Convex entire noncommutative functions are polynomials of degree two or less

الأصل · EN

This paper concerns matrix "convex" functions of (free) noncommuting variables, x = (x₁,, xg). 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 = (a₁,, ag) and x = (x₁,, xg) that is "analytic" and matrix convex in x on a "noncommutative open set" in a is a polynomial of degree two or less.