p-summing operators on injective tensor products of spaces

الأصل · EN

Let X,Y and Z be Banach spaces, and let ∏ₚ(Y,Z) (1≤ p<∞) denote the space of p-summing operators from Y to Z. We show that, if X is a $∞-space, then a bounded linear operator T: X ⊗εY Z is 1-summing if and only if a naturally associated operator T#: X ∏₁(Y,Z) is 1-summing. This result need not be true if X is not a $∞-space. For p>1, several examples are given with X=C[0,1] to show that T# can be p-summing without T being p-summing. Indeed, there is an operator T on C[0,1] ⊗εℓ₁ whose associated operator T# is 2-summing, but for all N∈, there exists an N-dimensional subspace U of C[0,1] ⊗εℓ₁ such that T restricted to U is equivalent to the identity operator on ℓⁿ∞. Finally, we show that there is a compact Hausdorff space K and a bounded linear operator T:C(K) ⊗εℓ₁ ℓ₂ for which T#:C(K) ∏₁(ℓ₁, ℓ₂) is not 2-summing.