The convex Positivstellensatz in a free algebra

Abstract

Given a monic linear pencil L in g variables, let PL=(PL(n))n∈N where PL(n):={X∈Sng∣L(X)⪰0}, and Sng is the set of g-tuples of symmetric n×n matrices. Because L is a monic linear pencil, each PL(n) is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form PL. The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative matrix-valued polynomial p is positive semidefinite on PL if and only if it has a weighted sum of squares representation with optimal degree bounds: p=s∗s+∑jfinitefj∗Lfj, where s,fj are matrices of noncommutative polynomials of degree no greater than deg(p)2. This noncommutative result contrasts sharply with the commutative setting, where there is no control on the degrees of s,fj and assuming only p nonnegative, as opposed to p strictly positive, yields a clean Positivstellensatz so seldom that such cases are noteworthy.