Spectral synthesis in Hilbert spaces of entire functions

Abstract

We give a survey of recent advances in the theory of spaces of entire functions related to the notion of spectral synthesis. In particular, we discuss a solution of a longstanding problem about spectral synthesis for systems of exponentials in $L^2(-\pi ,\pi )$ as well as its generalization to de Branges spaces of entire functions. In the de Branges space setting the problem can be related (via a functional model) to spectral theory of rank one perturbations of compact selfadjoint operators; this leads to unexpected examples of rank one perturbations which do not admit spectral synthesis. Related problems for Fock-type spaces are also considered.