The convex Positivstellensatz in a free algebra
Abstract
Given a monic linear pencil L in g variables let DL be its positivity domain, i.e., the set of all g-tuples X of symmetric matrices of all sizes making L(X) positive semidefinite. Because L is a monic linear pencil, DL is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form DL. The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative polynomial p is positive semidefinite on DL if and only if it has a weighted sum of squares representation with optimal degree bounds: p = s* s + Σj fj* L fj, where s, fj are vectors of noncommutative polynomials of degree no greater than 1/2 deg(p). 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.