Matrix Representations for Positive Noncommutative Polynomials

Abstract

In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict positivity fails.Specifically, we treat a ``symmetric'' polynomial q(x, h) in noncommuting variables, {x 1, . . . , } and {h 1, . . . , } for which q(X,H) is positive semidefinite whenever are tuples of selfadjoint matrices with ||X j || ≤ 1 but H j unconstrained. The representation we obtain is a Gram representation in the variables h where P q is a symmetric matrix whose entries are noncommutative polynomials only in x and V is a ``vector'' whose entries are polynomials in both x and h. We show that one can choose P q such that the matrix P q (X) is positive semidefinite for all ||X j || ≤ 1. The representation covers sum of square results ([Am. Math. (to appear); Linear Algebra Appl. 326 (2001), 193–203; Non commutative Sums of Squares, preprint]) when g x = 0. Also it allows for arbitrary degree in h, rather than degree two, in the main result of [Matrix Inequalities: A Symbolic Procedure to Determine Convexity Automatically to appear IOET July 2003] when restricted to x-domains of the type ||X j || ≤ 1.