Fernando Bombal

Unconditionally converging multilinear operators

We introduce a notion of unconditionally converging multilinear operator which allows to extend many of the results of the linear case to the multilinear case. We prove several characterizations of these multilinear operators (one of which seems to be new also in the linear case), which allow to considerably simplify the work with this kind of operators

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 L 1 (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.

On l 1 subspaces of Orlicz vector-valued function spaces

The purpose of this paper is to characterize the Orlicz vector-valued function spaces containing a copy or a complemented copy of l 1 . Pisier proved in [13] that if a Banach space E contains no copy of l 1 , then the space L p ( S , Σ, μ, E ) does not contain it either, for 1 p l 1 as a complemented subspace of L Φ ( E ). We obtain a complete characterization when E is a Banach lattice and only partial results in case of a general Banach space. We use here in a crucial way a result of E. Saab and P. Saab concerning the embedding of l 1 as a complemented subspace of C ( K, E ), the Banach space of all the E -valued continuous functions on the compact Hausdorff space K (see [14]). Finally, we use these results to characterize several classes of Banach spaces for which L Φ ( E ) has some Banach space properties, namely the reciprocal Dunford-Pettis property and Pelczynskis V property.

On the Dunford-Pettis property of the tensor product of () spaces

In this paper we characterize those compact Hausdorff spaces such that (and ) have the Dunford-Pettis Property, answering thus in the negative a question posed by Castillo and Gonzalez who asked if and have this property.

Regular multilinear operators on C(K) spaces

The purpose of this paper is to characterise the class of regular continuous multilinear operators on a product of C(K) spaces, with values in an arbitrary Banach space. This class has been considered recently by several authors in connection with problems of factorisation of polynomials and holomorphic mappings. We also obtain several characterisations of a compact dispersed space K in terms of polynomials and multilinear forms defined on C(K).

The Dieudonné property on C(K,E).

In this paper we prove that if E is a Banach space with separable dual, then the space C(K, E) of all continuous E-valued functions on a compact Haus-dorff topological space K has the Dieudonnep roperty.

Distinguished subsets in Vector Sequence Spaces

We characterize several classes of subsets in the Λ-sum (ΣEn)Λ of a sequence (En) of Banach spaces (where A is a Banach sequence ideal space), in terms of the corresponding classes in the spaces En. Several stability results are obtained.

Polynomial sequential continuity on C(K,E) spaces☆

We show that, for bounded sequences in C(K,E), the polynomial sequential convergence is not equivalent to the pointwise polynomial sequential convergence. We introduce several conditions on E under which different versions of the result are true when K is a scattered compact space. These conditions are related with some others appeared in the literature and they seem to be of independent interest.