Multilinear operators and weighted l1 norms

An m-linear operator T on a normed vector space V shall be called multiplicative if N being the given norm. Similarly, we shall say that T is Jordan-multiplicative if where Sm denotes the symmetric group on m elements. Our purpose is to provide necessary and sufficient conditions for the multiplicativity and the Jordan-multiplicativity of T, when V is finite dimensional and N is a weighted l1 norm. The results obtained in this note generalize some of those in [1] where V was an associative algebra and the multilinear operation was simply the product of m vectors.