Order Continuous Banach Lattice Research Articles

On the diagonal of Riesz operators on Banach lattices

This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete vector lattice E. We prove that the class of regular operators for which the diagonal coincides with the atomic diagonal is always a band in , which contains the band of abstract integral operators. If E is also a Banach lattice, then contains positive Riesz and positive AM-compact operators.

Representation theorems for regular operators

AbstractWe elaborate, strengthen, and generalize known representation theorems by different authors for regular operators on vector and Banach lattices. Our main result asserts, in particular, that every regular linear operator T acting from a vector lattice E with the principal projection property to a Dedekind complete vector lattice F, which is an ideal of some order continuous Banach lattice G, admits a unique representation , where is the sum of an absolutely order summable family of disjointness preserving operators and is an order narrow (= diffuse) operator. Our main contribution is waiver of the order continuity assumption on T. In proofs, we use new techniques that allow obtaining more general results for a wider class of orthogonally additive operators, which has somewhat different order structure than the linear subspace of linear operators.

On Bilinear Narrow Operators

In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator T:E×F→W defined on the Cartesian product of vector lattices E and F and taking values in a vector lattice W is narrow if the partial operators Tx and Ty are narrow for all x∈E,y∈F. We prove that, for order-continuous Köthe–Banach spaces E and F and a Banach space X, the classes of narrow and weakly function narrow bilinear operators from E×F to X are coincident. Then, we prove that every order-to-norm continuous C-compact bilinear regular operator T is narrow. Finally, we show that a regular bilinear operator T from the Cartesian product E×F of vector lattices E and F with the principal projection property to an order continuous Banach lattice G is narrow if and only if |T| is.

Completely additive and C-compact operators in lattice-normed spaces

In this article, we investigate some classes of dominated orthogonally additive operators in lattice-normed spaces. We say that an orthogonally additive operator T from a lattice-normed space (V, E) to a lattice-normed space (W, F) is completely additive if, for every order summable family of pairwise disjoint elements $$(v_{\alpha })_{\alpha \in \Lambda }$$ of V, the family $$(Tv_{\alpha })_{\alpha \in \Lambda }$$ is order summable in W. The first part of the article is devoted to completely additive operators. We prove that a dominated orthogonally additive operator $$T:V\rightarrow W$$ from a Banach–Kantorovich space V to a Banach–Kantorovich space W is completely additive if and only if so is its exact dominant $$\left\bracevert T\right\bracevert :E\rightarrow F$$ . One of the our main results asserts that an orthogonally additive map defined on a lateral ideal and dominated by a positive completely additive operator can be extended to a dominated completely additive operator defined on the whole space. In the second part of the article, we consider C-compact dominated orthogonally additive operators. We show that the C-compactness of a dominated orthogonally additive operator $$T:V\rightarrow W$$ from a decomposable lattice-normed space (V, E) to an order continuous Banach lattice W implies the C-compactness of its exact dominant $$\left\bracevert T\right\bracevert :E\rightarrow W$$ .

Unbounded absolutely weak Dunford–Pettis operators

In the present article, we expose various properties of unbounded absolutely weak Dunford?Pettis and unbounded absolutely weak compact operators on a Banach lattice E. In addition to their topological and lattice properties, we investigate relationships between M-weakly compact operators, L-weakly compact operators, and order weakly compact operators with unbounded absolutely weak Dunford-Pettis operators. We show that the square of any positive uaw-Dunford-Pettis (M-weakly compact) operator on an order continuous Banach lattice is compact. Many examples are given to illustrate the essential conditions.

A Note on Spectral Sublinearity for Collections of Positive Compact Operators on Banach Lattices

Abstract In this paper we consider the question when a triangularizable semigroup S of positive compact ideal-triangularizable operators on an order continuous Banach lattice X is ideal-triangularizable. We prove that triangularizability always implies ideal-triangularizability iff X contains at most one atom. Under this condition we connect ideal-triangularizability of S with spectral properties of S. Surprisingly, S is idealtriangularizable iff the spectral radius is subadditive on S iff the spectral radius is submultiplicative on S. We also consider a pair of positive compact operators A and T with the property that A has a T-stable spectrum.

Some loose ends on unbounded order convergence

The notion of almost everywhere convergence has been generalized to vector lattices as unbounded order convergence, which proves a very useful tool in the theory of vector and Banach lattices. In this short note, we establish some new results on unbounded order convergence that tie up some loose ends. In particular, we show that every norm bounded positive increasing net in an order continuous Banach lattice is uo-Cauchy and that every uo-Cauchy net in an order continuous Banach lattice has a uo-limit in the universal completion.

Choquet type L1-spaces of a vector capacity

Given a set function Λ with values in a Banach space X, we construct an integration theory for scalar functions with respect to Λ by using duality on X and Choquet scalar integrals. Our construction extends the classical Bartle–Dunford–Schwartz integration for vector measures. Since just the minimal necessary conditions on Λ are required, several L1-spaces of integrable functions associated to Λ appear in such a way that the integration map can be defined in them. We study the properties of these spaces and how they are related. We show that the behavior of the L1-spaces and the integration map can be improved in the case when X is an order continuous Banach lattice, providing new tools for (non-linear) operator theory and information sciences.

Unbounded norm topology in Banach lattices

A net (xα) in a Banach lattice X is said to un-converge to a vector x if ‖|xα−x|∧u‖→0 for every u∈X+. In this paper, we investigate un-topology, i.e., the topology that corresponds to un-convergence. We show that un-topology agrees with the norm topology iff X has a strong unit. Un-topology is metrizable iff X has a quasi-interior point. Suppose that X is order continuous, then un-topology is locally convex iff X is atomic. An order continuous Banach lattice X is a KB-space iff its closed unit ball BX is un-complete. For a Banach lattice X, BX is un-compact iff X is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence.

Dominated operators from lattice-normed spaces to sequence Banach lattices

We show that every dominated linear operator from a Banach–Kantorovich space over an atomless Dedekind-complete vector lattice to a sequence Banach lattice ℓp(Γ) or c0(Γ) is narrow. As a consequence, we obtain that an atomless Banach lattice cannot have a finite-dimensional decomposition of a certain kind. Finally, we show that the order-narrowness of a linear dominated operator T from a lattice-normed space V to the Banach space with a mixed norm (W,F) over an order-continuous Banach lattice F implies the order-narrowness of its exact dominant |T|.

$p$-variations of vector measures with respect to vector measures and integral representation of operators

In this paper we provide two representation theorems for two relevant classes of operators from any $p$-convex order continuous Banach lattice with weak unit into a Banach space: the class of continuous operators and the class of cone absolutely summing operators. We prove that they can be characterized as spaces of vector measures with finite $p$-semivariation and $p$-variation, respectively, with respect to a fixed vector measure. We give in this way a technique for representing operators as integrals with respect to vector measures.

Unbounded order convergence and application to martingales without probability

A net (xα)α∈Γ in a vector lattice X is unbounded order convergent (uo-convergent) to x if |xα−x|∧y→o0 for each y∈X+, and is unbounded order Cauchy (uo-Cauchy) if the net (xα−xα′)Γ×Γ is uo-convergent to 0. In the first part of this article, we study uo-convergent and uo-Cauchy nets in Banach lattices and use them to characterize Banach lattices with the positive Schur property and KB-spaces. In the second part, we use the concept of uo-Cauchy sequences to extend Doob's submartingale convergence theorems to a measure-free setting. Our results imply, in particular, that every norm bounded submartingale in L1(Ω;F) is almost surely uo-Cauchy in F, where F is an order continuous Banach lattice with a weak unit.

Convergence results for a class of pramarts and superpramarts in Banach spaces

We prove several results in the convergence of vector-valued pramarts and convex weakly compact valued pramarts-sub-superpramarts in a separable Banach space. We also establish several a.s. norm convergence of vector-valued pramarts in the dual of a separable Banach space. An unusual convergence result for bounded positive vector-valued submartingales in separable order continuous Banach lattice is provided via a renorming lattice norm and other related tools. 200 Mathematics Subject Classification: 28B20, 60F15, 60B12.

A lattice approach to narrow operators

It is known that if a rearrangement invariant function space E on (0,1) has an unconditional basis then each linear continuous operator on E is a sum of two narrow operators. On the other hand, the sum of two narrow operators in L1 is narrow. To find a general approach to these results, we extend the notion of a narrow operator to the case when the domain space is a vector lattice. Our main result asserts that the set Nr(E, F )o f all nar- row regular operators is a band in the vector lattice Lr(E, F) of all regular operators from a non-atomic order continuous Banach lattice E to an order continuous Banach lattice F. The band generated by the disjointness preserv- ing operators is the orthogonal complement to Nr(E, F )i nLr(E, F). As a consequence we obtain the following generalization of the Kalton-Rosenthal theorem: every regular operator T : E → F from a non-atomic Banach lattice E to an order continuous Banach lattice F has a unique representation as T = TD + TN where TD is a sum of an order absolutely summable family of disjointness preserving operators and TN is narrow. Mathematics Subject Classification (2000). Primary 47B65; secondary 47B38.

Radon–Nikodym property for the projective tensor product of Köthe function spaces

Let E be an order continuous Köthe function space (or an order continuous Banach lattice) and X be a dual Banach space. Then E ⊗ ̂ X , the projective tensor product of E and X, has the Radon–Nikodym property if and only if both E and X do.

Subspaces of Rearrangement-Invariant Spaces

AbstractWe prove a number of results concerning the embedding of a Banach lattice X into an r. i. space Y. For example we show that if Y is an r. i. space on [0, ∞) which is p-convex for some p > 2 and has nontrivial concavity then any Banach lattice X which is r-convex for some r > 2 and embeds into Y must embed as a sublattice. Similar conclusions can be drawn under a variety of hypotheses on Y; if X is an r. i. space on [0, 1] one can replace the hypotheses of r-convexity for some r > 2 by X ≠ L2.We also show that if Y is an order-continuous Banach lattice which contains no complemented sublattice lattice-isomorphic to ℓ2X is an order-continuous Banach lattice so that ℓ2 is not complementary lattice finitely representable in X and X is isomorphic to a complemented subspace of Y then X is isomorphic to a complemented sublattice of YN for some integer N.

WhenL1of a vector measure is an AL-space

We consider the space of real functions which are integrable with respect to a countably additive vector measure with values in a Banach space. In a previous paper we showed that this space can be any order continuous Banach lattice with weak order unit. We study a priori conditions on the vector measure in order to guarantee that the resulting Lι is order isomorphic to an AL-space. We prove that for separable measures with no atoms there exists a Co-valued measure that generates the same space of integrable functions. Introduction. Given a vector measure v we consider the space of classes of real functions which are integrable with respect to v in the sense of Lewis (L-l), denoted by Lι{y). In (C, Theorem 8) we showed that every order continuous Banach lattice with weak unit can be obtained as Lι(u) for a suitable vector measure v. In particular we have Hubert spaces as L 1 of a vector measure. A natural question arises. Under what conditions on the vector measure, or on the Banach space in which the measure takes its values, is the resulting L1 of the vector measure order isomorphic to an AL-space? Recall that a Banach lattice is an AL-space when the norm is additive for disjoint vectors. An order isomorphism is a linear isomorphism that preserves the lattice operations. So the question can be restated in the following way. When can Lx(u) be equivalently renormed so that endowed with the new norm and the same order is a Banach lattice where the norm is additive for disjoint functions? In § 1 we fix notation and basic definitions. In §2 we show the special role that the space Lι(\u\) plays in the problem we are studying, where \v is the variation of υ . It is shown in Proposition 2 that Lx(v) is an Jϊ?\-space, in the sense of Linden- strauss and Pelczynski (L-P), if and only if it is order isomorphic to Lι(\v\). From this it follows that bounded variation of the measure is a necessary condition. The conditions cannot be placed on the Banach space in which the measure takes its values. This is shown in Example 1, that also shows that neither bounded variation nor domination of the variation by the semivariation are sufficient conditions. In §3 we study measures with values in C(K) spaces. In Theorem 1

On (V*) sets and Pelczynski's property (V*)

The concept of (V*) set was introduced, as a dual companion of that of (V)-set, by Pelczynski in his important paper [14]. In the same paper, the so called properties (V) and (V*) are defined by the coincidence of the (V) or (V*) sets with the weakly relatively compact sets. Many important Banach space properties are (or can be) defined in the same way; that is, by the coincidence of two classes of bounded sets. In this paper, we are concerned with the study of the class of (V*) sets in a Banach space, and its relationship with other related classes. To this general study is devoted Section I. A (as far as we know) new Banach space property (we called it property weak (V*)) is defined, by imposing the coincidence of (V*) sets and weakly conditionally compact sets. In this way, property (V*) is decomposed into the conjunction of the weak (V*) property and the weak sequential completeness. In Section II, we specialize to the study of (V*) sets in Banach lattices. The main result in the section is that every order continuous Banach lattice has property weak (V*), which extends previous results of E. and P. Saab ([16]). Finally, Section III is devoted to the study of (V*) sets in spaces of Bochner integrable functions. We characterize a broad class of (V*) sets in L1(μ, E), obtaining similar results to those of Andrews [1], Bourgain [6] and Diestel [7] for other classes of subsets. Applications to the study of properties (V*) and weak (V*) are obtained. Extension of these results to vector valued Orlicz function spaces are also given.

Banach sublattices of weakL 1

It is proved that every separable, order-continuous Banach lattice is lattice isometric to a sublattice of the Banach envelope of weakL1. This result is also proved for certain non-separable Banach lattices with generalized basis, where the norm satisfies a “spreading invariance” condition.

On the Dieudonne Property for C(Ω, E)

In a recent paper, F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E), the Banach space of continuous functions from a compact Hausdorff space Q to E, has the Dieudonne property. They asked whether or not the result is still true if one only assumes that E does not contain a copy of l1. In this paper we give a positive answer to their question. As a corollary we show that if E is a subspace of an order continuous Banach lattice, then E has the Dieudonne property if and only if C(Q, E) has the same property. If E is a Banach space and Q is a compact Hausdorff space, then C(Q, E) will stand for the Banach space of the E-valued continuous functions on Q under the supremum norm. A Banach space E is said to have the Dieudonne property if for every Banach space F, any bounded linear operator T: E -> F that transforms weakly Cauchy sequences into weakly convergent sequences is weakly compact. In [3] F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E) has the Dieudonne property and they asked whether the same result is true when replacing the assumption that E* is separable by supposing only that 11 does not embed in E. In this paper we give a positive answer to their question. Recall that a topological space (X, -y) is said to be Polish if it is homeomorphic to a separable complete metric space and it is said to be analytic if it is the continuous image of a Polish space. A subset A of a topological space (X, y) is said to be coanalytic if its complement (X\A, y) is analytic. Finally A is said to be PCA if it is the continuous image of a coanalytic space. The notations and terminology used and not defined can be found in [5, 8, or 10]. In the proof of Lemma 3 we need the following two results. THEOREM 1 (M. SREBRNY [9]). Let X and Y be two analytic spaces and let F be a multivalued function from X to the subsets of Y, such that its graph is PCA and for which one can prove that for every x c X, F(x) :8 0 using only the axioms of ZFC. Then there exists a universally measurable map f: X -> Y such that f (x) c F(x) for every x E X. THEOREM 2 (I. ASSANI [1, 2]). Let E be a separable Banach space. The set of weakly Cauchy sequences is a coanalytic subset of EN. Received by the editors January 5, 1985. 1980 Mathematics Subject Classification. Primary 46G10, 46B22.