Recent developments in spectral synthesis for exponential systems and for non-self-adjoint operators

Abstract

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 (−π, π). 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.

On the completeness and Riesz basis property of root subspaces of boundary value problems for first order systems and applications

“…The spectral synthesis problem was originated by J. Wermer [71] and then studied by many authors (see [11,47,53] and references therein). We also mention recent preprints [3,4,5,6] devoted to the problems of completeness and spectral synthesis for singular perturbations of selfadjoint operators and systems of exponents and to problems of removal of spectrum.…”