Noncommutative plurisubharmonic polynomials part I: Global assumptions

We consider symmetric polynomials, p  , in the noncommutative (nc) free variables {x1,x2,…,xg}{x1,x2,…,xg}. We define the nc complex hessian of p   as the second directional derivative (replacing xTxT 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×nn×n matrices for every size n; i.e.,q(X,XT)[H,HT]≽0q(X,XT)[H,HT]≽0 for all X,H∈g(Rn×n)X,H∈(Rn×n)g for every n⩾1n⩾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 formequation(0.1)p=∑fjTfj+∑kjkjT+F+FT where the sums are finite and fjfj, kjkj, 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.