Purely non-atomic weak $L^p$ spaces

Leung, Denny

Purely non-atomic weak

L^{p}

spaces

Leung, Denny

Studia Mathematica, Tome 122 (1997), p. 55-66 / Harvested from The Polish Digital Mathematics Library

Access to full text
Full (PDF)

Résumé

Let (Ω,∑,μ) be a purely non-atomic measure space, and let 1 < p < ∞. If $L^{p, \infty} (Ω, \sum, μ)$ is isomorphic, as a Banach space, to $L^{p, \infty} (Ω^{'}, \sum^{'}, μ^{'})$ for some purely atomic measure space (Ω’,∑’,μ’), then there is a measurable partition $Ω = Ω_{1} \cup Ω_{2}$ such that $(Ω_{1}, Σ \cap Ω_{1}, μ |_{Σ \cap Ω_{1}})$ is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable $σ \subseteq Ω_{2}$ . In particular, $L^{p, \infty} (Ω, \sum, μ)$ is isomorphic to $ℓ^{p, \infty}$ .

Publié le : 1997-01-01

Zbl 0901.46025

EUDML-ID : urn:eudml:doc:216360

@article{bwmeta1.element.bwnjournal-article-smv122i1p55bwm,
     author = {Denny Leung},
     title = {Purely non-atomic weak $L^p$ spaces},
     journal = {Studia Mathematica},
     volume = {122},
     year = {1997},
     pages = {55-66},
     zbl = {0901.46025},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv122i1p55bwm}
}

Leung, Denny. Purely non-atomic weak $L^p$ spaces. Studia Mathematica, Tome 122 (1997) pp. 55-66. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv122i1p55bwm/

Bibliographie

[00000] [1] M. I. Kadec and A. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$ , Studia Math. 21 (1962), 161-176.

[00001] [2] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, 1974. | Zbl 0285.46024

[00002] [3] D. H. Leung, Isomorphism of certain weak $L^{p}$ spaces, Studia Math. 104 (1993), 151-160. | Zbl 0814.46015

[00003] [4] D. H. Leung, Isomorphic classification of atomic weak $L^{p}$ spaces, in: Interaction between Functional Analysis, Harmonic Analysis and Probability, N. J. Kalton, E. Saab and S. J. Montgomery-Smith (eds.), Marcel Dekker, 1996, 315-330. | Zbl 0836.46016

[00004] [5] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, 1977. | Zbl 0362.46013

[00005] [6] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, 1979. | Zbl 0403.46022

[00006] [7] H. P. Lotz, Weak* convergence in the dual of weak $L^{p}$ , unpublished manuscript. | Zbl 1201.46010

[00007] [8] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, 1974. | Zbl 0296.47023