Abstract
We study the setP X of scalarsp such thatL p is lattice-isomorphically embedded into a given rearrangement invariant (r.i.) function spaceX[0, 1]. Given 0<α≤β<∞, we construct a family of Orlicz function spacesX=L F[0, 1], with Boyd indicesα andβ, whose associated setsP X are the closed intervals [γ, β], for everyγ withα≤γ≤β. In particular forα>2, this proves the existence of separable 2-convex r.i. function spaces on [0,1] containing isomorphically scales ofL p-spaces for different values ofp. We also show that, in general, the associated setP X is not closed. Similar questions in the setting of Banach spaces with uncountable symmetric basis are also considered. Thus, we construct a family of Orlicz spaces ℓF(I), with symmetric basis and indices fixed in advance, containing ℓp(Γ-subspaces for differentp’s and uncountable Λ⊂I. In contrast with the behavior in the countable case (Lindenstrauss and Tzafriri [L-T1]), we show that the set of scalarsp for which ℓp(Γ) is isomorphic to a subspace of a given Orlicz space ℓF(I) is not in general closed.
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Supported in part by DGICYT grant PB 94-0243.
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Hernández, F.L., Rodriguez-Salinas, B. Lattice-embedding scales ofL p spaces into orlicz spaces. Isr. J. Math. 104, 191–220 (1998). https://doi.org/10.1007/BF02897064
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DOI: https://doi.org/10.1007/BF02897064