Commutative Jacobi fields in Fock space
type: Item
instance of: scholarly article
title: Commutative Jacobi fields in Fock space
zbMATH Open document ID: 0896.60019
DOI: 10.1007/BF01238217
author: Yurij M. Berezansky
published in: Integral Equations and Operator Theory
publication date: 1998-10-01T00:00:00Z
review text: Let \(H\) be a real separable Hilbert space with \(H=H_0\). By a rigging of Hilbert spaces, we have \(\Phi=\text{pr-}\lim_{r\in T}H_r\) and \(\Phi'= \text{ind-}\lim_{r\in T} H_{-r}\). Actually, \(\Phi'\) is the dual of \(\Phi\) and \(\Phi\subset H\subset \Phi'\) is a Gel'fand triple. A usual procedure to take the \(n\)-th symmetric tensor power of the above spaces with complexification gives the rigging of \(n\)-particle Fock spaces: \({\mathcal F}_n(\Phi) \subset {\mathcal F}_n(H) \subset {\mathcal F}_n (\Phi')\). A generalization of the white noise analysis was proposed by \textit{Yu. M. Berezanskij}, \textit{V. O. Livinskij} and \textit{E. V. Litvinov} [Ukr. Math. J. 46, No. 3, 183-203 (1994); translation from Ukr. Mat. Zh. 46, No. 3, 177-197 (1994; Zbl 0843.47012) and Methods Funct. Anal. Topol. 1, No. 1, 28-55 (1995; Zbl 0870.60037)], where a theory of distributions on an infinite-dimensional space is constructed in the case of a non-Gaussian measure. The spectral measure of some commuting family \(A=(\widetilde A(\varphi)\); \(\varphi \in\Phi)\) of selfadjoint operators \(\widetilde A(\varphi)\) plays an important role in this generalization. In fact, a Jacobi field is a family \((\widetilde A(\varphi)\); \(\varphi\in \Phi)\) of commuting selfadjoint operators \(\widetilde A(\varphi)\) acting in a Fock space \({\mathcal F} (H)\), having a Jacobi structure, and depending linearly on the test function \(\varphi\). While, the Fourier transform \(I\) connected with the family \(A\) maps the Fock space onto the space \(L^2\) on the spectrum of this family with respect to the measure \(\rho\). Moreover, the rigging of the Fock space is transferred to some rigging of the space \(L^2\), and the latter rigging generates the corresponding theory of generalized functions on this spectrum [cf. \textit{E. W. Litvinov}, ibid. 1, No. 1, 61-85 (1995; Zbl 0871.60034)]. Indeed, we have \[ (L^2)= L^2(\Phi',d\rho) =\bigoplus^\infty_{n=0} (L^2_n), \quad (L^2_n)= I\bigl({\mathcal F…
Mathematics Subject Classification ID: 60G20
Mathematics Subject Classification ID: 46F25
Mathematics Subject Classification ID: 46F20
Mathematics Subject Classification ID: 60J99
zbMATH DE Number: 1129330
zbMATH Keywords: Jacobi field
zbMATH Keywords: spectral representation
zbMATH Keywords: distribution on infinite-dimensional spaces
reviewed by: Isamu Dôku
MaRDI profile type: Publication
cites work: Asymptotic behavior of the Gelfand and the Bernstein widths of some classes of analytic functions
cites work: Infinite-dimensional analysis related to generalized translation operators
cites work: Q3126833
cites work: Q4202615
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description: scientific article; zbMATH DE number 1129330