Real Algebraic and Analytic Geometry
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Submission: 2011, March 24.
The main result of this paper establishes the perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set: suppose L is a monic linear pencil in g variables and let D_L be its positivity domain, i.e., the set of all g-tuples X of symmetric matrices of all sizes making L(X) positive semidefinite. Then a noncommutative polynomial p is positive semidefinite on D_L if and only if it has a weighted sum of squares representation with optimal degree bounds. Namely, p = s^* s + \sum_j f_j^* L f_j, where s, f_j are vectors of noncommutative polynomials of degree no greater than 1/2 deg(p). This result contrasts sharply with the commutative setting, where the degrees of s, f_j are vastly greater than deg(p) and assuming only p nonnegative yields a clean Positivstellensatz so seldom that the cases are noteworthy.
The main ingredient of the proof is an analysis of rank preserving extensions of truncated noncommutative Hankel matrices. It is proved that any such positive definite matrix M of "degree k" has, for each m>0, a positive semidefinite Hankel extension M_m of degree k+m and the same rank as M. .
Mathematics Subject Classification (2010): 47A57, 14P10, 47B35, 13J30, 46N10.
Keywords and Phrases: Positivstellensatz, Hankel matrix, flat extension, moment problem, rank preserving, noncommutative algebra, free positivity.
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