A function space model for canonical systems

Recently, a generalization to the Pontryagin space setting of the notion of canonical (Hamiltonian) systems which involves a finite number of inner singularities has been given. The spectral theory of indefinite canonical systems was investigated with help of an operator model. This model consists of a Pontryagin space boundary triple and was constructed in an abstract way. Moreover, the construction of this operator model involves a procedure of splitting-and-pasting which is technical but at the present stage of development in general inevitable. In this paper we provide an isomorphic form of this operator model which acts in a finite-dimensional extension of a function space naturally associated with the given indefinite canonical system. We give explicit formulae for the model operator and the boundary relation. Moreover, we show that under certain asymptotic hypotheses the procedure of splitting-and-pasting can be avoided by employing a limiting process. We restrict attention to the case of one singularity. This is the core of the theory, and by making this restriction we can significantly reduce the technical effort without losing sight of the essential ideas

A companion to coalgebraic weak bisimulation for action-type systems

We propose a coalgebraic definition of weak bisimulation for classes of coalgebras obtained from bifunctors in the category Set. Weak bisim-ilarity for a system is obtained as strong bisimilarity of a transformed system. The particular transformation consists of two steps: First, the behavior on actions is lifted to behavior on finite words. Second, the behavior on finite words is taken modulo the hiding of internal or in-visible actions, yielding behavior on equivalence classes of words closed under silent steps. The coalgebraic definition is validated by two cor-respondence results: one for the classical notion of weak bisimulation of Milner, another for the notion of weak bisimulation for generative probabilistic transition systems as advocated by Baier and Hermanns.

Coalgebraic Weak Bisimulation for Action-Type Systems

We propose a coalgebraic definition of weak bisimulation for classes of coalgebras obtained from bifunctors in the category Set. Weak bisimilarity for a system is obtained as strong bisimilarity of a transformed system. The particular transformation consists of two steps: First, the behavior on actions is lifted to behavior on finite words. Second, the behavior on finite words is taken modulo the hiding of internal or invisible actions, yielding behavior on equivalence classes of words closed under silent steps. The coalgebraic definition is validated by two correspondence results: one for the classical notion of weak bisimulation of Milner, another for the notion of weak bisimulation for generative probabilistic transition systems as advocated by Baier and Hermanns

Coalgebraic Weak Bisimulation for Action-Type Systems

We propose a coalgebraic definition of weak bisimulation for classes of coalgebras obtained from bifunctors in the category Set. Weak bisimilarity for a system is obtained as strong bisimilarity of a transformed system. The particular transformation consists of two steps: First, the behavior on actions is lifted to behavior on finite words. Second, the behavior on finite words is taken modulo the hiding of internal or invisible actions, yielding behavior on equivalence classes of words closed under silent steps. The coalgebraic definition is validated by two correspondence results: one for the classical notion of weak bisimulation of Milner, another for the notion of weak bisimulation for generative probabilistic transition systems as advocated by Baier and Hermanns

Bitangential interpolation in generalized Schur classes

Bitangential interpolation problems in the class of matrix valued functions in the generalized Schur class are considered in both the open unit disc and the open right half plane, including problems in which the solutions is not assumed to be holomorphic at the interpolation points. Linear fractional representations of the set of solutions to these problems are presented for invertible and singular Hermitian Pick matrices. These representations make use of a description of the ranges of linear fractional transformations with suitably chosen domains that was developed in a previous paper.Comment: Second version, corrected typos, changed subsection 5.6, 47 page

The class of n-entire operators

We introduce a classification of simple, regular, closed symmetric operators with deficiency indices (1,1) according to a geometric criterion that extends the classical notions of entire operators and entire operators in the generalized sense due to M. G. Krein. We show that these classes of operators have several distinctive properties, some of them related to the spectra of their canonical selfadjoint extensions. In particular, we provide necessary and sufficient conditions on the spectra of two canonical selfadjoint extensions of an operator for it to belong to one of our classes. Our discussion is based on some recent results in the theory of de Branges spaces.Comment: 33 pages. Typos corrected. Changes in the wording of Section 2. References added. Examples added. arXiv admin note: text overlap with arXiv:1104.476

De Branges spaces and Krein's theory of entire operators

This work presents a contemporary treatment of Krein's entire operators with deficiency indices $(1,1)$ 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

The Beurling--Malliavin Multiplier Theorem and its analogs for the de Branges spaces

Let $\omega$ be a non-negative function on $\mathbb{R}$. We are looking for a non-zero $f$ from a given space of entire functions $X$ satisfying $$(a) \quad|f|\leq \omega\text{\quad or\quad(b)}\quad |f|\asymp\omega.$$ The classical Beurling--Malliavin Multiplier Theorem corresponds to $(a)$ and the classical Paley--Wiener space as $X$. We survey recent results for the case when $X$ is a de Branges space \he. Numerous answers mainly depend on the behaviour of the phase function of the generating function $E$.Comment: Survey, 25 page