E. Jiménez Fernández

The Köthe Dual of an Abstract Banach Lattice

We analyze a suitable definition of Kothe dual for spaces of integrable functions with respect to vector measures defined on δ-rings. This family represents a broad class of Banach lattices, and nowadays it seems to be the biggest class of spaces supported by integral structures, that is, the largest class in which an integral representation of some elements of the dual makes sense. In order to check the appropriateness of our definition, we analyze how far the coincidence of the Kothe dual with the topological dual is preserved.

A Non-linear Approach to Signal Processing by Means of Vector Measure Orthogonal Functions

This research was supported by Ministerio de Ciencia e Innovacion, under project #MTM2009-14483-C02-02 (Spain).

Lattice Copies of ℓ2 in L1 of a Vector Measure and Strongly Orthogonal Sequences

Let m be an l2-valued (countably additive) vector measure and consider the space L2(m) of square integrable functions with respect to m. The integral with respect to m allows to define several notions of orthogonal sequence in these spaces. In this paper, we center our attention in the existence of strongly m-orthonormal sequences. Combining the use of the Kadec-Pelczynski dichotomy in the domain space and the Bessaga-Pelczynski principle in the range space, we construct a two-sided disjointification method that allows to prove several structure theorems for the spaces L1(m) and L2(m). Under certain requirements, our main result establishes that a normalized sequence in L2(m) with a weakly null sequence of integrals has a subsequence that is strongly m-orthonormal in L2(m∗), where m∗ is another l2-valued vector measure that satisfies L2(m) = L2(m∗). As an application of our technique, we give a complete characterization of when a space of integrable functions with respect to an l2-valued positive vector measure contains a lattice copy of l2.