Real Algebraic and Analytic Geometry
Submission: 2006, April 12.
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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