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$ J_{\lowercase{p},\lowercase{m}}$-inner dilations of matrix-valued functions that belong to the Carathéodory class and admit pseudocontinuation


Authors: D. Z. Arov and N. A. Rozhenko
Translated by: V. Vasyunin
Original publication: Algebra i Analiz, tom 19 (2007), nomer 3.
Journal: St. Petersburg Math. J. 19 (2008), 375-395
MSC (2000): Primary 47A56
DOI: https://doi.org/10.1090/S1061-0022-08-01002-9
Published electronically: March 21, 2008
MathSciNet review: 2340706
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Abstract | References | Similar Articles | Additional Information

Abstract: The class $ \ell^{p\times p}$ of matrix-valued functions $ c(z)$ holomorphic in the unit disk $ D=\{{z\in\mathbb{C}:\vert z\vert<1}\}$, having order $ p$, and satisfying $ \operatorname{Re}c(z)\ge 0$ in $ D$ is considered, as well as its subclass $ \ell^{p\times p}\Pi$ of matrix-valued functions $ c(z)\in \ell^{p\times p}$ that have a meromorphic pseudocontinuation $ c_-(z)$ to the complement $ D_e=\{z\in\mathbb{C}:1<\vert z\vert\le \infty\}$ of the unit disk with bounded Nevanlinna characteristic in $ D_e$.

For matrix-valued functions $ c(z)$ of class $ \ell^{p\times p}\Pi$ a representation as a block of a certain $ J_{p,m}$-inner matrix-valued function $ \theta(z)$ is obtained. The latter function has a special structure and is called the $ J_{p,m}$-inner dilation of $ c(z)$. The description of all such representations is given.

In addition, the following special $ J_{p,m}$-inner dilations are considered and described: minimal, optimal, $ *$-optimal, minimal and optimal, minimal and $ *$-optimal. Also, $ J_{p,m}$-inner dilations with additional properties are treated: real, symmetric, rational, or any combination of them under the corresponding restrictions on the matrix-valued function $ c(z)$. The results extend to the case where the open upper half-plane $ \mathbb{C}_+$ is considered instead of the unit disk $ D$. For entire matrix-valued functions $ c(z)$ with  $ \operatorname{Re}c(z) \ge 0$ in  $ \mathbb{C_+}$ and with Nevanlinna characteristic in $ \mathbb{C}_-$, the $ J_{p,m}$-inner dilations in  $ \mathbb{C}_+$ that are entire matrix-valued functions are also described.


References [Enhancements On Off] (What's this?)

  • 1. D. Z. Arov, Darlington’s method in the study of dissipative systems, Dokl. Akad. Nauk SSSR 201 (1971), no. 3, 559–562 (Russian); Russian transl., Soviet Physics Dokl. 16 (1971), 954–956 (1972). MR 0428098
  • 2. D. Z. Arov, Realization of matrix-valued functions according to Darlington, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 1299–1331 (Russian). MR 0357820
  • 3. D. Z. Arov, Realization of a canonical system with a dissipative boundary condition at one end of the segment in terms of the coefficient of dynamical compliance, Sibirsk. Mat. Ž. 16 (1975), no. 3, 440–463, 643 (Russian). MR 0473196
  • 4. -, Passive linear steady-state dynamical systems, Sibirsk. Mat. Zh. 20 (1979), no. 2, 211-228; English transl., Siberian Math. J. 20 (1979), no. 2, 149-162. MR 0530486 (80g:93031)
  • 5. -, Optimal and stable passive systems, Dokl. Akad. Nauk SSSR 247 (1979), no. 2, 265-268; English transl., Soviet Math. Dokl. 20 (1979), no. 4, 676-680. MR 0545346 (80k:93036)
  • 6. -, Stable dissipative linear stationary dynamical scattering systems, J. Operator Theory 2 (1979), no. 1, 95-126. (Russian) MR 0553866 (81g:47007)
  • 7. -, Functions of class $ \Pi$, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 135 (1984), 5-30; English transl., J. Soviet Math. 31 (1985), no. 1, 2645-2659. MR 0741691 (85h:47041)
  • 8. D. Z. Arov and M. A. Nudel′man, Conditions for the similarity of all minimal passive realizations of a given transfer function (scattering and resistance matrices), Mat. Sb. 193 (2002), no. 6, 3–24 (Russian, with Russian summary); English transl., Sb. Math. 193 (2002), no. 5-6, 791–810. MR 1957950, https://doi.org/10.1070/SM2002v193n06ABEH000657
  • 9. N. I. Ahiezer and I. M. Glazman, \cyr Teoriya lineĭnykh operatorov v Gil′bertovom prostranstve, Second revised and augmented edition, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0206710
  • 10. R. G. Douglas, H. S. Shapiro, and A. L. Shields, On cyclic vectors of the backward shift, Bull. Amer. Math. Soc. 73 (1967), 156–159. MR 0203465, https://doi.org/10.1090/S0002-9904-1967-11695-1
  • 11. Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. MR 0275190
  • 12. M. Rosenblum and J. Rovnyak, Hardy classes and operator theory, Oxford Univ. Press, Oxford, 1985. MR 0822228 (87e:47001)
  • 13. I. I. Privalov, Graničnye svoĭstva analitičeskih funkciĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). 2d ed.]. MR 0047765
  • 14. Yu. P. Ginzburg, On $ J$-nondilating operators in Hilbert space, Nauchn. Zap. Fiz.-Mat. Fak. Odessk. Gos. Ped. Inst. 22 (1958), no. 1, 13-20. (Russian )
  • 15. R. G. Douglas and J. William Helton, Inner dilations of analytic matrix functions and Darlington synthesis, Acta Sci. Math. (Szeged) 34 (1973), 61–67. MR 0322538
  • 16. Ulf Grenander and Gabor Szegö, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley-Los Angeles, 1958. MR 0094840
  • 17. V. E. Katsnelson and B. Kirstein, On the theory of matrix-valued functions belonging to the Smirnov class, Topics in interpolation theory (Leipzig, 1994) Oper. Theory Adv. Appl., vol. 95, Birkhäuser, Basel, 1997, pp. 299–350. MR 1473261
  • 18. A. Lindquist and M. Pavon, On the structure of state-space models for discrete-time stochastic vector processes, IEEE Trans. Automat. Control 29 (1984), no. 5, 418-432. MR 0748206 (85h:93071)
  • 19. A. Lindquist and G. Picci, On a condition for minimality of Markovian splitting subspaces, Systems Control Lett. 1 (1981/82), no. 4, 264-269. MR 0670210 (83m:93055)
  • 20. -, Realization theory for multivariate stationary Gaussian processes, SIAM J. Control Optim. 23 (1985), no. 6, 809-857. MR 0809539 (87a:93056)

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Additional Information

D. Z. Arov
Affiliation: Department of Physics and Mathematics, South-Ukrainian State Pedagogical University, Staroportofrankovskaya, 26, 650029, Odessa, Ukraine

N. A. Rozhenko
Affiliation: Department of Physics and Mathematics, South-Ukrainian State Pedagogical University, Staroportofrankovskaya, 26, 650029, Odessa, Ukraine

DOI: https://doi.org/10.1090/S1061-0022-08-01002-9
Keywords: Holomorphic matrix-valued functions, dilations, pseudocontinuation
Received by editor(s): November 9, 2006
Published electronically: March 21, 2008
Article copyright: © Copyright 2008 American Mathematical Society

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