On the nature of the de Branges Hamiltonian

Abstract

We prove the theorem announced by the author in 1995 in the paper "Criterion for discreteness of spectrum of singular canonical system" (Functional Analysis and Its Applications, Vol. 29, No. 3).

In developing the theory of Hilbert spaces of entire functions (we call them the Krein - de Branges spaces or, briefly, K-B spaces), L. de Branges arrived at some class of canonical equations of phase dimension 2. He proved that, for any given K-B space, there exists a canonical equation of the considered class such that it restores the chain of included K-B spaces. The Hamiltonians of such canonical equations are called the de Branges Hamiltonians. The following question arises: Under which conditions the Hamiltonian of some canonical equation should be a de Branges Hamiltonian. The basic theorem of the present paper together with Theorem 1 of the mentioned paper gives the answer to this question.