Box norm singular-value identity
= Box norm singular-value identity
{title2=$\|H\|_\square^4=\sum_j\sigma_j^4$}
For a <real-valued function> $H$ on a nonempty finite <Cartesian product>, let $T_Hv(x)=\mathbb E_yH(x,y)v(y)$ with uniform <inner products>. The <box norm> obeys $\|H\|_\square^4=\operatorname{Tr}((T_HT_H^*)^2)=\sum_j\sigma_j^4$, where $\sigma_j$ are its <singular values>. For a <biregular graph> this separates the constant component $\gamma^4$ from the other fourth powers.