Operator Calculus on Vector-Valued White Noise Functionals

Abstract

A general theory of operators on vector-valued white noise functionals is established in line with Hida′s white noise calculus. A basic role is played by an integral kernel operator which is a superposition of creation and annihilation operators with operator-valued distributions as integral kernel. The symbol of a continuous operator on vector-valued white noise functionals is characterized by its analyticity and boundedness. A significant consequence is that every continuous operator on vector-valued white noise functionals admits an infinite series expansion in terms of integral kernel operators with precise estimate of the convergence.