Image of the Spectral Measure of a Jacobi Field and the Corresponding Operators

Abstract.

By definition, a Jacobi field $J = (\tilde J(\phi ))_{\phi \in H_ + }$ is a family of commuting selfadjoint three-diagonal operators in the Fock space $\mathcal{F}(H).$ 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 $K^+:T_{-} \to H_{-}$ is a probability measure ρ _K on T₋. We obtain a family J _K of operators whose spectral measure is equal to ρ _K . We also obtain the chaotic decomposition for the space L²(T₋, d_{ρ K}).