Hilbert-Schmidt and multiple summing operators

We characterize the multiple p-summing operators for p a parts per thousand yen 2, in the situation where the range and some of the factors of the cartesian product are Hilbert spaces. As a consequence, we complete some recent results obtained in this area. Thus, we prove a coincidence result for multiple p-summing operators for p a parts per thousand yen 2, which has no linear analogue. Also, we give the necessary and sufficient conditions for some concrete bilinear operators to be multiple p-summing, p a parts per thousand yen 1.