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

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The $ O\Tau $-Continuous, Lebesgue, KB, and Levi Operators Between The oτ-continuous, Lebesgue, KB, and Levi operators between vector lattices and topological vector spaces May 6, 2021 Safak Alpay1, Eduard Emelyanov1,2, Svetlana Gorokhova 3 1 Department of Mathematics, Middle East Technical University, Ankara, Turkey 2 Sobolev Institute of Mathematics, Novosibirsk, Russia 3 Southern Mathematical Institute of the Russian Academy of Sciences, Vladikavkaz, Russia Abstract We investigate the oτ-continuous/bounded/compact and Lebesgue op- erators from vector lattices to topological vector spaces; the KB operators between locally solid lattices and topological vector spaces; and the Levi operators from locally solid lattices to vector lattices. The main idea of operator versions of notions related to vector lattices lies in redistributing topological and order properties of a topological vector lattice between the domain and range of an operator under investigation. Domination properties for these classes of operators are studied. keywords: vector lattice, topological vector space, locally solid lat- tice, Banach lattice, order convergence, domination property, adjoint op- erator MSC2020: 46A40, 46B42, 47L05 arXiv:2105.01810v1 [math.FA] 5 May 2021 1 Introduction and preliminaries In the present paper (unless otherwise stated) all vector spaces are supposed to be real, operators linear, vector topologies Hausdorff, and vector lattices Archimedean. For any vector lattice X, the Dedekind complete vec- tor lattice of all order bounded linear functionals on X is called order dual of X and is denoted by X∼. The order 1 ∼ ∼ continuous part Xn is a band of X . Some vector lat- tices may have trivial order duals, for example X∼ = {0} whenever X = Lp[0, 1] with 0 <p< 1 (cf. [2, Thm.5.24]). If T is an operator from a vector space X to a vector space Y , the algebraic adjoint T # is an operator from the algebraic dual Y # to X#, defined by (T #f)(x) := f(T x) for all x ∈ X and all f ∈ Y #. In the case when X and Y are vector lattices and T is order bounded, the restriction T ∼ of T # to the order dual Y ∼ of Y is called the order ad- joint of T . The operator T ∼ : Y ∼ → X∼ is not only order bounded, but even order continuous (cf. [3, Thm.1.73]). ∼ ∼ ∼ Clearly, T : Yn → Xn when T is order continuous. In the case when (X, ς) and (Y, τ) are topological vec- tor spaces and T is continuous, the restriction T ′ of T # to the topological dual Y ′ (= the collection of all ς-continuous linear functionals on Y ) is called the topological adjoint of T . For every locally solid lattice (X, ς), we have X′ ⊆ X∼ since ς-continuous functionals are bounded on ς-bounded subsets and hence are order bounded. The topological dual X′ of a locally convex-solid lattice (X, ς) is an ideal of X∼ (and hence X′ is Dedekind complete) (cf. [3, Thm.3.49]). Every Fr´echet lattice (X, ς) satisfies X′ = X∼ (cf. [2, Thm.5.23]). A net (xα)α∈A in a vector lattice X is said to be: a) order convergent (o-convergent) to x ∈ X, if there exists a net (zβ)β∈B in X such that zβ ↓ 0 and, for any β ∈ B, there exists αβ ∈ A with |xα − x| ≤ zβ for o all α ≥ αβ. In this case, we write xα −→ x; o b) uo-convergent to x ∈ X, if |xα − x| ∧ u −→ 0 for every u ∈ X+. A locally solid lattice (X, ς) is called: ς c) Lebesgue/σ-Lebesgue if xα ↓ 0 implies xα −→ 0 for ev- ery net/sequence xα in X. 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/sequen- ce 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 contin- uous 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 op- erator is Dunford-Pettis iff it takes weakly Cauchy se- quences 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 ev- ery 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 op- erators. 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 prop- erty 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 topolog- ical and order properties of a topological vector lattice between the domain and range of the operator under in- vestigation (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/sequen- ces 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 conver- gences in their domains/ranges in (c), (e), and (f).
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