On Banach lattices with Levi norms (original) (raw)

Some properties of the space of regular operators on atomic Banach lattices

Collectanea mathematica, 2011

Let L r (E, X ) denote the space of regular linear operators from a Banach lattice E to a Banach lattice X . In this paper, we show that if E is a separable atomic Banach lattice, then L r (E, X ) is reflexive if and only if both E and X are reflexive and each positive linear operator from E to X is compact; moreover, if E is a separable atomic Banach lattice such that E and E * are order continuous, then L r (E, X ) has the Radon-Nikodym property (respectively, is a KB-space) if and only if X has the Radon-Nikodym property (respectively, is a KB-space) and each positive linear operator from E to X is compact.

Some characterizations of Riesz spaces in the sense of strongly order bounded operators

Positivity, 2019

We investigate some properties of strongly order bounded operators. For example, we prove that if a Riesz space E is an ideal in E ∼∼ and F is a Dedekind complete Riesz space then for each ideal A of E, T is strongly order bounded on A if and only if T A is strongly order bounded. We show that the class of strongly order bounded operators satisfies the domination problem. On the other hand, we present two ways for decomposition of strongly order bounded operators, and we give some of their properties. Also, it is shown that E has order continuous norm or F has the b-property whenever each pre-regular operator form E into F is order bounded.

Multi-norms and Banach lattices

Dissertationes Mathematicae

In 2012, Dales and Polyakov introduced the concepts of multi-norms and dual multi-norms based on a Banach space. Particular examples are the lattice multi-norm p}¨} L n q and the dual lattice multi-norm p}¨} DL n q based on a Banach lattice. Here we extend these notions to cover 'p-multi-norms' for 1 ď p ď 8, where 8-multi-norms and 1-multi-norms correspond to multinorms and dual multi-norms, respectively. We shall prove two representation theorems. First we modify a theorem of Pisier to show that an arbitrary multi-normed space can be represented as ppY n , }¨} L n q : n P Nq, where Y is a closed subspace of a Banach lattice; we then give a version for certain p-multi-norms. Second, we obtain a dual version of this result, showing that an arbitrary dual multi-normed space can be represented as pppX{Y q n , }¨} DL n q : n P Nq, where Y is a closed subspace of a Banach lattice X; again we give a version for certain p-multi-norms. We shall discuss several examples of p-multi-norms, including the weak p-summing norm and its dual and the canonical lattice p-multi-norm based on a Banach lattice. We shall determine the Banach spaces E such that the p-sum power-norm based on E is a p-multi-norm. This relies on a famous theorem of Kwapień; we shall present a simplified proof of this result. We shall relate p-multi-normed spaces to certain tensor products. Our representation theorems depend on the notion of 'strong' p-multi-norms, and we shall define these and discuss when p-multi-norms and strong p-multi-norms pass to subspaces, quotients, and duals; we shall also consider whether these multi-norms are preserved when we interpolate between couples of p-multi-normed spaces. We shall discuss multi-bounded operators between p-multi-normed spaces, and identify the classes of these spaces in some cases, in particular for spaces of operators between Banach lattices taken with their canonical lattice p-multi-norms. Acknowledgements. The authors are grateful to the London Mathematical Society for the award of Scheme 2 grant 21202 that allowed Troitsky to come to Lancaster in May 2013; to the Fields Institute in Toronto, for invitations to Dales, Laustsen, and Troitsky to participate in the Thematic Program on Abstract Harmonic Analysis, Banach and Operator Algebras in March and April, 2014; to the Lorentz Center in Leiden for invitations to Dales, Laustsen, and Troitsky to participate in a meeting on Ordered Banach Algebras in July, 2014. Oikhberg acknowledges with thanks the support of the Simons Foundation Travel Grant 210060, and Troitsky acknowledges with thanks the support of an NSERC grant.

L-Limited and Almost L-Limited Sets in Dual Banach Lattices

Journal of Mathematical Extension, 2018

Following the concept of L-limited sets in dual Banach spaces introduced by Salimi and Moshtaghioun, we introduce the concept of almost L-limited sets in dual Banach lattices and then by a class of disjoint limited completely continuous operators on Banach lattices, we characterize Banach lattices in which almost L-limited subsets of their dual, coincide with L-limited sets.

Operators intoL 1 of a vector measure and applications to Banach lattices

Mathematische Annalen, 1992

Using compactness properties of bounded subsets of spaces of vector measure integrable functions and a representation theorem for q-convex Banach lattices, we prove a domination theorem for operators between Banach lattices. We generalize in this way several classical factorization results for operators between these spaces, as psumming operators.

Some characterizations of KB-operators on Banach lattices and ordered Banach spaces

Turkish Journal of Mathematics, 2020

We determine that two recent classes of KB-operators and weak KB-operators and the well-known class of b -weakly compact operators, from a Banach lattice into a Banach space, are the same. We extend our study to the ordered Banach space setting by showing that a weak chain-preserving operator between two ordered Banach spaces is a KB-operator if and only if it is a weak KB-operator.

Lattice norms on the unitization of a truncated normed Riesz space

Positivity, 2019

Truncated Riesz spaces was first introduced by Fremlin in the context of real-valued functions. An appropriate axiomatization of the concept was given by Ball. Keeping only the first Ball's Axiom (among three) as a definition of truncated Riesz spaces, the first named author and El Adeb proved that if E is truncated Riesz space then E ⊕ R can be equipped with a non-standard structure of Riesz space such that E becomes a Riesz subspace of E ⊕ R and the truncation of E is provided by meet with 1. In the present paper, we assume that the truncated Riesz space E has a lattice norm. and we give a necessary and sufficient condition for E ⊕R to have a lattice norm extending. . Moreover, we show that under this condition, the set of all lattice norms on E ⊕ R extending. has essentially a largest element. 1 and a smallest element. 0. Also, it turns out that any alternative lattice norm on E ⊕ R is either equivalent to. 1 or equals. 0. As consequences, we show that E ⊕ R is a Banach lattice if and only if E is a Banach lattice and we get a representation's theorem sustained by the celebrate Kakutani's Representation Theorem.

Strictly singular and power-compact operators on Banach lattices

preprint

Compactness of the iterates of strictly singular operators on Banach lattices is analyzed. We provide suitable conditions on the behavior of disjoint sequences in a Banach lattice, for strictly singular operators to be Dunford-Pettis, compact or have compact square. Special emphasis is given to the class of rearrangement invariant function spaces (in particular, Orlicz and Lorentz spaces). Moreover, examples of rearrangement invariant function spaces of xed arbitrary indices with strictly singular non power-compact operators are also presented.