Positivity Certificates in Free Analysis
By Dr. Abhay Jindal, University of Ljubljana
Madhava Hall, 3rd Floor, Main Building IISER
Abstract
A nonnegative polynomial in several real variables need not be a sum of squares of polynomials. Artin showed in 1927 that every such polynomial can nevertheless be written as a sum of squares of rational functions. In free analysis, the situation is strikingly more rigid. Helton proved in 2002 that every scalar-valued noncommutative polynomial positive on all self-adjoint matrix tuples is a sum of squares of noncommutative polynomials, with the optimal degree bound.
In this talk, I will explain how this rigidity persists for noncommutative polynomials with operator coefficients, both globally and on matrix convex sets. Time permitting, I will conclude with an application to positive operator-valued polynomials on free products of finite abelian groups. The talk is based on joint work with Igor Klep and Scott McCullough.