A positivstellensatz for non-commutative polynomials
HTML articles powered by AMS MathViewer
- by J. William Helton and Scott A. McCullough PDF
- Trans. Amer. Math. Soc. 356 (2004), 3721-3737 Request permission
Abstract:
A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the commutative case. A broader issue is, to what extent does real semi-algebraic geometry extend to non-commutative polynomials? Our “strict" Positivstellensatz is positive news, on the opposite extreme from strict positivity would be a Real Nullstellensatz. We give an example which shows that there is no non-commutative Real Nullstellensatz along certain lines. However, we include a successful type of non-commutative Nullstellensatz proved by George Bergman.References
- Jim Agler, On the representation of certain holomorphic functions defined on a polydisc, Topics in operator theory: Ernst D. Hellinger memorial volume, Oper. Theory Adv. Appl., vol. 48, Birkhäuser, Basel, 1990, pp. 47–66. MR 1207393
- Agler, Jim and McCarthy John Featured talk by McCarthy at SEAM in Athens GA, 2001.
- Ball, Joseph; Malakorn, Tanit; and Groenwalde, Gilbert, Conservative Structured Realizations, preprint.
- Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998. Translated from the 1987 French original; Revised by the authors. MR 1659509, DOI 10.1007/978-3-662-03718-8
- Helton, J. William, “Positive” noncommutative polynomials are sums of squares, Annals of Math. vol. 56, no. 2, 2002, pp. 675-694.
- Scott McCullough, Factorization of operator-valued polynomials in several non-commuting variables, Linear Algebra Appl. 326 (2001), no. 1-3, 193–203. MR 1815959, DOI 10.1016/S0024-3795(00)00285-8
- McCullough, Scott and Putinar, Mihai, Non-commutative Sums of Squares, preprint.
- Jan Stochel and Franciszek Hugon Szafraniec, The complex moment problem and subnormality: a polar decomposition approach, J. Funct. Anal. 159 (1998), no. 2, 432–491. MR 1658092, DOI 10.1006/jfan.1998.3284
- Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
Additional Information
- J. William Helton
- Affiliation: Department of Mathematics, University of California, San Diego, California 92093
- MR Author ID: 84075
- Email: helton@osiris.ucsd.edu
- Scott A. McCullough
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105
- MR Author ID: 220198
- Email: sam@math.ufl.edu
- Received by editor(s): January 6, 2003
- Received by editor(s) in revised form: June 5, 2003
- Published electronically: March 23, 2004
- Additional Notes: The first author was partially supported by the the NSF, DARPA and Ford Motor Co.
The second author was partially supported by NSF grant DMS-0140112 - © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 3721-3737
- MSC (2000): Primary 47A13
- DOI: https://doi.org/10.1090/S0002-9947-04-03433-6
- MathSciNet review: 2055751