Existence of zerofree functions N-associated to a de Branges Pontryagin space

Abstract

In the theory of de Branges Hilbert spaces of entire functions, so-called `functions associated to a space´ play an important role. In the present paper we deal with a generalization of this notion in two directions, namely with functions N-associated (N ∈ ℤ) to a de Branges Pontryagin space. Let a de Branges Pontryagin space P and N ∈ ℤ be given. Our aim is to characterize whether there exists a real and zerofree function N associated to P in terms of Kreĭn´s Q-function associated with the multiplication operator in P. The conditions which appear in this characterization involve the asymptotic distribution of the poles of the Q-function plus a summability condition. Although this question may seem rather abstract, its answer has a variety of nontrivial consequences. We use it to answer two questions arising in the theory of general (indefinite) canonical systems. Namely, to characterize whether a given generalized Nevanlinna function is the intermediate Weyl-coefficient of some system in terms of its poles and residues, and to characterize whether a given general Hamiltonian ends with a specified number of indivisible intervals in terms of the Weyl-coefficient associated to the system. In addition, we present some applications, e.g. dealing with admissible majorants in de Branges spaces or the continuation problem for hermitian indefinite functions.