Regular and positive noncommutative rational functions

Abstract

Call a noncommutative rational function $r$ regular if it has no singularities, i.e., $r(X)$ is defined for all tuples of self-adjoint matrices $X$. In this article regular noncommutative rational functions $r$ are characterized via the properties of their (minimal size) linear systems realizations $r=c^{\ast }{L}^{-1}b$. It is shown that $r$ is regular if and only if $L=A_0+{\sum }_j{A}_j{x}_j$ is privileged. Roughly speaking, a linear pencil $L$ is privileged if, after a finite sequence of basis changes and restrictions, the real part of $A_0$ is positive definite and the other $A_j$ are skew-adjoint. The second main result is a solution to a noncommutative version of Hilbert's 17th problem: a positive regular noncommutative rational function is a sum of squares.

Cited by in corpus (5)

• The free Grothendieck theorem
• Noncommutative polynomials describing convex sets
• Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem
• Noncommutative rational functions invariant under the action of a finite solvable group
• Factorization of noncommutative polynomials and Nullstellensätze for the free algebra