A 3x3 dilation counterexample

TL;DR

This paper constructs a specific example of four 3x3 commuting contractions that do not dilate to commuting isometries, despite satisfying the scalar von Neumann inequality, highlighting limitations in dilation theory.

Contribution

It provides a counterexample of four 3x3 commuting contractions that do not dilate to commuting isometries, expanding understanding of dilation limitations.

Findings

Four 3x3 commuting contractions do not dilate to commuting isometries.

These matrices satisfy the scalar von Neumann inequality.

Any three such contractions do dilate to commuting isometries.

Abstract

We define four 3x3 commuting contractions which do not dilate to commuting isometries. However they do satisfy the scalar von Neumann inequality. These matrices are all nilpotent of order 2. We also show that any three $3\times3$ commuting contractions which are scalar plus nilpotent of order 2 do dilate to commuting isometries.