Lattice copies of \(\ell^2\) in \(L^1\) of a vector measure and strongly orthogonal sequences
Summary: Let \textbf{m} be an \(\ell^2\)-valued (countably additive) vector measure and consider the space \(L^2\)(\textbf{m}) of square integrable functions with respect to \textbf{m}. The integral with respect to \textbf{m} allows to define several notions of orthogonal sequences in these spaces. In this paper, we center our attention on the existence of strongly \textbf{m}-orthonormal sequences. Combining the use of the Kadec-Pelczyński dichotomy in the domain space and the Bessaga-Pelczyński principle in the range space, we construct a two-sided disjointification method that allows to prove several structure theorems for the spaces \(L^1\)(\textbf{m}) and \(L^2\)(\textbf{m}). Under certain requirements, our main result establishes that a normalized sequence in \(L^2\)(\textbf{m}) with a weakly null sequence of integrals has a subsequence that is strongly \textbf{m}-orthonormal in \(L^2\)(\textbf{m}\(^\ast\)), where \textbf{m}\(^\ast\) is another \(\ell^2\)-valued vector measure that satisfies \(L^2(\mathbf{m}) = L^2(\text\textbf{m}^\ast)\). As an application of our technique, we give a complete characterization of when a space of integrable functions with respect to an \(\ell^2\)-valued positive vector measure contains a lattice copy of \(\ell^2\).
scientific article; zbMATH DE number 6102730
scientific article