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 Focktype 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2013
 arXiv:
 arXiv:1309.6915
 Bibcode:
 2013arXiv1309.6915B
 Keywords:

 Mathematics  Complex Variables;
 Mathematics  Functional Analysis
 EPrint:
 38 pages. Shortened text with streamlined proofs. This version is accepted for publication in "Geometric and Functional Analysis"