On trace-convex noncommutative polynomials (Q274135)

To each continuous function \(f: \mathbb{R}\to\mathbb{R}\) there is an associated trace function on \(n\times n\) real symmetric matrices \(\mathrm{Tr} f\). The classical Klein lemma states that \(f\) is convex if and only if \(\mathrm{Tr} f\) is convex. The authors present an algebraic strengthening of this lemma for univariate polynomials \(f\). Nameley, \(\mathrm{Tr} f\) is convex if and only if the noncommutative second directional derivative of \(f\) is a sum of Hermitian squares and commutators in a free algebra. They also give a localized version of this result.