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 $ρ_K$is a Levy noise measure.