Matrix Representations for Positive Noncommutative Polynomials

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 {x1, . . . , xgx} and {h1, . . . , hgh} for which q(X,H) is positive semidefinite whenever X = (X1, . . . , Xgx) and H = (H1, . . . , Hgh) are tuples of selfadjoint matrices with ‖Xj‖ ≤ 1 but Hj unconstrained. The representation we obtain is a Gram representation in the variables h q(x, h) = V (x)[h] Pq(x)V (x)[h], where Pq 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 Pq such that the matrix Pq(X) is positive semidefinite for all ‖Xj‖ ≤ 1. The representation covers sum of square results ([H],[M],[MP]) when gx = 0. Also it allows for arbitrary degree in h rather than degree two in the main result of [CHSY] when it is restricted to x-domains of the type ‖Xj‖ ≤ 1. ∗Partially supported by NSF, DARPA and Ford Motor Co. †Partially supported by NSF grant DMS-0140112 ‡Partially supported by NSF grant DMS-0100367