UNITARITY OF KUO'S FOURIER–MEHLER TRANSFORM

Abstract

Kuo's Fourier–Mehler transforms {ℱΘ} form a one-parameter automorphism group of the space of white noise distributions [Formula: see text]. The adjoint transforms [Formula: see text] form a one-parameter automorphism group of the space of test white noise functions [Formula: see text] but do not admit unitary extensions with respect to the L2-norm induced from the Gaussian measure with variance 1. We prove that [Formula: see text] admits an extension to a one-parameter group of unitary operators on the L2-space with respect to the Gaussian measure with variance 1/2.