The moment operators of phase space observables and their number margins

We show that the moment operators of any of the phase space observables B(C) )̄Z ↦ A¦s〉(Z)≔ 1π∫Z/LL> D¦s〉〈s¦Dz∗ dλ(z) ¬E L(H) associated with the number states ¦s〉 are the powers of the lowering operator. We also determine all the moment operators of the number margins of these observables. They are integer valued polynomials of the number operator, the degree of the polynomial being the degree of the moment and the integer coefficients depending on ¦s〉. In the Appendix we develop the necessary tools for integrating unbounded functions with respect to operator measures.