0] would be weakly compact.

Now, from Theorem 2.4, we derive two characterizations. The first one concerns Banach lattices whose topological duals have order continuous norms:

Corollary 2.7. Let E be a KB-space and X a non reflexive Banach space. Then the following assertions are equivalent:

1. each AM-compact operator T: E [right arrow] X is weakly compact.

2. the norm of E' is order continuous.

The second one concerns reflexive Banach spaces:

Corollary 2.8. Let X be a Banach space. Then the following assertions are equivalent:

1. each operator T: [l.sup.1] [right arrow] X is weakly compact.

2. X is reflexive.

If in Theorem 2.4, we take E and F are two Banach lattices, then we obtain the following characterization:

Theorem 2.9. Let E and F be two Banach lattices such that E is a KB-space. Then the following assertions are equivalent:

1. each AM-compact operator T: E [right arrow] F is weakly compact,

2. each positive AM-compact operator T: E [right arrow] F is weakly compact,

3. one of the following assertions holds:

(a) the norm of E' is order continuous,

(b) F is reflexive.

On the other hand, we observe that if E is a Banach lattice, the second power of an AM-compact operator T: E [right arrow] E is not necessary weakly compact. In fact, the identity operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is AM-compact but its second power [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not weakly compact.

In the following, we give a necessary and sufficient condition for which the second power operator of an AM-compact operator is always weakly compact.

Theorem 2.10. Let E be a KB-space. Then the following assertions are equivalent:

1. for all positive operators S and T from E into E with 0 [less than or equal to] S [less than or equal to] T and T is AM-compact, S is weakly compact,

2. each positive AM-compact operator T: E [right arrow] E is weakly compact,

3. for each positive AM-compact operator T: E [right arrow] E, the second power [T.sup.2] is weakly compact,

4. the norm of E' is order continuous.

Proof. (1) [??] (2) Let T: E [right arrow] E be a positive AM-compact operator. Since 0 [less than or equal to] T [less than or equal to] T, then by our hypothesis T is weakly compact.

(2) [??] (3) By our hypothesis T is weakly compact and hence [T.sup.2] is weakly compact.

(3) [??] (4) By way of contradiction, suppose that the norm of E' is not order continuous. We have to construct a positive AM-compact operator such that its second power is not weakly compact.

Since the norm of E is not order continuous, it follows from Theorem 2.4.14 and Proposition 2.3.11 of [14] that E contains a complemented copy of [l.sup.1] and there exists a positive projection P: E [right arrow] [l.sup.1].

Consider the operator T = i [omicron] P with i is the canonical injection of [l.sup.1] in E. Clearly the operator T is AM-compact but it is not weakly compact. Otherwise, the operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] would be weakly compact, and this is impossible. Hence, the operator [T.sup.2] = T is not weakly compact.

(4) [??] (1) Follows from Theorem 14.22 of [1].

Let us recall from [15] that an operator T from a Banach lattice E into a Banach space X is called of strong type B whenever T carries the band BE, generated by E in E", into X. Note that each weakly compact operator is of strong type B but the converse is false in general. In fact, the identity operator of the Banach lattice [L.sup.1] [0,1] is of strong type B but it is not weakly compact. And in [5] Alpay studied the weak compactness of operators of strong type B.

Also, if E is a Banach lattice, the second power of an operator of strong type B, T: E [right arrow] E, is not necessary weakly compact. In fact, the identity operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is of strong type B but its second power [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not weakly compact.

In the following result, we characterize Banach lattices on which the second power of each operator of strong type B is weakly compact.

Theorem 2.11. Let E be a Banach lattice. Then the following assertions are equivalent:

1. for all positive operators S and T from E into E with 0 [less than or equal to] S [less than or equal to] T and T is of strong type B, S is weakly compact,

2. each positive operator, of strong type B, is weakly compact,

3. for each positive operator of strong type B, T: E [right arrow] E, its second power [T.sup.2] is weakly compact,

4. the norm of E' is order continuous.

Proof. (1) [??] (2) Let T: E [right arrow] E be a positive operator of strong type B. Since 0 [less than or equal to] T [less than or equal to] T, then by our hypothesis T is weakly compact.

(2) [??] (3) Let T: E [right arrow] E be an operator of strong type B. By our hypothesis T is weakly compact and hence [T.sup.2] is weakly compact.

(3) [??] (4) Suppose that the norm of E' is not order continuous. Then it follows from Theorem 2.4.14 and Proposition 2.3.11 of [14] that E contains a complemented copy of [l.sup.1] and there exists a positive projection P: E [right arrow] [l.sup.1]. Consider the operator T = i [omicron] P with i is the canonical injection of [l.sup.1] in E. The operator T is of strong type B but it is not weakly compact. Otherwise, the operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] would be weakly compact, and this is impossible. Hence, the operator [T.sup.2] = T is not weakly compact.

(4) [??] (1) Follows from Proposition 3.2 of [5] and Theorem 5.31 of [1].

Recall from [11] that an operator T defined from a Banach lattice E into a Banach space X is said to be b-AM-compact provided that T maps b-order bounded subsets of E into relatively compact subsets of X. Note that this class of operators is larger than that of compact operators but smaller than that of AM-compact operators.

On the other hand, there exists an operator which is b-AM-compact but not weakly compact. In fact, the identity operator of the Banach lattice [l.sup.1] is b-AM-compact but it is not weakly compact.

The following result was claimed for b-weakly compact operators in [10] and for b-AM-compact in [12]. In one part of the proof we were misguided by an erroneous part of Proposition 2 in [4]. However, our claim is still true under the condition "the norm of E is order continuous".

Theorem 2.12. Let E be a Banach lattice with an order continuous norm and let X be a Banach space. Then the following assertions are equivalent:

1. each b-weakly compact operator T: E [right arrow] X is weakly compact,

2. each b-AM-compact operator T: E [right arrow] X is weakly compact,

3. one of the following assertions holds:

(a) the norm of E' is order continuous,

(b) X is reflexive.

Proof. (1) [??] (2) Since each b-AM-compact operator is b-weakly compact, it follows from the assertion 1 that each b-AM-compact operator is weakly compact.

(2) [??] (3) The proof of this implication follows by the same lines as in [10], it suffices to remark that the operator constructed in [10] is b-AM-compact but it is not weakly compact.

(3) [??] (1) Let T: E [right arrow] X be a b-weakly compact operator. Since the norm of E is order continuous, then T: E [right arrow] X is of strong type B. As the norm of E' is order continuous, it follows from Proposition 3.2 of [5] that T: E [right arrow] X is weakly compact.

Remark 2.13. The assumption "the norm of E is order continuous" is essential in Theorem 2.12. For instance, for p > 1 the operator [T.sub.p]: [X.sub.P] [right arrow] [c.sub.0] constructed in [13] is b-weakly compact but it is not weakly compact. However, the norm of ([X.sub.P]) is order continuous. Note that the norm of the Banach lattice [X.sub.P] is not order continuous. Otherwise, the operator [T.sub.p]: [X.sub.P] [right arrow] [c.sub.0] would be of strong type B and since the norm of ([X.sub.P])' is order continuous, it follows from Proposition 3.2 of [5] that the operator [T.sub.p]: [X.sub.P] [right arrow] [c.sub.0] is weakly compact.

Whenever E and F are two Banach lattices, then we obtain the following result:

Theorem 2.14. Let E and F be two Banach lattices such that the norm of E is order continuous. Then the following assertions are equivalent:

1. each b-weakly compact operator T: E [right arrow] F is weakly compact,

2. each b-AM-compact operator T: E [right arrow] F is weakly compact,

3. each positive b-AM-compact operator T : E [right arrow] F is weakly compact,

4. one of the following assertions holds:

(a) the norm of E' is order continuous,

(b) F is reflexive.

On the other hand, if E is a Banach lattice, the second power of a b-weakly compact operator T : E [right arrow] E is not necessary weakly compact. In fact, the identity operator Id(y is b-weakly compact but its second power [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not weakly compact.

The following result was stated as Theorem 2.8 in [10]:

Theorem 2.15. Let E be a Banach lattice with an order continuous norm. Then the following assertions are equivalent:

1. for all positive operators S and T from E into E with 0 [less than or equal to] S [less than or equal to] T and T is b-weakly compact, S is weakly compact,

2. each positive b-weakly compact operator T: E [right arrow] E is weakly compact,

3. for each positive b-weakly compact operator T: E [right arrow] E, the second power T2 is weakly compact,

4. the norm of E' is order continuous.

Proof. (1) [??] (2) Let T: E [right arrow] E be a positive b-weakly compact operator. Since 0 [less than or equal to] T [less than or equal to] T, then by our hypothesis T is weakly compact.

(2) [??] (3) Let T: E [right arrow] E be a b-weakly compact operator. By our hypothesis T is weakly compact and hence [T.sup.2] is weakly compact.

(3) [??] (4) Suppose that the norm of E' is not order continuous. Then it follows from Theorem 2.4.14 and Proposition 2.3.11 of [14] that E contains a complemented copy of [l.sup.1] and there exists a positive projection P: E [right arrow] [l.sup.1].

Consider the operator T = i [omicron] P with i is the canonical injection of [l.sup.1] in E. The operator T is b-weakly compact but it is not weakly compact. Otherwise, the operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] would be weakly compact, and this is impossible. Hence, the operator [T.sup.2] = T is not weakly compact.

(4) [??] (1) Let S and T be two positive operators from E into E with 0 [less than or equal to] S [less than or equal to] T and T is b-weakly compact. It follows from Corollary 2.9 of [3] that S is b-weakly compact. Since the norm of E is order continuous, then the operator S is of strong type B and since the norm of E' is order continuous, it follows from Proposition 3.2 of [5] that S is weakly compact.

We end this paragraph by proving a necessary condition for which a positive AM-compact operator is compact. In fact, we have the following Theorem:

Theorem 2.16. Let E be a Banach lattice. If each positive AM-compact operator T from E into E is compact, then E has an order continuous norm.

Proof. Assume that the norm of E is not order continuous, then it follows from Theorem 2.4.14 and Proposition 2.3.11 of [14] that E contains a sublattice isomorphs to [l.sup.1] and there exists a positive projection P from E into [l.sup.1].

Consider the operator product

i [omicron] P: E [right arrow] [l.sup.1] [right arrow] E

where i is the inclusion operator of [l.sup.1] in E. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the operator i [omicron] P is AM-compact which is not compact. If not its restriction to [l.sup.1], that we denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], would be compact and the product operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] will be compact. This presents a contradiction.

Remark 2.17. Note that there exist Banach lattices E and F and an AM-compact operator T from E into F which is not weakly compact, however

1. the norms of E' and F are order continuous,

2. E' is discrete and its norm is order continuous,

3. F is discrete and its norm is order continuous.

In fact, if we take E = F = [c.sub.0], the identity operator of [c.sub.0], is AM-compact but is not weakly compact.

Acknowledge. We would like to thank a lot the referee for reporting us that a result that we use in our proofs is wrong. Also, we thank him for his suggestion regarding the class of operators of strong type B.

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University Mohammed V-Souissi, Faculte des Sciences Economiques, Juridiques et Sociales, Departement d'Economie, B.P. 5295, SalaAljadida, Morocco. email:baqzzouz@hotmail.com

University Ibn Tofail, Faculte des Sciences, Departement de Mathematiques, B.P. 133, Kenitra, Morocco.

Received by the editors June 2011--In revised form in August 2011.

Communicated by F. Bastin.

2000 Mathematics Subject Classification: 46A40, 46B40, 46B42.