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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 completelypositive maps and operator algebras, operator systems and spaces, and serve asthe paradigm for matrix convex sets. The matricial feasibility set of an LMI iscalled a free spectrahedron.  In this article, the bianalytic maps between a very general class ofball-like free spectrahedra (examples of which include row or columncontractions, and tuples of contractions) and arbitrary free spectrahedra arecharacterized and seen to have an elegant algebraic form. They are all highlystructured rational maps. In the case that both the domain and codomain areball-like, these bianalytic maps are explicitly determined and the articlegives necessary and sufficient conditions for the existence of such a map witha specified value and derivative at a point. In particular, this leads to aclassification of automorphism groups of ball-like free spectrahedra. Theresults depend on a novel free Nullstellensatz, established only after newtools in free analysis are developed and applied to obtain fine detail,geometric in nature locally and algebraic in nature globally, about theboundary of ball-like free spectrahedra.

Bianalytic free maps between spectrahedra and spectraballs