Complete characterizations of Kadec-Klee properties in Orlicz spaces

Abstract

We study the connections between the Kadec-Klee property for local convergence in measure H`, the Kadec-Klee property for global onvergence in measure Hg and the ∆2-condition for Orlicz function spaces Lϕ equipped with either the Luxemburg norm k · kϕ or the Orlicz norm k · k0 ϕ. Nominally, we prove that for (Lϕ, k · kϕ) the conditions: ϕ satisfies an appropriate ∆2-condition and Lϕ ∈ H`, Lϕ ∈ Hg are equivalent, although Lϕ ∈ Hg is not equivalent to Eϕ ∈ Hg. In contrast, we also prove that, in the case of a non-atomic infinite measure space, properties H` and Hg for (Lϕ, k · k0 ϕ) do not coincide. More precisely, we prove that if ϕ vanishes only at zero, then both these properties coincide and they are equivalent to ϕ ∈ ∆2. However, if ϕ vanishes outside zero, then (Lϕ, k · k0 ϕ) ∈ Hg if and only if ϕ ∈ ∆2(∞). Since in the last case (Lϕ, k k0 ϕ) is not order continuous, properties H` and Hg differ. Analogous results are also proved for the subspace Eϕ of Lϕ. It is also worth mentioning that the criteria for Eϕ ∈ H` as well as for Eϕ ∈ Hg were not previously known. It follows from the criteria that the appropriate regularity ∆2-condition for ϕ is necessary for Eϕ ∈ H`, E ϕ 0 ∈ H`, Eϕ ∈ Hg and E ϕ 0 ∈ Hg although these spaces are order continuous for any ϕ.