Free Bianalytic Maps between Spectrahedra and Spectraballs in a Generic Setting

Abstract

Given a tuple E = (E 1 , . . . , E g ) of d×d matrices, the collection B E of those tuples of matrices X = (X 1 , . . . , X g ) (of the same size) such that E j ⊗ X j ≤ 1 is a spectraball. Likewise, given a tuple B = (B 1 , . . . , B g ) of e × e matrices the collection D B of tuples of matrices X = (X 1 , . . . , X g ) (of the same size) such thatj 0 is a free spectrahedron. Assuming E and B are irreducible, plus an additional mild hypothesis, there is a free bianalytic map p : B E → D B normalized by p(0) = 0 and p ′ (0) = I if and only if B E = B B and B spans an algebra. Moreover p is unique, rational and has an elegant algebraic representation.