Reflexivity and the Grothendieck property for positive tensor products of Banach lattices-II

Abstract

Let ϕ be an Orlicz function such that ϕ and its complementary function ϕ* satisfy the Δ₂-condition, let ℓ_ϕ be an Orlicz sequence space associated to ϕ, and let X be a Banach lattice. Then ℓ_ϕ ⊗_F X (respectively, ℓ_ϕ ∼⊗_i X), the Fremlin projective (respectively, the Wittstock injective) tensor product of ℓ_ϕ and X, has reflexivity or the Grothendieck property if and only if X has the same property and each positive linear operator from ℓ_ϕ (respectively, from ℓ_ϕ*) to X* (respectively, to X**) is compact.

*The author is partially supported by the NSF of China, Grant No. 10871213.

^† The author is supported by the NSF of China, Grant No. 10671048.