Limit formulas for norms of tensor power operators

TL;DR

This paper investigates the asymptotic behavior of tensor power operators between Banach spaces, establishing that their normalized operator norms converge to a well-known operator ideal norm, the 2-dominated norm.

Contribution

It provides a new limit formula connecting tensor power operator norms with the 2-dominated norm, enhancing understanding of tensor operators in Banach space theory.

Findings

Normalized tensor power operator norms converge to the 2-dominated norm.

The limit formula links tensor powers with standard operator ideals.

Results deepen the theoretical understanding of tensor operators in Banach spaces.

Abstract

Given an operator $\phi:X\rightarrow Y$ between Banach spaces, we consider its tensor powers $\phi^{\otimes k}$ 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 $\phi^{\otimes k}$ converges to the $2$-dominated norm $\gamma^*_2(\phi)$, one of the standard operator ideal norms.