On finite rank Hankel operators

For self-adjoint Hankel operators of finite rank, we find an explicit formula for the total multiplicity of their negative and positive spectra. We also show that very strong perturbations, for example, a perturbation by the Carleman operator, do not change the total number of negative eigenvalues of finite rank Hankel operators. As a by-product of our considerations, we obtain an explicit description of the group of unitary automorphisms of all bounded Hankel operators.