Noncommutative plurisubharmonic polynomials part I: Global assumptions

We consider symmetric polynomials, p, in the noncommutative (nc) free variables {x1, x2, . . . , xg}. We define the nc complex hessian of p as the second directional derivative (replacing xT by y) q x,xT h,hT := ∂2p ∂s∂t (x + th,y +sk) t,s=0 y=xT , k=hT . We call an nc symmetric polynomial nc plurisubharmonic (nc plush) if it has an nc complex hessian that is positive semidefinite when evaluated on all tuples of n×n matrices for every size n; i.e., q X,XT H,HT 0 for all X,H ∈ (Rn×n)g for every n 1. In this paper, we classify all symmetric nc plush polynomials as convex polynomials with an nc analytic change of variables; i.e., an nc symmetric polynomial p is nc plush if and only if it has the form p = f T j fj + kj kT j + F +FTwhere the sums are finite and fj , kj , F are all nc analytic. In this paper, we also present a theory of noncommutative integration for nc polynomials and we prove a noncommutative version of the Frobenius theorem. A subsequent paper (J.M. Greene, preprint [6]), proves that if the nc complex hessian, q, of p takes positive semidefinite values on an “nc open set” then q takes positive semidefinite values on all tuples X, H. Thus, p has the form in Eq. (0.1). The proof, in J.M. Greene (preprint) [6], draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions. © 2011 Elsevier Inc. All rights reserved.