Multilinear absolutely summing operators associated to a tensor norm

Let 1 <= p < infinity, beta(n) a tensor norm on the full n-fold tensor product. On X1 circle times center dot center dot center dot circle times X-n circle times Xn+1 we define the Saphar tensor norm associated with beta n by d(p)* (beta(n)) (u) = inf { l(p)* x(i)(n+1) (sup)(?alpha?p*<= 1) beta(n) ( Sigma(m)(i=1) alpha ix(i)(1) circle times center dot center dot center dot circle times x(i)(n) )} where the infimum is taken over all u= rin=1xi1 circle times center dot center dot center dot circle times xn+1 i . We introduce the concept of the multilinear(beta(n), p)-summing operator and prove that Pi beta np (X-1, ..., X-n; Y*), the class of all (beta n, p)-summing operators from X-1 x center dot center dot center dot x X-n into Y*, is isometrically isomorphic to X-1 circle times center dot center dot center dot circle times X-n circle times Y, d(p)* (beta(n)). Various other questions are also investigated.