p-Summing Operators on Injective Tensor Products of Spaces

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⊗̂ǫY −→ 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 psumming. Indeed, there is an operator T on C[0, 1]⊗̂ǫl1 whose associated operator T is 2-summing, but for all N ∈ N, there exists an N -dimensional subspace U of C[0, 1]⊗̂ǫl1 such that T restricted to U is equivalent to the identity operator on l∞. Finally, we show that there is a compact Hausdorff space K and a bounded linear operator T : C(K)⊗̂ǫl1 −→ l2 for which T : C(K) −→ ∏ 1(l1, l2) is not 2-summing. (∗) Research supported in part by an NSF Grant DMS 9001796 (∗∗) Research supported in part by an NSF Grant DMS 87500750 A.M.S. (1980) subject classification: 46B99 Introduction Let X and Y be Banach spaces, and let X⊗̂ǫY denote their injective tensor product. In this paper, we shall study the behavior of those operators on X⊗̂ǫY that are p-summing. If X , Y and Z are Banach spaces, then every p-summing operator T : X⊗̂ǫY −→ Z induces a p-summing linear operator T : X −→ ∏ p(Y, Z). This raises the following question: given two Banach spaces Y and Z, and 1 ≤ p < ∞, for what Banach spaces X is it true that a bounded linear operator T : X⊗̂ǫY −→ Z is p-summing whenever T : X −→ ∏ p(Y, Z) is p-summing? In [11], it was shown that whenever X = C(Ω) is a space of all continuous functions on a compact Hausdorff space Ω, then T : C(Ω)⊗̂ǫY −→ Z is 1-summing if and only if T : C(Ω) −→ ∏ 1(Y, Z) is 1-summing. We will extend this result by showing that this result still remains true if X is any £∞-space. We will also give an example to show that the result need not be true if X is not a £∞-space. For this, we shall exhibit a 2-summing operator T on l2⊗̂ǫl2 that is not 1-summing, but such that the associated operator T is 1-summing. The case p > 1 turns out to be quite different. Here, the £∞-spaces do not seem to play any important role. We show that for each 1 < p < ∞, there exists a bounded linear operator T : C[0, 1]⊗̂ǫl2 −→ l2 such that T : C[0, 1] −→ ∏ p(l2, l2) is p-summing, but such that T is not p-summing. We will also give an example that shows that, in general, the condition on T to be 2-summing is too weak to imply any good properties for the operator T at all. To illustrate this, we shall exhibit a bounded linear operator T on C[0, 1]⊗̂ǫl1 with values in a certain Banach space Z, such that T : C[0, 1] −→ ∏ 2(l1, Z) is 2-summing, but for any given N ∈ N, there exists a subspace U of C[0, 1]⊗̂ǫl1, with dimU = N , such that T restricted to U is equivalent to the identity operator on l∞. Finally, we show that there is a compact Hausdorff space K and a bounded linear operator T : C(K)⊗̂ǫl1 −→ l2 for which T : C(K) −→ ∏ 1(l1, l2) is not 2-summing. 1 I Definitions and Preliminaries Let E and F be Banach spaces, and let 1 ≤ q ≤ p < ∞. An operator T : E −→ F is said to be (p, q)-summing if there exists a constant C ≥ 0 such that for any finite sequence e1, e2, . . . , en in E, we have