Positivity of polynomials in matrix and operator variables
Abstract: Hilbert’s 17th problem asked whether every positive polynomial can be written as a quotient of sums of squares of polynomials. As many others on Hilbert’s famous list, this problem and its affirmative resolution by Artin started a thriving mathematical discipline, known as real algebraic geometry. At its core, it studies the interplay between polynomial inequalities and positivity (geometry) and sums of squares certifying such positivity (algebra). Apart from its pure mathematics appeal, this theory is the pillar of polynomial optimization, since sums of squares can be efficiently traced via semidefinite programming.
This talk reviews old and new results on positivity of noncommutative polynomials and their traces, in terms of their evaluations on matrices or operators. There are three natural setups to consider: positivity in matrix variables of a fixed size, positivity in matrix variables of arbitrary size, and positivity in operator variables acting on a separable Hilbert space. This talk compares the sums-of-squares certificates of positivity across these three setups, their shortcomings and open ends.