  We study spectral properties of nonselfadjoint rank one perturbations of
compact selfadjoint operators. The problems under consideration include
completeness of eigenvectors, relations between completeness of the perturbed
operator and its adjoint, and the spectral synthesis problem. We obtain new
criteria for completeness and spectral synthesis in this class as well as a
series of counterexamples which show that the spectral structure of rank one
perturbations is, in general, unexpectedly rich and complicated.
  A parallel spectral theory is developed for one-dimensional singular
perturbations of unbounded selfadjoint operators. Our approach is based on a
functional model for this class which translates the properties of operators to
completeness problems for systems of reproducing kernels and their
biorthogonals in some spaces of analytic (entire) functions.
