Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2017
Volume
Vol. 37
Number
No. 1
Description
Journal Volume
Opuscula Mathematica
Vol. 37 (2017)
Projects
Pages
Articles
Limit-point criteria for the matrix Sturm-Liouville operator and its powers
(2017) Brojtigam, Irina Nikolaevna
We consider matrix Sturm-Liouville operators generated by the formal expression $l[y]=-(P(y^{\prime}-Ry))^{\prime}-R^*P(y^{\prime}-Ry)+Qy,$ in the space $L^2_n(I)$, $I:=[0, \infty)$. Let the matrix functions $P:=P(x)$, $Q:=Q(x)$ and $R:=R(x)$ of order $n \in \mathbb{N}$ be defined on $I$, $P$ is a nondegenerate matrix, $P$ and $Q$ are Hermitian matrices for $x \in I$ and the entries of the matrix functions $P^{-1}$, $Q$ and $R$ are measurable on $I$ and integrable on each of its closed finite subintervals. The main purpose of this paper is to find conditions on the matrices $P$, $Q$ and $R$ that ensure the realization of the limit-point case for the minimal closed symmetric operator generated by $l^k[y]$ $k \in \mathbb{N}$. In particular, we obtain limit-point conditions for Sturm-Liouville operators with matrix-valued distributional coefficients.
The LQ/KYP problem for infinite-dimensional systems
(2017) Grabowski, Piotr
Our aim is to present a solution to a general linear-quadratic (LQ) problem as well as to a Kalman-Yacubovich-Popov (KYP) problem for infinite-dimensional systems with bounded operators. The results are then applied, via the reciprocal system approach, to the question of solvability of some Lur'e resolving equations arising in the stability theory of infinite-dimensional systems in factor form with unbounded control and observation operators. To be more precise the Lur’e resolving equations determine a Lyapunov functional candidate for some closed-loop feedback systems on the base of some properties of an uncontrolled (open-loop) system. Our results are illustrated in details by an example of a temperature of a rod stabilization automatic control system.
Towards theory of C-symmetries
(2017) Kužel', Sergìj Oleksandrovič; Sudìlovsʹka, Veronìka Igorìvna
The concept of $\mathcal{C}$-symmetry originally appeared in $\mathcal{PT}$-symmetric quantum mechanics is studied within the Krein spaces framework.
Seminormal systems of operators in Clifford environments
(2017) Martin, Mircea
The primary goal of our article is to implement some standard spin geometry techniques related to the study of Dirac and Laplace operators on Dirac vector bundles into the multidimensional theory of Hilbert space operators. The transition from spin geometry to operator theory relies on the use of Clifford environments, which essentially are Clifford algebra augmentations of unital complex $C^*$-algebras that enable one to set up counterparts of the geometric Bochner-Weitzenbock and Bochner-Kodaira-Nakano curvature identities for systems of elements of a $C^*$-algebra. The so derived self-commutator identities in conjunction with Bochner’s method provide a natural motivation for the definitions of several types of seminormal systems of operators. As part of their study, we single out certain spectral properties, introduce and analyze a singular integral model that involves Riesz transforms, and prove some self-commutator inequalities.
Eigenvalue asymptotics for the Sturm-Liouville operator with potential having a strong local negative singularity
(2017) Nursultanov, Medet; Rozenblioum, Grigori
We find asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the finite interval, with potential having a strong negative singularity at one endpoint. This is the case of limit circle in H. Weyl sense. We establish that, unlike the case of an infinite interval, the asymptotics for positive eigenvalues does not depend on the potential and it is the same as in the regular case. The asymptotics of the negative eigenvalues may depend on the potential quite strongly, however there are always asymptotically fewer negative eigenvalues than positive ones. By unknown reasons this type of problems had not been studied previously.