Authors Index,

Elliptic boundary-value problems in Hörmander spaces

Anna Anop, Tetiana Kasirenko

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 4, 295-310

We investigate general elliptic boundary-value problems in Hörmander inner product spaces that form the extended Sobolev scale. The latter consists of all Hilbert spaces that are interpolation spaces with respect to the Sobolev Hilbert scale. We prove that the operator corresponding to an arbitrary elliptic problem is Fredholm in appropriate couples of the Hörmander spaces and induces a collection of isomorphisms on the extended Sobolev scale. We obtain a local a priory estimate for generalized solutions to this problem and prove a theorem on their local regularity in the Hörmander spaces. We find new sufficient conditions under which generalized derivatives (of a given order) of the solutions are continuous.

Generalized solutions of Riccati equalities and inequalities

D. Z. Arov, M. A. Kaashoek, D. R. Pik

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 2, 95-116

The Riccati inequality and equality are studied for infinite dimensional linear discrete time stationary systems with respect to the scattering supply rate. The results obtained are an addition to and based on our earlier work on the Kalman-Yakubovich-Popov inequality in [6]. The main theorems are closely related to the results of Yu. M. Arlinskiĭ in [3]. The main difference is that we do not assume the original system to be a passive scattering system, and we allow the solutions of the Riccati inequality and equality to satisfy weaker conditions.

Poisson measure as a spectral measure of a family of commuting selfadjoint operators, connected with some moment problem

Yu. M. Berezansky

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 4, 311-329

It is proved that the Poisson measure is a spectral measure of some family of commuting selfadjoint operators acting on a space constructed from some generalization of the moment problem.

The investigation of Bogoliubov functionals by operator methods of moment problem

Yu. M. Berezansky, V. A. Tesko

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 1, 1-47

The article is devoted to a study of Bogoliubov functionals by using methods of the operator spectral theory being applied to the classical power moment problem. Some results, similar to corresponding ones for the moment problem, where obtained for such functionals. In particular, the following question was studied: under what conditions a sequence of nonlinear functionals is a sequence of Bogoliubov functionals.

On a generalization of the three spectral inverse problem

O. P. Boyko, O. M. Martynyuk, V. N. Pivovarchik

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 1, 74-80

We consider a generalization of the three spectral inverse problem, that is, for given spectrum of the Dirichlet-Dirichlet problem (the Sturm-Liouville problem with Dirichlet conditions at both ends) on the whole interval $[0,a]$, parts of spectra of the Dirichlet-Neumann and Dirichlet-Dirichlet problems on $[0,a/2]$ and parts of spectra of the Dirichlet-Newman and Dirichlet-Dirichlet problems on $[a/2,a]$, we find the potential of the Sturm-Liouville equation.

On nonsymmetric rank one singular perturbations of selfadjoint operators

Mykola Dudkin, Tetiana Vdovenko

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 2, 137-151

We consider nonsymmetric rank one singular perturbations of a selfadjoint operator, i.e., an expression of the form $\tilde A=A+\alpha\left\langle\cdot,\omega_1\right\rangle\omega_2$, $\omega_1\not=\omega_2$, $\alpha\in{\mathbb C}$, in a general case $\omega_1,\omega_2\in{\mathcal H}_{-2}$. Using a constructive description of the perturbed operator $\tilde A$, we investigate some spectral and approximations properties of $\tilde A$. The wave operators corresponding to the couple $A$, $\tilde A$ and a series of examples are also presented.

L-Dunford-Pettis property in Banach spaces

A. Retbi, B. El Wahbi

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 4, 387-392

In this paper, we introduce and study the concept of L-Dunford-Pettis sets and L-Dunford-Pettis property in Banach spaces. Next, we give a characterization of the L-Dunford-Pettis property with respect to some well-known geometric properties of Banach spaces. Finally, some complementability of operators on Banach spaces with the L-Dunford-Pettis property are also investigated.

Some results on order bounded almost weak Dunford-Pettis operators

Nabil Machrafi, Aziz Elbour, Mohammed Moussa

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 3, 256-265

We give some new characterizations of almost weak Dunford-Pettis operators and we investigate their relationship with weak Dunford-Pettis operators.

Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbb{R}^n$

M. Faierman

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 4, 330-345

We consider a spectral problem over $\mathbb{R}^n$ for a Douglis-Nirenberg system of differential operators under limited smoothness assumptions and under the assumption of parameter-ellipticity in a closed sector $\mathcal{L}$ in the complex plane with vertex at the origin. We pose the problem in an $L_p$ Sobolev-Bessel potential space setting, $1 < p < \infty$, and denote by $A_p$ the operator induced in this setting by the spectral problem. We then derive results pertaining to the Fredholm theory for $A_p$ for values of the spectral parameter $\lambda$ lying in $\mathcal{L}$ as well as results pertaining to the invariance of the Fredholm domain of $A_p$ with $p$.

Actions of finite groups and smooth functions on surfaces

Bohdan Feshchenko

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 3, 210-219

Let $f:M\to \mathbb{R}$ be a Morse function on a smooth closed surface, $V$ be a connected component of some critical level of $f$, and $\mathcal{E}_V$ be its atom. Let also $\mathcal{S}(f)$ be a stabilizer of the function $f$ under the right action of the group of diffeomorphisms $\mathrm{Diff}(M)$ on the space of smooth functions on $M,$ and $\mathcal{S}_V(f) = \{h\in\mathcal{S}(f)\,| h(V) = V\}.$ The group $\mathcal{S}_V(f)$ acts on the set $\pi_0\partial \mathcal{E}_V$ of connected components of the boundary of $\mathcal{E}_V.$ Therefore we have a homomorphism $\phi:\mathcal{S}(f)\to \mathrm{Aut}(\pi_0\partial \mathcal{E}_V)$. Let also $G = \phi(\mathcal{S}(f))$ be the image of $\mathcal{S}(f)$ in $\mathrm{Aut}(\pi_0\partial \mathcal{E}_V).$ Suppose that the inclusion $\partial \mathcal{E}_V\subset M\setminus V$ induces a bijection $\pi_0 \partial \mathcal{E}_V\to\pi_0(M\setminus V).$ Let $H$ be a subgroup of $G.$ We present a sufficient condition for existence of a section $s:H\to \mathcal{S}_V(f)$ of the homomorphism $\phi,$ so, the action of $H$ on $\partial \mathcal{E}_V$ lifts to the $H$-action on $M$ by $f$-preserving diffeomorphisms of $M$. This result holds for a larger class of smooth functions $f:M\to \mathbb{R}$ having the following property: for each critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to \mathbb{R}$ without multiple linear factors.

Non-autonomous interacting particle systems in continuum

Martin Friesen

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 3, 220-244

A conservative Feller evolution on continuous bounded functions is constructed from a weakly continuous, time-inhomogeneous transition function describing a pure jump process on a locally compact Polish space. The transition function is assumed to satisfy a Foster-Lyapunov type condition. The results are applied to interacting particle systems in continuum, in particular to general birth-and-death processes (including jumps). Particular examples such as the BDLP and Dieckmann-Law model are considered in the end.

Evolution of states and mesoscopic scaling for two-component birth-and-death dynamics in continuum

Martin Friesen, Oleksandr Kutoviy

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 4, 346-374

Two coupled spatial birth-and-death Markov evolutions on $\mathbb{R}^d$ are obtained as unique weak solutions to the associated Fokker-Planck equations. Such solutions are constructed by its associated sequence of correlation functions satisfying the so-called Ruelle-bound. Using the general scheme of Vlasov scaling we are able to derive a system of non-linear, non-local mesoscopic equations describing the effective density of the particle system. The results are applied to several models of ecology and biology.

Inverse moment problem for non-Abelian Coxeter double Bruhat cells

Michael Gekhtman

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 2, 117-136

We solve the inverse problem for non-Abelian Coxeter double Bruhat cells in terms of the matrix Weyl functions. This result can be used to establish complete integrability of the non-Abelian version of nonlinear Coxeter-Toda lattices in $GL_n$.

On approximation of solutions of operator-differential equations with their entire solutions of exponential type

V. M. Gorbachuk

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 3, 245-255

We consider an equation of the form $y'(t) + Ay(t) = 0, \ t \in [0, \infty)$, where $A$ is a nonnegative self-adjoint operator in a Hilbert space. We give direct and inverse theorems on approximation of solutions of this equation with its entire solutions of exponential type. This establishes a one-to-one correspondence between the order of convergence to $0$ of the best approximation of a solution and its smoothness degree. The results are illustrated with an example, where the operator $A$ is generated by a second order elliptic differential expression in the space $L_{2}(\Omega)$ (the domain $\Omega \subset \mathbb{R}^{n}$ is bounded with smooth boundary) and a certain boundary condition.

Hilbert space hypocoercivity for the Langevin dynamics revisited

Martin Grothaus, Patrik Stilgenbauer

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 2, 152-168

We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics via studying the longtime behavior of the strongly continuous contraction semigroup solving the associated Kolmogorov (backward) equation as an abstract Cauchy problem. This hypocoercivity result is proven in previous works before by Dolbeault, Mouhot and Schmeiser in the corresponding dual Fokker-Planck framework, but without including domain issues of the appearing operators. In our elaboration, we include the domain issues and additionally compute the rate of convergence in dependence of the damping coefficient. Important statements for the complete elaboration are the m-dissipativity results for the Langevin operator established by Conrad and the first named author of this article as well as the essential selfadjointness results for generalized Schrödinger operators by Wielens or Bogachev, Krylov and Röckner. We emphasize that the chosen Kolmogorov approach is natural. Indeed, techniques from the theory of (generalized) Dirichlet forms imply a stochastic representation of the Langevin semigroup as the transition kernel of diffusion process which provides a martingale solution to the Langevin equation. Hence an interesting connection between the theory of hypocoercivity and the theory of (generalized) Dirichlet forms is established besides.

Level sets of asymptotic mean of digits function for $4$-adic representation of real number

M. V. Pratsiovytyi, S. O. Klymchuk, O. P. Makarchuk

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 2, 184-196

We study topological, metric and fractal properties of the level sets $$S_{\theta}=\{x:r(x)=\theta\}$$ of the function $r$ of asymptotic mean of digits of a number $x\in[0;1]$ in its $4$-adic representation, $$r(x)=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits^{n}_{i=1}\alpha_i(x)$$ if the asymptotic frequency $\nu_j(x)$ of at least one digit does not exist, were $$ \nu_j(x)=\lim_{n\to\infty}n^{-1}\#\{k: \alpha_k(x)=j, k\leqslant n\}, \:\: j=0,1,2,3. $$

Fractional statistical dynamics and fractional kinetics

José Luís da Silva, Anatoly N. Kochubei, Yuri Kondratiev

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 3, 197-209

We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the fractional kinetics in terms of a non-linear Vlasov-type kinetic equation. As an application we study the intermittency of the fractional mesoscopic dynamics.

Dynamical systems of conflict in terms of structural measures

Volodymyr Koshmanenko, Inga Verygina

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 1, 81-93

We investigate the dynamical systems modeling conflict processes between a pair of opponents. We assume that opponents are given on a common space by distributions (probability measures) having the similar or self-similar structure. Our main result states the existence of the controlled conflict in which one of the opponents occupies almost whole conflicting space. Besides, we compare conflicting effects stipulated by the rough structural approximation under controlled redistributions of starting measures.

Foliations with all non-closed leaves on non-compact surfaces

Sergiy Maksymenko, Eugene Polulyakh

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 3, 266-282

Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $\Delta$ be a foliation on $X$ such that each leaf $\omega\in\Delta$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that the plane splits into a family of open strips foliated by parallel lines and glued along some boundary intervals. However W. Kaplan's construction depends on a choice of those intervals, and a foliation is described in a non-unique way. We propose a canonical cutting by open strips which gives a uniqueness of classifying invariant. We also describe topological types of closures of those strips under additional assumptions on $\Delta$.

On the generation of Beurling type Carleman ultradifferentiable $C_0$-semigroups by scalar type spectral operators

Marat V. Markin

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 2, 169-183

A characterization of the scalar type spectral generators of Beurling type Carleman ultradifferentiable $C_0$-semigroups is established, the important case of the Gevrey ultradifferentiability is considered in detail, the implementation of the general criterion corresponding to a certain rapidly growing defining sequence is observed.

A criterion for continuity in a parameter of solutions to generic boundary-value problems for higher-order differential systems

Vladimir Mikhailets, Aleksandr Murach, Vitalii Soldatov

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 4, 375-386

We consider the most general class of linear boundary-value problems for ordinary differential systems, of order $r\geq1$, whose solutions belong to the complex space $C^{(n+r)}$, with $0\leq n\in\mathbb{Z}.$ The boundary conditions can contain derivatives of order $l$, with $r\leq l\leq n+r$, of the solutions. We obtain a constructive criterion under which the solutions to these problems are continuous with respect to the parameter in the normed space $C^{(n+r)}$. We also obtain a two-sided estimate for the degree of convergence of these solutions.

On the finiteness of the discrete spectrum of a 3x3 operator matrix

Tulkin H. Rasulov

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 1, 48-61

An operator matrix $H$ associated with a lattice system describing three particles in interactions, without conservation of the number of particles, is considered. The structure of the essential spectrum of $H$ is described by the spectra of two families of the generalized Friedrichs models. A symmetric version of the Weinberg equation for eigenvectors of $H$ is obtained. The conditions which guarantee the finiteness of the number of discrete eigenvalues located below the bottom of the three-particle branch of the essential spectrum of $H$ is found.

Joint functional calculus in algebra of polynomial tempered distributions

S. V. Sharyn

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 1, 62-73

In this paper we develop a functional calculus for a countable system of generators of contraction strongly continuous semigroups. As a symbol class of such calculus we use the algebra of polynomial tempered distributions. We prove a differential property of constructed calculus and describe its image with the help of the commutant of polynomial shift semigroup. As an application, we consider a function of countable set of second derivative operators.

Homeotopy groups of rooted tree like non-singular foliations on the plane

Yu. Yu. Soroka

↓ Abstract   |   Article (.pdf)

Methods Funct. Anal. Topology 22 (2016), no. 3, 283-294

Let $F$ be a non-singular foliation on the plane with all leaves being closed subsets, $H^{+}(F)$ be the group of homeomorphisms of the plane which maps leaves onto leaves endowed with compact open topology, and $H^{+}_{0}(F)$ be the identity path component of $H^{+}(F)$. The quotient $\pi_0 H^{+}(F) = H^{+}(F)/H^{+}_{0}(F)$ is an analogue of a mapping class group for foliated homeomorphisms. We will describe the algebraic structure of $\pi_0 H^{+}(F)$ under an assumption that the corresponding space of leaves of $F$ has a structure similar to a rooted tree of finite diameter.

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