H. Woracek:

"Existance of zerofree functions $N$-associated to a de Branges Pontryagin space";

in: "ASC Report 23/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

In the theory of de Branges Hilbert spaces of entire functions, so-called

`functions associated to a spaceī 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 2 Z) to a de Branges Pontryagin space.

Let a de Branges Pontryagin space P and N 2 Z be given. Our aim

is to characterize whether there exists a real and zerofree function Nassociated

to P in terms of Kre˘ınīs 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.

de Branges space, Pontryagin space, associated function, canonical

http://dx.doi.org/10.1007/s00605-010-0203-2

http://www.asc.tuwien.ac.at/preprint/2009/asc23x2009.pdf

Created from the Publication Database of the Vienna University of Technology.