Free bianalytic maps between spectrahedra and spectraballs in a generic setting

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 $\Vert \sum E_j\otimes X_j\Vert \le 1$ 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 $I+\sum B_j\otimes X_j+\sum B_j^{\ast }\otimes X_j^{\ast }⪰0$ 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 $p^{\prime }(0)=I$ 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.

19 pages