# Authors Index, Vol. 17, 2011

### Dunford-Pettis property of the product of some operators

Belmesnaoui Aqzzouz, Othman Aboutafail, Aziz Elbour

Methods Funct. Anal. Topology **17** (2011), no. 4, 295-299

We establish a sufficient condition under which the product of an order bounded almost Dunford-Pettis operator and an order weakly compact operator is Dunford-Pettis. And we derive some consequences.

### On fine structure of singularly continuous probability measures and random variables with independent $\widetilde{Q}$-symbols

S. Albeverio, V. Koshmanenko, M. Pratsiovytyi, G. Torbin

Methods Funct. Anal. Topology **17** (2011), no. 2, 97-111

We introduce a new fine classification of singularly continuous probability measures on $R^1$ on the basis of spectral properties of such measures (topological and metric properties of the spectrum of the measure as well as local behavior of the measure on subsets of the spectrum). The theorem on the structural representation of any one-dimensional singularly continuous probability measure in the form of a convex combination of three singularly continuous probability measures of pure spectral type is proved.

We introduce into consideration and study a $\widetilde{Q}$-representation of real numbers and a family of probability measures with independent $\widetilde{Q}$-symbols. Topological, metric and fractal properties of the above mentioned probability distributions are studied in details. We also show how the methods of $\widetilde{P}-\widetilde{Q}$-measures can be effectively applied to study properties of generalized infinite Bernoulli convolutions.

### $\hat{g}$-closed sets in ideal topological spaces

J. Antony Rex Rodrigo, O. Ravi, A. Naliniramalatha

Methods Funct. Anal. Topology **17** (2011), no. 3, 274-280

Characterizations and properties of $\mathcal{I}_{\hat{g}}$-closed sets and $\mathcal{I}_{\hat{g}}$-open sets are given. A characterization of normal spaces is given in terms of $\mathcal{I}_{\hat{g}}$-open sets. Also, it is established that an $\mathcal{I}_{\hat{g}}$-closed subset of an $\mathcal{I}$-compact space is $\mathcal{I}$-compact.

### Complement on order weakly compact operators

Belmesnaoui Aqzzouz, Jawad H'mishane

Methods Funct. Anal. Topology **17** (2011), no. 2, 112-117

We generalize a result on the duality property for order weakly compact operators and use it to establish some characterizations of positive operators.

### On asymptotic behavior of the constants in generalized Khintchine's inequality

Methods Funct. Anal. Topology **17** (2011), no. 3, 244-251

We establish an asymptotic behavior of the constants in Khintchine's inequality for independent random variables of mean zero.

### A q-difference operator with discrete and simple spectrum

Miron B. Bekker, Martin J. Bohner, Hristo Voulov

Methods Funct. Anal. Topology **17** (2011), no. 4, 281-294

We continue our study of a $q$-difference version of a second-order differential operator which depends on a real parameter. This version was introduced in our previous article. For values of the parameter for which the difference operator is self adjoint, we show that the spectrum of the operator is discrete and simple. When $q$ approaches $1$, the spectrum fills the whole positive or negative semiaxis.

### On generalized selfadjoint operators on scales of Hilbert spaces

Yu. M. Berezansky, J. Brasche, L. P. Nizhnik

Methods Funct. Anal. Topology **17** (2011), no. 3, 193-198

We consider examples of generalized selfadjoint operators that act from a positive Hilbert space to a negative space. Such operators were introduced and studied in [1]. We give examples of selfadjoint operators on the principal Hilbert space $H_ 0$ that, being considered as operators from the positive space $H_ + \subset H_ 0$ into the negative space $H_ - \supset H_ 0$, are not essentially selfadjoint in the generalized sense.

### On the group of Lie-orthogonal operators on a Lie algebra

S. V. Bilun, D. V. Maksimenko, A. P. Petravchuk

Methods Funct. Anal. Topology **17** (2011), no. 3, 199-203

Finite dimensional Lie algebras over the field of complex numbers with a linear operator $T: L\to L$ such that $[T(x),T(y)]=[x,y]$ for all $x,y\in L$ are studied. The group of such non-degenerative linear operators on $L$ is considered. Some properties of this group and its relations with the group $\operatorname{Aut}(L)$ in the general linear group $GL(L)$ are described.

### On infinitesimal structure of a hypergroup that originates from a Lie group

A. A. Kalyuzhnyi, G. B. Podkolzin, Yu. A. Chapovsky

Methods Funct. Anal. Topology **17** (2011), no. 4, 319-329

We describe an infinitesimal algebra to a hypergroup constructed from a Lie group and a conditional expectation. We also prove a theorem on a decomposition of the conditional expectation into the product of a counital conditional expectation and the one that arises in the double coset construction.

### Positive operators on the Bergman space and Berezin transform

Methods Funct. Anal. Topology **17** (2011), no. 3, 204-210

Let $\mathbb{D}=\{{z\in\mathbb{C}:|z|<1}\}$ and $L^2_a(\mathbb{D})$ be the Bergman space of the disk. In thispaper we characterize the class $\mathcal{A}\subset L^\infty(\mathbb{D})$ such that if $\phi,\psi\in\mathcal{A},\alpha\geq 0$ and $0\leq\phi\leq\alpha\psi$ then there exist positive operators $S,T\in\mathcal{L}(L^2_a(\mathbb{D}))$ such that $\phi(z)=\widetilde{S}(z)\leq\alpha\widetilde{T}(z)=\alpha\psi(z)$ for all $z\in\mathbb{D}$. Further, we have shown that if $S$ and $T$ are two positive operators in $\mathcal{L}(L^2_a(\mathbb{D}))$ and $T$ is invertible then there exists a constant $\alpha\geq0$ such that $\widetilde{S}(z)\leq\alpha\widetilde{T}(z)$ for all $z\in\mathbb{D}$ and $\widetilde{S},\widetilde{T}\in\mathcal{A}$. Here $\mathcal{L}(L^2_a(\mathbb{D}))$ is the space of all boundedlinear operators from $L^2_a(\mathbb{D})$ into $L^2_a(\mathbb{D})$ and $\widetilde{A}(z)=\langle Ak_z,k_z\rangle$ is the Berezintrans form of $A\in\mathcal{L}(L^2_a(\mathbb{D}))$ and $k_z$ is thenormalized reproducing kernel of $L^2_a(\mathbb{D})$. Applications of these results are also obtained.

### On $(\bigwedge, \mu)$-closed sets in generalized topological spaces

Methods Funct. Anal. Topology **17** (2011), no. 2, 174-179

In this paper, we introduce and study $(\bigwedge ,\mu )$-open sets and $(\bigwedge ,\mu )$-closed sets via $\mu $-open and $\mu $-closed sets in generalized topological spaces. Moreover, we introduce some generalized separation axioms in generalized topological spaces.

### On equiangular configurations of subspaces of a Hilbert space

Yu. S. Samoilenko, Yulia Yu. Yershova

Methods Funct. Anal. Topology **17** (2011), no. 1, 84-96

In this paper, we find $\tau$, $0<\tau<1$, such that there exists an equiangular $(\Gamma, \tau)$-configuration of one-dimensional subspaces, and describe $(\Gamma, \tau)$-configurations that correspond to unicyclic graphs and to some graphs that have cyclomatic number satisfying $\nu(\Gamma) \geq 2$.

### Boundary problems and initial-boundary value problems for one class of nonlinear parabolic equations with Lévy Laplacian

Methods Funct. Anal. Topology **17** (2011), no. 2, 118-125

We develop a method to construct a solution to a boundary problem and an initial-boundary value problem in a fundamental domain of a Hilbert space for a class of nonlinear parabolic equations not containing explicitly the unknown function, $$\frac{\partial U(t,x)}{\partial t}=f(t,\Delta_LU(t,x)),$$ where $\Delta _L$ is the infinite dimensional Lévy Laplacian.

### Functional evolutions for homogeneous stationary death-immigration spatial dynamics

Methods Funct. Anal. Topology **17** (2011), no. 4, 300-318

We discover death-immigration non-equilibrium stochastic dynamics in the continuum also known as the Surgailis process. Explicit expression for the correlation functions is presented. Dynamics of states and their generating functionals are studied. Ergodic properties for the evolutions are considered.

### Multi-dimensional Schrödinger operators with point interactions

Methods Funct. Anal. Topology **17** (2011), no. 2, 126-143

We study two- and three-dimensional matrix Schrödinger operators with $m\in \Bbb N$ point interactions. Using the technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results obtained by the other authors in this field. For instance, we parametrize all self-adjoint extensions of the initial minimal symmetric Schrödinger operator by abstract boundary conditions and characterize their spectra. Particularly, we find a sufficient condition in terms of distances and intensities for the self-adjoint extension $H_{\alpha,X}^{(3)}$ to have $m'$ negative eigenvalues, i.e., $\kappa_-(H_{\alpha,X}^{(3)})=m'\le m$. We also give an explicit description of self-adjoint nonnegative extensions.

### Existence and uniqueness of mild solutions of second order semilinear differential equations in Banach space

Methods Funct. Anal. Topology **17** (2011), no. 1, 1-9

We consider the Cauchy problem for second order semilinear differential equations in Banach space. Sufficient conditions of local and global existence and uniqueness of mild solutions are presented.

### On one class of nonselfadjoint operators with a discrete spectrum

G. M. Gubreev, M. G. Volkova, A. A. Tarasenko

Methods Funct. Anal. Topology **17** (2011), no. 3, 211-218

In this work completely continious nondissipative operators with two-dimensional imaginary parts, acting in separable Hilbert space are studied. The criteria of completeness and unconditional basis property of root vectors of such operators are obtained. The results are formulated in terms of characteristic matrix-valued functions of nonselfadjoint operators and proved using analysis of functional models in de Branges spaces.

### Unconditional bases of de Branges spaces and interpolation problems corresponding to them

Methods Funct. Anal. Topology **17** (2011), no. 2, 144-149

In this paper the unconditional bases of de Branges spaces are constructed from the values of reproducing kernels. Appropriate problems of interpolation by entire functions are also considered. The paper is a continuation of papers [2, 3].

### G-Frames and operator valued-frames in Hilbert C*-modules

Sedighe Hosseini, Amir Khosravi

Methods Funct. Anal. Topology **17** (2011), no. 1, 10-19

g-frames and fusion frames in Hilbert C*-modules have been defined by the second author and B.~Khosravi in [15] and operator-valued frames in Hilbert C*-modules have been defined by Kaftal et al in [11]. We show that every operator-valued frame is a g-frame, we also show that in Hilbert C*-modules tensor product of orthonormal basis is an orthonormal basis and tensor product of g-frames is a g-frame, we get some relations between their g-frame operators, and we study tensor product of operator-valued frames in Hilbert C*-modules.

### Notes on Wick calculus on parametrized test functions spaces of Meixner white noise

Methods Funct. Anal. Topology **17** (2011), no. 2, 150-167

Using a general approach that covers the cases of Gaussian, Poissonian, Gamma, Pascal and Meixner measures, we construct elements of a Wick calculus on parametrized Kondratiev-type spaces of test functions; consider the interconnection between the extended stochastic integration and the Wick calculus; and give an example of a stochastic equation with a Wick-type nonlinearity. The main results consist in studying properties of a Wick product and Wick versions of holomorphic functions on the parametrized Kondratiev-type spaces of test functions. These results are necessary, in particular, in order to describe properties of solutions of stochastic equations with Wick type nonlinearities in the "Meixner analysis".

### Non-Archimedean unitary operators

Methods Funct. Anal. Topology **17** (2011), no. 3, 219-224

We describe a subclass of the class of normal operators on Banach spaces over non-Archimedean fields (A. N. Kochubei, J. Math. Phys. 51 (2010), article 023526) consisting of operators whose properties resemble those of unitary operators. In particular, an analog of Stone's theorem about one-parameter groups of unitary operators is proved.

### The infinite direct products of probability measures and structural similarity

Methods Funct. Anal. Topology **17** (2011), no. 1, 20-28

We show that any similar structure measure on the segment $[0,1]$ is an image-measure of the appropriate constructed infinite direct product of discrete probability measures.

### A note on equilibrium Glauber and Kawasaki dynamics for permanental point processes

Methods Funct. Anal. Topology **17** (2011), no. 1, 29-46

We construct two types of equilibrium dynamics of an infinite particle system in a locally compact metric space $X$ for which a permanental point process is a symmetrizing, and hence invariant measure. The Glauber dynamics is a birth-and-death process in $X$, while in the Kawasaki dynamics interacting particles randomly hop over $X$. In the case $X=\mathbb R^d$, we consider a diffusion approximation for the Kawasaki dynamics at the level of Dirichlet forms. This leads us to an equilibrium dynamics of interacting Brownian particles for which a permanental point process is a symmetrizing measure.

### Hardy type spaces on infinite dimensional group orbits

O. V. Lopushansky, M. V. Oleksienko

Methods Funct. Anal. Topology **17** (2011), no. 3, 225-234

In Hilbert Hardy spaces of complex analytic functions with infinitely many variables, defined on unitary orbits of locally compact second countable group, the Cauchy type integral formulas are established. Existence of radial boundary values is proved. Results are illustrated for a reduced Heisenberg group.

### Hill's potentials in Hörmander spaces and their spectral gaps

V. A. Mikhailets, V. M. Molyboga

Methods Funct. Anal. Topology **17** (2011), no. 3, 235-243

The paper deals with the Hill-Schrödinger operators with singular periodic potentials in the space $H^{\omega}(\mathbb{T})\subset H^{-1}(\mathbb{T})$. The authors exactly describe the classes of sequences being the lengths of spectral gaps of these operators. The functions $\omega$ may be nonmonotonic. The space $H^{\omega}(\mathbb{T})$ coincides with the Hörmander space $H_{2}^{\omega}(\mathbb{T})$ with the weight function $\omega(\sqrt{1+\xi^{2}})$ if $\omega$ is in the Avakumovich class $\mathrm{OR}$.

### General forms of the Menshov-Rademacher, Orlicz, and Tandori theorems on orthogonal series

Vladimir A. Mikhailets, Aleksandr A. Murach

Methods Funct. Anal. Topology **17** (2011), no. 4, 330-340

We prove that the classical Menshov--Rademacher, Orlicz, and Tandori theorems remain true for orthogonal series given in the direct integrals of measurable collections of Hilbert spaces. In particular, these theorems are true for the spaces $L_{2}(X,d\mu;H)$ of vector-valued functions, where $(X,\mu)$ is an arbitrary measure space, and $H$ is a real or complex Hilbert space of an arbitrary dimension.

### The Faddeev equation and essential spectrum of a Hamiltonian in Fock space

Mukhiddin I. Muminov, Tulkin H. Rasulov

Methods Funct. Anal. Topology **17** (2011), no. 1, 47-57

A Hamiltonian (model operator) $H$ associated to a quantum system describing three particles in interaction, without conservation of the number of particles, is considered. The Faddeev type system of equations for eigenvectors of $H$ is constructed. The essential spectrum of $H$ is described by the spectrum of the channel operator.

### On generalization of the Freudenthal's theorem for compact irreducible standard polyhedric representation for superparacompact complete metrizable spaces

Methods Funct. Anal. Topology **17** (2011), no. 1, 58-64

In this paper for superparacompact complete metrizable spaces, the Freudenthal's theorem for compact irreducible standard polyhedral representation is ge e alized. Furthermore, for superparacompact metric spaces the following is strengthened: 1) the Morita's theorem about universality of the product $Q^\infty\times B(\tau)$ of Hilbert cube $Q^\infty$ to generalized Baire space $B(\tau)$ of the weight $\tau$ in the space of all strongly metrizable spaces of weight $\le \tau$; 2) Nagata's theorem about universality of the product $\Phi^n\times B(\tau)$ of the universal $n$-dimensional compact $\Phi^n$ to $B(\tau)$ in the space of all strongly metrizable spaces $\le\tau$ and dimension $\operatorname{dim}X\le n.$

### Some class of real sequences having indefinite Hankel forms

Luis J. Navarro, Vladimir Strauss

Methods Funct. Anal. Topology **17** (2011), no. 1, 65-74

In this paper we generalize the results given in [14] about real sequences which are not necessarily positive (i.e, they are not sequences of power moments) but can be mapped, by a difference operator, into a power moment sequence. We prove by elementary methods that the integro-polynomial representation of such sequences remains after dropping the condition on its growth imposed in the mentioned article. Some additional results on the uniqueness of the representation are included.

### Elimination of Jacobi equation in extremal variational problems

Methods Funct. Anal. Topology **17** (2011), no. 4, 341-349

It is shown that the extremal problem for the one--dimensional Euler--Lagrange variational functional in ${C^1[a;b]}$ under a strengthened Legendre condition can be solved without using the Jacobi equation. In this case, exactly one of the two possible cases requires a restriction to the length of $[a;b]$, defined only by the form of the integrand. The result is extended to the case of compact extremum in ${H^1[a;b]}$.

### On *-representations of a class of algebras with polynomial growth related to Coxeter graphs

Methods Funct. Anal. Topology **17** (2011), no. 3, 252-273

For a Hilbert space $H$, we study configurations of its subspaces related to Coxeter graphs $\mathbb{G}_{s_1,s_2}$, $s_1,s_2\in\{4,5\}$, which are arbitrary trees such that one edge has type~$s_1$, another one has type~$s_2$ and the rest are of type~$3$. We prove that such irreducible configurations exist only in a finite dimensional $H$, where the dimension of $H$ does not exceed the number of vertices of the graph by more than twice. We give a description of all irreducible nonequivalent configurations; they are indexed with a continuous parameter. As an example, we study irreducible configurations related to a graph that consists of three vertices and two edges of type $s_1$ and $s_2$.

### On $C^*$-algebra generated by Fock representation of Wick algebra with braided coefficients

Methods Funct. Anal. Topology **17** (2011), no. 2, 168-173

We consider $C^*$-algebras $\mathcal{W}(T)$ generated by operators of Fock representations of Wick $*$-algebras with a braided coefficient operator $T$. It is shown that for any braided $T$ with $||T||<1$ one has the inclusion $\mathcal{W}(0)\subset\mathcal{W}(T)$. Conditions for existence of an isomorphism $\mathcal{W}(T)\simeq\mathcal{W}(0)$ are discussed.

### Strong base for fuzzy topology

Methods Funct. Anal. Topology **17** (2011), no. 4, 350-355

It is known that a base for a traditional topology, or for a $L$-topology, $\tau$, is a subset ${\mathcal B}$ of $\tau$ with the property that every element $G\in \tau$ can be written as a union of elements of ${\mathcal B}$. In the classical case it is equivalent to say that $G\in \tau$ if and only if for any $x\in G$ we have $B\in {\mathcal B}$ satisfying $x\in B \subseteq G$. This latter property is taken as the foundation for a notion of strong base for a $L$-topology. Characteristic properties of a strong base are given and among other results it is shown that a strong base is a base, but not conversely.

### One generalization of the classical moment problem

Methods Funct. Anal. Topology **17** (2011), no. 4, 356-380

Let $\ast_P$ be a product on $l_{fin}$ (a space of all finite sequences) associated with a fixed family $(P_n)_{n=0}^{\infty}$ of real polynomials on $\mathbb{R}$. In this article, using methods from the theory of generalized eigenvector expansion, we investigate moment-type properties of $\ast_P$-positive functionals on $l_{fin}.$ If $(P_n)_{n=0}^{\infty}$ is a family of the Newton polynomials $P_n(x)=\prod_{i=0}^{n-1}(x-i)$ then the corresponding product $\star=\ast_P$ is an analog of the so-called Kondratiev--Kuna convolution on a "Fock space". We get an explicit expression for the product $\star$ and establish a connection between $\star$-positive functionals on $l_{fin}$ and a one-dimensional analog of the Bogoliubov generating functionals (the classical Bogoliubov functionals are defined correlation functions for statistical mechanics systems).

### Polarization formula for $(p,q)$-polynomials on a complex normed space

T. V. Vasylyshyn, A. V. Zagorodnyuk

Methods Funct. Anal. Topology **17** (2011), no. 1, 75-83

The aim of this paper to give some analogues of polarization formulas and the polarization inequality for $(p,q)$-polynomials between complex normed spaces. Obtained results are useful for investigation of real-differentiable mappings on complex spaces.

### Factor representations of infinite semi-direct products

Methods Funct. Anal. Topology **17** (2011), no. 2, 180-192

In this article, we propose a new method to study unitary representations of inductive limits of locally compact groups. For the group of infinite upper triangular matrices, we construct a family of type III factorial representations. These results are complements to previous results of A. V. Kosyak, and Albeverio and Kosyak [1, 5].