Real Algebraic and Analytic Geometry Preprint Server

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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 sets´ 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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