The classical spectral theory of Jacobi matrices (which is linked to the theory of orthogonal polynomials) asserts that (i) every self-adjoint Jacobi matrix is uniquely determined by its spectral measure and (ii) any measure on the real line is a spectral measure of some self-adjoint Jacobi matrix. In this talk, I will discuss the extension of this theory to some class of non-self-adjoint Jacobi matrices. I will define the spectral data of a non-self-adjoint Jacobi matrix J in the spirit of the polar decomposition. The spectral data consists of a spectral measure (which encoodes information about |J|) and a certain function (we call it the phase function) which encodes some "rotations" in the Hilbert space. The main result is that (i) the pair (spectral measure, phase function) uniquely determines the non-self-adjoint Jacobi matrix and (ii) any such pair from a certain natural class corresponds to some Jacobi matrix.

This is recent joint work with Frantisek Stampach (Czech Technical University in Prague).