Multilinear forms of Hilbert type and some other distinguished forms (Q2509116)

The authors of the article under review provide new examples of bounded multilinear forms on the Hilbert spaces \(\ell_2\) and \(L_2(0,\infty)\). In a first step, they show that for \(m \geq 3\) the \(m\)-linear integral versions of the Hilbert matrix, the Schur matrix and the Bergman--Hilbert matrix are bounded non-compact \(m\)-linear forms on \(L_2(0, \infty)\), providing lower and upper estimates for the norms of these forms. From this, they obtain consequences for \(m\)-linear forms generated by certain measurable kernels, also giving conditions when such forms are Hilbert--Schmidt. The authors then carry over these results to \(m\)-linear forms on \(\ell_2\), in particular to the analogues of the classical matrix forms. Finally, the exact value of the norm of the permanent on the \(n\)-dimensional Euclidean space (real or complex) is determined.