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Free Bianalytic Maps between Spectrahedra and Spectraballs in a Generic Setting

  • Meric AugatEmail author
  • 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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Copyright information

© Springer Nature Switzerland AG 2019

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

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