Convex Entire Noncommutative Functions are Polynomials of Degree Two or Less

Abstract

This paper concerns matrix “convex” functions of (free) noncommuting variables, $${x = (x_1, \ldots, x_g)}$$ . It was shown in Helton and McCullough (SIAM J Matrix Anal Appl 25(4):1124–1139, 2004) that a polynomial in $${x}$$ which is matrix convex is of degree two or less. We prove a more general result: that a function of $${x}$$ that is matrix convex near $${0}$$ and also that is “analytic” in some neighborhood of the set of all self-adjoint matrix tuples is in fact a polynomial of degree two or less. More generally, we prove that a function $${F}$$ in two classes of noncommuting variables, $${a = (a_1, \ldots, a_{\tilde{g}})}$$ and $${x = (x_1, \ldots, x_g)}$$ that is both“analytic” and matrix convex in $${x}$$ on a “noncommutative open set” in $${a}$$ is a polynomial of degree two or less.