Asymptotic estimates on the von Neumann inequality for homogeneous polynomials

TL;DR

This paper investigates the asymptotic growth of the von Neumann inequality constant for homogeneous polynomials in multiple variables, providing precise behavior for certain cases and improving bounds for others.

Contribution

It determines the asymptotic behavior of the von Neumann inequality constant for fixed degree and q, answering a longstanding question for q=∞, and refines bounds for 2 ≤ q < ∞.

Findings

Exact asymptotic behavior for q=∞

Improved lower bounds for 2 ≤ q < ∞

Estimates for norms of homogeneous unimodular Steiner polynomials

Abstract

By the von Neumann inequality for homogeneous polynomials there exists a positive constant $C_{k,q}(n)$ such that for every $k$-homogeneous polynomial $p$ in $n$ variables and every $n$-tuple of commuting operators $(T_1, \dots, T_n)$ with $\sum_{i=1}^{n} \Vert T_{i} \Vert^{q} \leq 1$ we have \[ \|p(T_1, \dots, T_n)\|_{\mathcal L(\mathcal H)} \leq C_{k,q}(n) \; \sup\{ |p(z_1, \dots, z_n)| : \textstyle \sum_{i=1}^{n} \vert z_{i} \vert^{q} \leq 1 \}\,. \] For fixed $k$ and $q$, we study the asymptotic growth of the smallest constant $C_{k,q}(n)$ as $n$ (the number of variables/operators) tends to infinity. For $q = \infty$, we obtain the correct asymptotic behavior of this constant (answering a question posed by Dixon in the seventies). For $2 \leq q < \infty$ we improve some lower bounds given by Mantero and Tonge, and prove the asymptotic behavior up to a logarithmic factor. To achieve this we provide estimates of the norm of homogeneous unimodular Steiner polynomials, i.e. polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems.