Positive elements in the algebra of the quantum moment problem (Q1814299)

Let \(W\) be the Weyl algebra, i. e. the algebra with unit \textbf{1}, generators \(x\), \(p\) and relation \(xp-px=i\mathbf{1}\), where \(i=\sqrt{- 1}\). Let \(\tilde W\) be the algebra over the same commutation relation, but with generators \(x\), \(p\) and \((ax+i\mathbf{1})^{-1}\) for \(a\in R\). The algebra \(\tilde W\) is called the extended Weyl algebra. Both algebras, \(W\) and \(\tilde W\), may be realized on \(L^ 2(R)\) in the Schrödinger representation \(\sigma\). Let \({\mathcal U}_ 0=\sigma(W)\), and \({\mathcal U}=\sigma(\tilde W)\). It is known that there exists an operator \(T\in{\mathcal U}_ 0\) such that \(\langle f,Tf\rangle\geq0\) for all \(f\in{\mathcal S}\) (i.e. that \(T\) has positive spectrum), but \(T\) can not be represented as finite sum of operators of the form \(H^*H\) for \(H\in{\mathcal U}_ 0\) where \({\mathcal S}\) denotes the Schwartz space of functions on \(R\) and \(\langle f,g\rangle=\int_{-\infty}^ \infty \overline{f}(x)g(x)dx\). The authors prove that if \(A\in{\mathcal U}\) and \(A\) has positive spectrum then \(A\) is a finite sum of quadratic terms \(H^*H\) for a finite subset of elements \(H\) in \({\mathcal U}\). The proof of this result is based on a certain regularity property for the representations which are generated by states on \({\mathcal U}\), and this property is not in general shared by representations generated by states defined only on the subalgebra \({\mathcal U}_ 0\).