Free bianalytic maps between spectrahedra and spectraballs in a generic setting

TL;DR

This paper characterizes when a free bianalytic map exists between spectrahedra and spectraballs, showing it exists uniquely under certain irreducibility and algebraic spanning conditions, with an explicit algebraic form.

Contribution

It provides a complete characterization and explicit construction of free bianalytic maps between spectrahedra and spectraballs in a generic setting, under irreducibility and mild hypotheses.

Findings

Existence of a unique rational free bianalytic map under specified conditions.

Characterization of when spectraballs and spectrahedra are related via such maps.

Explicit algebraic form of the bianalytic map.

Abstract

Given a tuple $E=(E_1,\dots,E_g)$ of $d\times d$ matrices, the collection of those tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $\| \sum E_j\otimes X_j\|\le 1$ is called a spectraball $\mathcal B_E$. Likewise, given a tuple $B=(B_1,\dots,B_g)$ of $e\times e$ matrices the collection of tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $I + \sum B_j\otimes X_j +\sum B_j^* \otimes X_j^*\succeq 0$ is a free spectrahedron $\mathcal D_B$. Assuming $E$ and $B$ are irreducible, plus an additional mild hypothesis, there is a free bianalytic map $p:\mathcal B_E\to \mathcal D_B$ normalized by $p(0)=0$ and $p'(0)=I$ if and only if $\mathcal B_E=\mathcal B_B$ and $B$ spans an algebra. Moreover $p$ is unique, rational and has an elegant algebraic representation.