Radonification problem for cylindrical measures on tensor products of Banach spaces (Q1084626)

Let \(\lambda\) be an arbitrary cylindrical measure of type p \((1<p<\infty)\) defined on the tensor product \(E\otimes F\) of two Banach spaces (with respect to the duality with the space \(E'\otimes F')\). Let \(\alpha\) be a reasonable norm on \(G\otimes H\) for which the canonical embedding \(G{\hat \otimes}_{\alpha}H\to L(G',H)\) is one to one. An operator \(w_ 1\otimes w_ 2:\) \(E\otimes F\to G{\hat \otimes}_{\alpha}H\) is said to be p-Radonifying if \((w_ 1\otimes w_ 2)(\lambda)\) is a Radon probability on \(G{\hat \otimes}_{\alpha}H\) of order p. The main result of this paper is: \(w_ 1\otimes w_ 2\) is p- Radonifying if and only if it is \(\tilde p\)-summing. As an example, the tensor product of two p-summing operators is p-Radonifying whenever the norm \(\alpha\) satisfies \(\alpha \leq /d_ p\) or \(\alpha \leq g_ p\setminus\), and, the tensor product of a p-left-nuclear and p-summing operator is p-Radonifying, whenever \(\alpha \leq d_ p\).