Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals

Abstract

Let N be the number operator in the space (Formula presented.) 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 (Formula presented.), which is also a dense linear subspace of (Formula presented.), and then by taking its dual (Formula presented.), we obtain a real Gel’fand triple (Formula presented.). Using the well-known Minlos theorem, we prove that there exists a unique Gauss measure (Formula presented.) on (Formula presented.) such that its covariance operator coincides with N. We examine the properties of (Formula presented.), and, among others, we show that (Formula presented.) can be represented as a convolution of a sequence of Borel probability measures on (Formula presented.). Some other results are also obtained.