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...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2012/357210 |
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| author | E. Jiménez Fernández E. A. Sánchez Pérez |
| author_facet | E. Jiménez Fernández E. A. Sánchez Pérez |
| author_sort | E. Jiménez Fernández |
| collection | DOAJ |
| description | 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-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 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. |
| format | Article |
| id | doaj-art-de92a5f5ba6a424a81e2b7901f9e9e8f |
| institution | OA Journals |
| issn | 0972-6802 1758-4965 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-de92a5f5ba6a424a81e2b7901f9e9e8f2025-08-20T02:21:06ZengWileyJournal of Function Spaces and Applications0972-68021758-49652012-01-01201210.1155/2012/357210357210Lattice Copies of ℓ2 in L1 of a Vector Measure and Strongly Orthogonal SequencesE. Jiménez Fernández0E. A. Sánchez Pérez1Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera, s/n, 46022 Valencia, SpainInstituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera, s/n, 46022 Valencia, SpainLet 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-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 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.http://dx.doi.org/10.1155/2012/357210 |
| spellingShingle | E. Jiménez Fernández E. A. Sánchez Pérez Lattice Copies of ℓ2 in L1 of a Vector Measure and Strongly Orthogonal Sequences Journal of Function Spaces and Applications |
| title | Lattice Copies of ℓ2
in L1 of a Vector Measure and Strongly Orthogonal Sequences |
| title_full | Lattice Copies of ℓ2
in L1 of a Vector Measure and Strongly Orthogonal Sequences |
| title_fullStr | Lattice Copies of ℓ2
in L1 of a Vector Measure and Strongly Orthogonal Sequences |
| title_full_unstemmed | Lattice Copies of ℓ2
in L1 of a Vector Measure and Strongly Orthogonal Sequences |
| title_short | Lattice Copies of ℓ2
in L1 of a Vector Measure and Strongly Orthogonal Sequences |
| title_sort | lattice copies of l2 in l1 of a vector measure and strongly orthogonal sequences |
| url | http://dx.doi.org/10.1155/2012/357210 |
| work_keys_str_mv | AT ejimenezfernandez latticecopiesofl2inl1ofavectormeasureandstronglyorthogonalsequences AT easanchezperez latticecopiesofl2inl1ofavectormeasureandstronglyorthogonalsequences |