Shape, Scale, and Minimality of Matrix Ranges

TL;DR

This paper investigates the properties of matrix convex sets, focusing on their determination by initial levels, the uniqueness of operator tuples from their matrix ranges, and implications for dilation scales.

Contribution

It provides new insights into the extent a matrix convex set is determined by its first level, offers counterexamples to existing assumptions about minimality and uniqueness, and improves dilation bounds.

Findings

Quantifies the disparity between product operations on matrix convex sets.

Shows counterexamples where minimality does not imply uniqueness.

Provides improved bounds for dilation of contractions to commuting normal operators.

Abstract

We study containment and uniqueness problems concerning matrix convex sets. First, to what extent is a matrix convex set determined by its first level? Our results in this direction quantify the disparity between two product operations, namely the product of the smallest matrix convex sets over $K_i \subseteq \mathbb{C}^d$, and the smallest matrix convex set over the product of $K_i$. Second, if a matrix convex set is given as the matrix range of an operator tuple $T$, when is $T$ determined uniquely? We provide counterexamples to results in the literature, showing that a compact tuple meeting a minimality condition need not be determined uniquely, even if its matrix range is a particularly friendly set. Finally, our results may be used to improve dilation scales, such as the norm bound on the dilation of (non self-adjoint) contractions to commuting normal operators, both concretely and abstractly.