On Realizations of Rational Matrix Functions of Several Complex Variables

  • M. F. Bessmertnyĭ
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 134)


Holomorphic functions of one complex variable with a non-negative imaginary part, and the related classes of J-contractive matrix functions (contractive in a space with an indefinite metric, defined in a standard way by a hermitian matrix J such that J 2 = I) were actively studied by many mathematicians in the past years. A series of problems of mathematical analysis and its applications lead to the necessity of studying analogous classes of holomorphic functions of several complex variables. Holomorphic functions with a non-negative imaginary part in a tubular domain over a cone and in the polydisk were studied by V.S. Vladimirov (see [Vla2], [V1a5]–[V1a8]) and by Vladimirov and Drozhzhinov [29]. In bounded strictly star-shaped domains, and in particular in the classical symmetric domains, they were studied by L.A. Aizenberg and Sh.A. Dautov [1]. W. Rudin (see [Rud]) gives a “parametrical” representation of scalar rational inner functions in the polydisk \(\mathbb{D}^n \). The parameter is an arbitrary polynomial non-vanishing in \(\mathbb{D}^n \).


Rational Function Holomorphic Function Matrix Function English Trans Rational Matrix Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AD]
    AIZENBERG, L.A. AND SH.A. DAUTOV: Golomorfnye funktsii mnogikh peremen-nykh s neotritsatel’noi deistvitel’noi chast’yu. Sledy golomorfnykh i plyurigarmonicheskikh funktsiY na granitse Shilova. (Russian). Matern. Sbornik, (Novaya. Ser.) 99(141), pp. 342–355 (1976). English transl.: Holomorphic functions of several complex variables with non-negative real part. Traces of holomorphic and pluriharmonic functions on the Silov boundary. (English) Math. USSR, Sb. 28 (1976), 301–313.MathSciNetGoogle Scholar
  2. [Bosl]
    BOSE, N.K.: New techniques and results in multidimensional problems. Journal of the Franklin Inst., 301, no. 1/2 (Jan.—Feb. 1976), Special issue: Recent trends in systems theory, pp. 83–101.Google Scholar
  3. [Bos2]
    BOSE, N.K.: Problems and progress in multidimensional system theory. In[Multi], pp. 824–840.Google Scholar
  4. [EfPo]
    EFIMOV, A.V. and V.P. PoTAPOV: J-rastyagivayushchie matritsy-funktsii i ikh rol’ IT analiticheskoi teorii elektricheskikh tsepei (Russian). Uspekhi Matematicheskikh Nauk, 28:1 (1973), pp. 65–130. English transl: J-expanding matrix functions and their role in the analytical theory of electrical circuits. Russ. Math. Surveys, 28:1 (1973), pp. 69–140.Google Scholar
  5. [Ful]
    Fuxs, B.A.: Vvedenie v Teoriyu Analiticheskikh Funktsii Mnogikh Komplek-snykh Peremennykh (Russian). Fizmztgiz, Moscow 1962, 12 419 p. English transl.: Introduction to the Theory of Analytic Functions of Several Complex Variables. (Series: Translations of mathematical monographs, vol. 8). American Mathematical Society, Providence, R. I., 1963, x+388 p.Google Scholar
  6. [Fu2]
    Fuxs, B.A.: Spetsial’nye Glavy Teorii Analiticheskikh Funktsii Mnogikh Kom-pleksnykh peremennykh (Russian). Fizmztgiz, Moscow 1963, 427 p. English transl.: Special chapters in the theory of analytic functions of several complex variables. (Series: Translations of mathematical monographs, vol. 14). American Mathematical Society, Providence, R. 1.,1965, vi+357 p.Google Scholar
  7. [Hef]
    HEFER,H.: Zur Funktionentheorie mehrerer Veränderlichen. Uber die Zer-legung analytischer Funktionen and die Weilische Integraldarstellung. Math. Ann., 122 (1950), pp. 276–280.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [Kar] KARNI, SH.: Network theory; analysis and synthesis. (Allyn and Bacon seriesin electrical engineering). Allyn and Bacon, Boston 1966, xi+483 p. Russian transl.: Teoriya tsepei: Analiz i Sintez., Izdat. “Svyaz”, Moscow 1968, 368 p.Google Scholar
  9. [KFA]
    KALMAN, R., P. FALB and M. ARBIB: Topics in Mathematical Systems Theory, McGraw-Hill, New York, 1969. Russian Transl.: Oéerki po Matematiceskoi Teorii Sistem, Mir, Moskwa, 1971.Google Scholar
  10. [Kog]
    KOLA, T.: Synthesis of finite passive n-ports with prescribed positive real ma-trices of several variables. IEEE Trans. Circuit Theory CT 15 no.1 (March 1968), pp. 2–23.Google Scholar
  11. [Lan]
    LANG, S.: Algebra. Addison-Wesley Publishing Co., Inc., Reading, Mass. 1965 xvii+508 pp.Google Scholar
  12. [Liv]
    Livsid, M.S.(=LIVSHITS, M.S.): Operatory,Kolebaniya, Volny. Otkrytye Sis-temy (Russian). English transl. (edited by R. HERDEN.): Operators,oscillations, waves (open systems). (Translations of Mathematical Monographs, Vol. 34.) American Mathematical Society, Providence, R.I., 1973. vi+274 pp.Google Scholar
  13. [Me]
    MELAMUD, E.YA.: Ob odnom obobshchenii teoremy Darlingtona. [A certain generalization of Darlington’s theorem](Russian). Izvestiya Akad. Nauk Armyanskoi SSR, ser. Matem., 7:3 (1972), pp. 183–195, 226.MathSciNetGoogle Scholar
  14. [Muhl]
    Proceedings of the IEEE, 65:7 (June 1977). Special issue on multidimensional system. (BosE, N.K.-ed.), pp. 819–980.Google Scholar
  15. [Mult2]
    Multidimensional Systems Theory. (Progress, Directions and Open Problems in Multidimensional Systems). D. Reidel Publishing Company, Dordrecht•Boston•Lancaster 1985.5 Google Scholar
  16. [Pot 1]
    POTAPOV, V.P.: Mul’tiplikativnaya struktura J-rastyagivayushchikh matrits-funktsii (Russian). Trudy Moskovskogo Matematicheskogo Obshchestba, 4 (1955), pp. 125–236. English transi.: The multiplicative structure of J-contractive matrix functions. Amer. Math. Soc., Transi., (Ser. 2), 15 (1960), pp. 131–243.Google Scholar
  17. [Pot2]
    POTAPOV, V.P.: Drobno-lineinye preobrazovaniya matrits (Russian). In: Issle-dovaniya po Teorii Operatorov i ikh Prilozheniyam. [Studies in the Theory of Operators and their Applications], (MAROHENKO, V.A. and V.YA. GOLODETS — editors.) Naukova Dumka, Kiev 1979, pp. 75–97, 177. English transi.: Linear fractional transformations of matrices. Amer. Math. Soc. Translations (Ser.2) vol.138 (1988), (Seven papers translated from the Russian. Edited by BEN SILVER), American Mathematical Society, Providence, RI, 1988, viii+77 pp., pp. 21–35.MathSciNetGoogle Scholar
  18. [RUA]
    RUDIN, W.: Function theory in polydiscs. W.A. Benjamin, New York • Ams-terdam 1969, vii+188 pp. Russian transl.: Teoriya Funktsii v polikruge. Mir, Moscow 1974.Google Scholar
  19. [SeRe]
    SESHU, S. and M.B. REED: Linear Graphs and Electrical Networks. (Addison-Wesley series in the engineering sciences. Electrical and control systems). Addison-Wesley, Reading, MA, 1961. 315 p. Russian transl.: Lineinye grafy i elektricheskie tsepi. Izdat. “Vysshaya shkola”, Moskow 1971, 448 p.Google Scholar
  20. [Vial]
    VLADIMIROV, V.S.: Metody teorii funktsii mnogikh kompleksnykh peremen-nykh. (Russian). [Methods in the theory of functions of several complex variables] With a Foreword by N.N. Bogoljubov. Izdat. “Nauka”, Moscow 1964,1 411 pp. English transl: Methods of the theory of functions of many complex variables. Translation edited by LEON EHRENPREIS. The M.I.T. Press, Cambridge, MA•London 1966 xii+353 pp. French transi.: Les fonctions de plusieurs variables complexes et leur application à la théorie quantique des champs. (Travaux et Recherches Mathématiques, No. 14). Dunod, Paris 1967 xv+338 PP.Google Scholar
  21. [V1a2]
    VLADIMIROV, V.S.:Golomorfnye funktsii mnogikh peremennykh s neotritsa-tel’noi deistvitel’noi chast’yu v trubchatoi oblasti nad konusom. (Russian). Matem. Sbornik, 79 (1969), pp. 128–152. English transi.: Holomorphic functions with non-negative imaginary part in a tubular domain overacone. Math. USSR, Sbornik, 8(1969), 125–146.zbMATHGoogle Scholar
  22. [V1a3]
    Vladimirov, V.S.: Lineinye passivnye sistemy (Russian). Teoret. i Mat. Fiz. 1 (1969), pp. 67–94. English transi.: Linear passive systems. Theoret. and Math. Phys. (Consultants Bureau), 1 (1969), pp. 51–72.zbMATHGoogle Scholar
  23. [V1a4]
    VLADIMIROV, V.S.: Mnogomernye lineinye passivnye sistemy [Multidimen-sional passive linear systems] (Russian). In: Mekhanika Sploshnoi Sredy i Rodstvennye Problemy Analiza. [Continuum mechanics and related problems of analysis] (Russian). (On the occasion of the eightieth birthday of Academician N.I. Mushelisvili), pp. 121–134. Izdat. “Nauka”, Moscow, 1972.Google Scholar
  24. [5]
    VLADIMIROV, V.S.: Golomorfnye funktsii s polozhitel’noi mnimoi chast’yu vtrube budushchego (Russian). Matem. Sbornuk (Nov. Ser.), 93 (135) (1974), pp. 3–17. English transi.: Holomorphic functions with positive imaginary part in the future tube. Math. USSR, Sbornik 22 (1974), pp. 1–16.Google Scholar
  25. [V1a6]
    VLADIMIROV, V.S.: Golomorfnye funktsii s polozhitel’noi mnimoi chast’yu vtrube budushchego. II (Russian). Matern. Sbornuk (Nov. Ser.)94 (136), (1974), pp. 3–17. English transl.: Holomorphic functions with positive imaginary part in the future tube. II. Math. USSR, Sbornik 22 (1974), pp. 1–16.Google Scholar
  26. [V1a7]
    VLADIMIROV, V.S.: Golomorfnye funktsii s polozhitel’noi mnimoi chast’yu vtrube budushchego. III (Russian). Matem. Sbornuk (Nov. Ser.), 98 (140)(1975), pp. 292–297. English transi.: Holomorphic functions with positive imag- inary part in the future tube. III. Math. USSR, Sbornik 27 (1975), pp. 263–268.Google Scholar
  27. [V1a8] VLADIMIROV, V.S.: Golomorfnye funktsii s polozhitel’noi mnimoi chast’yu vtrube budushchego. IV (Russian). Matem. Sbornuk (Nov. Ser.), 104 (146), (1977), pp. 341–370. English transi.: Holomorphic functions with positive imag- inary part in the future tube. IV. Math. USSR, Sbornik 33 (1977), pp. 301–325.Google Scholar
  28. [V1a9]
    VLADIMIROV, V.S.: Obobshennye Funktsii v Matematicheskoi fizike. (Second edition). (Russian). Nauka, Moscow 1979. English transl.: Generalized functions in mathematical physics. Moscow: Mir Publishers. 362 p. French transl.: Distributions en physique mathématique. 280 pp. Moscou, Editions Mir 1979. Italian transi.: Le distribuzioni nella fisica matematica. 320 pp. Mosca: Edizioni Mir 1981.Google Scholar
  29. [VlaDr] VLADIMIROV, V.S. and Yu.N. DROZHZHINOV (=Ju.N. DR022INov): Golomorfnye funktsii v polikruge s neotritsatel’noi mnimoi chast’yu. (Russian). Matem. zametki, 15 (1974), pp. 55--61. English transl.: Holomorphic functions in a poly-circle with non-negative imaginary part. Math. Notes, 15, (1974), 31–34.Google Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • M. F. Bessmertnyĭ
    • 1
  1. 1.Department of Mathematics Faculty of PhysicsKharkov National UniversityKharkovUkraina

Personalised recommendations