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An Analysis and Extension of V.P. Potapov’s Approach to Interpolation Problems with Applications to the Generalized Bi-Tangential Schur-Nevanlinna-Pick Problem and J-Inner-Outer Factorization

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Matrix and Operator Valued Functions

Part of the book series: Operator Theory Advances and Applications ((OT,volume 72))

Abstract

There exist several approaches to the generalization and unification of various problems of the Nevanlinna-Pick type [1–4, 5, 9, 10, 12, 14, 28, 29, 31, 35, 38, 40].

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References

  1. V.M. Adamjan, D.Z. Arov and M.G. Krein: Infinite block Hankel matrices and related continuation problems. Amer. Math. Soc. Transl. 111:2(1978).

    Google Scholar 

  2. V.M. Adamjan: Non-degenerate unitary intertwinings of semi-unitary operators (Russian), Funkt. Anal. i Prilozhen, 7:4 (1973), 1–16; English transl. Functional Anal. Appl. 7 (1973), 255–267.

    Article  MathSciNet  Google Scholar 

  3. V.M. Adamjan, D.Z. Arov and M.G. Krein: Infinite Hankel matrices and generalized Caratheodory-Fejer and I. Schur problems. (Russian), Funkt. Anal. i Prilozhen, 2:4(1968), 1–17.

    Article  MathSciNet  Google Scholar 

  4. V.M. Adamjan, D.Z. Arov and M.G. Krein: Bounded operators that commute with a contraction of class C00 of unit rank of non-unitary. (Russian), Funkt. Anal. i Prilozhen, 3:3(1969), 86–87.

    MathSciNet  Google Scholar 

  5. N.I. Akhiezer: The Classical Moment Problem. Oliver and Boyd, Edinburgh, 1965.

    MATH  Google Scholar 

  6. D.Z. Arov: Regular J-inner matrix-functions and related continuation problems. Operator Theory: Advances and Applications, 43 (1990), 63–87.

    MathSciNet  Google Scholar 

  7. D.Z. Arov: γ-derived matrices, J-inner matrix-functions and related extrapolation problems. (Russian), Teor. Funk., Funkt. Anal. i ikh Prilozhen, (Kharkov), 51 (1989), 61–67; 52 (1989), 103–109; 53 (1990), 57–65; English transl. J. Soviet Math. 52 (1990), 3487–3491; 52 (1990) 3421–3425; 58 (1992), 532–537.

    MATH  Google Scholar 

  8. D.Z. Arov and L.Z. Grossman: Scattering matrices in the extension theory of isometrical operators. Soviet Math. Dokl., 27 (1983), 573–578.

    Google Scholar 

  9. J.A. Ball and J.W. Helton: A Beurling-Lax theorem for Lie group U(m,n) which contains most classical interpolation problems. J. Oper. Theory 9 (1983), 107–142.

    MathSciNet  MATH  Google Scholar 

  10. L. de Branges: Hilbert Spaces of Entire Functions. Prentice Hall, Englewood Cliffs, NJ, 1968.

    MATH  Google Scholar 

  11. L. de Branges and J. Rovnyak: Canonical models in quantum scattering theory. In: “Perturbation Theory and Its Application in Quantum Mechanics”, Wiley, New York, 359–391, 1966.

    Google Scholar 

  12. H. Dym: J contractive matrix functions, reproducing kernel Hilbert spaces and interpolation. CBMS Regional Conf. Ser. in Math., 71, Amer. Math. Soc., Providence, R.I., 1989.

    Google Scholar 

  13. A.V. Efimov and V.P. Potapov: J-expansive matrix-functions and their role in the analytic theory of electrical circuits. (Russian), Uspehi Mat. Nauk 28 (1974), 65–130; English transl. in Russian Math. Surveys, 28 (1974).

    MathSciNet  Google Scholar 

  14. J.W. Helton: Operator theory, analytic functions, matrices and electrical engineering. CBMS Regional Conf., Series in Math., No. 68, American Math. Soc., Providence, R.I., 1987.

    Google Scholar 

  15. T.S. Ivanchenko and L.A. Sahnovich: Operator approach to investigation of interpolation problems. (Russian), 1985, deposited in Ukr NIINTI, No. 701-Uk D85.

    Google Scholar 

  16. T.S. Ivanchenko and L.A. Sahnovich: Operator-theoretic approach to the V. P. Potapov scheme of investigation of interpolation problems. (Russian), Ukr. Math. Jour., 39:5 (1987), 573–578.

    Google Scholar 

  17. V.E. Katsnelson: Continuous analogs of Hamburger-Nevanlinna theorem and fundamental matrix inequality of classical problems. (Russian), Teor. Funk., Funkt. Anal. i ikh Prilozhen. (Kharkov), 36 (1981), 31–48; 37 (1982), 31–58; 39 (1983), 61–73; 40 (1983), 79–80; English transl. in Amer. Math. Soc. Transl. (2) 136 (1987), 49–108.

    MathSciNet  Google Scholar 

  18. V.E. Katsnelson: The fundamental matrix inequality of the problem of positive-definite kernel decomposition on the elementary products. (Russian), Dokl. Akad. Nauk Ukr. SSSR, Ser. A 2 (1984), 10–13.

    MathSciNet  Google Scholar 

  19. V.E. Katsnelson: Regularization of the fundamental matrix inequality of the problem of positive-definite kernel decomposition on the elementary products. (Russian), Dokl. Akad. Nauk Ukr. SSSR, Ser. A, 3 (1984), 6–8.

    MathSciNet  Google Scholar 

  20. V.E. Katsnelson, A. Ya. Kheifets and P.M. Yuditskii: The abstract interpolation problem and extension theory of isometric operators. (Russian), in: Operator in Spaces of Functions and Problems in Function Theory: Collected scientific papers, Kiev, Naukova Dumka, 1987, 83–96.

    Google Scholar 

  21. A.Ya. Kheifets: Parseval equality in abstract interpolation problem and coupling of open systems. (Russian), Teor. Funk., Funkt. Anal. i ikh Prilozhen, (Kharkov), 49 (1988), 112–120; 50 (1988), 98–103; English transl. in Journal of Soviet Math., 49 (1990), 1114–1120; N6, 1307–1310.

    MathSciNet  Google Scholar 

  22. A.Ya. Kheifets: Generalized bi-tangential Schur-Nevanlinna-Pick problem, related Parseval equality and scattering operator. (Russian), deposited in VINITI 11.05.1989, No. 3108–889 Dep., 1’60, 1989.

    Google Scholar 

  23. A.Ya. Kheifets: Generalized bi-tangent Shur-Nevanlinna-Pick problem and related Parseval equality. (Russian), Teor. Funk., Funkt. Anal. i ikh Prilozhen, (Kharkov), 54 (1990), 89–96; English transl. in Journal of Soviet Math.

    Google Scholar 

  24. A.Ya. Kheifets: Nevanlinna-Adamjan-Arov-Krein theorem in semidetermined case. (Russian), Teor. Funk., Funkt. Anal. i ikh Prilozhen (Kharkov), 56 (1991), 128–137; English transl. in Journal of Soviet Math.

    Google Scholar 

  25. A. Ya. Kheifets and P.M. Yuditskii: Interpolation of operator commuting with shorted shift by I. Schur class functions. (Russian), Teor. Funk., Funkt. Anal. i ikh Prilozhen (Kharkov), 40 (1983), 129–136.

    MathSciNet  Google Scholar 

  26. I.V. Kovalishina: Analytic theory of a class of interpolation problems. (Russian), Math. SSSR Izv., 22 (1984), 419–463.

    Article  MATH  Google Scholar 

  27. I.V. Kovalishina and V.P. Potapov: Indefinite metric in Nevanlinna-Pick problem. (Russian), Dokl. Akad. Nauk Armjan. - SSR, ser. mat., 9:1(1974), 3–9; Private translation by T. Ando, Sapporo, 1982.

    Google Scholar 

  28. M.G. Krein: On generalized moment problem. (Russian), Dokl. Akad. Nauk USSR, 44:6(1944), 233–243.

    Google Scholar 

  29. M.G. Krein: On Hermitian operators with direction functionals. (Russian), Shorn. Trudov Inst. Mat. Ukr., 10 (1948), 83–105.

    MathSciNet  Google Scholar 

  30. M.G. Krein: Basic considerations on the representation theory of Hermitian operators with defect index (m, m). (Russian), Ukr. Mat. Zhurnal, 1:2(1949), 3–66.

    MathSciNet  Google Scholar 

  31. M.S. Livshits: On one application Hermitian operators theory to generalized moment problem. (Russian), Dokl. Akad. Nauk. USSR, 44 (1944), 3–7.

    MATH  Google Scholar 

  32. S.H. Naboko: Absolutely continuous spectrum of non dissipative operator and functional model. (Russian), Zap. Nauchn. Sem. LOMI, 65 (1976), 90–102; 73 (1977), 118–135.

    MathSciNet  MATH  Google Scholar 

  33. N.K. Nikolskii: Treatise on the Shift Operator; (with an Appendix by S.V. Hruscev and V.V. Peller), Springer Verlag, Heidelberg, 1986.

    Book  Google Scholar 

  34. N.K. Nikolskii and V.I. Vasyunin: A unified approach to function models and the transcription problem. Operator Theory: Advances and Application 41 (1989), 405–434.

    MathSciNet  Google Scholar 

  35. A.A. Nudelman: On a generalization of classical interpolation problems. Sov. Math. Dokl. 23 (1961), 125–128.

    Google Scholar 

  36. V.P. Potapov: Multiplicative structure of J-nonexpending matrix-function. (Russian), Dokl. Mosk. Mat. Obsc. (1955), 125–236.

    Google Scholar 

  37. M. Rosenblum and J. Rovnyak: Hardy Classes and Operator Theory. Oxford University Press, 1986.

    Google Scholar 

  38. D. Sarason: Generalized interpolation in H∞. Trans. Amer. Math. Soc. 127:2 (1967), 179–203.

    MathSciNet  MATH  Google Scholar 

  39. D. Sarason: Operator theoretic aspect of the Nevanlinna-Pick interpolation problem in operator and function theory. Lancaster Conference Notes, Reidel, Dordrecht, Higtham, Mass., 1985, 279–314.

    Google Scholar 

  40. B.Sz.-Nagy and C. Foias: Harmonic analysis of operators in Hilbert space. North-Holland, Amsterdam, 1970.

    Google Scholar 

  41. P.M. Yuditskii: Lifting problem. (Russian), deposited in Ukr. NIINTI 18.04.1983, no. 311-Uk-D83.

    Google Scholar 

  42. P.M. Yuditskii: On one problem connected with Nehari problem. (Russian), deposited in Ukr. NIINTI 25.11.1983, no. 1336-Uk-D83.

    Google Scholar 

  43. P.M. Yuditskii: On reconstruction of j-contractive analytic matrix-function by its j-form of boundary values and related interpolation problem. (Russian), Teor. Funk. Funkt. Anal. i ikh Prilozhen (Kharkov), 44 (1985), 141–143; English transl. in Journal of Soviet Math.

    Google Scholar 

  44. P.M. Yuditskii: Inner-outer factorization of j-expanding invertible matrix-functions. (Russian), Teor. Funk. Funkt. Anal. i ikh Prilozhen (Kharkov), 46 (1986), 132–136; English transl. in Journal of Soviet Math.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematical Department, Kharkov State University, Kharkov, 310077, Ukraine

    A. Ya. Kheifets

  2. Mathematical Division of the Institut for Low, Temperature Physics and Engineering, Lenin Av. 47, Kharkov, 310164, Ukraine

    P. M. Yuditskii

Authors
  1. A. Ya. Kheifets
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  2. P. M. Yuditskii
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Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Israel

    I. Gohberg  & L. A. Sakhnovich  & 

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© 1994 Springer Basel AG

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Kheifets, A.Y., Yuditskii, P.M. (1994). An Analysis and Extension of V.P. Potapov’s Approach to Interpolation Problems with Applications to the Generalized Bi-Tangential Schur-Nevanlinna-Pick Problem and J-Inner-Outer Factorization. In: Gohberg, I., Sakhnovich, L.A. (eds) Matrix and Operator Valued Functions. Operator Theory Advances and Applications, vol 72. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8532-4_6

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  • DOI: https://doi.org/10.1007/978-3-0348-8532-4_6

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9667-2

  • Online ISBN: 978-3-0348-8532-4

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