Bianalytic free maps between spectrahedra and spectraballs

TL;DR

This paper characterizes bianalytic maps between general ball-like free spectrahedra and arbitrary free spectrahedra, revealing their algebraic structure and classifying automorphisms using new free Nullstellensatz tools.

Contribution

It provides a complete algebraic characterization of bianalytic maps between free spectrahedra and introduces a novel free Nullstellensatz for boundary analysis.

Findings

Bianalytic maps are highly structured rational maps.

Explicit classification of automorphisms of ball-like free spectrahedra.

Development of a new free Nullstellensatz for boundary analysis.

Abstract

Linear matrix inequalities (LMIs) are ubiquitous in real algebraic geometry, semidefinite programming, control theory and signal processing. LMIs with (dimension free) matrix unknowns are central to the theories of completely positive maps and operator algebras, operator systems and spaces, and serve as the paradigm for matrix convex sets. The matricial feasibility set of an LMI is called a free spectrahedron. In this article, the bianalytic maps between a very general class of ball-like free spectrahedra (examples of which include row or column contractions, and tuples of contractions) and arbitrary free spectrahedra are characterized and seen to have an elegant algebraic form. They are all highly structured rational maps. In the case that both the domain and codomain are ball-like, these bianalytic maps are explicitly determined and the article gives necessary and sufficient conditions for the existence of such a map with a specified value and derivative at a point. In particular, this leads to a classification of automorphism groups of ball-like free spectrahedra. The results depend on a novel free Nullstellensatz, established only after new tools in free analysis are developed and applied to obtain fine detail, geometric in nature locally and algebraic in nature globally, about the boundary of ball-like free spectrahedra.