IUMJ

Title: Non-commutative partial matrix convexity

Authors: Damon M. Hay, J. William Helton, Adrian Lim and Scott McCullough

Issue: Volume 57 (2008), Issue 6, 2815-2842

Abstract:

Let $p$ be a polynomial in the non-commuting variables $(a,x)=(a_1,\dots,a_{g},x_1,\dots,x_{g})$. If $p$ is convex in the variables $x$, then $p$ has degree two in $x$ and moreover, $p$ has the form $$p = L + \Lambda ^T \Lambda,$$ where $L$ has degree at most one in $x$ and $\Lambda$ is a (column) vector which is linear in $x$, so that $\Lambda^T\Lambda$ is a both sum of squares and homogeneous of degree two. Of course the converse is true also. Further results involving various convexity hypotheses on the $x$ and $a$ variables separately are presented.