Noncommutative Plurisubharmonic Polynomials Part II: Local Assumptions

TL;DR

This paper proves that noncommutative plurisubharmonic polynomials defined locally are actually globally of a specific form, extending previous results that required global assumptions.

Contribution

It shows that local noncommutative plurisubharmonicity implies a global polynomial structure, generalizing prior global-only results.

Findings

Local noncommutative plurisubharmonicity implies global form

Polynomials have a sum-of-squares structure with analytic components

Uses Gram matrix representation to establish the main result

Abstract

We say that a symmetric noncommutative polynomial in the noncommutative free variables (x_1, x_2, ..., x_g) is noncommutative plurisubharmonic on a noncommutative open set if it has a noncommutative 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 a noncommutative polynomial is noncommutative plurisubharmonic on a noncommutative open set, then the polynomial is actually noncommutative plurisubharmonic everywhere and has the form p = \sum f_j^T f_j + \sum k_j k_j^T + F + F^T where the sums are finite and f_j, k_j, F are all noncommutative analytic. In the paper by Greene, Helton, and Vinnikov, it is shown that if p is noncommutative plurisubharmonic everywhere, then p has the form above. In other words, the paper by Greene, Helton, and Vinnikov makes a global assumption while the current paper makes a local assumption, but both reach the same conclusion. This paper uses a Gram-like matrix representation of noncommutative polynomials. A careful analysis of this Gram matrix plus the main theorem in the paper by Greene, Helton, and Vinnikov ultimately force the form in the equation above.