On the Lattice Properties of Almost L-Weakly and Almost M-Weakly Compact Operators

Abstract

We establish the domination property and some lattice approximation properties for almost L-weakly and almost M-weakly compact operators. Then, we consider the linear span of positive almost L-weakly (resp., almost M-weakly) compact operators and give results about when they form a Banach lattice and have an order continuous norm.

1. Introduction and Notation

In this article, we denote real Banach spaces by and and real Banach lattices by and . The closed unit ball and the norm dual of are denoted by and , respectively. denotes the positive cone of , i.e., . Let . The positive part, the negative part, and the modulus of are given by , , and , respectively. For all with , the order interval between and is denoted by . We write for the solid hull of a set . By an operator , we mean a bounded linear mapping. The space of all operators from into is denoted by . If is an operator, its adjoint is defined by for each and for each . The space of all regular operators from into is denoted by . If is an operator with modulus, then the regular norm of is given by . For any unexplained notion and terminology, we refer to [1, 2].

Recently, the class of compact and related operators was studied extensively (for instance in [3, 4]). In [3], some identities and estimates for the Hausdorff measures of noncompactness of some operators on the fractional sets of sequences of fractional orders were established, and some classes of compact operators on the fractional sets of sequences were characterized. Also, necessary and sufficient conditions for the class of compact matrix operators from the fractional sets of sequences into the set of bounded sequences were given. In [4], power bounded -isometric Banach space operator was shown to be polaroid, and the polaroid property for -quasi left -invertible operators was proved. In approximation theory, many authors studied some estimates on the positive linear operators with an emphasis on the Kantorovich operators, Durrmeyer-Bernstein operators, and exponential type operators [5–7]. These types of operators have nice and interesting convergence properties. The approximation process by the sequence of positive linear operators for integrable or continuous functions was presented [5–7]. The common properties of these studies and the present work are positivity of linear operators, uniform convergence of sequences, and Banach lattices (e.g., and C[0,1]).

The class of compact (resp., weakly compact) operators does not satisfy the domination property [1, 2]. In other words, if two positive operators between Banach lattices satisfy and is compact (resp., weakly compact), then is not necessarily compact (resp., weakly compact). Also, a compact operator (resp., weakly compact operator) between Banach lattices need not possess a modulus ([1], p. 277). In [8], Meyer-Nieberg introduced the classes of L-weakly and M-weakly compact operators to overcome some difficulties in studying compact and weakly compact operators. Recall that an operator is called L-weakly compact if is an L-weakly compact set, i.e., every disjoint sequence in the solid hull of converges to zero in norm. An operator is called M-weakly compact if for each norm bounded disjoint sequence in , we have . Note that is L-weakly compact (resp., M-weakly compact) if and only if its adjoint is M-weakly compact (resp., L-weakly compact) ([2], Proposition 3.6.11). In contrast to compact and weakly compact operators, the domination property holds for L-weakly (resp., M-weakly) compact operators ([9], Proposition 2.1). However, an L-weakly (resp., M-weakly) compact operator need not have a modulus ([10], Theorem 2.2). In order to study these operator classes as vector lattices, Bayram and Wickstead worked in the linear span of positive L-weakly (resp., M-weakly) compact operators between Banach lattices [9]. For more details about these classes of operators, we refer to [1, 2, 8–10].

The class of almost L-weakly (resp., almost M-weakly) compact operators was introduced in [11] as a generalization of that of L-weakly (resp., M-weakly) compact operators. Recall that an operator is called almost L-weakly compact if maps relatively weakly compact subsets of onto L-weakly compact subsets of , and an operator is called almost M-weakly compact if for each disjoint sequence in and for each weakly convergent sequence in , we have . Every L-weakly (resp., M-weakly) compact operator is almost L-weakly (resp., almost M-weakly) compact, but the converse is not true in general [11]. For example, the identity operator is almost L-weakly compact but not L-weakly compact, and the identity operator is almost M-weakly compact but not M-weakly compact ([11], p. 1435). Note that an operator is almost M-weakly compact if and only if its adjoint is almost L-weakly compact, and an operator is almost L-weakly compact whenever its adjoint is almost M-weakly compact ([11], Theorem 2.5). The relationship between almost L-weakly (resp., almost M-weakly) compact operators and other classes of operators (e.g., compact operators, weakly compact operators, L-weakly, and M-weakly compact operators) was studied in the literature [11–13].

In the sequel, we will use the following notations:

is the space of L-weakly compact operators from into

is the space of M-weakly compact operators from into

is the space of almost L-weakly compact operators from into

is the space of almost M-weakly compact operators from into

In this paper, our aim is to study the vector lattice properties of almost L-weakly and almost M-weakly compact operators. First, we will show that both classes of operators satisfy the domination property. To establish this, we will make use of the class of almost Dunford-Pettis operators, which was introduced by Sanchez [14]. Next, we will present some lattice approximation properties for almost L-weakly and almost M-weakly compact operators. Then, we will consider the spaces and and investigate whether these spaces are Banach lattices with the regular norm. Here, we will make use of the domination property and obtain analogous results to that of [9]. Finally, we will present some necessary and sufficient conditions for and to have an order continuous regular norm.

Let us recall some definitions and well-known facts. A Banach lattice is said to have an order continuous norm if for each net in with , we have . Here, the notation means that is decreasing, its infimum exists, and . By , we denote the maximal (order) ideal of on which the induced norm is order continuous. Note that is closed, and every L-weakly compact subset of is contained in ([2], p. 212). Let be a lattice seminorm on . A subset of is called approximately order bounded with respect to if for every there exists such that , where ([2], p. 73). We have if and only if for all ([2], p. 73). An operator is called semicompact if for each there exists such that , i.e., is an approximately order bounded set in . Let be an operator. The lattice seminorm on is given by ([2], p. 192). From [14, 15], an operator is called almost Dunford-Pettis if for every weakly null sequence consisting of pairwise disjoint elements in , we have .

2. Main Results

2.1. Domination Property

In the following result, we show that the class of almost L-weakly compact operators satisfies the domination property.

Theorem 1. Let and be Banach lattices and be positive operators satisfying . If is almost L-weakly compact, then is also almost L-weakly compact.

Proof. Let be a positive almost L-weakly compact operator. It follows from Proposition 2.3 of [11] that is almost Dunford-Pettis. It is known that the class of almost Dunford-Pettis operators satisfies the domination property ([16], Corollary 2.3). Hence, the operator is almost Dunford-Pettis. Now, let be a relatively weakly compact subset and . Since is almost Dunford-Pettis, the set is approximately order bounded with respect to the lattice seminorm ([16], Proposition 2.1). So, there exists some such that for all . On the other hand, since is almost L-weakly compact, we have ([12], Proposition 1). By using the fact that and is a solid set, it is easy to see that . Put . Then, for all , we have By using (1) and (2), we obtain for all . From the identity , we see that . Since , the set is approximately order bounded in . Therefore, is an L-weakly compact set ([2], Proposition 3.6.2). As a result, is an almost L-weakly compact operator.

The domination property for almost M-weakly compact operators follows easily from Theorem 1.

Theorem 2. Let and be Banach lattices and be positive operators satisfying . If is almost M-weakly compact, then is also almost M-weakly compact.

Proof. Clearly, we have . Since is almost M-weakly compact, its adjoint is almost L-weakly compact ([11], Theorem 2.5). Theorem 1 yields that the operator is also almost L-weakly compact. Using Theorem 2.5 of [11] again, we conclude that is almost M-weakly compact.

2.2. Lattice Approximation Properties

The class of almost L-weakly compact operators satisfies the following lattice approximation property.

Proposition 3. Let be an almost L-weakly compact operator from a Banach space into a Banach lattice . If is a relatively weakly compact subset of , then for each , there exists lying in the ideal generated by such that for all .

Proof. Let be a relatively weakly compact set. Consider the solid hull of and call it . Clearly, the set is bounded and solid. Since is almost L-weakly compact, each disjoint sequence in converges to zero in norm. Define the identity operator and let . If is an arbitrary disjoint sequence in , then Let . It follows from Theorem 4.36 of [1] that there exists some lying in the ideal generated by such that for all . Since , we have for all . As the ideal generated by is equal to the ideal generated by , we obtain the desired result.

For the class of almost M-weakly compact operators, we first give a characterization and obtain from that a lattice approximation property.

Proposition 4. A positive operator between Banach lattices is almost M-weakly compact if and only if given any relatively weakly compact set and a disjoint sequence in , the sequence converges uniformly to zero on the solid hull of .

Proof. For the forward implication, we give a similar proof to that of ([1], Theorem 5.100). Assume that is almost M-weakly compact. Let be a relatively weakly compact set and be a disjoint sequence in . Fix . First, we claim that there exist some and such that for all and for all .
To see this, assume on the contrary that (6) is false. So, for each and for each , there exist and such that . By induction, we can see that there exist a sequence and a subsequence of such that for all . Put and . Clearly, we have for all . We define and note that is a disjoint sequence in the solid hull of ([1], Lemma 4.35). From Theorem 4.34 of [1], we see that in . Moreover, is a disjoint sequence with positive terms in . As is almost M-weakly compact, we have . Now, define the constant sequence in . Clearly, is weakly convergent. Since is almost M-weakly compact, we obtain . From the inequality , we have a contradiction. Thus, (6) is true. We pick and such that (6) holds. Define the constant sequence in . As is almost M-weakly compact and is weakly convergent, . So there is some such that for all . Let be arbitrary. Then, there is some with . We have for all . This proves that converges uniformly to zero on the solid hull of .
For the converse, let be a disjoint sequence in and be a weakly convergent sequence in . Put . Then, is a relatively weakly compact subset of . By assumption, the sequence converges uniformly to zero on the solid hull of ; that is, . For all , we have So , and hence, is almost M-weakly compact.

Corollary 5. Let be a positive almost M-weakly compact operator between Banach lattices and be a relatively weakly compact set. Then, for each , there exists satisfying for all and for all .

Proof. Define the seminorm on by ([1], Theorem 5.100 (2)). Then, is continuous on . From Proposition 4, we have for each disjoint sequence in . Since is a solid set, Theorem 4.36 of [1] yields that for each there exists satisfying for all and for all .

2.3. The Spaces and

In this section, we study the Banach lattice properties of and . In ([9], Theorem 2.2), Bayram and Wickstead proved that is always a Dedekind complete Banach lattice with the regular norm. In a similar way, we show that the same is true for .

Theorem 6. Let and be Banach lattices. Then, , equipped with the regular norm, is a Dedekind complete Banach lattice.

Proof. Let . Then, , where . Put . Clearly, . Since , , and is Dedekind complete, it follows that has a modulus in and hence in . As , Theorem 1 yields that . Thus, is a vector lattice. As is Dedekind complete, is also Dedekind complete. Now, we show that is closed in with respect to the regular norm. To this end, assume that is in the closure of . Then, there exists a sequence in such that . Since has a modulus for each , Theorem 2.1 of [17] implies that has a modulus in and that . Therefore, . Since for all and is closed in ([11], Proposition 2.1 (1)), we have . This shows that and are also in (by Theorem 1), and so is . As a result, is a Dedekind complete Banach lattice.

Assuming that is Dedekind complete, we have the following result for almost M-weakly compact operators. The proof is similar to the proof of the above theorem with the corresponding results for almost M-weakly compact operators (Theorem 2 and Proposition 2.1 (2) of [11]).

Theorem 7. Let and be Banach lattices with Dedekind complete. Then, , equipped with the regular norm, is a Dedekind complete Banach lattice.

Now, we investigate conditions under which and have an order continuous norm, respectively. For , we have exactly the same necessary and sufficient condition with ([9], Theorem 3.1).

Theorem 8. Let and be Banach lattices with . Then, the regular norm on is order continuous if and only if has an order continuous norm.

Proof. Assume that has an order continuous norm. By Theorem 1, for all , the order intervals are the same in and in . Let . Since has an order continuous norm, is M-weakly compact ([12], Theorem 4). As and the norm of is order continuous, we conclude that the norm on is order continuous ([2], Proposition 3.6.19). Conversely, suppose that the norm on is order continuous. Fix and let be a net in such that . Define by . Then, , and hence, by assumption. So . This shows that the norm of is order continuous.

We now turn our attention to .

Proposition 9. Let and be Banach lattices with . If the regular norm on is order continuous, then has an order continuous norm.

Proof. Let be a net in such that . Fix and define by . It is easily seen that each is M-weakly compact (hence almost M-weakly compact). Then, for each , . Thus, . As has an order continuous norm, . As a result, .

The condition that having an order continuous norm is not sufficient for to have an order continuous regular norm. We demonstrate this fact in the following example.

Example 10. We consider and note that has an order continuous norm. We claim that does not have an order continuous regular norm. Suppose on the contrary it has. Let be the identity operator. Then, is a positive almost M-weakly compact operator ([12], p. 143). The order interval is the same in and in (by Theorem 2). So the norm is order continuous on . By Proposition 3.6.19 of [2], we infer that the identity operator is M-weakly compact, a contradiction ([12], p. 143).

Below we give some sufficient conditions under which admits an order continuous regular norm.

Proposition 11. Let and be Banach lattices. If one of the following conditions holds, then has an order continuous regular norm. (i) and have an order continuous norm.(ii) is reflexive.(iii) is an AM-space with unit, and has an order continuous norm.

Proof. Let . We note that in each part ((i)–(iii)) has an order continuous norm. By Proposition 3.6.19 of [2], it is sufficient to show that is M-weakly compact. Suppose (i) holds. Since is almost M-weakly compact, its adjoint is almost L-weakly compact. By ([12], Corollary 4,) is L-weakly compact, and hence, is M-weakly compact. Now, suppose (ii) holds. Again, is almost L-weakly compact. Since is reflexive (because is reflexive), is L-weakly compact. Consequently, is M-weakly compact. Finally, suppose (iii) holds. Since is an AM-space with unit, is semicompact ([1], p. 339, Ex. 15). As has an order continuous norm, is L-weakly compact ([2], Corollary 3.6.14) and hence weakly compact. It follows from Theorem 5.62 of [1] that is M-weakly compact, and we are done.

We end with a question: “What is a necessary and sufficient condition for the Banach lattice to have an order continuous regular norm?”

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Conflicts of Interest

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