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Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field

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Assume that K +: H T is a bounded operator, where H and T are Hilbert spaces and ρ is a measure on the space H . Denote by ρK the image of the measure ρ under K +. We study the measure ρK under the assumption that ρ is the spectral measure of a Jacobi field and obtain a family of operators whose spectral measure is equal to ρK. We also obtain an analog of the Wiener-Itô decomposition for ρK. Finally, we illustrate the results obtained by explicit calculations carried out for the case, where ρK is a Lévy noise measure.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 6, pp. 744–763, June, 2007.

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Berezansky, Y.M., Pulemyotov, A.D. Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field. Ukr Math J 59, 811–832 (2007). https://doi.org/10.1007/s11253-007-0052-x

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  • DOI: https://doi.org/10.1007/s11253-007-0052-x

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