Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals

Abstract

Let N be the number operator in the space H of real-valued square-integrable Bernoulli functionals. In this paper, we further pursue properties of N from a probabilistic perspective. We first construct a nuclear space G, which is also a dense linear subspace of H, and then by taking its dual G*, we obtain a real Gel’fand triple G⊂H⊂G*. Using the well-known Minlos theorem, we prove that there exists a unique Gauss measure γN on G* such that its covariance operator coincides with N. We examine the properties of γN, and, among others, we show that γN can be represented as a convolution of a sequence of Borel probability measures on G*. Some other results are also obtained.