Journal of the Mathematical Society of Japan

A role of Bargmann-Segal spaces in characterization and expansion of operators on Fock space

Un Cig JI and Nobuaki OBATA

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A rigged Hilbert space formalism is introduced to study Fock space operators. The symbols of continuous operators on a rigged Fock space are characterized in terms of Bargmann-Segal spaces and complex Gaussian integrals. In particular, characterizations of bounded operators and of operators of Hilbert-Schmidt class on the middle Fock space are obtained. As an application we establish an operator version of chaotic expansion (Wiener-Itô expansion) and describe a relation to the Fock expansion in terms of the Wick exponential of the number operator. As another application we discuss regularity property of a solution to a normal-ordered white noise differential equation generalizing a quantum stochastic differential equation.

Article information

J. Math. Soc. Japan, Volume 56, Number 2 (2004), 311-338.

First available in Project Euclid: 3 October 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46G20: Infinite-dimensional holomorphy [See also 32-XX, 46E50, 46T25, 58B12, 58C10]
Secondary: 46F25: Distributions on infinite-dimensional spaces [See also 58C35] 60H40: White noise theory 81S25: Quantum stochastic calculus

Bargmann-Segal space Gaussian analysis Fock space integral kernel operator operator symbol chaotic expansion quantum stochastic differential equation white noise differential equation


Cig JI, Un; OBATA, Nobuaki. A role of Bargmann-Segal spaces in characterization and expansion of operators on Fock space. J. Math. Soc. Japan 56 (2004), no. 2, 311--338. doi:10.2969/jmsj/1191418632.

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