Convex entire noncommutative functions are polynomials of degree two or less

TL;DR

This paper extends the characterization of matrix convex functions in noncommutative variables, showing such functions are necessarily polynomials of degree two or less under broad conditions.

Contribution

It generalizes previous results by proving that matrix convexity and analyticity imply the function is a low-degree polynomial in noncommutative variables.

Findings

Matrix convex functions near zero are degree two or less polynomials.

Functions analytic in a neighborhood and matrix convex are degree two or less polynomials.

Generalization to functions of two classes of noncommuting variables.

Abstract

This paper concerns matrix "convex" functions of (free) noncommuting variables, $x = (x_1, \ldots, x_g)$. Helton and McCullough showed 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 "analytic" and matrix convex in $x$ on a "noncommutative open set" in $a$ is a polynomial of degree two or less.