let $A\in\mathbb{C}^{m\times n}$ and define $S=AA^\dagger$. The (square of the) operator/spectral norm of $A$ can be written as $$ \sigma_0^2 = \lim_{n\rightarrow\infty}\frac{u^TS^{n+1}u}{u^TS^nu} $$ where $u$ is any vector that is not orthogonal to the first left singular vector of $A$ (first as in corresponding to the largest singular value of $A$).
Question: can I make a tensor $\Gamma_n$ $=$ "$\frac{S^{n+1}}{S^n}$" (please notice the quotation marks) such that $$\sigma_0^2=\lim_{n\rightarrow\infty}\Gamma_n uuuu$$ where by $\Gamma_n uuuu$ we mean the contraction $\sum_{ijkl}(\Gamma_n)_{ijlk}u_iu_ju_ku_l$?