Regular and positive noncommutative rational functions

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 talk regular noncommutative rational functions r will be characterized via the properties of their (minimal size) linear systems realizations r = c∗L−1b. Our main result states that r is regular if and only if L = A0 + ∑ j Ajxj is privileged. Roughly speaking, a linear pencil L is privileged if, after a finite sequence of basis changes and restrictions, the real part of A0 is positive definite and the other Aj are skewadjoint. Afterwards I will speak about a solution to a noncommutative version of Hilbert’s 17th problem: a positive regular noncommutative rational function is a sum of squares. The talk is based on the joint work with I. Klep and J. E. Pascoe.