A Strict Positivstellensatz for the Weyl Algebra

Abstract

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 $b_1,...,b_d$ in $W(d)$ such that $b_{1}c{b}_1^{\ast }+...+b_{d}c{b}_d^{\ast }$ is a finite sum of squares.

17 pages, condition (i) in Theorem 1.1 implies (ii). Assumption (ii) can be omitted