# Authors Index, Vol. 16, 2010

### Israel Gohberg

Methods Funct. Anal. Topology **16** (2010), no. 4, 289-290

### GC-Fusion frames

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

Methods Funct. Anal. Topology **16** (2010), no. 2, 112-119

In this paper we introduce the generalized continuous version of fusion frame, namely $gc$-fusion frame. Also we get some new results about Bessel mappings and perturbation in this case.

### On mixing and completely mixing properties of positive $L^1$-contractions of finite real W* -algebras

Methods Funct. Anal. Topology **16** (2010), no. 3, 259-263

We consider a non-commutative real analogue of Akcoglu and Sucheston's result about the mixing properties of positive L$^1$-contractions of the L$^1$-space associated with a measure space with probability measure. This result generalizes an analogous result obtained for the L$^1$-space associated with a finite (complex) W$^*$-algebras.

### Boundary problems for the wave equation with the Lévy Laplacian in Shilov's class

S. Albeverio, Ya. I. Belopolskaya, M. N. Feller

Methods Funct. Anal. Topology **16** (2010), no. 3, 197-202

We present solutions to some boundary value and initial-boundary value problems for the "wave" equation with the infinite dimensional L\'evy Laplacian $\Delta _L$ $$\frac{\partial^2 U(t,x)}{\partial t^2}=\Delta_LU(t,x)$$ in the Shilov class of functions.

### The strong Hamburger moment problem and related direct and inverse spectral problems for block Jacobi-Laurent matrices

Yurij M. Berezansky, Mykola E. Dudkin

Methods Funct. Anal. Topology **16** (2010), no. 3, 203-241

In this article we propose an approach to the strong Hamburger moment problem based on the theory of generalized eigenvectors expansion for a selfadjoint operator. Such an approach to another type of moment problems was given in our works earlier, but for strong Hamburger moment problem it is new. We get a sufficiently complete account of the theory of such a problem, including the spectral theory of block Jacobi-Laurent matrices.

### Hereditary properties of hyperspaces

Methods Funct. Anal. Topology **16** (2010), no. 1, 1-5

In this paper, we investigate hereditary properties of hyperspaces. Our basic cardinals are the Suslin hereditary number, the hereditary $\pi$-weight, the Shanin hereditary number, the hereditary density, the hereditary cellularity. We prove that the hereditary cellularity, the hereditary $\pi$-weight, the Shanin hereditary number, the hereditary density, the hereditary cellularity for any Eberlein compact and any Danto space and their hyperspaces coincide.

### Harmonic analysis on a locally compact hypergroup

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

Methods Funct. Anal. Topology **16** (2010), no. 4, 304-332

We propose a new axiomatics for a locally compact hypergroup. On the one hand, the new object generalizes a DJS-hypergroup and, on the other hand, it allows to obtain results similar to those for a unimodular hypecomplex system with continuous basis. We construct a harmonic analysis and, for a commutative locally compact hypergroup, give an analogue of the Pontryagin duality theorem.

### Maharam traces on von Neumann algebras

Methods Funct. Anal. Topology **16** (2010), no. 2, 101-111

Traces $\Phi$ on von Neumann algebras with values in complex order complete vector lattices are considered. The full description of these traces is given for the case when $\Phi$ is the Maharam trace. The version of Radon-Nikodym-type theorem for Maharam traces is established.

### A spectral decomposition in one class of non-selfadjoint operators

G. M. Gubreev, M. V. Dolgopolova, S. I. Nedobachiy

Methods Funct. Anal. Topology **16** (2010), no. 2, 140-157

In this paper, a class of special finite dimensional perturbations of Volterra operators in Hilbert spaces is investigated. The main result of the article is finding necessary and sufficient conditions for an operator in a chosen class to be similar to the orthogonal sum of a dissipative and an anti-dissipative operators with finite dimensional imaginary parts.

### Singularly perturbed normal operators

Methods Funct. Anal. Topology **16** (2010), no. 4, 298-303

We give an effective description of finite rank singular perturbations of a normal operator by using the concepts we introduce of an admissible subspace and corresponding admissible operators. We give a description of rank one singular perturbations in terms of a scale of Hilbert spaces, which is constructed from the unperturbed operator.

### Dimension stabilization effect for the block Jacobi-type matrix of a bounded normal operator with the spectrum on an algebraic curve

Oleksii Mokhonko, Sergiy Dyachenko

Methods Funct. Anal. Topology **16** (2010), no. 1, 28-41

Under some natural assumptions, any bounded normal operator in an appropriate basis has a three-diagonal block Jacobi-type matrix. Just as in the case of classical Jacobi matrices (e.g. of self-adjoint operators) such a structure can be effectively used. There are two sources of difficulties: rapid growth of blocks in the Jacobi-type matrix of such operators (they act in $\mathbb C^1\oplus\mathbb C^2\oplus\mathbb C^3\oplus\cdots$) and potentially complicated spectra structure of the normal operators. The aim of this article is to show that these two aspects are closely connected: simple structure of the spectra can effectively bound the complexity of the matrix structure. The main result of the article claims that if the spectra is concentrated on an algebraic curve the dimensions of Jacobi-type matrix blocks do not grow starting with some value.

### Regularization of singular Sturm-Liouville equations

Andrii Goriunov, Vladimir Mikhailets

Methods Funct. Anal. Topology **16** (2010), no. 2, 120-130

The paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' + qy$$ with the coefficients $$q = Q', \quad 1/p, Q/p, Q^2/p \in L_1, $$ where the derivative of the function $Q$ is understood in the sense of distributions. Due to a new regularization, the corresponding operators are correctly defined as quasi-differentials. Their resolvent approximation is investigated and all self-adjoint and maximal dissipative extensions and generalized resolvents are described in terms of homogeneous boundary conditions of the canonical form.

### Systems of one-dimensional subspaces of a Hilbert space

R. V. Grushevoy, Yu. S. Samoilenko

Methods Funct. Anal. Topology **16** (2010), no. 2, 131-139

We study systems of one-dimensional subspaces of a Hilbert space. For such systems, symmetric and orthoscalar systems, as well as graph related configurations of one-dimensional subspaces have been studied.

### In Memoriam Israel Gohberg

Methods Funct. Anal. Topology **16** (2010), no. 4, 291-297

### The integration by parts formula in the Meixner white noise analysis

Methods Funct. Anal. Topology **16** (2010), no. 1, 6-16

Using a general approach that covers the cases of Gaussian, Poissonian, Gamma, Pascal and Meixner measures on an infinite- dimensional space, we construct a general integration by parts formula for analysis connected with each of these measures. Our consideration is based on the constructions of the extended stochastic integral and the stochastic derivative that are connected with the structure of the extended Fock space.

### Operators defined on $L_1$ which "nowhere" attain their norm

I. V. Krasikova, V. V. Mykhaylyuk, M. M. Popov

Methods Funct. Anal. Topology **16** (2010), no. 1, 17-27

Let $E$ be either $\ell_1$ of $L_1$. We consider $E$-unattainable continuous linear operators $T$ from $L_1$ to a Banach space $Y$, i.e., those operators which do not attain their norms on any subspace of $L_1$ isometric to $E$. It is not hard to see that if $T: L_1 \to Y$ is $\ell_1$-unattainable then it is also $L_1$-unattainable. We find some equivalent conditions for an operator to be $\ell_1$-unattainable and construct two operators, first $\ell_1$-unattainable and second $L_1$-unattainable but not $\ell_1$-unattainable. Some open problems remain unsolved.

### On $J$-self-adjoint extensions of the Phillips symmetric operator

S. Kuzhel, O. Shapovalova, L. Vavrykovych

Methods Funct. Anal. Topology **16** (2010), no. 4, 333-348

$J$-self-adjoint extensions of the Phillips symmetric operator $S$ are %\break studied. The concepts of stable and unstable $C$-symmetry are introduced in the extension theory framework. The main results are the following: if ${A}$ is a $J$-self-adjoint extension of $S$, then either $\sigma({A})=\mathbb{R}$ or $\sigma({A})=\mathbb{C}$; if ${A}$ has a real spectrum, then ${A}$ has a stable $C$-symmetry and ${A}$ is similar to a self-adjoint operator; there are no $J$-self-adjoint extensions of the Phillips operator with unstable $C$-symmetry.

### On the $H$-ring structure of infinite Grassmannians

Methods Funct. Anal. Topology **16** (2010), no. 3, 242-258

The $H$-ring structure of certain infinite dimensional Grassmannians is discussed using various algebraic and analytical methods but avoiding cellular arguments. These methods allow us to treat these Grassmannians in a greater generality.

### Hyperspaces of closed limit sets

Methods Funct. Anal. Topology **16** (2010), no. 2, 158-166

We study Michael's lower semifinite topology and Fell's topology on the collection of all closed limit subsets of a topological space. Special attention is given to the subfamily of all maximal limit sets.

### Functions on surfaces and incompressible subsurfaces

Methods Funct. Anal. Topology **16** (2010), no. 2, 167-182

Let $M$ be a smooth connected compact surface, $P$ be either a real line $\mathbb R$ or a circle $S^1$. Then we have a natural right action of the group $D(M)$ of diffeomorphisms of $M$ on $C^\infty(M,P)$. For $f\in C^\infty(M,P)$ denote respectively by $S(f)$ and $O(f)$ its stabilizer and orbit with respect to this action. Recently, for a large class of smooth maps $f:M\to P$ the author calculated the homotopy types of the connected components of $S(f)$ and $O(f)$. It turned out that except for few cases the identity component of $S(f)$ is contractible, $\pi_i O(f)=\pi_i M$ for $i\geq3$, and $\pi_2 O(f)=0$, while $\pi_1 O(f)$ it only proved to be a finite extension of $\pi_1D_{Id}M\oplus\mathbb Z^{l}$ for some $l\geq0$. In this note it is shown that if $\chi(M)<0$, then $\pi_1O(f)=G_1\times\cdots\times G_n$, where each $G_i$ is a fundamental group of the restriction of $f$ to a subsurface $B_i\subset M$ being either a $2$-disk or a cylinder or a Mobius band. For the proof of main result incompressible subsurfaces and cellular automorphisms of surfaces are studied.

### Spectral problem for figure-of-eight graph of Stieltjes strings

Methods Funct. Anal. Topology **16** (2010), no. 4, 349-358

We describe the spectrum of the problem generated by the Stieltjes string recurrence relations on a figure-of-eight graph. The continuity and the force balance conditions are imposed at the vertex of the graph. It is shown that the eigenvalues of such (main) problem are interlaced with the elements of the union of sets of eigenvalues of the Dirichlet problems generated by the parts of the string which correspond to the loops of the figure-of-eight graph. Also the eigenvalues of the main problem are interlaced with the elements of the union of sets of eigenvalues of the periodic problems generated by the same parts of the string.

### Direct and inverse problems for generalized Pick matrix

Methods Funct. Anal. Topology **16** (2010), no. 4, 359-382

All matrix modifications of classical Nevanlinna-Pick interpolation problem with a finite number of nonreal nodes which can be investigated by V. P. Potapov method are described.

### On the number of negative eigenvalues of a Schrodinger operator with $\delta$ interactions

Methods Funct. Anal. Topology **16** (2010), no. 1, 42-50

We give necessary and sufficient conditions for a one-dimensional Schrodinger operator to have the number of negative eigenvalues equal to the number of negative intensities in the case of $\delta$ interactions.

### On the number of negative eigenvalues of a multi-dimensional Schrodinger operator with point interactions

Methods Funct. Anal. Topology **16** (2010), no. 4, 383-392

We prove that the number $N$ of negative eigenvalues of a Schr\"odinger operator $L$ with finitely many points of $\delta$-interactions on $\mathbb R^{d}$ (${d}\le3$) is equal to the number of negative eigenvalues of a certain class of matrix $M$ up to a constant. This $M$ is expressed in terms of distances between the interaction points and the intensities. As applications, we obtain sufficient and necessary conditions for $L$ to satisfy $N=m,n,n$ for ${d}=1,2,3$, respectively, and some estimates of the minimum and maximum of $N$ for fixed intensities. Here, we denote by $n$ and $m$ the numbers of interaction points and negative intensities, respectively.

### Strong compact properties of the mappings and K-Radon-Nikodym property

Methods Funct. Anal. Topology **16** (2010), no. 2, 183-196

For mappings acting from an interval into a locally convex space, we study properties of strong compact variation and strong compact absolute continuity connected with an expansion of the space into subspaces generated by the compact sets. A description of strong $K$-absolutely continuous mappings in terms of indefinite Bochner integral is obtained. A special class of the spaces having $K$-Radon-Nikodym property is obtained. A relation between the $K$-Radon-Nikodym property and the classical Radon-Nikodym property is considered.

### Algebraically admissible cones in free products of $*$-algebras

Methods Funct. Anal. Topology **16** (2010), no. 1, 51-56

It was proved in~\cite{Pop09b} that a $*$-algebra is $C^*$-representable, i.e., $*$-isomorphic to a self-adjoint subalgebra of bounded operators acting on a Hilbert space if and only if there is an algebraically admissible cone in the real space of Hermitian elements of the algebra such that the algebra unit is an Archimedean order unit. In the present paper we construct such cones in free products of $C^*$-representable $*$-algebras generated by unitaries. We also express the reducing ideal of any algebraically bounded $*$-algebra with corepresentation $\mathcal F/\mathcal J$ where $\mathcal F$ is a free algebra as a closure of the ideal $\mathcal J$ in some universal enveloping $C^*$-algebra.

### On decompositions of the identity operator into a linear combination of orthogonal projections

Methods Funct. Anal. Topology **16** (2010), no. 1, 57-68

In this paper we consider decompositions of the identity operator into a linear combination of $k\ge 5$ orthogonal projections with real coefficients. It is shown that if the sum $A$ of the coefficients is closed to an integer number between $2$ and $k-2$ then such a decomposition exists. If the coefficients are almost equal to each other, then the identity can be represented as a linear combination of orthogonal projections for $\frac{k-\sqrt{k^2-4k}}{2} < A < \frac{k+\sqrt{k^2-4k}}{2}$. In the case where some coefficients are sufficiently close to $1$ we find necessary conditions for the existence of the decomposition.

### Inverse theorems in the theory of approximation of vectors in a Banach space with exponential type entire vectors

Methods Funct. Anal. Topology **16** (2010), no. 1, 69-82

An arbitrary operator $A$ on a Banach space $X$ which is a generator of a $C_0$-group with a certain growth condition at infinity is considered. A relationship between its exponential type entire vectors and its spectral subspaces is found. Inverse theorems on the 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 operator $A$, and the $k$-module of continuity with respect to $A$ are established. Also, a generalization of the Bernstein-type inequality is obtained. The results allow to obtain Bernstein-type inequalities in weighted $L_p$ spaces.

### Unitarization of Schur representations of a poset corresponding to $\widetilde{E_8}$

Methods Funct. Anal. Topology **16** (2010), no. 3, 264-270

We prove that every Schur representation of a poset corresponding to $\widetilde{E_8}$ can be unitarized with some character.

### Positive definite kernels satisfying difference equations

Methods Funct. Anal. Topology **16** (2010), no. 1, 83-100

We study positive definite kernels $K = (K_{n,m})_{n,m\in A}$, $A=\mathbb Z$ or $A=\mathbb Z_+$, which satisfy a difference equation of the form $L_n K = \overline L_m K$, or of the form $L_n \overline L_m K = K$, where $L$ is a linear difference operator (here the subscript $n$ ($m$) means that $L$ acts on columns (respectively rows) of $K$). In the first case, we give new proofs of Yu.M. Berezansky results about integral representations for $K$. In the second case, we obtain integral representations for $K$. The latter result is applied to strengthen one our result on abstract stochastic sequences. As an example, we consider the Hamburger moment problem and the corresponding positive matrix of moments. Classical results on the Hamburger moment problem are derived using an operator approach, without use of Jacobi matrices or orthogonal polynomials.

### A description of all solutions of the matrix Hamburger moment problem in a general case

Methods Funct. Anal. Topology **16** (2010), no. 3, 271-288

We describe all solutions of the matrix Hamburger moment problem in a general case (no conditions besides solvability are assumed). We use the fundamental results of A. V. Shtraus on the generalized resolvents of symmetric operators. All solutions of the truncated matrix Hamburger moment problem with an odd number of given moments are described in an "almost nondegenerate" case. Some conditions of solvability for the scalar truncated Hamburger moment problem with an even number of given moments are given.