Free analysis, convexity and LMI domains

TL;DR

This paper explores free analytic maps on noncommutative domains, establishing rigidity results and conditions under which noncommutative domains are convex or LMI domains, with implications for systems engineering.

Contribution

It introduces a free version of a classical theorem, showing biholomorphic equivalences imply linear maps and convexity properties in noncommutative domains.

Findings

Proper free analytic maps are one-to-one and biholomorphic when domains are equal in variables.

Between biholomorphic bounded circular domains, a linear biholomorphism exists.

Bounded circular noncommutative domains biholomorphic to LMI domains are themselves LMI domains.

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 D' in 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 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 (LMI domains) 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.