# Volume 59, № 5, 2007

### Mark Grigorievich Krein (to the centenary of his birth)

Adamyan V. M., Arov D. Z., Berezansky Yu. M., Gorbachuk M. L., Gorbachuk V. I., Mikhailets V. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 579-587

### A generalization of an extended stochastic integral

Berezansky Yu. M., Tesko V. A.

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 588–617

We propose a generalization of an extended stochastic integral to the case of integration with respect to a broad class of random processes. In particular, we obtain conditions for the coincidence of the considered integral with the classical Itô stochastic integral.

### Passive impedance systems with losses of scattering channels

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 618–649

A new model of the passive impedance system with minimal losses of scattering channels and with bilaterally stable evolution semigroup is studied. In the case of discrete time, the passive linear stationary bilaterally stable impedance system $\Sigma$ is considered as a part of some minimal scattering-impedance lossless transmission system, that has a $(\tilde{J}_1, \tilde{J}_2)$-unitary system operator and a bilaterally $(J_1, J_2)$-inner (in certain weak sense) transmission function in the unit disk 22-block of which coincides with the impedance matrix of system $\Sigma$, belongs to the Caratheodory class, and has a pseudocontinuation. If the external space of the system $\Sigma$ is infinite-dimensional, then instead of the last mentioned property, we consider more complicated necessary and sufficient conditions on the impedance matrix of the system $\Sigma$. Different kinds of passive bilaterally stable impedance realizations with minimal losses of scattering channels (minimal, optimal, *-optimal, minimal and optimal, minimal and *-optimal) are studied.

### Phragmén-Lindelöf theorem for solutions of elliptic differential equations in a banach space

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 650–657

For a second-order elliptic differential equation considered on a semiaxis in a Banach space, we show that if the order of growth of its solution at infinity is not higher than the exponential order, then this solution tends exponentially to zero at infinity.

### On the nature of the de Branges Hamiltonian

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 658–678

We prove the theorem announced by the author in 1995 in the paper "Criterion for discreteness of spectrum of singular canonical system" (Functional Analysis and Its Applications, Vol. 29, No. 3).

In developing the theory of Hilbert spaces of entire functions (we call them the Krein - de Branges spaces or, briefly, *K-B* spaces),
L. de Branges arrived at some class of canonical equations of phase dimension 2. He proved that, for any given K-B space, there exists a canonical
equation of the considered class such that it restores the chain of included *K-B* spaces. The Hamiltonians of such canonical equations are called the de Branges Hamiltonians.
The following question arises:
Under which conditions the Hamiltonian of some canonical equation should be a de Branges Hamiltonian. The basic theorem of the present paper together with Theorem 1 of the mentioned paper gives the answer to this question.

### Improved scales of spaces and elliptic boundary-value problems. III

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 679–701

We study elliptic boundary-value problems in improved scales of functional Hilbert spaces on smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The local smoothness of a solution of an elliptic problem in an improved scale is investigated. We establish a sufficient condition under which this solution is classical. Elliptic boundary-value problems with parameter are also studied.

### On spectra of a certain class of quadratic operator pencils with one-dimensional linear part

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 702–716

We consider a class of quadratic operator pencils that occur in many problems of physics. The part of such a pencil linear with respect to the spectral parameter describes viscous friction in problems of small vibrations of strings and beams. Patterns in the location of eigenvalues of such pencils are established. If viscous friction (damping) is pointwise, then the operator in the linear part of the pencil is one-dimensional. For this case, rules in the location of purely imaginary eigenvalues are found.

### On indecomposable and transitive systems of subspaces

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 717–720

We prove that the indecomposability of a system of subspaces of a finite-dimensional Hilbert space implies the transitivity of this system under the condition of the linear coherence of the corresponding system of orthogonal projectors.