A note on Dunford-Pettis-like properties in projective tensor spaces

Abstract

Let ℓ_p ⊗ _π X (resp. ℓ_p ⊗_{Δ p} X) denote the projective (resp. natural tensor product) of ℓ_p and a Banach space X. We study p-(V ^∗) sets, Dunford-Pettis sets, and limited sets in ℓ_p ⊗ _π X (resp. ℓ_p ⊗_{Δ p} X). Some classes of operators T on ℓ_p ⊗ _π X (resp. ℓ_p ⊗_{Δ p} X) are characterized in terms of the representing operators T_n . We study pseudo weakly compact, limited completely continuous, DP q-convergent, limited q-convergent operators on ℓ_p ⊗ _π X (resp. ℓ_p ⊗_{Δ p} X). Applications to some properties on these spaces are given.