# Authors Index, Vol. 13, 2007

### $pg$-frame in Banach spaces

M. R. Abdollahpour, M. H. Faroughi, A. Rahimi

Methods Funct. Anal. Topology **13** (2007), no. 3, 201-210

For extending the concepts of $p$-frame, frame for Banach spaces and atomic decomposition, we will define the concept of $pg$-frame and $g$-frame for Banach spaces, by which each $f\in X$ ($X$ is a Banach space) can be represented by an unconditionally convergent series $f=\sum g_{i}\Lambda_{i},$ where $\{\Lambda_{i}\}_{i\in J}$ is a $pg$-frame, $\{g_{i}\}\in(\sum\oplus Y_{i}^{*})_{l_q}$ and $\frac{1}{p}+\frac{1}{q}=1$. In fact, a $pg$-frame $\{\Lambda_{i}\}$ is a kind of an overcomplete basis for $X^{*}.$ We also show that every separable Banach space $X$ has a $g$-Banach frame with bounds equal to $1.$

### Non-negative perturbations of non-negative self-adjoint operators

Methods Funct. Anal. Topology **13** (2007), no. 2, 103-109

Let $A$ be a non-negative self-adjoint operator in a Hilbert space $\mathcal{H}$ and $A_{0}$ be some densely defined closed restriction of $A_{0}$, $A_{0}\subseteq A eq A_{0}$. It is of interest to know whether $A$ is the unique non-negative self-adjoint extensions of $A_{0}$ in $\mathcal{H}$. We give a natural criterion that this is the case and if it fails, we describe all non-negative extensions of $A_{0}$. The obtained results are applied to investigation of non-negative singular point perturbations of the Laplace and poly-harmonic operators in $\mathbb{L}_{2}(\mathbf{R}_{n})$.

### Inverse spectral problems for coupled oscillating systems: reconstruction from three spectra

S. Albeverio, R. Hryniv, Ya. Mykytyuk

Methods Funct. Anal. Topology **13** (2007), no. 2, 110-123

We study an inverse spectral problem for a compound oscillating system consisting of a singular string and $N$~masses joined by springs. The operator $A$ corresponding to this system acts in $L_2(0,1)\times C^N$ and is composed of a Sturm--Liouville operator in $L_2(0,1)$ with a distributional potential and a Jacobi matrix in~$C^N$ that are coupled in a special way. We solve the problem of reconstructing the system from three spectra---namely, from the spectrum of $A$ and the spectra of its decoupled parts. A complete description of possible spectra is given.

### The Efimov effect for a model operator associated with the Hamiltonian of a non conserved number of particles

Sergio Albeverio, Saidakhmat N. Lakaev, Tulkin H. Rasulov

Methods Funct. Anal. Topology **13** (2007), no. 1, 1-16

A model operator associated with the energy operator of a system of three non conserved number of particles is considered. The essential spectrum of the operator is described by the spectrum of a family of the generalized Friedrichs model. It is shown that there are infinitely many eigenvalues lying below the bottom of the essential spectrum, if a generalized Friedrichs model has a zero energy resonance.

### The $\varepsilon_{\infty}$-product of a $b$-space by a quotient bornological space

Methods Funct. Anal. Topology **13** (2007), no. 3, 211-222

We define the $\varepsilon_{\infty }$-product of a Banach space $G$\ by a quotient bornological space $E\mid F$ that we denote by $G\varepsilon _{\infty }(E\mid F)$, and we prove that $G$ is an $% \mathcal{L}_{\infty }$-space if and only if the quotient bornological spaces $G\varepsilon _{\infty }(E\mid F)$ and $% (G\varepsilon E)\mid (G\varepsilon F)$ are isomorphic. Also, we show that the functor $\mathbf{.\varepsilon }_{\infty }\mathbf{.}:\mathbf{Ban\times qBan\longrightarrow qBan}$ is left exact. Finally, we define the $\varepsilon _{\infty }$-product of a b-space by a quotient bornological space and we prove that if $G$ is an $% \varepsilon $b-space\ and $E\mid F$ is a quotient bornological space, then $(G\varepsilon E)\mid (G\varepsilon F)$ is isomorphic to $G\varepsilon _{\infty }(E\mid F)$.

### On non-densely defined invariant Hermitian contractions

Methods Funct. Anal. Topology **13** (2007), no. 3, 223-235

We consider a non-densely defined Hermitian contractive operator which is unitarily equivalent to its linear-fractional transformation. We show that such an operator always admits self-adjoint extensions which are also unitarily equivalent to their linear-fractional transformation.

### The investigation of a generalized moment problem associated with correlation measures

Yu. M. Berezansky, D. A. Mierzejewski

Methods Funct. Anal. Topology **13** (2007), no. 2, 124-151

The classical power moment problem can be viewed as a theory of spectral representations of a positive functional on some classical commutative algebra with involution. We generalize this approach to the case where the algebra is a special commutative algebra of functions on the space of multiple finite configurations.

If the above-mentioned functional is generated by a measure on the space of usual finite configurations then this measure is a correlation measure for a probability spectral measure on the space of infinite configurations. The latter measure is practically arbitrary, so that we have a connection between this complicated measure and its correlation measure defined on more simple objects that are finite configurations. The paper gives an answer to the following question: when this latter measure is a correlation measure for a complicated measure on infinite configurations? (Such measures are essential objects of statistical mechanics).

### Generalized Krein algebras and asymptotics of Toeplitz determinants

A. Böttcher, A. Karlovich, B. Silbermann

Methods Funct. Anal. Topology **13** (2007), no. 3, 236-261

We give a survey on generalized Krein algebras $K_{p,q}^{\alpha,\beta}$ and their applications to Toeplitz determinants. Our methods originated in a paper by Mark Krein of 1966, where he showed that $K_{2,2}^{1/2,1/2}$ is a Banach algebra. Subsequently, Widom proved the strong Szego limit theorem for block Toeplitz determinants with symbols in $(K_{2,2}^{1/2,1/2})_{N\times N}$ and later two of the authors studied symbols in the generalized Krein algebras $(K_{p,q}^{\alpha,\beta})_{N\times N}$, where $\lambda:=1/p+1/q=\alpha+\beta$ and $\lambda=1$. We here extend these results to $0< \lambda <1$. The entire paper is based on fundamental work by Mark Krein, ranging from operator ideals through Toeplitz operators up to Wiener-Hopf factorization.

### On solutions to "almost everywhere" - Euler-Lagrange equation in Sobolev space $W_2^1$

Methods Funct. Anal. Topology **13** (2007), no. 3, 262-266

It is known, that if the Euler--Lagrange variational equation is fulfilled everywhere in classical case $C^1$ then it's solution is twice continuously differentiable. The present note is devoted to the study of a similar problem for the Euler--Lagrange equation in the Sobolev space $W_{2}^{1}$.

### A description of characters on the infinite wreath product

Methods Funct. Anal. Topology **13** (2007), no. 4, 301-317

Let $\mathfrak{S}_\infty$ be the infinity permutation group and $\Gamma$ an arbitrary group. Then $\mathfrak{S}_\infty$ admits a natural action on $\Gamma^\infty$ by automorphisms, so one can form a semidirect product $\Gamma^\infty \times \mathfrak{S}_\infty$, known as the wreath product $\Gamma\wr\mathfrak{S}_\infty$ of $\Gamma$ by $\mathfrak{S}_{\infty}$. We obtain a full description of unitary $I\!I_1-$factor-representations of $\Gamma\wr\mathfrak{S}_\infty$ in terms of finite characters of $\Gamma$. Our approach is based on extending Okounkov's classification method for admissible representations of $\mathfrak{S}_\infty\times\mathfrak{S}_\infty$. Also, we discuss certain examples of representations of type $I\!I\!I$, where the modular operator of Tomita-Takesaki expresses naturally by the asymptotic operators, which are important in the theory of characters of $\mathfrak{S}_\infty$.

### Dedication [To the memory of Mark Krein on his 100th anniversary]

Methods Funct. Anal. Topology **13** (2007), no. 2, 101-102

### Operator-valued integral of vector-function and bases

Methods Funct. Anal. Topology **13** (2007), no. 4, 318-328

In the present paper we are going to introduce an operator-valued integral of a square modulus weakly integrable mappings the ranges of which are Hilbert spaces, as bounded operators. Then, we shall show that each operator-valued integrable mapping of the index set of an orthonormal basis of a Hilbert space $H$ into $H$ can be written as a multiple of a sum of three orthonormal bases.

### On holomorphic solutions of the heat equation with a Volterra operator coefficient

Sergey Gefter, Anna Vershynina

Methods Funct. Anal. Topology **13** (2007), no. 4, 329-332

Let $A$ be a bounded operator on a Hilbert space and $g$ a vector-valued function, which is holomorphic in a neighborhood of zero. The question about existence of holomorphic solutions of the Cauchy problem $\left\{ \begin{array}{ll} \displaystyle\frac{\partial u}{\partial t}= A\displaystyle\frac{\partial^{2}u}{\partial x^2}\\ u(0,x)=g(x) \\ \end{array} \right.$ is considered in the paper.

### On cloused ideals of entire functions of finite gamma-growth

K. G. Malyutin, V. O. Gerasimenko

Methods Funct. Anal. Topology **13** (2007), no. 3, 279-283

We extend the result of Beurling on the closure in $H^p$ of the linear manifold $F(z)\cdot $$ \{$polynomials of $z\}$ to the classes of entire functions of finite gamma-growth.

### Direct theorems in the theory of approximation of Banach space vectors by exponential type entire vectors

Methods Funct. Anal. Topology **13** (2007), no. 3, 267-278

For an arbitrary operator $A$ on a Banach space $X$ which is the generator of a $C_0$--group with certain growth condition at infinity, direct theorems on connection between the degree of smoothness of a vector $x\in X$ with respect to the operator $A$, the rate of convergence to zero of the best approximation of $x$ by exponential type entire vectors for the operator $A$, and the $k$-module of continuity are established. The results allow to obtain Jackson-type inequalities in a number of classic spaces of periodic functions and weighted $L_p$ spaces.

### Development of the Markov moment problem approach in the optimal control theory

S. Yu. Ignatovich, G. M. Sklyar

Methods Funct. Anal. Topology **13** (2007), no. 4, 386-400

The paper is a survey of the main ideas and results on using of the Markov moment problem method in the optimal control theory. It contains a version of the presentation of the Markov moment approach to the time-optimal control theory, linear and nonlinear.

### On $*$-representations of algebras given by graphs

Methods Funct. Anal. Topology **13** (2007), no. 1, 17-27

*-algebras given by trees generated by projections with Temperley-Lieb type relations are considered in this work. Formulas of representations are built by algorithms for *-algebras, associated with the Dynkin diagrams, and we get estimates for the parameters at which non-trivial *-representations of the *-algebras exist.

### Generalized selfadjoinness of differentiation operator on weight Hilbert spases

Methods Funct. Anal. Topology **13** (2007), no. 4, 333-337

We consider examples of operators that act in some Hilbert rigging from positive Hilbert space into the negative one. For the first derivative operator we investigate a ``generalized'' selfadjointness in the sense of weight Hilbert riggings of the spaces $L^2([0,1])$ and $L^2(\mathbb{R})$. We will show that an example of the operator $i \frac{d}{dt}$ in some rigging scales, which is selfadjoint in usual case and not generalized selfadjoint, can not be constructed.

### On an extended stochastic integral and the Wick calculus on the connected with the generalized Meixner measure Kondratiev-type spaces

Methods Funct. Anal. Topology **13** (2007), no. 4, 338-379

We introduce an extended stochastic integral and construct elements of the Wick calculus on the Kondratiev-type spaces of regular and nonregular gene alized functions, study the interconnection between the extended stochastic integration and the Wick calculus, and consider examples of stochastic equations with Wick-type nonlinearity. Our researches are based on the general approach that covers the Gaussian, Poissonian, Gamma, Pascal and Meixner analyses.

### On stability and instability of small motions of hydrodynamical systems

Methods Funct. Anal. Topology **13** (2007), no. 2, 152-168

We give a short survey of an operator approach to some linear evolution and spectral problems of hydrodynamics: small motions and normal oscillations of a heavy or capillary fluid, a partially filled cavity in a moving or immovable vessel. The main attention is given to the problem of stability and instability of these hydromechanical systems with an infinite number of degrees of freedom.

### $p$-Adic fractional differentiation operator with point interactions

Methods Funct. Anal. Topology **13** (2007), no. 2, 169-180

Finite rank point perturbations of the $p$-adic fractional differentiation operator $D^{\alpha}$ are studied. The main attention is paid to the description of operator realizations (in $L_2(\mathbb{Q}_p)$) of the heuristic expression $D^{\alpha}+\sum_{i,j=1}^{n}b_{ij}\delta_{x_i}$ in a form that is maximally adapted for the preservation of physically meaningful relations to the parameters $b_{ij}$ of the singular potential.

### On $\mu$-scale invariant operators

K. A. Makarov, E. Tsekanovskii

Methods Funct. Anal. Topology **13** (2007), no. 2, 181-186

We introduce the concept of a $\mu$-scale invariant operator with respect to a unitary transformation in a separable complex Hilbert space. We show that if a nonnegative densely defined symmetric operator is $\mu$-scale invariant for some $\mu>0$, then both the Friedrichs and the Krein-von Neumann extensions of this operator are also $\mu$-scale invariant.

### Classification of noncompact surfaces with boundary

A. O. Prishlyak, K. I. Mischenko

Methods Funct. Anal. Topology **13** (2007), no. 1, 62–66

We give a topological classification of noncompact surfaces with any number of boundary components.

### Spectral measure of commutative Jacobi field equipped with multiplication structure

Methods Funct. Anal. Topology **13** (2007), no. 1, 28-42

The article investigates properties of the spectral measure of the Jacobi field constructed over an abstract Hilbert rigging $H_-\supset H\supset L\supset H_+.$ Here $L$ is a real commutative Banach algebra that is dense in $H.$ It is shown that with certain restrictions, the Fourier transform of the spectral measure can be found in a similar way as it was done for the case of the Poisson field with the zero Hilbert space $L^2(\Delta,d u).$ Here $\Delta$ is a Hausdorff compact space and $ u$ is a probability measure defined on the Borel $\sigma$-algebra of subsets of $\Delta.$ The article contains a formula for the Fourier transform of a spectral measure of the Jacobi field that is constructed over the above-mentioned abstract rigging.

### On quadruples of linearly connected projections and transitive systems of subspaces

Yulia Moskaleva, Vasyl Ostrovskyi, Kostyantyn Yusenko

Methods Funct. Anal. Topology **13** (2007), no. 1, 43-49

We study conditions under which the images of irreducible quadruples of linearly connected projections give rise to all transitive systems of subspaces in a finite dimensional Hilbert space.

### The set of discontinuity points of separately continuous functions on the products of compact spaces

Methods Funct. Anal. Topology **13** (2007), no. 3, 284-295

We solve the problem of constructing separately continuous functions on the product of compact spaces with a given set of discontinuity points. We obtain the following results. 1. For arbitrary \v{C}ech complete spaces $X$, $Y$, and a separable compact perfect projectively nowhere dense zero set $E\subseteq X\times Y$ there exists a separately continuous function $f:X\times Y\to\mathbb R$ the set of discontinuity points, which coincides with $E$. 2. For arbitrary \v{C}ech complete spaces $X$, $Y$, and nowhere dense zero sets $A\subseteq X$ and $B\subseteq Y$ there exists a separately continuous function $f:X\times Y\to\mathbb R$ such that the projections of the set of discontinuity points of $f$ coincides with $A$ and $B$, respectively. We construct an example of Eberlein compacts $X$, $Y$, and nowhere dense zero sets $A\subseteq X$ and $B\subseteq Y$ such that the set of discontinuity points of every separately continuous function $f:X\times Y\to\mathbb R$ does not coincide with $A\times B$, and a $CH$-example of separable Valdivia compacts $X$, $Y$ and separable nowhere dense zero sets $A\subseteq X$ and $B\subseteq Y$ such that the set of discontinuity points of every separately continuous function $f:X\times Y\to\mathbb R$ does not coincide with $A\times B$.

### Superstable criterion and superstable bounds for infinite range interaction I: two-body potentials

Methods Funct. Anal. Topology **13** (2007), no. 1, 50-61

A continuous infinite system of point particles interacting via two-body infinite-range potential is considered in the framework of classical statistical mecha ics. We propose some new criterion for interaction potentials to be superstable and give a very transparent proof of the Ruelle's uniform bounds for a family of finite volume correlation functions. It gives a possibility to prove that for any temperature and chemical activity there exists at least one Gibbs state. This article is a generalization of the work \cite{Re98} for the case of infinite range interaction potential.

### The square-transform of Hermite-Biehler functions. A geometric approach

Vyacheslav Pivovarchik, Harald Woracek

Methods Funct. Anal. Topology **13** (2007), no. 2, 187-200

We investigate the subclass of symmetric indefinite Hermite-Biehler functions which is obtained from positive definite Hermite-Biehler functions by means of the square-transform. It is known that functions of this class can be characterized in terms of location of their zeros. We give another, more elementary and geometric, proof of this result. The present proof employs a `shifting-of-zeros' perturbation method. We apply our results to obtain information on the eigenvalues of a concrete boundary value problems.

### Another form of separation axioms

S. Athisaya Ponmani, M. Lellis Thivagar

Methods Funct. Anal. Topology **13** (2007), no. 4, 380-385

It is the object of this paper to introduce the $(1, 2)^*$pre-$D_k$ axioms for $k$ = $0$, $1$, $2$.

### On models of function type for a special class of normal operators in Krein spaces and their polar representation

Methods Funct. Anal. Topology **13** (2007), no. 1, 67-82

The paper is devoted to a function model representation of a normal operator $N$ acting in a Krein space. We assume that $N$ and its adjoint operator $N^{\#}$ have a common invariant subspace $L_{+}$ which is a maximal nonnegative subspace and has a representation as a sum of a finite-dimensional neutral subspace and a uniformly positive subspace. For $N$ we construct a model representation as the multiplication operator by a scalar function acting in a suitable function space. This representation is applied to the problem of existence of a polar representation for normal operators of $D_{\kappa}^+$-class.

### About one class of Hilbert space uncoditional bases

A. A. Tarasenko, M. G. Volkova

Methods Funct. Anal. Topology **13** (2007), no. 3, 296-300

Let a sequence $\left\{v_k \right\}^{+\infty}_{-\infty}\in l_2$ and a real sequence $\left\{\lambda_k \right\}^{+\infty}_{-\infty}$ such that $\left\{\lambda_k^{-1} \right\}^{+\infty}_{-\infty}\in l_2$, and an orthonormal basis $\left\{e_k \right\}^{+\infty}_{-\infty}$ of a Hilbert space be given. We describe a sequence $M=\left\{\mu_k \right\}^{+\infty}_{-\infty}$, $M\cap \mathbb{R}=\varnothing$, such that the families $$ f_k = \sum\limits_{j\in\mathbb{Z}} {v_j\left(\lambda_j-\bar{\mu}_k \right)^{-1}}e_k, \quad k\in \mathbb{Z} $$ form an unconditional basis in $\mathfrak{H}$.

### On the Gauss-Manin connection in cyclic homology

Methods Funct. Anal. Topology **13** (2007), no. 1, 83-94

Getzler constructed a flat connection in the periodic cyclic homology, called the Gauss-Manin connection. In this paper we define this connection, and its monodromy, at the level of the periodic cyclic complex. The construction does not depend on an associator, and provides an explicit structure of a DG module over an auxiliary DG algebra. This paper is, to a large extent, an effort to clarify and streamline our work [4] with Yu.L. Daletsky.

### On Whitney constants for differentiable functions

Methods Funct. Anal. Topology **13** (2007), no. 1, 95-100

Some estimates of the constants in Whitney inequality for the differentiable functions are obtained.