Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2016
Volume
Vol. 36
Number
No. 6
Description
Journal Volume
Opuscula Mathematica
Vol. 36 (2016)
Projects
Pages
Articles
Characterizations of rectangular (para)-unitary rational functions
(2016) Alpay, Daniel; Jørgensen, Palle E.T.; Lewkowicz, Izchak
We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle: (i) through the realization matrix of Schur stable systems, (ii) the Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters, (iii) through the (not necessarily reducible) Matrix Fraction Description (MFD). In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infinity). A special attention is devoted to exploring the gap between the square and rectangular cases.
Eigenvalue estimates for operators with finitely many negative squares
(2016) Behrndt, Jussi; Möws, Roland; Trunk, Carsten
Let $A$ and $B$ be selfadjoint operators in a Krein space. Assume that the resolvent difference of $A$ and $B$ is of rank one and that the spectrum of $A$ consists in some interval $I\subset\mathbb{R}$ of isolated eigenvalues only. In the case that $A$ is an operator with finitely many negative squares we prove sharp estimates on the number of eigenvalues of $B$ in the interval $I$. The general results are applied to singular indefinite Sturm-Liouville problems.
On locally Hilbert spaces
(2016) Gheondea, Aurelian
This is an investigation of some basic properties of strictly inductive limits of Hilbert spaces, called locally Hilbert spaces, with respect to their topological properties, the geometry of their subspaces, linear functionals and dual spaces.
Minimal realizations of generalized Nevanlinna functions
(2016) Hassi, Seppo; Wietsma, Hendrik Luit
Minimal realizations of generalized Nevanlinna functions that carry the information on their generalized poles of nonpositive type in an explicit form are established. These realizations are based on a modification of the basic canonical factorization of generalized Nevanlinna functions whereby the non-minimality problems in realizations that are based directly on the canonical factorization are circumvented.
Dispersion estimates for spherical Schrödinger equations - the effect of boundary conditions
(2016) Holzleitner, Markus; Kostenko, Aleksej; Teschl, Gerald
We investigate the dependence of the $L^1\to L^{\infty}$ dispersive estimates for one-dimensional radial Schrödinger operators on boundary conditions at $0$. In contrast to the case of additive perturbations, we show that the change of a boundary condition at zero results in the change of the dispersive decay estimates if the angular momentum is positive, $l\in (0,1/2)$. However, for nonpositive angular momenta, $l\in (-1/2,0]$, the standard $O(|t|^{-1/2})$ decay remains true for all self-adjoint realizations.