Free analysis, convexity and LMI domains

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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} }

2011-09-05T08:03:18Z Free analysis, convexity and LMI domains 2011-09-05T08:03:18Z Klep, Igor 2011 eng Helton, J. William terms-of-use McCullough, Scott Helton, J. William Klep, Igor McCullough, Scott This paper concerns the geometry of noncommutative domains and analytic free<br />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)<br />variables and form a canonical class of mappings from one noncommutative domain<br />D in say g variables to another noncommutative domain ~D in ~g variables.<br /><br />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.

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