## Non-commutative inequalities

#### Event Title

Regular and positive noncommutative rational functions

Crow 206

#### Start Date

7-22-2016 6:00 PM

#### End Date

22-7-2016 6:20 PM

#### Description

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 $r=c^* L^{-1}b$. Our main result states 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. 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.

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COinS

Jul 22nd, 6:00 PM Jul 22nd, 6:20 PM

Regular and positive noncommutative rational functions

Crow 206

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 $r=c^* L^{-1}b$. Our main result states 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. 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.