Image of the spectral measure of a Jacobi field and the corresponding operators

Abstract

By definition, a Jacobi field $J=(J(\phi))_{\phi\in H_+}$ is a family of commuting selfadjoint three-diagonal operators in the Fock space $\mathcal F(H)$. The operators $J(\phi)$ are indexed by the vectors of a real Hilbert space $H_+$. The spectral measure $\rho$ of the field $J$ is defined on the space $H_-$ of functionals over $H_+$. The image of the measure $\rho$ under a mapping $K^+:T_-\to H_-$ is a probability measure $\rho_K$ on $T_-$. We obtain a family $J_K$ of operators whose spectral measure is equal to $\rho_K$. We also obtain the chaotic decomposition for the space $L^2(T_-,d\rho_K)$.