# Volume 60, № 4, 2008

### Myroslav L’vovych Horbachuk (on his 70th birthday)

Adamyan V. M., Berezansky Yu. M., Khruslov E. Ya., Kochubei A. N., Kuzhel' S. A., Marchenko V. O., Mikhailets V. A., Nizhnik L. P., Ptashnik B. I., Rofe-Beketov F. S., Samoilenko A. M., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 439–442

### Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 443–452

We investigate the Abel summability of a system of eigenfunctions and associated functions of Bitsadze-Samarskii-type boundary-value problems for elliptic equations in a rectangle. These problems are reduced to a boundary-value problem for elliptic operator differential equations with an operator in boundary conditions in the corresponding spaces and are studied by the method of operator differential equations.

### Integration of a modified double-infinite Toda lattice by using the inverse spectral problem

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 453–469

An approach to finding a solution of the Cauchy problem for a modified double-infinite Toda lattice by using the inverse spectral problem is given.

### On conditions under which the sum of self-adjoint operators with given integer spectra is a scalar operator

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 470–477

We describe the set ∑_{M1 ,...,Mn } и of parameters γ for which there exists a decomposition of
the operator γ*I _{H}* into a sum of

*n*self-adjoint operators with the spectra belonging to the sets

*M*

_{1 },...,

*M*.

_{n }The description of this set is performed for

*M*= {0,1, ...,

_{i }*k*} in the case of

_{i }*n*≥ 4 and in some cases for

*n*= 3.

### Distributed-order calculus: An operator-theoretic interpretation

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 478–486

Within the Bochner-Phillips functional calculus and Hirsch functional calculus, we describe the operators of distributed-order differentiation and integration as functions of the classical operators of differentiation and integration, respectively.

### On infinite-rank singular perturbations of the Schrödinger operator

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 487–496

Schrodinger operators with infinite-rank singular potentials $\sum^\infty_{i,j=1}b_{i,j}(\psi_j,\cdot)\psi_i$ are studied under the condition that singular elements $\psi_j$ are $\xi_j(t)$-invariant with respect to scaling transformations in ${\mathbb R}^3$.

### Elliptic boundary-value problem in a two-sided improved scale of spaces

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 497–520

We study a regular elliptic boundary-value problem in a bounded domain with smooth boundary. We prove that the operator of this problem is a Fredholm one in a two-sided improved scale of functional Hilbert spaces and that it generates there a complete collection of isomorphisms. Elements of this scale are Hörmander-Volevich-Paneyakh isotropic spaces and some their modi.cations. An *a priori* estimate for a solution is obtained and its regularity is investigated.

### Nonisospectral flows on semiinfinite unitary block Jacobi matrices

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 521–544

It is proved that if the spectrum and spectral measure of a unitary operator generated by a semiinfinite block Jacobi matrix $J(t)$ vary appropriately,
then the corresponding operator $\textbf{J}(t)$ satisfies the generalized
Lax equation $\dot{\textbf{J}}(t) = \Phi(\textbf{J}(t), t) + [\textbf{J}(t), A(\textbf{J}(t), t)]$,
where $\Phi(\lambda, t)$ is a polynomial in $\lambda$ and $\overline{\lambda}$ with $t$-dependent coefficients and $A(J(t), t) = \Omega + I + \frac12 \Psi$ is a skew-symmetric matrix.

The operator $J(t$) is analyzed in the space ${\mathbb C}\oplus{\mathbb C}^2\oplus{\mathbb C}^2\oplus...$.
It is mapped into the unitary operator of multiplication $L(t)$ in the isomorphic space $L^2({\mathbb T}, d\rho)$, where ${\mathbb T} = {z: |z| = 1}$.
This fact enables one to construct an efficient algorithm for solving the block lattice of differential equations generated by the Lax equation.
A procedure that allows one to solve the corresponding Cauchy problem by the Inverse-Spectral-Problem method is presented.

The article contains examples of block difference-differential lattices and the corresponding flows that are analogues of the Toda and van Moerbeke lattices
(from self-adjoint case on ${\mathbb R}$)
and some notes about applying this technique for Schur flow (unitary case on ${\mathbb T}$ and OPUC theory).

### On the *-representation of one class of algebras associated with Coxeter graphs

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

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 545–556

We investigate *-representations of a class of algebras that are quotient algebras of the Hecke algebras associated with Coxeter graphs. A description of all unitarily nonequivalent irreducible *-representations of finite-dimensional algebras is given. We prove that only trees that have at most one edge of type *s* > 3 define algebras of finite Hilbert type for all values of parameters.

### Characterization of the rate of convergence of one approximate method for the solution of an abstract Cauchy problem

Kashpirovskii A. I., Torba S. M.

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 557–563

We consider an approximate method for the solution of the Cauchy problem for an operator differential equation. This method is based on the expansion of an exponential in orthogonal Laguerre polynomials. We prove that the fact that an initial value belongs to a certain space of smooth elements of the operator *A* is equivalent to the convergence of a certain weighted sum of integral residuals. As a corollary, we obtain direct and inverse theorems of the theory of approximation in the mean.

### A locally compact quantum group of triangular matrices

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 564–576

We construct a one parameter deformation of the group of 2 × 2 upper triangular matrices with determinant 1 using the twisting construction.
An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a non-trivial way.
Also, we give a complete description of the dual *C**-algebra and the dual comultiplication.