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

Authors: Barış Akay; Ömer Gök
Journal: Journal of Function Spaces
 (2021)

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## 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.


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## Body

## 1. Introduction and Notation

In this article, we denote real Banach spaces byX and Y and real Banach lattices by E and F. The closed unit ball and the norm dual of X are denoted by BX and X′, respectively. E+ denotes the positive cone of E, i.e., E+=x∈E:x≥0. Let x∈E. The positive part, the negative part, and the modulus of x are given by x+=x∨0, x−=−x∨0, and x=−x∨x, respectively. For all x,y∈E with x≤y, the order interval between x and y is denoted by x,y=z∈E:x≤z≤y . We write SolA for the solid hull of a set A⊆E. By an operator T:X⟶Y, we mean a bounded linear mapping. The space of all operators from X into Y is denoted by LX,Y. If T:X⟶Y is an operator, its adjoint T′:Y′⟶X′ is defined by T′fx=fTx for each f∈Y′ and for each x∈X. The space of all regular operators from E into F is denoted by LrE,F. If T:E⟶F is an operator with modulus, then the regular norm of T is given by Tr=T. 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 m-isometric Banach space operator was shown to be polaroid, and the polaroid property for n-quasi left m-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., L10,1 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 S,T:E⟶F between Banach lattices satisfy 0≤S≤T and T is compact (resp., weakly compact), then S 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 T:X⟶F is called L-weakly compact if TBX is an L-weakly compact set, i.e., every disjoint sequence in the solid hull of TBX converges to zero in norm. An operator T:E⟶Y is called M-weakly compact if for each norm bounded disjoint sequence xn in E, we have Txn⟶0. Note that T:E⟶F is L-weakly compact (resp., M-weakly compact) if and only if its adjoint T′ 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 T:X⟶F is called almost L-weakly compact if T maps relatively weakly compact subsets of X onto L-weakly compact subsets of F, and an operator T:E⟶Y is called almost M-weakly compact if for each disjoint sequence xn in BE and for each weakly convergent sequence fn in Y′, we have fnTxn⟶0. 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 I:ℓ1⟶ℓ1 is almost L-weakly compact but not L-weakly compact, and the identity operator I:ℓ∞⟶ℓ∞ is almost M-weakly compact but not M-weakly compact ([11], p. 1435). Note that an operator T:E⟶Y is almost M-weakly compact if and only if its adjoint T′ is almost L-weakly compact, and an operator T:X⟶F is almost L-weakly compact whenever its adjoint T′ 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:WLE,F is the space of L-weakly compact operators from E into FWME,F is the space of M-weakly compact operators from E into FAWLE,F is the space of almost L-weakly compact operators from E into FAWME,F is the space of almost M-weakly compact operators from E into FWLrE,F=T1−T2:T1,T2∈WLE,F+WMrE,F=T1−T2:T1,T2∈WME,F+AWLrE,F=T1−T2:T1,T2∈AWLE,F+AWMrE,F=T1−T2:T1,T2∈AWME,F+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 AWLrE,F and AWMrE,F 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 AWLrE,F and AWMrE,F to have an order continuous regular norm.Let us recall some definitions and well-known facts. A Banach latticeE is said to have an order continuous norm if for each net xα in E with xα↓0, we have xα⟶0. Here, the notation xα↓0 means that xα is decreasing, its infimum exists, and infxα=0. By Ea, we denote the maximal (order) ideal of E on which the induced norm is order continuous. Note that Ea is closed, and every L-weakly compact subset of E is contained in Ea ([2], p. 212). Let ρ be a lattice seminorm on E. A subset A of E is called approximately order bounded with respect to ρ if for every ε>0 there exists u∈E+ such that A⊆−u,u+εBρ, where Bρ=x∈E:ρx≤1 ([2], p. 73). We have A⊆−u,u+εBρ if and only if ρx−u+≤ε for all x∈A ([2], p. 73). An operator T:X⟶F is called semicompact if for each ε>0 there exists u∈F+ such that TBX⊆−u,u+εBF, i.e., TBX is an approximately order bounded set in F. Let T:E⟶Y be an operator. The lattice seminorm qT on E is given by qTx=supTy:y≤x ([2], p. 192). From [14, 15], an operator T:E⟶Y is called almost Dunford-Pettis if for every weakly null sequence xn consisting of pairwise disjoint elements in E, we have Txn⟶0.


## 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.
LetE and F be Banach lattices and S,T:E⟶F be positive operators satisfying 0≤S≤T. If T is almost L-weakly compact, then S is also almost L-weakly compact.Proof.
LetT:E⟶F be a positive almost L-weakly compact operator. It follows from Proposition 2.3 of [11] that T 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 S is almost Dunford-Pettis. Now, let W⊆E be a relatively weakly compact subset and ε>0. Since S is almost Dunford-Pettis, the set W is approximately order bounded with respect to the lattice seminorm qS ([16], Proposition 2.1). So, there exists some u∈E+ such that
(1)Sx−u+≤qSx−u+≤ε,for all x∈W. On the other hand, since T is almost L-weakly compact, we have TE⊆Fa ([12], Proposition 1). By using the fact that 0≤S≤T and Fa is a solid set, it is easy to see that SE⊆Fa. Put z=Su∈F+a. Then, for all x∈W, we have
(2)Sx−z+=Sx−Su+≤Sx−Su+≤Sx−u+≤Sx−u+.
By using (1) and (2), we obtain Sx−z+≤ε for all x∈W. From the identity Sx=Sx∧z+Sx−z+, we see that SW⊆−z,z+εBF. Since z∈F+a, the set SW is approximately order bounded in Fa. Therefore, SW is an L-weakly compact set ([2], Proposition 3.6.2). As a result, S is an almost L-weakly compact operator.The domination property for almost M-weakly compact operators follows easily from Theorem1.Theorem 2.
LetE and F be Banach lattices and S,T:E⟶F be positive operators satisfying 0≤S≤T. If T is almost M-weakly compact, then S is also almost M-weakly compact.Proof.
Clearly, we have0≤S′≤T′. Since T is almost M-weakly compact, its adjoint T′ is almost L-weakly compact ([11], Theorem 2.5). Theorem 1 yields that the operator S′ is also almost L-weakly compact. Using Theorem 2.5 of [11] again, we conclude that S 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.
LetT:X⟶F be an almost L-weakly compact operator from a Banach space X into a Banach lattice F. If W is a relatively weakly compact subset of X, then for each ε>0, there exists uε∈F+ lying in the ideal generated by TW such that
(3)Tx−uε+<ε,for all x∈W.Proof.
LetW⊆X be a relatively weakly compact set. Consider the solid hull of TW and call it A. Clearly, the set A is bounded and solid. Since T is almost L-weakly compact, each disjoint sequence in A converges to zero in norm. Define the identity operator I:F⟶F and let px=x. If xn is an arbitrary disjoint sequence in A, then pIxn=xn⟶0. Let ε>0. It follows from Theorem 4.36 of [1] that there exists some u∈E+ lying in the ideal generated by A such that
(4)Iy−u+<ε,for all y∈A. Since TW⊆A, we have
(5)Tx−u+<ε,for all x∈W. As the ideal generated by TW is equal to the ideal generated by A, 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 operatorT:E⟶F between Banach lattices is almost M-weakly compact if and only if given any relatively weakly compact set V⊆F′ and a disjoint sequence xn in BE, the sequence Txn converges uniformly to zero on the solid hull of V.Proof.
For the forward implication, we give a similar proof to that of ([1], Theorem 5.100). Assume that T is almost M-weakly compact. Let V⊆F′ be a relatively weakly compact set and xn⊆E+ be a disjoint sequence in BE. Fix ε>0. First, we claim that there exist some g∈F′+ and k∈ℕ+ such that
(6)f−g+Txn<ε,for all f∈V and for all n>k.
To see this, assume on the contrary that (6) is false. So, for each g∈F′+ and for each k∈ℕ+, there exist f∈V and m>k such that f−g+Txm≥ε. By induction, we can see that there exist a sequence fn⊆V and a subsequence zn of xn such that
(7)fn+1−4n∑i=1n‍fi+Tzn≥ε,for all n. Put f=∑n=1∞‍2−nfn and hn=fn+1−4n∑i=1n‍fi+. Clearly, we have hnTzn≥ε for all n. We define
(8)gn=fn+1−4n∑i=1n‍fi−2−nf+,and note that gn is a disjoint sequence in the solid hull of V ([1], Lemma 4.35). From Theorem 4.34 of [1], we see that gn⟶w0 in F′. Moreover, zn is a disjoint sequence with positive terms in BE. As T is almost M-weakly compact, we have gnTzn⟶0. Now, define the constant sequence in=f in F′. Clearly, in is weakly convergent. Since T is almost M-weakly compact, we obtain inTzn=fTzn⟶0. From the inequality 0≤hn≤gn+2−nf, we have
(9)0<ε≤hnTzn≤gnTzn+2−nfTzn⟶0,a contradiction. Thus, (6) is true. We pick g∈F′+ and k∈ℕ+ such that (6) holds. Define the constant sequence jn=g in F′. As T is almost M-weakly compact and jn is weakly convergent, jnTxn=gTxn⟶0. So there is some m>k such that gTxn<ε for all n≥m. Let h∈SolV be arbitrary. Then, there is some f∈V with h≤f. We have
(10)hTxn≤hTxn≤fTxn≤f−g+Txn+gTxn≤ε+ε=2ε,for all n≥m. This proves that Txn converges uniformly to zero on the solid hull of V.
For the converse, letxn be a disjoint sequence in BE and fn be a weakly convergent sequence in F′. Put V=fn:n∈ℕ. Then, V is a relatively weakly compact subset of F′. By assumption, the sequence Txn converges uniformly to zero on the solid hull of V; that is, supf∈SolVfTxn⟶0. For all n, we have
(11)fnTxn≤supf∈VfTxn≤supf∈SolVfTxn.
SofnTxn⟶0, and hence, T is almost M-weakly compact.Corollary 5.
LetT:E⟶F be a positive almost M-weakly compact operator between Banach lattices and V⊆F′ be a relatively weakly compact set. Then, for each ε>0, there exists uε∈E+ satisfying
(12)fTx−uε+<ε,for all x∈BE and for all f∈V.Proof.
Define the seminormρ on E by ρx=supfx:f∈V ([1], Theorem 5.100 (2)). Then, ρ is continuous on E. From Proposition 4, we have ρTxn⟶0 for each disjoint sequence xn in BE. Since BE is a solid set, Theorem 4.36 of [1] yields that for each ε>0 there exists uε∈E+ satisfying
(13)fTx−uε+<ε,for all f∈V and for all x∈BE.


### 2.3. The SpacesAWLrE,F and AWMrE,F

In this section, we study the Banach lattice properties ofAWLrE,F and AWMrE,F. In ([9], Theorem 2.2), Bayram and Wickstead proved that WLrE,F is always a Dedekind complete Banach lattice with the regular norm. In a similar way, we show that the same is true for AWLrE,F.Theorem 6.
LetE and F be Banach lattices. Then, AWLrE,F, equipped with the regular norm, is a Dedekind complete Banach lattice.Proof.
LetT∈AWLrE,F. Then, T=T1−T2, where T1,T2∈AWLrE,F+. Put U=T1+T2. Clearly, ±T≤U. Since TE⊆Fa, UE⊆Fa, and Fa is Dedekind complete, it follows that T has a modulus in LrE,Fa and hence in LrE,F. As T≤U, Theorem 1 yields that T∈AWLrE,F. Thus, AWLrE,F is a vector lattice. As Fa is Dedekind complete, AWLrE,F is also Dedekind complete. Now, we show that AWLrE,F is closed in LrE,F with respect to the regular norm. To this end, assume that T is in the closure of AWLrE,F. Then, there exists a sequence Tn in AWLrE,F such that Tn−Tr⟶0. Since Tn has a modulus for each n, Theorem 2.1 of [17] implies that T has a modulus in LrE,F and that Tn−Tr⟶0. Therefore, Tn−T⟶0. Since Tn∈AWLE,F for all n and AWLE,F is closed in LE,F ([11], Proposition 2.1 (1)), we have T∈AWLE,F. This shows that T+ and T− are also in AWLE,F (by Theorem 1), and so is T. As a result, AWLrE,F is a Dedekind complete Banach lattice.Assuming thatF 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.
LetE and F be Banach lattices with F Dedekind complete. Then, AWMrE,F, equipped with the regular norm, is a Dedekind complete Banach lattice.Now, we investigate conditions under whichAWLrE,F and AWMrE,F have an order continuous norm, respectively. For AWLrE,F, we have exactly the same necessary and sufficient condition with ([9], Theorem 3.1).Theorem 8.
LetE and F be Banach lattices with Fa≠0. Then, the regular norm on AWLrE,F is order continuous if and only if E′ has an order continuous norm.Proof.
Assume thatE′ has an order continuous norm. By Theorem 1, for all T∈AWLrE,F+, the order intervals 0,T are the same in AWLrE,F and in LrE,F. Let T∈AWLrE,F+. Since E′ has an order continuous norm, T is M-weakly compact ([12], Theorem 4). As TE⊆Fa and the norm of Fa is order continuous, we conclude that the norm on 0,T is order continuous ([2], Proposition 3.6.19). Conversely, suppose that the norm on AWLrE,F is order continuous. Fix 0≠y∈F+a and let fα be a net in E′ such that fα↓0. Define Tα:E⟶F by Tαx=fαxy. Then, Tα↓0, and hence, Tα⟶0 by assumption. So Tα=fαy⟶0. This shows that the norm of E′ is order continuous.We now turn our attention toAWMrE,F.Proposition 9.
LetE and F be Banach lattices with E′a≠0. If the regular norm on AWMrE,F is order continuous, then F has an order continuous norm.Proof.
Letyα be a net in F such that yα↓0. Fix 0≠f∈E′a and define Tα:E⟶F by Tαx=fxyα. It is easily seen that each Tα is M-weakly compact (hence almost M-weakly compact). Then, for each x∈E+, Tαx↓0. Thus, Tα↓0. As AWMrE,F has an order continuous norm, Tα=fyα⟶0. As a result, yα⟶0.The condition thatF having an order continuous norm is not sufficient for AWMrE,F to have an order continuous regular norm. We demonstrate this fact in the following example.Example 10.
We considerAWMrc0,c0 and note that c0 has an order continuous norm. We claim that AWMrc0,c0 does not have an order continuous regular norm. Suppose on the contrary it has. Let I:c0⟶c0 be the identity operator. Then, I is a positive almost M-weakly compact operator ([12], p. 143). The order interval 0,I is the same in AWMrE,F and in LrE,F (by Theorem 2). So the norm is order continuous on 0,I. By Proposition 3.6.19 of [2], we infer that the identity operator I is M-weakly compact, a contradiction ([12], p. 143).Below we give some sufficient conditions under whichAWMrE,F admits an order continuous regular norm.Proposition 11.
LetE and F be Banach lattices. If one of the following conditions holds, then AWMrE,F has an order continuous regular norm.
(i)
E′ and F′′ have an order continuous norm.(ii)
F is reflexive.(iii)
E is an AM-space with unit, and F has an order continuous norm.Proof.
LetT∈AWMrE,F+. We note that in each part ((i)–(iii)) F has an order continuous norm. By Proposition 3.6.19 of [2], it is sufficient to show that T is M-weakly compact. Suppose (i) holds. Since T is almost M-weakly compact, its adjoint T′ is almost L-weakly compact. By ([12], Corollary 4,) T′ is L-weakly compact, and hence, T is M-weakly compact. Now, suppose (ii) holds. Again, T′ is almost L-weakly compact. Since F′ is reflexive (because F is reflexive), T′ is L-weakly compact. Consequently, T is M-weakly compact. Finally, suppose (iii) holds. Since E is an AM-space with unit, T is semicompact ([1], p. 339, Ex. 15). As F has an order continuous norm, T is L-weakly compact ([2], Corollary 3.6.14) and hence weakly compact. It follows from Theorem 5.62 of [1] that T is M-weakly compact, and we are done.We end with a question: “What is a necessary and sufficient condition for the Banach latticeAWMrE,F to have an order continuous regular norm?”


## 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.
LetE and F be Banach lattices and S,T:E⟶F be positive operators satisfying 0≤S≤T. If T is almost L-weakly compact, then S is also almost L-weakly compact.Proof.
LetT:E⟶F be a positive almost L-weakly compact operator. It follows from Proposition 2.3 of [11] that T 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 S is almost Dunford-Pettis. Now, let W⊆E be a relatively weakly compact subset and ε>0. Since S is almost Dunford-Pettis, the set W is approximately order bounded with respect to the lattice seminorm qS ([16], Proposition 2.1). So, there exists some u∈E+ such that
(1)Sx−u+≤qSx−u+≤ε,for all x∈W. On the other hand, since T is almost L-weakly compact, we have TE⊆Fa ([12], Proposition 1). By using the fact that 0≤S≤T and Fa is a solid set, it is easy to see that SE⊆Fa. Put z=Su∈F+a. Then, for all x∈W, we have
(2)Sx−z+=Sx−Su+≤Sx−Su+≤Sx−u+≤Sx−u+.
By using (1) and (2), we obtain Sx−z+≤ε for all x∈W. From the identity Sx=Sx∧z+Sx−z+, we see that SW⊆−z,z+εBF. Since z∈F+a, the set SW is approximately order bounded in Fa. Therefore, SW is an L-weakly compact set ([2], Proposition 3.6.2). As a result, S is an almost L-weakly compact operator.The domination property for almost M-weakly compact operators follows easily from Theorem1.Theorem 2.
LetE and F be Banach lattices and S,T:E⟶F be positive operators satisfying 0≤S≤T. If T is almost M-weakly compact, then S is also almost M-weakly compact.Proof.
Clearly, we have0≤S′≤T′. Since T is almost M-weakly compact, its adjoint T′ is almost L-weakly compact ([11], Theorem 2.5). Theorem 1 yields that the operator S′ is also almost L-weakly compact. Using Theorem 2.5 of [11] again, we conclude that S 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.
LetT:X⟶F be an almost L-weakly compact operator from a Banach space X into a Banach lattice F. If W is a relatively weakly compact subset of X, then for each ε>0, there exists uε∈F+ lying in the ideal generated by TW such that
(3)Tx−uε+<ε,for all x∈W.Proof.
LetW⊆X be a relatively weakly compact set. Consider the solid hull of TW and call it A. Clearly, the set A is bounded and solid. Since T is almost L-weakly compact, each disjoint sequence in A converges to zero in norm. Define the identity operator I:F⟶F and let px=x. If xn is an arbitrary disjoint sequence in A, then pIxn=xn⟶0. Let ε>0. It follows from Theorem 4.36 of [1] that there exists some u∈E+ lying in the ideal generated by A such that
(4)Iy−u+<ε,for all y∈A. Since TW⊆A, we have
(5)Tx−u+<ε,for all x∈W. As the ideal generated by TW is equal to the ideal generated by A, 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 operatorT:E⟶F between Banach lattices is almost M-weakly compact if and only if given any relatively weakly compact set V⊆F′ and a disjoint sequence xn in BE, the sequence Txn converges uniformly to zero on the solid hull of V.Proof.
For the forward implication, we give a similar proof to that of ([1], Theorem 5.100). Assume that T is almost M-weakly compact. Let V⊆F′ be a relatively weakly compact set and xn⊆E+ be a disjoint sequence in BE. Fix ε>0. First, we claim that there exist some g∈F′+ and k∈ℕ+ such that
(6)f−g+Txn<ε,for all f∈V and for all n>k.
To see this, assume on the contrary that (6) is false. So, for each g∈F′+ and for each k∈ℕ+, there exist f∈V and m>k such that f−g+Txm≥ε. By induction, we can see that there exist a sequence fn⊆V and a subsequence zn of xn such that
(7)fn+1−4n∑i=1n‍fi+Tzn≥ε,for all n. Put f=∑n=1∞‍2−nfn and hn=fn+1−4n∑i=1n‍fi+. Clearly, we have hnTzn≥ε for all n. We define
(8)gn=fn+1−4n∑i=1n‍fi−2−nf+,and note that gn is a disjoint sequence in the solid hull of V ([1], Lemma 4.35). From Theorem 4.34 of [1], we see that gn⟶w0 in F′. Moreover, zn is a disjoint sequence with positive terms in BE. As T is almost M-weakly compact, we have gnTzn⟶0. Now, define the constant sequence in=f in F′. Clearly, in is weakly convergent. Since T is almost M-weakly compact, we obtain inTzn=fTzn⟶0. From the inequality 0≤hn≤gn+2−nf, we have
(9)0<ε≤hnTzn≤gnTzn+2−nfTzn⟶0,a contradiction. Thus, (6) is true. We pick g∈F′+ and k∈ℕ+ such that (6) holds. Define the constant sequence jn=g in F′. As T is almost M-weakly compact and jn is weakly convergent, jnTxn=gTxn⟶0. So there is some m>k such that gTxn<ε for all n≥m. Let h∈SolV be arbitrary. Then, there is some f∈V with h≤f. We have
(10)hTxn≤hTxn≤fTxn≤f−g+Txn+gTxn≤ε+ε=2ε,for all n≥m. This proves that Txn converges uniformly to zero on the solid hull of V.
For the converse, letxn be a disjoint sequence in BE and fn be a weakly convergent sequence in F′. Put V=fn:n∈ℕ. Then, V is a relatively weakly compact subset of F′. By assumption, the sequence Txn converges uniformly to zero on the solid hull of V; that is, supf∈SolVfTxn⟶0. For all n, we have
(11)fnTxn≤supf∈VfTxn≤supf∈SolVfTxn.
SofnTxn⟶0, and hence, T is almost M-weakly compact.Corollary 5.
LetT:E⟶F be a positive almost M-weakly compact operator between Banach lattices and V⊆F′ be a relatively weakly compact set. Then, for each ε>0, there exists uε∈E+ satisfying
(12)fTx−uε+<ε,for all x∈BE and for all f∈V.Proof.
Define the seminormρ on E by ρx=supfx:f∈V ([1], Theorem 5.100 (2)). Then, ρ is continuous on E. From Proposition 4, we have ρTxn⟶0 for each disjoint sequence xn in BE. Since BE is a solid set, Theorem 4.36 of [1] yields that for each ε>0 there exists uε∈E+ satisfying
(13)fTx−uε+<ε,for all f∈V and for all x∈BE.


## 2.3. The SpacesAWLrE,F and AWMrE,F

In this section, we study the Banach lattice properties ofAWLrE,F and AWMrE,F. In ([9], Theorem 2.2), Bayram and Wickstead proved that WLrE,F is always a Dedekind complete Banach lattice with the regular norm. In a similar way, we show that the same is true for AWLrE,F.Theorem 6.
LetE and F be Banach lattices. Then, AWLrE,F, equipped with the regular norm, is a Dedekind complete Banach lattice.Proof.
LetT∈AWLrE,F. Then, T=T1−T2, where T1,T2∈AWLrE,F+. Put U=T1+T2. Clearly, ±T≤U. Since TE⊆Fa, UE⊆Fa, and Fa is Dedekind complete, it follows that T has a modulus in LrE,Fa and hence in LrE,F. As T≤U, Theorem 1 yields that T∈AWLrE,F. Thus, AWLrE,F is a vector lattice. As Fa is Dedekind complete, AWLrE,F is also Dedekind complete. Now, we show that AWLrE,F is closed in LrE,F with respect to the regular norm. To this end, assume that T is in the closure of AWLrE,F. Then, there exists a sequence Tn in AWLrE,F such that Tn−Tr⟶0. Since Tn has a modulus for each n, Theorem 2.1 of [17] implies that T has a modulus in LrE,F and that Tn−Tr⟶0. Therefore, Tn−T⟶0. Since Tn∈AWLE,F for all n and AWLE,F is closed in LE,F ([11], Proposition 2.1 (1)), we have T∈AWLE,F. This shows that T+ and T− are also in AWLE,F (by Theorem 1), and so is T. As a result, AWLrE,F is a Dedekind complete Banach lattice.Assuming thatF 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.
LetE and F be Banach lattices with F Dedekind complete. Then, AWMrE,F, equipped with the regular norm, is a Dedekind complete Banach lattice.Now, we investigate conditions under whichAWLrE,F and AWMrE,F have an order continuous norm, respectively. For AWLrE,F, we have exactly the same necessary and sufficient condition with ([9], Theorem 3.1).Theorem 8.
LetE and F be Banach lattices with Fa≠0. Then, the regular norm on AWLrE,F is order continuous if and only if E′ has an order continuous norm.Proof.
Assume thatE′ has an order continuous norm. By Theorem 1, for all T∈AWLrE,F+, the order intervals 0,T are the same in AWLrE,F and in LrE,F. Let T∈AWLrE,F+. Since E′ has an order continuous norm, T is M-weakly compact ([12], Theorem 4). As TE⊆Fa and the norm of Fa is order continuous, we conclude that the norm on 0,T is order continuous ([2], Proposition 3.6.19). Conversely, suppose that the norm on AWLrE,F is order continuous. Fix 0≠y∈F+a and let fα be a net in E′ such that fα↓0. Define Tα:E⟶F by Tαx=fαxy. Then, Tα↓0, and hence, Tα⟶0 by assumption. So Tα=fαy⟶0. This shows that the norm of E′ is order continuous.We now turn our attention toAWMrE,F.Proposition 9.
LetE and F be Banach lattices with E′a≠0. If the regular norm on AWMrE,F is order continuous, then F has an order continuous norm.Proof.
Letyα be a net in F such that yα↓0. Fix 0≠f∈E′a and define Tα:E⟶F by Tαx=fxyα. It is easily seen that each Tα is M-weakly compact (hence almost M-weakly compact). Then, for each x∈E+, Tαx↓0. Thus, Tα↓0. As AWMrE,F has an order continuous norm, Tα=fyα⟶0. As a result, yα⟶0.The condition thatF having an order continuous norm is not sufficient for AWMrE,F to have an order continuous regular norm. We demonstrate this fact in the following example.Example 10.
We considerAWMrc0,c0 and note that c0 has an order continuous norm. We claim that AWMrc0,c0 does not have an order continuous regular norm. Suppose on the contrary it has. Let I:c0⟶c0 be the identity operator. Then, I is a positive almost M-weakly compact operator ([12], p. 143). The order interval 0,I is the same in AWMrE,F and in LrE,F (by Theorem 2). So the norm is order continuous on 0,I. By Proposition 3.6.19 of [2], we infer that the identity operator I is M-weakly compact, a contradiction ([12], p. 143).Below we give some sufficient conditions under whichAWMrE,F admits an order continuous regular norm.Proposition 11.
LetE and F be Banach lattices. If one of the following conditions holds, then AWMrE,F has an order continuous regular norm.
(i)
E′ and F′′ have an order continuous norm.(ii)
F is reflexive.(iii)
E is an AM-space with unit, and F has an order continuous norm.Proof.
LetT∈AWMrE,F+. We note that in each part ((i)–(iii)) F has an order continuous norm. By Proposition 3.6.19 of [2], it is sufficient to show that T is M-weakly compact. Suppose (i) holds. Since T is almost M-weakly compact, its adjoint T′ is almost L-weakly compact. By ([12], Corollary 4,) T′ is L-weakly compact, and hence, T is M-weakly compact. Now, suppose (ii) holds. Again, T′ is almost L-weakly compact. Since F′ is reflexive (because F is reflexive), T′ is L-weakly compact. Consequently, T is M-weakly compact. Finally, suppose (iii) holds. Since E is an AM-space with unit, T is semicompact ([1], p. 339, Ex. 15). As F has an order continuous norm, T is L-weakly compact ([2], Corollary 3.6.14) and hence weakly compact. It follows from Theorem 5.62 of [1] that T is M-weakly compact, and we are done.We end with a question: “What is a necessary and sufficient condition for the Banach latticeAWMrE,F to have an order continuous regular norm?”



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