Limit formulas for norms of tensor power operators

Abstract

Given an operator $\ddot{I}•:X\to Y$ between Banach spaces, we consider its tensor powers $\ddot{I}{•}^{\otimes k}$ 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 $k$th root, the operator norm of $\ddot{I}{•}^{\otimes k}$ converges to the $2$-dominated norm $γ^*_2(ϕ)$, one of the standard operator ideal norms.

11 pages, no figures