Shape, Scale, and Minimality of Matrix Ranges

We study the matrix range $\mathcal{W}(T)$ of an operator tuple $T$, which is the closed and bounded matrix convex set containing all images of $T$ under unital completely positive maps into matrix algebras. To what extent does $\mathcal{W}(T)$ determine $T$? This problem is considered for minimal tuples, which by definition admit no proper summand with the same matrix range as the original. We clarify results in the literature by showing that a minimal compact tuple need not be determined uniquely by its matrix range. The counterexamples constructed are such that the matrix range $\mathcal{W}(T)$ is the smallest matrix convex set lying over its scalar level $K$, denoted $\mathcal{W}^{\text{min}}(K)$. On the other hand, if a compact tuple $T$ satisfies $\mathcal{W}(T) = \mathcal{W}^{\text{min}}(K)$, then the shape of $K$ is determined, even if $T$ is not normal. For non-compact tuples, we show that similar non-uniqueness constructions apply to $\mathcal{W}^{\text{min}}(K)$ if there are sufficiently many isolated extreme points of $K$. In addition, we consider containment problems for graded products of matrix convex sets, leading to a potentially stricter norm bound on the dilation of (non self-adjoint) contractions to commuting normal operators. This claim is paired with an independent, explicit dilation procedure that concretely improves previous bounds.