KOPS - The Institutional Repository of the University of Konstanz
# Free analysis, convexity and LMI domains

Type of Publication: | Working Paper/Technical Report |

URI (citable link): | http://nbn-resolving.de/urn:nbn:de:bsz:352-152820 |

Author: | Helton, J. William; Klep, Igor; McCullough, Scott |

Year of publication: | 2011 |

Series: | Konstanzer Schriften in Mathematik ; 286 |

Summary: |
This paper concerns the geometry of noncommutative domains and analytic free
maps. These maps are free analogs of classical analytic functions in several complex variables, and are de ned in terms of noncommuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (noncommuting) variables and form a canonical class of mappings from one noncommutative domain D in say g variables to another noncommutative domain ~D in ~g variables. This article contains rigidity results paralleling those in the commutative world of several complex variables { particularly, in the case that the domains are circular and bounded. For instance, we show that proper free maps are one-to-one. Furthermore, between two freely biholomorphic bounded oncommutative domains there exists a linear biholomorphism. Because of its role in systems engineering, convexity is a major topic. Hence of particular interest is the case of domains de ned by a linear matrix inequality, or LMI domains. Our main theorem yields the following nonconvexi cation result: If a bounded circular noncommutative domain D is freely biholomorphic to a bounded circular LMI domain, then D is itself an LMI domain. |

Subject (DDC): | 510 Mathematics |

Keywords: | noncommutative set and function, analytic map, proper map, rigidity, linear matrix inequality, several complex variables, free analysis, free real algebraic geometry |

Link to License: | Terms of use |

Checksum:
MD5:0123ee6b3b46e51d33875c72d87c0e41

HELTON, J. William, Igor KLEP, Scott MCCULLOUGH, 2011. Free analysis, convexity and LMI domains

@techreport{Helton2011analy-15282, series={Konstanzer Schriften in Mathematik}, title={Free analysis, convexity and LMI domains}, year={2011}, number={286}, author={Helton, J. William and Klep, Igor and McCullough, Scott} }

286 Helton.pdf | 156 |