Publikation: The convex Positivstellensatz in a free algebra
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2011
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Zusammenfassung
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
DL := ⋃_(n∊N){X ∈ (Sℝnxn)g | L(X)≥0}.
Then a noncommutative polynomial p is positive semide nite on DL if and only if it has a
weighted sum of squares representation with optimal degree bounds. Namely,
p = ss + ∑_j^finite ƒjLƒj
where s; ƒj are vectors of noncommutative polynomials of degree no greater than deg(p)/2 .
This result contrasts sharply with the commutative setting, where the degrees of s; ƒj 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 de nite matrix Mk of "degree k" has, for each m ≥ 0, a positive semide nite Hankel extension ~Mk+m of degree
k + m and the same rank as Mk
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510 Mathematik
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Positivstellensatz, at extension, moment problem, rank preserving, noncommutative algebra, free positivity.
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HELTON, J. William, Igor KLEP, Scott MCCULLOUGH, 2011. The convex Positivstellensatz in a free algebraBibTex
@techreport{Helton2011conve-15285,
year={2011},
series={Konstanzer Schriften in Mathematik},
title={The convex Positivstellensatz in a free algebra},
number={288},
author={Helton, J. William and Klep, Igor and McCullough, Scott}
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<dcterms:abstract xml:lang="eng">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></dcterms:abstract>
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