Image of the spectral measure of a Jacobi field and the corresponding operators (Q814389)

Let \(H\) be a Hilbert space and \[ {\mathcal F}(H)=\bigoplus^\infty_{n=0} {\mathcal F}_n(H)\tag{1} \] the corresponding Fock space. The Jacobi field \(J= (\widetilde J(\varphi))_{\varphi\in H_+}\) is considered as a family of commuting selfadjoint operators which have a three-diagonal structure with respect to the decomposition (1). These operators are assumed to linearly and continuously depend on the indexing parameter \(\varphi\in H_+\). The corresponding Fourier transform is a unitary operator between the Fock space \({\mathcal F}(H)\) and the space \(L^2(H_-,dp)\). The measure \(\rho\) on \(H_-\) is called the spectral measure of \(J\). The Jacobi field with the Gaussian spectral measure is the classical free field in quantum field theory. The problem of finding an operator family with given spectral measure often arises in applications. Let \(\rho\) be the spectral measure of the field \(J\) and consider a mapping \(K^+:H_-\to T_-\), where \(T_-\) is a certain Hilbert space. This mapping takes \(\rho\) to the measure \(\rho_K\) on \(T_-\). The main result of the paper is to obtain a family \(J_K\) of operators whose spectral measure equals \(\rho_K\), or in other words to track the changes of the Jacobi field caused by mapping of its spectral measure. If \(K^+\) is an invertible operator, then \(J_K\) is isomorphic to the initial family \(J\). Also, the chaotic decomposition of the space \(L^2(T_-,d\rho_K)\) is derived through the orthogonalization of polynomials on \(T_-\).