Non-commutative Partial Matrix Convexity

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 + ΛΛ, where L has degree at most one in x and Λ is a (column) vector which is linear in x, so that ΛΛ 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.