White Noise Calculus and Fock Space
White Noise Calculus is a distribution theory on Gaussian space, proposed by T. Hida in 1975. This approach enables us to use pointwise defined creation and annihilation operators as well as the well-established theory of nuclear space.This self-contained
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1577
Nobuaki Obata
White Noise Calculus and Fock Space
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Nobuaki Obata Department of Mathematics School of Science Nagoya University Nagoya, 464-01, Japan
Mathematics Subject Classification (199 I): 46F25, 46E50, 47A70, 47B38,47D30, 47D40, 60H99, 60J65
ISBN 3-540-57985-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57985-0 Springer-Verlag New York Berlin Heidelberg CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany
SPIN: 10130019
46/3140-5432] 0 - Printed on acid-free paper
Contents Introduction 1 Prerequisites
1.1 1.2 1.3 1.4 1.5 1.6
Locally convex spaces in general . Countably Hilbert spaces . . . . . Nuclear spaces and kernel theorem Standard CH-spaces of functions. Bochner-Minlos theorem .. Further notational remarks. Bibliographical notes
2 White Noise Space
2.1 Gaussian measure . 2.2 Wick-ordered polynomials . . . . . . . . . . . 2.3 Wiener-Ito-Segal isomorphism and Fock space Bibliographical notes . 3 White Noise Functionals
3.1 3.2 3.3 3.4 3.5 3.6
Standard Construction Continuous version theorem S-transform . Contraction of tensor products. Wiener product . . . . . . Characterization theorems Bibliographical notes
4 Operator Theory
4.1 4.2 4.3 4.4 4.5 4.6
Hida's differential operator. Translation operators . . Integral kernel operators Symbols of operators Fock expansion . . . Some examples ... Bibliographical notes
vii 1 1
3 7 11
16 17 18
19 19
23
28 32 33 33 38 48 53 58 65 69 71 71
76 79 88
98 .100 .107
CONTENTS
VI
5 Toward Harmonic Analysis 5.1 First order differential operators . 5.2 Regular one-parameter transformation group 5.3 Infinite dimensional Laplacians .. 5.4 Infinite dimensional rotation group 5.5 Rotation-invariant operators . . . . 5.6 Fourier transform . . . . . . . . . . 5.7 Intertwining property of Fourier transform Bibliographical notes . . . . . . . . . . . . 6 Addendum
6.1 Integral-sum kernel operators 6.2 Reduction to finite degree of freedom 6.3 Vector-valued white noise functionals Appendices
A B C
Polarization formula . . . . . . Hermite polynomials . . . . . . Norm estimates of contractions
109 · 109 · 118
· 121 · 126 · 132 · 140 · 145 .149
151 · 151 · 154
· 159 167 · 167 · 168 .169
References
171
Index
181
Introduction The white noise calculus (or analysis) was launched out by Hida [lJ in 1975 with his lecture notes on
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