On a Question of Pietsch about Hilbert–Schmidt Multilinear Mappings

In 1983, Pietsch asked if, for n ≥ 3 and all Hilbert spaces E1 En, the vector space of the scalar valued absolutely r r1 rn -summing multilinear mappings on E1 × · · · × En coincides with the vector space of the n-linear Hilbert–Schmidt functionals on E1 × · · · × En, for some choice of r r1 rn ∈ 0 +∞ , satisfying 1/r ≤ 1/r1 +· · ·+1/rn. We show that the answer to this question is no. Moreover, we show that the same question, for n ≥ 2 and mappings with values in infinite dimensional Hilbert spaces, has the answer no.