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

Abstract

Let X be a Banach lattice and p, p be real numbers such that 1< p, p <∞ and 1/ p + 1/ p = 1. Then p⊗F X (respectively, p⊗i X ), the Fremlin projective (respectively, the Wittstock injective) tensor product of p and X , has reflexivity or the Grothendieck property if and only if X has the same property and each positive linear operator from p (respectively, from p ) to X * (respectively, to X * * ) is compact.

Grothendieck ordered Banach spaces with an interpolation property

“…Also, for any w r we have x * n+1 (w r ) ≤ x * n+1 (u rk r ) and by (10) we have u rk r ≤ r+k r i=r y i . Therefore…”

“…Trivial examples of Grothendieck spaces are the reflexive spaces and of non-Grothendieck the non-reflexive, separable spaces. For some recent results on Grothendieck spaces, independent from this article, we refer to [10], [2] and [14]. In [14], Theorem 15, the following cone characterization is proved, which unfortunately cannot be applied, at least directly in this article.…”

Cone characterization of Grothendieck spaces and Banach spaces containing c 0

“…In Banach-lattices we have: any σ -Dedekind complete AM-space with a unit is a Grothendieck space, see in [2,Theorem 4.44]. For a study of Grothendieck spaces and different equivalent definitions we refer to [6] and for some resent results on Grothendieck spaces we refer to [1,10].…”

Lattice sequence spaces and summing mappings

“…In [9,7] and [3] the authors introduces the space of positive strongly p-summable sequences ℓ π p (X) (this space introduced by Cohen for Banach space [10]) and the space of positive unconditionally p-summable sequences ℓ ε,0 p (X). In this paper we use it together with the space of positive weakly p-summable sequences to define some classes of positive summing operators and characterize the classes of positive strongly (p, q)summing operators, positive (p, q)-summing, and the positive Cohen (p, q)-nuclear operators.…”

“…Consider the case where E is replaced by a Banach lattice X. The space of positive weakly p-summable sequences was introduced in [9,7] by…”