The positive approximation property of Banach lattices

In this paper we study the positive approximation property (p.a.p.) of Banach lattices. The main results give some characterizations of the p.a.p. and the bounded p.a.p. Some perturbation results on positive operators, which are of interest in other contexts, too, are proved.     

The strong Schur property in Banach lattices

We prove that in the class of discrete Banach lattices the strong Schur property is equivalent to the disjoint strong Schur property (Theorem 3.1). Roughly speaking the strong Schur property holds iff an appropriate condition concerning sequences with positive pairwise disjoint terms is satisfied.      

The Fatou property in p-convex Banach lattices

New features of the Banach function space , that is, the space of all ν-scalarly pth power integrable functions (with 1?p<∞ and ν any vector measure), are presented. The Fatou property plays an essential role and leads to a new representation theorem for a large class of abstract p-convex Banach lattices.     

Dual banach lattices and Banach lattices with the Radon-Nikodym property

We construct a separable dual Banach latticeE such that no non-trivial order interval of its dual is weakly compact. HenceE has the Radon-Nikodym property without being in some sense a dual in a natural way. The final draft of this paper was written while the author held a grant from NATO to visit the Ohio State University.     

On the dual positive Schur property in Banach lattices

The paper contains several characterizations of Banach lattices $E$ with the dual positive Schur property (i.e., $0 \le f_n \xrightarrow {\sigma (E^*,E)} 0$ implies $\Vert f_n\Vert \rightarrow 0$ ) and various examples of spaces having this property. We also investigate relationships between the dual positive Schur property, the positive Schur property, the positive Grothendieck property and the weak Dunford–Pettis property.     

The classification of injective Banach lattices

LetE be a 1-injective Banach lattice,X any Banach space andT: E ← X a norm bounded linear operator. Then eitherT is an isomorphism on some copy ofl _∞ inE or for all σ > 0 there is φ ∈E ₊ ^′ such that ‖Tu‖≦φ (|u|)+σ ‖u‖ for allu ∈E. We deduce the theorem that: A norm order continuous injective Banach lattice is order isomorphic to an (AL)-space.     

Banach lattices with the fatou property and optimal domains of kernel operators

New features of the Banach function space L¹_w(v), that is, the space of all v-scalarly integrable functions (with v any vector measure), are exposed. The Fatou property plays an essential role and leads to a new representation theorem for a large class of abstract Banach lattices. Applications are also given to the optimal domain of kernel operators taking their values in a Banach function space.     

Lipschitz classes and the Hardy-Littlewood property

We study the geometry of plane domains and the uniform Hölder continuity properties of analytic functions.With 1 Figure     

Free complex Banach lattices

The construction of the free Banach lattice generated by a real Banach space is extended to the complex setting. It is shown that for every complex Banach space E there is a complex Banach lattice ${\text{FBL}}_{\mathbb{C}}[E]$ containing a linear isometric copy of E and satisfying the following universal property: for every complex Banach lattice $X_{\mathbb{C}}$, every operator $T:E\to X_{\mathbb{C}}$ admits a unique lattice homomorphic extension $\stackrel{?}{T}:{\text{FBL}}_{\mathbb{C}}[E]\to X_{\mathbb{C}}$ with $6\stackrel{?}{T}6=6T6$. The free complex Banach lattice ${\text{FBL}}_{\mathbb{C}}[E]$ is shown to have analogous properties to those of its real counterpart. However, examples of non-isomorphic complex Banach spaces E and F can be given so that ${\text{FBL}}_{\mathbb{C}}[E]$ and ${\text{FBL}}_{\mathbb{C}}[F]$ are lattice isometric. The spectral theory of induced lattice homomorphisms on ${\text{FBL}}_{\mathbb{C}}[E]$ is also explored.     

The positive aspects of smoothness in Banach lattices

Let X be a Banach lattice, and let ${x\in X{\setminus}\{0\}}$ . We study the structure of the set Grad(x), of all supporting functionals of x. If X is a Dedekind σ-complete Banach lattice, there is an isometry from Grad(x) onto Grad(|x|); hence the elements x and |x| are smooth simultaneously. And if, additionally, X* is strictly monotone then Grad(|x|) consists of positive functionals. As a by-product of our results we obtain that an arbitrary Banach lattice X is strictly monotone whenever its dual X* is smooth.     

Measurable bundles of Banach lattices

The aim of this paper is to outline a portion of the theory of liftable bundles of Banach lattices and to give some applications to representation of dominated operators.