Let \(F, F_1, \dots , F_n\) and \(G\) be Hilbert spaces. For \(0 < s < \infty,\) set \(\ell_s(F) = \{(x_j)_j \subset F : \|(x_j)\|_s \) (\( = (\sum_{j=1}^\infty \|x_j\|^s)^{\frac{1}{s}})\)) \(< \infty\},\) and let \(\ell_s^w(F) = \{(x_j)_j \subset F : (\varphi(x_j)) \in \ell_s\) for every \(\varphi \in F^\prime\}\) endowed with the usual norm \(\|(x_j)\|_{w,s} = \sup_{\varphi \in B_{F^\prime}} \|(\varphi(x_j))\|_s.\) The Banach space \(\ell_\infty(F) = \ell^w_\infty(F)\) is defined analogously. For \(r,r_1,\dots ,r_n \in (0,\infty),\frac{1}{r} \leq \frac{1}{r_1} + \dots + \frac{1}{r_n},\) a continuous \(n\)-linear operator \(T:F_1 \times \dots \times F_n \to G\) is said to be absolutely \((r;r_1,\dots ,r_n)\)-summing if for every \((x_j^1) \in \ell_{r_1}^w(F_1),\dots ,(x_j^n) \in \ell_{r_n}^w(F_n),\) we have \((T(x_j^1,\dots ,x_j^n)) \in \ell_r(G).\) The space of such \(n\)-linear operators is denoted \({\mathcal L}_{as}^{r;r_1,\dots ,r_n}(F_1,\dots ,F_n;G).\) Also, \(T\) is said to be Hilbert-Schmidt (\(T \in {\mathcal L}_{\text{HS}}(F_1,\dots ,F_n;G))\) if for each orthonormal basis \((e_j^1) \subset F_1,\dots , (e_j^n) \subset F_n,\) \(\|T\|_{\text{HS}} \equiv (\sum_{j_1,\dots ,j_n} \|T(e_{j_1}^1,\dots ,e_{j_n}^n)\|^2)^{\frac{1}{2}} < \infty.\)
The main direction of this paper is a discussion of a nearly twenty year old problem of A. Pietsch [“Ideals of multilinear functionals”, Teubner-Texte Math. 67, 185-199 (1984; Zbl 0562.47037)]: For \(n \geq 3,\) is it possible to find \(r, r_1, \dots , r_n\) as above so that \({\mathcal L}_{\text{HS}}(F_1,\dots ,F_n;\mathbb{K}) = {\mathcal L}_{as}^{r;r_1,\dots ,r_n}(F_1,\dots ,F_n;\mathbb{K})\) for all Hilbert spaces \(F_1,\dots ,F_n?\)
The author shows by means of a number of examples that the answer to this question is ‘no.’
(Reviewer’s remark: The author has noticed that Example 5.3 in this paper is not complete; a full correction is available directly from the author).