The convex Positivstellensatz in a free algebra

Given a monic linear pencil L in g variables, let PL=(PL(n))n∈N where PL(n):={X∈Sng∣L(X)⪰0}, and Sng is the set of g-tuples of symmetric n×n matrices. Because L is a monic linear pencil, each PL(n) is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic s...

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Bibliographic Details
Published inAdvances in mathematics (New York. 1965) Vol. 231; no. 1; pp. 516 - 534
Main Authors Helton, J. William, Klep, Igor, McCullough, Scott
Format Journal Article
LanguageEnglish
Published Elsevier Inc 10.09.2012
Subjects
Free convexity
Free positivity
Free real algebraic geometry
Linear matrix inequality
Moment problem
Positivstellensatz
secondary
Free positivity
Free convexity
Free real algebraic geometry
Positivstellensatz
Linear matrix inequality
Moment problem
primary
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Summary:Given a monic linear pencil L in g variables, let PL=(PL(n))n∈N where PL(n):={X∈Sng∣L(X)⪰0}, and Sng is the set of g-tuples of symmetric n×n matrices. Because L is a monic linear pencil, each PL(n) is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form PL. The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative matrix-valued polynomial p is positive semidefinite on PL if and only if it has a weighted sum of squares representation with optimal degree bounds: p=s∗s+∑jfinitefj∗Lfj, where s,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,fj and assuming only p nonnegative, as opposed to p strictly positive, yields a clean Positivstellensatz so seldom that such cases are noteworthy.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2012.04.028