Characterization of Hermite–Biehler functions in terms of canonical systems

Louis de Branges in 1968 asked which canonical systems correspond to arbitrary (not necessarily regular) Hermite-Biehler entire functions. In the important partial case of diagonal hamiltonians an answer to the de Branges question follows from the results of M. G. Krein on the spectra of strings. In the general situation the question is equivalent to the problem of characterization of the hamiltonians corresponding to canonical systems with discrete spectrum. It is naturally included into the more general problem of description of systems with discrete spectrum in the Schatten–von Neumann classes $S^p$ ($p>1$).
The talk is an exposition of a recent solution of the characterization problem obtained jointly with H. Woracek (TU Wien). In particular, we will answer the initial question of de Branges.