Spectral synthesis in de Branges spaces

Abstract

We solve completely the spectral synthesis problem for reproducing kernels in the de Branges spaces \({\mathcal{H}(E)}\) . Namely, we describe the de Branges spaces \({\mathcal{H}(E)}\) such that every complete and minimal system of reproducing kernels \({\{k_\lambda\}_{\lambda \in \Lambda}}\) with complete biorthogonal \({\{g_\lambda\}_{\lambda \in \Lambda}}\) admits the spectral synthesis, i.e., \({f \in \overline{\rm Span} \{(f, g_\lambda) k_\lambda : \lambda \in \Lambda\}}\) for any f in \({\mathcal{H}(E)}\) . Surprisingly, this property takes place only for two essentially different classes of de Branges spaces: spaces with finite spectral measure and spaces which are isomorphic to Fock-type spaces of entire functions. The first class goes back to de Branges himself, while special cases of de Branges spaces of the second class appeared in the literature only recently; we give a complete characterisation of this second class in terms of the spectral data for \({\mathcal{H}(E)}\) .