Bianalytic free maps between spectrahedra and spectraballs

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 result leads to a classification of automorphism groups of ball-like free spectrahedra. The proofs 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.

Reinhardt Free Spectrahedra

“…The results of [AHKM,HKMV20] characterize bianalytic maps between P A and P B under certain generic irreducibility hypotheses on the tuples A and B. They also characterize bianalytic maps between spectraballs absent any additional hypotheses.…”

“…The articles [AHKM,HKMV20] characterize bianalytic maps between free spectrahedra under hypotheses that are, in the sense of algebraic geometry, generic; and bianalytic maps between spectraballs. Free spectrahedra satisfying a Reinhardt condition represent arguably the simplest class of free spectrahedra not covered by these results.…”

Hyper-Reinhardt Free Spectrahedra

“…More generally, the results of [AHKM,HKMV20] characterize bianalytic maps between free spectrahedra P B and P C under certain generic irreducibility hypotheses on the tuples B and C. They also characterize bianalytic maps between spectraballs absent any additional hypotheses. A canonical class of free spectrahedra not covered by these results are those with circular symmetry.…”

Free potential functions

“…A free set Ω is said to be a free domain if each Ω [n] is open. We note that while our definition requires Ω to be closed under simultaneous conjugation by similarities, in certain settings it is desirable to assume the weaker condition of Ω being closed under simultaneous conjugation by unitaries, see [JKM + 19,HKMV19].…”