A test for the existence of Gohberg-Krein representations in terms of multiparameter Wiener processes
type: Item
instance of: scholarly article
title: A test for the existence of Gohberg-Krein representations in terms of multiparameter Wiener processes
zbMATH Open document ID: 0576.60044
DOI: 10.1016/0022-1236(85)90066-7
author: John Daughtry
author: Bruce Dearden
published in: Journal of Functional Analysis
publication date: 1985-01-01T00:00:00Z
review text: \textit{C. Bromley} and \textit{G. Kallianpur} [Appl. Math. Optimization 6, 361-376 (1980; Zbl 0443.60034)] extended to certain Gaussian random fields a formula for the calculation of likelihood ratios from a special type of nonanticipative representation, a ''Gohberg-Krein representation''. They also made a penetrating study of the existence of such representations. The paper being summarized uses the methods of Bromley and Kallianpur to obtain necessary and sufficient conditions for the existence of Gohberg-Krein representations in terms of multiparameter Wiener processes. As an application, it demonstrates the nonexistence of a Gohberg-Krein representation of \(W_{s,t}-st W_{1,1}\) in terms of the two-parameter Wiener process \(W_{s,t}\) with (s,t)\(\in [0,a]\times [0,b]\) for \(0<a<1\), \(0<b<1.\) From the point of view of operator theory, the starting point of the investigation is the theory of operator factorization along a chain of subspaces as developed by \textit{I. C. Gohberg} and \textit{M. G. Kreĭn} [Theory and applications of Volterra operators in Hilbert space. (Russian) (1967; Zbl 0168.120); English translation: Am. Math. Soc. (1970)]. In the special setting of the random field problem under investigation, a necessary and sufficient condition for a ''Gohberg-Krein factorization'' along one coordinate nest of subspaces to be a Gohberg- Krein factorization along the (commuting) nest determined by any other time parameter is given. The first-named author has generalized this result to a special class of nests in any Hilbert space in a paper to appear in Indiana Univ. Math. J.
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 47A68
zbMATH DE Number: 3921639
zbMATH Keywords: necessary and sufficient conditions for the existence of
zbMATH Keywords: Gohberg-Krein representations
zbMATH Keywords: multiparameter Wiener processes
zbMATH Keywords: ''Gohberg-Krein factorization''
zbMATH Keywords: necessary and sufficient conditions for the existence of Gohberg-Krein representations
MaRDI profile type: Publication
full work available at URL: 0022-1236(85)90066-7
OpenAlex ID: W2048665415
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description: scientific article; zbMATH DE number 3921639