Reconstruction of operators from their normally ordered moments for a single boson mode

New formulae are derived for the coefficients of normally ordered operator expansions for a single mode, which depend on the matrix elements in the Fock state representation. The closely connected problem of the reconstruction of operators F by the set of normally ordered moments Tr(a Dagger k a l f), (k, l=0, 1, 2, . . .), is solved. For this purpose, a new set of operators a k,l that is orthonormal to the set of normally ordered powers of boson operators a Dagger k a l is introduced and investigated. Furthermore, a symmetric and a Hermitian scalar product are defined in the linear space of linear operators for the boson mode. The duality obtained in the expansion and reconstruction formulae of the operators can be understood more deeply by considering not only the usual Hilbert space but also a rigged Hilbert space of linear operators for the boson mode.