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 all $M$-bases of reproducing kernels (i.e., complete and minimal systems $\{k_{\lambda }{\}}_{\lambda \in \Lambda }$ with complete biorthogonal $\{g_{\lambda }{\}}_{\lambda \in \Lambda }$) are strong $M$-bases (i.e., every mixed system $\{k_{\lambda }{\}}_{\lambda \in \Lambda \setminus \tilde{\Lambda }}\cup \{g_{\lambda }{\}}_{\lambda \in \tilde{\Lambda }}$ is also complete). 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, the second class appeared in a recent work of A. Borichev and Yu. Lyubarskii. Moreover, we are able to give a complete characterisation of this second class in terms of the spectral data for $\mathcal{H}(E)$. In addition, we obtain some results about possible codimension of mixed systems for a fixed de Branges space $\mathcal{H}(E)$, and prove that any minimal system of reproducing kernels in $\mathcal{H}(E)$ is contained in an exact system of reproducing kernels.

38 pages. Shortened text with streamlined proofs. This version is accepted for publication in "Geometric and Functional Analysis"

References in corpus (1)

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