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arXiv · 1411.6636

On trace-convex noncommutative polynomials

Igor Klep·Scott A. McCullough·Christopher S. Nelson
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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.

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Igor Klep, Scott A. McCullough, Christopher S. Nelson. 2014-11-24. On trace-convex noncommutative polynomials. https://doi.org/10.1307/mmj%2F1457101814

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