Regular and positive noncommutative rational functions

TL;DR

This paper characterizes regular noncommutative rational functions through their linear system realizations and proves that positive regular functions can be expressed as sums of squares, extending classical results to the noncommutative setting.

Contribution

It provides a characterization of regular noncommutative rational functions via privileged linear pencils and solves a noncommutative Hilbert's 17th problem for positive regular functions.

Findings

Regular noncommutative rational functions are characterized by privileged linear pencils.

Positive regular functions can be expressed as sums of squares.

The paper extends classical positivity results to the noncommutative rational function setting.

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 $r=c^* L^{-1}b$. It is shown that $r$ is regular if and only if $L=A_0+\sum_jA_j x_j$ 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.