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

Abstract

By definition, a Jacobi field $J=(J(\varphi ){)}_{\varphi \in H_{+}}$ is a family of commuting selfadjoint three-diagonal operators in the Fock space $\mathcal{F}(H)$. The operators $J(\varphi )$ 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 $L^2(T_{-},d{\rho }_K)$.