Non-Commutative Partial Matrix Convexity

Abstract

Let p be a polynomial in the non-commuting variables (a,x)=(a1,...,aga,x1,...,xgx). If p is convex in the variables x, then p has degree two in x and moreover, p has the form p = L + T , where L has degree at most one in x and is a (column) vector which is linear in x, so that T 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.