Overdetermined conservative 2D Systems, Invariant in One Direction and a Generalization of Potapov's theorem
| dc.creator | Melnikov, Andrey | |
| dc.creator | Vinnikov, Victor | |
| dc.date | 2008-12-20 | |
| dc.date | 2008-12-23 | |
| dc.date.accessioned | 2026-07-07T12:21:06Z | |
| dc.date.available | 2026-07-07T12:21:06Z | |
| dc.description | This work is a direct continuation of the authors work arXiv:0812.3779v1. A special case of conservative overdetermined time invariant 2D systems is developed and studied. Defining transfer function of such a systems we obtain a class CI of inner functions $S(λ,t_2)$, which are identity for $λ=\infty$, satisfy certain regularity assumptions and intertwines solutions of ODEs with a spectral parameter $λ$. Using translation model, we prove that every function in the class CI can be realized as a transfer function of a certain vessel. The highlight of this theory is a generalization Potapov's theorem, which gives a very special formula for such a function in the form of multiplication of Blacke-Potapov products, corresponding to the discrete spectrum of certain system operator $A_1(t_2)$ and of multiplicative integral, corresponding to the continuous spectrum of $A_1(t_2)$. This theorem is proved under a slightly more restrictive assumptions, then the development of the whole theory. Namely, we suppose that the derivative of the transfer function is a continuous function of $t_2$ for almost all $λ$. At the last part zero/pole interpolation problem for matrix functions in CI is considered and a realization theorem of such functions appeared in arXiv:0812.3779v1 (theorem 8.1) is reproved. Hermitian case is also analyzed and the corresponding realization theorem is proved. | |
| dc.description | 54 pages | |
| dc.identifier | https://arxiv.org/abs/0812.3970 | |
| dc.identifier | http://arxiv.org/abs/0812.3970 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/213263 | |
| dc.subject | Functional Analysis | |
| dc.subject | Spectral Theory | |
| dc.subject | 47N20, 47N70, 47E05, 26B30, 45D05, 34K06 | |
| dc.title | Overdetermined conservative 2D Systems, Invariant in One Direction and a Generalization of Potapov's theorem | |
| dc.type | text |