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

Abstract

Assume that $K^+: H_- \rightarrow T_-$ is a bounded operator, where $H_—$ and $T_—$ are Hilbert spaces and $p$ is a measure on the space $H_—$. Denote by $\rho_K$ the image of the measure $\rho$ under $K^+$. This paper aims to study the measure $\rho_K$ assuming $\rho$ to be the spectral measure of a Jacobi field. We obtain a family of operators whose spectral measure equals $\rho_K$. We also obtain an analogue of the Wiener – Ito decomposition for $\rho_K$. Finally, we illustrate the results obtained by carrying out the explicit calculations for the case, where $\rho_K$is a Levy noise measure.