Weak compactness of almost limited operators

The paper is devoted to the relationship between almost limited operators and weakly compacts operators. We show that if $F$ is a $\sigma $-Dedekind complete Banach lattice then, every almost limited operator $T:E\rightarrow F $ is weakly compact if and only if $E$ is reflexive or the norm of $F$ is order continuous. Also, we show that if $E$ is a $\sigma $-Dedekind complete Banach lattice then the square of every positive almost limited operator $ T:E\rightarrow E$ is weakly compact if and only if the norm of $E$ is order continuous.


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 and B + E := B E ∩ E + is the positive part of B E . We will use the term operator T : X → Y between two Banach spaces to mean a bounded linear mapping. We refer to [1,5] for unexplained terminology of the Banach lattice theory and positive operators.
Let us recall that 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, sup x∈A |f n (x)| → 0. An operator T : X → Y is said to be limited whenever T (B X ) is a limited set in Y , equivalently, whenever T * (f n ) → 0 for every weak * null sequence (f n ) ⊂ Y * .
Recently, the authors of [2] considered the disjoint version of limited sets by introducing the class of almost limited sets in Banach lattices. From [2] 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.
From [4], an operator T : X → E is called almost limited if T (B X ) is an almost limited set in E, equivalently, T * (f n ) → 0 for every disjoint weak * null sequence (f n ) ⊂ E * . Note that an almost limited operator need not be weakly compact. In fact, the identity operator of the Banach lattice ℓ ∞ is almost limited but it is not weakly compact.
In this paper, we characterize pairs of Banach lattices E, F for which every almost limited operator T : E → F is weakly compact. More precisely, we will prove that if F is a σ-Dedekind complete Banach lattice then, every almost limited operator T : E → F is weakly compact if and only if E is reflexive or the norm of F is order continuous (Theorem 2.5). Next, we will prove that if E is a σ-Dedekind complete Banach lattice then the square of every positive almost limited operator T : E → E is weakly compact if and only if the norm of E is order continuous (Theorem 2.9). As consequences, we will give some interesting results.

Main results
Let us recall that a Banach lattice E is said to have the dual positive Schur property if f n → 0 for every weak * null sequence (f n ) ⊂ (E * ) + , equivalently, f n → 0 for every weak * null sequence (f n ) ⊂ (E * ) + consisting of pairwise disjoint terms (Proposition 2.3 of [8]). A Banach lattice E has the property (d) whenever |f n |∧|f m | = 0 and f n w * → 0 in E * imply |f n | w * → 0. It should be noted, by Proposition 1.4 of [8], that every σ-Dedekind complete Banach lattice has the property (d) but the converse is not true in general. In fact, the Banach lattice ℓ ∞ /c 0 has the property (d) but it is not σ-Dedekind complete [8,Remark 1.5].
Our first result shows that we can restrict sequences appearing in the definition of almost limited operator T : X → E to positive disjoint sequences if the Banach lattice E has the property (d).
The next result follows immediately from Proposition 2.3 of [8] combined with Proposition 2.1.

Corollary 2.2. A Banach lattice E with the property (d) has the dual positive Schur property if and only if the identity operator on E is almost limited.
The following result shows that if a positive almost limited operator T : E → F has its range in a Banach lattice with the property (d), then every positive operator S : E → F that it dominates (i.e., 0 ≤ S ≤ T ) is also almost limited. Proposition 2.3. Let E and F be two Banach lattices such that F has the property (d). If a positive operator S : E → F is dominated by an almost limited operator, then S itself is almost limited.
The next remark will be useful in further considerations.  Our next major result characterizes pairs of Banach lattices E, F for which every positive almost limited operator T : E → F is weakly compact.
Theorem 2.5. Let E and F be two Banach lattices such that F is σ-Dedekind complete. Then the following assertions are equivalent: (1) Every almost limited operator T : E → F is weakly compact.
(2) Every positive almost limited operator T : E → F is weakly compact.
(3) One of the following statements is valid: The norm of F is order continuous. Proof.
(2) ⇒ (3) Assume by way of contradiction that E is not reflexive and the norm of F is not order continuous. We have to construct a positive almost limited operator T : E → F which is not weakly compact.
Indeed, since the norm of F is not order continuous, then by Corollary 2.4.3 of [5] we may assume that ℓ ∞ is a closed sublattice of F . As E is not reflexive then E * is not reflexive, and hence the closed unit ball B E * of E * is not weakly compact. So, from B E * ⊂ B + E * − B + E * , we see that B + E * is not weakly compact. Then, by the Eberlein-Šmulian theorem one can find a sequence (f n ) in B + E * which does not have any weakly convergent subsequence. Consider the positive operator T : E → ℓ ∞ ⊂ F defined by for all x ∈ E. By Remark 2.4(2b) T is an almost limited operator. But T is not weakly compact. In fact, if T were weakly compact then T * : So, if e n is the usual basis element in ℓ 1 then T * (e n ) = f n so that (f n ) would have a weakly convergent subsequence. This contradicts the choice of (f n ). Therefore, T is not weakly compact, as desired.
(a) ⇒ (1) In this case, every operator from E into F is weakly compact. (b) ⇒ (1) By Theorem 4.2 of [4] we see that T is L-weakly compact, and by Theorem 5.61 of [1] T is well weakly compact.
By a similar proof as the previous theorem, we obtain the following result.
Theorem 2.6. Let X a Banach space and F a σ-Dedekind complete Banach lattice. Then the following assertions are equivalent: (1) Every almost limited operator T : X → F is weakly compact.
(2) One of the following statements is valid: (a) X is reflexive. (b) The norm of F is order continuous.
As a consequence of Theorem 2.5, we obtain an operator characterization of order continuity of the norm of a σ-Dedekind complete Banach lattice.
Corollary 2.7. Let E be a σ-Dedekind complete Banach lattice. Then the following statements are equivalent: (1) Every almost limited operator T from E into E is weakly compact.
(2) Every positive almost limited operator T from E into E is weakly compact.
(3) The norm of E is order continuous.
The following result characterize Banach lattice E for which every positive almost limited operator T : E → E has a weakly compact square. (2) ⇒ (3) Assume by way of contradiction that the norm of E is not order continuous. So, by Theorem 4.14 of [1] there exists a disjoint sequence (u n ) ⊂ E + satisfying u n = 1 and 0 ≤ u n ≤ u for all n and for some u ∈ E + . We can now proceed analogously to the proof of Proposition 0.5.5 of [7]. Let g n ∈ E * + be of norm one and such that g n (u n ) = u n = 1 and let P n be the band projection onto {u n } dd , where {u n } dd is the band generated by {u n }. If f n = g n • P n , then f n ∧ f m = 0 for n = m, sup n f n ≤ 1 and f n (u m ) = δ nm . Hence the operator S : ℓ ∞ → E defined by ∞ n=1 t n u n denotes the order limit of the sequence of the partial sums m n=1 t n u n for each (t n ) ∞ n=1 ∈ ℓ ∞ . Also, let R : E → ℓ ∞ be the positive operator defined by So, by Remark 2.4(2b), the positive operator T = S • R : E → F defined by is almost limited. But T is not weakly compact. In fact, let x n = n k=1 u k for each n, and note that 0 ≤ x n ↑≤ u. Clearly T (u n ) = u n , and hence T (x n ) = x n for all n. If x is a weak limit of a subsequence of (x n ), then it is easy to see that x n ↑ x and x n w → x must hold. By Theorem 3.52 of [1] we have x n − x → 0, and hence x n+1 − x n → 0. But this contradicts x n+1 − x n = u n+1 = 1 for all n. Thus (x n ) has no weakly convergent subsequence, and hence T is not weakly compact, as desired.
Finally, note that a weakly compact operator T : X → F need not be almost limited. In fact, the identity operator of the Banach lattice ℓ 2 is weakly compact but it is not almost limited. However, if F has the positive Schur property, then the two class coincide. The details follow. Proof. The "if" part follows from Theorem 2.6. For the "only if" part, assume that T : X → F is weakly compact. It follows from Theorem 3.4 of [3] that T is L-weakly compact, and hence T is almost limited [4,Theorem 4.2].