On a question of Pietsch about Hilbert-Schmidt multilinear mappings

In 1983, Pietsch asked if, for n greater than or equal to 3 and all Hilbert spaces E-1,..., E-n, the vector space of the scalar valued absolutely (r;r (1),..., r(n))-summing multilinear mappings on E-1 x ... x E-n coincides wi th the vector space of the n-linear Hilbert-Schmidt functionals on E-1 x... x E-n, for some choice of r, r(1),..., r(n) is an element of ]0, +infinity] , satisfying 1/r less than or equal to 1/r(1) + ... + 1/r(n). We show that the answer to this question is no. Moreover, we show that the same question , for n greater than or equal to 2 and mappings with values in infinite dim ensional Hilbert spaces, has the answer no. (C) 2001 Academic Press.