# Volume 59, № 6, 2007

### Singularly perturbed self-adjoint operators in scales of Hilbert spaces

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 723–743

Finite rank perturbations of a semi-bounded self-adjoint operator $A$ are studied in the scale of Hilbert spaces associated with $A$. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and singular perturbations of $A$ by the same formula. As an application the one-dimensional Schrodinger operator with generalized zero-range potential is considered in the Sobolev space $W_2^p(R),\quad p \in N$.

### Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field

Berezansky Yu. M., Pulemyotov A. D.

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 744–763

Assume that $K^+: H_- \rightarrow T_-$ is a bounded operator, where $H_—$ and $T_—$ are Hilbert spaces and $p$ is a measure on the space $H_—$. Denote by $\rho_K$ the image of the measure $\rho$ under $K^+$. This paper aims to study the measure $\rho_K$ assuming $\rho$ to be the spectral measure of a Jacobi field. We obtain a family of operators whose spectral measure equals $\rho_K$. We also obtain an analogue of the Wiener – Ito decomposition for $\rho_K$. Finally, we illustrate the results obtained by carrying out the explicit calculations for the case, where $\rho_K$is a Levy noise measure.

### Rational matrix functions associated with the Nevanlinna-Pick problem in the class *S* [*a, b*] and orthogonal on a compact interval

Dyukarev Yu. M., Serikova I. Yu.

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 764–770

We consider the interpolation Nevanlinna-Pick problem with infinitely many interpolation nodes in the class *S*[*a, b*] and rational matrix functions associated with this problem and orthogonal on the segment [*a, b*]. We obtain a criterion of complete indeterminacy of the Nevanlinna-Pick problem in terms of orthogonal rational matrix functions.

### Reconstruction of the spectral type of limiting distributions in dynamical conflict systems

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 771–784

We establish the conditions of reconstruction of pure spectral types (pure point, pure absolutely continuous, or pure singularly continuous spectra) in the limiting distributions of dynamical systems with compositions of alternative conflict. In particular, it is shown that the point spectrum can be reconstructed starting from the states with pure singularly continuous spectra.

### Singularly perturbed periodic and semiperiodic differential operators

Mikhailets V. A., Molyboga V. M.

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 785–797

Qualitative and spectral properties of the form sums $$S_{±}(V) := D^{2m}_{±} + V(x),\quad m ∈ N,$$ are studied in the Hilbert space $L_2(0, 1)$. Here, $(D_{+})$ is a periodic differential operator, $(D_{-})$ is a semiperiodic differential operator, $D_{±}: u ↦ −iu′$, and $V(x)$ is an arbitrary 1-periodic complex-valued distribution from the Sobolev spaces $H_{per}^{−mα},\; α ∈ [0, 1]$.

### Elliptic pseudodifferential operators in the improved scale of spaces on a closed manifold

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 798–814

We study linear elliptic pseudodifferential operators in the improved scale of functional Hilbert spaces on a smooth closed manifold. Elements of this scale are isotropic Hörmander-Volevich-Paneyakh spaces. We investigate the local smoothness of a solution of an elliptic equation in the improved scale. We also study elliptic pseudodifferential operators with parameter.

### Extension of the Stieltjes moment sequence to the left and related problems of the spectral theory of inhomogeneous string

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 815–825

For a nonhomogeneous string with the known mass distribution (the full mass is assumed to be infinite),
the known finite length, and the unknown spectral measure $d\sigma(t)$, we construct an analogous string with spectral measure $d\sigma(t)/t$.
This allows to calculate the moments of all negative orders of the measure $d\sigma(t)$.
The mechanical interpretation of the Stieltjes investigations on the moment problem proposed by M. G. Krein enables one to solve the following problem: for given
Stieltjes moment sequence with unique solution, calculate the moments of negative orders.
This problem is equivalent to the following one: establish the asymptotic behavior of the associate Stieltjes function near zero if its asymptotic behavior near infinity is given.

### On the growth of deformations of algebras associated with Coxeter graphs

Popova N. D., Samoilenko Yu. S., Strilets O. V.

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 826–837

We investigate a class of algebras that are deformations of quotient algebras of group algebras of Coxeter groups. For algebras from this class, a linear basis is found by using the “diamond lemma.” A description of all finite-dimensional algebras of this class is given, and the growth of infinite-dimensional algebras is determined.

### Direct and inverse theorems of approximate methods for the solution of an abstract Cauchy problem

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 838–852

We consider an approximate method for the solution of the Cauchy problem for an operator differential equation based on the expansion of the exponential function in orthogonal Laguerre polynomials. For an initial value of finite smoothness with respect to the operator *A*, we prove direct and inverse theorems of the theory of approximation in the mean and give examples of the unimprovability of the corresponding estimates in these theorems. We establish that the rate of convergence is exponential for entire vectors of exponential type and subexponential for Gevrey classes and characterize the corresponding classes in terms of the rate of convergence of approximation in the mean.

### On a criterion for the uniform boundedness of a *C*_{0}-semigroup of operators in a Hilbert space

Gomilko A. M., Wróbel I., Zemanek J.

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 853-858

Let $T(t),\quad t ≥ 0$, be a $C_0$-semigroup of linear operators acting in a Hilbert space $H$ with norm $‖·‖$. We prove that $T(t)$ is uniformly bounded, i.e., $‖T(t)‖ ≤ M, \quad t ≥ 0$, if and only if the following condition is satisfied: $$\sup_{t > 0} \frac1t ∫_0^t∥(T(s)+T^{∗}(s))x ∥^2ds < ∞$$ forall $x ∈ H$, where $T^{*}$ is the adjoint operator.

### Commutants of some classes of operators associated with shift operators

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 859–864

Commutants of some classes of operators associated with shift operators are described in the space of entire functions.