Limit formulas for norms of tensor power operators

Given an operator $\varphi :X\to Y$ between Banach spaces, we consider its tensor powers ${\varphi }^{\otimes k}$ as operators from the k-fold injective tensor product of X to the k-fold projective tensor product of Y. We show that after taking the kth root, the operator norm of ${\varphi }^{\otimes k}$ converges to the 2-dominated norm ${\gamma }_2^{⁎}\left(\varphi \right)$, one of the standard operator ideal norms.