A Strict Positivstellensatz for the Weyl Algebra

Let $c$ be an element of the Weyl algebra $W(d)$ which is given by a strictly positive operator in the Schr"odinger representation. It is shown that, under some conditions, there exist elements $b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_d$ in $W(d)$ such that $b_1 c b_1^* + … + b_d c b_d^*$ is a finite sum of squares.