Abstract

This paper is devoted to the relationship between almost limited operators and weakly compact operators. We show that if is a -Dedekind complete Banach lattice, then every almost limited operator is weakly compact if and only if is reflexive or the norm of is order continuous. Also, we show that if is a -Dedekind complete Banach lattice, then the square of every positive almost limited operator is weakly compact if and only if the norm of is order continuous.

1. Introduction

Throughout this paper will denote real Banach spaces, and will denote real Banach lattices. is the closed unit ball of and is the positive part of . We will use the term operator between two Banach spaces to mean a bounded linear mapping. We refer to [1, 2] for unexplained terminology of the Banach lattice theory and positive operators.

Let us recall that a norm bounded set in a Banach space is called limited, if every null sequence in converges uniformly to zero on ; that is, . An operator is said to be limited whenever is a limited set in , equivalently, whenever for every null sequence .

Recently, the authors of  [3] considered the disjoint version of limited sets by introducing the class of almost limited sets in Banach lattices. From [3] a norm bounded subset of a Banach lattice is said to be almost limited, if every disjoint null sequence in converges uniformly to zero on .

From [4], an operator is called almost limited if is an almost limited set in , equivalently, for every disjoint null sequence . Note that an almost limited operator need not be weakly compact. In fact, the identity operator of the Banach lattice is almost limited but it is not weakly compact.

In this paper, we characterize pairs of Banach lattices , for which every almost limited operator is weakly compact. More precisely, we will prove that if is a -Dedekind complete Banach lattice, then every almost limited operator is weakly compact if and only if is reflexive or the norm of is order continuous (Theorem 5). Next, we will prove that if is a -Dedekind complete Banach lattice, then the square of every positive almost limited operator is weakly compact if and only if the norm of is order continuous (Theorem 9). As consequences, we will give some interesting results.

2. Main Results

Let us recall that a Banach lattice is said to have the dual positive Schur property if for every null sequence , equivalently, for every null sequence consisting of pairwise disjoint terms (Proposition 2.3 of [5]). A Banach lattice has the property whenever and in imply . It should be noted, by Proposition 1.4 of [5], that every -Dedekind complete Banach lattice has the property but the converse is not true in general. In fact, the Banach lattice has the property but it is not -Dedekind complete [5, Remark 1.5].

Our first result shows that we can restrict sequences appearing in the definition of almost limited operator to positive disjoint sequences if the Banach lattice has the property .

Proposition 1. An operator from a Banach space into a Banach lattice with the property is almost limited if and only if for every null sequence in consisting of positive and pairwise disjoint elements.

Proof. The “only if” part is trivial. For the “if” part, let be a disjoint null sequence. As has the property , . Using the inequalities and , we see that and are disjoint null sequences of . So, from our hypothesis, we see that and . This implies that , and hence is almost limited.

The next result follows immediately from Proposition 2.3 of [5] combined with Proposition 1.

Corollary 2. A Banach lattice with the property has the dual positive Schur property if and only if the identity operator on is almost limited.

The following result shows that if a positive almost limited operator has its range in a Banach lattice with the property , then every positive operator that it dominates (i.e., ) is also almost limited.

Proposition 3. Let and be two Banach lattices such that has the property . If a positive operator is dominated by an almost limited operator, then itself is almost limited.

Proof. Let be two operators such that and is almost limited. Let be a disjoint sequence in such that . As is almost limited, . Using the inequalities , we see that for all , from which we get . Now, by Proposition 1,   is well almost limited.

The next remark will be useful in further considerations.

Remark 4. We have the following.(1)Consider the scheme of operators . It is easy to see that if is an almost limited operator, then is likewise almost limited.(2)Consider the scheme of operators .(a)If is an almost limited operator, then is not necessarily almost limited. In fact, by a result in [6], there exists a nonregular operator , which is certainly not compact. So by Proposition 4.3 of [4], is not almost limited. If is the identity operator on , then is almost limited but is not almost limited.(b)However, if has the dual positive Schur property (e.g., ) and has the property and is positive, then is an almost limited operator. In fact, according to Proposition 1, let be a positive disjoint null sequence. Clearly holds in . Since has the dual positive Schur property, then , and hence , as desired.

Our next major result characterizes pairs of Banach lattices , for which every positive almost limited operator is weakly compact.

Theorem 5. Let and be two Banach lattices such that is -Dedekind complete. Then the following assertions are equivalent.

(1) Every almost limited operator is weakly compact.(2) Every positive almost limited operator is weakly compact.(3) One of the following statements is valid:(a) is reflexive;(b) the norm of is order continuous.

Proof. The proof is obvious.
Assume by way of contradiction that is not reflexive and the norm of is not order continuous. We have to construct a positive almost limited operator which is not weakly compact.
Indeed, since the norm of is not order continuous, then by Corollary 2.4.3 of [2] we may assume that is a closed sublattice of . As is not reflexive, then is not reflexive, and hence the closed unit ball of is not weakly compact. So, from , we see that is not weakly compact. Then, by the Eberlein-Šmulian theorem one can find a sequence in which does not have any weakly convergent subsequence. Consider the positive operator defined by for all . By Remark 4(2b) is an almost limited operator. But is not weakly compact. In fact, if were weakly compact, then would weakly be compact. Note that for every . So, if is the usual basis element in , then so that would have a weakly convergent subsequence. This contradicts the choice of . Therefore, is not weakly compact, as desired.
In this case, every operator from into is weakly compact.
By Theorem 4.2 of [4] we see that is -weakly compact, and by Theorem 5.61 of [1] is well weakly compact.

By a similar proof as the previous theorem, we obtain the following result.

Theorem 6. Let be a Banach space and let be a -Dedekind complete Banach lattice. Then the following assertions are equivalent.(1)Every almost limited operator is weakly compact.(2)One of the following statements is valid:(a) is reflexive;(b)the norm of is order continuous.

As a consequence of Theorem 5, we obtain an operator characterization of order continuity of the norm of a -Dedekind complete Banach lattice.

Corollary 7. Let be a -Dedekind complete Banach lattice. Then the following statements are equivalent.(1)Every almost limited operator from into is weakly compact.(2)Every positive almost limited operator from into is weakly compact.(3)The norm of is order continuous.

Another consequence of Theorem 5 is the following result.

Corollary 8. For a Banach lattice , the following statements are equivalent.(1)Every positive operator from to an arbitrary infinite dimensional AM-space is weakly compact.(2)Every positive operator is weakly compact.(3) is reflexive.

Proof. and are obvious.
The proof follows from Theorem 5.

The following result characterizes Banach lattice for which every positive almost limited operator has a weakly compact square.

Theorem 9. Let be a -Dedekind complete Banach lattice. Then the following statements are equivalent.(1)Every positive almost limited operator from into is weakly compact.(2)For every positive almost limited operator from into , the operator is weakly compact.(3)The norm of is order continuous.

Proof. The proof is obvious.
Assume by way of contradiction that the norm of is not order continuous. So, by Theorem 4.14 of [1], there exists a disjoint sequence satisfying and for all and for some . We can now proceed analogously to the proof of Proposition 0.5.5 of [7]. Let be of norm one such that and let be the band projection onto , where is the band generated by . If , then for , , and . Hence the operator defined by is a lattice isomorphism from into , where denotes the order limit of the sequence of the partial sums for each . Also, let be the positive operator defined by So, by Remark 4(2b), the positive operator defined by is almost limited. But is not weakly compact. In fact, let for each , and note that . Clearly , and hence for all . If is a weak limit of a subsequence of , then it is easy to see that and must hold. By Theorem 3.52 of [1] we have , and hence . But this contradicts for all . Thus has no weakly convergent subsequence, and hence is not weakly compact, as desired.
The proof follows from Theorem 5.

Finally, note that a weakly compact operator need not be almost limited. In fact, the identity operator of the Banach lattice is weakly compact but it is not almost limited. However, if has the positive Schur property, then the two classes coincide. The details follow.

Proposition 10. An operator from a Banach space to a Banach lattice with the positive Schur property is weakly compact if and only if it is almost limited.

Proof. The “if” part follows from Theorem 6. For the “only if” part, assume that is weakly compact. It follows from Theorem 3.4 of [8] that is -weakly compact, and hence is almost limited [4, Theorem 4.2].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.