MOMENTS AND COMMUTATORS OF PROBABILITY MEASURES

Abstract

Let a0, a- and a+ be the preservation, annihilation, and creation operators of a probability measure μ on ℝd, respectively. The operators a0 and [a-, a+] are proven to uniquely determine the moments of μ. We discuss the question: "What conditions must two families of operators satisfy, in order to ensure the existence of a probability measure, having finite moments of any order, so that, its associated preservation operators and commutators between the annihilation and creation operators are the given families of operators?" For the case d = 1, a satisfactory answer to this question is obtained as a simple condition in terms of the Szegö–Jacobi parameters. For the multidimensional case, we give some necessary conditions for the answer to this question. We also give a table with the associated preservation and commutator between the annihilation and creation operators, for some of the classic probability measures on ℝ.