Orthogonal tensors and their extremal spectral properties

In most tensor spaces, the sharp estimate for the ratio between spectral and Frobenius norm is not known. However, using an adequate generalization of orthogonality, we can show that in those spaces in which orthogonal tensors exist, the extremal ratio can be determined and is attained only for multiples of such tensors. The existence of a real orthogonal third-order tensor is equivalent to a composition formula for bilinear forms (Hurwitz problem). We also present an inductive construction for certain tensors of order higher than three. Interestingly, orthogonal tensors (in the proposed sense) do not exist in higher-order complex spaces.