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Limit formulas for norms of tensor power operators

Guillaume Aubrun, Alexander Müller-Hermes

Abstract

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

Limit formulas for norms of tensor power operators

Abstract

Given an operator between Banach spaces, we consider its tensor powers as operators from the -fold injective tensor product of to the -fold projective tensor product of . We show that after taking the th root, the operator norm of converges to the -dominated norm , one of the standard operator ideal norms.

Paper Structure

This paper contains 13 sections, 16 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on tensor norms and operator ideals
  3. Tensor norms
  4. 2-summing norm
  5. The Hilbert space factorization norm and its trace dual
  6. Proof of the main identities in finite dimensions
  7. Injective-to-Hilbertian regularization
  8. Injective-to-projective regularization
  9. Extension to infinite-dimensional spaces
  10. Discussion
  11. Tensor radii
  12. The Hilbert space factorization property
  13. The random tensors argument

Key Result

Proposition 3.1

Let $X_1$, $X_2$ be normed spaces, $H_1$, $H_2$ be Hilbert spaces, all of finite dimension. For every operators $\phi_1:X_1 \to H_1$ and $\phi_2 : X_2 \to H_2$, the operator satisfies $\pi_2(\phi_1 \otimes \phi_2)=\pi_2(\phi_1)\pi_2(\phi_2)$.

Theorems & Definitions (30)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • ...and 20 more