# Arov D. Z.

### 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

### 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.

### Criterion of unitary similarity of minimal passive scattering systems with a given transfer function

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 147-156

We establish necessary and sufficient conditions under which all minimal passive scattering systems that have a given transfer operator function are unitarily equivalent. These conditions can be significantly simplified in special cases important for applications, in particular, in the case where a transfer function is rational and in a more general case where this function is pseudoextendable.