Lattice copies of l(2) in L-1 of a vector measure and strongly orthogonal sequences

Abstract

Let m be an l(2)-valued (countably additive) vector measure and consider the space L-2(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 L-1(m) and L-2(m). Under certain requirements, our main result establishes that a normalized sequence in L-2(m) with a weakly null sequence of integrals has a subsequence that is strongly m-orthonormal in L-2(m*), where m* is another l(2)-valued vector measure that satisfies L-2(m) = L-2(m*). As an application of our technique, we give a complete characterization of when a space of integrable functions with respect to an l(2)-valued positive vector measure contains a lattice copy of l(2).