Abstract
A symmetric non-commutative polynomial p when evalu- ated at a tuple of operators on a flnite dimensional, real Hilbert space H has a value which is a symmetric operator. We show that any such polynomial which takes positive semideflnite values on the variety Z of spherical isometries is represented as a sum of squares of polynomials plus a residual part vanishing on Z. Here by spherical isometries we mean tuples A =( A 1;A2;:::;An) of operators on H such that A T A1 + :::+A TAn =I: This observation improves prior theorems known only for strictly pos- itive polynomials. It is known that for commutative polynomials the result is false.
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