Cascade connections of
multiparametric  linear  systems and conservative realization of a decomposable 
inner operator-valued function on bidisk (in Russian)

D. C. Kalyuzhnyi-Verbovetskyi

We introduce the notion of cascade connection of multiparametric discrete time-invariant linear dynamical systems with unit delay. This allows us to construct the explicit example of conservative realization of a decomposable operator-valued function $\theta (z_1, z_2)=\theta_2(z_2)\theta_1(z_1)$, where $\theta_1$ and $\theta_2$ are inner operator-valued functions on the unit disk, vanishing at zero, and discuss some properties of such realization.

1. Аров Д. З. Пассивные линейные стационарные динамические системы // Сиб. мат. ж. – 1979. – Т. 20, № 2. – С. 211–228.

2. Бродский В. М., Шварцман Я. С. Инвариантные подпространства сжатия и факторизации характеристических функций // Теория функций, функц. анализ и их прилож. – 1975. – Вып. 22. – С. 15–35.

3. Бродский М. С. Унитарные узлы и их характеристические функции // Успехи мат. наук. – 1978. – Т. 33, № 4. – С. 141–168.

4. Золотарёв В. А. Схема рассеяния Лакса-Филлипса на группах и функциональная модель алгебры Ли // Мат. сб. – 1992. – Т. 183, № 5. – С. 115–144.

5. Сёкефальви-Надь Б., Фояш Ч. Гармонический анализ операторов в гильбертовом пространстве: Пер. с фр. – М.: Мир, 1970. – 430 с.

6. Agler J. On the representation of certain holomorphic functions defined on a polydisc // Topics in Operator Theory: Ernst D. Hellinger Memorial Volume (L. de Branges, I. Gohberg, and J. Rovnyak, editors). - Oper. Theory and Appl. – V. 48. – Basel: Birkhäuser-Verlag, 1990. – P. 47–66.

7. Agler J., McCarthy J. Nevanlinna-Pick interpolation on the bidisk // J. Reine Angew. Math. – 1999. – V. 506. – P. 191–204.

8. Ball J.A., Cohen N. De Branges-Rovnyak operator models and systems theory: a survey // Topics in Matrix and Operator Theory (H. Bart, I. Gohberg, and M.A. Kaashoek, editors). – Oper. Theory Adv. Appl. – V. 50. – Basel: Birkhäuser-Verlag, 1991. – P. 93–136.

9. Ball J. A., Li W. S., Timotin D., Trent T. A commutant lifting theorem on the polydisk // Preprint.

10. Ball J. A., Trent T. Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables // J. Funct. Anal. – 1998. – V. 157. – P. 1–61.

11. Ball J. A., Vinnikov V. Realization and interpolation for multipliers on reproducing kernel Hilbert spaces // Preprint.

12. Cotlar M., Sadosky C. Integral representations of bounded Hankel forms defined in scattering systems with a multiparametric evolution group // Oper. Theory Adv. Appl. - 1988. - V. 35. - P. 357–375.

13. Cotlar M., Sadosky C. Transference of metrics induced by unitary coupling, a Sarason theorem for the bidimensional torus, and a Sz.-Nagy–Foias theorem for two pairs of dilations // J. Funct. Anal. – 1993. – V. 111. – P. 473–488.

14. Cotlar M., Sadosky C. Nehari and Nevanlinna-Pick problems and holomorphic extensions in the polydisk in terms of restricted BMO // J. Funct. Anal. – 1994. – V. 124. – P. 205–210.

15. Kalyuzhniy D.S. Multiparametric dissipative linear stationary dynamical scattering systems: Discrete case // J. Operator Theory. - 2000. - V. 43, № 2. - P. 427–460.

16. Kalyuzhniy D. S. Multiparametric dissipative linear stationary dynamical scattering systems: Discrete case, II: Existence of conservative dilations // Integral Equations Operator Theory. – 2000. – V. 36, № 1. – P. 107–120.

17. Kalyuzhniy D.S. On the notions of dilation, controllability, observability, and minimality in the theory of dissipative scattering linear nD systems // Proceedings CD of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS 2000). – June 19–23, 2000. - Perpignan (France). – 6 pp.

18. Khan~D.C. ($\pm$)-regular factorization of transfer functions and passive scattering systems for cascade couplings // J. Oper. Theory. -- 1994. -- V.~32. -- P. 1--16.

19. Livsic M.S., Kravitsky N., Markus A.S., Vinnikov V. Theory of Commuting Nonselfadjoint Operators // Mathematics and Its Applications. – V. 332. – Dordrecht: Kluwer, 1995.

20. Livsic M.S., Waksman L. Commuting Nonselfadjoint Operators in Hilbert Space // Lect. Notes in Math. – V. 1272. – Berlin: Springer-Verlag, 1987. – 115 p.

	Cascade connections of multiparametric linear systems and conservative realization of a decomposable inner operator-valued function on bidisk (in Russian)
Author	D. C. Kalyuzhnyi-Verbovetskyi d.kv@paco.net Odessa State Academy of Civil Engineering and Architecture 4 Didrikhsona St., Odesa 65029, Ukraine.
Abstract	We introduce the notion of cascade connection of multiparametric discrete time-invariant linear dynamical systems with unit delay. This allows us to construct the explicit example of conservative realization of a decomposable operator-valued function $\theta (z_1, z_2)=\theta_2(z_2)\theta_1(z_1)$, where $\theta_1$ and $\theta_2$ are inner operator-valued functions on the unit disk, vanishing at zero, and discuss some properties of such realization.
Keywords	cascade connection, multiparametric discrete time-invariant linear dynamical systems, unit delay, conservative realization, inner operator-valued functions, unit disk, vanishing at zero, properties of realization
DOI	doi:10.30970/ms.15.1.65-76
Reference	1. Аров Д. З. Пассивные линейные стационарные динамические системы // Сиб. мат. ж. – 1979. – Т. 20, № 2. – С. 211–228. 2. Бродский В. М., Шварцман Я. С. Инвариантные подпространства сжатия и факторизации характеристических функций // Теория функций, функц. анализ и их прилож. – 1975. – Вып. 22. – С. 15–35. 3. Бродский М. С. Унитарные узлы и их характеристические функции // Успехи мат. наук. – 1978. – Т. 33, № 4. – С. 141–168. 4. Золотарёв В. А. Схема рассеяния Лакса-Филлипса на группах и функциональная модель алгебры Ли // Мат. сб. – 1992. – Т. 183, № 5. – С. 115–144. 5. Сёкефальви-Надь Б., Фояш Ч. Гармонический анализ операторов в гильбертовом пространстве: Пер. с фр. – М.: Мир, 1970. – 430 с. 6. Agler J. On the representation of certain holomorphic functions defined on a polydisc // Topics in Operator Theory: Ernst D. Hellinger Memorial Volume (L. de Branges, I. Gohberg, and J. Rovnyak, editors). - Oper. Theory and Appl. – V. 48. – Basel: Birkhäuser-Verlag, 1990. – P. 47–66. 7. Agler J., McCarthy J. Nevanlinna-Pick interpolation on the bidisk // J. Reine Angew. Math. – 1999. – V. 506. – P. 191–204. 8. Ball J.A., Cohen N. De Branges-Rovnyak operator models and systems theory: a survey // Topics in Matrix and Operator Theory (H. Bart, I. Gohberg, and M.A. Kaashoek, editors). – Oper. Theory Adv. Appl. – V. 50. – Basel: Birkhäuser-Verlag, 1991. – P. 93–136. 9. Ball J. A., Li W. S., Timotin D., Trent T. A commutant lifting theorem on the polydisk // Preprint. 10. Ball J. A., Trent T. Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables // J. Funct. Anal. – 1998. – V. 157. – P. 1–61. 11. Ball J. A., Vinnikov V. Realization and interpolation for multipliers on reproducing kernel Hilbert spaces // Preprint. 12. Cotlar M., Sadosky C. Integral representations of bounded Hankel forms defined in scattering systems with a multiparametric evolution group // Oper. Theory Adv. Appl. - 1988. - V. 35. - P. 357–375. 13. Cotlar M., Sadosky C. Transference of metrics induced by unitary coupling, a Sarason theorem for the bidimensional torus, and a Sz.-Nagy–Foias theorem for two pairs of dilations // J. Funct. Anal. – 1993. – V. 111. – P. 473–488. 14. Cotlar M., Sadosky C. Nehari and Nevanlinna-Pick problems and holomorphic extensions in the polydisk in terms of restricted BMO // J. Funct. Anal. – 1994. – V. 124. – P. 205–210. 15. Kalyuzhniy D.S. Multiparametric dissipative linear stationary dynamical scattering systems: Discrete case // J. Operator Theory. - 2000. - V. 43, № 2. - P. 427–460. 16. Kalyuzhniy D. S. Multiparametric dissipative linear stationary dynamical scattering systems: Discrete case, II: Existence of conservative dilations // Integral Equations Operator Theory. – 2000. – V. 36, № 1. – P. 107–120. 17. Kalyuzhniy D.S. On the notions of dilation, controllability, observability, and minimality in the theory of dissipative scattering linear nD systems // Proceedings CD of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS 2000). – June 19–23, 2000. - Perpignan (France). – 6 pp. 18. Khan~D.C. ($\pm$)-regular factorization of transfer functions and passive scattering systems for cascade couplings // J. Oper. Theory. -- 1994. -- V.~32. -- P. 1--16. 19. Livsic M.S., Kravitsky N., Markus A.S., Vinnikov V. Theory of Commuting Nonselfadjoint Operators // Mathematics and Its Applications. – V. 332. – Dordrecht: Kluwer, 1995. 20. Livsic M.S., Waksman L. Commuting Nonselfadjoint Operators in Hilbert Space // Lect. Notes in Math. – V. 1272. – Berlin: Springer-Verlag, 1987. – 115 p.
Pages	65-76
Volume	15
Issue	1
Year	2001
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Cascade connections of multiparametric linear systems and conservative realization of a decomposable inner operator-valued function on bidisk (in Russian)