Asymptotic Tensor Powers of Banach Spaces

TL;DR

This paper investigates the asymptotic behavior of tensor powers of Banach spaces and operators, introducing the tensor radius concept to characterize Euclidean spaces and compute asymptotic norms.

Contribution

It defines the tensor radius for spaces and operators, characterizes Euclidean spaces via tensor radius, and computes this for symmetric spaces like erspaces.

Findings

Euclidean spaces have tensor radius equal to their dimension

Tensor radius for erspaces er spaces is computed

Tensor radius of operators with Euclidean domain or range equals their nuclear norm

Abstract

We study the asymptotic behaviour of large tensor powers of normed spaces and of operators between them. We define the tensor radius of a finite-dimensional normed space $X$ as the limit of the sequence $A_k^{1/k}$, where $A_k$ is the equivalence constant between the projective and injective norms on $X^{\otimes k}$. We show that Euclidean spaces are characterized by the property that their tensor radius equals their dimension. Moreover, we compute the tensor radius for spaces with enough symmetries, such as the spaces $\ell_p^n$. We also define the tensor radius of an operator $T$ as the limit of the sequence $B_k^{1/k}$, where $B_k$ is the injective-to-projective norm of $T^{\otimes k}$. We show that the tensor radius of an operator whose domain or range is Euclidean is equal to its nuclear norm, and give some evidence that this property might characterize Euclidean spaces.