Qingying Bu

OBSERVATIONS ABOUT THE PROJECTIVE TENSOR PRODUCT OF BANACH SPACES, I—

Abstract A sequential description of the projective tensor product lp⊗X, is given. This description allows us to show the inclusion lp⊗X, into lp⊗X, is a semi-embedding. As a consequence, if X has the Radon-Nikodym property so does lp⊗X, A discussion of difficulties with Lp(0, 1) ⊗X, follows.

Abstract M- and Abstract L-Spaces of Polynomials on Banach Lattices

In this paper we use the norm of bounded variation to study multilinear operators and polynomials on Banach lattices. As a result, we obtain when all continuous multilinear operators and polynomials on Banach lattices are regular. We also provide new abstract M- and abstract L-spaces of multilinear operators and polynomials and generalize all the results by Grecu and Ryan, from Banach lattices with an unconditional basis to all Banach lattices.

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 ℓϕ be an Orlicz sequence space associated to ϕ, and let X be a Banach lattice. Then ℓϕ ⊗F X (respectively, ℓϕ ∼⊗i X), the Fremlin projective (respectively, the Wittstock injective) tensor product of ℓϕ and X, has reflexivity or the Grothendieck property if and only if X has the same property and each positive linear operator from ℓϕ (respectively, from ℓϕ*) 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.

ON BANACH SPACES VERIFYING GROTHENDIECK'S THEOREM

The projective tensor product � 2 ˆ ⊗ X of � 2 with any Banach space X sits inside the space Rad(X )o f all almost unconditionally summable sequences in X .I fX is of cotype 2 and u : X −→ Y is 2-summing, then u takes Rad(X )i nto� 2 ˆ ⊗ Y .C onsequently, if X is of cotype 2, then every operator from X to � 2 is 1-summing if and only if � 1 ˇ ⊗ X ⊆ � 2 ˆ ⊗ X .I nt his case, each 2-summing operator from � 2 to X is nuclear, and X does not have non-trivial type provided that dim X = ∞.

Diagonals of injective tensor products of Banach lattices with bases

Let E be a Banach lattice with a 1-unconditional basis

Polynomials on Banach lattices and positive tensor products

\{e_i: i \in \mathbb {N}\}

Types of Radon-Nikodym properties for the projective tensor product of Banach spaces

{ei:i∈N}. Denote by