Recent developments in spectral synthesis for exponential systems and   for non-self-adjoint operators
  We survey recent results concerning the hereditary completeness of some special systems of functions and the spectral synthesis problem for a related class of linear operators. We present a solution of the spectral synthesis problem for systems of exponentials in $L^2(-\pi, \pi)$. Analogous results are obtained for the systems of reproducing kernels in the de Branges spaces of entire functions. We also apply these results (via a functional model) to the spectral theory of rank one perturbations of compact self-adjoint operators. 
Anton Baranov, Yurii Belov, Alexander Borichev, Dmitry Yakubovich
math.FA math.CV math.SP