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The von Neumann inequality for $3\times3$ matrices in the unit Euclidean ball

Dariusz Piekarz

TL;DR

This work proves that the constant $c_{d,3}$ in von Neumann's inequality for $d$-tuples of commuting, row contractive $3\times3$ matrices on the unit Euclidean ball is independent of $d$, equaling $c_{2,3}$. The approach extends the 2×2 case via Pick interpolation on the ball, employing the Kosi\'nski–Zwonek extremal 3-point classification to reduce the $d$-variable problem to fixed-coordinate or lower-dimensional instances. The authors provide a numerical bound $1.11767\le c_{d,3}\le 3.14626$, and conjecture the optimal value is near $2/\sqrt{3}$, based on computational evidence. Overall, the paper advances multivariable operator theory by linking ball Pick interpolation to a d-independent von Neumann constant for the 3×3 case and offering concrete computable estimates.

Abstract

It is shown that the constant $c_{d,3}$ in von Neumann's inequality for d-tuples of commutative and row contractive $3\times3$ matrices, as proved by Hartz, Richter, and Shalit in [2], is independent of the size of the d-tuple. A numerical estimation of the constant is provided.

The von Neumann inequality for $3\times3$ matrices in the unit Euclidean ball

TL;DR

This work proves that the constant in von Neumann's inequality for -tuples of commuting, row contractive matrices on the unit Euclidean ball is independent of , equaling . The approach extends the 2×2 case via Pick interpolation on the ball, employing the Kosi\'nski–Zwonek extremal 3-point classification to reduce the -variable problem to fixed-coordinate or lower-dimensional instances. The authors provide a numerical bound , and conjecture the optimal value is near , based on computational evidence. Overall, the paper advances multivariable operator theory by linking ball Pick interpolation to a d-independent von Neumann constant for the 3×3 case and offering concrete computable estimates.

Abstract

It is shown that the constant in von Neumann's inequality for d-tuples of commutative and row contractive matrices, as proved by Hartz, Richter, and Shalit in [2], is independent of the size of the d-tuple. A numerical estimation of the constant is provided.
Paper Structure (4 sections, 8 theorems, 43 equations)

This paper contains 4 sections, 8 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Case of a $d$-tuple of $2\times 2$ commuting, row contractive matrices
  3. Proof of Theorem \ref{['main_thm:1']}: Constant $c_{d,3}=c_{2,3}$
  4. Proof of Theorem \ref{['main_thm:1']}: Estimation of the $c_{2,3}$ constant

Key Result

Theorem 1.1

There exists the smallest constant $c_{d,n}\geq0,$ such that for any $d-$tuple of commuting, row contractive, $n\times n$ matrices and for any polynomial $p\in\mathbb{C}[z_{1},...,z_{d}],$ one has where $d,n\geq2.$ Additionally, $c_{d,n}>1$ if $n\geq3,\ d\geq2$ and $c_{d,2}=1,$ if $d\geq2.$

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['NP2:1']}
  • proof : Proof of Proposition \ref{['vN2:1']}
  • Theorem 3.1: Kosiński, Zwonek
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 4 more