Singular Values of Trilinear Forms

Abstract

Let $T:\CH1 \times \CH2 \times \CH3 \ra \C $ be a trilinear form, where $\CH1$, $\CH2$, $\CH3$ are separable Hilbert spaces. In the hypothesis that at least two of the three spaces are finite \dimn/al we show that the norm square $\lam=\Norm T{}2$ is a root of a certain algebraic equation, usually of very high degree, which we baptize the millennial equation, because it is an analogue of the secular equation in the bilinear case. More generally, as indicated in the title, we can consider singular values of a trilinear form and their squares too satisfy the same equation. We work out the binary case (all three spaces are two dimensional). Even in this case the situation is complex, so, in the absence of any genuine results, we content ourselves with advancing a number of conjectures suggested by computer experiments. Finally, we connect the singular values of a trilinear form with the critical values of an associated family of a one parameter family of bilinear forms. Also here we have to offer mainly only experimental evidence.