Authors Index, Vol. 25, 2019

On eigenvalues of banded matrices

O. Akhiiezer, O. Dunaievska

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 2, 98-103

In the paper, asymptotics for eigenvalues of Hermitian, compact operators, generated by infinite, banded matrices is obtained in terms of the asymptotics of their matrix entries. Analogues for banded matrices of Gershgorin's disks theory are discussed.

Measure of noncompactness, essential approximation and defect pseudospectrum

Aymen Ammar, Aref Jeribi, Kamel Mahfoudhi

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 1, 1-11

The scope of the present research is to establish some findings concerning the essential approximation pseudospectra and the essential defect pseudospectra of closed, densely defined linear operators in a Banach space, building upon the notion of the measure of noncompactness. We start by giving a refinement of the definition of the essential approximation pseudospectra and that of the essential defect pseudospectra by means of the measure of noncompactness. From these characterizations we shall deduce several results and we shall give sufficient conditions on the perturbed operator to have its invariance.

Operators preserving orthogonality on Hilbert $\it{K}(H)$-modules

R. G. Sanati, E. Ansari-piri, M. Kardel

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 2, 189-194

In this paper, we study the class of orthogonality preserving operators on a Hilbert $\it{K(H)}$-module $W$ and show that an operator $T$ on $W$ is orthogonality preserving if and only if it is orthogonality preserving on a special dense submodule of $W$. Then we apply this fact to show that an orthogonality preserving operator $T$ is normal if and only if $T^*$ is orthogonality preserving.

On a localization of the spectrum of a complex Volterra operator

Miron B. Bekker, Joseph A. Cima

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 1, 12-14

A complex Volterra operator with the symbol $g=\log{(1+u(z))}$, where $u$ is an analytic self map of the unit disk $\mathbb D$ into itself is considered. We show that the spectrum of this operator on $H^p(\mathbb D)$, $1\le p<\infty$, is located in the disk $\{\lambda:|\lambda+p/2|\leq p/2\}$.

On maximal multiplicity of eigenvalues of finite-dimensional spectral problem on a graph

Olga Boiko, Olga Martynyuk, Vyacheslav Pivovarchik

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 2, 104-117

Recurrence relations of the second order on the edges of a metric connected graph together with boundary and matching conditions at the vertices generate a spectral problem for a self-adjoint finite-dimensional operator. This spectral problem describes small transverse vibrations of a graph of Stieltjes strings. It is shown that if the graph is cyclically connected and the number of masses on each edge is not less than 3 then the maximal multiplicity of an eigenvalue is $\mu+1$ where $\mu$ is the cyclomatic number of the graph. If the graph is not cyclically connected and each edge of it bears at least one point mass then the maximal multiplicity of an eigenvalue is expressed via $\mu$, the number of edges and the number of interior vertices in the tree obtained by contracting all the cycles of the graph into vertices.

Köthe-Orlicz vector-valued weakly sequence spaces of difference operators

Anu Choudhary, Kuldip Raj

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 2, 161-176

In the present article, we propose vector-valued weakly null, weakly convergent and weakly bounded sequences over n-normed spaces associated with infinite matrix, Musielak-Orlicz function and difference operator. We make an effort to study some algebraic and topological properties of these sequence spaces. Further, we shall investigate some inclusion relations between newly formed sequence spaces.

Yuriy Makarovich Berezansky

Editorial Board

Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 2, 97-97

Weak-coupling limit for ergodic environments

Martin Friesen, Yuri Kondratiev

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 2, 118-133

The main aim of this work is to establish an averaging principle for a wide class of interacting particle systems in the continuum. This principle is an important step in the analysis of Markov evolutions and is usually applied for the associated semigroups related to backward Kolmogorov equations, c.f. [27]. Our approach is based on the study of forward Kolmogorov equations (a.k.a. Fokker-Planck equations). We describe a system evolving as a Markov process on the space of finite configurations, whereas its rates depend on the actual state of another (equilibrium) process on the space of locally finite configurations. We will show that ergodicity of the environment process implies the averaging principle for the solutions of the coupled Fokker-Planck equations.

The Welland inequality on hypergroups

M. G. Hajibayov

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 2, 134-141

The Welland inequality for fractional integrals on hypergroups with quasi-metric and Haar measure is proved. This inequality gives pointwise estimates of fractional integrals by fractional maximal operators.

Complex moment problem and recursive relations

K. Idrissi, E. H. Zerouali

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 1, 15-34

We introduce a new methodology to solve the truncated complex moment problem. To this aim we investigate recursive doubly indexed sequences and their characteristic polynomials. A characterization of recursive doubly indexed \emph{moment} sequences is given. A simple application gives a computable solution to the complex moment problem for cubic harmonic characteristic polynomials of the form $z^3+az+b\overline{z}$, where $a$ and $b$ are arbitrary real numbers. We also recapture a recent result due to Curto-Yoo given for cubic column relations in $M(3)$ of the form $Z^3=itZ+u\overline{Z}$ with $t,u$ real numbers satisfying some suitable inequalities. Furthermore, we solve the truncated complex moment problem with column dependence relations of the form $Z^{k+1}= \sum\limits_{0\leq n+ m \leq k} a_{nm} \overline{Z}^n Z^m$ ($a_{nm} \in \mathbb{C}$).

Abstract formulation of the Cole-Hopf transform

Yoritaka Iwata

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 2, 142-151

Operator representation of Cole-Hopf transform is obtained based on the logarithmic representation of infinitesimal generators. For this purpose the relativistic formulation of abstract evolution equation is introduced. Even independent of the spatial dimension, the Cole-Hopf transform is generalized to a transform between linear and nonlinear equations defined in Banach spaces. In conclusion a role of transform between the evolution operator and its infinitesimal generator is understood in the context of generating nonlinear semigroup.

A gentle introduction to James’ weak compactness theorem and beyond

Warren B. Moors

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 1, 35-83

The purpose of this paper is twofold: firstly, to provide an accessible proof of James' weak compactness theorem that is able to be taught in a first-year graduate class in functional analysis and secondly, to explore some of the latest and possible future extensions and applications of James' theorem.

On isometries satisfying deformed commutation relations

Olha Ostrovska, Roman Yakymiv

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 2, 152-160

We consider an $C^*$-algebra $\mathcal{E}_{1,n}^q$, $q\le 1$, generated by isometries satisfying $q$-deformed commutation relations. For the case $|q|<1$, we prove that $\mathcal E_{1,n}^q \simeq\mathcal E_{1,n}^0=\mathcal O_{n+1}^0$. For $|q|=1$ we show that $\mathcal E_{1,n}^q$ is nuclear and prove that its Fock representation is faithul. In this case we also discuss the representation theory, in particular construct a commutative model for representations.

Subscalarity of $k$-quasi-class $A$ operators

M. H. M. Rashid

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 2, 177-188

In this paper, we show that every $k$-quasi-class $A$ operator has a scalar extension and give some spectral properties of the scalar extensions of $k$-quasi-class $A$ operators. As a corollary, we get that such an operator with rich spectrum has a nontrivial invariant subspace.

Boundary triples for integral systems on the half-line

D. Strelnikov

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 25 (2019), no. 1, 84-96

Let $P$, $Q$ and $W$ be real functions of locally bounded variation on $[0,\infty)$ and let $W$ be non-decreasing. In the case of absolutely continuous functions $P$, $Q$ and $W$ the following Sturm-Liouville type integral system: \begin{equation} \label{eq:abs:is} J\vec{f}(x)-J\vec{a} = \int_0^x \begin{pmatrix}\lambda dW-dQ & 0\\0 & dP\end{pmatrix} \vec{f}(t), \quad J = \begin{pmatrix}0 & -1\\1 & 0\end{pmatrix} \end{equation} (see [5]) is a special case of so-called canonical differential system (see [16, 20, 24]). In [27] a maximal $A_{\max}$ and a minimal $A_{\min}$ linear relations associated with system (1) have been studied on a compact interval. This paper is a continuation of [27] , it focuses on a study of $A_{\max}$ and $A_{\min}$ on the half-line. Boundary triples for $A_{\max}$ on $[0,\infty)$ are constructed and the corresponding Weyl functions are calculated in both limit point and limit circle cases at $\infty$.

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