An inverse spectral problem for non-self-adjoint Jacobi matrices

We consider the class of bounded symmetric Jacobi matrices $J$ with positive off-diagonal elements and complex diagonal elements. With each matrix $J$ from this class, we associate the spectral data, which consists of a pair $(\nu ,\psi )$. Here $\nu$ is the spectral measure of $\mathrm{\mid }J\mathrm{\mid }=\sqrt{J^{\ast }J}$ and $\psi$ is a $\mathit{\text{phase}}\mathit{\text{function}}$ on the real line satisfying $\mathrm{\mid }\psi \mathrm{\mid }\le 1$ almost everywhere with respect to the measure $\nu$. Our main result is that the map from $J$ to the pair $(\nu ,\psi )$ is a bijection between our class of Jacobi matrices and the set of all spectral data.