Lattice copies of \(\ell^2\) in \(L^1\) of a vector measure and strongly orthogonal sequences
type: Item
instance of: scholarly article
title: Lattice copies of \(\ell^2\) in \(L^1\) of a vector measure and strongly orthogonal sequences
zbMATH Open document ID: 1264.46021
DOI: 10.1155/2012/357210
published in: Journal of Function Spaces and Applications
publication date: 2012-11-07T00:00:00Z
review text: 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 \(\ell^2\).
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 46B25
zbMATH DE Number: 6102730
zbMATH Keywords: vector measure
zbMATH Keywords: Kadec-Pelczyński dichotomy
zbMATH Keywords: space of integrable functions
zbMATH Keywords: lattice copy of \(\ell^2\)
Wikidata QID: Q58908213
author: Eduardo Jiménez Fernández
author: E. A. Sánchez-Pérez
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/357210
OpenAlex ID: W2032246021
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description: scientific article; zbMATH DE number 6102730