Real Algebraic and Analytic Geometry |

A Nichtnegativstellensatz for polynomials in noncommuting variables.

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Submission: 2006, April 12.

*Abstract:
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
setsī 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.*

Mathematics Subject Classification (2000): 11E25, 13J30, 47L07.

Keywords and Phrases: noncommutative polynomials, Nichtnegativstellensatz, sums of squares, semialgebraic sets, contractive operators.

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