Shift Operators Contained in Contractions, Schur Parameters and Pseudocontinuable Schur Functions

Part of the Operator Theory: Advances and Applications book series (OT, volume 165)


The main goal of the paper is to study the properties of the Schur parameters of the noninner functions of the Schur class S which admit a pseudocontinuation. To realize this aim we construct a model of completely nonunitary contraction in terms of Schur parameters of its characteristic function (see Chapters 2 and 3). By means of the constructed model a quantitative criterion of pseudocontinuability is established (see Chapter 4 and Sections 5.1 and 5.2). The properties of the Schur parameter sequences of pseudocontinuable noninner Schur functions are studied (see Sections 5.3 and 5.4).


Shift coshift contraction unitary colligation characteristic operator function Schur function Schur parameters pseudocontinuability of Schur functions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Adamjan, V.M., Arov, D.Z.: On the unitary couplings of isometric operators, Math. Issled. Kishinev I(2)(1966), 3–64 (in Russian).MathSciNetGoogle Scholar
  2. [2]
    Akhiezer, N.J., Glasman, J.M.: Theory of linear operators in Hilbert space, 2nd ed., Nauka, Moscow (1966) (in Russian). English transl.: Frederick Ungar, New York, 1961.Google Scholar
  3. [3]
    Arov, D.Z.: Darlington realization of matrix-valued functions, Izv. Akad. Nauk SSSR, Ser. Mat.37(1973), 1299–1331(in Russian). English transl.: Math. USSR Izvestija 7(1973),1295–1326.zbMATHMathSciNetGoogle Scholar
  4. [4]
    Arov, D.Z.: On unitary coupling with loss, Funk. Anal. i ego Prilozh. 8(4)(1974), 5–22 (in Russian).zbMATHMathSciNetGoogle Scholar
  5. [5]
    Arov, D.Z.: Stable dissipative linear stationary dynamical scattering systems, J. Operator Theory 2(1979), 95–126 (in Russian); English transl. in Operator Theory: Advances and Applications, v. 134, Birkhäuser Verlag, Basel-Boston-Berlin(2002), 99–136.zbMATHMathSciNetGoogle Scholar
  6. [6]
    Bakonyi, M., Constantinescu, T.: Schur’s algorithm and several applications, Pitman Research Notes, v. 261, 1992.Google Scholar
  7. [7]
    Bertin, M.J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M., Schreiber, J.P.: Pisot and Salem numbers, Birkhäuser Basel-Boston-Berlin, 1992.zbMATHGoogle Scholar
  8. [8]
    Boiko, S.S., Dubovoy, V.K.: On some extremal problem connected with the suboperator of the scattering through inner channels of the system, Dopovidi NAN Ukr. 4(1997), 8–11.Google Scholar
  9. [9]
    Boiko, S.S., Dubovoy, V.K., Fritzsche, B., Kirstein, B.: Contractions, defect functions and scattering theory Ukrain. Math. J. 49(1997), 481–489 (in Russian).Google Scholar
  10. [10]
    Boiko, S.S., Dubovoy, V.K., Fritzsche, B., Kirstein, B.: Shift operators contained in contractions and pseudocontinuable matrix-valued Schur functions, Math. Nachr. 278, No. 7–8(2005), 784–807.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Boiko, S.S., Dubovoy, V.K., Kheifets, A.Ja.: Measure Schur complements and spectral functions of unitary operators with respect to different scales, Operator Theory: Advances and Applications, v. 123, Birkhäuser Verlag, Basel-Boston-Berlin(2001), p. 89–138.Google Scholar
  12. [12]
    Brodskii, M.S.: Unitary operator colligations and their characteristic functions, Uspekhi Math. Nauk 33, 4(202)(1978), 141–168 (in Russian); English transl.: Russian Math. Surveys, 33(4)(1987), 159–191.zbMATHMathSciNetGoogle Scholar
  13. [13]
    Cima, J.A., Ross, W.T.: The backward shift on the Hardy space, Math. surveys and monographs, V. 79(2000).Google Scholar
  14. [14]
    Constantinescu, T.: On the structure of the Naimark dilation, J. Operator Theory, 12(1984), 159–175.zbMATHMathSciNetGoogle Scholar
  15. [15]
    Constantinescu, T.: Schur parameters, factorization and dilation problems, Operator Theory, Advances and Applications, v. 82, Birkhäuser Basel-Boston-Berlin, 1996.Google Scholar
  16. [16]
    DeWilde, P.,: Roomy scattering matrix synthesis, Technical Report, Berkeley, (1971).Google Scholar
  17. [17]
    Douglas, R.G., Shapiro, H.S., Shields, A.L.: Cyclic vectors and invariant subspaces for the backward shift, Ann. Inst. Fourier 20(1971),37–76.MathSciNetGoogle Scholar
  18. [18]
    Douglas, R.G., Helton, J.W.: Inner dilations of analytic matrix functions and Darlington synthesis, Acta Sci. Math. (Szeged) 34(1973), 61–67.MathSciNetzbMATHGoogle Scholar
  19. [19]
    Dubovoy, V.K.: Indefinite metric in Schur’s interpolation problem for analytic functions, Teor. Funkcii, Funkcional. Anal. i Prilozen (Kharkov), part I: 37(1982), 14–26; part II: 38(1982), 32–39; part III: 41(1984), 55–64; part IV: 42(1984), 46–57; part V: 45(1986), 16–26; part VI: 47(1987), 112–119 (in Russian); English transl.: part I: II. Ser., Am. Math. Soc. 144(1989), 47–60; part II: II. Ser., Am. Math. Soc. 144(1989), 61–70; part IV: Oper. Theory: Adv. Appl. 95(1997), 93–104; part V: J. Sov. Math. 48, No. 4(1990), 376–386; part VI: J. Sov. Math. 48, No. 6(1990), 701–706.Google Scholar
  20. [20]
    Dubovoy, V.K., Mohammed, R.K.: Defect functions of holomorphic contractive matrix functions, regular extensions and open systems, Math. Nachr. 160(1993), 69–110.MathSciNetCrossRefGoogle Scholar
  21. [21]
    Dubovoy, V.K.: Schur’s parameters and pseudocontinuation of contractive holomorphic functions in the unit disk, Dopovidi NAN Ukr., v. 2(1998), 24–29.Google Scholar
  22. [22]
    Dubovoy, V.K.: On a contraction operator model constructed by using Schur’s parameters of its characteristic function, Dopovidi NAN Ukr., v. 3(1998), 7–10.Google Scholar
  23. [23]
    Dubovoy, V.K.: Some criteria for the pseudocontinuability of contractive holomorphic functions in the unit disc in terms of its Schur parameters, Dopovidi NAN Ukr., v. 7(2004),13–19.Google Scholar
  24. [24]
    Dubovoy, V.K., Fritzsche, B., Kirstein B.: Matricial Version of the Classical Schur Problem, Teubner-Texte zur Mathematik Bd. 129, Teubner, Stuttgart-Leipzig 1992.Google Scholar
  25. [25]
    Foias, C., Frazho, A.E.: The commutant lifting approach to interpolation problems, Operator Theory: Advances and Applications, v. 44, Birkhäuser Basel-Boston-Berlin, 1990.Google Scholar
  26. [26]
    Geronimus, Ya.L.: On polynomials orthogonal on the circle, on trigonometric moment problem, and on allied Carathéodory and Schur functions, Mat. Sb., v. 15(1944), 99–130 (in Russian).zbMATHMathSciNetGoogle Scholar
  27. [27]
    Gragg, W.B.: Positive definite Toeplitz matrices, the Arnoldi process for isometric operators and Gaussian quadrature on the unit circle, Numerical methods of linear algebra (Gos. Univ. Moskow), 1982, 16–32 (in Russian); English transl.: J. Comput. Appl. Math., 46(1993), 183–198.Google Scholar
  28. [28]
    Koosis, P.: Introduction to H p spaces, Cambridge Univ. Press, Cambridge etc. 1998.zbMATHGoogle Scholar
  29. [29]
    Nikolski, N.K.: Operators, functions and systems: an easy reading, Math. surveys and monographs, v.92, Contents: v. 1, Hardy, Hankel and Toeplitz (2002); v. 93, Contents: v. 2, Model operators and systems, 2002.Google Scholar
  30. [30]
    Ross, W.T., Shapiro, H.S.: Generalized Analytic Continuation, Amer. Math. Soc., Providence, RI, University Lecture Series, v. 25(2002).Google Scholar
  31. [31]
    Schur, I.: Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind, J. reine und angew. Math.,part I: 147(1917), 205–232; part II: 148(1918), 122–145.Google Scholar
  32. [32]
    Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 1:Classical Theory, Amer. Math. Soc. Colloq. Publ., Providence, RI, v. 54(2004).Google Scholar
  33. [33]
    Sz.-Nagy, B., Foias, C.: Harmonic analysis of operators in Hilbert space, North Holland Publishing Co., Amsterdam-Budapest, 1970.Google Scholar
  34. [34]
    Teplyaev, A.V.: Continuous analogues of random orthogonal polynomials on the circle, Dokl. Akad. Nauk SSSR, v. 320(1991), 49–53 (in Russian); English transl.: Soviet. Math. Dokl. 44(1992), 407–411.zbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  1. 1.Department of Mathematics and MechanicsState UniversityKharkovUkraine

Personalised recommendations