Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field - Berezansky, Yu. M.; Pulemyotov, A. D. (Ukrains’kyi Matematychnyi Zhurnal, 2007)
Assume that K^+: H_- → T_- is a bounded operator, where H_— and T_— are Hilbert spaces and p is a measure on the space H_—. Denote by ρ_K the image of the measure ρ under K^+. This paper aims to study the measure ρ_K assuming ρ to be the spectral measure of a Jacobi field. We obtain a family of operators whose spectral measure equals ρ_K. We also obtain an analogue of the Wiener – Ito decomposition for ρ_K. Finally, we illustrate the results obtained by carrying out the explicit calculations for the case, where ρ_Kis a Levy noise measure.