A Nichtnegativstellensatz for polynomials in noncommuting variables

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2007
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Israel Journal of Mathematics. 2007, 161(1), pp. 17-27. ISSN 0021-2172. Available under: doi: 10.1007/s11856-007-0070-2
Zusammenfassung

Helton recently proved that a symmetric polynomial in noncommuting variables is positive semidefinite on all bounded self-adjoint Hilbert space operators if and only if it is a sum of hermitian squares. We characterize the polynomials which are nowhere negative semidefinite on certain `bounded basic closed semialgebraic setsĀ“ of bounded Hilbert space operators. The obtained representation for these polynomials involves multipliers analogous to the representation known from the classical commutative Positivstellensatz. It is still an open problem if a noncommutative version of Hilbert's 17th problem holds.

Zusammenfassung in einer weiteren Sprache
Fachgebiet (DDC)
510 Mathematik
Schlagwƶrter
noncommutative polynomials, Nichtnegativstellensatz, sums of squares, semialgebraic sets, contractive operators.
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Zitieren
ISO 690KLEP, Igor, Markus SCHWEIGHOFER, 2007. A Nichtnegativstellensatz for polynomials in noncommuting variables. In: Israel Journal of Mathematics. 2007, 161(1), pp. 17-27. ISSN 0021-2172. Available under: doi: 10.1007/s11856-007-0070-2
BibTex
@article{Klep2007Nicht-15642,
  year={2007},
  doi={10.1007/s11856-007-0070-2},
  title={A Nichtnegativstellensatz for polynomials in noncommuting variables},
  number={1},
  volume={161},
  issn={0021-2172},
  journal={Israel Journal of Mathematics},
  pages={17--27},
  author={Klep, Igor and Schweighofer, Markus}
}
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