Radonification theorem for F-cylindrical probabilities (Q1089585)

Let E,F,G be Banach spaces. An operator W:E\(\otimes F\to G\) is \(\bar r{}_ p\)-nuclear, if it has a representation of the form \(W(u)=\sum v_ j([u,x_ j'])\), \(u\in E\otimes F\), for some \(\{x_ j'\}\subset E'\), \(\{v_ j\}\subset L(F,G)\), such that it holds \[ (\sum \| x_ j'\|^ p)^{1/p}\cdot \sup_{\| z'\| \leq 1}(\sum \| v_ j'z'\|^{p'})^{1/p'}<\infty. \] In the present paper, the following theorem is proved: Let f be reflexive and \(1\leq p<\infty\). If W:E\(\otimes F\to G\) is \(\bar r{}_ p\)-nuclear and \(\ell\) F-cylindrical probability on \(E\otimes F\) of type (p,F), then W(\(\lambda)\) is a Radon probability on G of order p. As an example, if w:E\(\to G\) is p-left nuclear, then \(w\otimes 1_ F:E\otimes F\to G{\hat \otimes}_{d_ p}F\) is p-radonifying, for \(1\leq p<\infty\) and reflexive F.