Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

$ 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
Full-text PDF Free Access

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; English transl., Soviet Phys. Dokl. 16 (1971), 954-956 (1972). MR 0428098 (55:1127)
  • 2. -, Realization of matrix-valued functions according to Darlington, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), no. 6, 1299-1331; English transl. in Math. USSR-Izv. 7 (1973). MR 0357820 (50:10287)
  • 3. -, 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. Zh. 16 (1975), no. 3, 440-463; English transl., Siberian Math. J. 16 (1975), no. 3, 335-352. MR 0473196 (57:12872)
  • 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 or resistance matrices), Mat. Sb. 193 (2002), no. 6, 3-24; English transl., Sb. Math. 193 (2002), no. 5-6, 791-810. MR 1957950 (2003k:47020)
  • 9. N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, 2nd ed., ``Nauka'', Moscow, 1966; English transl. from 3rd ed., Vols. I, II, Monogr. and Stud. in Math., vols. 9, 10, Pitman, Boston, MA-London, 1981. MR 0206710 (34:6527); MR 0615736 (83i:47001a); MR 0615737 (83i:47001b)
  • 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 (34:3316)
  • 11. B. Sz.-Nagy and C. Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publ. Co., Amsterdam-London, 1970. MR 0275190 (43:947)
  • 12. M. Rosenblum and J. Rovnyak, Hardy classes and operator theory, Oxford Univ. Press, Oxford, 1985. MR 0822228 (87e:47001)
  • 13. I. I. Privalov, Boundary properties of analytic functions, 2nd ed., ``GITTL'', Moscow-Leningrad, 1950. (Russian) MR 0047765 (13:926h)
  • 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. W. Helton, Inner dilations of analytic matrix functions and Darlington synthesis, Acta Sci. Math. (Szeged) 34 (1973), 61-67. MR 0322538 (48:900)
  • 16. U. Grenander and G. Szegö, Toeplitz forms and their applications, Univ. California Press, Berkeley-Los Angeles, 1958. MR 0094840 (20:1349)
  • 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) (H. Dym, B. Fritsche, V. Katsnelson, and B. Kirstein, eds.), Oper. Theory Adv. Appl., vol. 95, Birkhäuser, Basel, 1997, pp. 299-350. MR 1473261 (98m:47016)
  • 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)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 47A56

Retrieve articles in all journals with MSC (2000): 47A56


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

American Mathematical Society