Bianalytic maps between free spectrahedra

Linear matrix inequalities (LMIs) \(I_d + \sum _{j=1}^g A_jx_j + \sum _{j=1}^g A_j^*x_j^* \succeq 0\) play a role in many areas of applications. The set of solutions of an LMI is a spectrahedron. LMIs in (dimension-free) matrix variables model most problems in linear systems engineering, and their solution sets are called free spectrahedra. Free spectrahedra are exactly the free semialgebraic convex sets. This paper studies free analytic maps between free spectrahedra and, under certain (generically valid) irreducibility assumptions, classifies all those that are bianalytic. The foundation of such maps turns out to be a very small class of birational maps we call convexotonic. The convexotonic maps in g variables sit in correspondence with g-dimensional algebras. If two bounded free spectrahedra \({\mathcal {D}}_A\) and \({\mathcal {D}}_B\) meeting our irreducibility assumptions are free bianalytic with map denoted p, then p must (after possibly an affine linear transform) extend to a convexotonic map corresponding to a g-dimensional algebra spanned by \((U-I)A_1,\ldots ,(U-I)A_g\) for some unitary U. Furthermore, B and UA are unitarily equivalent. The article also establishes a Positivstellensatz for free analytic functions whose real part is positive semidefinite on a free spectrahedron and proves a representation for a free analytic map from \({\mathcal {D}}_A\) to \({\mathcal {D}}_B\) (not necessarily bianalytic). Another result shows that a function analytic on any radial expansion of a free spectrahedron is approximable by polynomials uniformly on the spectrahedron. These theorems are needed for classifying free bianalytic maps.