The convex Positivstellensatz in a free algebra


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HELTON, J. William, Igor KLEP, Scott MCCULLOUGH, 2011. The convex Positivstellensatz in a free algebra

@techreport{Helton2011conve-15285, series={Konstanzer Schriften in Mathematik}, title={The convex Positivstellensatz in a free algebra}, year={2011}, number={288}, author={Helton, J. William and Klep, Igor and McCullough, Scott} }

eng Helton, J. William The convex Positivstellensatz in a free algebra 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 DL be its positivity domain<br /><br />D<sub>L</sub> := ⋃_(n∊N){X ∈ (Sℝ<sup>nxn</sup>)<sup>g</sup> | L(X)≥0}.<br /><br /><br />Then a noncommutative polynomial p is positive semide nite on DL if and only if it has a<br />weighted sum of squares representation with optimal degree bounds. Namely,<br /><br />p = s*s + ∑_j^finite ƒ*<sub>j</sub>Lƒ<sub>j</sub><br /><br /><br />where s; ƒ<sub>j</sub> are vectors of noncommutative polynomials of degree no greater than deg(p)/2 .<br />This result contrasts sharply with the commutative setting, where the degrees of s; ƒ<sub>j</sub> are vastly greater than deg(p) and assuming only p nonnegative yields a clean Positivstellensatz so seldom that the cases are noteworthy.<br /><br />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 de nite matrix M<sub>k</sub> of "degree k" has, for each m ≥ 0, a positive semide nite Hankel extension ~M<sub>k+m</sub> of degree<br />k + m and the same rank as M<sub>k</sub> Klep, Igor terms-of-use 2011-09-05T10:50:35Z Klep, Igor McCullough, Scott 2011 McCullough, Scott 2011-09-05T10:50:35Z Helton, J. William

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