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 selfadjoint 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 skewadjoint. 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2016
 arXiv:
 arXiv:1605.03188
 Bibcode:
 2016arXiv160503188K
 Keywords:

 Mathematics  Rings and Algebras
 EPrint:
 J. Lond. Math. Soc. 95 (2017) 613632