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Free Analysis, Convexity and LMI Domains

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Mathematical Methods in Systems, Optimization, and Control

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 222))

Abstract

This paper concerns free analytic maps on noncommutative domains. These maps are free analogs of classical holomorphic functions in several complex variables, and are defined in terms of noncommuting variables amongst which there are no relations – they are free variables. Free analytic 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 \(\tilde{D}\) in \(\tilde{g}\) variables. Motivated by determining the possibilities for mapping a nonconvex noncommutative domain to a convex noncommutative domain, this article focuses on rigidity results for free analytic maps. Those obtained to date parallel and are often stronger than those in several complex variables. For instance, a proper free analytic map between noncommutative domains is one-one and, if \(\tilde{g} = g\), free biholomorphic. Making its debut here is a free version of a theorem of Braun-Kaup-Upmeier: between two freely biholomorphic bounded circular noncommutative domains there exists a linear biholomorphism. An immediate consequence is the following nonconvexification result: if two bounded circular noncommutative domains are freely biholomorphic, then they are either both convex or both not convex. Because of their roles in systems engineering, linear matrix inequalities (LMIs) and noncommutative domains defined by an LMI (LMIdom ains) are of particular interest. As a refinement of above the nonconvexification result, if a bounded circular noncommutative domain D is freely biholomorphic to a bounded circular LMI domain, then D is itself an LMI domain.

Mathematics Subject Classification. Primary: 46L52, 47A56, 32A05, 46G20; secondary: 47A63, 32A10, 14P10.

J. William Helton, Research supported by NSF grants DMS-0700758, DMS-0757212, and the Ford Motor Co.

Igor Klep, Supported by the Faculty Research Development Fund (FRDF) of The University of Auckland (project no. 3701119). Partially supported by the Slovenian Research Agency (project no. J1-3608 and program no. P1-0222). The article was written while the author was visiting the University of Konstanz.

Scott McCullough, Research supported by the NSF grant DMS-0758306.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, USA

    J. William Helton

  2. Department of Mathematics, The University of Auckland, Auckland, New Zealand

    Igor Klep

  3. Department of Mathematics, University of Florida, Gainesville, FL, 32611-8105, USA

    Scott McCullough

Authors
  1. J. William Helton
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  2. Igor Klep
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  3. Scott McCullough
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Correspondence to J. William Helton .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Harry Dym

  2. , Department of Mechanical and, University of California San Diego, Gilman Drive 9500, La Jolla, 92093-0411, California, USA

    Mauricio C. de Oliveira

  3. College of Letters & Science, Dept. Mathematics, University of California, Santa Barbara, Santa Barbara, 93106, California, USA

    Mihai Putinar

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Helton, J.W., Klep, I., McCullough, S. (2012). Free Analysis, Convexity and LMI Domains. In: Dym, H., de Oliveira, M., Putinar, M. (eds) Mathematical Methods in Systems, Optimization, and Control. Operator Theory: Advances and Applications, vol 222. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0411-0_15

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