%0 Journal Article
%T Spectral synthesis in de Branges spaces
%+ Saint Petersburg State University (SPBU)
%+ National Research University Higher School of Economics [St. Petersburg]
%+ Chebyshev Laboratory
%+ Institut de Mathématiques de Marseille (I2M)
%A Baranov, Anton
%A Belov, Yurii
%A Borichev, Alexander
%Z 38 pages. Shortened text with streamlined proofs. This version is accepted for publication in "Geometric and Functional Analysis"
%< avec comité de lecture
%@ 1016-443X
%J Geometric And Functional Analysis
%I Springer Verlag
%V 25
%N 2
%P 417-452
%8 2015
%D 2015
%Z 1309.6915
%R 10.1007/s00039-015-0322-y
%Z Mathematics [math]/Complex Variables [math.CV]
%Z Mathematics [math]/Functional Analysis [math.FA]Journal articles
%X 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.
%G English
%2 https://hal.science/hal-01258039/document
%2 https://hal.science/hal-01258039/file/1309.6915.pdf
%L hal-01258039
%U https://hal.science/hal-01258039
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-