Spectral synthesis in de Branges spaces

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 $\{k_λ\}_{λ\inΛ}$ with complete biorthogonal $\{g_λ\}_{λ\inΛ}$) are strong $M$-bases (i.e., every mixed system $\{k_λ\}_{λ\inΛ\setminus\tilde Λ} \cup\{g_λ\}_{λ\in \tilde Λ}$ 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.

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