A test for the existence of Gohberg-Krein representations in terms of multiparameter Wiener processes
\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.
necessary and sufficient conditions for the existence of
necessary and sufficient conditions for the existence of Gohberg-Krein representations
scientific article; zbMATH DE number 3921639