Bianalytic free maps between spectrahedra and spectraballs

J. William Helton, Igor Klep, Scott McCullough, Jurij Volčič

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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 completely positive maps and operator algebras, operator systems and spaces, and serve as the paradigm for matrix convex sets. The matricial feasibility set of an LMI is called a free spectrahedron. In this article, the bianalytic maps between a very general class of ball-like free spectrahedra (examples of which include row or column contractions, and tuples of contractions) and arbitrary free spectrahedra are characterized and seen to have an elegant algebraic form. They are all highly structured rational maps. In the case that both the domain and codomain are ball-like, these bianalytic maps are explicitly determined and the article gives necessary and sufficient conditions for the existence of such a map with a specified value and derivative at a point. In particular, this result leads to a classification of automorphism groups of ball-like free spectrahedra. The proofs depend on a novel free Nullstellensatz, established only after new tools in free analysis are developed and applied to obtain fine detail, geometric in nature locally and algebraic in nature globally, about the boundary of ball-like free spectrahedra.

Original languageEnglish
Article number108472
JournalJournal of Functional Analysis
Issue number11
StatePublished - 15 Jun 2020
Externally publishedYes


  • Bianalytic map
  • Birational map
  • Linear matrix inequality (LMI)
  • Spectrahedron

ASJC Scopus subject areas

  • Analysis


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