Authors Index,
On eigenvalues of banded matrices
MFAT 25 (2019), no. 2, 98-103
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.
Limited and Dunford-Pettis operators on Banach lattices
Khalid Bouras, Abdennabi EL Aloui, Aziz Elbour
MFAT 25 (2019), no. 3, 205-210
205-210
This paper is devoted to investigation of conditions on a pair of Banach lattices $E; F$ under which every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited. Mainly, it is proved that if every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited, then the norm on $E'$ is order continuous or $F$ is finite dimensional. Also, it is proved that every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited, if one of the following statements is valid:
1) The norm on $E^{\prime }$ is order continuous, and $F^{\prime }$ has weak$^{\ast }$ sequentially continuous lattice operations.
2) The topological dual $E^{\prime }$is discrete and its norm is
order continuous.
3) The norm of $E^{\prime }$ is order continuous and the lattice
operations in $E^{^{\prime }}$ are weak$^{\ast }$ sequentially continuous.
4) The norms of $E$ and of $E^{\prime }$ are order continuous.
Essential approximate point and essential defect spectrum of a sequence of linear operators in Banach spaces
Toufik Heraiz, Aymen Ammar, Aref Jeribi
MFAT 25 (2019), no. 4, 373-380
373-380
This paper is devoted to an investigation of the relationship between the essential approximate point spectrum (respectively, the essential defect spectrum) of a sequ\-ence of closed linear operators $(T_n)_{n\in\mathbb{N}}$ on a Banach space $X$, and the essential approximate point spectrum (respectively, the essential defect spectrum) of a linear operator $T$ on $X$, where $(T_n)_{n\in\mathbb{N}}$ converges to $T$, in the case of convergence in generalized sense as well as in the case of the convergence compactly
Measure of noncompactness, essential approximation and defect pseudospectrum
Aymen Ammar, Aref Jeribi, Kamel Mahfoudhi
MFAT 25 (2019), no. 1, 1-11
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
MFAT 25 (2019), no. 2, 189-194
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
MFAT 25 (2019), no. 1, 12-14
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 $\{\lambda:|\lambda+p/2|\leq p/2\}$.
Space of configurations and the special measures on it
MFAT 25 (2019), no. 3, 197-204
197-204
The article is devoted to an exact account of initial results about configurations and measures on them, starting from the concept of a unique topologization of the space of all configurations, including both finite and infinite cases (not as it is made in the classical works).
On maximal multiplicity of eigenvalues of finite-dimensional spectral problem on a graph
Olga Boiko, Olga Martynyuk, Vyacheslav Pivovarchik
MFAT 25 (2019), no. 2, 104-117
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.
Analyticity and other properties of functionals $I\left(f, p\right)=\int_{A}|f(t)|^p dt$ and $n(f,p)=\left(\frac{1}{\mu(A)}\int_{A}|f(t)|^p dt\right)^{\frac{1}{p}}$ as functions of variable $p$
MFAT 25 (2019), no. 4, 339-359
339-359
We showed that for each function $f(t)$, which is not equal to zero almost everywhere in the Lebesgue measurable set, functionals $I\left(f,z\right)=\int_A{{|f(t)|}^z dt}$ as functions of a complex variable $z=p+iy$ are continuous on the domain and analytic on a set of all inner points of this domain. The functions $I(f,p)$ as functions of a real variable $ p $ are strictly convex downward and log-convex on the domain. We proved that functionals $n(f,p)$ as functions of a real variable $p$ are analytic at all inner points of the interval, in which the function $n(f,p)\neq 0$ except the point $p=0$, continuous and strictly increasing on this interval.
Köthe-Orlicz vector-valued weakly sequence spaces of difference operators
MFAT 25 (2019), no. 2, 161-176
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.
Scattering problem for Dirac system with nonlocal potentials
MFAT 25 (2019), no. 3, 211-218
211-218
For a Dirac system on the half-axis, we obtain an explicit expression for the scattering operator in terms of a nonlocal potential.
Eigenvalues and virtual levels of a family of 2×2 operator matrices
Tulkin H. Rasulov, Elyor B. Dilmurodov
MFAT 25 (2019), no. 3, 273-281
273-281
In the present paper we consider a family of $2 \times 2$ operator matrices ${\mathcal A}_\mu(k),$ $k \in {\mathbb T}^3:=(-\pi, \pi]^3,$ $\mu>0,$ associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice ${\mathbb Z}^3,$ interacting via creation and annihilation operators. We prove that there is a value $\mu_0$ of the parameter $\mu$ such that only for $\mu=\mu_0$ the operator ${\mathcal A}_\mu(\overline{0})$ has a virtual level at the point $z=0=\min\sigma_{\rm ess}({\mathcal A}_\mu(\overline{0}))$ and the operator ${\mathcal A}_\mu(\overline{\pi})$ has a virtual level at the point $z=18=\max\sigma_{\rm ess}({\mathcal A}_\mu(\overline{\pi}))$, where $\overline{0}:=(0,0,0), \overline{\pi}:=(\pi,\pi,\pi) \in {\mathbb T}^3.$ The absence of the eigenvalues of ${\mathcal A}_\mu(k)$ for all values of $k$ under the assumption that $\mu=\mu_0$ is shown. The threshold energy expansions for the Fredholm determinant associated to ${\mathcal A}_\mu(k)$ are obtained.
Characterization of Schur parameter sequences of polynomial Schur functions
Vladimir K. Dubovoy, Bernd Fritzsche, Bernd Kirstein
MFAT 25 (2019), no. 4, 289-310
289-310
A function is called a Schur function if it is holomorphic in the open unit disk and bounded by one. In the paper, the Schur parameters of polynomial Schur functions are characterized.
Three spectra problems for star graph of Stieltjes strings
MFAT 25 (2019), no. 4, 311-323
311-323
The (main) spectral problem for a star graph with three edges composed of Stieltjes strings is considered with the Dirichlet conditions at the pendant vertices. In addition we consider the Dirichlet-Neumann problem on the first edge (Problem 2) and the Dirichlet-Dirichlet problem on the union of the second and the third strings (Problem 3). It is shown that the spectrum of the main problem interlace (in a non-strict sense) with the union of spectra of Problems 2 and 3. The inverse problem lies in recovering the masses of the beads (point masses) and the lengths of the intervals between them using the spectra of the main problem and of Problems 2 and 3. Conditions on three sequences of numbers are proposed sufficient to be the spectra of the main problem and of Problems 2 and 3, respectively.
Problem of determining a multidimensional thermal memory in a heat conductivity equation
D. K. Durdiev, Zh. Zh. Zhumayev
MFAT 25 (2019), no. 3, 219-226
219-226
We consider a multidimensional integro-differential equation of heat conductivity with time-convolution integral in the right hand-side. The direct problem is represented by the Cauchy problem of determining the temperature of the medium for a known initial distribution of heat. We study the inverse problem of determining the kernel, in the integral part, that depends on time and spatial variables, if a solution of the direct problem is known on the hyperplane $x_n=0$ for $t>0.$ With a use of the resolvent of the kernel, this problem is reduced to a study of a more convenient inverse problem. The later problem is replaced with an equivalent system of integral equations with respect to the unknown functions and, using a contractive mapping, we prove that the direct and the inverse problems have unique solutions.
Yuriy Makarovich Berezansky
MFAT 25 (2019), no. 2, 97-97
97-97
Anatoly Naumovich Kochubei (to 70th birthday anniversary)
MFAT 25 (2019), no. 3, 195-196
195-196
Yuriy M. Arlinskii (to 70th birthday anniversary)
MFAT 25 (2019), no. 4, 287-288
287-288
Weak-coupling limit for ergodic environments
Martin Friesen, Yuri Kondratiev
MFAT 25 (2019), no. 2, 118-133
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.
Some results on almost Banach-Saks operators
N. Hafidi, J. H'Michane, M. Sarih
MFAT 25 (2019), no. 3, 227-235
227-235
We introduce and study a new class of operators that we call almost Banach-Saks operators. We characterize Banach lattices under which each operator is almost Banach-Saks. Furthermore, we study the relationship between this class and other classes of operators, some other interesting results are also obtained.
The Welland inequality on hypergroups
MFAT 25 (2019), no. 2, 134-141
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
MFAT 25 (2019), no. 1, 15-34
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
MFAT 25 (2019), no. 2, 142-151
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.
On the $F$-contraction properties of multivalued integral type transformations
MFAT 25 (2019), no. 3, 282-288
282-288
The main purpose of this work is to extend the properties of multivalued transformations to the integral type transformations and to obtain the existence of fixed points under $F$-contraction. In addition, the results of this study were evaluated with some interesting example.
Point spectrum in conflict dynamical systems with fractal partition
V. Koshmanenko, O. Satur, V. Voloshyna
MFAT 25 (2019), no. 4, 324-338
324-338
We discuss the spectral problem for limit distributions of conflict dynamical systems on spaces subjected to fractal divisions. Conditions ensuring the existence of the point spectrum are established in two cases, the repulsive and the attractive interactions between the opponents. A key demand is the strategy of priority in a single region.
Unbounded translation invariant operators on commutative hypergroups
Vishvesh Kumar, N. Shravan Kumar, Ritumoni Sarma
MFAT 25 (2019), no. 3, 236-247
236-247
Let $K$ be a commutative hypergroup. In this article, we study the unbounded translation invariant operators on $L^p(K),\, 1\leq p \leq \infty.$ For $p \in \{1,2\},$ we characterize translation invariant operators on $L^p(K)$ in terms of the Fourier transform. We prove an interpolation theorem for translation invariant operators on $L^p(K)$ and we also discuss the uniqueness of the closed extension of such an operator on $L^p(K)$. Finally, for $p \in \{1,2\},$ we prove that the space of all closed translation invariant operators on $L^p(K)$ forms a commutative algebra over the field of complex numbers. We also prove Wendel's theorem for densely defined closed linear operators on $L^1(K).$
The Fourier transform on 2-step Lie groups
MFAT 25 (2019), no. 3, 248-272
248-272
In this paper, we study the Fourier transform on finite dimensional $2$-step Lie groups in terms of its canonical bilinear form (CBF) and its matrix coefficients. The parameter space of these matrix coefficients $\tilde{g}$, endowed with a distance $\rho_E$ which exchanges the regularity of a function with the decay of its Fourier matrix coefficients (cf. the Riemann-Lebesgue lemma for the classical Fourier transform) is however not complete. We compute explicitly its completion $\hat{g}$; the lack of completeness appears exactly when the CBF has nonmaximal rank. We provide an example for which partial degeneracy (partial rank loss) of the canonical form occurs, as opposed to the full degeneracy at the origin. We also compute the kernel $(w,\hat{w}) \mapsto \Theta(w,\hat{w})$ of the matrix-coefficients Fourier transform, the analogue for the $2$-step groups of the classical Fourier kernel $(x,\xi) \mapsto e^{i \langle \xi, x\rangle}$.
Non-autonomous systems on Lie groups and their topological entropy
MFAT 25 (2019), no. 4, 360-372
360-372
In the present paper we introduce and study the topological entropy of non-autonomous dynamical systems and define the non-autonomous dynamical system on Lie groups and manifolds. Our main purpose is to estimate the topological entropy of the non-autonomous dynamical system on Lie groups. We show that the topological entropy of the non-autonomous dynamical system on Lie groups and induced Lie algebra are equal under topological conjugacy, and a method to estimate the topological entropy of non-autonomous systems on Lie groups is given. To illustrate our results, some examples are presented. Finally some discussions and comments about positive entropy on nil-manifold Lie groups for non-autonomous systems are presented.
A gentle introduction to James’ weak compactness theorem and beyond
MFAT 25 (2019), no. 1, 35-83
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
MFAT 25 (2019), no. 2, 152-160
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
MFAT 25 (2019), no. 2, 177-188
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.
Approximation by Fourier sums in classes of differentiable functions with high exponents of smoothness
A. S. Serdyuk, I. V. Sokolenko
MFAT 25 (2019), no. 4, 381-387
381-387
We find asymptotic equalities for the exact upper bounds of approximations by Fourier sums of Weyl-Nagy classes $W^r_{\beta,p}, 1\le p\le\infty,$ for rapidly growing exponents of smoothness $r$ $(r/n\rightarrow\infty)$ in the uniform metric. We obtain similar estimates for approximations of the classes $W^r_{\beta,1}$ in metrics of the spaces $L_p, 1\le p\le\infty$.
Boundary triples for integral systems on the half-line
MFAT 25 (2019), no. 1, 84-96
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 $\infty$.