On trace-convex noncommutative polynomials

TL;DR

This paper extends Klein's lemma to univariate polynomials, characterizing trace convexity through algebraic conditions involving noncommutative derivatives expressed as sums of hermitian squares and commutators.

Contribution

It provides an algebraic characterization of trace convexity for univariate polynomials using noncommutative derivatives and sums of hermitian squares, strengthening classical results.

Findings

Trace convexity of univariate polynomials is equivalent to noncommutative second derivatives being sums of hermitian squares.

The paper offers a localized version of the algebraic characterization.

The results deepen understanding of convexity in noncommutative polynomial settings.

Abstract

To each real continuous function f there is an associated trace function on real symmetric matrices Tr f. The classical Klein lemma states that f is convex if and only if Tr f is convex. In this note we present an algebraic strengthening of this lemma for univariate polynomials f: 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. We also give a localized version of this result.