# Free Bianalytic Maps between Spectrahedra and Spectraballs in a Generic Setting

• Meric Augat
• J. William Helton
• Igor Klep
• Scott McCullough
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 272)

## Abstract

Given a tuple E = (E1, . . . , Eg) of d × d matrices, the collection $$\mathcal{B}_{E}$$ of those tuples of matrices X = (X1, . . . , Xg) (of the same size) such that $$\|\sum {E}_{j} \otimes {X}_{j}\| \leq {1}$$ is a spectraball. Likewise, given a tuple B = (B1, . . . , Bg) of e × e matrices the collection $$\mathcal{D}_{B}$$ of tuples of matrices X = (X1, . . . , Xg) (of the same size) such that $${I} + \sum {B}_{j} \otimes {X}_{j} + {\sum} {B}_{j}^{\ast} \otimes {X}_{j}^{\ast} \succeq {0}$$ is a free spectrahedron. Assuming E and B are irreducible, plus an additional mild hypothesis, there is a free bianalytic map $${p} : \mathcal{B}_{E} \rightarrow \mathcal{D}_{B}$$ normalized by p(0) = 0 and p’(0) = I if and only if $$\mathcal{B}_{E} = \mathcal{B}_{B}$$ and B spans an algebra. Moreover p is unique, rational and has an elegant algebraic representation.

## Keywords

Bianalytic map birational map linear matrix inequality (LMI) spectrahedron convex set Positivstellensatz free analysis real algebraic geometry

## Mathematics Subject Classification (2010)

47L25 32H02 13J30 (Primary) 14P10 52A05 46L07 (Secondary)

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## Authors and Affiliations

• Meric Augat
• 1
Email author
• J. William Helton
• 2
• Igor Klep
• 3
• Scott McCullough
• 1
1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA
2. 2.Department of MathematicsUniversity of CaliforniaSan DiegoUSA
3. 3.Department of MathematicsThe University of AucklandAucklandNew Zealand