Noncommutative plurisubharmonic polynomials part II: Local assumptions

Abstract

We say that a symmetric noncommutative (nc) polynomial is nc plurisubharmonic (nc plush) on an nc open set if it has an nc complex hessian that is positive semidefinite when evaluated on open sets of matrix tuples of sufficiently large size. In this paper, we show that if an nc polynomial is nc plurisubharmonic on an nc open set then the polynomial is actually nc plurisubharmonic everywhere and has the form (0.1)p=∑fjTfj+∑kjkjT+F+FT where the sums are finite and fj, kj, F are all nc analytic.Greene et al. (2011) [1] has shown that if p is nc plurisubharmonic everywhere then p has the form in Eq. (0.1). In other words, [1] makes a global assumption while the current paper makes a local assumption, but both reach the same conclusion. We show that if p is nc plurisubharmonic on an nc “open set” (local) then p is, in fact, nc plurisubharmonic everywhere (global) and has the form expressed in Eq. (0.1).This paper requires a technique that is not used in [1]. We use a Gram-like vector and matrix representation (called the border vector and middle matrix) for homogeneous degree 2 nc polynomials. We then analyze this representation for the nc complex hessian on an nc open set and positive semidefiniteness forces a very rigid structure on the border vector and middle matrix. This rigid structure plus the theorems in [1] ultimately force the form in Eq. (0.1).