42 research outputs found
De Branges spaces and Krein's theory of entire operators
This work presents a contemporary treatment of Krein's entire operators with
deficiency indices and de Branges' Hilbert spaces of entire functions.
Each of these theories played a central role in the research of both renown
mathematicians. Remarkably, entire operators and de Branges spaces are
intimately connected and the interplay between them has had an impact in both
spectral theory and the theory of functions. This work exhibits the
interrelation between Krein's and de Branges' theories by means of a functional
model and discusses recent developments, giving illustrations of the main
objects and applications to the spectral theory of difference and differential
operators.Comment: 37 pages, no figures. The abstract was extended. Typographical errors
were corrected. The bibliography style was change
Two-dimensional Hamiltonian systems
This survey article contains various aspects of the direct and inverse spectral problem for twodimensional Hamiltonian systems, that is, two dimensional canonical systems of homogeneous differential equations of the form Jy'(x) = -zH(x)y(x); x â [0;L); 0 < L †â; z â C; with a real non-negative definite matrix function H â„ 0 and a signature matrix J, and with a standard boundary condition of the form y1(0+) = 0. Additionally it is assumed that Weyl's limit point case prevails at L. In this case the spectrum of the canonical system is determined by its Titchmarsh-Weyl coefficient Q which is a Nevanlinna function, that is, a function which maps the upper complex half-plane analytically into itself. In this article an outline of the Titchmarsh-Weyl theory for Hamiltonian systems is given and the solution of the direct spectral problem is shown. Moreover, Hamiltonian systems comprehend the class of differential equations of vibrating strings with a non-homogenous mass-distribution function as considered by M.G. Krein. The inverse spectral problem for two{dimensional Hamiltonian systems was solved by L. de Branges by use of his theory of Hilbert spaces of entire functions, showing that each Nevanlinna function is the Titchmarsh-Weyl coefficient of a uniquely determined normed Hamiltonian. More detailed results of this connection for e.g. systems with a semibounded or discrete or finite spectrum are presented, and also some results concerning spectral perturbation, which allow an explicit solution of the inverse spectral problem in many cases
The Beurling--Malliavin Multiplier Theorem and its analogs for the de Branges spaces
Let be a non-negative function on . We are looking for a
non-zero from a given space of entire functions satisfying The
classical Beurling--Malliavin Multiplier Theorem corresponds to and the
classical Paley--Wiener space as . We survey recent results for the case
when is a de Branges space \he. Numerous answers mainly depend on the
behaviour of the phase function of the generating function .Comment: Survey, 25 page
The exponential type of the fundamental solution of an indefinite Hamiltonian system
The fundamental solution of a Hamiltonian system whose Hamiltonian H is positive definite and locally integrable is an entire function of exponential type. Its exponential type can be computed as the integral over . We show that this formula remains true in the indefinite (Pontryagin space) situation, where the Hamiltonian is permitted to have finitely many inner singularities. As a consequence, we obtain a statement on non-cancellation of exponential growth for a class of entire matrix functions
Spectral theorem for definitizable normal linear operators on Krein spaces
The final publication is available at Springer via https://doi.org/10.1007/s00020-016-2288-z.In the present note a spectral theorem for normal definitizable linear operators on Krein spaces is derived by developing a functional calculus ÏâŠÏ(N) which is the proper analogue of ÏâŠâ«ÏdE in the Hilbert space situation. This paper is the first systematical study of definitizable normal operators on Krein spaces