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Functions of class II

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Abstract

We consider the class II of operator functions f(z), meromorphic for ¦z¦⊋1, which admit a representation in the form f(z)=f −12 (z)f1(z), where f1(z) is an operatorvalued holomorphic function and f2(z) a scalar-valued bounded holomorphic function, such that the strong limits

and

coincide a.e. on the unit circle ¦ζ¦=1. We prove that any function of class II can be represented as a block of some J-inner function of class II. We describe all such representations. The results found are applied to the question of realizations of functions of class II as transfer functions of linear systems.

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Literature cited

  1. D. Z. Arov, “Darlington's method in the investigation of dissipative systems,” Dokl. Akad. Nauk SSSR,201, No. 3, 559–562 (1971);

    Google Scholar 

  2. D. Z. Arov, “Realization of a matrix-function according to Darlington,” Izv. Akad. Nauk SSSR, Ser. Mat.,37, No. 6, 1299–1331 (1973);

    Google Scholar 

  3. D. Z. Arov, “Unitary couplings with losses,” Funkts. Anal. Prilozhen.,8, No. 4, 5–22 (1974);

    Google Scholar 

  4. D. Z. Arov, “Realization of a canonical system with dissipative boundary condition on one end of a segment with respect to the coefficient of dynamic pliability,” Sib. Mat. Zh.,16, No. 3, 540–563 (1975).

    Google Scholar 

  5. G. Ts. Tumarkin, “Description of a class of functions admitting approximation by fractions with fixed poles,” Izv. Akad. Nauk Arm. SSR, Ser. Mat.,1, No. 2, 89–105 (1966).

    Google Scholar 

  6. R. G. Douglas, H. S. Shapiro, and A. L. Shields, “On cyclic vectors of the backward shift,” Bull. AMS,73, No. 1, 156–159 (1967).

    Google Scholar 

  7. M. Rosenblum and J. Rovnyak, “The factorization problem for nonnegative operatorvalued function,” Bull. AMS,77, No. 3, 287–318 (1971);

    Google Scholar 

  8. M. Rosenblum and J. Rovnyak, “Factorization of operatorvalued entire function,” Indiana Univ. Math. J.,20, No. 2, 157–173 (1970).

    Article  Google Scholar 

  9. M. G. Krein, “Indefinite case of the Sturm-Liouville boundary problem in the interval (0; +∞),” Izv. Akad. Nauk SSSR, Ser. Mat.,16, 293–324 (1952);

    Google Scholar 

  10. M. G. Krein, “Entire functions of exponential type,” Izv. Akad. Nauk SSSR, Ser. Mat.,11, No. 4, 309–326 (1947);

    Google Scholar 

  11. M. G. Krein, “Entire matrix-functions of exponential type,” Ukr. Mat. Zh.,3, No. 2, 164–173 (1951).

    Google Scholar 

  12. P. A. Fuhrmann, “Realization theory in Hilbert space for a class of transfer functions,” J. Funct. Anal.,18, No. 4, 338–349 (1975).

    Article  Google Scholar 

  13. V. P. Potapov, “Multiplicative structure of-nonexpanding matrix-functions,” Tr. Mosk. Mat. O-va,4, 125–236 (1955).

    Google Scholar 

  14. D. Z. Arov and L. A. Simakova, “Boundary values of convergent sequences of-compressing matrix-functions,” Mat. Zametki,19, No. 4, 491–500 (1976).

    Google Scholar 

  15. A. V. Efimov and V. P. Potapov, “-Expanding matrix-functions and their role in the analytic theory of electric circuits,” Usp. Mat. Nauk,1, 65–130 (1973).

    Google Scholar 

  16. Yu. P. Ginzburg, “-Nonexpanding operators in Hilbert space,” Nauchn. Zap. Odessk. Pedagog. Inst.,22, No. 1, 13–20 (1958).

    Google Scholar 

  17. B. S.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, Elsevier (1971).

  18. D. Z. Arov, “Passive linear stationary dynamical systems,” Sib. Mat. Zh.,20, No. 2, 211–228 (1979).

    Google Scholar 

  19. Yu. P. Ginzburg, “Multiplicative representations of operator-functions of restricted form,” Usp. Mat. Nauk,22, No. 1, 163–165 (1967).

    Google Scholar 

  20. M. G. Krein and I. E. Ovcharenko, “Inverse problems for canonical differential equations,” Dokl. Akad. Nauk Ukr. SSR, 14–18 (1982).

  21. D. Z. Arov and M. G. Krein, “Finding the minimum entropy in indefinite problems of extension,” Funkts. Anal. Prilozhen.,15, No. 2, 61–64 (1981).

    Google Scholar 

  22. M. G. Krein, “Continuous analogs of propositions about polynomials, orthogonal on the unit circle,” Dokl. Akad. Nauk SSSR,105, No. 4, 637–640 (1956).

    Google Scholar 

  23. P. Dewilde and H. Dym, “Lossless chain scattering matrices and optimum linear prediction: the vector case,” Circuit Theory Appl.,9, 135–175 (1981).

    Google Scholar 

  24. V. M. Brodskii, “Operator nodes and their characteristic functions,” Dokl. Akad. Nauk SSSR,198, No. 1, 16–19 (1971).

    Google Scholar 

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Authors
  1. D. Z. Arov
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta.im. V. A. Steklova AN SSSR, Vol. 135, pp. 5–30, 1984.

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Arov, D.Z. Functions of class II. J Math Sci 31, 2645–2659 (1985). https://doi.org/10.1007/BF02107246

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