On the relation between Radonifying mappings and kernels of probability measures on Banach spaces (Q1067572)

Let E, F be Banach spaces with the duals \(E^*\), \(F^*\), and \(\mu\) a Radon probability measure on E. Denote by \(\tau_{\mu}\) the vector topology on \(E^*\) induced by the convergence in probability on \(L^ 0(E,\mu)\). The kernel of \(\mu\) is defined as the dual of \((E^*,\tau_{\mu})\) and denoted by \(K_{\mu}\). Note that \(K_{\mu}\) is a linear subspace of E. Let T be a continuous linear mapping from F into E. Then we consider the following problem: Are the following two conditions equivalent? (1) There exists a Radon probability measure \(\mu\) on E such that \(K_{\mu}\) contains T(F). (2) \(T^*: E^*\to F^*\) is Radonifying, that is, it takes each continuous cylindrical measure on \(E^*\) into a Radon measure on \(F^*.\) It is shown that the answer is yes if E is isomorphic to a subspace of some \(L^ p\) \((1<p<\infty)\), or, if E is of type 2. Moreover, the other equivalent conditions are given in terms of quasi-invariant measures and p-summing operators. Some applications are also given.