The $ O\Tau $-Continuous, Lebesgue, KB, and Levi Operators Between

Home , Banach lattice, Band (order theory), Order convergence, Order dual, Order unit

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d) pre-Lebesgue if 0 ≤ xn ↑≤ x in X implies that xn is a ς-Cauchy sequence in X.

2 e) Levi/σ-Levi if every increasing ς-bounded net/sequence in X+ has a supremum in X. f) Fatou/σ-Fatou if the topology ς has a base at zero consisting of order/σ-order closed solid sets.

The assumption xα ↓ 0 on the net xα in c) can be replaced o by xα −→ 0. A Lebesgue lattice (X, ς) is Dedekind complete iff order intervals of X are ς-complete [29, Prop.3.16]. Ev- ery Lebesgue lattice is pre-Lebesgue [2, Thm.3.23] and Fa- tou [2, Lem.4.2]. Furthermore, (X, ς) is pre-Lebesgue iff the topological completion (X,ˆ ςˆ) of (X, ς) is Lebesgue [2, Thm.3.26] iff (X,ˆ ςˆ) is pre-Lebesgue and Fatou [2, Thm.4.8]. By [2, Thm.3.22], d) is equivalent to each of the following two conditions: ′ d ) if 0 ≤ xα ↑≤ x holds in X, then xα is a ς-Cauchy net; d′′) every order bounded disjoint sequence in X is ς- convergent to zero. The next well known fact follows directly from d′). Proposition 1.1. Every ς-complete pre-Lebesgue lattice (X, ς) is Dedekind complete. A normed lattice (X, k·k) is called Kantorovich-Banach or KB-space if every norm bounded upward directed set in X+ converges in the norm. Each KB-space is a Levi lattice with order continuous complete norm; each order continuous Levi normed lattice is Fatou; and each Levi normed lattice is Dedekind complete. Lattice-normed versions of KB-spaces were recently studied in [8, 9, 11]. g) We call a locally solid lattice (X, ς) by a KB/σ-KB lattice if every increasing ς-bounded net/sequence in X+ is ς-convergent. Clearly, each KB/σ-KB lattice is Levi/σ-Levi and each Levi/σ-Levi lattice is Dedekind complete/σ-complete. Recall that a continuous operator T : h) between two Banach spaces is said to be Dunford- Pettis if T takes weakly null sequences to norm null

3 sequences. It is well known that every weakly compact operator on L1(µ) is Dunford-Pettis and that an operator is Dunford-Pettis iﬀ it takes weakly Cauchy sequences to norm convergent sequences [3, Thm.5.79]. i) from a Banach lattice X to a Banach space Y is called M-weakly compact if kT xnkY → 0 holds for every norm bounded disjoint sequence xn in X. j) from a Banach space Y to a Banach lattice X is called L-weakly compact whenever kxnkX → 0 holds for every disjoint sequence xn in the solid hull of T (UY ), where UY is the closed unit ball of Y . An operator T from a vector lattice X to a topological vector space (Y, τ) is called τ k) oτ-continuous/σoτ-continuous if T xα −→ 0 for every o net/sequence xα such that xα −→ 0 [27]. Replacement of o-null nets/sequences by uo-null ones above gives the deﬁnitions of uoτ-continuous/σuoτ-continuous operators. L-/M-weakly compact operators are weakly compact (cf. [3, Thm.5.61]) and the norm limit of a sequence of L-/M- weakly compact operators is again L-/M-weakly compact (cf. [3, Thm.5.65]). For further unexplained terminology and notions, we refer to [2, 3, 4, 5, 25, 28, 33, 34]. Various versions of Banach lattice properties like a property to be a KB-space were investigated recently (see, e.g., [1, 6, 7, 8, 9, 12, 13, 14, 17, 18, 20, 21, 22, 23, 25, 27, 29, 30, 31]). In the present paper we continue the study of operator versions of several topological/order properties, focusing on locally solid lattices. The main idea behind operator versions consists in a redistribution of topological and order properties of a topological vector lattice between the domain and range of the operator under investigation (like in the case of Dunford-Pettis and L-/M- weakly compact operators). As the order convergence is not topological in general [13, 26], the most important op-

4 erator versions emerge when both o- and ς-convergences are involved simultaneously. Definition 1.1. Let T be an operator from a vector lattice X to a topological vector space (Y, τ). We say that: τ (a) T is τ-Lebesgue/στ-Lebesgue if T xα −→ 0 for every net/ sequence xα such that xα ↓ 0; T is quasi τ- Lebesgue/ στ-Lebesgue if T xα is τ-Cauchy for every net/sequence xα in X+ satisfying xα ↑≤ x ∈ X. If there is no confusion with the choice of the topology τ on Y , we call τ-Lebesgue operators by Lebesgue etc. (b) T is oτ-bounded/oτ-compact if T [0, x] is a τ-bounded/ τ-totally bounded subset of Y for each x ∈ X+. If additionally X =(X, ς) is a locally solid lattice, (c) T is KB/σ-KB if, for every ς-bounded increasing net/sequence xα in X+, there exists (not necessarily τ unique) x ∈ X such that T xα −→ T x. (d) T is quasi KB/quasi σ-KB if T takes ς-bounded increasing nets/sequences in X+ to τ-Cauchy nets. If X and Y are vector lattices with (X, ς) locally solid, (e) T is Levi/σ-Levi if, for every ς-bounded increasing net/sequence xα in X+, there exists (not necessarily o unique) x ∈ X such that T xα −→ T x. (f) T is quasi Levi/quasi σ-Levi if T takes ς-bounded increasing nets/sequences in X+ to o-Cauchy nets. Replacement of decreasing o-null nets/sequences by uo- null ones in (a) and o-convergent (o-Cauchy) nets/sequences by (uo-Cauchy) uo-convergent ones in (e) and (f) above gives the definitions of uoτ-continuous and of(quasi) uo-Levi operators respectively. In our approach, we focus on: ∗ modification of nets/sets of operators domains in (a), (b), (d), (e), and (f); ∗∗ information which operators provide about convergences in their domains/ranges in (c), (e), and (f).

5 Remark 1.1. a) The identity operator in a locally solid lattice (Y, τ) is Lebesgue/KB/Levi iff (Y, τ) is Lebes- gue/KB/Levi. This motivates the terminology. Cle- arly, every oτ-continuous/σoτ-continuous operator is τ-Lebesgue/στ-Lebesgue. By Lemma 2.1, any regular operator to a locally solid lattice is Lebesgue/σ- Lebesgue iff T is oτ-continuous/σoτ-continuous; and every regular Lebesgue operator to a locally solid lattice is quasi Lebesgue by Theorem 2.2. b) Each regular operator from a vector lattice X to a locally solid lattice (Y, τ) is oτ-bounded by Proposition 2.1. In the case of normed range space Y , oτ-bounded operators are also known as interval-bounded (cf. [28, Def.3.4.1]). Like in [28, Lem.3.4.2], each oτ-bounded operator T from a vector lattice X to a topological vector space (Y, τ) possesses adjoint T ◦ : Y ′ → X∼ given by T ◦y′ = y′ ◦ T for all y′ ∈ Y ′. c) In the case when X is also a topological vector lattice, the τ-continuity of operator T is not assumed in (b) of Definition 1.1. For example, the rank one dis- T x ∞ x e c , continuous operator := (Pk=1 k) 1 in ( 00 k·k) is oτ-compact and oτ-continuous yet not compact. Each continuous operator T from a discrete Dedekind complete locally convex Lebesgue lattice to a topological vector space is oτ-compact by the Kawai theorem (cf. [2, Cor.6.57)]. Every Dunford-Pettis operator from a Banach lattice to a Banach space is o-weakly-compact (cf. [3, Thm 5.91]). d) Each KB-operator is quasi KB; each quasi (σ-) KB- operator is quasi (σ-) Lebesgue; and each continuous operator from a KB-space to a topological vector space is KB. It is well known that the identity operator I in a Banach lattice is KB iff I is σ-KB iff I is quasi KB. Proposition 1.2 shows that the notions of quasi KB and quasi σ-KB operator coincide. The most important reason for using the term KB-

6 operator for (c) of Definition 1.1 is existence of limits of ς-bounded increasing nets. Some authors (see, e.g., [7, 12, 31]) use the term “KB-operator” for (d) of Definition 1.1 which is slightly confusing because it says nothing about existence of limits of topologically bounded increasing nets and all continuous finite-rank operators in every Banach lattice satisfy (d). Each order bounded operator from a Banach lattice to a KB- space is quasi KB. However, if we take X =(c00, k·kl) and Y =(c00, k·kp) with l,p ∈ [1, ∞], the identity operator I is quasi KB iff l ≤ p< ∞. e) It is clear that every compact operator T from a Banach lattice X to a Banach space Y is oτ-compact. However, compact operator need not to be Lebesgue (cf. Example 3.1). In particular, oτ-compact operator need not to be oτ-continuous.

Example 1.1. Let (c(R), k·k∞) be the Banach lattice of all R-valued functions on R such that for every f ∈ c(R) there exists af ∈ R for which the set {r ∈ R : |f(r) − af | ≥ ε} is finite for each ε > 0. Then the identity operator I in c(R) is σoτ-continuous and quasi σ-Lebesgue yet neither Lebesgue nor quasi Lebesgue. Example 1.2. Recall that, for a nonempty set H, the vector space c00(H) of all finitely supported R-valued functions on H is a Dedekind complete vector lattice. Furthermore, any vector space X is linearly isomorphic to c00(H), where H is a Hamel basis for X. As each order interval of c00(H) lies in a finite-dimensional subspace of c00(H), every operator from c00(H) to any topological vector space (Y, τ) is oτ-continuous, oτ-bounded, and oτ-compact. On the other hand, the identity operator in (c00, k·k∞) is neither quasi σ-KB nor quasi σ-Levi.

Remark 1.2. It is well known that, if a net yα of a topological vector space (Y, τ) is not τ-Cauchy, then there exist U ∈ τ(0) and an increasing sequence αn such that yαn+1 − yαn 6∈ U for each n (see, e.g., [2, Lem.2.5]).

7 Proposition 1.2. An operator T from a locally solid lattice (X, ς) to a topological vector space (Y, τ) is quasi KB iﬀ T is quasi σ-KB. Proof. The necessity is trivial. For the suﬃciency, suppose T is not quasi KB. Then there exists a ς-bounded increasing net xα in X+ such that T xα is not τ-Cauchy in Y . It follows from Remark 1.2 that for some increasing sequence αn and some U ∈ τ(0) N T xαn+1 − T xαn 6∈ U (∀n ∈ ) (1)

Since the sequence xαn is increasing and ς-bounded, the condition (1) implies that T is not quasi σ-KB. Corollary 1.1. Every ς-complete σ-KB lattice (X, ς) is a KB lattice. Proof. The identity operator I on X is σ-KB and hence quasi σ-KB. By Proposition 1.2, I is quasi KB. Then every ς-bounded increasing net in X+ is ς-Cauchy, and hence is ς-convergent. Remark 1.3. The following fact (cf., e.g., [10, Prop.1.1]) is well known:

a) ρ(yα,y) → 0 in a metric space (Y, ρ) iﬀ, for every

subnet yαβ of the net yα, there exists a (not necessary increasing) sequence βk of indices with ρ(yα ,y) → 0. βk Application of a) to Cauchy nets/sequences in (Y, ρ) gives that:

b) a net/sequence yα of a metric space (Y, ρ) is Cauchy

iﬀ, for every subnet/subsequence yαβ of yα, there exists a sequence βk of indices such that the sequence yα is ρ-Cauchy. βk Application of b) and Proposition 1.2 gives that, for an operator T from a locally solid lattice (X, ς) to a metric vector space (Y, ρ), the following are equivalent: (i) T is quasi KB;

8 (ii) Every ς-bounded increasing sequence xn in X+ has a

subsequence xnk such that T xnk is ρ-Cauchy in Y . Another application of b) to an operator T from a locally solid lattice (X, ς) to a metric vector space (Y, ρ) gives the equivalence of the following conditions: (i)′ T is KB; ′ (ii) For any ς-bounded increasing net xα in X+, there exist an element x ∈ X and a sequence βk of indices such that ρ(T xα , T x) → 0. βk In many cases, like for (quasi) KB or Levi operators, the lattice structure in the domain/range of operator can be relaxed to the ordered space structure [7, 11, 19, 21], or combined with the lattice-norm structure [1, 8, 9, 11]. Such generalizations are not included in the present paper. In Section 2 we investigate operators whose domains and/or ranges are locally solid lattices. Section 3 is de- voted to operators between Banach lattices.

2 Operators between locally solid lattices

In this section, we study mostly operators whose domains and/or ranges are locally solid lattices. Observe ﬁrst that the sets LLeb(X,Y ), Loτ (X,Y ), Loτb(X,Y ), and Loτc(X,Y ) of Lebesgue, oτ-continuous, oτ-bounded, and oτ-compact operators respectively from a vector lattice X to a topological vector space (Y, τ) are vector spaces and:

(∗) Loτ (X,Y ) ⊆ LLeb(X,Y );

(∗∗) Loτc(X,Y ) ⊆ Loτb(X,Y ) since every totally bounded subset in Y is bounded. Theorem 2.1. Let X be a vector lattice and (Y, τ) be a locally convex Lebesgue lattice which is either Dedekind complete or τ-complete. Then Loτc(X,Y ) T Lr(X,Y ) is a band of the lattice Lr(X,Y ) of all regular operators from X to Y .

9 Proof. Since every Lebesgue lattice is pre-Lebesgue by [2, Thm.3.23], and every topologically complete pre-Lebesgue lattice is Dedekind complete by Proposition 1.1, the lattice (Y, τ) is Dedekind complete in either case. By [3, Thm.5.10], for each x ∈ X+, the set

C(x)= {T ∈ Lb(X,Y ): T [0, x] is τ-totally bounded} is a band of the Dedekind complete lattice Lb(X,Y ) of all order bounded operators from X to Y . Since Lr(X,Y )= L (X,Y ), the set L (X,Y ) L (X,Y )= C(x) is b oτc T r Tx∈X+ also a band of Lr(X,Y ) as desired. The following two propositions might be known. We include their elementary proofs as we did not find appro- priate references. Proposition 2.1. Each regular operator T from a vector lattice X to a locally solid lattice (Y, τ) is oτ-bounded. Proof. Without lost of generality, assume T ≥ 0. Let x ∈ X+, U ∈ τ(0), and V be a solid τ-neighborhood of zero of Y with V ⊆ U. Then T x ∈ λV for all λ ≥ λ0 for some λ0 > 0 and hence T [0, x] ⊆ [0, T x] ⊆ λV ⊆ λU for all λ ≥ λ0. Since U ∈ τ(0) was arbitrary, T is oτ- bounded. Proposition 2.2. An order continuous regular operator T from a vector lattice X to a locally convex-solid lattice (Y, τ) is Lebesgue iff T is weakly Lebesgue. Proof. Without lost of generality, we can suppose T ≥ 0. The necessity is trivial. For the sufficiency, assume that T is weakly Lebesgue (i.e., T : X → Y is Lebesgue with ′ respect to the weak topology σ(Y,Y ) on Y ). Let xα ↓ 0 ′ in X. Then T xα ↓ 0 in Y . Since T is σ(Y,Y )-Lebesgue, σ(Y,Y ′) T xα −−−−→ 0. By using Dini-type arguments (cf. [3, τ Thm.3.52]), we conclude T xα −→ 0, as desired.

10 We do not know any example of an order continuous oτ- bounded weakly Lebesgue operator from a vector lattice to a locally solid lattice that is not Lebesgue. Proposition 2.3. A weakly continuous regular operator T from a Lebesgue (σ-Lebesgue) lattice (X, ς) to a locally solid lattice (Y, τ) is Lebesgue (σ-Lebesgue). Proof. As arguments are similar, we restrict ourselves to the Lebesgue case. Without lost of generality, assume T ≥ ς 0. Let xα ↓ 0 in X. Since (X, ς) is Lebesgue, xα −→ 0 and σ(X,X′) hence xα −−−−→ 0. The weak continuity of T implies ′ σ(Y,Y ) τ T xα −−−−→ 0. Since T xα ↓ 0 in Y , it follows T xα −→ 0, as in the proof of Proposition 2.2. Theorem 2.2. Each regular Lebesgue operator T from a vector lattice X to a locally solid lattice (Y, τ) is quasi Lebesgue. Proof. Without lost of generality, we can suppose T ≥ 0. Let xα ↑≤ x ∈ X, so uα := x − xα ↓≥ 0. Therefore (uα − z)α;z∈L ↓ 0, where L = {z ∈ X :(∀α ∈ A)[z ≤ uα]} is the upward directed set of lower bounds of the net uα (cf. [16, Thm.1.2]). Since T is Lebesgue and T ≥ 0, τ (Tuα − T z)α;z∈L −→ 0 & (Tuα − T z)α;z∈L ↓≥ 0. (2)

It follows from (2), that (Tuα − T z)α;z∈L ↓ 0. Then Tuα and hence T xα is τ-Cauchy, as desired. We do not know any example of an oτ-bounded Lebesgue operator which is not quasi Lebesgue. In contrast to The- orem 3.24 of [2], the converse of Theorem 2.2 is false even if Y = R due to Example 3.1. Recall that an Archimedean vector lattice X is called laterally (σ-) complete whenever every (countable) subset of pairwise disjoint vectors of X+ has a supremum. Every laterally complete vector lattice X contains a weak order unit and every band of such an X is a principal band (cf. [2, Thm.7.2]). Furthermore, by the Veksler – Geiler

11 theorem (cf. [2, Thm.7.4]), if a vector lattice X is laterally (σ-) complete, then X satisfies the (principal projection) projection property. A vector lattice that is both laterally and Dedekind (σ- ) complete is referred to as universally (σ-) complete. It follows from [2, Thm.7.4] that a vector lattice X is universally complete iff X is Dedekind σ-complete and laterally complete iff X is uniformly complete and laterally complete (cf. [2, Thm.7.5]). Similarly, a laterally σ-complete vector lattice X is Dedekind σ-complete iff X is uniformly complete. A universal completion of a vector lattice X is a laterally and Dedekind complete vector lattice Xu which contains X as an order dense sublattice. Every vector lattice has unique universal completion (cf. [2, Thm.7.21]). A laterally complete vector lattice X is discrete iff X is lattice isomorphic to RS for some nonempty set S iff X admits a Hausdorff locally convex-solid Lebesgue topol- ∼ ogy iff the space Xn of order continuous functionals on X separates X (cf. [2, Thm.7.48]); in each of these cases X = Xu. Definition 2.1. A topological vector lattice (Y, τ) is called τ-laterally (σ-) complete whenever every τ-bounded (countable) subset of pairwise disjoint vectors of X+ has a supremum. Every laterally (σ-) complete topological vector lattice (Y, τ) is τ-laterally (σ-) complete. Every Dedekind complete AM-space X with an order unit is τ-laterally complete with respect to the norm. Example 2.1. Let X be a vector lattice of real functions on R such that each f ∈ X may differ from a constant say af on a countable subset of R and f − af IR ∈ ℓ1(R) for each f ∈ X. (i) The vector lattice X is not Dedekind σ-complete, as fn := IR\{1,2,...,n} ↓≥ 0 yet inf fn does not exist in X. n∈N

12 (ii) The vector lattice X is a Banach space with respect to the norm kfk := |af | + kf − af IRk1. (iii) The vector lattice X is not τ-laterally σ-complete with respect to the norm topology on X. Indeed, the norm bounded countable set of pairwise disjoint orths en = I{n} of X+ has no supremum in X. (iv) The identity operator I on X is not σ-Lebesgue, as 1 I 1 I n+1 n R\{1,2,...,n} ↓ 0 in X yet k n R\{1,2,...,n}k = n 6→ 0. Proposition 2.4. Any Levi (σ-Levi) lattice is τ-laterally (σ-) complete and Dedekind (σ-) complete. Proof. We consider the Levi case only, because the σ-Levi case is similar. The Dedekind completeness of X is trivial. Let D be a τ-bounded subset of pairwise disjoint positive vectors of a Levi lattice (Y, τ). Then the collection D∨ of suprema of ﬁnite subsets of D forms an increasing τ- bounded net indexed by the set Pfin(D) of all ﬁnite subsets of D directed by inclusion. Since (Y, τ) is Levi, D∨ ↑ d for some d ∈ Y . It follows from sup D = sup D∨ = d that (Y, τ) is τ-laterally complete. Example 2.2. The locally solid lattice (RS, τ), where τ is the product topology on the vector lattice RS of real functions on a set S is locally convex, Lebesgue, Levi, and universally complete. Proposition 2.5. Each regular operator T from a laterally σ-complete vector lattice X to a σ-Lebesgue lattice (Y, τ) is σ–Lebesgue. Proof. Without lost of generality, we can suppose T ≥ 0. Let xn ↓ 0 in X. By Theorem 7.8 of [2], T is σ-order continuous, and hence T xn ↓ 0 in Y . Since (Y, τ) is σ– τ Lebesgue, T xn −→ 0. We do not know whether or not every positive operator T from a laterally complete vector lattice X to a Lebesgue lattice (Y, τ) is Lebesgue.

13 Each laterally σ-complete locally solid lattice is σ-Le- besgue and pre-Lebesgue [2, Thm.7.49]. Proposition 2.6. Let T be a continuous operator from a laterally σ-complete locally solid lattice (X, ς) to a topological vector space (Y, τ). Then T is σ-Lebesgue in each of the following cases: (i) X is discrete. (ii) (X, ς) is metrizable.

Proof. Let xn ↓ 0 in X. In case (i), (X, ς) is σ-Lebesgue by [2, Thm.7.49]. The result follows because T is ςτ- continuous. In case (ii), the result follows from (i) as every inﬁnite dimensional laterally σ-complete metrizable locally solid lattice is lattice isomorphic to RN. Since (X,uς) is pre-Lebesgue iﬀ (X, ς) is pre-Lebesgue [30, Cor.4.3], the statement of Proposition 2.6 is also true when (X, ς) is replaced by (X,uς). Remark 2.1. Let T be a positive operator from a locally solid lattice (X, ς) to a laterally σ-complete locally solid lattice (Y, τ) and let 0 ≤ xα ↑≤ x in X. Since (Y, τ) is pre-Lebesgue by [2, 7.49], it follows from 0 ≤ T xα ↑≤ T x that T xα is τ-Cauchy in Y and hence T is quasi Lebesgue. Remark 2.2. Let T be a continuous operator from a laterally σ-complete locally solid lattice (X, ς) to a locally solid lattice (Y, τ).

a) Suppose T ≥ 0 and 0 ≤ xα ↑≤ x in X. Since (X, ς) is pre-Lebesgue by [2, 7.49] then xα is ς-Cauchy in X and hence T xα is τ-Cauchy in Y meaning that T is quasi Lebesgue.

b) Suppose (X, ς) is Fatou, and xα ↓ 0 in X. Then by [2, ς τ 7.50] (X, ς) is Lebesgue, so xα −→ 0 and hence T xα −→ 0 meaning that T is Lebesgue. c) Suppose ς is a metrizable locally solid topology and xα ↓ 0 in X. Then (X, ς) is Lebesgue by [2, Thm.7.55].

14 ς τ Thus xα −→ 0 and hence T xα −→ 0 meaning that T is Lebesgue. d) If(X, ς) is a Fréchet lattice then ς is uniquely defined and every regular T : (X, ς) → (Y, τ) is continuous (cf. [2, Thm.5.19 and Thm.5.21]). In particular, every regular operator from a laterally σ-complete Fréchet lattice X(ς) to a locally solid lattice (Y, τ) is Lebesgue. Proposition 2.7. Any continuous operator T from a locally solid lattice (X, ς) to a topological vector space (Y, τ) is oτ-bounded.

Proof. Let x ∈ X+, U ∈ τ(0), and V be a solid ς-neighborhood of zero of X with V ⊆ T −1U. Then there exists λ0 > 0 such that x ∈ λV for all λ ≥ λ0 and hence [0, x] ⊆ −1 λV ⊆ λT (U) for all λ ≥ λ0. Thus T [0, x] ⊆ λU for all λ ≥ λ0, which shows that T is oτ-bounded. It is worth mentioning here that an oτ-bounded operator T from Dedekind σ-complete vector lattice X to a normed lattice Y is σ-Lebesgue iff T is order-weakly compact (cf. [27, Cor.1]). The following lemma can be considered as an extension of [27, Lem.1] to locally solid lattices. Lemma 2.1. A regular operator T from a vector lattice X to a locally solid lattice (Y, τ) is Lebesgue/σ-Lebesgue iff T is oτ-continuous/σoτ-continuous. Proof. Without lost of generality, we suppose T ≥ 0. As the σ-Lebesgue case is similar, we consider only Lebesgue operators. The sufficiency is routine. For the necessity, o assume T is Lebesgue and xα −→ 0 in X. Take a net zβ in X with zβ ↓ 0 such that, for each β, there exists αβ with |xα| ≤ zβ for α ≥ αβ. It follows from T ≥ 0 that |T xα| ≤ T |xα| ≤ T zβ for α ≥ αβ. As T is Lebesgue, τ τ T zβ −→ 0, which implies T xα −→ 0, because the topology τ is locally solid. Corollary 2.1. An order bounded operator T from a vector lattice to a Dedekind complete locally solid lattice is

15 Lebesgue/σ-Lebesgue iﬀ T is oτ-continuous/σoτ-continuous. Now we investigate some properties of adjoint operators. Recall that, for a locally solid lattice (X, ς), the absolute weak topology |σ|(X∼,X) on X∼ is the locally convex-solid topology generated by the collection of Riesz seminorms {ρx(f) = |f|(|x|): x ∈ X}. The locally solid lattice (X∼, |σ|(X∼,X)) is Lebesgue, Levi, and Fatou [24, Prop.81C]. We begin with the following technical lemma. Lemma 2.2. Let T :(X, ς) → (Y, τ) be an order bounded continuous operator from a locally solid lattice (X, ς) to a Dedekind complete locally solid lattice (Y, τ). Then the topological adjoint T ′ :(Y ′, |σ|(Y ∼,Y )) → (X′, |σ|(X∼,X)) is also continuous. ∼ |σ|(Y ,Y ) ∼ Proof. Indeed, for fα −−−−−−→ 0 in Y , it follows from

′ ′ ′ ′ ρx(T fα)= h|T fα|, |x|i ≤ h|T ||fα|, |x|i ≤ h|T | |fα|, |x|i =

h|fα|, |T ||x|i = |fα|(|T ||x|)= ρ|T ||x|(fα) → 0 (3) ∼ ′ |σ|(X ,X) that T fα −−−−−−→ 0. The second inequality in (3) follows from the existence of the modulus of T and from the observation ±T ≤ |T | ⇒ ±T ′ ≤ |T |′ (cf. [3, p.67]) which implies |T ′|≤|T |′. Theorem 2.3. Let T : X → Y be an order bounded operator from a vector lattice X to a Dedekind complete vector lattice Y , and T ′ : (Y ∼, |σ|(Y ∼,Y )) → (X∼, |σ|(X∼,X)) the correspondent topological adjoint of T . Then T ′ is oτ- bounded, oτ-continuous, Levi, and KB. Proof. The oτ-boundedness of T ′ follows from Lemma 2.2 by Proposition 2.7. Since T is regular, without lost of generality, we can suppose T ≥ 0 (and hence T ′ ≥ 0). o ∼ ′ ′ o Let fα −→ 0 in Y . As T is order continuous, T fα −→ 0 in X∼, and since (X∼, |σ|(X∼,X)) is a Lebesgue lattice, ∼ ′ |σ|(X ,X) ′ we obtain T fα −−−−−−→ 0 and hence T is oτ-continuous.

16 ∼ Let fα be a positive |σ|(Y ,Y )-bounded increasing net ∼ ∼ ∼ in Y . Since (Y , |σ|(Y ,Y )) is Levi, fα ↑ f for some ∼ ′ ′ ′ ∼ f ∈ Y . As T is order continuous, T fα ↑ T f in X , in particular T ′ is Levi. Since the locally solid lattice ∼ ∼ ∼ ′ |σ|(X ,X) ′ (X , |σ|(X ,X)) is Lebesgue, T fα −−−−−−→ T f and hence T ′ : (Y ∼, |σ|(Y ∼,Y )) → (X∼, |σ|(X∼,X)) is KB.

∼ ∼ The natural embedding J of a vector lattice X to (Xn )n ∼ is defined by (Jx)(f)= f(x) for all f ∈ Xn , x ∈ X. The mapping J is an order continuous lattice homomorphism (cf. e.g. [3, Thm.1.70]). By the Nakano theorem (cf. [3, Thm.1.71]), J is one-to-one and onto (i.e. X is perfect) ∼ iff Xn separates X and whenever a net xα of X+ satisfies ∼ xα ↑ and sup f(xα) < ∞ for each 0 ≤ f ∈ Xn , then there exists some x ∈ X satisfying xα ↑ x in X. Lemma 2.3. Let T be an order bounded operator from a vector lattice X to a vector lattice Y . The second order ∼∼ ∼ ∼ adjoint T , while restricted to (Xn ) , ∼∼ ∼ ∼ ∼ ∼ ∼ T ∼ ∼ ∼ ∼ ∼ ((Xn ) , |σ|((Xn ) ,Xn )) −−→ ((Y ) , |σ|((Yn ) ,Yn )) is continuous. ∼ ∼ ∼ ∼ ∼ |σ|((Xn ) ,Xn ) Proof. Let (Xn ) ∋ xα −−−−−−−−−→ 0. By use of the fact |T ∼∼|≤|T ∼|∼ it follows

∼∼ ∼ ∼ ∼ h|T xα|, |y|i≤ h|T | |xα|, |y|i≤ h|xα|, |T ||y|i (4)

∼ ∼ ∼ ∼ |σ|((Xn ) ,Xn ) for all y ∈ Yn . Since xα −−−−−−−−−→ 0, it follows from (4) ∼ ∼ ∼ ∼∼ |σ|((Yn ) ,Yn ) ∼∼ that T xα −−−−−−−−→ 0 and hence the operator T is ∼ ∼ ∼ ∼ ∼ ∼ |σ|((Xn ) ,Xn ) to |σ|((Yn ) ,Yn ) continuous. Theorem 2.4. Let T be an order bounded operator from a vector lattice X to a vector lattice Y . Then ∼∼ ∼ ∼ ∼ ∼ ∼ T ∼ ∼ ∼ ∼ ∼ ((Xn )n , |σ|((Xn )n ,Xn )) −−→ ((Yn )n , |σ|((Yn )n ,Yn )). is a continuous, Lebesgue, Levi, and KB operator.

17 Proof. The continuity follows from Lemma 2.3 by restrict- ∼∼ ∼ ∼ ing of the second order adjoint T to (Xn )n . ∼ ∼ ∼∼ Let xα ↓ 0 in (Xn )n . As T is order continuous, ∼∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ T xα ↓ 0 in (Yn )n . Since ((Yn )n , |σ|((Yn )n ,Yn )) is ∼ ∼ ∼ ∼∼ |σ|((Yn )n ,Yn ) Lebesgue, T xα −−−−−−−−→ 0 showing that the operator is Lebesgue. ∼ ∼ ∼ Let xα be a positive increasing |σ|((Xn )n ,Xn )-bounded ∼ ∼ ∼ ∼ ∼ ∼ ∼ net in (Xn )n . Since ((Xn )n , |σ|((Xn )n ,Xn )) is Levi, the ∼ ∼ net xα has a supremum in (Xn )n , say x. Thus xα ↑ x ∼∼ ∼∼ ∼∼ and T xα ↑ T x as T is order continuous. Thus the operator is Levi. ∼ ∼ ∼ ∼ ∼ ∼ ∼ Since ((Xn )n , |σ|((Xn )n ,Xn )) is Levi, xα ↑ x in (Xn )n . ∼∼ o ∼∼ Then T xα −→ T x by order continuity of the operator ∼∼ ∼ ∼ ∼ ∼ ∼ T . As ((Yn )n , |σ|((Yn )n ,Yn )) is a Lebesgue lattice, ∼ ∼ ∼ ∼∼ |σ|((Yn )n ,Yn ) ∼∼ T xα −−−−−−−−→ T x and the operator is KB. A vector sublattice Z of a vector lattice X is called regular, if the embedding of Z into X preserves arbitrary suprema and inﬁma. Order ideals and order dense vector sublattices of a vector lattice X are regular [2, Thm.1.23]. ∼ ∼ Corollary 2.2. Let X = Xn , and Y be a vector lattice. ∼ Then each order bounded operator T : X → (Y, |σ|(Y,Yn )) is Lebesgue.

Proof. Let xα ↓ 0 in X. Since the natural embedding J : X → X∼∼ is one-to-one and J(X) is a regular sublattice ∼∼ ∼∼ of X (cf. [2, Thm.1.67]), then T = T ◦J and Jxα ↓ 0 ∼ ∼ ∼ ∼∼ ∼∼ |σ|((Xn )n ,Xn ) ∼∼ in X . Thus T xα = T (Jxα) −−−−−−−−−→ 0 since T is Lebesgue by Theorem 2.4.

We also mention the following question. Let T be Lebes- gue, oτ-continuous, oτ-bounded, oτ-compact, KB, or Levi. Does T ∼∼ satisfy the same property? The answer is nega- tive, when T maps a Banach lattice X to a Banach lattice Y , in the following cases:

18 (∗) for Lebesgue and for oτ-continuous operators, since the identity I : c0 → c0 is oτ-continuous yet its second order adjoint I : ℓ∞ → ℓ∞ is not even Lebesgue; (∗∗) for oτ-compact operators, since the natural embed- ∼∼ ∗∗ ding J : c0 → ℓ∞ is oτ-compact but J : ℓ∞ → ℓ∞ is not oτ-compact. Next we discuss the domination properties of positive operators. Let T and S be positive operators between vector lattices X and Y satisfying 0 ≤ S ≤ T . Question 2.1. When does the assumption that T is Lebes- gue, oτ-continuous, oτ-bounded, oτ-compact, KB, or Levi imply that S has the same property? Question 2.1 has positive answers in the following cases: Trivially, for Lebesgue; σ-Lebesgue; and oτ-bounded operators from a vector lattice to a locally solid lattice. For oτ-compact operators from a vector lattice X to a locally convex Lebesgue lattice (Y, τ) which is either Dedekind complete or τ-complete by Theorem 2.1. For o|σ|(Y,Y ′)-compact operators from a Banach lattice X to a Banach lattice Y [3, Thm.5.11]. For KB/σ-KB lattice homomorphisms between locally solid lattices by Corollary 2.3. For quasi KB operators between locally solid lattices by Theorem 2.6. For quasi Levi operators from a locally solid lattice to a vector lattice by Theorem 2.7. We remind that the modulus |T | of an order bounded disjointness preserving operator T between vector lattices X and Y exists and satisﬁes |T ||x| = |T |x|| = |T x| for all x ∈ X [3, Thm.2.40]; there exist lattice homomorphisms R1, R2 : X → Y with T = R1 − R2 (cf. [3, Exer.1, p.130]. Theorem 2.5. Let T be an order bounded disjointness preserving KB/σ-KB operator between locally solid lattices (X, ς) and (Y, τ). If |S|≤|T | then S is KB/σ-KB.

19 Proof. Take a ς-bounded increasing net/sequence xα in τ X+. Then T (xα − x) −→ 0 for some x ∈ X. It follows from τ |S(xα − x)|≤|S||xα − x|≤|T ||xα − x| = |T xα − T x| −→ 0 τ that Sxα −→ Sx, as desired. Since 0 ≤ S ≤ T with a lattice homomorphism T implies that S is also a lattice homomorphism, the next result is a direct consequence of Theorem 2.5. Corollary 2.3. Let T be a KB/σ-KB lattice homomorphism between locally solid lattices (X, ς) and (Y, τ). Then each S satisfying 0 ≤ S ≤ T is also a KB/σ-KB lattice homomorphism. Theorem 2.6. Let T be a positive quasi KB operator between locally solid lattices (X, ς) and (Y, τ). Then each operator S : X → Y with 0 ≤ S ≤ T is also quasi KB.

Proof. Let 0 ≤ S ≤ T and xα be an increasing ς-bounded net in X+. Since T ≥ 0 then T xα ↑. Since T is quasi KB then T xα is τ-Cauchy. Take a U ∈ τ(0) and a solid neighborhood V ∈ τ(0) with V − V ⊆ U. There exists α0 satisfying T (xα − xβ) ∈ V for all α, β ≥ α0. In particular,

T (xα − xα0 ) ∈ V for all α ≥ α0. Since 0 ≤ S ≤ T and V is solid, S(xα − xα0 ) ∈ V for all α ≥ α0. Thus, we obtain

Sxα −Sxβ = S(xα −xα0 )−S(xβ −xα0 ) ∈ V −V ⊆ U (5) for all α, β ≥ α0. Since U ∈ τ(0) was taken arbitrary, (5) implies that Sxα is also τ-Cauchy. Theorem 2.7. Let T be a positive quasi Levi operator from a locally solid lattice (X, ς) to a vector lattice Y . Then each operator S : X → Y with 0 ≤ S ≤ T is also quasi Levi.

Proof. Let T : X → Y be quasi Levi and xα be a positive ς-bounded increasing net in (X, ς). Then T xα is o-Cauchy in Y . So, there exists a net zβ ↓ in Y such that, for each

20 β, there exists αβ such that |T xα1 − T xα2 | ≤ zβ for all α1,α2 ≥ αβ. Choosing α1,α2 ≥ αβ for a ﬁxed αβ we have

Sxα1 − Sxα2 ≤ S(xα1 − xαβ ) ≤ T (xα1 − xαβ ) ≤ zβ, and similarly

Sxα2 − Sxα1 ≤ S(xα2 − xαβ ) ≤ T (xα2 − xαβ ) ≤ zβ.

Thus |Sxα1 − Sxα2 | ≤ zβ for all α1,α2 ≥ αβ showing that the operator S is also quasi Levi. A locally solid lattice (X, ς) is a regular sublattice of its ς-completion Xˆ iff every ς-Cauchy net xα in X+ such that o ς xα −→ 0 in X satisfies xα −→ 0 [2, Thm.2.41]. Lemma 2.4. Let Z be an order dense and majorizing sublattice of a vector lattice X and T be an oτ-continuous operator from X to a topological vector space (Y, τ). Then the restriction T |Z : Z → Y is also oτ-continuous. Proof. Under the assumptions of the lemma, for each net o o zα in Z, zα −→ 0 in Z iff zα −→ 0 in X, by [25, Thm.2.8]. Therefore the result follows directly from the definition of an oτ-continuous operator. Situation is even better for uo-null nets. Let Z be a regular uo uo sublattice of X then xα −→ 0 in Z iff xα −→ 0 in X by [25, Thm.3.2], and hence, for a uoτ-continuous operator T from a vector lattice X to a topological vector space (Y, τ), the restriction T |Z : Z → Y is also uoτ-continuous. Lemma 2.5. Let T be a regular Lebesgue operator from a vector lattice X to a locally solid lattice (Y, τ). If Z is an order dense and majorizing sublattice of X, then the restriction T |Z : Z → Y is oτ-continuous. Proof. T is oτ-continuous by Lemma 2.1. It follows from Lemma 2.4 that T |Z : Z → Y is oτ-continuous. Since a vector lattice X is order dense and majorizing in Xδ, the next result follows immediately from Lemma 2.5.

21 Proposition 2.8. If an operator T from the Dedekind completion Xδ of a vector lattice X to a locally solid lattice (Y, τ) is Lebesgue then the operator T |X is oτ-continuous. Each order continuous lattice homomorphism T from a Fatou lattice (X, ς) to a Lebesgue lattice (Y, τ) is ςτ- continuous on order intervals of X by [2, Thm.4.23]. Since T is also Lebesgue under the given conditions, the following can be considered as a generalization of [2, Thm.4.23]. Theorem 2.8. Let (X, ς) be a Fatou lattice, (Y, τ) a Lebes- gue lattice, and T : X → Y a Lebesgue lattice homomorphism. Then T is ςτ-continuous when restricted to order bounded subsets in X. Proof. Let N be a base of solid neighborhoods for τ(0). Since T is a Lebesgue lattice homomorphism, the family −1 {T (V )}V ∈N is a base of solid neighborhoods of zero for a Lebesgue (not necessary Hausdorff) topology ζ on X. Since ς is Fatou and ζ is Lebesgue, then on any order bounded subset of X the topology induced by ς is finer than the topology induced by ζ due to Theorem 4.21 of [2]. Clearly, T : X → Y is ζτ-continuous. Thus, T is also ςτ-continuous on order bounded subsets of X. Recall that, for a locally solid lattice (X, ς), a vector vˆ ∈ Xˆ+ is called an upper element of X whenever there ςˆ exists an increasing net uα in X+ such that uα → vˆ (cf. [2, Def.5.1]). In a metrizable locally solid lattice (X, ς) upper elements can be described by sequences: a vectorv ˆ ∈ Xˆ+ is an upper element of X iff there exists an increasing ςˆ sequence un in X+ such that un → vˆ (cf. [2, Thm.5.2]). For a metrizable σ-Lebesgue lattice (X, ς) its topological completion (X,ˆ ςˆ) is also σ-Lebesgue (cf. [2, Thm.5.36]). Theorem 2.9. Let (X, ς) be a metrizable locally solid lattice, (Y, τ) a σ-Lebesgue lattice, and T : (X, ς) → (Y, τ) a positive ςτ-continuous σ-Lebesgue operator. Then the unique extension Tˆ :(X,ˆ ςˆ) → (Y,ˆ τˆ) is positive, ςˆτˆ-continuous, and σ-Lebesgue.

22 Proof. Clearly, Tˆ is positive andς ˆτˆ-continuous. To show that Tˆ is σ-Lebesgue, assumex ˆn ↓ 0 in (X,ˆ ςˆ). Let W ∈ τ(0). We have to show Tˆxˆn ∈ W for large enough n. Take a solid neighborhood W1 ∈ τ(0) with W1 + W1 ⊆ W and choose a neighborhood V ∈ ς(0) with T (V ) ⊆ W1. Let ∞ {Vn}n=1 be a base for ς(0) satisfying Vn+1 + Vn+1 ⊆ Vn for all natural n and also V1 + V1 ⊆ V . At this point we borrow the second paragraph of the proof of [2, Thm.5.36]. For each n, letv ˆn ∈ Xˆ be an upper element of X such thatx ˆn ≤ vˆn andv ˆn − xˆn ∈ V n; see [2, Thm.5.4]. Put n wˆn := ∧i=1vˆn for all n and note that eachw ˆn is an upper element of X,w ˆn ↓ andw ˆn − xˆn ∈ V n. Sincex ˆn ↓ 0, it follows from [2, Thm.2.21(e)] thatw ˆn ↓ 0 also holds in Xˆ. Thus, we can assume without loss in generality thatx ˆn is a sequence of upper elements of X. Take un ∈ X+ with un ≤ xˆn andx ˆn − un ∈ V n. Application of [2, Thm.2.21(e)] to the netsx ˆn ↓ 0 and n ςˆ zn := ∧i=1un ↓ withx ˆn − zn −→ 0 gives zn ↓ 0 in X. As T is τ σ-Lebesgue, T zn −→ 0. So, there exists some n1 such that