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

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By definition, a Jacobi field J=(J(ϕ))ϕ∈ ₕ₊ is a family of commuting selfadjoint three-diagonal operators in the Fock space F(H). The operators J(ϕ) are indexed by the vectors of a real Hilbert space H+. The spectral measure ρ of the field J is defined on the space H- of functionals over H+. The image of the measure ρ under a mapping K+:T-→ H- is a probability measure ρₖ on T-. We obtain a family Jₖ of operators whose spectral measure is equal to ρₖ. We also obtain the chaotic decomposition for the space L²(T-,dρₖ).