p-summing operators on injective tensor products of spaces

Abstract

Let X, Y and Z be Banach spaces, and let Πp(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→Π1(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∈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 l∞N. 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)→Π1(l1,l2) is not 2-summing.