The convex Positivstellensatz in a free algebra

Given a monic linear pencil LL in gg variables, let PL=(PL(n))n∈NPL=(PL(n))n∈N where PL(n):={X∈Sng∣L(X)⪰0}, and Sng is the set of gg-tuples of symmetric n×nn×n matrices. Because LL is a monic linear pencil, each PL(n)PL(n) is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form PLPL. The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative matrix-valued polynomial pp is positive semidefinite   on PLPL if and only if it has a weighted sum of squares representation with optimal degree bounds: p=s∗s+∑jfinitefj∗Lfj, where s,fjs,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,fjs,fj and assuming only pp nonnegative, as opposed to pp strictly positive, yields a clean Positivstellensatz so seldom that such cases are noteworthy.