White noise calculus and Fock space
type: Item
instance of: scholarly article
title: White noise calculus and Fock space
zbMATH Open document ID: 0814.60058
author: Nobuaki Obata
published in: Lecture Notes in Mathematics
publication date: 1994-06-26T00:00:00Z
review text: In the beginning, Hida had started his investigations on white noise calculus with strong interest in infinite-dimensional rotation groups and related functional analysis. The author stands on this view point and gives us a bird's eye view of white noise calculus. Let \(E \subset H \subset E'\) be a Gel'fand triplet. The group \(O(E;H) = \{g \in \text{GL} (E)\); \(| g \xi | = | \xi |\), \(\forall \xi \in E\}\) is called the infinite-dimensional rotational group of this triplet. An operator which commutes with this rotation group is called a Laplacian. There exist three well-known Laplacians, Gross Laplacian \(\Delta_ G\), its dual operator \(\Delta_ G^*\) and the number operator \(N\). Then, \(\{\Xi \in {\mathcal L} ((E), (E)^*)\); Laplacian\} is generated by \(\Delta_ G^*\), \(\Delta_ G\) and \(N\), and \(\{\Xi \in {\mathcal L} ((E), (E))\); Laplacian\} is generated by \(\Delta_ G\) and \(N\). Here \((E)\) is the space of testing functionals and \((E)^*\) is the space of Hida distributions. The infinite-dimensional Fourier transformation is introduced and some of its properties are shown.
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 46F25
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 47L60
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 47A70
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47L10
Mathematics Subject Classification ID: 46-02
Mathematics Subject Classification ID: 47-02
zbMATH DE Number: 595993
zbMATH Keywords: white noise calculus
zbMATH Keywords: infinite-dimensional Laplacians
zbMATH Keywords: harmonic analysis of infinitely many variables
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0073952
OpenAlex ID: W109843813
DOI: 10.1007/BFB0073952
description: scientific article; zbMATH DE number 595993