Unitarity of Generalized Fourier–Gauss Transforms

Abstract

A generalized Fourier–Gauss transform is an operator acting in a Boson Fock space and is formulated as a continuous linear operator acting on the space of test white noise functions. It does not admit, in general, a unitary extension with respect to the norm of the Boson Fock space induced from the Gaussian measure with variance 1 but is extended to a unitary isomorphism if the Gaussian measure is replaced with the ones with different covariance operators. As an application, unitarity of a generalized dilation is discussed.

Keywords:

• Boson Fock space
• Fourier–Gauss transform
• Generalized dilation
• Generalized Fourier–Gauss transform
• Kuo's Fourier transform
• Unitarity
• White noise theory

Mathematics Subject Classification (2000):

• Primary 60H40
• Secondary 46F25

ACKNOWLEDGMENTS

This study was supported by Grant No. R05-2002-000-00142-0 from the Basic Research Program of the Korea Science & Engineering Foundation, by Grant-in-Aid for Scientific Research No. 15340039 of JSPS and by the program “R & D Support Scheme for Funding Selected IT Proposals” of the Ministry of Public Management, Home Affairs, Posts and Telecommunications.