Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A positivstellensatz for non-commutative polynomials

Authors: J. William Helton and Scott A. McCullough
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 3721-3737
MSC (2000): Primary 47A13
Published electronically: March 23, 2004
MathSciNet review: 2055751
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [A] Agler, Jim, On the representation of certain holomorphic functions defined on a polydisc. Topics in operator theory: Ernst D. Hellinger memorial volume, 47-66, Oper. Theory Adv. Appl., 48, Birkhauser, Basel, 1990. MR 93m:47013
  • [AM] Agler, Jim and McCarthy John Featured talk by McCarthy at SEAM in Athens GA, 2001.
  • [BMprep] Ball, Joseph; Malakorn, Tanit; and Groenwalde, Gilbert, Conservative Structured Realizations, preprint.
  • [BCR] Bochnak, Jacek, Costi, Michel and Roy, Marie-Francoise, Real Algebraic Geometry, Springer, 1991, pp. 430. MR 2000a:14067
  • [H] Helton, J. William, ``Positive'' noncommutative polynomials are sums of squares, Annals of Math. vol. 56, no. 2, 2002, pp. 675-694.
  • [M] McCullough, Scott, Factorization of operator-valued polynomials in several non-commuting variables. Linear Algebra Appl. 326 (2001), no. 1-3, 193-203. MR 2002f:47035
  • [MP] McCullough, Scott and Putinar, Mihai, Non-commutative Sums of Squares, preprint.
  • [PV] Putinar, Mihai and Vasilescu, Florian-Horia, Solving moment problems by dimensional extension. Annals of Math. (2) 149 (1999), no. 3, 1087-1107. MR 2001c:47023b
  • [R] Rudin, Walter, Functional analysis, second edition, International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. xviii+424 pp. MR 92k:46001

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47A13

Retrieve articles in all journals with MSC (2000): 47A13

Additional Information

J. William Helton
Affiliation: Department of Mathematics, University of California, San Diego, California 92093

Scott A. McCullough
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105

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
Article copyright: © Copyright 2004 American Mathematical Society