Geometric properties of some Calderón-Lozanovskiǐ spaces and Orlicz-Lorentz spaces


Geometry of Calderón-Lozanovskiǐ spaces Eφ in the case of a σ-finite measure and a Banach function space with the Fatou property is studied. It is proved that if the space of order continuous elements Ea ≠ {0} and φ ∉ ΔE2, then Eφ contains an order isometric copy of l∞. If Ea ≠ E then it is proved that Eφ contains an order almost isometric copy of l∞ for every Orlicz function φ. Under the assumption that E is uniformly monotone it is proved that ∈0(Eφ) ≤ ∈0(Lφ), where ∈0(Eφ) and ∈0(Lφ) stand respectively for the characteristic of convexity of Eφ and Lφ (the Orlicz space). As a consequence of this inequality, the characteristic of convexity of Orlicz-Lorentz space Λφ,ω is computed in the case when the Orlicz function φ is strictly convex. This generalizes the criterion for uniform rotundity of Λφ,ω given in [Ka3]. Criteria for strict monotonicity, local uniform monotonicity and uniform monotonicity of Orlicz-Lorentz spaces Λφ,ω are also given. Finally, uniform non-squareness, B-convexity and superreflexivity of Λφ,ω are studied.

Publication Title

Houston Journal of Mathematics

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